Research Article An Iterated Local Search Algorithm for a...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 659297 7 pageshttpdxdoiorg1011552013659297

Research ArticleAn Iterated Local Search Algorithm for a PlaceScheduling Problem

Shicheng Hu1 Zhaoze Zhang1 Qingsong He1 and Xuedong Sun2

1 School of Economics and Management Harbin Institute of Technology Weihai 264209 China2 School of Software Sun Yat-sen University Guangzhou 510275 China

Correspondence should be addressed to Xuedong Sun sunxdys163com

Received 15 August 2013 Accepted 9 September 2013

Academic Editor Yunqiang Yin

Copyright copy 2013 Shicheng Hu 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

We study the place scheduling problem which has many application backgrounds in realities For the block manufacturing projectwith special manufacturing platform requirements we propose a place resource schedule problem First the mathematical modelfor the place resource schedule problem is given On the basis of resource-constrained project scheduling problem and packingproblem we develop a hybrid heuristic method which combines priority rules and three-dimensional best fit algorithm in whichthe priority rules determine the scheduling order and the three-dimensional best fit algorithm solves the placement After thismethod is used to get an initial solution the iterated local search is employed to get an improvement Finally we use a set ofsimulation data to demonstrate the steps of the proposed method and verify its feasibility

1 Introduction

One of the main works in ship manufacturing is hull sectionprocessing This process is always implemented on the blockmanufacturing platform which usually takes a long timeto finish Because of the limited area it cannot meet therequirement of the shipbuilding enterprises Effective use oflimited resources has become a concern for the shipbuildingenterprises

For utilization of the platform in blockmanufacturing wepropose a place scheduling problem It includes two relatedproblems One is resource constrained project schedulingproblem (RCPSP) and the other is bin packing problem

The RCPSP is described as an individual project thatincludes 119899 activities these activities cannot stop during itsprocessing time There are two kinds of restrictions duringthe process One is precedence constraints It means thatactivity 119895 cannot start before its predecessor activity in 119875119895The other one is resource constraints To make the activitywork smoothly the activity 119895 needs 119896119895119903 unit resources duringits process We decide the finish time of every activity underthe priority and resource constraints to get the shortestproject makespan Hartmann proposes algorithm for static

job scheduling problem which belongs to NP-hard problem[1] Kelley Alvar-Valdes and Tamarit give approximate andexact methods [2 3] Alcaraz and Maroto develop a heuristicmethod based on priority rules [4] It consists of two partsa scheduling generation method and a priority rule Kolischsummarizes the scheduling generation methods into twocategories serial and parallel methods [5] Serial scheduleis implemented by the addition of activities while parallelschedule is produced with the increasing of the time Bakerconsiders that every activity should only be scheduled oncefor a single path [6]Mingozzi et al propose aRCPSPproblembased on a newmathematicalmodel and use an exactmethod[7]

Packing problem is about how to put more boxes intoa bin in which the bottom area is limited We can dividethe packing problem into two categories for discussion two-dimensional and three-dimensional packing problems Fortwo-dimensional packing problem Burke et al propose anew placement heuristic for the rectangular cutting issues[8] Egeblad and Pisinger provide a new method for thetwo-dimensional packing problem and extend to three-dimensional packing problem [9] In three-dimensionalpacking the method of 3BF is based on searching the list in

2 Mathematical Problems in Engineering

0m + 1

Figure 1 A network diagram

order Similar to 2BF algorithm it searches dynamically tofind suitable boxes to be put in the available space The 3BFdefines that blank area is at the farthest point where the boxesare to be putThe 3BF is raised byMartello et al [10] it solvesthe problem of how to select suitable boxes and the pointcannot be blocked by other boxes

This paper is organized as follows Section 2 gives amathematical model Section 3 proposes an iterated localsearch algorithm Section 4 presents a set of simulationexamples to demonstrate the method Section 5 comes upwith a summary for this paper

