Research Article Robust Satellite Scheduling Approach for...

Research ArticleRobust Satellite Scheduling Approach forDynamic Emergency Tasks

Xuejun Zhai12 Xiaonan Niu12 Hong Tang12 Lixin Wu3 and Yonglin Shen4

1State Key Laboratory of Earth Surface Processes and Resource Ecology Beijing Normal University Beijing 100875 China2Key Laboratory of Environment Change and Natural Disaster Ministry of Education Beijing Normal UniversityBeijing 100875 China3School of Environment Science and Spatial Informatics China University of Mining and Technology Xuzhou 221116 China4Faculty of Information Engineering China University of Geosciences Wuhan 430074 China

Correspondence should be addressed to Hong Tang hongtangbnueducn

Received 30 June 2015 Accepted 15 October 2015

Academic Editor Xiaobo Qu

Copyright copy 2015 Xuejun Zhai 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

Earth observation satellites play a significant role in rapid responses to emergent events on the Earthrsquos surface for exampleearthquakes In this paper we propose a robust satellite scheduling model to address a sequence of emergency tasks in which boththe profit and robustness of the schedule are simultaneously maximized in each stage Both the multiobjective genetic algorithmNSGA2 and rule-based heuristic algorithm are employed to obtain solutions of the model NSGA2 is used to obtain a flexible andhighly robust initial scheduleWhen every set of emergency tasks arrives a combined algorithm calledHA-NSGA2 is used to adjustthe initial schedule The heuristic algorithm (HA) is designed to insert these tasks dynamically to the waiting queue of the initialschedule Then the multiobjective genetic algorithm NSGA2 is employed to find the optimal solution that has maximum revenueand robustness Meanwhile to improve the revenue and resource utilization we adopt a compact taskmerging strategy consideringthe duration of task execution in the heuristic algorithm Several experiments are used to evaluate the performance of HA-NSGA2All simulation experiments show that the performance of HA-NSGA2 is significantly improved

1 Introduction

Earth observing satellites (EOSs) orbiting the Earth are ableto collect images of specified areas of the Earthrsquos surface atthe request of customers by using observation sensors [1]EOSs not only play key roles in application for environ-ment surveillance reconnaissance resource investigationand other fields but also support many important servicessuch as remote sensing navigation geodesy and monitoring[2] As a result many countries especially China tend toincrease their investments to develop associated techniquesand EOSs For example China plans to launch four smalloptical satellites and four small SAR satellites to form anatural disaster-monitoring constellation [3] Although thenumber of EOSs is continuously increasing they are stillunable to satisfy the requirements of various customers [4]Therefore it is very important to develop effective methodsof satellites observation scheduling to make full use of scarce

satellite resources to better satisfy the demands of customersThe satellite observation scheduling problem (SOSP) is toreasonably assign satellite resources and time windows toEarth observation tasks on the precondition of satisfyingcomplex constraints [5] Additionally SOSP can be seen as akind of multidimensional knapsack problem that is NP-hard[6]

From the previous survey the satellite observation tasksscheduling problem can be divided into two classes that isstatic tasks scheduling and dynamic tasks scheduling Overthe past decades the satellite scheduling imaging problemhasbeen intensively investigated [7ndash10] Regarding static singlesatellite scheduling Lin et al employed mathematical pro-grammingmethods such as Tabu search Lagrange relaxationand liner search to solve the scheduling problem and acquireda near-optimal schedule [11ndash13] Wolfe and Sorensen pro-posed three corresponding algorithms including a dispatchalgorithm a look-ahead algorithm and a genetic algorithm to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 482923 20 pageshttpdxdoiorg1011552015482923

2 Mathematical Problems in Engineering

model the imaging satellite scheduling problem [14] Gabreland Murat presented two upper bound procedures basedon the graph theory and a column generation technique[15] Vasquez and Hao formulated the satellite observationscheduling problem into a generalized version of the well-known knapsack model and used the Tabu search methodto solve the problem [16] Furthermore they proposed apartition-based approachUPPB to obtain tight upper boundsto evaluate the TS search algorithm [17] Several exactmethods such as depth first branch and bound or Russiandolls search and some approximate methods such as greedysearch and Tabu search were used by Bensana et al to solvethe scheduling problem of the SPOT5 satellite They viewedthe scheduling problem as a value constraint satisfactionproblem or an integer linear programming problem [6]

Although the single satellite scheduling problem can besolved perfectly by the above methods with the developmentof the EOS system these methods are unable to satisfy theneeds of multisatellite system scheduling In later studiesmany researchers adopted a heuristic algorithm to solve themultisatellite scheduling problem such as Zweben et al [18]Frank et al [19] Bianchessi and Righini [20] Wang andReinelt [21] and Marinelli et al [22] Bianchessi and Righiniproposed a FIFO heuristic and a look-ahead insert heuristicalgorithm to solve the scheduling problem [20] Frank etal used a constraint based interval planning framework tomodel the problem and proposed a heuristic for guidingthis search procedure Wang and Reinelt considered imagedownloads and proposed a priority-based conflict-avoidedheuristic algorithm [21] With the development of the intel-ligent optimization algorithm many researchers proposedusing intelligent algorithms to solve the complex schedulingproblem The intelligent algorithms mainly include geneticalgorithm and ant colony optimization algorithm Barbulescuet al [23] Wang et al [24] Baek et al [25] Chen et al [26]and Mansour and Dessouky [7] used the genetic algorithm(GA) to address the satellite scheduling problem Barbulescuet al proposed Genitor a genetic algorithm and provedthat it performs well for a broad range of problem instances[23] A multiobjective earth observation satellite schedulingmethod called SPEA2 based on the strength of the Paretoevolutionary algorithm was proposed by Wang et al [24]Additionally Mansour and Dessouky solved the satellitescheduling problem by developing a genetic algorithm con-sidering two objectives that is the profits and the numberof acquired photographs [7] Ant colony optimization (ACO)was also used to address this problem Based on ant colonyoptimization Zhang et al [27] and Liu et al [28] consideredmultisatellite resources and task merging respectively Wuet al presented a novel two-phase based scheduling methodwith the consideration of task clustering [29] Moreover Qiuet al proposed the first finish first schedule with discardtask moving back (FFFS-DTMB) and accommodate discardtask predicting coexistence with task moving back (ADTPC-DTMB) algorithms to solve the problem [30] Globus et aldeveloped an evolutionary algorithm to solve this problemand compared it with existing algorithms [31 32] From thesurvey of static tasks scheduling we find that the heuristicalgorithm is shown to perform well for simple problems and

the genetic algorithm is proven to be suitable for large scalemore complex problems [23]

Static scheduling assumes that all imaging tasks havebeen submitted before scheduling and once the schedulingscheme is produced it is immutable until all tasks havebeen finished In practice because emergent events usuallyoccur unexpectedly with uncertainties of occurrence timeand number traditional static scheduling cannot attend tothese emergent tasks in time It is suggested that dynamicscheduling copes with these unexpected factors

Until now there are a few research efforts towards thedynamic scheduling on Earth observation satellites Pem-berton and Greenwald described the problem of dynamicscheduling and discussed contingency conditions [33]Kramer and Smith proposed a repair-based search methodfor the oversubscribed scheduling problem [34] Verfaillieand Schiex developed an approach employing local adjust-ment to solve the dynamic constraint satisfaction problem(DCSP) [35] Considering the tradeoff between the perfor-mance and degree of adjustmentWang et al proposed a rule-based heuristic algorithm to solve the dynamic schedulingproblem [36] Also Wang et al established a multiobjectivemathematic programming model for the dynamic real-timescheduling of EOSs and proposed a dynamic real-timescheduling algorithm called DMTRH to solve the problem[37] Through analyzing the main constraints Dishan et alconstructed an integer programming model and proposed aRolling Horizon (RH) strategy with a mixed triggering modeto schedule the dynamic tasks and common tasks together[2]

To the best of our knowledge EOSs play an importantrole in the process of disaster relief For example when anearthquake occurs new emergency tasks will be submitted tothe satellite observation system to acquire images that containdisaster information According to the needs of disaster reliefmore and more different emergency tasks will arrive So farjust a few dynamic schedulingmethods consider dealing withthe emergency tasks After analyzing the dynamic propertiesof satellite scheduling Wang et al proposed an optimizationmodel and two heuristic algorithms to solve the problem [38]However they did not consider the task merging method inthe heuristic algorithms Wu et al adopted the ant colonyoptimization plus iteration local search (ACO-ILS) approachto resolve the multisatellite emergency tasks and commontasks scheduling problem [4] However they did not accountfor the uncertainty of emergency tasks Moreover Wang etal presented a dynamic scheduling algorithm called TMBSR-DES which comprehensively considers task merging back-ward shift and rehabilitation [39] The task merging strategyin TMBSR-DES only combines new tasks The robustnessof schedule has not been considered in all of the abovealgorithms

To sumup the studies on dynamic emergency schedulingmentioned above have the following shortages

(1) Some dynamic schedulingmethods ignored the time-liness of emergency tasks

(2) The real-time scheduling method [2 37 38] took intoaccount the timeliness of dynamic emergency tasks

Mathematical Problems in Engineering 3

but it did not consider the influence of the commontasks on satellites

(3) Most of studies overlooked the robustness of dynamicsatellite scheduling

In this paper we propose a robust scheduling approachoriented to both dynamic tasks and common tasks Weestablish a robust satellite scheduling model to maximize theprofit and robustness of the schedule To solve the model wedesign a combined algorithm HA-NSGA2 that is made upof multiobjective genetic algorithm NSGA2 and rule-basedheuristic algorithm NSGA2 is to acquire a flexible and highlyrobustness initial schedule in the scheduling of commontasks Then a heuristic algorithm is designed to insert moreemergency tasks dynamically Meanwhile to improve therevenue and resource utilization we adopt a compact taskmerging strategy considering the duration of task executionin the heuristic algorithm Based on the heuristic algorithmNSGA2 is employed to optimize the solution so as to keephigh continuous robustness for new emergency tasks

The major contributions of this paper are summarized asfollows (1)The robustness is considered in the whole processof common tasks and dynamic emergency tasks schedulingto improve the scheduling ability of resisting disturbances (2)

To leavemore opportunities for emergency tasks and improvethe revenue and resource utilization a compact task mergingstrategy is used (3) Through considering the advantagesof the heuristic algorithm and NSGA2 comprehensively wepropose an efficient solution to deal with the emergency tasksdynamically

The remainder of this paper is organized as follows Thecharacteristics of emergency tasks and the solution frame-work are described in Section 2 In Section 3 we present therobust scheduling model We introduce the correspondingsolution to the model including the multiobjective geneticalgorithm rule-based heuristic algorithm and neighborhoodoperator in detail in Section 4 The simulation results andperformance analysis through comparison with other meth-ods are given in Section 5 Section 6 concludes the paperwithsome future research directions

