Post on 23-Feb-2020
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 708495 16 pageshttpdxdoiorg1011552013708495
Research ArticleA Genetic Algorithm for Task Scheduling on NoCUsing FDH Cross Efficiency
Song Chai Yubai Li Jian Wang and Chang Wu
School of Communication and Information Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China
Correspondence should be addressed to Song Chai stschaigmailcom
Received 5 August 2013 Accepted 7 November 2013
Academic Editor Hao-Chun Lu
Copyright copy 2013 Song Chai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A CrosFDH-GA algorithm is proposed for the task scheduling problem on the NoC-based MPSoC regarding the multicriterionoptimization First of all four common criterions namely makespan data routing energy average link load and workload balanceare extracted from the task scheduling problem on NoC and are used to construct the DEA DMU model Then the FDH analysisis applied to the problem and a FDH cross efficiency formulation is derived for evaluating the relative advantage among schedulesolutions Finally we introduce the DEA approach to the genetic algorithm and propose a CrosFDH-GA scheduling algorithm tofind the most efficient schedule solution for a given scheduling problemThe simulation results show that our FDH cross efficiencyformulation effectively evaluates the performance of schedule solutions By conducting comparative simulations our CrosFDH-GAproposal produces more metrics-balanced schedule solution than other multicriterion algorithms
1 Introduction
The scheduling problem has long been a research hotspotsince its proposal in the 1950s as the Job-shop schedulingproblem [1] After the computer technology emerged in the1940s the scheduling problem also found its position in thecomputer science region as the task scheduling problem onthe uniprocessor in the 1960s [2] the multiprocessor in the1970s [3] the distributed computing in the 1980s [4] andthe grid computing in the early 21st century [5] Now thechip fabrication technology has brought us to the single-chip multicore era [6] The presence of chip multiprocessor(CMP) especially theNoC (network-on-chip)-basedMPSoC[7] brings new challenges to the task scheduling algorithmdesign
In NoC solution the idea of introducing the networkinfrastructure to the chip design along with the newly arisenconcept of green communication [8] makes the goal ofscheduling algorithm change from single-objective optimiza-tion on makespan to performing optimization simultane-ously onmultiple metrics not only the traditional makespanbut also energy [9] and NoC criterions [10] and some ofthe optimizations of these metrics are even in conflict with
each other So the goal of scheduling algorithm design leansforward to balancing these multiple metrics
On the other hand DEA is a nonparametric techniquethat is used to measure the relative efficiency of multi-input multioutput DMUs (decisionmaking units) It was firstpresented byCharnes Cooper andRhodes as theCCRmodelin 1976 [11] then developed into several variations based ondifferent RTS (return-to-scale) assumptions The concept ofefficiency in DEA gives us a reasonable standard to maketrade-off between multiple metrics
In this paper a FDH DEAmodel of NoC task schedulingis constructed and a FDH cross efficiency formulation isproposed based on peer appraisal for further assessmentof the relative advantages of DMUs Then the proposedDEA approach is introduced to the genetic algorithm and aCrosFDH-GA scheduling algorithm is proposed for the taskscheduling problem on NoC to find the most efficient andbalanced schedule solution
The rest of this paper is organized as follows Section 2summarizes the related work of this paper Section 3 for-mulates the task scheduling problem on NoC the FDHand cross efficiency FDH formulation is given in Section 4Section 5 presents our CrosFDH-GA scheduling algorithms
2 Mathematical Problems in Engineering
R R R R
R
R
RRRR
R
R R R
RR
PE
PE PE PE
PEPE PE
PE
PE
PE PE
PEPE PE
PE PE
(0 0) (1 0) (2 0) (3 0)
(1 1) (2 1) (3 1)
(1 2) (2 2) (3 2)
(0 1)
(0 2)
(0 3) (1 3) (2 3) (3 3)
(a) A 4 times 4 2D mesh NoC
East Inport
FIFO
Dec
oder
Req
ReqAck
Ack
East Outport
Arbiter
Sel
WestOutport
North Outport
North Inport
South Outport
South Inport
Local OutportLocal Inport
MU
X
West Inport
full in
east in
full out
west out
(b) Microstructure of NoC node
Figure 1 Architecture of NoC-based MPSoC
simulations results and discussion are given in Section 6Section 7 concludes the paper
2 Related Works
The multicriterion scheduling algorithms for CMPs havebeen widely researched in recent years In [12] a schedulingalgorithm is proposed for multicore processors to avoidresource contention aswell as to reduce energy consumptionA modified genetic algorithm which incorporates bacterio-logical algorithm is proposed in [13] to maximize the systemreliability and reduce makespan In [14] the optimization ofmakespan and workload balance is addressed and a NSGA-II based schedule algorithm is proposed for multicore-basedgrid Amultiobjective evolutionary algorithm (MOEA) basedschedule heuristic is proposed for the joint optimization ofperformance energy and temperature on multicore proces-sors in [15]
As for the DEArsquos application in the field of task schedulingonmulticore a FDH-based evaluationmethod for the assess-ment of schedule heuristics is proposed in [16] Althoughboth [16] and ourwork adoptDEAFDHmodel as the analytictool ourwork introduces the concept of cross efficiency to theFDHmodel and uses FDH cross efficiency to rank schedulesMoreover the incorporation of DEA evaluation method intometaheuristic also distinguishes itself from the work in [16]
3 Problem Formulation
31 Task Model In this paper tasks are modeled usingdirected acyclic graphs (DAGs) A DAG 119866 = (119881 119864) is anacyclic graph where 119881 is the set of nodes which represent thetasks and 119864 is the set of edges in which an element 119890
119894119895denotes
the communication from task 119894 to task 119895 The edge indicatesthe precedent relation between two tasks
Each node V119894and edge 119890
119894119895are associated with a weight
denoted by119862119894and119879119894119895 respectivelyWeight119862
119894is the computa-
tional load required by a processing Element (PE) to executetask 119894 and119879
119894119895is the data transmission load between task 119894 and
task 119895 In our work both 119862119894and 119879
119894119895are presented using time
unit (cycles)
32 Network-on-Chip Hardware The target hardware is a 2Dmesh NoC-based MPSoC as illustrated in Figure 1(a) EachPE is connected to a router and routers are interconnectedwith each other through bidirection links Data is transferredthrough NoC in the form of packets
PEs are homogenous processor cores with local datacache If two consequential tasks are scheduled to the samePE the successor task reads the predecessorrsquos data directlyfrom the data cache of the PE without routing in NoC
The microstructure of a NoC router is shown in Fig-ure 1(b) The router has five Inports and Outports corre-sponding to five directions of East West North South andLocal The decoder in the Inport scans the first flit of theFIFO for any incoming packet If the decoder detects thehead flit of a packet it performs XY routing algorithm andsends request signal to the arbiter of the correspondingOutport If the arbiter receives multiple request signals thecontention is solved using Round-Robin arbitration Thegranted Inport then forwards the packet to the downstreamrouter Wormhole routing is adopted to minimize the bufferrequirement as well as the packet latency [17] The backpressure mechanism is also employed to further reduce end-to-end delay [18]
The energy model of a NoC is presented by the BitEnergy proposed in [19] Analytically the average energyconsumption of transmitting one bit from node 119894 to node 119895is calculated by
119864119894119895
bit = 119899hops sdot 119864119873bit + (119899hops minus 1) sdot 119864119871bit (1)
Mathematical Problems in Engineering 3
where 119864119873bit
and 119864119871bit
represent the energy consumed on thenode and on the link respectively and 119899hops is the number ofnodes the bit passes on its way from node 119894 to node 119895
33 Monitored Metrics In this paper four common metricsof NoC are extracted and monitored for the assessmentof schedule solution and they are makespan data routingenergy average link load [20] and workload balance [21]
Makespan (the 119872 metric) which is the amount of timerequired by a NoC to finish entire tasks in a DAG followingthe instruction of a schedule is the timemetric of a schedulewhile the data routing Energy (the 119864 metric) is the totalamount of energy dissipated in each NoC component duringthe execution of the DAG
The average Link load (the 119871 metric) is calculated byadding up all the data transmission time (in cycles) on eachlink and then dividing it by the number of links (for the NoCin Figure 1(a) there are 48 links) The 119871 metric representshow busy the NoC is during the execution and a higheraverage link load metric implies a higher possibility of linkcontentions A good schedule is supposed to reduce the 119871metric
Finally the workload Balance (the 119861metric) is defined tobe the inverse coefficient of variant of the total workload oneach processor as shown in (2) The loadprc(119899) is the actualload on processor 119899 and the loadave is the average load The119861metric reflects the load balance of the processors
workload balance =loadave
(sum119899(loadave minus loadprc (119899))
2
)
12
(2)
4 DEA Evaluation of Schedules
41 A Brief Review of DEA Data envelopment analysis(DEA) is a nonparametric technique that is widely used tomeasure the relative efficiency among many-input many-output decision-making units (DMUs) which in our contextare the schedules The efficiency of a DMU is defined as theweighted sum of its output divided by the weighted sum ofits input The essence of DEA is that it allows each DMU tochoose a particular set of weight coefficients which favors itsown efficiency under the constraint that the efficiencies of allDMUs calculated by this set of coefficients do not exceed 1A DMU is ldquoefficientrdquo if the efficiency calculated by DEA is 1otherwise the DMU is marked as ldquoinefficientrdquo The followingLinear Programing (LP) problem is the CCR model of DEA(CCR multiplier form)
max 119885 = 119906119879
sdot 119910119894
st
V sdot 119909119894= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
V 119906 ge 0
(3)
where 119909119894= (119909
11989411199091198942
sdot sdot sdot 119909119894119898)119879
isin 119877119898 and 119910
119894=
(1199101198941
1199101198942
sdot sdot sdot 119910119894119904)119879
isin 119877119904 are the input and out-
put vectors of DMU 119894 V = (V1
V2
sdot sdot sdot V119898) and
119906 = (11990611199062
sdot sdot sdot 119906119904) are the coefficient (multiplier)
vectors of the inputs and outputs 119899 is the number of DMUsand the objective function 119885 is the efficiency of DMU 119894
The result of applying DEA is a classification amongDMUs as efficient group or inefficient group The efficientDMUs forman efficient frontier on themulti-inputmultiout-put space that envelops all inefficient DMUs The projectionof inefficient DMUon the efficient frontier is the hypotheticalefficient unit which is a linear combination of the efficientDMUs An inefficient DMU can also be converted to an effi-cient DMU by proportionally scaling down by the value of itsefficiency in the inputs and maintaining its original outputsThis interpretation of DEA efficiency is corresponding to theenvelop form of DEA which is the dual problem of (3) (CCRenvelop form)
min120579120582
119885 = 120579
st
120579119909119894minus 119883 sdot 120582 ge 0
119910119894minus 119884 sdot 120582 le 0
120582 ge 0
(4)
where 119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 =
(11991011199102
sdot sdot sdot 119910119899) are the input and output matrices
120582 = (12058211205822
sdot sdot sdot 120582119899)119879
isin 119877119899 is a nonnegative vector and
120579 is the efficiency of DMU 119894The way that the efficient frontier is generated differen-
tiates between DEA models which imply different returnsto scale assumptions There are four basic returns to scaleassumption constant returns to scale (CRS) correspondingto the CCR model [11] variable returns to scale (VRS)corresponding to the BCC model [22] increasing returns toscale and decreasing returns to scale
In this paper we focus on a special case of DEA namelythe free disposal hull (FDH) [23] In VRS FDH formulationof DEA each DMU is evaluated by comparing itself to otherDMUs on a one-on-one basis and a DMU is consideredefficient only when no other DMU dominates it
In most cases the idea of DEA that let DMU specifyits own weight coefficients to show its maximum advantageis desirable However in some extreme scenes a DMU canldquocheatrdquo a high efficiency score by weighting a single input ora single output and setting the rest weight coefficients closeto 0 This can happen when some DMUs have a particularlysmall input or particularly large output in our words theseDMUs have unbalanced metrics and these ldquomavericksrdquoneed to be depreciated Moreover although DEA effectivelydiscriminates between efficient and inefficient DMUs it doesnot further assess the relative advantages among the efficientones
One solution to the above questions is to introduce crossefficiency in the efficiencymeasurementThe concept of crossefficiency corresponding to the simple efficiency implied byoriginal DEA is the peer-appraisal equivalent of DEArsquos self-appraisal process The cross efficiency of a certain DMU isthe efficiency value calculated by using weight coefficientsderived by other DMUs If a DMU is a maverick or hasunbalanced metrics then its cross efficiency value derivedfrom other DMUrsquos coefficients is not likely to be high
4 Mathematical Problems in Engineering
Two widely accepted crossefficiency formulations theaggressive and benevolent formulations were proposedin [24] based on CCR Model Both formulations add asecondary goal to the normal DEA efficiency calculation(maximizing reference DMUrsquos efficiency) the aggressiveformulation minimizes target DMUrsquos crossefficiency whilethe benevolent formulation maximizes its efficiency Giventhat the simple DEA efficiency of DMU 119894 is 119864
119894119894 then the
crossefficiency of DMU 119895 evaluated by DMU 119894 is defined by(CCR crossefficiency benevolent formulation)
max 119864119895119894= 119906119879
sdot 119910119895
st
V sdot 119909119895= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
119864119894119894sdot V sdot 119909119894minus 119906 sdot 119910
119894= 0
V 119906 ge 0
(5)
42 FDHDEA Evaluation of Schedules In order to apply dataenvelopment analysis to the schedule evaluation the multi-input multioutput DMUmodel needs to be defined using theschedule metrics namelymakespan (119872) routing energy (119864)average link load (119871) and workload balance (119861) proposed inSection 3
The classification of metrics as inputs and outputs followsa simple ldquorule of thumbrdquo [16] If the value of a metric islarger-is-better then it is an output otherwise the metric isan input As a result of this classification the DMUmodel ofour schedule evaluation is as follows 119909 = (119872 119864 119871)
119879 arethe inputs and 119910 = 119861 is the output