2 Place Scheduling Problem

The project consists of 119898 activities119872 = 1 2 119898 Everyactivity should be processed on the required platform thatis place We define it as 119903119894119903 = 119886119894119903 times 119887119894119903 where 119886119894119903 and119887119894119903 denote the length of the place 119903 that activity 119894 requiresrespectively The duration that activity 119894 requires is definedas 119905119894 119878119894 and 119865119894 represent the start and finish time of theactivity 119894 respectively In the project there are 119899 kinds ofplace recourses and every place resource is given by 119877119894119903 =119860 119894119903 times 119861119894119903 where 119860 119894119903 and 119861119894119903 denote the length and width ofthe available place 119903 respectively There are 119899 types of places119873 = 1 2 119899 All activities have priority constraints 119876119895is the predecessors of 119895 We assume that the start time of theproject is 0 We decide the start time of the activities so thatthe makespan of the project is minimized

Formodeling this problem two dummynodes 0 and119898+1are introduced 0 is the predecessor of all the activities and119898 + 1 is the successor of all the activities in the projectThe duration and required place resource of the two dummynodes are 0 Obviously the finish time of 119898 + 1 is themakespan of the project so the problem can be formulatedas follows

minimize119865119898+1 (1)

Priorities between activities can be defined as

119878119894 + 119905119894 le 119878119895 119894 isin 119876119895 (2)

We can show it by a network diagram as in Figure 1

We regard an activity as a cuboid and it is defined as119908119894119903 =119886119894119903 times 119887119894119903 times 119905119894

119908119894119903 cap 119908119895119903 = 120593 forall119894 119895 isin 119872

119894 lt 119895 forall119903 isin 119873

(3)

[119892119894119887119894119903 + (1 minus 119892119894) 119886119894119903]

le 119909119894 + [119892119894119887119894119903 + (1 minus 119892119894) 119886119894119903]

le 119861119903

(4)

[119892119894119886119894119903 + (1 minus 119892119894) 119887119894119903]

le 119910119894 + [119892119894119887119894119903 + (1 minus 119892119894) 119887119894119903]

le 119860119903

(5)

119892119894 isin 0 1 (6)

Constraint (3) denotes that any two cuboids cannotoverlap Constraints (4) and (5) denote that any cuboidshould be finished within the available place Constraint (6)means that the cuboid can rotate ninety degrees which isrepresented by two values

3 Iterated Local Search Algorithm

31 The Initial Solution Method For the initial solution wedevelop a hybrid heuristic method of project schedulingproblem and packing problem to solve it Each activity isabstracted as a cuboid and is put in the available placeAs we put the cuboid into the space we divide differentplaces according to different heights Every place is a two-dimensional area

There aremanymethods of how to place the rectangleWewill use best fit [8] heuristic algorithm to solve our problemWe cannot just rely on the size of the rectangle to judge whichis more suitable because cuboids have their sequences Butin the Best Fit method the standard to select the cuboid to bemoved only relies on the lengths and widths of the unselectedcuboids Inspired by this we should consider the sequencebefore we select the cuboids using the Best Fit method Weplace the larger bottom area first if they are equal select theone with the higher height

Before placing it we can regard the bottom area as anobject in a two-dimensional packing problem After packingwe can use some methods of the three-dimensional packingWe can put them according to some rules Length widthand height can be interchanged when we put them But inour problem the height is the duration so only length andwidth can be interchanged This is different from the three-dimensional packing There are many methods regardingthe three-dimensional packing problem We mainly use thealgorithmofCrainic et al which is based on the extremepoint[13] Its main objective is to determine the placement Weplace the bottom-left-rear point on the selected placementThe extreme point is represented as 119909 119910 119905 We select thelocation based on the extreme point

Mathematical Problems in Engineering 3

Table 1 Priority rules

Priority rule Paper V(119895)

MTS Alvar-Valdes and Tamarit [3] |119878119895|

LST Alvar-Valdes and Tamarit [3] LFT119895 minus 119889119895GRPW Alvar-Valdes and Tamarit [3] 119889119895 + sum119894isin119904119895

119889119894

WRUP Ulusoy and Ozdamar [11] 07

10038161003816100381610038161003816119878119895

10038161003816100381610038161003816+ 03sum119903isin119877 119896119895119903119870119903