2 Problem Description

Earth observing satellites (EOSs) orbiting the Earth collectimages of the Earthrsquos surface As shown in Figure 1(a) a stripof EOSs can be formed on the ground by the subsatellitepoint of satellite and the field of view of the sensor slew-ing angle of the sensor and observation duration Satellitemission scheduling allocates the limited satellite resources toobserving tasks efficiently and reasonably Figure 1(b) showsthat each schedule is a sequence of tasks ordered in time foran EOS

The EOSs work in a complex environment in which thereexist many dynamic factors as follows

(i) The change in task properties the change in userrequirements or the reasonable task attributes willlead to a change in task priority When the priorityof the task changes we cancel the task before it is

performed Then the task which is viewed as anemergency task is added to the emergency task set

(ii) The change in the state of satellites sometimescertain satellites may be out of use because of mal-functionmemory or energy shortage and so on As aresult the initial schedule cannot be executed contin-uously These tasks which are affected by the satellitescan be seen as emergency tasks to be inserted

(iii) The arrival of emergency tasks according to the actualrequirement users may insert some new tasks whena scheduling scheme is executed Specifically thereare some new incoming emergency tasks caused byemergent events for example earthquakes fire andlandslides Take an earthquake as an example Thereis an urgent need to reasonably utilize the existingsatellites to rapidly image the affected area duringa short time period Once the emergency tasks aresubmitted and scheduled EOSs can acquire remotesensing images of disaster area quickly which canprovide rapid and effective information for quickinvestigation and assessment of earthquake damage

(iv) Uncertainty of weather conditions some imagingtasks may not be completed because of a changein weather conditions To satisfy user requirementsthese tasks must be rearranged

The four casesmentioned above can be seen as emergencytasks to be inserted into a scheduling scheme Thus thedynamic scheduling problem can be described as a unifiedform of inserting new tasks In this paper we researchdynamic scheduling focus onnew incoming emergency tasksThe general method is to produce a temporary schedule andthen to adjust the schedule as quickly as possible Whendynamically adjusting the initial schedule the solution stabil-ity is an important problem In fact the satellite application isa complex process and the orders can only be uploaded withthe special equipment within limited visible time windowsso excessive changes may cause great operational trouble[36] Hence we should not only consider arranging moreemergency tasks dynamically but also decrease the distur-bances to the initial schedule as much as possible Moreoverthe uncertainty of arrival time is an important propertyof new tasks Thus we should consider the timeliness ofemergency tasks in dynamic scheduling Therefore the goalof our research is to arrange emergency tasks as more aspossible in smaller disturbances considering the timeliness ofthe emergency tasks As a result the optimization objectivesin this paper are revenue and robustness

In this paper we propose a dynamic scheduling approachthat can deal with the emergency tasks in real time dynami-cally It is actually a two-stagemethod to produce a temporaryschedule and then to adjust the schedule as quickly as possibleas is shown in Figure 2 The robust scheduling approachproposed in the paper is oriented to both dynamic emergencytasks and common tasks We describe the flowchart ofthe solution in Figure 3 Before scheduling pretreatment isconducted to compute the available opportunities for eachtask Then we establish a robust satellite scheduling model


Satellite S1 Satellite S1

Satellite S1Satellite S2

Satellite S2T1

S1

T3S1

T4S1T1

S2

T2S2

T3S2 T4

S2

T5S2

(a) Multisatellite observation schematic diagram

Time

Time

d1

d1

d2

d2

d3

d3d4d5

T1S1 T3

S1 T4S1

T1S2 T2

S2 T3S2 T4

S2T5S2

ws1

ws1

ws2

ws3

ws2ws3

ws4ws5

we1

we2

we1we2

we3

we3

we4we5

minusmsgS1

msgS1

Slew

ing

angl

e

minusmsgS2

msgS2

Slew

ing

angl

e

(b) The satellitesrsquo schedule

Figure 1 Illustration of satellite observing activity

Pretreatment Establish the scheduling model

Generate the initial schedule

Adjust the solution

Update the schedule

Add emergency tasks

Figure 2 The framework of solution to the dynamic scheduling problem

Pretreatment

Common tasks Satellite resources

NSGA2 is employed to generate the initial schedule

Emergency tasks

Update the state of executing schedule

Insert the new tasks with rule-based heuristic algorithm

Output the adjusted schedule

Optimize the solution with NSGA2

Establish the multiobjective robust scheduling model

Figure 3 The flowchart of the dynamic scheduling approach

with two objectives considering both revenue and robustnessTo optimize objectives of the model the multiobjectivegenetic algorithm NSGA2 is used to obtain robust solutions

We can acquire an initial schedule with high revenue andstronger ability to accept emergency tasks With the arrivalof emergency tasks we update the state of the tasks in thecurrent executing schedule To arrange the emergency tasksdynamically the rule-based heuristic algorithmwith compacttask merging strategy is designed Thus we can get sometemporary adjusted schedule Lastly NSGA2 is adopted tofurther improve the solution by searching the global space Itis worth mentioning that we design a neighborhood operatorto initialize the population in NSGA2

3 Model

The satellite scheduling amounts to a reasonable arrangementof satellites sensors time windows and sensor slewing anglefor observation tasks to maximize one or more objectives forexample the overall observation profit while being subject torelated constraints As a result the satellite scheduling prob-lem usually consists of five parts tasks satellite resourcesopportunities objectives and constraints A detailed intro-duction of the model is as follows

31 Tasks In this paper we concentrate on two types of tasksthe common task and emergency task Common imagingtasks are usually submitted to the planning system in advanceThe characteristics of emergency tasks are different fromthose of common tasks Emergency events usually occur


unexpectedly with uncertainties of occurrence time andnumber Thus the emergency task number and arrival timesare not known a priori Moreover dynamic tasks need to becompleted in time because users want to obtain observationresults within certain time limits

(i) Common task let 119879 = 1199051 1199052 119905

119873 be the common

task set where 119873 is the number of common tasksWe assume that all tasks are independent aperiodicand nonpreemptive Each task is associated with theweight 119901

119894 and the indispensable duration of task

execution 119889119894

(ii) Emergency task let 119879119864 = 119905119873+1

119905119873+2

119905119873+119873

119864

be the emergency task set Each emergency task ispresented by 119905

119899= (119901119899 119889119899 at119899 dt119899) where 119901

119899is the

weight 119889119899is the duration at

119899is the arrival time and

dt119899is the deadline

In addition to make the denotation of common tasksconsistent with that of emergency tasks we assume that eachcommon task has an arrival time at

119899= 0 and a deadline

dt119899

= 119905119891

119878

Considering that many tasks are commonly submittedtogether thus we assume in this paper that the new tasksarrive in batch style Let 119879

119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 be the

scheduling time set where 119905119894119878(119894 ge 1) is the 119894th dynamic

scheduling time 1199050119878is the initial scheduling time 119890 is the

total batch of emergency tasks and 119905119891

119878is the end time of

schedulingImaging tasks can be divided into two types point target

and area targetThe type depends on the size of the target areaand the attributes of the satellite such as the field of view andheight of satellite The point target is small so that it can beobserved by single observations trip The area target whichcovers a wide geographical area cannot be photographed ina scene Usually the area target can be segmented into severalpoint targets to be imaged In this paper we focus on dealingwith point targets

32 Satellite Resources As for the resources we consider aset 119878 = 119904

1 1199042 119904119872 where 119872 is the satellite number Weassume that the sensors of the satellites considered in ourstudy are able to slew laterally Each resource 119904119895 isin 119878 is denotedby 119904119895 = (Δ120579119895 Δ119889119895 sl119895 st119895msg119895 orb119895 su119895 sd119895 duty119895) whereΔ120579119895 Δ119889119895 sl119895 msg119895 orb119895 su119895 sd119895 and duty119895 are the field ofview the longest opening time slewing rate attitude stabilitytime maximum slewing angle the flight time in each orbitthe start-up time of sensor the retention time of shutdownand the longest imaging time in each orbit

33 Available Opportunities For available opportunities letAO = ao1

1 ao21 ao119895

1 ao12 ao119895

2 ao119872

119873 be the oppor-

tunity set of every task on each satellite resource For ao119895119894

isin

AO ao119895119894

= ao1198951198941

ao1198951198942

ao1198951198943

ao119895119894119896119894119895

represents the availableopportunities of task 119905

119894on resource 119904119895 where 119896

119894119895represents

the number of available opportunities Giving an availableopportunity ao119895

119894119896isin ao119895

119894 it is represented by ao119895

119894119896=

[ws119895119894119896

we119895119894119896

] 120579119895

119894119896 where ws119895

119894119896and we119895

119894119896are denoted by the

start time and end time of the window 119882119895

119894119896 and 120579

119895

119894119896is the

swing angle The set of time windows of task 119894 on satellite 119895

is 119882119895

119894= ⋃119896isin[1119870119894119895]

119882119895

119894119896

34 OptimizationObjectives Weassume that 119905119896119878is the current

scheduling time and then the model can be described asfollows

The primary objective is to maximize the revenue of allscheduled tasks

max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (1)

where 119909119895

119894119896is the decision variable whether the task 119894 is

scheduled

119909119895

119894119896=

1 if task 119894 is executed by satellite 119895 in the 119896th time window

0 otherwise(2)

The second objective is called the neighborhood-basedrobust indicator [40]The robustness based on neighborhoodis proposed from the advances in robust optimization ofcontinuous functions [41 42] The idea is that robust optimaare located on broad peaks (plateaus) in the fitness landscapewhereas brittle optima are located on narrow peaks as shownin Figure 4 A tradeoff is made between the broadness andheight of the peaks such that a low and broad peak maybe preferable to a narrow and high peak The problem ofsatellite scheduling is a discrete problem There are someschedules which are different sequence of the scheduledtasks ordered in time but have the same revenue If only

considering the revenue any one of these schedules is afeasible solution However dynamic task scheduling not onlyrequires arranging more emergency tasks but also requiresreducing the frequency and difficulty of adjustment as muchas possible Thus the property of resisting disturbances andself-adjusting is very important for dynamic task scheduling

Hence we define the neighborhood-based robustness asthe total revenue of scheduled tasks that can be reassigned toother timeslots in the scheme directly It is used to measurethe ability of a scheduling scheme to rearrange scheduledtasks The higher value of the neighborhood-based robustindicator is the more possible it is to rearrange scheduled


Search-space

Obj

ectiv

e

f(x)

f(x1)

f(y)f(y1)

x1 y1x y

+Δ +Δ

Figure 4 The idea in the robust optimization technique forcontinuous problem [47]

tasksTherefore wemaximize the value of the neighborhood-based robust indicator

max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (3)

where 119910119894is the parameter that indicates whether the task 119905

119894isin

SS can be rearranged in another timeslot

119910119894

=

1 if task 119905119894can be rearranged in another timeslot

0 otherwise

(4)

