A Novel Memetic Algorithm Based on Decomposition for...

Research ArticleA Novel Memetic Algorithm Based on Decomposition forMultiobjective Flexible Job Shop Scheduling Problem

ChunWang Zhicheng Ji and YanWang

Engineering Research Center of IoT Technology Applications Ministry of Education Jiangnan University Wuxi 214122 China

Correspondence should be addressed to Yan Wang wangyan88jiangnaneducn

Received 12 May 2017 Revised 1 November 2017 Accepted 9 November 2017 Published 29 November 2017

Academic Editor Josefa Mula

Copyright copy 2017 Chun Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel multiobjective memetic algorithm based on decomposition (MOMAD) is proposed to solve multiobjective flexible jobshop scheduling problem (MOFJSP) which simultaneously minimizes makespan total workload and critical workload Firstlya population is initialized by employing an integration of different machine assignment and operation sequencing strategiesSecondly multiobjective memetic algorithm based on decomposition is presented by introducing a local search to MOEADThe Tchebycheff approach of MOEAD converts the three-objective optimization problem to several single-objective optimizationsubproblems and the weight vectors are grouped by K-means clustering Some good individuals corresponding to different weightvectors are selected by the tournament mechanism of a local search In the experiments the influence of three different aggregationfunctions is first studied Moreover the effect of the proposed local search is investigated Finally MOMAD is compared with eightstate-of-the-art algorithms on a series of well-known benchmark instances and the experimental results show that the proposedalgorithm outperforms or at least has comparative performance to the other algorithms

1 Introduction

The job shop scheduling problem (JSP) is one of the mostimportant and difficult problems in the field of manufac-turing which processes a set of jobs on a set of machinesEach job consists of a sequence of successive operations andeach operation is allowed to process on a unique machineDifferent from JSP which one operation is merely allowedto process on a specific machine the flexible job shopscheduling problem (FJSP) permits one operation processedby any machine from its available machine set Since FJSPneeds to assign operations to their suited machine as well assequence those operations assigned on the same machine itis a complex NP-hard optimization problem [1]

The existing literatures [2ndash5] about solving single-objective FJSP (SOFJSP) over the past decades mainlyconcentrated on minimizing one specific objective such asmakespan However in practical manufacturing processsingle-objective optimization cannot fully satisfy the produc-tion requirements since many optimized objectives are usu-ally in conflict with each other In recent years multiobjective

flexible job shop scheduling problem (MOFJSP) has receivedmuch attention and until now many algorithms have beendeveloped to solve this kind of problem These methods canbe classified into two groups one is a priori approach and theother is Pareto approach

Multiple objectives are usually linearly combined into asingle one by weighted sum approach in the a priori methodwhich can be illustrated as 119865 = sum푀푖=1 120582푖119891푖 where sum푀푖=1 120582푖 =1 0 le 120582푖 le 1 However we can get only one or severalPareto solutions by using this approach which may not wellreflect the tradeoffs among different objectives and it wouldbe difficult to assign an appropriate weight for each problemEven more important the performance of the algorithmdeteriorates when solving the problems contains nonconcavePareto front (PF) The Pareto approach mainly focuses onsearching the Pareto set (PS) of optimization problems bycomparing two solutions based on Pareto dominance relation[6] A solution x is said to dominate solution y iff x is notworse than y in all objectives and there exists at least oneobjective in which x is better than y xlowast is called a Paretooptimal solution iff there is no solution x isin Ω that dominates

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2857564 20 pageshttpsdoiorg10115520172857564

2 Mathematical Problems in Engineering

xlowast All the Pareto optimal solutions constitute the PS andPF is the mapped vector of PS in the objective space SincePareto approach can achieve a set of Pareto solutions ratherthan a specific one it has receivedmuchmore attention than apriori approach and is recognized to bemore suitable to solveMOFJSP