LFT Davis and Patterson [12] LFT119895MSLK Davis and Patterson [12] LFT119895 minus EFT1015840119895

We give the description on how to select the position asfollows

(1) Put the first activity on the origin (0 0 0) and then itwill produce three new extreme points

(2) Put the second activity by our 119905 minus 119910 minus 119909 rule whichmeans that the smaller 119905 among the extreme pointsis preferred Then the smaller 119910 will be consideredFinally the 119909will be selectedMeanwhile we considerrotation when we place the boxes

(3) If an activity cannot match the placement we will useanother activity In theworse condition if all activitiescannot match the placement we will find anotherplacement to find a bigger 119905 as a placement

(4) If the extreme point of 119905119894 cannot meet the activityrsquosrequirement we will delete this point In other wordswe will not search for it again in the next stage

Priority rule based scheduling is an important method toRCPSP It is straightforward and easy to use There are manypriorities the better ones are presented in Table 1

We cannot use these rules directly but we can learn someideas like MTS It is sorted by the minimum of the latestfinish time We use it in the improvement algorithm In ourrule the larger bottom area will be considered first if they arethe same the longer duration (ie 119905) will be considered

It is closely related to priority and schedule generationscheme (SGS) SGS gets a feasible schedule by local searchLocal searchmeans arranging part of activities in stageThereare two different SGSs We call them serial and parallel Thearrangement is described as follows

(1) We select the activities from the network graphaccording to the priority rule Then we put themin the set 119863119892 which means that activities can bearranged

(2) We will select an activity from119863119892 The larger bottomarea will be considered first if they are the same thelonger duration (ie 119905) will be considered

(3) We put the activities into the set 119878119892 which means thatactivities have been scheduledThis is one cycleThena new cycle will begin

(4) According to the rule we arrange new activities whichare in the set 119863119892 on the extreme point If it cannotbe placed we will consider the rotation If not the

second in the set 119863119892 will be placed The process willnot finish until all the activities are scheduled

There are some kinds of placement We consider puttingthe cuboid along the 119909-axis or the 119910-axis

32 Optimization Solution Method For the initial solutionwe solve it by a rule so the result is not necessarily thebest Iterated local search will be used to improve the initialsolution by searching the local neighborhood to producelocal optimum

321 Local Search Savelsbergh proposes the ideas of localsearch to solve the time window problem [14] Voudouris andTsang apply local search in the traveling salesman problem[15] Schaerf and Meisels use local search in the employeetimetabling problems [16] Marco et al reviews the localsearch and introduces specific applications [17] The idea ofthe local search is to search for the neighborhood of theinitial solution and then iterate constantly Local search isnot only attractive but also very successful in practice Inrecent research combinatorial optimization and continuousoptimization problems can be applied by local search Thereis essential difference in search space type between combi-natorial optimization problems and continuous optimizationproblems The space of combinatorial optimization is finitebut the space of the continuous optimization is infinite Weuse the space of the combinatorial optimization and then thelocal search will be used to find the optimal

Lucio et al use the local search in a network planningalgorithm that incorporates uncertain traffic demands [18]Cabido et al propose local search driven by adaptive scaleestimation on Gpus [19] Funke et al use it in the vehiclerouting and scheduling problems [20] Tricoire proposesmultidirectional local search [21] Ricca and Simeone applyit in political districting [22] Mills et al propose an extendedguided local search to the quadratic assignment problem [23]

322 Iterated Local Search Iterated local search is widelyused because of its simplicity and efficiency It starts thesearch from the initial solution So before we use it aninitial solution should be obtained In general amore optimalsolution can be found based on several initial solutions Fornondeterministic polynomial (NP) problem it is difficult toenumerate all circumstances so we find a generally better


A B

A

FEDC

C D B E F

Old

New

Figure 2 The demonstration of local search

solution From an initial solution we can find a bettersolution by constantly search It may take more time

The key to the iterated local search is searching the localneighborhood constantly Then we decide whether it is thesolution we want It is as follows

(1) Get an initial solution(2) Start the local search from the initial solution(3) Get another solution by perturbation(4) Start the local search from the recent solution(5) Compare and choose a better solution(6) Repeat until the termination condition is met