35 Constraints There are some constraints that are used toanalyze the complex restricted condition in the model

(1) Each task 119905119894must be executed in an available opportu-

nity ao119895119894119896 ao119895119894119896

isin ao119895119894

Hence we have the available opportunity constraint

1198621

119909119895

119894119896(ts119895119894

minus ws119895119894119896

) ge 0

119909119895

119894119896(ts119895119894

+ 119889119894minus we119895119894119896

) le 0

119886119894le 119905119896119878

le ts119895119894

le dt119894

(5)

where ts119895119894denotes the start time of task 119905

119894 Actually for a

given scheduled task 119905119894on 119904119895 its starting observation time ts119895

119894

and observation angle 120579119895

119894are determined All scheduled tasks

form the current schedule SS = ⋃119895isin[1119872]

119879119895 where 119879119895 is asequence of scheduled tasks on satellite 119895

(2) Each task only needs to be executed once and theexecution process does not involve preemptive service So wehave the following uniqueness constraint

1198622

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896le 1 (6)

(3) There should be an adequate transmission time forsensor in any two adjacent tasks 119905

119906 119905V assigned to the same

satellite 119904119895 to finish the actions such as shutting down

NSGA2

Static scheduling

7 2 8 225 14 613Initial schedule

Rule-based heuristic algorithm

NSGA2

Temporary adjusted schedule

Updated schedule

Update the state based on thedynamic scheduling time

94

8 225 14 613 94

8 225 14 613 9433 36 3237

Adjusted schedule8 225 14 632 9433 13 3637

middot middot middot




Figure 5 The flowchart of the algorithm

swinging the observation angle stabilizing gesture and start-up So we have the switch time constraint

1198623

te119895119906

+ sd119895 + sl119895 times 10038161003816100381610038161003816120579119895

V minus 120579119895

119906

10038161003816100381610038161003816+ st119895 + su119895 le ts119895V (7)

(4) Another constraint is the total imaging time of anysatellite which should be less than the allowable longestimaging time of its sensor during every time period Let thescheduling period be [119905119896

119878 119891] Hereby we have the imaging

time constraint

1198624

sum

119894isin119879119895

119905119887

119889119894le duty119895

(8)

where 119879119895

119905119887denotes a sequence of scheduled tasks on satellite

119895 which flies during the time span [119905119887 119905119887

+ orb119895] where 119905119887

isin

[119905119896119878 119891 minus orb119895]

4 Algorithms

In this section we introduce the dynamic emergency tasksscheduling algorithm HA-NSGA2 proposed in this paperThe algorithm which is actually a combined algorithm aimsat improving the efficiency of dynamic emergency tasksscheduling The flowchart of the algorithm is depicted inFigure 5 Firstly themultiobjective genetic algorithmNSGA2with two objectives that is revenue and robustness is usedfor common tasks scheduling to obtain the optimal solution


With the arrival of emergency tasks the rule-based heuristicalgorithm is applied to insert these emergency tasks into theinitial schedule as much as possible To improve the imagingefficiency as much as possible we propose a compact taskmerging method that considers task execution duration inthe rule-based heuristic algorithm If two or more targetsare geographically adjacent they may be combined intoa compact composite task by the compact task mergingmethod The details of compact task merging strategy aredescribed in Appendix B Based on the rule-based heuristicalgorithm the multiobjective genetic algorithm NSGA2 isemployed to find the optimal solution The neighborhoodoperator which is embedded in NSGA2 aims at generatingthe initial population

41 Multiobjective Genetic Algorithm Satellites observationscheduling problem has been proven to be NP-completeThus it is impossible to find polynomial time algorithms forfinding optimal solutions Genetic algorithms are the mostsuccessful algorithms for efficiently dealing with the com-plexity of computationally hard problems from the network-ing domain [23 43] Before introducing the multiobjectivegenetic algorithm NSGA2 we firstly make clear some termssuch as ParetoDominance Pareto optimality ParetoOptimalSet Pareto Front Nondominated Sorting and crowdingdistance which will be referred to in the algorithm We givethe associative definitions in Appendix A

As shown in Figure 6 we present the main process ofNSGA2 First we initialize the parameters and generatethe initial population 119875father After computing the fitnessfunction of 119875father we make use of the binary tournamentselection mechanism single point crossover operator andsingle point mutation operator for the current individualsuntil the number of the next generation population 119875childreaches the size 119872 Then we set the total population as119876 = 119875father cup 119875child by combining 119875father and 119875child Nextthe nondominated sorting algorithm is used to divide thepopulation 119876 into different Pareto Front 119865rank with the rankof each individualWe choose the individuals from119865rank untilthe number of the next population |119875next| satisfies the requestranking from small to large If |119875next| is greater than 119872 thenwe sort the current 119865rank in order according to the crowdingdistance and choose the better individuals as 119875next Finally ifthe result satisfies the termination conditions then we outputthe nondominated solution set Otherwise we let 119875father =

119875next and repeat the processThe main parts of the genetic algorithm will be described

in detail including initialization selection operator crossoveroperator mutation operator nondominated sorting and thecomputation of crowding distance

411 Initialization Unlike local search techniques thegenetic algorithmuses a population of individuals giving thusthe search a larger scope and chances to find a better solutionIn this paper the satellite scheduling problem is encodedinto a set of chromosomes Each chromosome composed ofan imaging task sequence of different satellites represents asolution and is assigned a fitness value to determine howgood each chromosome is The structure of a chromosome

is shown in Figure 7 In the algorithm some parameters suchas the size of population 119872 the maximum iterative number119879 the probability of crossover 119901crossover and mutation proba-bility 119901mutation must be setThe initial population is generatedby random strategy

412 Selection Operator Several selection strategies havebeen proposed in the previous literatures such as roulettewheel selection and binary tournament selection We adoptbinary tournament selection in this paperThemain steps areto choose two individuals randomly from a population andto select the individual that dominates the others

413 CrossoverOperator Crossover operator plays an impor-tant role in the genetic algorithm The best genetic informa-tion of parents can be transmitted to offspring during gen-erations of the evolution process by the crossover operatorAfter parent chromosomes have been selected from the pop-ulation we set the crossover probability 119901crossover to allow themajority of chromosomes to mate and pass to their offspringFigure 8 shows an example of the crossover According to thecharacteristics of the satellite scheduling problem we crossthe parent chromosomes to generate child chromosomes andthen delete the same genes in common chromosome Finallywe attempt to arrange the tasks differently for both parentchromosomes in child chromosomes

414 Mutation Operator Mutation operator intends to keepthe variety of the population and improve the individualsby small local perturbations Additionally the mutationoperator provides a component of randomness in the neigh-borhood of the individuals of the population Thereforegenetic algorithm has the ability to avoid falling prematurelyinto local optima and eventually escapes from them duringthe search process For example if we set the probability119901mutation = 02 then the overall procedure of the randommutation operator on the chromosome is shown in Figure 9We choose a scheduled task that has another window ran-domly such as 119905

8 Then the task is rearranged in another

available window

415 Nondominated Sorting The main procedure of non-dominated sorting is described in Algorithm 1 in detail

416 Crowding Distance Computation The main proce-dure of computing the crowding distance is described inAlgorithm 2 in detail

42 Rule-Based Heuristic Algorithm The solution withmaxi-mum revenue and robustness which is produced by NSGA2is selected as the initial schedule When the schedule isexecuted a number of emergency tasks arrive In this casethe current executing schedule must be modified to arrangethese emergency tasks

When a batch of emergency tasks arrives based on thecurrent scheduling time tasks may have different statesfinished task running task waiting task and new task In theexample shown in Figure 10 119905119896

119878is the current scheduling time

If task 119894 has been executed before the current time then task


Initialization parameter

Generate the initial

Select operator Cross operator Mutation operator

Calculate the fitness of

Conduct nondominated sorting to get nondominated

Satisfy the terminal conditions

Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M

i = i + 1 |Pnext| = MPfather = Pnext

Fitness function of Pnext

Pchild = Pchild cup new individuals

Q = Pchild cup Pfather

Pnext = Pfather cup Fi

|Pnext| lt M

Output Qfinal while Qfinal = F1

front F = F1 F2 F3 i = 1

Crowding distance Fi

Figure 6 The main process of NSGA-2

119894 is a finished task while te119894lt 119905119896119878 If ts119894lt 119905119896119878

lt te119894 then task 119894 is

a running task Task 119894 is a waiting task if ts119894gt 119905119896119878 If task 119894 is an

emergent task that has not been planned then task 119894 is a newtask

According to the analysis on the state of the tasksthe waiting tasks in the current schedule can be adjustedFocusing on the new emergency tasks we assign these

tasks to the current schedule In this paper we employa rule-based heuristic algorithm considering compact taskmerging (CTM-DAHA) to insert the new tasks dynamically[44] The main idea of the heuristic algorithm is to assignthe most conflict-free time window to the task most inneed With CTM-DAHA we obtain the temporary adjustedschedules


7 612908 1294912908 12927 19

Satellite 1Satellite 2

Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji

Figure 7 The structure of chromosome

To describe the dynamic adjusting rule-based heuristicalgorithm considering compact task merging clearly we firstgive some definitions

Definition 1 Theneed of the task indicates how badly the taskneeds to be performed [30 44]

119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)

Definition 2 The contention of the task window is measuredby counting the number of tasks that need the time windowweighted by the weights of the tasks

119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)

where 119879(119908) is the set of tasks that could be executed at anymoment within time window 119908 and 119901(119905

119894) is the weight of the

tasks 119905119894

The dynamic adjusting rule-based heuristic algorithmCTM-DAHA is presented as shown in Figure 11 [44] Toaccommodate emergency tasks in the initial scheme firstlywe update the state of tasks in the schedule Then all newtasks are inserted into a waiting queue according to the needsof the tasks119873(119905

119894) fromhigh to lowThe timewindows of each

task are ranked from low to high according to the contention119862(119908) For a task we first judge whether it can merge with anyother existing task in the current schedule If all timewindowsof the task fail to be merged we attempt to insert the taskdirectly if it does not conflict with any other task If directinsertion fails we judge whether the task can be inserted byrearranging the conflict tasks If direct insertion and insertionby rearranging do not succeed some tasks may be deletedfor inserting an emergency task Then the entire process willrepeat until the waiting queue is empty Take for examplea current executing schedule as shown in Figure 12(a) Withthe arrival of emergency tasks that is 119879

119864 = 1199058 1199059 11990510

11990511

they should be arranged in time Task 119905

8can be combined

with 1199052by compact task merging Task 119905

9can be inserted

directly Task 11990510is inserted into the schedule by rearranging

the conflict task 1199055 Task 119905

7is deleted from the schedule so

that task 11990511with higher weight can be inserted Figure 12(b)

shows the planning task list of the adjusted schedule

43 Neighborhood Operator The solution obtained by CTM-DAHA still needs to be optimized Based on the tempo-rary adjusted schedule we use NSGA2 to find the optimalsolution Different from the multiobjective genetic algorithmdescribed in Section 41 we design a neighborhood operatorto generate the initial population