Moreover in the rest of this paper the term ldquoschedulerdquo isreferring to a schedule that is discriminable using the four-metric classification If two schedules have identical metricsthey are regarded as the same schedule at least they are notdiscriminable under current metrics
With the schedulingDMUmodel the observed schedulesare defined as follows
Definition 1 (schedule set) The observed schedules or theschedule setΓ = 120574
1 120574
119894 120574
119899 = (119909
1 1199101) (119909
119894 119910119894)
(119909119899 119910119899) is the set of 119899 observed schedules where each
schedule 120574119894is defined by the input vector 119909
119894= (119872119894119864119894119871119894)119879
and output scalar 119910119894= 119861119894
Another concept that is relevant to DEA is the possi-ble production set The possible production set is the spaceenveloped by the efficient frontier on the multi-input mul-tioutput space Normally the possible production set isunknown in the DEA and needs to be constructed using theobserved DMUs The FDH possible production set postulateswere proposed in [23] Here under our context we restatethese postulates as the following axiom
Axiom 1 (possible schedule set) The possible schedule set(PSS) of a schedule set Γ satisfies the following
Postulate I PSS contains all schedules in Γ
Postulate II PSS contains unobserved schedule 120574119894if
(i) 120574119895= (119909119895 119910119895) isin Γ and 119909
119894= (119872119894119864119894119871119894) ge 119909119895or
(ii) 119910119894= 119861119894le 119910119895
Postulate II is the free disposal postulate which suggestsa free disposal hull of Γ [25] Together with the deterministPostulate I they define the FDH possible schedule set ofour schedule efficiency analysis Now we formally introduceFDH DEA to the schedule evaluation and define the efficientschedule as follows
Definition 2 (efficient schedule and efficient schedule set)In a schedule set Γ a schedule 120574
119894is called efficient if the
optimization problem (6) has an optimal solution of 119885lowast = 1otherwise 120574
119894is inefficientThe set of all efficient schedules in Γ
is called the efficient schedule set denoted by Γ119890 and likewise
the inefficient schedule set is denoted by Γ119894
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (1 sdot 119904+ + 119904minus)
st
120579119909119894minus 119883 sdot 120582 minus 119904
+
= 0119910119894minus 119884 sdot 120582 + 119904
minus
= 0
1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899
119904+
119904minus
ge 0
(6)
where 119909119894= (119872
119894119864119894119871119894)119879 and 119910
119894= 119861119894are the input vector
and output scalar of 120574119894 119899 is the number of schedules in Γ
119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 = (119910
11199102
sdot sdot sdot 119910119899) are
the input and output matrices of Γ 120582 = (12058211205822
sdot sdot sdot 120582119899)119879
is a binary vector in 119877119899 119904+ and 119904minus are the nonnegative slackvariables representing input excess and output shortfall 120576is a non-Archimedean infinitesimal constant and 120579 is theefficiency of 120574
119894
Optimization problem (6) is aMixed-Integer Programing(MIP)The binary vector 120582 and the constraint 1sdot120582 = 1 enforcea one-on-one comparison between target schedule 120574
119894and all
the schedules in Γ to search for a reference schedule 120574119903that
minimizes 120579The first two constraints of problem (6) can be simplified
to (7) by removing the slack variables Obviously the problemhas feasible solutions when 119897 = 119894 and 120579 = 1 in this situationFrom this point on if a reference schedule 120574
119903is found with
120579 lt 1 that means that 120574119903produces output 119910
119903at least as same
as 120574119894 with the inputs nomore than 120579119909
119894 which is a scale-down
from 119909119894 Then this makes 120574
119894inefficient
120579119909119894ge 119909119897
119910119894le 119910119897 119897 = 1 2 119899
(7)
However only 120579 = 1 is not enough for a schedule to beefficient Consider an efficient schedule 120574 = (119872 119864 119871 119861)
and an inefficient schedule 120574 = (119872 + Δ119898
119864 119871 119861) whichis distinguished from 120574 by a small input excess Δ
119898in the
makespan the constraint (7) holding for 120574 also holds for120574 thus 120579 = 1 for 120574 The slacks variables 119904+ and 119904minus areintroduced to remove these schedules with input excess andoutput shortfall The nonzero 119904+ and 119904minus force the objectivefunction in (6) to be less than 1
Mathematical Problems in Engineering 5
Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows
Definition 3 (dominant schedule) A schedule 120574119886is said to
dominate 120574119887if
(I) each component of 119909119886= (119872
119886119864119886119871119886)119879 is not
greater than that of 119909119887
(II) 119910119886= 119861119886is not less than 119910
119887
(IIIa) at least of component of 119909119886is less than the corre-
sponding one of 119909119887(dominates in input) or
(IIIb) 119910119886is greater than 119910
119887(dominates in output)
Then the relationship between efficient schedule anddominant schedule is given inTheorem 4
Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574
119894is called efficient if and only if no other schedule
dominates it
Proof If Part Suppose there is a schedule 120574119895that dominates
120574119894The efficiency of 120574
119894is calculated by solving the following
optimization problem
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (119904+
119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861)
st
120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872
119899) sdot 120582 minus 119904
+
119872= 0
120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904
+
119864= 0
120579 sdot 119871119894minus (11987111198712sdot sdot sdot 119871
119899) sdot 120582 minus 119904
+
119871= 0
119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904
minus
119861= 0
sum120582119897= 1 120582
119897isin 0 1 119897 = 1 2 119899
(8)
Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted
to
119872119894minus119872119895minus 119904+
119872= 0
119864119894minus 119864119895minus 119904+
119864= 0
119871119894minus 119871119895minus 119904+
119871= 0
119861119894minus 119861119895+ 119904minus
119861= 0
(9)
From the domination relation we have 119872119894ge 119872
119895
119864119894ge 119864119895 119871119894ge 119871119895 and 119861
119894le 119861119895 and at least one of above
inequations holds strictly This means at least one of 119904+119872 119904+119864
119904+
119871 and 119904minus
119861in (4) is not zero So the objective function119885 in (8)
has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861) lt 1
and schedule solution 120574119894is not efficient
Only If Part Suppose schedule 120574119894is not efficient then there
must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal
value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we
have
120579lowast
sdot 119872119894= 119872119895+ 119904+lowast
119872
120579lowast
sdot 119864119894= 119864119895+ 119904+lowast
119864
120579lowast
sdot 119871119894= 119871119895+ 119904+lowast
119871
119861119894= 119861119895minus 119904minuslowast
119861
(10)
The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast
119864+ 119904+lowast
119871+ 119904minuslowast
119861) lt 1
suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574
119894is dominated by 120574
119895
Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4
Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies
(1) 119872119894lt 119872119897 or
(2) 119864119894lt 119864119897 or
(3) 119871119894lt 119871119897 or
(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)
then it is in the efficient schedule set Γ119890
Proof If 120574119894satisfies one of the above conditions then there is
no schedule dominating 120574119894 From Theorem 4 120574
119894is efficient
Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule
Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ
119890
Proof The schedules in Γ119890are dominant ones and removing
any inefficient schedule will not change the dominant posi-tion of the elements in Γ
119890 Thus Γ
119890remains unchanged
43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required
FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally
6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
R R R R
R
R
RRRR
R
R R R
RR
PE
PE PE PE
PEPE PE
PE
PE
PE PE
PEPE PE
PE PE
(0 0) (1 0) (2 0) (3 0)
(1 1) (2 1) (3 1)
(1 2) (2 2) (3 2)
(0 1)
(0 2)
(0 3) (1 3) (2 3) (3 3)
(a) A 4 times 4 2D mesh NoC
East Inport
FIFO
Dec
oder
Req
ReqAck
Ack
East Outport
Arbiter
Sel
WestOutport
North Outport
North Inport
South Outport
South Inport
Local OutportLocal Inport
MU
X
West Inport
full in
east in
full out
west out
(b) Microstructure of NoC node
Figure 1 Architecture of NoC-based MPSoC
simulations results and discussion are given in Section 6Section 7 concludes the paper
2 Related Works
The multicriterion scheduling algorithms for CMPs havebeen widely researched in recent years In [12] a schedulingalgorithm is proposed for multicore processors to avoidresource contention aswell as to reduce energy consumptionA modified genetic algorithm which incorporates bacterio-logical algorithm is proposed in [13] to maximize the systemreliability and reduce makespan In [14] the optimization ofmakespan and workload balance is addressed and a NSGA-II based schedule algorithm is proposed for multicore-basedgrid Amultiobjective evolutionary algorithm (MOEA) basedschedule heuristic is proposed for the joint optimization ofperformance energy and temperature on multicore proces-sors in [15]
As for the DEArsquos application in the field of task schedulingonmulticore a FDH-based evaluationmethod for the assess-ment of schedule heuristics is proposed in [16] Althoughboth [16] and ourwork adoptDEAFDHmodel as the analytictool ourwork introduces the concept of cross efficiency to theFDHmodel and uses FDH cross efficiency to rank schedulesMoreover the incorporation of DEA evaluation method intometaheuristic also distinguishes itself from the work in [16]
3 Problem Formulation
31 Task Model In this paper tasks are modeled usingdirected acyclic graphs (DAGs) A DAG 119866 = (119881 119864) is anacyclic graph where 119881 is the set of nodes which represent thetasks and 119864 is the set of edges in which an element 119890
119894119895denotes
the communication from task 119894 to task 119895 The edge indicatesthe precedent relation between two tasks
Each node V119894and edge 119890
119894119895are associated with a weight
denoted by119862119894and119879119894119895 respectivelyWeight119862
119894is the computa-
tional load required by a processing Element (PE) to executetask 119894 and119879
119894119895is the data transmission load between task 119894 and
task 119895 In our work both 119862119894and 119879
119894119895are presented using time
unit (cycles)
32 Network-on-Chip Hardware The target hardware is a 2Dmesh NoC-based MPSoC as illustrated in Figure 1(a) EachPE is connected to a router and routers are interconnectedwith each other through bidirection links Data is transferredthrough NoC in the form of packets
PEs are homogenous processor cores with local datacache If two consequential tasks are scheduled to the samePE the successor task reads the predecessorrsquos data directlyfrom the data cache of the PE without routing in NoC
The microstructure of a NoC router is shown in Fig-ure 1(b) The router has five Inports and Outports corre-sponding to five directions of East West North South andLocal The decoder in the Inport scans the first flit of theFIFO for any incoming packet If the decoder detects thehead flit of a packet it performs XY routing algorithm andsends request signal to the arbiter of the correspondingOutport If the arbiter receives multiple request signals thecontention is solved using Round-Robin arbitration Thegranted Inport then forwards the packet to the downstreamrouter Wormhole routing is adopted to minimize the bufferrequirement as well as the packet latency [17] The backpressure mechanism is also employed to further reduce end-to-end delay [18]
The energy model of a NoC is presented by the BitEnergy proposed in [19] Analytically the average energyconsumption of transmitting one bit from node 119894 to node 119895is calculated by
119864119894119895
bit = 119899hops sdot 119864119873bit + (119899hops minus 1) sdot 119864119871bit (1)
Mathematical Problems in Engineering 3
where 119864119873bit
and 119864119871bit
represent the energy consumed on thenode and on the link respectively and 119899hops is the number ofnodes the bit passes on its way from node 119894 to node 119895
33 Monitored Metrics In this paper four common metricsof NoC are extracted and monitored for the assessmentof schedule solution and they are makespan data routingenergy average link load [20] and workload balance [21]
Makespan (the 119872 metric) which is the amount of timerequired by a NoC to finish entire tasks in a DAG followingthe instruction of a schedule is the timemetric of a schedulewhile the data routing Energy (the 119864 metric) is the totalamount of energy dissipated in each NoC component duringthe execution of the DAG
The average Link load (the 119871 metric) is calculated byadding up all the data transmission time (in cycles) on eachlink and then dividing it by the number of links (for the NoCin Figure 1(a) there are 48 links) The 119871 metric representshow busy the NoC is during the execution and a higheraverage link load metric implies a higher possibility of linkcontentions A good schedule is supposed to reduce the 119871metric
Finally the workload Balance (the 119861metric) is defined tobe the inverse coefficient of variant of the total workload oneach processor as shown in (2) The loadprc(119899) is the actualload on processor 119899 and the loadave is the average load The119861metric reflects the load balance of the processors
workload balance =loadave
(sum119899(loadave minus loadprc (119899))
2
)
12
(2)
4 DEA Evaluation of Schedules
41 A Brief Review of DEA Data envelopment analysis(DEA) is a nonparametric technique that is widely used tomeasure the relative efficiency among many-input many-output decision-making units (DMUs) which in our contextare the schedules The efficiency of a DMU is defined as theweighted sum of its output divided by the weighted sum ofits input The essence of DEA is that it allows each DMU tochoose a particular set of weight coefficients which favors itsown efficiency under the constraint that the efficiencies of allDMUs calculated by this set of coefficients do not exceed 1A DMU is ldquoefficientrdquo if the efficiency calculated by DEA is 1otherwise the DMU is marked as ldquoinefficientrdquo The followingLinear Programing (LP) problem is the CCR model of DEA(CCR multiplier form)
max 119885 = 119906119879
sdot 119910119894
st
V sdot 119909119894= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
V 119906 ge 0
(3)
where 119909119894= (119909
11989411199091198942
sdot sdot sdot 119909119894119898)119879
isin 119877119898 and 119910
119894=
(1199101198941
1199101198942
sdot sdot sdot 119910119894119904)119879
isin 119877119904 are the input and out-
put vectors of DMU 119894 V = (V1
V2
sdot sdot sdot V119898) and
119906 = (11990611199062
sdot sdot sdot 119906119904) are the coefficient (multiplier)