For combinatorial optimization problems the candidatefeasible solutions in the neighborhood of the feasible solutionare countable Local optimum can be defined as forall119904 isin 119873(119904

119897)

We get 119891(119904) le 119891(119904119897) The 119891 is the objective function We

can enumerate all the candidate solutions and check them ifthey are better than the current solution If there is no bettersolution the current solution is the optimal

It is very important to learn the concept of neighborhoodfor the local search If a feasible solution 1199041015840can change into afeasible solution 119904 by a step a feasible solution 1199041015840 can be calleda neighboring feasible solution 119904 Neighborhood is the set ofthe feasible solution 119904 It begins by local search to find localoptimum The initial solution changes after being constantlyperturbed We can find a new local optimum this way Thepurpose of perturbation is to find different solutions Thenwe compare them to find a better one

The local search is demonstrated in Figure 2There are sixactivities A B C D E and FThe old stands for the previoussolution The new means the current solution after the localsearch Local search is based on the old solution the new oneis the one among the neighboring feasible solutions of the old

In Figure 2 the B is inserted into the front of theD to forma new solution In our problem we can consider the activitiesthat start at the same time to exchange to get a new sequence

4 Simulation Experiment

We use a set of simulation data to show our algorithm Thisset of data is adapted from PSPLIB by Kolisch and Sprecher[24] in which any two resources are set as the length andwidth of a place respectively One of the generated instancesis described as follows

There are two places the length and width of place 1 are3 and 2 are 3 and 2 for place 2 The activities of odd numberswork on place 1The activities of even numbers work on place2 The network diagram is given in Figure 3

1

2

3

4

5

10

7 15 13

6

8

9

14

15

16

17

18

12

Figure 3 The network diagram

Table 2 The parameters of activities

Activity Duration Length width1 0 0 02 8 2 13 4 2 24 6 2 15 3 2 26 8 2 17 5 2 18 6 3 19 2 3 210 3 2 111 7 2 212 2 2 213 7 3 114 9 2 115 4 3 216 6 2 117 2 2 118 0 0 0

The data of activities is presented in Table 2The arrangement steps described as follows

(1) At first the lowest time is 0 the activities that can bearranged are 2 3 4 5 and the odd activities of place1 are 3 5 The bottom area of them is the same sowe compare their duration We arrange activity 3 atfirst According to our rule we put bottom left rear ofactivity 3 on the (0 0 0) Then it produces three newextreme points see Figure 4

(2) At time 0 there is not enough space for activity 5 sowe will consider it later On place 2 both activity 2and activity 4 can be arranged We lay activity 2 firston the (0 0 0) Then it produces three new extremepoints (2 0 0) (0 1 0) (0 0 8) see Figure 5


x1

y1

t1

3

4

Figure 4 The status while activity 3 is put

(3) At time 0 activity 4will be put in order After rotationit lays on the (2 0 0) see Figure 6

(4) There is still available space at time 0 but it cannotmeet the activity needsWe should find another spaceAccording to our rule we search for the space fromtime 4 At time 4 there are some activities to bearranged 5 6 7 which include unscheduled activi-ties at time 0 According to the area and duration theactivity sequence of place 1 is 5 7 We put activity 5on the (0 0 4) see Figure 7

(5) All activities that start from time 4 are finished Weselect time 6 as the start searching time based onthe arranged activities At time 6 8 9 can be putBut 8 9 cannot match the space so we continueto search for the next lowest time According to thearranged activities the next earliest time is time 7Check the network diagram 14 15 can be joined inincluding 8 9 On place 1 we arrange activity 15 firstbased on our rule see Figure 8

(6) There is not enough space to put activity 9 at time 7We will consider it later On the place 2 8 14 can bearranged Activity 8 should be put first but it cannotmatch the placement So we lay the activity 14 on(2 0 7) see Figure 9

(7) After we arrange all the activities at time 7 the nextlowest time will be found to start the placement Stepswill continue until all the activities are arranged