The main idea of the neighborhood operator is to searchall neighborhoods of the current schedule As shown inFigure 13 the current schedule SS is represented by the redpoint and the yellow points denote the neighborhoods of SSSpecifically we change the locus of a task tomove it to anotherposition as shown in Figure 14

The procedure of the neighborhood operator is describedas shown in Algorithm 3

After initializing the population using the neighbor-hood operator the next serial of operation is selectioncrossover mutation and nondominated sorting as expressedin Section 41 Finally using the multiobjective genetic algo-rithm NSGA2 we obtain the optimal solution as the newadjusted schedule

5 Experimental Simulation and Discussion

In this section we evaluate the performance of the pro-posed algorithm HA-NSGA2 To reveal the advantages ofthe algorithm we compare it with RBHA-DTSP [36] andCTM-DAHA [44]The two methods both use multiobjectivegenetic algorithm to make the initial schedule for commontasks To evaluate the advantage of the compact task mergingmethod (CTM) we perform experiments to compare thescheduling results obtained by HA-NSGA2 with the two taskmerging strategies traditional task merging strategy (TTM)and compact task merging method (CTM) Specifically weembed the two task merging methods into HA-NSGA2respectively so as to prove the effectiveness of compact taskmerging strategy

(i) RBHA-DTSP the basic idea of RBHA-DTSP is tocarry out some amount of iterative repair searchinside each emergency taskrsquos available opportunitiesWith the repair search for a given emergency tasksome rules are adopted to decide which tasks shouldbe retracted [36]

(ii) CTM-DAHA this is a dynamic adjusting rule-basedheuristic algorithm considering compact task merg-ing (CTM-DAHA) which is designed to insert theemergency tasks into the initial schedule

To evaluate the result we use the following performancemetrics

10 Mathematical Problems in EngineeringSa

telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

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t1

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t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

Father_segment_1 Father_segment_2

Offspring_segment_1 Offspring_segment_2 Offspring_segment_1 Offspring_segment_2

Offspring_segment_1 Offspring_segment_2

t3

t3 t2

t8

t8

t2t2

t8

t15

t3

Figure 8 Crossover operator

Emergency task revenue (ETR) can be described as

ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)

Satisfaction ratio (SR) is defined as SR = total numberof tasks that can be arranged in the schemetotal number oftasks

Scheduled tasks robustness (STR) can be defined as

STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)

In fact the satellite application is a complex process andthe orders can only be uploaded with special equipmentwithin limited visible time windows so it is important tomake the least adjustment to keep the stability of the scheduleas well as to insert more tasks to keep the efficiency of the


Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

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1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

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147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

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t1

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t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11

Figure 9 Mutation operator

New taskWaiting task Finished task

Running task

Time

Task 1 Task 3

tkminus1S tkS tfS

ts1 te1 ts2 te2 ts3 te3ws4 we4

Figure 10 The states of tasks

schedule Therefore the main measurement of perturbationneeds to be considered

Perturbation measurement (PM) is a metric used todescribe the distance between the adjusted scheme and theinitial scheme PM can be defined with the total number ofinitial tasks that change in the adjusted scheme comparedwith the initial scheme

51 Simulation Method and Parameters Without lossof generality we acquire the imaging targets randomlyin the area latitude minus180∘sim180∘ and longitude minus65∘ sim65∘The scheduling period is 24 h which can be denoted by[0 86400] The number of common tasks is 200 In additionthere are five batches of new coming emergency tasks andeach batch is given the same size 50 The arrival timeof a batch 119905

119896

119878is a random positive number distributed

in [1199050119878 119905119891

119878] In the experiment we simulate 5 batches of

emergency tasks and 119879119878is set as 119879

119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =

0 59 13300 26100 41800 65000 86400 Moreover thedeadline of an emergency task is the end time of scheduling86400 Without loss of generality the weights of alltasks are randomly distributed in [1 10] Two sensors ondifferent satellites are simulated to accomplish the tasks Theparameters of sensors are shown in Table 1 The availableopportunities including time windows and slewing anglesassociated with tasks are computed using STK (Satellite

Table 1 Parameters of satellites

Parameters Satellite 1199041 Satellite 1199042

Average height (km) 780099 505984Orbital inclination (degree) 985 97421Local descending node time (am) 1030 1030Argument of perigee 90 90Eccentricity ratio (degree) 00011 0Nodical period (min) 10038 97716Maximum slewing angle (degree) plusmn32 plusmn32Field of view (degree) 84 6

Tool Kit) The time window is removed if its span is shorterthan the duration of the corresponding task or its finishedtime is later than the deadline Moreover we assume that themaximum time for each satellite to open its sensor once is60 seconds and that the longest imaging time in any periodtime of orb119895 is 150 seconds

52The Simulation Result We carry out two groups of exper-iments in this sectionWe investigate the performance impactof different algorithms that is RBHA-DTSP CTM-DAHAand HA-NSGA2 in the first group The experimental resultsare depicted in Figure 15 The second group of experimentsis designed to evaluate the efficiency of the task mergingmethod CTM and TTM

Figure 15(a) shows the result of the revenue of scheduledemergency tasks in each adjusted schedule Among thealgorithms HA-NSGA2 shows the best scheduling revenueThe explanation is that HA-NSGA2 optimizes the robustnessand uses the compact task merging strategy In contrastCTM-DAHA does not improve the robustness and RBHA-DTSP does not consider taskmerging Based on Figure 15(b)we can see that with more batches of emergency tasks therobustness of each adjusted schedule decreases This canbe attributed to the fact that with the time of schedulingincreasing the scheduling period becomes shorter Thismeans that the tasksrsquo available opportunities in a scheduledecrease and the number of tasks which can be rearrangeddecreases As a result the robustness becomes smaller Inaddition in each scheduling HA-NSGA2 can generate amore robust schedule except the last scheduling In the lastscheduling the schedule obtained by HA-NSGA2 shows thelowest robustness because HA-NSGA2 scheduled more tasksand less spare time is left for other tasks to be rearrangedAs a result the robustness is worse Figure 15(c) presentsthe satisfaction ratio of every scheduling result It is foundthat the satisfaction ratio of all algorithms decreases as thearrival of more emergency tasks This can be attributedto the fact that the number of scheduled tasks decreasescompared with the increasing emergency tasks MoreoverHA-NSGA2 shows the highest satisfaction ratio among all ofthe algorithms Table 2 describes the number of scheduledemergency tasks and the number of adjusted tasks in theadjusted schedules It is found that there is little difference ofdisturbance among the algorithms On the whole the ability


Is task merging successful

Is direct insertion successful

Is inserting by rearranging successful

No

Construct a compact composite task

Yes

YesInserting by deleting Rearrange the conflict tasks

No new task

Yes

No

Choose a new task to insert

No Yes

Emergency task set TE

Remove task from TE

No

The current schedule SS

Set SS998400 = SS

Update the adjusted schedule SS998400

Output SS998400

Update the state of SS998400

Figure 11 The process of inserting a new task

Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)

Figure 12 (a) The current executing schedule (b) The adjusted schedule

dealing with emergency tasks of HA-NSGA2 is superior tothat of the other algorithms

Figure 16 shows the performance impact of the twotask merging methods Figure 16(a) demonstrates that withthe increase in scheduling time the compact task merg-ing method (CTM) can get a higher total emergency taskrevenue This result indicates that more emergency taskscan be arranged by the compact task merging method andmore opportunities can be left for other emergency tasksFrom Figure 16(b) it is found that robustness of the scheduleof the compact task merging method is lower than that

of the traditional task merging strategy This result canbe attributed to two factors First the algorithm with thecompact task merging strategy can arrange more emergencytasks compared with traditional task merging method Theother is the composite task which is unable to participatein the robustness computing because there is only one syn-thetic available window acquired by compact task mergingMoreover the number of composite tasks by compact taskmerging is more than that by traditional task merging So thecompact task merging strategy is better than the traditionalmerging strategy in the available windows and the satellite


0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space

Figure 13 The neighborhods of the current schedule

Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3

Neighborhood schedule

128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12

Figure 14 Illustration of neighborhood operator

Table 2 The perturbation measurements of different algorithms

Batch RBHA-DTSP CTM-DAHA HA-NSGA2119873 PM 119873 PM 119873 PM

1 47 2 47 2 48 42 46 5 46 5 47 53 41 3 41 1 42 14 43 3 46 4 47 45 43 1 42 1 44 1Note119873 the number of scheduled emergency tasks

resources left Figure 16(c) depicts that the compact taskmerging method can arrange more emergencies and obtaina more satisfactory schedule Table 3 describes the way ofadjusting of emergency tasks and the distance of the adjustedscheme from the initial scheme In the same perturbationmeasurement HA-NSGA2 with the compact task mergingmethod can arrange more emergency tasks Also CTMcan merge more tasks than CTM The result proves that

Table 3 The perturbation measurements of different task mergingstrategies

Batch Method 119873 TM NI NIR NID PM

1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1

Note 119873 the number of scheduled emergency tasks PM the number ofadjusted tasks in the schedules NITM the number of emergency tasksinserted by task merging NI the number of emergency tasks inserteddirectly NIR the number of emergency tasks inserted by rearranging andNID the number of emergency tasks inserted by deleting

the compact task merging strategy has an advantage overthe existing traditional method The compact task mergingmethod can improve the chances of inserting emergencytasks

53 Discussion Based on the extensive experiments we canevaluate the performance of HA-NSGA2 proposed in thepaper Using the algorithm HA-NSGA2 with two objectivesthat is the revenue and robustness we can obtain thescheduling scheme that has higher revenue and strongerability to resist disturbance in emergency tasks schedulingHA-NSGA2 algorithm can insert more emergency taskswhile keeping the high revenue as is shown in Figure 15Moreover HA-NSGA2 can significantly reduce the schedul-ing disturbance in the process of scheduling emergency tasksThe experiments prove that the HA-NSGA2 is capable offinding optimal solutions with the ability to resist disturbanceand to address emergency tasks efficiently Moreover thecompact task merging strategy can improve the chances ofmerging multiple tasks and save the satellite resources So themethod proposed in this paper is more practical and satisfiesthe requests of the users

6 Conclusion and Future Work

To solve dynamic scheduling problem for emergency tasksin a multisatellite observation system we propose a mixedalgorithm namedHA-NSGA2 to improve the ability of emer-gency tasks scheduling from two aspects In our algorithmtwo optimization objectives that is revenue and robustnessare considered continuously Firstly we schedule commontasks by using themultiobjective genetic algorithmNSGA2 toobtain an initial solution with high revenue and robustnessThen the heuristic algorithm which embeds a compact taskmerging strategy is used to handle emergency tasks Finallyto optimize the solution NAGA2 which embeds a novelneighborhood operator to generate the initial population