vectors of the inputs and outputs 119899 is the number of DMUsand the objective function 119885 is the efficiency of DMU 119894
The result of applying DEA is a classification amongDMUs as efficient group or inefficient group The efficientDMUs forman efficient frontier on themulti-inputmultiout-put space that envelops all inefficient DMUs The projectionof inefficient DMUon the efficient frontier is the hypotheticalefficient unit which is a linear combination of the efficientDMUs An inefficient DMU can also be converted to an effi-cient DMU by proportionally scaling down by the value of itsefficiency in the inputs and maintaining its original outputsThis interpretation of DEA efficiency is corresponding to theenvelop form of DEA which is the dual problem of (3) (CCRenvelop form)
min120579120582
119885 = 120579
st
120579119909119894minus 119883 sdot 120582 ge 0
119910119894minus 119884 sdot 120582 le 0
120582 ge 0
(4)
where 119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 =
(11991011199102
sdot sdot sdot 119910119899) are the input and output matrices
120582 = (12058211205822
sdot sdot sdot 120582119899)119879
isin 119877119899 is a nonnegative vector and
120579 is the efficiency of DMU 119894The way that the efficient frontier is generated differen-
tiates between DEA models which imply different returnsto scale assumptions There are four basic returns to scaleassumption constant returns to scale (CRS) correspondingto the CCR model [11] variable returns to scale (VRS)corresponding to the BCC model [22] increasing returns toscale and decreasing returns to scale
In this paper we focus on a special case of DEA namelythe free disposal hull (FDH) [23] In VRS FDH formulationof DEA each DMU is evaluated by comparing itself to otherDMUs on a one-on-one basis and a DMU is consideredefficient only when no other DMU dominates it
In most cases the idea of DEA that let DMU specifyits own weight coefficients to show its maximum advantageis desirable However in some extreme scenes a DMU canldquocheatrdquo a high efficiency score by weighting a single input ora single output and setting the rest weight coefficients closeto 0 This can happen when some DMUs have a particularlysmall input or particularly large output in our words theseDMUs have unbalanced metrics and these ldquomavericksrdquoneed to be depreciated Moreover although DEA effectivelydiscriminates between efficient and inefficient DMUs it doesnot further assess the relative advantages among the efficientones
One solution to the above questions is to introduce crossefficiency in the efficiencymeasurementThe concept of crossefficiency corresponding to the simple efficiency implied byoriginal DEA is the peer-appraisal equivalent of DEArsquos self-appraisal process The cross efficiency of a certain DMU isthe efficiency value calculated by using weight coefficientsderived by other DMUs If a DMU is a maverick or hasunbalanced metrics then its cross efficiency value derivedfrom other DMUrsquos coefficients is not likely to be high
4 Mathematical Problems in Engineering
Two widely accepted crossefficiency formulations theaggressive and benevolent formulations were proposedin [24] based on CCR Model Both formulations add asecondary goal to the normal DEA efficiency calculation(maximizing reference DMUrsquos efficiency) the aggressiveformulation minimizes target DMUrsquos crossefficiency whilethe benevolent formulation maximizes its efficiency Giventhat the simple DEA efficiency of DMU 119894 is 119864
119894119894 then the
crossefficiency of DMU 119895 evaluated by DMU 119894 is defined by(CCR crossefficiency benevolent formulation)
max 119864119895119894= 119906119879
sdot 119910119895
st
V sdot 119909119895= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
119864119894119894sdot V sdot 119909119894minus 119906 sdot 119910
119894= 0
V 119906 ge 0
(5)
42 FDHDEA Evaluation of Schedules In order to apply dataenvelopment analysis to the schedule evaluation the multi-input multioutput DMUmodel needs to be defined using theschedule metrics namelymakespan (119872) routing energy (119864)average link load (119871) and workload balance (119861) proposed inSection 3
The classification of metrics as inputs and outputs followsa simple ldquorule of thumbrdquo [16] If the value of a metric islarger-is-better then it is an output otherwise the metric isan input As a result of this classification the DMUmodel ofour schedule evaluation is as follows 119909 = (119872 119864 119871)
119879 arethe inputs and 119910 = 119861 is the output
Moreover in the rest of this paper the term ldquoschedulerdquo isreferring to a schedule that is discriminable using the four-metric classification If two schedules have identical metricsthey are regarded as the same schedule at least they are notdiscriminable under current metrics
With the schedulingDMUmodel the observed schedulesare defined as follows
Definition 1 (schedule set) The observed schedules or theschedule setΓ = 120574
1 120574
119894 120574
119899 = (119909
1 1199101) (119909
119894 119910119894)
(119909119899 119910119899) is the set of 119899 observed schedules where each
schedule 120574119894is defined by the input vector 119909
119894= (119872119894119864119894119871119894)119879
and output scalar 119910119894= 119861119894
Another concept that is relevant to DEA is the possi-ble production set The possible production set is the spaceenveloped by the efficient frontier on the multi-input mul-tioutput space Normally the possible production set isunknown in the DEA and needs to be constructed using theobserved DMUs The FDH possible production set postulateswere proposed in [23] Here under our context we restatethese postulates as the following axiom
Axiom 1 (possible schedule set) The possible schedule set(PSS) of a schedule set Γ satisfies the following
Postulate I PSS contains all schedules in Γ
Postulate II PSS contains unobserved schedule 120574119894if
(i) 120574119895= (119909119895 119910119895) isin Γ and 119909
119894= (119872119894119864119894119871119894) ge 119909119895or
(ii) 119910119894= 119861119894le 119910119895
Postulate II is the free disposal postulate which suggestsa free disposal hull of Γ [25] Together with the deterministPostulate I they define the FDH possible schedule set ofour schedule efficiency analysis Now we formally introduceFDH DEA to the schedule evaluation and define the efficientschedule as follows
Definition 2 (efficient schedule and efficient schedule set)In a schedule set Γ a schedule 120574
119894is called efficient if the
optimization problem (6) has an optimal solution of 119885lowast = 1otherwise 120574
119894is inefficientThe set of all efficient schedules in Γ
is called the efficient schedule set denoted by Γ119890 and likewise
the inefficient schedule set is denoted by Γ119894
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (1 sdot 119904+ + 119904minus)
st
120579119909119894minus 119883 sdot 120582 minus 119904
+
= 0119910119894minus 119884 sdot 120582 + 119904
minus
= 0
1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899
119904+
119904minus
ge 0
(6)
where 119909119894= (119872
119894119864119894119871119894)119879 and 119910
119894= 119861119894are the input vector
and output scalar of 120574119894 119899 is the number of schedules in Γ
119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 = (119910
11199102
sdot sdot sdot 119910119899) are
the input and output matrices of Γ 120582 = (12058211205822
sdot sdot sdot 120582119899)119879
is a binary vector in 119877119899 119904+ and 119904minus are the nonnegative slackvariables representing input excess and output shortfall 120576is a non-Archimedean infinitesimal constant and 120579 is theefficiency of 120574
119894
Optimization problem (6) is aMixed-Integer Programing(MIP)The binary vector 120582 and the constraint 1sdot120582 = 1 enforcea one-on-one comparison between target schedule 120574
119894and all
the schedules in Γ to search for a reference schedule 120574119903that
minimizes 120579The first two constraints of problem (6) can be simplified
to (7) by removing the slack variables Obviously the problemhas feasible solutions when 119897 = 119894 and 120579 = 1 in this situationFrom this point on if a reference schedule 120574
119903is found with
120579 lt 1 that means that 120574119903produces output 119910
119903at least as same
as 120574119894 with the inputs nomore than 120579119909
119894 which is a scale-down
from 119909119894 Then this makes 120574
119894inefficient
120579119909119894ge 119909119897
119910119894le 119910119897 119897 = 1 2 119899
(7)
However only 120579 = 1 is not enough for a schedule to beefficient Consider an efficient schedule 120574 = (119872 119864 119871 119861)
and an inefficient schedule 120574 = (119872 + Δ119898
119864 119871 119861) whichis distinguished from 120574 by a small input excess Δ
119898in the
makespan the constraint (7) holding for 120574 also holds for120574 thus 120579 = 1 for 120574 The slacks variables 119904+ and 119904minus areintroduced to remove these schedules with input excess andoutput shortfall The nonzero 119904+ and 119904minus force the objectivefunction in (6) to be less than 1
Mathematical Problems in Engineering 5
Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows
Definition 3 (dominant schedule) A schedule 120574119886is said to
dominate 120574119887if
(I) each component of 119909119886= (119872
119886119864119886119871119886)119879 is not
greater than that of 119909119887
(II) 119910119886= 119861119886is not less than 119910
119887
(IIIa) at least of component of 119909119886is less than the corre-
sponding one of 119909119887(dominates in input) or
(IIIb) 119910119886is greater than 119910
119887(dominates in output)
Then the relationship between efficient schedule anddominant schedule is given inTheorem 4
Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574
119894is called efficient if and only if no other schedule
dominates it
Proof If Part Suppose there is a schedule 120574119895that dominates
120574119894The efficiency of 120574
119894is calculated by solving the following
optimization problem
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (119904+
119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861)
st
120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872
119899) sdot 120582 minus 119904
+
119872= 0
120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904
+
119864= 0
120579 sdot 119871119894minus (11987111198712sdot sdot sdot 119871
119899) sdot 120582 minus 119904
+
119871= 0
119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904
minus
119861= 0
sum120582119897= 1 120582
119897isin 0 1 119897 = 1 2 119899
(8)
Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted
to
119872119894minus119872119895minus 119904+
119872= 0
119864119894minus 119864119895minus 119904+
119864= 0
119871119894minus 119871119895minus 119904+
119871= 0
119861119894minus 119861119895+ 119904minus
119861= 0
(9)
From the domination relation we have 119872119894ge 119872
119895
119864119894ge 119864119895 119871119894ge 119871119895 and 119861
119894le 119861119895 and at least one of above
inequations holds strictly This means at least one of 119904+119872 119904+119864
119904+
119871 and 119904minus
119861in (4) is not zero So the objective function119885 in (8)
has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861) lt 1
and schedule solution 120574119894is not efficient
Only If Part Suppose schedule 120574119894is not efficient then there
must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal
value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we
have
120579lowast
sdot 119872119894= 119872119895+ 119904+lowast
119872
120579lowast
sdot 119864119894= 119864119895+ 119904+lowast
119864
120579lowast
sdot 119871119894= 119871119895+ 119904+lowast
119871
119861119894= 119861119895minus 119904minuslowast
119861
(10)
The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast
119864+ 119904+lowast
119871+ 119904minuslowast
119861) lt 1
suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574
119894is dominated by 120574
119895
Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4
Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies
(1) 119872119894lt 119872119897 or
(2) 119864119894lt 119864119897 or
(3) 119871119894lt 119871119897 or
(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)
then it is in the efficient schedule set Γ119890
Proof If 120574119894satisfies one of the above conditions then there is
no schedule dominating 120574119894 From Theorem 4 120574
119894is efficient
Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule
Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ
119890
Proof The schedules in Γ119890are dominant ones and removing
any inefficient schedule will not change the dominant posi-tion of the elements in Γ
119890 Thus Γ
119890remains unchanged
43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required
FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally
6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
where 119864119873bit
and 119864119871bit
represent the energy consumed on thenode and on the link respectively and 119899hops is the number ofnodes the bit passes on its way from node 119894 to node 119895
33 Monitored Metrics In this paper four common metricsof NoC are extracted and monitored for the assessmentof schedule solution and they are makespan data routingenergy average link load [20] and workload balance [21]
Makespan (the 119872 metric) which is the amount of timerequired by a NoC to finish entire tasks in a DAG followingthe instruction of a schedule is the timemetric of a schedulewhile the data routing Energy (the 119864 metric) is the totalamount of energy dissipated in each NoC component duringthe execution of the DAG
The average Link load (the 119871 metric) is calculated byadding up all the data transmission time (in cycles) on eachlink and then dividing it by the number of links (for the NoCin Figure 1(a) there are 48 links) The 119871 metric representshow busy the NoC is during the execution and a higheraverage link load metric implies a higher possibility of linkcontentions A good schedule is supposed to reduce the 119871metric
Finally the workload Balance (the 119861metric) is defined tobe the inverse coefficient of variant of the total workload oneach processor as shown in (2) The loadprc(119899) is the actualload on processor 119899 and the loadave is the average load The119861metric reflects the load balance of the processors
workload balance =loadave
(sum119899(loadave minus loadprc (119899))
2
)
12
(2)
4 DEA Evaluation of Schedules
41 A Brief Review of DEA Data envelopment analysis(DEA) is a nonparametric technique that is widely used tomeasure the relative efficiency among many-input many-output decision-making units (DMUs) which in our contextare the schedules The efficiency of a DMU is defined as theweighted sum of its output divided by the weighted sum ofits input The essence of DEA is that it allows each DMU tochoose a particular set of weight coefficients which favors itsown efficiency under the constraint that the efficiencies of allDMUs calculated by this set of coefficients do not exceed 1A DMU is ldquoefficientrdquo if the efficiency calculated by DEA is 1otherwise the DMU is marked as ldquoinefficientrdquo The followingLinear Programing (LP) problem is the CCR model of DEA(CCR multiplier form)
max 119885 = 119906119879
sdot 119910119894
st
V sdot 119909119894= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