Improvement Process The sequence of the activities is veryimportant to complete the duration So we use ILS to disturbcurrent searching point which can cause random selection inthe local optimumThe ILS includes three stepsThe first stepis producing the initial solution the second step is searchingin the neighbor to get a local optimum and the third is aloop which has three parts The first part is disturbing thelocal optimum to get a new sequence The second part issearching a solution based on the new sequence The third

t2

8

2

y2

x2


4

y2

x2

t2

8

2

6


x1

y1

t1

4

3

5

7



7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12


part is compared with previous solution to decide which isbetter until the termination condition is met

As in our example we swap two activities that start at thesame time during the local search The others are arrangedaccording to our normal rule The perturbation of ILS in ourexample is that any two activities are swapped at any time inthe local optimumThe activities that have been swapped willnot be consideredThe others are scheduled according to ournormal rule

The initial solution of this example is 27 After ILS theoptimal solution is 21 producing about 22 improvementFor the set of instances the average improvement is 21

5 Conclusion

For the proposed place scheduling problem we have devel-oped a hybrid heuristic method to generate the initialsolution In order to produce local optimal solution wehave designed ILS method to create improvement for theinitial solution In the improvement the random disturbancehas an impact on the result So the improvement based onrandomness will be the future research

Conflict of InterestsWe declare that we have no financial and personal relation-ships with other people or organizations that can inappropri-ately influence our work and we also declare that there is noconflict of interests regarding the publication of this paper

Acknowledgments

This work was supported in part by NSFC under Grant no2012-62000-4103036 NSFS under Grant no ZR2012FM006and SDPW under Grant no IMZQWH010016 The authorsthank the 3 reviewers and handling editor for their meticu-lous reading of the paper and their constructive commentswhich greatly improved the paper

References

[1] S Hartmann ldquoA competitive genetic algorithm for resource-constrained project schedulingrdquo Naval Research Logistics vol45 no 7 pp 733ndash750 1998

[2] J E Kelley ldquoThe critical-path method resources planning andschedulingrdquo Industrial Scheduling no 13 pp 347ndash365 1963

[3] R Alvar-Valdes and J M Tamarit ldquoHeuristic algorithms forresource-constrained project scheduling a review and empir-ical analysisrdquo Advances in Project Scheduling no 5 pp 113ndash1341989

[4] J Alcaraz and C Maroto ldquoA robust genetic algorithm forresource allocation in project schedulingrdquo Annals of OperationsResearch vol 102 no 1ndash4 pp 83ndash109 2001

[5] R Kolisch ldquoSerial and parallel resource-constrained projectscheduling methods revisited theory and computationrdquo Euro-pean Journal of Operational Research vol 90 no 2 pp 320ndash3331996

[6] R Baker Introduction to Scheduling and Sequencing JohnWileyamp Sons New York NY USA 1974

[7] A Mingozzi V Maniezzo S Ricciardelli and L Bianco ldquoAnexact algorithm for the resource-constrained project schedulingproblem based on a new mathematical formulationrdquo Manage-ment Science vol 44 no 5 pp 714ndash729 1998

[8] E K Burke G Kendall and G Whitwell ldquoA new placementheuristic for the orthogonal stock-cutting problemrdquoOperationsResearch vol 52 no 4 pp 655ndash671 2004

[9] J Egeblad and D Pisinger ldquoHeuristic approaches for the two-and three-dimensional knapsack packing problemrdquo Computersand Operations Research vol 36 no 4 pp 1026ndash1049 2009

[10] S Martello D Pisinger and D Vigo ldquoThree-dimensional binpacking problemrdquo Operations Research vol 48 no 2 pp 256ndash267 2000

[11] G Ulusoy and L Ozdamar ldquoA heuristic scheduling algorithmfor improving the duration and net present value of a projectrdquoInternational Journal of Operations and Production Manage-ment vol 15 no 1 pp 89ndash98 1995

[12] E W Davis and J H Patterson ldquoA comparison of heuristic andoptimum solutions in resource-constrained project schedul-ingrdquoManagement Science vol 21 no 8 pp 944ndash955 1975

[13] T G Crainic G Perboli and R Tadei ldquoExtreme point-based heuristics for three-dimensional bin packingrdquo INFORMSJournal on Computing vol 20 no 3 pp 368ndash384 2008