Revenue of emergency tasks

The batch of emergency tasks

RBHA-DTSPCTM-DAHAHA-NSGA2

1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)

Figure 15 The performance of different dynamic scheduling algorithms

is adopted to search the optimal result in the local scopeExtensive experimental simulation proves that the robustnessis an important factor in dynamic scheduling The algorithmHA-NSGA2 proposed in this paper optimizes the robustnessand obtains the more robust scheduling results with thehighest revenue compared with other algorithms Moreoverthe compact task merging strategy can improve the chancesof merging multiple tasks Therefore we conclude that therobustness of the schedule has considerable significance inthe dynamic scheduling problem

In the future our studies will address the following issuesfirst we will conduct some research on area targets andconsider more critical issues such as resource availabilitydata dimension and communication In addition we willimprove our scheduling model by considering more con-straints such as memory capacity energy consumption andspeed of upload Finally we will extend our experiments inreal scene

Appendices

To help the readers understand we make some terms clearthat will be referred to in NSGA2 and describe the compacttask merging strategy in detail in these appendices InAppendix A we introduce some definitions of the multiob-jective genetic algorithm NSGA2 Compact task mergingstrategy is presented in Appendix B

A The Associative Definitions for NSGA2

Some fundamental definitions of the multiobjective geneticalgorithm NSGA2 are given in the following [44 45]

Definition for Pareto Dominance For any u isin R119898 and k isin

R119898 u dominates k (in symbols u ≺ k) if and only if forall119894 =

1 119898 u le k and exist119894 isin 1 119898 u lt k


for each 119901 isin 119876 every individual in population 119901 isin 119876

119878119901

= let the set of individuals dominated by 119901 be null119899119901

= 0 the number of individuals dominating 119901 be 0for each 119902 isin 119876

if (119901 ≺ 119902) then if 119901 dominates 119902

119878119901

= 119878119901

cup 119902 add 119902 into 119878119901

else if (119901 ≻ 119902) then if 119902 dominates 119901

119899119901

= 119899119901

+ 1end forend forfor each 119901 isin 119876

if (119899119901

== 0) then119901rank = 1 the rank of 119901 is 1st1198651

= 1198651

cup 119901 add 119901 into the set of the first non-dominated front 1198651

end for119894 = 1 initialize the rank of the non-dominated frontswhile (119865

119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902

minus 1 the number of individuals dominating 119901 is reduced by oneif (119899119902

== 0) then if there is no individual dominating 119902

119902rank = 119894 + 1 the rank of 119902 plus one119865119894+1

= 119865119894+1

cup 119902 add 119902 into the non-dominated front 119865119894+1

end forend for

119894 = 119894 + 1end while

Algorithm 1

119868 = |119865rank| number of solutions in 119865rank

for each 119901 isin 119865rank

119865rank[119901]distance = 0 initialize the distanceend forfor each objective 119898

119865rank119898 = sort(119865rank 119898) sort using each objective valuefor each 119894 = 2 rarr 119868 minus 1

119865rank119898[119894]distance = (119865rank119898[119894 + 1] minus 119865rank119898[119894 minus 1])(119865maxrank119898 minus 119865

minrank119898) crowding-distance of individual 119894 inm th objective

end forend forfor each 119901 isin 119865rank

for each objective 119898

119865rank[119901]distance = 119865rank[119901]distance + 119865rank119898[119901]distance crowding-distance of 119901

end forend for

Algorithm 2


(1) According to the arrival time of a batch of emergency tasks select the waiting tasks in the current schedule SS as 119879119908

(2) for all 119894 isin [1 2 |119879119908|] do(3) for all 119895 isin [1 119872] do(4) compute all available opportunities for task 119894 on satellite 119895

(5) end for(6) compute the number of available opportunities |119882

119894| for task 119894

(7) end for(8) let the neighborhood set of the current schedule SS119873 be SS119873 = SS(9) while 119879119908 = do(10) take the serial number 119894 of the first task in 119879

119908

(11) if |119882119894| le 1 do

(12) remove 119905119894from 119879119908 and then go to Step (9)

(13) end if(14) else(15) try to change the locus of task 119894 in the chromosome(16) if succeed update the chromosome and add the chromosome to SS119873(17) remove 119905

119894from 119879

119908 and then go to Step (9)(18) end if(19) else(20) remove 119905

119894from 119879

119908 and then go to Step (9)(21) end while(22) output the neighborhood set of the current schedule SS119873

Algorithm 3

Definition for Pareto Optimality The idea that a candidatesolution x isin Ω is the optimal solution of Pareto means thatx isin Ω makes 119865(119909

1015840) ≺ 119865(119909) x is a 119870-dimensional decisionvariable in the decision space and 119865(119909) is an objective space

Definition for Pareto Optimal Set Consider

119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)

Definition for Pareto Front Consider

119875119865 = 119865 (119909) | 119909 isin 119875 (A2)

Definition for Nondominated Sorting Nondominated sortingis used to confirm the rank of each individual in the wholepopulation according to the dominated space For exampleforall119909119894

isin Ω we can compute a dominated set of 119909119894 119875119909119894

= 119909 isin

Ω | forall1199091015840 isin Ω st 119865(1199091015840) ≺ 119865(119909119894) The size of 119875

119909119894is the rank of

119909119894 Rank(119909

119894) = |119875

119909119894| We divide the population into different

Pareto Front with the rank of each individual Ω = 1198651

cup 1198652

cup

sdot sdot sdot cup 119865max rank

Definition for Crowding Distance The crowding distanceapproaches aim to obtain a uniform spread of solutionsalong a similar Pareto Front without using a fitness sharingparameter In this paper there are two objectives that isrevenue and robustness Firstly we rank all individuals inorder according to the revenue and then compute the distanceof each individual in the sequence by formula dis 119896(119909

119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896

minus119891min119896

)Thus the crowding distancecan be defined as

Distance (119909119894) =

119891revenue (119909119894+1

) minus 119891revenue (119909119894minus1

)

119891maxrevenue minus 119891min

revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)

119891maxrobust minus 119891min

robust

(A3)

Definition for Fitness Function The determination of anappropriate fitness function is crucial to the performance ofthe genetic algorithm Thus we should construct optimiza-tion objectives with a weight sum to represent the fitnessfunction [46]

119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times Rearrange (119905

119894)

(A4)

where 119865 is the fitness and 119908revenue and 119908robustness are weights

B The Compact Task Merging Strategy

The compact task merging strategy which is embedded inthe rule-based heuristic algorithm is used to improve theimaging efficiency If two or more targets are geographicallyadjacent we can rationally tune the slewing angle and the



CTMTTM

The batch of emergency tasks1 2 3 4 5

200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098

099Satisfaction ratio


CTMTTM

(c)

Figure 16 The performances of different task merging methods

sj

120579i

ti

Δ120579

120579j

ti998400

Nadir of satellite sj

(a)

di

dj

wsi

wsj

wei

wej

(wsj + dj) minus (wei minus di)

(b)

Figure 17 Task merging constraints


observation duration of the sensor to enable an observationstrip to cover them In other words tasks in the same swath ofa sensor may be merged into one composite task To improvethe imaging opportunities of the new tasks a task mergingstrategy is required Suppose there are two tasks 119905

119894and 119905119895that

could be imaged by satellite 119895 And ao119894

= [ws119894we119894] 120579119894 isin

AO119895119894is an available opportunity of task 119905

119894 Accordingly ao

119895=

[ws119895we119895] 120579119895 isin AO119895

119895is an available opportunity of task

119905119895 Since the length of a visible time window must be larger

than the observation duration of task there often exists someunnecessary time to finish merged tasks according to thetraditional task merging strategy Therefore the duration oftask execution is an important factor in task merging Byconsidering the duration of task execution we employ anew compact task merging method to construct the compactcomposite tasks in this paper

When a task merging mechanism is embedded intothe schedule scheme we must judge when two tasks canbe combined into a composite task and determine how toconstruct a composite task

Without loss of generality between two tasks 119905119894and 119905119895 the

window start time ws119894of task 119905

119894is assumed to be no later than

that of task 119905119895in the following

TheoremB1 Two feasible tasks 119905119894and 119905119895can be combined into

a compact composite task 119905119894119895if and only if they satisfy

(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)

Equation (B1) is the timewindow constraint As shown inFigure 17(b) we illustrate the case where two time windowsof tasks intersect

Equation (B2) is the slewing angle constraint It is shownin Figure 17(a) that tasks 119905

119894and 119905119895must be located in the same

swath of the sensor

Theorem B2 If two feasible tasks 119905119894and 119905119895can be merged

into a compact composite task 119905119894119895 then its time window 119882

119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)

Its indispensable duration of task execution should be

119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)

and the slewing angle is given by

120579119894119895

=

max120579119894minus

Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Basic ResearchProgram of China (no 2011CB707102) the National NaturalScience Foundation of China (no 41571334) and the Funda-mental Research Funds for the Central Universities

References

[1] N Bianchessi J-F Cordeau J Desrosiers G Laporte andV Raymond ldquoA heuristic for the multi-satellite multi-orbitand multi-user management of Earth observation satellitesrdquoEuropean Journal of Operational Research vol 177 no 2 pp750ndash762 2007

[2] Q Dishan H Chuan L Jin and M Manhao ldquoA dynamicscheduling method of earth-observing satellites by employingrolling horizon strategyrdquoThe Scientific World Journal vol 2013Article ID 304047 11 pages 2013


[3] PWang G Reinelt P Gao andY Tan ldquoAmodel a heuristic anda decision support system to solve the scheduling problem of anearth observing satellite constellationrdquo Computers amp IndustrialEngineering vol 61 no 2 pp 322ndash335 2011

[4] G Wu M Ma J Zhu and D Qiu ldquoMulti-satellite observationintegrated scheduling method oriented to emergency tasks andcommon tasksrdquo Journal of Systems Engineering and Electronicsvol 23 no 5 pp 723ndash733 2012

[5] N G Hall andM J Magazine ldquoMaximizing the value of a spacemissionrdquo European Journal of Operational Research vol 78 no2 pp 224ndash241 1994

[6] E Bensana G Verfaillie J Agnese N Bataille and D Blum-stein ldquoExact amp INEXACT methods for daily management ofearth observation satellitrdquo in Proceedings of the Space MissionOperations and Ground Data Systems (SpaceOps rsquo96) pp 394ndash507 1996

[7] M A A Mansour and M M Dessouky ldquoA genetic algorithmapproach for solving the daily photograph selection problem ofthe SPOT5 satelliterdquo Computers amp Industrial Engineering vol58 no 3 pp 509ndash520 2010