V 119906 ge 0
(3)
where 119909119894= (119909
11989411199091198942
sdot sdot sdot 119909119894119898)119879
isin 119877119898 and 119910
119894=
(1199101198941
1199101198942
sdot sdot sdot 119910119894119904)119879
isin 119877119904 are the input and out-
put vectors of DMU 119894 V = (V1
V2
sdot sdot sdot V119898) and
119906 = (11990611199062
sdot sdot sdot 119906119904) are the coefficient (multiplier)
vectors of the inputs and outputs 119899 is the number of DMUsand the objective function 119885 is the efficiency of DMU 119894
The result of applying DEA is a classification amongDMUs as efficient group or inefficient group The efficientDMUs forman efficient frontier on themulti-inputmultiout-put space that envelops all inefficient DMUs The projectionof inefficient DMUon the efficient frontier is the hypotheticalefficient unit which is a linear combination of the efficientDMUs An inefficient DMU can also be converted to an effi-cient DMU by proportionally scaling down by the value of itsefficiency in the inputs and maintaining its original outputsThis interpretation of DEA efficiency is corresponding to theenvelop form of DEA which is the dual problem of (3) (CCRenvelop form)
min120579120582
119885 = 120579
st
120579119909119894minus 119883 sdot 120582 ge 0
119910119894minus 119884 sdot 120582 le 0
120582 ge 0
(4)
where 119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 =
(11991011199102
sdot sdot sdot 119910119899) are the input and output matrices
120582 = (12058211205822
sdot sdot sdot 120582119899)119879
isin 119877119899 is a nonnegative vector and
120579 is the efficiency of DMU 119894The way that the efficient frontier is generated differen-
tiates between DEA models which imply different returnsto scale assumptions There are four basic returns to scaleassumption constant returns to scale (CRS) correspondingto the CCR model [11] variable returns to scale (VRS)corresponding to the BCC model [22] increasing returns toscale and decreasing returns to scale
In this paper we focus on a special case of DEA namelythe free disposal hull (FDH) [23] In VRS FDH formulationof DEA each DMU is evaluated by comparing itself to otherDMUs on a one-on-one basis and a DMU is consideredefficient only when no other DMU dominates it
In most cases the idea of DEA that let DMU specifyits own weight coefficients to show its maximum advantageis desirable However in some extreme scenes a DMU canldquocheatrdquo a high efficiency score by weighting a single input ora single output and setting the rest weight coefficients closeto 0 This can happen when some DMUs have a particularlysmall input or particularly large output in our words theseDMUs have unbalanced metrics and these ldquomavericksrdquoneed to be depreciated Moreover although DEA effectivelydiscriminates between efficient and inefficient DMUs it doesnot further assess the relative advantages among the efficientones
One solution to the above questions is to introduce crossefficiency in the efficiencymeasurementThe concept of crossefficiency corresponding to the simple efficiency implied byoriginal DEA is the peer-appraisal equivalent of DEArsquos self-appraisal process The cross efficiency of a certain DMU isthe efficiency value calculated by using weight coefficientsderived by other DMUs If a DMU is a maverick or hasunbalanced metrics then its cross efficiency value derivedfrom other DMUrsquos coefficients is not likely to be high
4 Mathematical Problems in Engineering
Two widely accepted crossefficiency formulations theaggressive and benevolent formulations were proposedin [24] based on CCR Model Both formulations add asecondary goal to the normal DEA efficiency calculation(maximizing reference DMUrsquos efficiency) the aggressiveformulation minimizes target DMUrsquos crossefficiency whilethe benevolent formulation maximizes its efficiency Giventhat the simple DEA efficiency of DMU 119894 is 119864
119894119894 then the
crossefficiency of DMU 119895 evaluated by DMU 119894 is defined by(CCR crossefficiency benevolent formulation)
max 119864119895119894= 119906119879
sdot 119910119895
st
V sdot 119909119895= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
119864119894119894sdot V sdot 119909119894minus 119906 sdot 119910
119894= 0
V 119906 ge 0
(5)
42 FDHDEA Evaluation of Schedules In order to apply dataenvelopment analysis to the schedule evaluation the multi-input multioutput DMUmodel needs to be defined using theschedule metrics namelymakespan (119872) routing energy (119864)average link load (119871) and workload balance (119861) proposed inSection 3
The classification of metrics as inputs and outputs followsa simple ldquorule of thumbrdquo [16] If the value of a metric islarger-is-better then it is an output otherwise the metric isan input As a result of this classification the DMUmodel ofour schedule evaluation is as follows 119909 = (119872 119864 119871)
119879 arethe inputs and 119910 = 119861 is the output
Moreover in the rest of this paper the term ldquoschedulerdquo isreferring to a schedule that is discriminable using the four-metric classification If two schedules have identical metricsthey are regarded as the same schedule at least they are notdiscriminable under current metrics
With the schedulingDMUmodel the observed schedulesare defined as follows
Definition 1 (schedule set) The observed schedules or theschedule setΓ = 120574
1 120574
119894 120574
119899 = (119909
1 1199101) (119909
119894 119910119894)
(119909119899 119910119899) is the set of 119899 observed schedules where each
schedule 120574119894is defined by the input vector 119909
119894= (119872119894119864119894119871119894)119879
and output scalar 119910119894= 119861119894
Another concept that is relevant to DEA is the possi-ble production set The possible production set is the spaceenveloped by the efficient frontier on the multi-input mul-tioutput space Normally the possible production set isunknown in the DEA and needs to be constructed using theobserved DMUs The FDH possible production set postulateswere proposed in [23] Here under our context we restatethese postulates as the following axiom
Axiom 1 (possible schedule set) The possible schedule set(PSS) of a schedule set Γ satisfies the following
Postulate I PSS contains all schedules in Γ
Postulate II PSS contains unobserved schedule 120574119894if
(i) 120574119895= (119909119895 119910119895) isin Γ and 119909
119894= (119872119894119864119894119871119894) ge 119909119895or
(ii) 119910119894= 119861119894le 119910119895
Postulate II is the free disposal postulate which suggestsa free disposal hull of Γ [25] Together with the deterministPostulate I they define the FDH possible schedule set ofour schedule efficiency analysis Now we formally introduceFDH DEA to the schedule evaluation and define the efficientschedule as follows
Definition 2 (efficient schedule and efficient schedule set)In a schedule set Γ a schedule 120574
119894is called efficient if the
optimization problem (6) has an optimal solution of 119885lowast = 1otherwise 120574
119894is inefficientThe set of all efficient schedules in Γ
is called the efficient schedule set denoted by Γ119890 and likewise
the inefficient schedule set is denoted by Γ119894
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (1 sdot 119904+ + 119904minus)
st
120579119909119894minus 119883 sdot 120582 minus 119904
+
= 0119910119894minus 119884 sdot 120582 + 119904
minus
= 0
1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899
119904+
119904minus
ge 0
(6)
where 119909119894= (119872
119894119864119894119871119894)119879 and 119910
119894= 119861119894are the input vector
and output scalar of 120574119894 119899 is the number of schedules in Γ
119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 = (119910
11199102
sdot sdot sdot 119910119899) are
the input and output matrices of Γ 120582 = (12058211205822
sdot sdot sdot 120582119899)119879
is a binary vector in 119877119899 119904+ and 119904minus are the nonnegative slackvariables representing input excess and output shortfall 120576is a non-Archimedean infinitesimal constant and 120579 is theefficiency of 120574
119894
Optimization problem (6) is aMixed-Integer Programing(MIP)The binary vector 120582 and the constraint 1sdot120582 = 1 enforcea one-on-one comparison between target schedule 120574
119894and all
the schedules in Γ to search for a reference schedule 120574119903that
minimizes 120579The first two constraints of problem (6) can be simplified
to (7) by removing the slack variables Obviously the problemhas feasible solutions when 119897 = 119894 and 120579 = 1 in this situationFrom this point on if a reference schedule 120574
119903is found with
120579 lt 1 that means that 120574119903produces output 119910
119903at least as same
as 120574119894 with the inputs nomore than 120579119909
119894 which is a scale-down
from 119909119894 Then this makes 120574
119894inefficient
120579119909119894ge 119909119897
119910119894le 119910119897 119897 = 1 2 119899
(7)
However only 120579 = 1 is not enough for a schedule to beefficient Consider an efficient schedule 120574 = (119872 119864 119871 119861)
and an inefficient schedule 120574 = (119872 + Δ119898
119864 119871 119861) whichis distinguished from 120574 by a small input excess Δ
119898in the
makespan the constraint (7) holding for 120574 also holds for120574 thus 120579 = 1 for 120574 The slacks variables 119904+ and 119904minus areintroduced to remove these schedules with input excess andoutput shortfall The nonzero 119904+ and 119904minus force the objectivefunction in (6) to be less than 1
Mathematical Problems in Engineering 5
Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows
Definition 3 (dominant schedule) A schedule 120574119886is said to
dominate 120574119887if
(I) each component of 119909119886= (119872
119886119864119886119871119886)119879 is not
greater than that of 119909119887
(II) 119910119886= 119861119886is not less than 119910
119887
(IIIa) at least of component of 119909119886is less than the corre-
sponding one of 119909119887(dominates in input) or
(IIIb) 119910119886is greater than 119910
119887(dominates in output)
Then the relationship between efficient schedule anddominant schedule is given inTheorem 4
Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574
119894is called efficient if and only if no other schedule
dominates it
Proof If Part Suppose there is a schedule 120574119895that dominates
120574119894The efficiency of 120574
119894is calculated by solving the following
optimization problem
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (119904+
119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861)
st
120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872
119899) sdot 120582 minus 119904
+
119872= 0
120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904
+
119864= 0
120579 sdot 119871119894minus (11987111198712sdot sdot sdot 119871
119899) sdot 120582 minus 119904
+
119871= 0
119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904
minus
119861= 0
sum120582119897= 1 120582
119897isin 0 1 119897 = 1 2 119899
(8)
Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted
to
119872119894minus119872119895minus 119904+
119872= 0
119864119894minus 119864119895minus 119904+
119864= 0
119871119894minus 119871119895minus 119904+
119871= 0
119861119894minus 119861119895+ 119904minus
119861= 0
(9)
From the domination relation we have 119872119894ge 119872
119895
119864119894ge 119864119895 119871119894ge 119871119895 and 119861
119894le 119861119895 and at least one of above
inequations holds strictly This means at least one of 119904+119872 119904+119864
119904+
119871 and 119904minus
119861in (4) is not zero So the objective function119885 in (8)
has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861) lt 1
and schedule solution 120574119894is not efficient
Only If Part Suppose schedule 120574119894is not efficient then there
must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal
value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we
have
120579lowast
sdot 119872119894= 119872119895+ 119904+lowast
119872
120579lowast
sdot 119864119894= 119864119895+ 119904+lowast
119864
120579lowast
sdot 119871119894= 119871119895+ 119904+lowast
119871
119861119894= 119861119895minus 119904minuslowast
119861
(10)
The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast
119864+ 119904+lowast
119871+ 119904minuslowast
119861) lt 1
suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574
119894is dominated by 120574
119895
Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4
Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies
(1) 119872119894lt 119872119897 or
(2) 119864119894lt 119864119897 or
(3) 119871119894lt 119871119897 or
(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)
then it is in the efficient schedule set Γ119890
Proof If 120574119894satisfies one of the above conditions then there is
no schedule dominating 120574119894 From Theorem 4 120574
119894is efficient
Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule
Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ
119890
Proof The schedules in Γ119890are dominant ones and removing
any inefficient schedule will not change the dominant posi-tion of the elements in Γ
119890 Thus Γ
119890remains unchanged
43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required
FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally
6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Two widely accepted crossefficiency formulations theaggressive and benevolent formulations were proposedin [24] based on CCR Model Both formulations add asecondary goal to the normal DEA efficiency calculation(maximizing reference DMUrsquos efficiency) the aggressiveformulation minimizes target DMUrsquos crossefficiency whilethe benevolent formulation maximizes its efficiency Giventhat the simple DEA efficiency of DMU 119894 is 119864
119894119894 then the
crossefficiency of DMU 119895 evaluated by DMU 119894 is defined by(CCR crossefficiency benevolent formulation)
max 119864119895119894= 119906119879
sdot 119910119895
st
V sdot 119909119895= 1
V sdot 119909119897minus 119906 sdot 119910
119897ge 0 119897 = 1 2 119899
119864119894119894sdot V sdot 119909119894minus 119906 sdot 119910
119894= 0
V 119906 ge 0
(5)
42 FDHDEA Evaluation of Schedules In order to apply dataenvelopment analysis to the schedule evaluation the multi-input multioutput DMUmodel needs to be defined using theschedule metrics namelymakespan (119872) routing energy (119864)average link load (119871) and workload balance (119861) proposed inSection 3
The classification of metrics as inputs and outputs followsa simple ldquorule of thumbrdquo [16] If the value of a metric islarger-is-better then it is an output otherwise the metric isan input As a result of this classification the DMUmodel ofour schedule evaluation is as follows 119909 = (119872 119864 119871)
119879 arethe inputs and 119910 = 119861 is the output
Moreover in the rest of this paper the term ldquoschedulerdquo isreferring to a schedule that is discriminable using the four-metric classification If two schedules have identical metricsthey are regarded as the same schedule at least they are notdiscriminable under current metrics
With the schedulingDMUmodel the observed schedulesare defined as follows
Definition 1 (schedule set) The observed schedules or theschedule setΓ = 120574
1 120574
119894 120574
119899 = (119909
1 1199101) (119909
119894 119910119894)