[14] M W P Savelsbergh ldquoLocal search in routing problems withtime windowsrdquo Annals of Operations Research vol 4 no 1 pp285ndash305 1985


[15] C Voudouris and E Tsang ldquoGuided local search and its appli-cation to the traveling salesman problemrdquo European Journal ofOperational Research vol 113 no 2 pp 469ndash499 1999

[16] A Schaerf and A Meisels ldquoSolving employee timetablingproblems by generalized local searchrdquo in Advances in ArtificialIntelligence vol 1792 of Lecture Notes in Computer Science pp380ndash389 Springer New York NY USA 2000

[17] A Marco D O Montes C Carlos and N Ferrante ldquoLocalsearchrdquo in Handbook of Memetic Algorithms vol 24 pp 21ndash252012

[18] G F Lucio M J Reed and I D Henning ldquoGuided local searchas a network planning algorithm that incorporates uncertaintraffic demandsrdquo Computer Networks vol 51 no 11 pp 3172ndash3196 2007

[19] R Cabido A S Montemayor J J Pantrigo and B R PayneldquoMultiscale and local search methods for real time regiontracking with particle filters local search driven by adaptivescale estimation on GPUsrdquo Machine Vision and Applicationsvol 21 no 1 pp 43ndash58 2009

[20] B Funke T Grunert and S Irnich ldquoLocal search for vehiclerouting and scheduling problems review and conceptual inte-grationrdquo Journal of Heuristics vol 11 no 4 pp 267ndash306 2005

[21] F Tricoire ldquoMulti-directional local searchrdquo Computers andOperations Research vol 39 no 12 pp 3089ndash3101 2012

[22] F Ricca and B Simeone ldquoLocal search algorithms for politicaldistrictingrdquo European Journal of Operational Research vol 189no 3 pp 1409ndash1426 2008

[23] P Mills E Tsang and J Ford ldquoApplying an extended guidedlocal search to the quadratic assignment problemrdquo Annals ofOperations Research vol 118 no 1ndash4 pp 121ndash135 2003

[24] R Kolisch and A Sprecher ldquoPSPLIBmdasha project schedulingproblem libraryrdquo European Journal of Operational Research vol96 no 1 pp 205ndash216 1997

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Algebra

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Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


0m + 1

Figure 1 A network diagram

order Similar to 2BF algorithm it searches dynamically tofind suitable boxes to be put in the available space The 3BFdefines that blank area is at the farthest point where the boxesare to be putThe 3BF is raised byMartello et al [10] it solvesthe problem of how to select suitable boxes and the pointcannot be blocked by other boxes

This paper is organized as follows Section 2 gives amathematical model Section 3 proposes an iterated localsearch algorithm Section 4 presents a set of simulationexamples to demonstrate the method Section 5 comes upwith a summary for this paper

2 Place Scheduling Problem

The project consists of 119898 activities119872 = 1 2 119898 Everyactivity should be processed on the required platform thatis place We define it as 119903119894119903 = 119886119894119903 times 119887119894119903 where 119886119894119903 and119887119894119903 denote the length of the place 119903 that activity 119894 requiresrespectively The duration that activity 119894 requires is definedas 119905119894 119878119894 and 119865119894 represent the start and finish time of theactivity 119894 respectively In the project there are 119899 kinds ofplace recourses and every place resource is given by 119877119894119903 =119860 119894119903 times 119861119894119903 where 119860 119894119903 and 119861119894119903 denote the length and width ofthe available place 119903 respectively There are 119899 types of places119873 = 1 2 119899 All activities have priority constraints 119876119895is the predecessors of 119895 We assume that the start time of theproject is 0 We decide the start time of the activities so thatthe makespan of the project is minimized

Formodeling this problem two dummynodes 0 and119898+1are introduced 0 is the predecessor of all the activities and119898 + 1 is the successor of all the activities in the projectThe duration and required place resource of the two dummynodes are 0 Obviously the finish time of 119898 + 1 is themakespan of the project so the problem can be formulatedas follows

minimize119865119898+1 (1)