[8] D Habet M Vasquez and Y Vimont ldquoBounding the optimumfor the problem of scheduling the photographs of an agileearth observing satelliterdquo Computational Optimization andApplications vol 47 no 2 pp 307ndash333 2010

[9] L Barbulescu A E Howe J P Watson and L D WhitleyldquoSatellite range scheduling a comparison of genetic heuristicand local searchrdquo in Parallel Problem Solving from NaturemdashPPSN VII vol 2439 of Lecture Notes in Computer Science pp611ndash620 Springer Berlin Germany 2002

[10] M Lemaıtre G Verfaillie F Jouhaud J-M Lachiver andN Bataille ldquoSelecting and scheduling observations of agilesatellitesrdquo Aerospace Science and Technology vol 6 no 5 pp367ndash381 2002

[11] W-C Lin and S-C Chang ldquoHybrid algorithms for satelliteimaging schedulingrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics vol 3 pp 2518ndash2523 IEEE Waikoloa Village Hawaii USA October 2005

[12] W-C Lin D-Y Liao C-Y Liu and Y-Y Lee ldquoDaily imagingscheduling of an earth observation satelliterdquo IEEE Transactionson Systems Man and Cybernetics Part A Systems and Humansvol 35 no 2 pp 213ndash223 2005

[13] W-C Lin and D-Y Liao ldquoA tabu search algorithm for satelliteimaging schedulingrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo04) pp1601ndash1606 October 2004

[14] W J Wolfe and S E Sorensen ldquoThree scheduling algorithmsapplied to the earth observing systems domainrdquo ManagementScience vol 46 no 1 pp 148ndash168 2000

[15] V Gabrel and C Murat ldquoMathematical programming for earthobservation satellite mission planningrdquo in Operations Researchin Space and Air pp 103ndash122 Springer New York NY USA2003

[16] M Vasquez and J-K Hao ldquoA lsquologic-constrainedrsquo knapsackformulation and a tabu algorithm for the daily photographscheduling of an earth observation satelliterdquo ComputationalOptimization and Applications vol 20 no 2 pp 137ndash157 2001

[17] M Vasquez and J-K Hao ldquoUpper bounds for the SPOT 5 dailyphotograph scheduling problemrdquo Journal of CombinatorialOptimization vol 7 no 1 pp 87ndash103 2003

[18] M Zweben E Davis B Daun and M J Deale ldquoSchedulingand rescheduling with iterative repairrdquo IEEE Transactions onSystemsMan andCybernetics vol 23 no 6 pp 1588ndash1596 1993

[19] J Frank A Jonsson R Morris and D E Smith ldquoPlanning andscheduling for fleets of earth observing satellitesrdquo in Proceedingsof the 6th International Symposium on Artificial IntelligenceRobotics Automation and Space Montreal Canada June 2001

[20] N Bianchessi and G Righini ldquoPlanning and scheduling algo-rithms for the COSMO-SkyMed constellationrdquo Aerospace Sci-ence and Technology vol 12 no 7 pp 535ndash544 2008

[21] P Wang and G Reinelt ldquoA heuristic for an earth observingsatellite constellation scheduling problem with download con-siderationsrdquo Electronic Notes in Discrete Mathematics vol 36pp 711ndash718 2010

[22] F Marinelli S Nocella F Rossi and S Smriglio ldquoA Lagrangianheuristic for satellite range scheduling with resource con-straintsrdquo Computers amp Operations Research vol 38 no 11 pp1572ndash1583 2011

[23] L Barbulescu J-P Watson L D Whitley and A E HoweldquoScheduling space-ground communications for the air forcesatellite control networkrdquo Journal of Scheduling vol 7 no 1 pp7ndash34 2004

[24] J Wang N Jing J Li and Z H Chen ldquoA multi-objectiveimaging scheduling approach for earth observing satellitesrdquo inProceedings of the 9th Annual Conference on Genetic and Evolu-tionary Computation (GECCO rsquo07) pp 2211ndash2218 London UK2007

[25] S-W Baek S-M Han K-R Cho et al ldquoDevelopment ofa scheduling algorithm and GUI for autonomous satellitemissionsrdquo Acta Astronautica vol 68 no 7-8 pp 1396ndash14022011

[26] Y Chen D ZhangM Zhou andH Zou ldquoMulti-satellite obser-vation scheduling algorithm based on hybrid genetic particleswarm optimizationrdquo in Advances in Information Technologyand Industry Applications vol 136 of Lecture Notes in ElectricalEngineering pp 441ndash448 Springer Berlin Germany 2012

[27] Z Zhang N Zhang and Z Feng ldquoMulti-satellite controlresource scheduling based on ant colony optimizationrdquo ExpertSystems with Applications vol 41 no 6 pp 2816ndash2823 2014

[28] X Liu B Bai Y Chen and Y Feng ldquoMulti satellites schedulingalgorithm based on task merging mechanismrdquo Applied Mathe-matics and Computation vol 230 pp 687ndash700 2014

[29] G Wu J Liu M Ma and D Qiu ldquoA two-phase schedulingmethod with the consideration of task clustering for earthobserving satellitesrdquo Computers amp Operations Research vol 40no 7 pp 1884ndash1894 2013

[30] D Qiu L Zhang J Zhu andH Li ldquoFFFS-DTMB and ADTPC-DTMB algorithmin multi-satellites mission planningrdquo ActaAeronautica et Astronautica Sinica vol 30 pp 2178ndash2184 2009

[31] A Globus J Crawford J Lohn and A Pryor ldquoA comparisonof techniques for scheduling earth observing satellitesrdquo inProceedings of the 16th Conference on Innovative Applications ofArtificial Intelligence (IAAI rsquo04) pp 836ndash843 AAAI San JoseCalif USA July 2004

[32] A Globus J Crawford J Lohn and A Pryor ldquoSchedulingearth observing satellites with evolutionary algorithmsrdquo inProceedings of the International Conference on Space MissionChallenges for Information Technology (SMC-IT rsquo03) PasadenaCalif USA July 2003

[33] J C Pemberton and L G Greenwald ldquoOn the need for dynamicscheduling of imaging satellitesrdquo International Archives of Pho-togrammetry Remote Sensing and Spatial Information Sciencesvol 34 pp 165ndash171 2002

[34] L A Kramer and S F Smith ldquoMaximizing flexibility aretraction heuristic for oversubscribed scheduling problemsrdquo


in Proceedings of the 18th International Joint Conference onArtificial Intelligence (IJCAI rsquo03) pp 1218ndash1223 August 2003

[35] G Verfaillie and T Schiex ldquoSolution reuse in dynamic con-straint satisfaction problemsrdquo inProceedings of the 12thNationalConference on Artificial Intelligence (AAAI rsquo94) pp 307ndash312Seattle Wash USA July 1994

[36] J Wang J Li and Y Tan ldquoStudy on heuristic algorithm fordynamic scheduling problem of earth observing satellitesrdquoin Proceedings of the 8th ACIS International Conference onSoftware Engineering Artificial Intelligence Networking andParallelDistributed Computing (SNPD rsquo07) pp 9ndash14 QingdaoChina July 2007

[37] J Wang X Zhu L T Yang J Zhu and M Ma ldquoTowardsdynamic real-time scheduling for multiple earth observationsatellitesrdquo Journal of Computer and System Sciences vol 81 no1 pp 110ndash124 2015

[38] M Wang G Dai and M Vasile ldquoHeuristic scheduling algo-rithm oriented dynamic tasks for imaging satellitesrdquoMathemat-ical Problems in Engineering vol 2014 Article ID 234928 11pages 2014

[39] J Wang X Zhu D Qiu and L T Yang ldquoDynamic schedulingfor emergency tasks on distributed imaging satellites withtask mergingrdquo IEEE Transactions on Parallel and DistributedSystems vol 25 no 9 pp 2275ndash2285 2014

[40] B-C Bai Y-Z Ci and Y-W Chen ldquoDynamic task mergingin multi-satellites observing schedulingrdquo Journal of SystemSimulation vol 21 no 9 pp 2646ndash2649 2009

[41] J Branke ldquoCreating robust solutions by means of evolutionaryalgorithmsrdquo in Parallel Problem Solving from NaturemdashPPSN Vvol 1498 of Lecture Notes in Computer Science pp 119ndash128Springer Berlin Germany 1998

[42] S Tsutsui and A Ghosh ldquoGenetic algorithms with a robustsolution searching schemerdquo IEEE Transactions on EvolutionaryComputation vol 1 no 3 pp 201ndash208 1997

[43] F Xhafa J Sun A Barolli A Biberaj and L Barolli ldquoGeneticalgorithms for satellite scheduling problemsrdquo Mobile Informa-tion Systems vol 8 no 4 pp 351ndash377 2012

[44] X Niu H Tang L Wu R Deng and X Zhai ldquoImaging-duration embedded dynamic scheduling of Earth observationsatellites for emergent eventsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 731734 31 pages 2015

[45] T Mao Z Xu R Hou andM Peng ldquoEfficient satellite schedul-ing based on improved vector evaluated genetic algorithmrdquoJournal of Networks vol 7 no 3 pp 517ndash523 2012

[46] B Sun W Wang X Xie and Q Qin ldquoSatellite missionscheduling based on genetic algorithmrdquo Kybernetes vol 39 8pp 1255ndash1261 2010

[47] M T Jensen ldquoGenerating robust and flexible job shop schedulesusing genetic algorithmsrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 3 pp 275ndash288 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


model the imaging satellite scheduling problem [14] Gabreland Murat presented two upper bound procedures basedon the graph theory and a column generation technique[15] Vasquez and Hao formulated the satellite observationscheduling problem into a generalized version of the well-known knapsack model and used the Tabu search methodto solve the problem [16] Furthermore they proposed apartition-based approachUPPB to obtain tight upper boundsto evaluate the TS search algorithm [17] Several exactmethods such as depth first branch and bound or Russiandolls search and some approximate methods such as greedysearch and Tabu search were used by Bensana et al to solvethe scheduling problem of the SPOT5 satellite They viewedthe scheduling problem as a value constraint satisfactionproblem or an integer linear programming problem [6]