(119909119899 119910119899) is the set of 119899 observed schedules where each
schedule 120574119894is defined by the input vector 119909
119894= (119872119894119864119894119871119894)119879
and output scalar 119910119894= 119861119894
Another concept that is relevant to DEA is the possi-ble production set The possible production set is the spaceenveloped by the efficient frontier on the multi-input mul-tioutput space Normally the possible production set isunknown in the DEA and needs to be constructed using theobserved DMUs The FDH possible production set postulateswere proposed in [23] Here under our context we restatethese postulates as the following axiom
Axiom 1 (possible schedule set) The possible schedule set(PSS) of a schedule set Γ satisfies the following
Postulate I PSS contains all schedules in Γ
Postulate II PSS contains unobserved schedule 120574119894if
(i) 120574119895= (119909119895 119910119895) isin Γ and 119909
119894= (119872119894119864119894119871119894) ge 119909119895or
(ii) 119910119894= 119861119894le 119910119895
Postulate II is the free disposal postulate which suggestsa free disposal hull of Γ [25] Together with the deterministPostulate I they define the FDH possible schedule set ofour schedule efficiency analysis Now we formally introduceFDH DEA to the schedule evaluation and define the efficientschedule as follows
Definition 2 (efficient schedule and efficient schedule set)In a schedule set Γ a schedule 120574
119894is called efficient if the
optimization problem (6) has an optimal solution of 119885lowast = 1otherwise 120574
119894is inefficientThe set of all efficient schedules in Γ
is called the efficient schedule set denoted by Γ119890 and likewise
the inefficient schedule set is denoted by Γ119894
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (1 sdot 119904+ + 119904minus)
st
120579119909119894minus 119883 sdot 120582 minus 119904
+
= 0119910119894minus 119884 sdot 120582 + 119904
minus
= 0
1 sdot 120582 = 1 120582119897isin 0 1 119897 = 1 2 119899
119904+
119904minus
ge 0
(6)
where 119909119894= (119872
119894119864119894119871119894)119879 and 119910
119894= 119861119894are the input vector
and output scalar of 120574119894 119899 is the number of schedules in Γ
119883 = (11990911199092
sdot sdot sdot 119909119899) and 119884 = (119910
11199102
sdot sdot sdot 119910119899) are
the input and output matrices of Γ 120582 = (12058211205822
sdot sdot sdot 120582119899)119879
is a binary vector in 119877119899 119904+ and 119904minus are the nonnegative slackvariables representing input excess and output shortfall 120576is a non-Archimedean infinitesimal constant and 120579 is theefficiency of 120574
119894
Optimization problem (6) is aMixed-Integer Programing(MIP)The binary vector 120582 and the constraint 1sdot120582 = 1 enforcea one-on-one comparison between target schedule 120574
119894and all
the schedules in Γ to search for a reference schedule 120574119903that
minimizes 120579The first two constraints of problem (6) can be simplified
to (7) by removing the slack variables Obviously the problemhas feasible solutions when 119897 = 119894 and 120579 = 1 in this situationFrom this point on if a reference schedule 120574
119903is found with
120579 lt 1 that means that 120574119903produces output 119910
119903at least as same
as 120574119894 with the inputs nomore than 120579119909
119894 which is a scale-down
from 119909119894 Then this makes 120574
119894inefficient
120579119909119894ge 119909119897
119910119894le 119910119897 119897 = 1 2 119899
(7)
However only 120579 = 1 is not enough for a schedule to beefficient Consider an efficient schedule 120574 = (119872 119864 119871 119861)
and an inefficient schedule 120574 = (119872 + Δ119898
119864 119871 119861) whichis distinguished from 120574 by a small input excess Δ
119898in the
makespan the constraint (7) holding for 120574 also holds for120574 thus 120579 = 1 for 120574 The slacks variables 119904+ and 119904minus areintroduced to remove these schedules with input excess andoutput shortfall The nonzero 119904+ and 119904minus force the objectivefunction in (6) to be less than 1
Mathematical Problems in Engineering 5
Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows
Definition 3 (dominant schedule) A schedule 120574119886is said to
dominate 120574119887if
(I) each component of 119909119886= (119872
119886119864119886119871119886)119879 is not
greater than that of 119909119887
(II) 119910119886= 119861119886is not less than 119910
119887
(IIIa) at least of component of 119909119886is less than the corre-
sponding one of 119909119887(dominates in input) or
(IIIb) 119910119886is greater than 119910
119887(dominates in output)
Then the relationship between efficient schedule anddominant schedule is given inTheorem 4
Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574
119894is called efficient if and only if no other schedule
dominates it
Proof If Part Suppose there is a schedule 120574119895that dominates
120574119894The efficiency of 120574
119894is calculated by solving the following
optimization problem
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (119904+
119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861)
st
120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872
119899) sdot 120582 minus 119904
+
119872= 0
120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904
+
119864= 0
120579 sdot 119871119894minus (11987111198712sdot sdot sdot 119871
119899) sdot 120582 minus 119904
+
119871= 0
119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904
minus
119861= 0
sum120582119897= 1 120582
119897isin 0 1 119897 = 1 2 119899
(8)
Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted
to
119872119894minus119872119895minus 119904+
119872= 0
119864119894minus 119864119895minus 119904+
119864= 0
119871119894minus 119871119895minus 119904+
119871= 0
119861119894minus 119861119895+ 119904minus
119861= 0
(9)
From the domination relation we have 119872119894ge 119872
119895
119864119894ge 119864119895 119871119894ge 119871119895 and 119861
119894le 119861119895 and at least one of above
inequations holds strictly This means at least one of 119904+119872 119904+119864
119904+
119871 and 119904minus
119861in (4) is not zero So the objective function119885 in (8)
has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861) lt 1
and schedule solution 120574119894is not efficient
Only If Part Suppose schedule 120574119894is not efficient then there
must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal
value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we
have
120579lowast
sdot 119872119894= 119872119895+ 119904+lowast
119872
120579lowast
sdot 119864119894= 119864119895+ 119904+lowast
119864
120579lowast
sdot 119871119894= 119871119895+ 119904+lowast
119871
119861119894= 119861119895minus 119904minuslowast
119861
(10)
The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast
119864+ 119904+lowast
119871+ 119904minuslowast
119861) lt 1
suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574
119894is dominated by 120574
119895
Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4
Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies
(1) 119872119894lt 119872119897 or
(2) 119864119894lt 119864119897 or
(3) 119871119894lt 119871119897 or
(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)
then it is in the efficient schedule set Γ119890
Proof If 120574119894satisfies one of the above conditions then there is
no schedule dominating 120574119894 From Theorem 4 120574
119894is efficient
Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule
Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ
119890
Proof The schedules in Γ119890are dominant ones and removing
any inefficient schedule will not change the dominant posi-tion of the elements in Γ
119890 Thus Γ
119890remains unchanged
43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required
FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally
6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Definition 1 implies the dominancePareto optimality ofan efficient schedule A dominant schedule is defined asfollows
Definition 3 (dominant schedule) A schedule 120574119886is said to
dominate 120574119887if
(I) each component of 119909119886= (119872
119886119864119886119871119886)119879 is not
greater than that of 119909119887
(II) 119910119886= 119861119886is not less than 119910
119887
(IIIa) at least of component of 119909119886is less than the corre-
sponding one of 119909119887(dominates in input) or
(IIIb) 119910119886is greater than 119910
119887(dominates in output)
Then the relationship between efficient schedule anddominant schedule is given inTheorem 4
Theorem 4 (dominance and efficiency) In a schedule set Γa schedule 120574
119894is called efficient if and only if no other schedule
dominates it
Proof If Part Suppose there is a schedule 120574119895that dominates
120574119894The efficiency of 120574
119894is calculated by solving the following
optimization problem
min120579120582119904+119904minus
119885 = 120579 minus 120576 sdot (119904+
119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861)
st
120579 sdot 119872119894minus (11987211198722sdot sdot sdot 119872
119899) sdot 120582 minus 119904
+
119872= 0
120579 sdot 119864119894minus (11986411198642sdot sdot sdot 119864119899) sdot 120582 minus 119904
+
119864= 0
120579 sdot 119871119894minus (11987111198712sdot sdot sdot 119871
119899) sdot 120582 minus 119904
+
119871= 0
119861119894minus (11986111198612sdot sdot sdot 119861119899) sdot 120582 + 119904
minus
119861= 0
sum120582119897= 1 120582
119897isin 0 1 119897 = 1 2 119899
(8)
Let120582119895= 1 and 120579 = 1 then constraints of (8) are converted
to
119872119894minus119872119895minus 119904+
119872= 0
119864119894minus 119864119895minus 119904+
119864= 0
119871119894minus 119871119895minus 119904+
119871= 0
119861119894minus 119861119895+ 119904minus
119861= 0
(9)
From the domination relation we have 119872119894ge 119872
119895
119864119894ge 119864119895 119871119894ge 119871119895 and 119861
119894le 119861119895 and at least one of above
inequations holds strictly This means at least one of 119904+119872 119904+119864
119904+
119871 and 119904minus
119861in (4) is not zero So the objective function119885 in (8)
has a feasible solution of 1198851015840 = 1 minus 120576 sdot (119904+119872+ 119904+
119864+ 119904+
119871+ 119904minus
119861) lt 1
and schedule solution 120574119894is not efficient
Only If Part Suppose schedule 120574119894is not efficient then there
must exist a set of 120579lowast 120582lowast 119904+lowast and 119904minuslowast that makes the optimal
value of (8) be 119885lowast lt 1 Assuming 120582119895= 1 in 120582lowast from (8) we
have
120579lowast
sdot 119872119894= 119872119895+ 119904+lowast
119872
120579lowast
sdot 119864119894= 119864119895+ 119904+lowast
119864
120579lowast
sdot 119871119894= 119871119895+ 119904+lowast
119871
119861119894= 119861119895minus 119904minuslowast
119861
(10)
The optimal value of 119885lowast = 120579lowast minus 120576 sdot (119904+lowast119872+ 119904+lowast
119864+ 119904+lowast
119871+ 119904minuslowast
119861) lt 1
suggests that (I) 120579lowast = 1 and at least one of the 119904+lowast and 119904minuslowast isnot zero or (II) 120579lowast lt 1 Either of the above situations impliesthat schedule 120574
119894is dominated by 120574
119895
Theorem 4 relates the relatively abstract concept ofFDH efficiency to the concept of dominance Moreover thefollowing two corollaries are deduced fromTheorem 4
Corollary 5 In a schedule set Γ if a schedule 120574119894satisfies
(1) 119872119894lt 119872119897 or
(2) 119864119894lt 119864119897 or
(3) 119871119894lt 119871119897 or
(4) 119861119894gt 119861119897 (119897 = 1 2 119899 119897 = 119894)
then it is in the efficient schedule set Γ119890
Proof If 120574119894satisfies one of the above conditions then there is
no schedule dominating 120574119894 From Theorem 4 120574
119894is efficient
Corollary 5 points out that the schedule with the smallestmakespan or the smallest energy consumption or the smallestqueuing time or the best workload balance is an efficientschedule
Corollary 6 In a schedule set Γ removing any inefficientschedule from Γ does not change the elements in Γ
119890
Proof The schedules in Γ119890are dominant ones and removing
any inefficient schedule will not change the dominant posi-tion of the elements in Γ
119890 Thus Γ
119890remains unchanged
43 FDH Cross Evaluation of Schedules In this section across evaluation process is proposed based on peer-appraisalFDH DEA for further assessment of schedules DEA calcu-lates the efficiency of a DMU by allowing the DMU to choosea scenario that is best for itself Although this basis is plausiblein most situations some ldquomaverickrdquo DMUs especially theDMUs with a single small input or a single large output mayldquocheatrdquo DEA to achieve high score by valuing its only strengthand depreciating other metrics These unbalanced DMUsmust be devaluated during further assessment Moreover anassessment of relative advantages among efficient DMUs isalso required
FDHmodel is aMIP problem in nature In order to deriveits peer-appraisal variation the dual problem of the FDHDEA in (6) is needed to construct the formulation Normally
6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
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6 Mathematical Problems in Engineering
it is difficult to write the dual problem of a MIP however byexploring the particularity of the vector 120582 Agrell has provenin [26] that theMIP problem in FDHmodel can be simplifiedto a LP problem Using his result the FDH envelop form in(6) is reduced to the LP problem given in
min120579119897120582119904+119904minus
119885 = sum
119897
120579119897+ 120576 sdot sum
119897
(1 sdot 119904+119897+ 119904minus
119897)
st
120579119897119909119894minus 119909119897sdot 120582119897minus 119904+
119897= 0 119897 = 0 1 119899
(119910119894minus 119910119897) sdot 120582119897+ 119904minus
119897= 0 119897 = 0 1 119899
1 sdot 120582 = 1120582 119904+
119897 119904minus
119897ge 0
(11)
The dual problem of (11) is (FDH multiplier form equiv-alent)
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(12)
LP problem (12) is the multiplier form equivalent ofFDH model It also reveals the economic meaning of FDHCoefficients (119906 V) are the prices for output 119910 and input 119909Theprofit of DMU i under the price system (119906
119897 V119897) is calculated by
119906119897sdot 119910119894minus V119897sdot 119909119894 and the input cost of target DMU is normalized
V119897sdot 119909119894= 1 The second constraint of (11) is equivalent to
(119906119897sdot 119910119894minus 119911 sdot V
119897sdot 119909119894) minus (119906
119897sdot 119910119897minus V119897sdot 119909119897) ge 0 (13)
which suggests a nonnegative profit difference between theinput-scaled DMU 119894 and DMU 119897 The upper bound of scalefactor 119911 calculated by letting 119894 = 119897 is 119911 = 1 which indicatesa scaling down of DMU 119894rsquos input FDH scans all the DMUsto find a reference DMU and a price system with the largestscale-down factor 119911
Based on the FDH multiplier form equivalent given in(12) we now define the peer-appraisal FDH cross efficiencyas follows
Definition 7 (FDH cross efficiency benevolent formulation)Given that the FDH efficiency of schedule 120574
119894is 119864119894119894 the FDH
cross efficiency 119864119895119894of 120574119895evaluated by 120574