Because the three objectives makespan total workloadand critical workload are conflicted with each other it isbetter to handle this model with knowledge about their PFMultiobjective evolutionary algorithm (MOEA) is a kind ofmature global optimization method with high robustnessand wide applicability Due to the fact that MOEAs havelow requirements on the optimization problem itself andhigh ability to obtain multiple Pareto solutions during eachrun they are suitable for solving multiobjective optimizationproblems (MOPs) The multiobjective evolutionary algo-rithm based on decomposition (MOEAD) that integratesmathematical programming with evolutionary algorithm(EA) can obtain a set of Pareto solutions by aggregat-ing multiple objectives into different single-objectives withmany predefined weight vectors [7] MOEAD has showngreat superiority on continuous optimization problems [8ndash12] thus it is necessary to investigate its performance onmultiobjective combinatorial optimization problems (MO-COPs) such as MOFJSP To the best of our knowledge in theliterature reported although MOEAD has been applied todifferent kinds of multiobjective scheduling problems such asmultiobjective flow shop scheduling problem (MOFSP) [13]multiobjective permutation flow shop scheduling problem(MOPFSP) [14] multiobjective stochastic flexible job shopscheduling problem (MOSFJSP) [15] and multiobjective jobshop scheduling problem (MOJSP) [16] there is seldomcorresponding application on MOFJSP

The primary aim of this paper is to solve MOFJSPin a decomposition manner by proposing a multiobjectivememetic algorithm based on decomposition (MOMAD)hybridizing MOEAD with local search With the purpose ofmaking the proposed algorithmmore applicable four aspectsare studied (1) integration of different machine assignmentand operation sequencing strategies are presented to con-struct the initial population (2) objective normalization isincorporated into Tchebycheff approach to convert an MOPinto a number of single-objective optimization subproblems(3) all weight vectors are divided into a few groups based onK-means clustering some superior individuals correspond-ing to different weight vector groups are selected by usinga selection mechanism (4) local search based on movingcritical operations is applied on selected individuals Toevaluate the effectiveness of the proposed algorithm somebenchmark instances are tested with three purposes (1)investigating the effects of different aggregation functions andvalidating the effectiveness of local search (2) analyzing theinfluence of the key parameters on the performance of thealgorithm (3) comparing the performance of MOMAD withother state-of-the-art algorithms for solving MOFJSP

The rest of the paper is organized as follows Next sectionpresents a short overview of the existing related work InSection 3 the background knowledge of MOFJSP is intro-duced Section 4 introduces the framework of MOMADThe

implementation details of the proposed MOMAD includinggenetic global search and problem specific local search aredescribed in Section 5 Afterwards experimental studies areprovided in Section 6 Finally Section 7 concludes this paperand outlines some avenues for future research

2 Related Work

As mentioned above there are two main methods to solveMOFJSP a priori approach and Pareto approach As fora priori approach Xia and Wu [17] discussed a hybridalgorithm where particle swarm optimization (PSO) andsimulated annealing (SA)were employed in global search andlocal search respectively A bottleneck shifting-based localsearch was incorporated into genetic global search by Gaoet al [18] Zhang et al [19] introduced an effective hybridPSO algorithm combined with tabu search (TS) Xing etal [20] used ten different weight vectors to collect effectivesolution sets A hybrid TS (HTS) algorithm was structuredby combining adaptive rules with two neighborhoods Inthis algorithm three weight coefficients 1205821 1205822 and 1205823with different settings were given to test different problems[21] An effective estimation of distribution algorithm wasproposed by Wang et al [22] in which the new individualswere generated by sampling a probability model

Contrary to the a priori approach a PS can be obtainedby using Pareto approach and the tradeoffs among differentobjectives can be presented The integration of fuzzy logicand EA was proposed by Kacem et al [23] A guide localsearch was incorporated into EA to enhance the convergence[24] With the aim of keeping population diversity immuneand entropy principle were adopted in multiobjective geneticalgorithm (MOGA) [25] Two memetic algorithms (MAs)were respectively proposed both of which integrate non-dominated sorting genetic algorithm II (NSGA-II) [6] witheffective local search techniques [26 27] Several effectiveneighborhood approaches were used in variable neighbor-hood search to enhance the convergence ability in a hybridPareto-based local search (PLS) [28] Chiang and Lin [29]proposed a simple and effective evolutionary algorithmwhich only requires two parameters Both the neighborhoodsof machine assignment and operation sequence are consid-ered in Xiong et al [30] An effective Pareto-based EDAwas proposed by Wang et al [31] A novel path-relinkingmultiobjective TS algorithm was proposed in [32] in whicha routing solution is identified by problem-specific neighbor-hood search and is then further refined by the TS with back-jump tracking for a sequencing decision In addition to thesuccessful use of EA several swarm intelligence algorithmshave also been widely used for global search PSOs were usedas global search algorithms in [33ndash36] Besides shuffled frogleaping [37] and artificial bee colony [38]were integratedwithlocal search in related hybrid algorithms