Priorities between activities can be defined as

119878119894 + 119905119894 le 119878119895 119894 isin 119876119895 (2)

We can show it by a network diagram as in Figure 1

We regard an activity as a cuboid and it is defined as119908119894119903 =119886119894119903 times 119887119894119903 times 119905119894

119908119894119903 cap 119908119895119903 = 120593 forall119894 119895 isin 119872

119894 lt 119895 forall119903 isin 119873

(3)

[119892119894119887119894119903 + (1 minus 119892119894) 119886119894119903]

le 119909119894 + [119892119894119887119894119903 + (1 minus 119892119894) 119886119894119903]

le 119861119903

(4)

[119892119894119886119894119903 + (1 minus 119892119894) 119887119894119903]

le 119910119894 + [119892119894119887119894119903 + (1 minus 119892119894) 119887119894119903]

le 119860119903

(5)

119892119894 isin 0 1 (6)

Constraint (3) denotes that any two cuboids cannotoverlap Constraints (4) and (5) denote that any cuboidshould be finished within the available place Constraint (6)means that the cuboid can rotate ninety degrees which isrepresented by two values

3 Iterated Local Search Algorithm

31 The Initial Solution Method For the initial solution wedevelop a hybrid heuristic method of project schedulingproblem and packing problem to solve it Each activity isabstracted as a cuboid and is put in the available placeAs we put the cuboid into the space we divide differentplaces according to different heights Every place is a two-dimensional area

There aremanymethods of how to place the rectangleWewill use best fit [8] heuristic algorithm to solve our problemWe cannot just rely on the size of the rectangle to judge whichis more suitable because cuboids have their sequences Butin the Best Fit method the standard to select the cuboid to bemoved only relies on the lengths and widths of the unselectedcuboids Inspired by this we should consider the sequencebefore we select the cuboids using the Best Fit method Weplace the larger bottom area first if they are equal select theone with the higher height

Before placing it we can regard the bottom area as anobject in a two-dimensional packing problem After packingwe can use some methods of the three-dimensional packingWe can put them according to some rules Length widthand height can be interchanged when we put them But inour problem the height is the duration so only length andwidth can be interchanged This is different from the three-dimensional packing There are many methods regardingthe three-dimensional packing problem We mainly use thealgorithmofCrainic et al which is based on the extremepoint[13] Its main objective is to determine the placement Weplace the bottom-left-rear point on the selected placementThe extreme point is represented as 119909 119910 119905 We select thelocation based on the extreme point






119889119894


10038161003816100381610038161003816119878119895

10038161003816100381610038161003816+ 03sum119903isin119877 119896119895119903119870119903





















A B

A

FEDC

C D B E F

Old

New






119897)









1

2

3

4

5

10

7 15 13

6

8

9

14

15

16

17

18

12








x1

y1

t1

3

4








t2

8

2

y2

x2


4

y2

x2

t2

8

2

6


x1

y1

t1

4

3

5

7



7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12





5 Conclusion



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119889119894


10038161003816100381610038161003816119878119895

10038161003816100381610038161003816+ 03sum119903isin119877 119896119895119903119870119903





















A B

A

FEDC

C D B E F

Old

New






119897)









1

2

3

4

5

10

7 15 13

6

8

9

14

15

16

17

18

12








x1

y1

t1

3

4








t2

8

2

y2

x2


4

y2

x2

t2

8

2

6


x1

y1

t1

4

3

5

7



7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12





5 Conclusion



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A B

A

FEDC

C D B E F

Old

New






119897)









1

2

3

4

5

10

7 15 13

6

8

9

14

15

16

17

18

12








x1

y1

t1

3

4








t2

8

2

y2

x2


4

y2

x2

t2

8

2

6


x1

y1

t1

4

3

5

7



7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12





5 Conclusion



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1

y1

t1

3

4








t2

8

2

y2

x2


4

y2

x2

t2

8

2

6


x1

y1

t1

4

3

5

7



7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12





5 Conclusion



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










7

11

15

7

9

x1

y1

t1

43

5


4

14

2

6

x2

y2

t2

6

16

12





5 Conclusion



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Iterated Local Search Algorithm for a...

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