119894is the optimal value
of 119911 in
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(14)
Definition 7 is the FDH correspondence of the peer-appraisal CCR (benevolent formulation) in [24] which isreviewed in Section 41 The third constraint LP in (14)ensures that the efficiency of DMU i calculated by coefficients(119906119897 V119897) is the FDH efficiency 119864
119894119894(primary goal) and under
this constraint (14) searches for the best efficiency value of119864119895119894(secondary goal)The following two theorems reveal the relation between
FDH efficiency and FDH cross efficiency
Theorem 8 The cross efficiency of schedule 120574119894evaluated by
itself is its FDH efficiency
Proof Assume that the cross efficiency of schedule 120574119894evalu-
ated by itself is 119864119894119894and the FDH efficiency (simple efficiency)
is 119864The calculation of 119864
119894119894is solving the following LP
max119906119897V119897119911
119911
st
V119897sdot 119909119894= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894) le 0
119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(15)
Compared with the FDH multiplier form equivalent in(12) the LP in (15) has extra constraints of
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864 sdot 119909
119894)
= 119906119897sdot (119910119897minus 119910119894) minus V119897sdot 119909119897+ 119864 le 0 119897 = 1 2 119899
(16)
where 119864 is the optimal value of (12) Since 119911 = 119864 satisfies (12)the extra constraints of (16) always hold true That means 119864is also the optimal value of (15) thus 119864 = 119864
119894119894
Theorem 9 The cross efficiency 119864119895119894of schedule 120574j evaluated by
schedule 120574119894does not exceed the value of its simple efficiency 119864
119895119895
Proof It is obvious that the optimal solution in (14) is afeasible solution of calculating schedule 120574
119895rsquos FDH efficiency
using (12)
max119906119897V119897119911
119911
st
V119897sdot 119909119895= 1 119897 = 1 2 119899
119906119897sdot (119910119897minus 119910119895) minus V119897sdot 119909119897+ 119911 le 0 119897 = 1 2 119899
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(17)
Thus the optimal value of 119911 in (14) is not greater than theoptimal value of 119911 in (17)
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Using the peer-appraisal FDH proposed in (14) the crossefficiency of a DMU is defined as follows
Definition 10 (cross efficiencymatrix average cross efficiencyand the most efficient schedule) In a schedule set Γ with 119899schedules the cross efficiency matrix is defined by
119864cross =(
1198641111986412 1198641119899
1198642111986422 1198642119899
11986411989911198641198992 119864119899119899
) (18)
where 119864119895119894
is the FDH cross efficiency of 120574119895evaluated by
120574119894using (14) The average cross efficiency of DMU 119894 is
the average value of its cross efficiencies evaluated by allDMUs in Γ defined by 119864
119894= sum119899
119897=1119864119894119897119899 The most efficient
schedule (120574MES) is the schedule with the largest average crossefficiency
The cross efficiency matrix 119864cross is constructed to calcu-late the cross efficiency of each DMUThe diagonal elementsin 119864cross are the self-appraisal FDH efficiencies and the restof the elements are the peer-appraisal FDH efficiencies Theelements in 119894th rowof119864cross are the efficiencies ofDMU 119894 ratedby peers and the elements in 119894th column are the efficienciesof peers rated by DMU 119894 The cross efficiency of DMU 119894 isthe average value of the 119894th row And 120574MES is the best (bothefficient and well metrics balanced) schedule in Γ under ourevaluation system
Corollary 11 The average cross efficiency 119864119895of a schedule 120574
119895
which is defined in Definition 10 is not greater than its simpleefficiency 119864
119895119895
Proof Using the result of previous theorem the cross effi-ciency 119864
119895119894of schedule 120574
119895evaluated by arbitrary schedule 120574
119894is
not greater than the 119864119895119895 Thus 119864
119895 which is the average value
of 119864119895119894 is not greater than the 119864
119895119895
Then the relation between the FDH efficiency and themost efficient schedule is given in the following theorem
Theorem 12 The most efficient schedule of a schedule set Γ isan efficient schedule
Proof Suppose the most efficient schedule is an inefficientschedule and then there exits an efficient schedule thatdominates it Formally let 120574
120572= (119909
120572 119910120572) be the most
efficient schedule of Γ where 119909120572= (119872 119864 119871)
119879 is theinput vector and 119910
120572= 119861 is the output scalar Let 119911lowast
120572be
the cross efficiency of schedule 120574120572evaluated by schedule 120574
119894
and 119906lowast119897= 119906lowast
119897119887 Vlowast119897= (Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 119897 = 1 2 119899 are
the optimal coefficients Let 120574120573be a dominant schedule of
120574120572First assume that 120574
120573dominates 120574
120572in output (balance) and
120574120573= (119909120573 119910120573) where 119909
120573= 119909120572= (119872 119864 119871)
119879 and 119910120573=
119861 + Δ119861 has a small increment in balance Now we calculate
the cross efficiency of schedule 120574120573evaluated by schedule 120574
119894as
followsmax119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897sdot 119909120572= 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot [119910119897minus (119861 + Δ119861)] minus V
119897sdot 119909119897+ 119911120573
= 119906119897sdot (119910119897minus 119910120572) minus V119897sdot 119909119897+ (119911120573minus 119906119897119861sdot Δ119861) le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894) le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(19)
Compared with the calculation of the cross efficiency ofschedule 120574
120572evaluated by schedule 120574
119894 it is easy to verify that
120573= 119911lowast
120572+ max
119897(119906lowast
119897119861sdot Δ119861) 119906lowast
119897 Vlowast119897 119897 = 1 2 119899 is a feasible
solution of the above LP Then the optimal value of 119911120573is not
less than 119911lowast120572+ 119906lowast
119897119861sdot max119897(Δ119861) which is greater than 119911lowast
120572 That
means the cross efficiency of schedule 120574120573evaluated by an
arbitrary schedule 120574119894is greater than the 120574
120572 which contradicts
the assumption of 120574120572being the most efficient schedule
Then assume that 120574120573dominates 120574
120572in input Without
loss of generality we assume that 120574120573
dominates 120574120572
inmakespan and 120574
120573= (119909120573 119910120573) where 119910
120573= 119861 and 119909
120573=
(119872 minus Δ119872 119864 119871)119879 has a small decrement in makespan
Then the cross efficiency of schedule 120574120573evaluated by schedule
120574119894is calculated by solving the following LP
max119906119897V119897119911120573
119911120573
st
V119897sdot 119909120573= V119897119872sdot (119872 minus Δ119872) + V
119897119864sdot 119864 + V
119897119871sdot 119871 = 1
119906119897sdot (119910119897minus 119910120573) minus V119897sdot 119909119897+ 119911120573
= 119906119897119861sdot (119910119897minus 119910120573)
minus (V119897119872sdot 119909119897119872+ V119897119864sdot 119909119897119864+ V119897119871sdot 119909119897119871) + 119911120573le 0
119906119897sdot (119910119897minus 119910119894) minus V119897sdot (119909119897minus 119864119894119894sdot 119909119894)
= 119906119897sdot (119910119897minus 119910119894)
minus [V119897119872sdot (119909119897119872minus 119864119894119894sdot 119909119894119872)
+V119897119864sdot (119909119897119864minus 119864119894119894sdot 119909119894119864)
+V119897119871sdot (119909119897119871minus 119864119894119894sdot 119909119894119871)] le 0
119906119897ge 120576 V
119897ge 120576 sdot 1 119897 = 1 2 119899
(20)
Let V1015840119897
= (V1015840119897119872
V1015840119897119864
V1015840119897119871) = (Vlowast
119897119872(1 + Δ119872 sdot
Vlowast119897119872))(Vlowast119897119872
Vlowast119897119864
Vlowast119897119871) 1199061015840119897= (Vlowast119897119872(1 + Δ119872 sdot Vlowast
119897119872))119906lowast
119897 and
1199111015840
120573= 119911lowast
120572sdot (1 + Δ119872 sdot max
119897(Vlowast119897119872)) By substituting V1015840
119897 1199061015840119897 and
1199111015840
120573in (20) it is easy to verify that they are a feasible solution
of (20) Then the optimal solution of the optimal value of 119911120573
is not less than 119911lowast120572sdot (1 + Δ119872 sdotmax
119897(Vlowast119897119872)) which contradicts
the assumption of 120574120572being the most efficient schedule
Finally if 120574120573dominates 120574
120572in multiple metrics say 120574
120573=
(119872 minus Δ119872 119864 119871 119861 + Δ119861) intermediary schedules of 120574119894=
(119872 minus Δ119872 119864 119871 119861) and 120574119894= (119872 minus Δ119872 119864 119871 119861 + Δ119861)
can be constructed to prove that 120574120572is not the most efficient
schedule using the previous results
Now we will prove the unit invariance property of ourpeer-appraisal FDH proposal as well as the original FDHmodel
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Theorem 13 (unit invariance property) The values of optimalgoal in (5) and (12) are independent of the units that inputs andoutputs are measured in
Proof First we prove that FDH efficiency (5) is unit invariantThe FDHmodel in (5) is equivalent to the LP in (11) So if (11)is unit invariant then (5) is too
In a schedule set Γ = 120574119897= (119909119897 119910119897) | 119897 = 1 2 119899 let
119909119897= (119872
119897119864119897119871119897)119879 and 119910
119897= 119861119897be the input vector and
output scalar and let 119911lowast 119906lowast119897= (119906lowast
119897119872119906lowast
119897119864119906lowast
119897119871) and Vlowast
119897be the
optimal value for (12)Under the new unit system the inputs and outputs now
are the original values multiplied by conversion coefficientsdenoted by 1199091015840
119897= (119872
1015840
1198971198641015840
1198971198711015840
119897)119879
= (120572119872119872119897120572119864119864119897120572119871119871119897)119879
and 1199101015840119897= 120573119910119897 Then applying (11) to the new schedule set Γ1015840 =
1205741015840
119897= (1199091015840
119897 1199101015840
119897) | 119897 = 1 2 119899 obviously we have a feasible
solution of 1015840 = 119911lowast 1015840119897= (119906lowast
119897119872120572119872
119906lowast
119897119864120572119864119906lowast
119897119871120572119871) and
V1015840119897= Vlowast119897120573 which is transformed from the original problem
Therefore we have the optimal value of new problem 1199111015840lowast
ge
1015840 Suppose 1199111015840lowast gt
1015840 and the weight coefficients are1199061015840lowast
119897= (1199061015840lowast
1198971198721199061015840lowast
1198971198641199061015840lowast
119897119871) and V1015840lowast
119897under this circumstance
Converting theweight coefficients to the original unit systemwe have
119897= (1205721198721199061015840lowast
1198971198721205721198641199061015840lowast
1198971198641205721198711199061015840lowast
119897119871) and V
119897= 120573V1015840119897 which
also satisfy the constraints of original problem in (12) Thissuggests that 1199111015840lowast
119897 and V
119897are also a feasible solution of
original problem and 1199111015840lowast gt 1015840 = 119911lowast contradicts the optimalassumption of 119911lowast So 1199111015840lowast =
1015840 is the only possibility Thisproves that the FDH efficiency is not affected by the units thatinputs and outputs are measured in
Using this result the unit invariance of peer-appraisalFDH can be proved in the same manner
5 Efficient Scheduling Using CrosFDH-GA
51 Basic Design Themost straightforward way to introduceour FDH cross evaluation method to the genetic algorithm(GA) is to calculate the average cross efficiency of all indi-viduals in the pool and use the efficiency value as the fitnessof each individual The problem of this simple solution is thehigh computational requirement of DEA calculation
For a genetic algorithmwith 1000 population the calcula-tion of FDH simple efficiency of a single individual is to solvea LP with 4001 variables (119906
119897119861 V119897119872 V119897119864 v119897119871 119897 = 1 2 1000
and 119911) and 2000 constraints using (12) and the calculation ofFDH simple efficiency of all individuals is to solve 1000 suchLPs
Then the calculation of FDH cross efficiency of an indi-vidual evaluated by another individual is to solve a LP with4001 variables and 3000 constraints using (14) To calculatethe average cross efficiency of an individual solving 999 suchLPs is required And to calculate the average cross efficiencyof all individuals it requires repeating the process for 1000times
In our computing configuration (Intel i5-3210M 4GbRAM 32 bit Win7 VS2010 and GLPK 447) the calculationof a single FDH simple efficiency and a single FDH crossefficiency in the above DEA-GA implementation takes about36 seconds and 62 seconds on average The calculation of
Bottom Bottom Bottom BottomMetapopulation
Subpopulation MGoal makespan
Subpopulation EGoal energy
Subpopulation LGoal link
Subpopulation BGoal balance
Top 10Top 10Top 10Top 10
DEA-ready pool
Top 10 cross efficiency
Figure 2 The Subpopulation updating process
an average cross efficiency requires over 10 minutes TheDEA solving time of 1000 individuals in a single generationis estimated to be about 7 days If the GA runs for 50generations the whole solving time is near a year which isbeyond acceptable level
In this paper we propose a solution to this problemusing a ldquodivide-and-conquerrdquo methodThe whole population(Metapopulation) is divided in to 4 subpopulations Subpop-ulationM Subpopulation E Subpopulation L and Subpopula-tion B each of which experiences its own evolution towardsa single optimization goal (Makespan Energy average Linkload andworkload Balance correspondingly) In each gener-ation after the algorithm evaluates the performance of everyindividual the elites of each subpopulation are selected andregrouped as the DEA-ready pool for the DEA evaluationprocess The basic idea is that according to Theorem 4the more preeminent a schedule is in one metric the lesslikely it is dominated by another schedule Then the topperformers in the DEA-ready pool are duplicated to eachsubpopulation and replace the bottom individuals and thesubpopulations continue to evolve The process is shown inFigure 2
52 Genetic Operations In our proposal a chromosome oran individual represents one schedule solutionThe structureof a chromosome is an array with the size of processorsnumber and the value of its element 119888ℎ119903[119894] = 119895 representsthat task 119894 is assigned to processor 119895
The evolution of the four subpopulations is independenteach of which goes through a complete series of geneticoperationsThebasic framework of genetic algorithm is basedon the proposal in [27] as follows
Step 1 (initialization) The chromosome is randomly createdand added to the subpopulation When the number ofpopulation reaches subpopulation size algorithm goes to thenext step
Step 2 (evaluation) Performance of each individual in thepool is evaluated