Besides the successful using in many scheduling prob-lems MOEADs have also been widely dedicated to otherMO-COPs A novel NBI-style Tchebycheff approach wasused in MOEAD to solve portfolio management MOP withdisparately scaled objectives [39] Mei et al [40] developedan effective MA by combining MOEAD with extended

Mathematical Problems in Engineering 3

neighborhood search to solve capacitated arc routing prob-lem Hill climbing SA and evolutionary gradient searchwere respectively embedded into EDA for solving multipletraveling salesmen problem (MTSP) in a decompositionmanner [41] A hybrid MOEA was established by combiningant colony optimization with MOEAD [42] and then itwas adopted to solve multiobjective 0-1 knapsack problem(MOKP) and MTSP respectively Then aiming at the sametwo problems Ke and Zhang proposed a hybridization ofdecomposition and PLS [43]

As mentioned before MOEAD is a kind of popularMOEA which is very suitable for solving MO-COPs such asscheduling problem In this paper a MOMAD is proposedthat integrates MOEAD algorithm with local search toenrich the tool-kit for solving MOFJSP

3 Related Background Knowledge

31 Problem and Objective of MOFJSP The MOFJSP canbe formulated as follows There are a set of 119899 jobs J =1198691 1198692 119869푛 and119898machinesM = 11987211198722 119872푚 eachjob 119869푖 (119894 = 1 2 119899) contains one or more operations to beprocessed in accordance with the predetermined sequenceEach operation can be processed on any machine among itscorresponding operable machine set119872푖푗 isin M The problemis defined as T-FJSP iff 119872푖푗 = M otherwise it is calledP-FJSP [44] MOFJSP not only assigns suitable processingmachines for each operation but also determines the mostreasonable processing sequence of operations assigned on thesame machine in order to simultaneously optimize severalobjectives

The following constraints should be satisfied in theprocess

(1) At a certain time a machine can process one opera-tion at most and one operation can be processed byonly one machine at a certain moment

(2) Each operation cannot be interrupted once processed(3) All jobs and machines are available at time 0(4) Different jobs share the same priority(5) There exists no precedence constraint among the

operations of different jobs but there exist precedenceconstraints among the operations belonging to thesame job

An instance of P-FJSP with three jobs and threemachinesis illustrated in Table 1 Let 119862푖푗 and 119901푖푗푘 be the completiontime of operation 119874푖푗 and its processing time on machine119896 respectively 119862푖 denotes the completion time of job 119869푖Three considered objectives are makespan total workloadand critical workload which are formulated as follows

min F = (1198911 1198912 1198913)푇 1198911 119862max = max 119862푖 | 119894 = 1 2 119899 1198912 119882푇 = 푚sum

푘=1

푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘

Table 1 An instance of P-FJSP with 3 jobs on 3 machines

Job Operation Machine1198721 1198722 11987231198691 11987411 2 - 411987412 6 3 -11987413 - 2 2

1198692 11987421 3 3 411987422 - 2 511987423 - 3 2

1198693 11987431 4 3 211987432 4 - 211987433 3 4 3

1198913 119882max = max1le푘le푚

푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘 119906푖푗푘=

1 if machine 119896 is selected for operation 119874푖푗0 otherwise(1)

32 Disjunctive Graph Model Disjunctive graph model 119866 =(119881119880 119864) has been adapted for representing feasible schedulesof FJSP 119881 denotes nodes set and each of them representsan operation The virtual starting and ending operations arerepresented by two virtual nodes 0 and lowast respectively 119880 isthe set of conjunctive arcs which connect adjacent operationsof the same job and each arc indicates the precedenceconstraint within the same job 119864 is the set of disjunctive arcscorresponding to the adjacent operations scheduled on thesamemachine119864 = ⋃푚푘=1 119864푘 where119864푘 denotes the disjunctivearcs set of machine 119896 The weight value above the node 119874푖푗denotes119901푖푗푘 and the selectedmachine to operate119874푖푗 is labeledunder the node 119874푖푗 1199010 = 119901lowast = 0

Figure 1 shows the disjunctive graph of a feasible solutioncorresponding to the instance shown in Table 1 in whichevery disjunctive arc confirms a direction This graph iscalled digraph In digraph 119866 the longest path from node119886 to node 119887 is termed as critical path the length of whichdenotes the makespan of corresponding schedule Besideseach operation on critical path is called critical operation InFigure 1 there is one critical path 0 rarr 11987411 rarr 11987412 rarr 11987433 rarrlowast whose length equals 13