Although the single satellite scheduling problem can besolved perfectly by the above methods with the developmentof the EOS system these methods are unable to satisfy theneeds of multisatellite system scheduling In later studiesmany researchers adopted a heuristic algorithm to solve themultisatellite scheduling problem such as Zweben et al [18]Frank et al [19] Bianchessi and Righini [20] Wang andReinelt [21] and Marinelli et al [22] Bianchessi and Righiniproposed a FIFO heuristic and a look-ahead insert heuristicalgorithm to solve the scheduling problem [20] Frank etal used a constraint based interval planning framework tomodel the problem and proposed a heuristic for guidingthis search procedure Wang and Reinelt considered imagedownloads and proposed a priority-based conflict-avoidedheuristic algorithm [21] With the development of the intel-ligent optimization algorithm many researchers proposedusing intelligent algorithms to solve the complex schedulingproblem The intelligent algorithms mainly include geneticalgorithm and ant colony optimization algorithm Barbulescuet al [23] Wang et al [24] Baek et al [25] Chen et al [26]and Mansour and Dessouky [7] used the genetic algorithm(GA) to address the satellite scheduling problem Barbulescuet al proposed Genitor a genetic algorithm and provedthat it performs well for a broad range of problem instances[23] A multiobjective earth observation satellite schedulingmethod called SPEA2 based on the strength of the Paretoevolutionary algorithm was proposed by Wang et al [24]Additionally Mansour and Dessouky solved the satellitescheduling problem by developing a genetic algorithm con-sidering two objectives that is the profits and the numberof acquired photographs [7] Ant colony optimization (ACO)was also used to address this problem Based on ant colonyoptimization Zhang et al [27] and Liu et al [28] consideredmultisatellite resources and task merging respectively Wuet al presented a novel two-phase based scheduling methodwith the consideration of task clustering [29] Moreover Qiuet al proposed the first finish first schedule with discardtask moving back (FFFS-DTMB) and accommodate discardtask predicting coexistence with task moving back (ADTPC-DTMB) algorithms to solve the problem [30] Globus et aldeveloped an evolutionary algorithm to solve this problemand compared it with existing algorithms [31 32] From thesurvey of static tasks scheduling we find that the heuristicalgorithm is shown to perform well for simple problems and

the genetic algorithm is proven to be suitable for large scalemore complex problems [23]

Static scheduling assumes that all imaging tasks havebeen submitted before scheduling and once the schedulingscheme is produced it is immutable until all tasks havebeen finished In practice because emergent events usuallyoccur unexpectedly with uncertainties of occurrence timeand number traditional static scheduling cannot attend tothese emergent tasks in time It is suggested that dynamicscheduling copes with these unexpected factors

Until now there are a few research efforts towards thedynamic scheduling on Earth observation satellites Pem-berton and Greenwald described the problem of dynamicscheduling and discussed contingency conditions [33]Kramer and Smith proposed a repair-based search methodfor the oversubscribed scheduling problem [34] Verfaillieand Schiex developed an approach employing local adjust-ment to solve the dynamic constraint satisfaction problem(DCSP) [35] Considering the tradeoff between the perfor-mance and degree of adjustmentWang et al proposed a rule-based heuristic algorithm to solve the dynamic schedulingproblem [36] Also Wang et al established a multiobjectivemathematic programming model for the dynamic real-timescheduling of EOSs and proposed a dynamic real-timescheduling algorithm called DMTRH to solve the problem[37] Through analyzing the main constraints Dishan et alconstructed an integer programming model and proposed aRolling Horizon (RH) strategy with a mixed triggering modeto schedule the dynamic tasks and common tasks together[2]

To the best of our knowledge EOSs play an importantrole in the process of disaster relief For example when anearthquake occurs new emergency tasks will be submitted tothe satellite observation system to acquire images that containdisaster information According to the needs of disaster reliefmore and more different emergency tasks will arrive So farjust a few dynamic schedulingmethods consider dealing withthe emergency tasks After analyzing the dynamic propertiesof satellite scheduling Wang et al proposed an optimizationmodel and two heuristic algorithms to solve the problem [38]However they did not consider the task merging method inthe heuristic algorithms Wu et al adopted the ant colonyoptimization plus iteration local search (ACO-ILS) approachto resolve the multisatellite emergency tasks and commontasks scheduling problem [4] However they did not accountfor the uncertainty of emergency tasks Moreover Wang etal presented a dynamic scheduling algorithm called TMBSR-DES which comprehensively considers task merging back-ward shift and rehabilitation [39] The task merging strategyin TMBSR-DES only combines new tasks The robustnessof schedule has not been considered in all of the abovealgorithms

To sumup the studies on dynamic emergency schedulingmentioned above have the following shortages

(1) Some dynamic schedulingmethods ignored the time-liness of emergency tasks

(2) The real-time scheduling method [2 37 38] took intoaccount the timeliness of dynamic emergency tasks





















Satellite S2T1

S1

T3S1

T4S1T1

S2

T2S2

T3S2 T4

S2

T5S2


Time

Time

d1

d1

d2

d2

d3

d3d4d5

T1S1 T3

S1 T4S1

T1S2 T2

S2 T3S2 T4

S2T5S2

ws1

ws1

ws2

ws3

ws2ws3

ws4ws5

we1

we2

we1we2

we3

we3

we4we5

minusmsgS1

msgS1

Slew

ing

angl

e

minusmsgS2

msgS2

Slew

ing

angl

e





Adjust the solution

Update the schedule

Add emergency tasks


Pretreatment



Emergency tasks









3 Model





(i) Common task let 119879 = 1199051 1199052 119905






119905119873+2

119905119873+119873

119864


119899= (119901119899 119889119899 at119899 dt119899) where 119901

119899is the






dt119899

= 119905119891

119878


119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 be the










1 ao21 ao119895

1 ao12 ao119895

2 ao119872



isin

AO ao119895119894

= ao1198951198941

ao1198951198942

ao1198951198943

ao119895119894119896119894119895





119894119896isin ao119895


119894119896=

[ws119895119894119896

we119895119894119896

] 120579119895

119894119896 where ws119895

119894119896and we119895



119894119896 and 120579

119895

119894119896is the


is 119882119895

119894= ⋃119896isin[1119870119894119895]

119882119895

119894119896




max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (1)

where 119909119895


scheduled

119909119895

119894119896=


0 otherwise(2)





Search-space

Obj

ectiv

e

f(x)

f(x1)

f(y)f(y1)

x1 y1x y

+Δ +Δ



max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (3)


119894isin


119910119894

=


0 otherwise

(4)



nity ao119895119894119896 ao119895119894119896

isin ao119895119894


1198621

119909119895

119894119896(ts119895119894

minus ws119895119894119896

) ge 0

119909119895

119894119896(ts119895119894

+ 119889119894minus we119895119894119896

) le 0

119886119894le 119905119896119878

le ts119895119894

le dt119894

(5)




119894






1198622

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896le 1 (6)




NSGA2

Static scheduling



NSGA2


Updated schedule


94

8 225 14 613 94

8 225 14 613 9433 36 3237








1198623

te119895119906

+ sd119895 + sl119895 times 10038161003816100381610038161003816120579119895

V minus 120579119895

119906

10038161003816100381610038161003816+ st119895 + su119895 le ts119895V (7)



time constraint

1198624

sum

119894isin119879119895

119905119887

119889119894le duty119895

(8)

where 119879119895



+ orb119895] where 119905119887

isin

[119905119896119878 119891 minus orb119895]

4 Algorithms














available window














Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Satellite S2T1

S1

T3S1

T4S1T1

S2

T2S2

T3S2 T4

S2

T5S2


Time

Time

d1

d1

d2

d2

d3

d3d4d5

T1S1 T3

S1 T4S1

T1S2 T2

S2 T3S2 T4

S2T5S2

ws1

ws1

ws2

ws3

ws2ws3

ws4ws5

we1

we2

we1we2

we3

we3

we4we5

minusmsgS1

msgS1

Slew

ing

angl

e

minusmsgS2

msgS2

Slew

ing

angl

e





Adjust the solution

Update the schedule

Add emergency tasks


Pretreatment



Emergency tasks









3 Model





(i) Common task let 119879 = 1199051 1199052 119905






119905119873+2

119905119873+119873

119864


119899= (119901119899 119889119899 at119899 dt119899) where 119901

119899is the






dt119899

= 119905119891

119878


119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 be the










1 ao21 ao119895

1 ao12 ao119895

2 ao119872



isin

AO ao119895119894

= ao1198951198941

ao1198951198942

ao1198951198943

ao119895119894119896119894119895





119894119896isin ao119895


119894119896=

[ws119895119894119896

we119895119894119896

] 120579119895

119894119896 where ws119895

119894119896and we119895



119894119896 and 120579

119895

119894119896is the


is 119882119895

119894= ⋃119896isin[1119870119894119895]

119882119895

119894119896




max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (1)

where 119909119895


scheduled

119909119895

119894119896=


0 otherwise(2)





Search-space

Obj

ectiv

e

f(x)

f(x1)

f(y)f(y1)

x1 y1x y

+Δ +Δ



max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (3)


119894isin


119910119894

=


0 otherwise

(4)



nity ao119895119894119896 ao119895119894119896

isin ao119895119894


1198621

119909119895

119894119896(ts119895119894

minus ws119895119894119896

) ge 0

119909119895

119894119896(ts119895119894

+ 119889119894minus we119895119894119896

) le 0

119886119894le 119905119896119878

le ts119895119894

le dt119894

(5)




119894






1198622

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896le 1 (6)




NSGA2

Static scheduling



NSGA2


Updated schedule


94

8 225 14 613 94

8 225 14 613 9433 36 3237








1198623

te119895119906

+ sd119895 + sl119895 times 10038161003816100381610038161003816120579119895

V minus 120579119895

119906

10038161003816100381610038161003816+ st119895 + su119895 le ts119895V (7)



time constraint

1198624

sum

119894isin119879119895

119905119887

119889119894le duty119895

(8)

where 119879119895



+ orb119895] where 119905119887

isin

[119905119896119878 119891 minus orb119895]

4 Algorithms














available window














Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(i) Common task let 119879 = 1199051 1199052 119905






119905119873+2

119905119873+119873

119864


119899= (119901119899 119889119899 at119899 dt119899) where 119901

119899is the






dt119899

= 119905119891

119878


119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 be the










1 ao21 ao119895

1 ao12 ao119895

2 ao119872



isin

AO ao119895119894

= ao1198951198941

ao1198951198942

ao1198951198943

ao119895119894119896119894119895





119894119896isin ao119895


119894119896=

[ws119895119894119896

we119895119894119896

] 120579119895

119894119896 where ws119895

119894119896and we119895



119894119896 and 120579

119895

119894119896is the


is 119882119895

119894= ⋃119896isin[1119870119894119895]

119882119895

119894119896




max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (1)

where 119909119895


scheduled

119909119895

119894119896=


0 otherwise(2)





Search-space

Obj

ectiv

e

f(x)

f(x1)

f(y)f(y1)

x1 y1x y

+Δ +Δ



max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (3)


119894isin


119910119894

=


0 otherwise

(4)



nity ao119895119894119896 ao119895119894119896

isin ao119895119894


1198621

119909119895

119894119896(ts119895119894

minus ws119895119894119896

) ge 0

119909119895

119894119896(ts119895119894

+ 119889119894minus we119895119894119896

) le 0

119886119894le 119905119896119878

le ts119895119894

le dt119894

(5)




119894






1198622

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896le 1 (6)