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 1 Two randomly generated chromosomes
Chromosome Index1 2 3 4 5 6 7 8 9
chr1 3 1 3 2 2 3 1 2 3chr2 2 2 1 3 1 1 1 1 3
Step 3 (selection) Chromosomes are ordered according totheir subpopulationrsquos optimization goal and the top sel ratiochromosomes directly enter the next generationrsquos pool
Step 4 (crossover) Two randomchromosomes chr1 and chr2are selected from current pool and two new chromosomesnchr1 and nchr2 are generated by swapping middle part ofthe chromosome array This step produces population ofcros ratiolowast subpopulation size
Step 5 (mutation) A random chromosome is picked andvalues of two random positions of the chromosome areswapped to produce a new chromosome The populationgenerated in the mutation step accounts for mut ratio ofsubpopulation size
Step 6 (termination) GA is terminated after certain numberof generations If GAdoes notmeet its terminal condition thealgorithm iterates to Step 2 and repeats the whole process
For example consider a scheduling problem of 9 tasks(task 1simtask 9) scheduling to 3 processors (processor 1simprocessor 3) Two randomly generated chromosomes chr1and chr2 are listed in Table 1 Following the previousdefinition chr1 represents that task 2 and task 7 are scheduledto processor 1 tasks 4 5 and 8 are scheduled to processor2 and tasks 1 3 6 and 9 are scheduled to processor 3Chromosome chr2 represents that tasks 3 5 6 7 and 8 arescheduled to processor 1 task 1 and task 2 are scheduled toprocessor 2 and task 4 and task 9 are scheduled to processor3
In the crossover operation two randomly selected chro-mosomes chr1 and chr2 swap their middle part of chromo-some to form two new chromosomes nchr1 and nchr2 asshown in Figure 3
In the mutation operation a new chromosome nchr1is generated by randomly picking a chromosome chr1 andswapping two arbitrary positions in chr1 as shown in Figure 4
53 Cross Evaluation of Individuals The elites of each sub-population are selected to form a DEA-ready pool Then theDEA approach is applied to the individuals in this pool
First the FDH simple efficiency of every individual inthe pool is calculated and the inefficient individuals areremoved According to Corollary 5 the removal of inefficientDMUs does not change the dominant position of the efficientones Then the average cross efficiencies of the remainingindividuals are calculated and the individual with the largestvalue of average cross efficiency is marked as the mostefficient schedule
chr1
chr2
nchr1
nchr2
1 1
11 11 1
2 2 2
22
3 3 3 3
33
1 1 12 2 2
2
2 3 3
1111 3333
Crossover
Swap
Figure 3 The crossover operation
chr1
nchr1
1 12 2 223
1 1 2 2 23 3 3 3
33
SwapArbitraryposition 1
Arbitraryposition 2
Mutation
Figure 4 The mutation operation
The reason for the removal of inefficient DMUs is three-fold first of all as proven in Theorem 12 we know thatthe most efficient schedule which we are pursuing is notan inefficient schedule secondly the removal of inefficientschedules eliminates the influence of these obviously defectedschedules on the coefficients of the following calculation ofFDH cross efficiency (otherwise an inefficiency schedulewould become a constraint in the calculation of FDH crossefficiency according to (14)) finally it further reduces thecomputational demand of our algorithm
Pseudocode 1 shows cross evaluation process in ouralgorithm
6 Computational Experiments and Discussion
61 Simulation Results of FDH Cross Evaluation FormulationIn this section we extract 40 actual schedule solutions fromour simulation and use our proposed FDH cross evaluationas well as other DEA methods to analyze the performanceof these DMUs The chosen schedules are from the 5thgeneration of our CrosFDH-GA for a schedule problemof scheduling 100 tasks onto a 4 times 4 mesh NoC Top 10best schedules in each subpopulation are grouped as ourschedule setThe schedules are listed in Table 2 and for moreintuitive observation all the metrics shown are preprocessedby dividing the value by the average value of each metric inthe set As we have proved inTheorem 13 this normalizationprocess will not affect the value of the DMUrsquos efficiency
CCR efficiency (CCR) BCC efficiency (BCC) FDHefficiency (FDH) CCR super efficiency (Super CCR) CCRaverage Cross efficiency (Cros CCR) and FDH average Cross
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
BEGIN SCHEDULE-EVALUATION(1) FOR each schedule 120574
119897in the DEA-ready pool Γ DO
(11) Calculate 119864119897119897using FDH DEA
(2) END FOR(3) Regroup efficient schedule (119864
119897119897= 1) as Γ
119890
(4) FOR each schedule 120574119895in Γ119890
(41) FOR each schedule 120574119894= 120574119895in Γ119890DO
(411) Calculate the peer-appraisal FDH 119864119895119894
(42) END FOR(43) Calculate the average cross efficiency 119864
119895of 120574119895
(5) END FOR(6) Sorting the cross efficiency of schedules in Γ
119890
(7) Mark the schedule with the largest average cross efficiency as 120574MESEND SCHEDULE-EVALUATION
Pseudocode 1 Pseudocode of cross evaluation of DEA-ready pool
efficiency (Cros FDH) of each DMU are also calculated andlisted in Table 2 All the DEA formulations are solved usingGLPK [28]
Two kinds of FDH cross efficiencies Cros FDH (All)and Cros FDH (Eff) are presented in Table 2 The differencebetween these two is as follows Cros FDH (All) calculates theFDH cross efficiency using all 40 schedules while Cros FDH(Eff) only takes into account the FDH efficient ones whichis suggested in Section 53 The value of CrosFDH (Eff) forinefficient schedules is not calculated and marked with ldquomdashrdquo
Moreover a maverick index (MI) which is suggested in[24] is calculated for each FDH cross efficiency MI for DMU119894 is calculated by MI
119894= (119864119894119894minus 119864119894)119864119894119894
MI measures the difference between a DMUrsquos simpleefficiency and its cross efficiency The larger MI value impliesthat the DMU is more likely to be a maverick that ldquocheatsrdquoa high simple efficiency by choosing a particular set ofcoefficients that favors its only strength and depreciatingother metrics in our words a metric-unbalanced DMU
From Table 2 we observe that among the 3 simple effi-ciency formulations CCR has the smallest efficient schedulenumber of 2 while BCC has 9 efficient schedules and FDHhas 13 Both efficient schedules in CCR are efficient in BCCand FDH all 9 efficient schedules in BCC are also FDHefficient and FDH has 4 extra schedules schedule 4 schedule15 schedule 18 and schedule 19 than BCC The differencebetween the efficient schedule numbers of these three DEAmodels is caused by the different shapes of efficient frontierwhich is generated by different constraints in the modelformulations The convex-shaped efficient frontier of BCCin the multi-input multioutput space contains more DMUsthan the CCR efficient frontier while the staircase-like FDHefficient frontier has the largest number of efficient DMUson it Moreover the value of CCR efficiency of a scheduleis generally the smallest one among the three models andBCC is generally smaller than the FDH efficiency In fact aspointed out in [29] the FDH efficiencies are generally higherthan CCR and BCC efficiencies
ThreeDMUrankingmethods CCR super efficiency CCRcross efficiency and FDH cross efficiency are also compared
in Table 2 As observed in the table CCR super efficiencyand CCR cross efficiency are consistent with the CCR simpleefficiency and both CCR ranking methods mark schedule 32as the best schedule For the FDH average cross efficiency itis easy to tell that CrosFDH (all) is more discriminating thanthe previous methods The FDH average cross efficiencies of13 FDH efficient schedules vary from 09934 (schedule 20ranks 1 in 40 schedules) to 0905 (schedule 4 ranks 35 in 40schedules) The reason for an efficient schedule 4 achievingsuch a low average cross efficiency score is explained by itsMIvalue schedule 4 has a high MI of 0105 which suggests thatit ldquocheatsrdquo in the FDH simple efficiency calculation A closelook on its metrics reveals that it compromises too much onthe 119864 and 119871metricThe same situation happens with schedule31 (ranks 25 in 40) and schedule 32 (ranks 24 in 40) in whichboth are classified as CCR efficient schedules and schedule 32even being marked as the ldquobestrdquo schedule under CCR superefficiency and CCR cross efficiency analysis MIs of schedule31 and 32 are 00568 and 00553 the relevantly highMIs implythat they aremore likely to bemavericks And after examiningtheirmetrics it is shown that they are bothmetric unbalancedbecause they trade too much performance on the 119864 and 119871metric for the 119861metric
As to the two different implementations of the FDHaverage cross efficiency the Cros FDH (All) and Cros FDH(Eff) deliver very similar results on the DMU ranking Top 3schedules under Cros FDH (All) are schedules 20 17 and 18Comparing to the top 3 ranking of Cros FDH (Eff) schedules17 20 and 18 only a slight change of order exists whichvalidates the process of inefficient DMU removal in the crossevaluation of individuals proposed in Section 53
Moreover the average values of four metrics andMI (All)in each subpopulation are calculated and listed in Table 3The most interesting results are in the Subpopulation B Theaverage workload balance of Subpopulation B is almost astwice as the average value of the rest three subpopulationsHowever the performance of other metrics is not so goodin the Subpopulation B and is 12 154 and 16 largerthan the average value of the rest three subpopulations inmakespan energy and average link load respectively This
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table2DEA
analysisresults
of40
samples
chedules
Schedu
leSubp
op
Makespan
(inpu
t1)
Energy
(inpu
t2)
Link
(inpu
t3)
Balance
(outpu
t)CC
RBC
CFD
HSuperC
CRCr
osCC
RCr
osFD
H(A
ll)MI(All)
CrosFD
H(Eff)
MI(Eff
)
1M
09478
09695
09368
09657
06624
10000
10000
06624
06503
09878
00123
09917
00084
2M
09650
10624
10396
08970
05545
09821
09821
05545
05497
09371
004
80mdash
mdash3
M09683
09993
10154
07146
04522
09788
09788
04522
044
9209484
00320
mdashmdash
4M
09693
11701
11363
14241
08520
09933
1000
008520
08065
09050
01050
09123
00961
5M
09696
10624
10295
09227
05759
09775
09775
05759
05694
09368
00435
mdashmdash
6M
09732
10587
10557
06595
040
1409739
09739
040
1403991
09245
00534
mdashmdash
7M
09732
10736
10738
064
5303862
09739
09739
03862
03847
09184
006
05mdash
mdash8
M09748
11070
10759
09227
05511
09723
09723
05511
05472
09181
00590
mdashmdash
9M
09758
09955
09973
07551
04865
09721
09739
04865
04813
094
8800265
mdashmdash
10M
09758
09250
09429
07274
04957
09953
09963
04957
04889
09822
00144
mdashmdash
11E
1004
809064
09368
060
0204143
09918
09918
04143
040
55096
4900279
mdashmdash
12E
1004
809175
09288
060
0204143
09828
09918
04143
040
55096
4900279
mdashmdash
13E
1004
809138
09469
05826
03989
09839
09841
03989
03900
09611
00239
mdashmdash
14E
10103
09138
09409
05487
03757
09837
09837
03757
03689
09573
00276
mdashmdash
15E
10482
09138
09348
07274
05000
09875
10000
05000
04900
09490
00538
09704
00305
16E
1004
809138
09550
05759
03943
09839
09841
03943
03830
09599
00252
mdashmdash
17E
09745
09138
09187
09227
064
5410
000
1000
0064
5406340
09929
00072
09995
000
0518
E09745
09212
09227
09507
06620
09979
1000
006620
06504
09915
00086
09985
00015
19E
09761
09212
09248
09507
066
0609966
1000
0066
0606504
09915
00086
09985
00015
20E
09722
09212
09227
10153
07070
1000
010
000
07070
06947
09934
000
6609992
000
0821
L10
488
09287
09066
08850
06273
10000
10000
06273
06096
09631
00383
09896
00105
22L
09888
08990
09127
06695
04714
10000
10000
04714
04628
09819
00184
09903
000
9823
L10
087
09584
09147
09657
06784
10000
10000
06784
066
0109755
00251
09927
00074
24L
09729
09287
09147
05965
04191
10000
10000
04191
04106
09821
00183
09937
000
6325
L10
488
09435
09227
08734
06082
09850
09956
06082
05921
09550
004
26mdash
mdash26
L09791
09435
09227
08734
06082
09950
09956
06082
05955
09835
00124
mdashmdash
27L
10133
09175
09227
08412
05858
09924
09960
05858
05739
09701
00266
mdashmdash
28L
10133
09250
09248
09507
066
0609932
09978
066
06064
6809743
00242
mdashmdash
29L
09729
09435
09288
06320
04373
09929
09993
04373
04289
09771
00227
mdashmdash
30L
10495
09510
09288
07209
04988
09785
09892
04988
04857
09414
00507
mdashmdash
31B
09709
10550
10759
16743
10000
10000
10000
10013
09991
09463
00568
09535
004
8832
B09722
10475
10759
16743
10000
10000
10000
10071
09999
09476
00553
09550
00471
33B
10504
1166
411665
15996
08831
09220
09243
08831
08785
08682
006
46mdash
mdash34
B10
103
11590
11665
15996
09181
09586
09610
09181
08843
08848
00861
mdashmdash
35B
10221
11367
11403
15996
09076
09476
09499
09076
08996
08907
006
66mdash
mdash36
B10
279
10698
10859
15996
09465
09747
09907
09465
09408
09196
00774
mdashmdash
37B
10279
10698
10879
15341
09061
09590
09889
09061
09009
09185
00766
mdashmdash
38B
10504
11590
11585
15341
08510
09199
09287
08510
08477
08713
00659
mdashmdash
39B
10504
11293
11202
15341
08800
09314
09604
08800
08730
08893
00800
mdashmdash
40B
10534
10996
10799
15341
09129
09661
09963
09129
09011
09078
00975
mdashmdash
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 3 The average value of metrics of each subpopulation
Subpopulation Averagemakespan
Averageenergy
Averagelinkload
Averagebalance
AverageMI (All)
M 09618 09473 09399 08465 00455E 09885 09138 09298 08077 00217L 10491 09398 09177 08029 00279B 10121 10773 10779 16042 00727
phenomenon indicates that the optimization on the119861metricsis in conflict with other metrics especially with 119864 and 119871metrics The assertion is also supported by the MI (All)The average MI in Subpopulation B is 1293 larger thanthe average value of the rest three subpopulations whichsuggests that the schedules in Subpopulation B are extremelymetric unbalanced compared with the schedules in othersubpopulations Thus in order to achieve a highly metric-balanced schedule solution compromise must be made onthe 119861metric
62 Simulation Results of CrosFDH-GA Scheduling Algorithm
621 Simulation Setup In this section comparative simula-tions are made to evaluate the performance of our CrosFDH-GA scheduling algorithm Twenty DAGs tg1simtg20 are gener-ated usingTask Graph For Free (TGFF) [30]The task numberof generated DAGs varies from 50 to 100 Along with tworeal-world application solving laplace equation using Gauss-Seidel algorithm [31] and molecular dynamic coding [32] atotal of 22 task sets are simulated
The control groups of our simulation are four GAs withdifferent global objective functions and they are multipli-cation and division (MD) weighted sum (WS) weightedexponential sum (WES) and exponential weighted criterion(EWC) based on global criterion method [33] The fitnessfunctions are presented as follows
fitnessMD = 119872minus1
sdot 119864minus1
sdot 119871minus1
sdot 119861
fitnessWS = 119908119872119872minus1
+ 119908119864119864minus1
+ 119908119871119871minus1
+ 119908119861119861
fitnessWES = 119908119872119872minus2
+ 119908119864119864minus2
+ 119908119871119871minus2
+ 1199081198611198612
fitnessEWC = (119890119908119872 minus 1) sdot 119890
119872minus1
+ (119890119908119864 minus 1) sdot 119890
119864minus1
+ (119890119908119871 minus 1) sdot 119890
119871minus1
+ (119890119908119861 minus 1) sdot 119890
119861
(21)
where 119908119872 119908119864 119908119871 and 119908
119861are the weight coefficients of the
corresponding metric In our implementation 119908119872 119908119864 119908119871
and119908119861are all set to 025 which means there is no preference