Given that the local search in the following section canbe well described some concepts based on digraph 119866 aredenoted here Suppose ℎ is a node in119866 and its correspondingoperation is119874ℎ119872(119866 ℎ) denotes the corresponding machineto process 119874ℎ 119878퐸(119866 ℎ) and 119862퐸(119866 ℎ) denote its earlieststarting time and earliest completion time Let 119901ℎ푀(퐺ℎ) bethe processing time of operation 119874ℎ on 119872(119866 ℎ) the lateststarting time 119878퐿(119866 ℎ 119862max(119866)) and latest completion time


(1) Generate a set of weight vectors Λ larr 12058211205822 120582푁(2) Generate weight vector groups 1205771 1205772 120577푛119888 (3) Get the neighborhood 119861(119894) of each weight vector 120582푖 119894 = 1 2 119873 where 120582푖1 120582푖2 120582푖119879 are the 119879closest weight vectors to 120582푖(4) 1198750 larr Initialize the population( )(5) Initialize Idealpoint zlowast = (119911lowast1 119911lowast2 119911lowast푀)푇(6) Find the non-dominated solutions in initial population to construct the archive 119860119903119888(7) while the termination criterion is not satisfied do(8) for 119894 = 1 2 119873 do(9) if rand(1) lt 120575 then(10) 119875119875 larr 119861(119894)(11) else(12) 119875119875 larr 1 2 119873(13) end if(14) Randomly select two different indexes 119896 119897 from 119875119875 (119896 = 119894 119897 = 119894)(15) y larr GeneticOperators (x푘 x푙)(16) UpdateIdealPoint (y zlowast)(17) Objective Normalization(18) Update Current Population(19) Update External Archive 119860119903119888(20) end for(21) 119875 larr LocalSearch(119875)(22) end while

Algorithm 1 Framework of the proposed MOMAD

Disjunctive arcsConjunctive arcs

0

24 6

323

223

O O12

O21

O31 O32 O33

O22 O23

O13

M3

M3

M3

M1

M1

M1

M2

M2

M2

lowast

Figure 1 Illustration of the disjunctive graph

119862퐿(119866 ℎ 119862max(119866)) without delaying the required makespan119862max(119866) can be calculated by

119862퐸 (119866 ℎ) = 119878퐸 (119866 ℎ) + 119901ℎ푀(퐺ℎ)119862퐿 (119866 ℎ 119862max (119866)) = 119878퐿 (119866 ℎ 119862max (119866)) + 119901ℎ푀(퐺ℎ) (2)

The predecessor and successor operation scheduled onthe same machine right before or after 119874ℎ are denoted as119875119872(119866 ℎ) and 119878119872(119866 ℎ) respectively In addition 119875119869(119866 ℎ)and 119878119869(119866 ℎ) are the predecessor and successor operation of119874ℎ in the same job respectively 119874ℎ is a critical operation ifand only if 119862퐸(119866 ℎ) = 119862퐿(119866 ℎ 119862max(119866))

4 Framework of the Proposed MOMAD

The framework ofMOMADalgorithm is formed by hybridiz-ing MOEAD with local search which is given in Algo-rithm 1 First a set of uniformly distributed weight vectors12058211205822 120582푁 is generated by Das and Dennisrsquos approach[45] where each vector 120582푖 corresponds to subproblem 119894 Nextall the weight vectors are divided into 119899푐 groups by K-meansclustering After calculating the Euclidean distance betweenany two weight vectors the neighborhood 119861(119894) of 120582푖 is setby gathering 119879 closest weight vectors Then the populationcontaining 119873 solutions is initialized The ideal point vectorzlowast is obtained by calculating the infimum found so far of 119891푖The archive Arc is established by founding the nondominatedsolutions in initial population In Steps (9)ndash(13) the twomating parent solutions x푘 and x푙 are chosen from119875119875 formedby 119861(119894)with the probability 120575 or by whole population with theprobability 1 minus 120575 Then the new solution y is generated bycrossover and mutation and finally y is used to update zlowast