NSGA2

Static scheduling



NSGA2


Updated schedule


94

8 225 14 613 94

8 225 14 613 9433 36 3237








1198623

te119895119906

+ sd119895 + sl119895 times 10038161003816100381610038161003816120579119895

V minus 120579119895

119906

10038161003816100381610038161003816+ st119895 + su119895 le ts119895V (7)



time constraint

1198624

sum

119894isin119879119895

119905119887

119889119894le duty119895

(8)

where 119879119895



+ orb119895] where 119905119887

isin

[119905119896119878 119891 minus orb119895]

4 Algorithms














available window














Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Search-space

Obj

ectiv

e

f(x)

f(x1)

f(y)f(y1)

x1 y1x y

+Δ +Δ



max

119873+119873119864

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (3)


119894isin


119910119894

=


0 otherwise

(4)



nity ao119895119894119896 ao119895119894119896

isin ao119895119894


1198621

119909119895

119894119896(ts119895119894

minus ws119895119894119896

) ge 0

119909119895

119894119896(ts119895119894

+ 119889119894minus we119895119894119896

) le 0

119886119894le 119905119896119878

le ts119895119894

le dt119894

(5)




119894






1198622

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896le 1 (6)




NSGA2

Static scheduling



NSGA2


Updated schedule


94

8 225 14 613 94

8 225 14 613 9433 36 3237








1198623

te119895119906

+ sd119895 + sl119895 times 10038161003816100381610038161003816120579119895

V minus 120579119895

119906

10038161003816100381610038161003816+ st119895 + su119895 le ts119895V (7)



time constraint

1198624

sum

119894isin119879119895

119905119887

119889119894le duty119895

(8)

where 119879119895



+ orb119895] where 119905119887

isin

[119905119896119878 119891 minus orb119895]

4 Algorithms














available window














Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















available window














Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Yes

No

No

Yes

No

No

Yes

Yes

population Pfather

Pfather

|Pchild| = M






|Pnext| lt M












7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










7 612908 1294912908 12927 19


Chromosom

e

Satelliten

ta

tb

tc

td

tj

ti

tu

Task i wej

ikdi piwsj

iktsji teji




119873 (119905119894) =

119901119894

1003816100381610038161003816AO1198941003816100381610038161003816 (9)


119862 (119908) = sum119905119894isin119879(119908)

119901 (119905119894) (10)



tasks 119905119894




119864 = 1199058 1199059 11990510

11990511


8can be combined


9can be inserted
















telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










telli

te 1

19 6

Sate

llite

2

Crossover point

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

Sate

llite

1Sa

telli

te 2

128 128 147 153

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

t2

t8

t5

t14

t13

t9

t5

t15

t13

t9

t5

t15

t14

t9

t2

t8

t5

t14

t13

t9

t3

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t4

t7

t1

t12

t3

t4

t7

t1

t12

62 62 68 85 6 7

146 146 156 170 10 9

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

23 23 42 48 19 6

146 146 156 170 10 9

331 331 338 355 7 8

454 454 465 470 11 1

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10

t5

t13

t15

t9

t5

t13

t13t14

t9

t5

t15

t9

t3

t5

t13

t15

t9

t2

t4

t7

t8

t1

t12

t2

t4

t7

t8

t1

t12

t4

t7

t1

t12

t2

t4

t7

t8

t1

t12

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 132 149 6 7

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

292 292 305 315 13 8

622 622 630 649 8 9

539 539 551 552 12 6

788 788 801 815 13 10

128 128 147 153 19 6

205 205 216 231 11 1 292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

597 597 617 638 20 10

62 62 68 85 6 7

622 622 630 649 8 9

128 128 147 153 19 6

205 205 216 231 11 1

292 292 305 315 13 8

365 365 376 388 11 9

539 539 551 552 12 6

788 788 801 815 13 10

62 62 68 85 6 7

146 146 156 170 10 8

331 331 338 355 7 8

597 597 617 638 20 10

675 675 682 716 7 3

126 126 136 149 10 10

292 292 305 315 13 8

622 622 630 649 8 9

23 23 33 48 10 7

146 146 159 170 13 8

331 331 338 355 7 8

454 454 466 470 12 6

675 675 682 716 7 3

539 539 551 552 12 6

788 788 801 815 13 10

597 597 617 638 20 10




t3

t3 t2

t8

t8

t2t2

t8

t15

t3



ETR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894 (11)



STR =

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895

119894119896times 119910119894 (12)



Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Chromosome

Sate

llite

1

128 153 19

216 231 11

305 315 13

11

551 552 12

Sate

llite

2

801 815 10

62 62 68 85 6 7

156 170 10

338 355 7

617 638 20

682 716 7

Chromosome

Sate

llite

1

147 6

305 13

551 12

801 10

Sate

llite

2

62 62 68 85 6 7

156 9

338 355 7

617 10

682 7

643 1

Mutation

765 9

t2

t8

t5

t14

t13

t9

t2

t5

t13

t9

t8

147128 6

1205 205

292 292 8

365 365 376 388 9

788 788 13

539 539 6

t3

t4

t7

t1

t12

t3

t4

t7

t1

t12

t14

9146 146

8331 331

10597 597

675 675 3

128 128 153 19

292 292 315 8

539 539 552 6

632632 658 11

788 788 815 13

146 146 170 10

331 331 8

597 597 638 20

675 675 716 3

754 754 777 11



Running task

Time

Task 1 Task 3

tkminus1S tkS tfS






119896


in [1199050119878 119905119891



119878= 1199050119878 1199051119878 119905119890

119878 119905119891

119878 =












No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













No


Yes


No new task

Yes

No


No Yes


Remove task from TE

No


Set SS998400 = SS


Output SS998400



Time

t1

t2t3

t4 t5t6

t7

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(a)Time

t1

t28t3

t9 t4t10 t6

t5 t11

tkS tfS

minusmsg

msg

Slew

ing

angl

e

(b)






0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10

12

14

Obj

ectiv

e

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 331Search-space


Current schedule

Sate

llite

1

147 6

216 231 11

305 315 13

376 11

12

801 815 13

Sate

llite

2

62 62 68 85 6 7

10

8

617 638 20

3

Sate

llite

1

147 6

11

116 126 8

9

539 539 552 12

801 13

Sate

llite

2

62 62 68 85 6 7

10

338 355 7

617 638 20

3


128 128 153 19

205 205 1

292 292 8

365 365 388 9

539 539 551 552 6

788 788 10

146 146 156 170 9

331 331 338 355 7

597 597 10

675 675 682 716 7

128 128 153 19

205 205 216 231 1

365 365 376 388 11

551 6

788 788 815 10

103 103 13

146 146 156 170 9

331 331 8

597 597 10

675 675 682 716 7

t2

t8

t5

t14

t13

t9

t2

t8

t14

t13

t9

t3

t4

t7

t1

t12

t5

t3

t4

t7

t1

t12








1 CTM 48 4 42 1 1 4TTM 47 1 44 1 1 4

2 CTM 47 4 38 3 2 5TTM 46 1 41 2 2 6

3 CTM 42 3 38 1 0 1TTM 41 2 38 1 0 2

4 CTM 47 7 36 1 3 4TTM 46 1 39 1 4 6

5 CTM 44 3 40 0 1 1TTM 42 1 43 0 0 1










1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 2 3 4 5200

250

300

350

400

Reve

nue

(a)

Robustness

Robu

stnes

s



1 2 3 4 50

100

200

300

400

500

600

700

800

900

(b)

Satisfaction ratio

Satis

fact

ion

ratio

088

09

092

094

096

098

1



1 2 3 4 5

(c)




Appendices









119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878119901




119878119901

= 119878119901

cup 119902 add 119902 into 119878119901


119899119901

= 119899119901


if (119899119901


= 1198651



119894= )

for each 119901 isin 119865119894

for each 119902 isin 119878119901

119899119902

= 119899119902




= 119865119894+1


end forend for

119894 = 119894 + 1end while

Algorithm 1










end forend for

Algorithm 2





119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894| for task 119894


119908

(11) if |119882119894| le 1 do



119894from 119879


119894from 119879


Algorithm 3




119875 = 119909 isin Ω | notexist1199091015840

isin Ω st 119865 (1199091015840) ≺ 119865 (119909) (A1)


119875119865 = 119865 (119909) | 119909 isin 119875 (A2)



= 119909 isin



119909119894 Rank(119909

119894) = |119875



cup 1198652

cup



119894) =

(119891(119909119894+1

)minus119891(119909119894+1

))(119891max119896



Distance (119909119894) =

119891revenue (119909119894+1


)


revenue

+119891robust (119909

119894+1) minus 119891robust (119909

119894minus1)


robust

(A3)


119865 = 119908revenue

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119909119895

119894119896119901119894

+ 119908robustness

119873

sum119894=1

119872

sum119895=1

119870119894119895

sum119896=1

119901119894times 119909119895


119894)

(A4)






CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











CTMTTM


200

250

300

350

400

Reve

nue

(a)

Robustness

Reve

nue


0

200

400

600

800

1000

CTMTTM

(b)

Reve

nue

089

09

091

092

093

094

095

096

097

098



CTMTTM

(c)


sj

120579i

ti

Δ120579

120579j

ti998400


(a)

di

dj

wsi

wsj

wei

wej


(b)




119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and 119905119895that


= [ws119894we119894] 120579119894 isin



119895=

[ws119895we119895] 120579119895 isin AO119895











(119908119904119895

+ 119889119895) minus (119908119890

119894minus 119889119894) le Δ119889 (B1)

10038161003816100381610038161003816120579119894minus 120579119895

10038161003816100381610038161003816le Δ120579 (B2)




swath of the sensor



119894119895=

[119908119904119894119895

119908119890119894119895

] should range from

119908119904119894119895

=

119908119890119894minus 119889119894 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min 119908119904119895max (119908119904

119894 119908119904119895

+ 119889119895

minus 119889119894) 119890119897119904119890

(B3)

to

119908119890119894119895

=

119908119895

+ 119889119895 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

min (119908119890119894minus 119889119894 119908119890119895

minus 119889119895) + max (119889

119894 119889119895) 119890119897119904119890

(B4)


119889119894119895

=

119908119904119895

+ 119889119895

minus (119908119890119894minus 119889119894) 119894119891

10038161003816100381610038161003816119882119894cap 119882119895

10038161003816100381610038161003816le min (119889

119894 119889119895)

max (119889119894 119889119895) 119890119897119904119890

(B5)


120579119894119895

=


Δ120579119904

2 0 119894119891 120579

119894ge 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

min120579119894+

Δ120579119904

2 0 119894119891 120579

119894lt 0

10038161003816100381610038161205791198941003816100381610038161003816 ge

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816

max120579119895

minusΔ120579119904

2 0 119894119891 120579

119895ge 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

min120579119895

+Δ120579119904

2 0 119894119891 120579

119895lt 0

10038161003816100381610038161003816120579119895

10038161003816100381610038161003816gt

10038161003816100381610038161205791198941003816100381610038161003816

(B6)



Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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