between four metricsMoreover the sel ratio cros ratio and mut ratio of GA
are 02 04 and 04 and GA terminates its iteration after 50generationsThepopulation of four global objective function-based GAs is 1000The subpopulation size of CrosFDH-GA is250 which ensures that total population is 1000 individuals
All the output schedules are simulated under a System Cbased cycle-accurate NoC simulator which is a wormhole-routing modified version of [34] The implemented NoCsimulator is a 4 times 4 mesh NoC with the router structureillustrated in Figure 3 The link width is 16 bit and the FIFOdepth is one flit 119883119884 routing is used to forward packets andRR arbitration is adopted to solve contentions The NoCsimulator also integrates Orion 20 [35] to measure the actualrouting energy during execution of a DAG
622 Results and Discussion The simulation results of 22task sets which consist of 20 DAGs generated by TGFF andtwo real-world applications of solving Laplace equation (LE)using Gauss-Seidel algorithm andmolecular dynamic coding(MDC) under four global criterion GAs and our CrosFDHGA are illustrated in Figure 5 All the shown makespan andenergy metrics are the actual results measured in our NoCsimulator
As shown in Figure 5 it is observed that the four globalcriterion GAs render similar performance on the 119872 119864 119871and 119861 metrics while the proposed CrosFDH GA exhibits itsdifferent tendency on the optimization of metrics As shownin Figure 5(a) all five scheduling algorithms demonstratesame level of performance on the makespan The mainperformance difference exists in the optimization of theenergy (Figure 5(b)) average link load (Figure 5(c)) andworkload balance metrics (Figure 5(d)) The four globalcriterion GAs always output the schedule solution withthe best workload balance (in tg10 MD-GA and WS-GAdo not find the schedule solution with the best 119861 metricwithin 50 generations as the WES-GA and EWC-GA do)as shown in Figure 5(d) On the other hand our proposalalways compromises on the 119861 metrics and trades for betteroptimization on the 119864 and 119871metrics
To be more specific we normalize the 119872 119864 119871 and 119861metrics of the five scheduling algorithms to the CrosFDHcorrespondence in 22 task sets and calculate the average valueof thesemetrics for each scheduling algorithmThe results areshown in Table 4
From the table the four global criterion GAs haveeight times as much 119861 metric as CrosFDH-GA on averageHowever our proposal has better performance on both119864 and119871metrics The average energy of our algorithm is about 36smaller and the average link load is about 29 smaller thanthe rest algorithms on averageThis tendency of optimizationin the CrosFDH-GA is explained by the previous analysis ofschedulersquos FDH cross efficiency which suggests that a bettermetric-balanced schedule should trade the 119861metric for the 119864and 119871metrics
Moreover in our observation schedule solutions thatshow similar performance to the output schedule of thefour global criterion GAs exists in the Subpopulation B ofthe CrosFDH-GA However these schedule solutions aredepreciated during the peer-appraising process of FDH crossefficiency which supports the conclusion that the outputschedules of the global criterion GAs are metric unbalanced
Figure 5(e) illustrates the solving time (measured inseconds) of five algorithms and for illustrative presentation
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
0100020003000400050006000
Mak
espa
n (c
ycle
s)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(a) Makespan
14121
080604020
Ener
gy (J
)
times10minus7
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(b) Energy
6005004003002001000
Aver
age l
ink
load
(flits
)
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(c) Average link load
76543210W
orkl
oad
bala
nce
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20
(d) Workload balance
4
3
2
1
0
CrosFDHMDWS
WESEWC
LEM
DCtg1
tg2
tg3
tg4
tg5
tg6
tg7
tg8
tg9
tg10
tg11
tg12
tg13
tg14
tg15
tg16
tg17
tg18
tg19
tg20log10
(sol
ving
tim
e)
(e) Solving time
Figure 5 The comparative results of CrosFDH MD WS WES and EWC-GA
Table 4The average value of metrics of each scheduling algorithm
Algorithms Averagemakespan
Averageenergy
Averagelink load
Averagebalance
CrosFDH-GA 10000 10000 10000 10000MD-GA 09776 13451 14852 80536WS-GA 09992 14074 15757 80536WES-GA 10213 14214 15883 82483EWC-GA 09786 14323 16036 82483
the data in Figure 5(e) are preprocessed by applying log10to
the solving time of each algorithm According to the figureMD-GA has the smallest solving time in the five algorithmsThe WS-GA WES-GA and EWC-GA have 44 45 and546 more solving times than the MD-GA on averageThe CROSFDH-GA has the largest solving time which isnearly 109 times larger than MD-GA The reason of such
long solving time is caused by the high computation loadintroduced by DEA analysis In the CROSFDH-GA 902of the solving time is used to calculate the average FDH crossefficiency 83 of the solving time is used to calculate FDHsimple efficiency and the rest of the algorithm consumed only15 of the solving time
In MD-GA WS-GA WES-GA and EWC-GA a clearincreasing trend of solving time is observed as the scaleof the scheduling problem (task number) rises as shownin Figure 6(a) However such trend is not observed inCROSFDH-GA as shown in Figure 6(b)
The reason of this phenomenon in CROSFDH-GA is thatthe solving time of CROSFDH-GA is largely determined bythe calculation time of average FDH cross efficiency and thecalculation time of average FDH cross efficiency is dependingon the size of the schedule set which in our CROSFDH-GA isthe number of efficiency schedules in the DEA-ready pool ineach generation Thus the solving time of CROSFDH-GA isnot directly related to the task number of a schedule problem
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
40 50 60 70 80 90 10010
12
14
16
18
20
22
24
26
28
30
Solv
ing
time (
s)
Task number
MD-GAWS-GA
WES-GAEWC-GA
(a) MD- WS- WES- and EWC-GA
40 50 60 70 80 90 100500
1000
1500
2000
2500
3000
3500
4000
Task number
Solv
ing
time o
f DEA
-GA
(s)
(b) CROSFDH-GA
Figure 6 The relation of solving time and task number
but to the number of efficient schedules in the DEA-readypool in each generation
Figure 7(a) shows the relation between number of effi-cient schedules in the DEA-ready pool and the average solv-ing time of the average FDH cross efficiency in a generationAs shown in the figure the solving time rises rapidly as theefficient schedule number increases Moreover Figure 7(b)gives a statistic result of efficient schedule number duringall 22 simulations As observed in the figure most of thegenerations during the simulation have the efficient schedulenumber that lies between 30 and 70 which requires about 7to 100 seconds average FDH cross efficiency solving time
Finally Figure 8 demonstrates the trend of how fourmetrics converge during the iterationThe illustrated metricsare the best ones in the corresponding subpopulations in eachgeneration and are all normalized to the final value of the50th generation Figure 8 shows that the119872metric is the firstconverged metric followed by the 119864 metric and 119871 metric
10 20 30 40 50 60 70 80 900
50
100
150
200
250
300
Number of efficient schedules
Aver
age s
olvi
ng ti
me (
seco
nd)
(a) Solving time and efficient number
Number of efficient schedules10 20 30 40 50 60 70 80 90
0
1
2
3
4
Prop
ortio
n of
effici
ent n
umbe
r ()
45
35
25
15
05
(b) Efficient number distribution
Figure 7 Efficient schedule number in CROSFDH-GA
which both converge in the same pace And the 119861 metric isthe last converged metric
7 Conclusion
In this paper a FDH cross efficiency formulation as well as aCrosFDH-GA algorithm is proposed for the task schedulingproblem on the NoC-based MPSoC Four common metricsnamely makespan routing energy average link load andworkload balance are used to construct the multi-inputmultioutput DMUmodel After using FDH simple efficiencyto eliminate inefficient (dominated) schedules the peer-appraisal FDH cross efficiency is introduced to rankingschedules during which the maverick (metric-unbalanced)schedules are depreciatedThen a FDH cross efficiency-basedgenetic algorithm with four subpopulations each of whichoptimizes a single metric is proposed for solving actualscheduling problem on NoC According to our simulationresults the proposed FDH cross efficiency effectively distin-guishes the schedule solutions according to the balance oftheir metrics and our CrosFDH always outputsmoremetric-balanced schedules than other global criterion GAs
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
0 5 10 15 20 25 30 35 40 45 50Generations
Nor
mal
ized
met
rics
16
14
12
1
08
06
04
02
MakespanEnergy Workload balance
Average link load
Figure 8 The trend of convergence of four metrics
Conflict of Interests
The authors certify that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (61201005) the Fundamental ResearchFunds for the Central Universities (NoZYGX2011J006and NoZYGX2012J001) and the National Major Projects(2011ZX03003-003-04)
References
[1] A SManne ldquoOn the job-shop scheduling problemrdquoOperationsResearch vol 8 no 2 pp 219ndash223 1960
[2] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 no 9 pp 707ndash713 1966
[3] R R Muntz and E G Coffman Jr ldquoPreemptive scheduling ofreal-time tasks onmultiprocessor systemsrdquo Journal of the ACMvol 17 no 2 pp 324ndash338 1970
[4] T Malone ldquoEnterprise a market-like task scheduler for dis-tributed computing environmentsrdquo The Ecology of Computa-tion pp 177ndash205 1988
[5] X He X Sun and G Von Laszewski ldquoQoS guided Min-Minheuristic for grid task schedulingrdquo Journal of Computer Scienceand Technology vol 18 no 4 pp 442ndash451 2003
[6] D Geer ldquoIndustry trends chip makers turn to multicoreprocessorsrdquo Computer vol 38 no 5 pp 11ndash13 2005
[7] W J Dally and B Towles ldquoRoute packets not wires on-chipinterconnection networksrdquo in Proceedings of the 38th DesignAutomation Conference pp 684ndash689 June 2001
[8] W Vereecken W Van Heddeghem D Colle M Pickavet andP Demeester ldquoOverall ICT footprint and green communicationtechnologiesrdquo in Proceedings of the 4th International Symposiumon Communications Control and Signal Processing (ISCCSPrsquo10) pp 1ndash6 March 2010
[9] J Huang C Buckl A Raabe and A Knoll ldquoEnergy-awaretask allocation for network-on-chip based heterogeneous mul-tiprocessor systemsrdquo in Proceedings of the 19th International
Euromicro Conference on Parallel Distributed and Network-Based Processing (PDP rsquo11) pp 447ndash454 February 2011
[10] B Ge N Jing W He and Z Mao ldquoContention and energyaware mapping for real-time applications on Network-on-Chiprdquo in Proceedings of the International Conference in SoCDesign Conference (ISOCC rsquo12) pp 72ndash76 2012
[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978
[12] A Merkel J Stoess and F Bellosa ldquoResource-consciousscheduling for energy efficiency on multicore processorsrdquo inProceedings of the 5th ACM EuroSys Conference on ComputerSystems (EuroSys rsquo10) pp 153ndash166 April 2010
[13] O Sathappan P Chitra P Venkatesh and M Prabhu ldquoMod-ified genetic algorithm for multiobjective task scheduling onheterogeneous computing systemrdquo International Journal ofInformation Technology Communications and Convergence vol1 no 1 pp 146ndash158 2011
[14] J M N Abad S K Shekofteh H Tabatabaee and M Mehrne-jad ldquoCoreIIScheduler scheduling tasks in a multi-core-basedgrid using NSGA-II techniquerdquo in Intelligent Informatics pp507ndash518 Springer New York NY USA 2013
[15] H F Sheikh and I Ahmad ldquoFast algorithms for thermalconstrained performance optimization in DAG scheduling onmulti-core processorsrdquo in Proceedings of the International GreenComputing Conference (IGCC rsquo11) pp 1ndash6 July 2011
[16] A J Ruiz-Torres and F J Lopez ldquoUsing the FDH formulationof DEA to evaluate a multi-criteria problem in parallel machineschedulingrdquo Computers and Industrial Engineering vol 47 no2-3 pp 107ndash121 2004
[17] F Samman T Hollstein and M Glesner ldquoNew theory fordeadlock-free multicast routing in wormhole-switched virtual-channelless networks-on-chiprdquo IEEE Transactions on Paralleland Distributed Systems vol 22 no 4 pp 544ndash557 2011
[18] E Bolotin I Cidon R Ginosar and A Kolodny ldquoQNoC QoSarchitecture and design process for network on chiprdquo Journal ofSystems Architecture vol 50 no 2-3 pp 105ndash128 2004
[19] T T Ye L Benini and G De Micheli ldquoAnalysis of power con-sumption on switch fabrics in network routersrdquo in Proceedingsof the 39thAnnualDesignAutomationConference (DAC rsquo02) pp524ndash529 June 2002
[20] E Carvalho N Calazans and F Moraes ldquoHeuristics fordynamic task mapping in NoC-based heterogeneous MPSoCsrdquoin Proceedings of the 18th IEEEIFIP International Workshop onRapid System Prototyping (RSP rsquo07) pp 34ndash40 May 2007
[21] M Zakarya N Dilawar M A Khattak and H MaqssodldquoEnergy efficient workload balancing algorithm for real-timetasks over multi-corerdquo World Applied Sciences Journal vol 22no 10 pp 1431ndash1439 2013
[22] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984
[23] D Deprins L Simar and H Tulkens ldquoMeasuring labor-efficiency in post officesrdquo in Public Goods EnvironmentalExternalities and Fiscal Competition pp 285ndash309 SpringerNew York NY USA 2006
[24] J Doyle and R Green ldquoEfficiency and cross-efficiency in DEAderivations meanings and usesrdquo Journal of the OperationalResearch Society vol 45 no 5 pp 567ndash578 1994
[25] D McFadden ldquoCost revenue and profit functionsrdquo inHistoy ofEconomic Thought Chapters vol 1 1978
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[26] P J Agrell and J Tind ldquoA dual approach to nonconvex frontiermodelsrdquo Journal of Productivity Analysis vol 16 no 2 pp 129ndash147 2001
[27] O Sinnen L A Sousa and F E Sandnes ldquoToward a realistictask scheduling modelrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 3 pp 263ndash275 2006
[28] A Makhorin ldquoGLPK (GNU linear programming kit)rdquo 2006[29] H O Fried C A K Lovell and S S Schmidt The Measure-
ment of Productive Efficiency and Productivity Growth OxfordUniversity Press Oxford UK 2008
[30] R P Dick D L Rhodes and W Wolf ldquoTGFF task graphs forfreerdquo in Proceedings of the 1998 6th International Workshop onHardwareSoftware Codesign pp 97ndash101 March 1998
[31] M-Y Wu and D D Gajski ldquoHypertool a programming aidformessage-passing systemsrdquo IEEE Transactions on Parallel andDistributed Systems vol 1 no 3 pp 330ndash343 1990
[32] S Kim and J Browne ldquoA general approach to mapping ofparallel computation upon multiprocessor architecturesrdquo inProceedings of the International Conference on Parallel Process-ing 1988
[33] R T Marler and J S Arora ldquoSurvey of multi-objective opti-mization methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004
[34] C Song W Chang L Yubai and Y Zhongming ldquoA NoCsimulation and verification platform based on systemCrdquo inProceedings of the International Conference on Computer Scienceand Software Engineering (CSSE rsquo08) pp 423ndash426 December2008
[35] A B Kahng L Bin L Peh and K Samadi ldquoOrion 20 a fastand accurate NoC power and area model for early-stage designspace explorationrdquo in Proceedings of the conference on DesignAutomation and Test in Europe (DATE rsquo09) pp 423ndash428 April2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of