Steps (17)ndash(21) contain the updating and local searchphase The objective normalization is first adopted beforepopulation updating Suppose 120582푗 = (1205821푗 1205822푗 120582푀푗 )푇 is119895th weight vector and 119911nad푖 is the largest value of 119891푖 in thecurrent population then y is compared with the solutionsfrom 119875119875 one by one and the one that has poorer fitnessin terms of (3) will be replaced by y It should be notedthat the updating procedure will be terminated as soon asthe predefined maximal replacing number 119899푟 which benefitsfrom keeping population diversity is reached or 119875119875 is emptyAfterwards the updating of Arc is held If no solutions in Arcdominate y then copy y into Arc and remove all the repeatedand dominated solutions Finally after selecting the super


233 2 21 212 1 31 3 2 2 1 1 3

Machine selection part (MS)

Job 1 Job 2 Job 3

M1 M3 M3J1 J1 J1J3 J3 J3J2 J2 J2M2M1

Operation sequence part (OS)

O22 O32 O12 O13 O23 O33O21O31O11

Figure 2 Chromosome encoding

solutions in current population the local search is applied toget some improved solutions and then they are rejected intothe population to ameliorate it

119892푡푒 (x | 120582푗 zlowast) = max1le푖le푀

120582푖푗 100381610038161003816100381610038161003816100381610038161003816119891푖 (x) minus 119911lowast푖119911nad푖 minus 119911lowast푖

100381610038161003816100381610038161003816100381610038161003816 (3)

5 Detailed Description of Exploration andExploitation in MOMAD

51 Chromosome Encoding and Decoding In MOMAD achromosome coding consists of two parts machine selec-tion (MS) part and operation sequence (OS) part whichare encoded by machine selection and operation sequencevector respectively Each integer in MS vector represents thecorresponding machine assigned for each operation in turnAs for OS vector each gene is directly encoded with thejob number When compiling the chromosome from left toright the 119896th occurrence of the job number refers to the119896th operation of the corresponding job Figure 2 shows achromosome encoding of an P-FJSP instance which is shownin Table 1 1198723 is selected to process operation 11987411 and 1198721is selected to process operation 11987421 The operation sequencecan be interpreted as 11987411 rarr 11987431 rarr 11987421 rarr 11987422 rarr 11987432 rarr11987412 rarr 11987413 rarr 11987423 rarr 11987433

Since it has been verified that the optimal schedule existsin active schedule [3] the greedy inserting algorithm [25]is employed for chromosome encoding to make sure theoperation is positioned in the earliest capable time of itsassigned machine It should be noted that operation 119874푖푗 maybe started earlier than 119874푙푘 while 119874푙푘 appears before 119874푖푗 in theOS In order tomake the encoding be able to reflect the actualprocessing sequence the operation sequence in original OSwill be reordered in the light of their actual starting time afterdecoding

52 Population Initialization Population initialization playsan important role in MOMAD performance since a highquality initial population with more diversity can avoidfalling into premature convergence Here four machineassignment rules are used to produce themachine assignmentvectors for MOFJSP The first two rules are global selection

and local selection proposed by Zhao et al [46] The thirdrule prefers to select a machine from the candidate machineset at random The aim of the last rule is assigning eachoperation to a machine with the minimum processing timeIn our MOMAD for machine assignment initialization 50of individuals are generated by rule 1 10 of individuals areformed with rule 4 and rule 2 and rule 3 take share of the restof the population

Once the machines are assigned to each operation thesequence of operations should be considered next Amixtureof four operation sequencing rules is employed to generatethe initial operation sequencing vectors The probabilities ofusing four operation sequencing rules are set as 03 02 03and 02 respectively

Operation sequencing rules are as follows

(1) Most Work Remaining [47] The operations whichhave the most remaining processing time will be putinto the operation sequencing vector first(2) Most Number of Operations Remaining (MOR)[47] The operations which have the most subsequentoperations in the same job will be preferentially takeninto account(3) Shortest Processing Time (SPT) [48] The opera-tions with the shortest processing time will be firstlyprocessed(4) Random Dispatching It randomly orders theoperations on each machine

53 Exploration Using Genetic Operators The problem-specific crossover and mutation operators are applied toproduce the offspring both of which are performed on eachvector independently since the encoding of one chromosomehas two components

Crossover For theMS uniform crossover [3] is adopted Firstof all a subset of 119903 isin [1 119863] positions is uniformly chosen atrandom where 119863 equals the total number of all operationsThen two new individuals are generated by changing thegene between parent chromosomes corresponding to theselected positions With respect to OS vector the precedencepreserving order-based (POX) [3] crossover is applied


(1) for 119895 = 1 to 119899푐 do(2) Randomly select a weight 120582 from the weight vector set 120577푖(3) Select all solutions x1 x2 x푛119894 respectively correspond to weight vectors 12058211205822 120582ni whichbelong to 120577푖(4) (MSOS) larrTournament Selection (x1 x2 x푛119894 )(5) 119864耠푗 larr LocalSearchForIndividual ((MSOS)120582푖)(6) UpdateNeighborhood (119864耠푗 119861(120582))(7) 119860119903119888 larr UpdateArchive (119864耠푗 119860119903119888)(8) end for

Algorithm 2 LocalSearch (119875)

Mutation As for MS two positions are chosen arbitrarilyand the genes are replaced by changing the different machinefrom their corresponding machine set With regard to OSthemutation is achieved by exchanging two randomly chosenoperations

54 Exploitation Using Local Search It is widely acceptedthat integrating local search can effectually improve theperformance of EAs and it is an effective strategy to solveFJSP [31] so it is of great importance to design an effectlocal search method to make MOMAD keep good balancebetween exploration and exploitation First of all with theaim of saving the computational complexity as well as main-taining the population diversity only a number of lfloor119873 times 119875lsrfloorindividuals are selected from entire evolutionary populationto apply local search in each generation Then the followingtwo critical issues will be introduced in detail in the next twosubsections

(1) How to select appropriate solutions to apply localsearch

(2) Which local search method will be used

541 Selection of Individuals for Local Search When selectinga candidate individual each time at first a weight vector 120582 ischosen from the vector set 120577푖 at randomThen all incumbentsolutions corresponding to different weight vectors whichbelong to 120577푖 are selected Next all the selected individualsare compared according to (3) with the weight vector 120582 andthe one which has the best fitness is selected as a candidatesolution to apply local search Finally an improved solutionobtained by local search is adopted to update neighborhoodand archive Denote 119899푖 as the number of weight vectors whichbelong to 120577푖 The basic framework of local search can besummarized as Algorithm 2

542 Description of Local Search Method Considering thatmakespan is the most important and the hardest objectiveto be optimized among the three optimization objectivesthe local search is performed on the decoded scheduleof one chromosome rather than the chromosome itselfSuppose there are a set of 119899119888(119866) critical operations 119888(119866) =cor1 cor2 cor푛푐(퐺) and let 119862max(119866) be the makespan of

119866 Then the following theorems based on disjunctive graphare summarized as follows [2]

(1) It has been proven that the makespan can only bereduced by moving critical operations 119874푖푗 isin 119888(119866)

(2) Suppose 119866minus푖 is obtained by deleting one criticaloperation cor푖 in 119866 Let 120601푘 be the set of operationsprocessed on119872푘 which are sorted by the increasingof starting time (note that cor푖 notin 120601푘) in 119866minus푖 then twosubsequences of 120601푘 denoted as 119877푘 and 119871푘 are definedas follows

119877푘 = V isin 120601푘 | 119878퐸 (119866 V) + 119901V푘 gt 119878퐸 (119866minus푖 cor푖) 119871푘 = V isin 120601푘 | 119878퐿 (119866 V 119862max (119866))lt 119878퐿 (119866minus푖 cor푖 119862max (119866))

(4)

It has been verified that a feasible schedule 119866耠 canbe achieved by inserting cor푖 into a position V isinΥ푘 where Υ푘 contains all positions before all theoperations of 119877푘119871푘 and after all the operations of119871푘119877푘

(3) The insertion of moving cor푖 on position V mustsatisfy the following constraint

max 119862퐸 (119866minus푖 119875119872 (119866minus푖 V)) 119862퐸 (119866minus푖 119875119869 (cor푖))+ 119901cor119894 푘 le min 119878퐿 (119866minus푖 V 119862max (119866)) 119878퐿 (119866minus푖 119878119869 (cor푖) 119862max (119866))

(5)

When considering an action of finding a position on119872푘for cor푖 to insert into 119866minus푖 which is denoted as cor푖 997891rarr 119872푘only the positions in Υ푘 should be taken into account Insertcor푖 into one position V in Υ푘 if V satisfies (5) and otherpositions in Υ푘 will no longer be considered Suppose119872cor119894is the alternative machine set to operate cor푖 119899cor119894 denotesthe total actions of cor푖 997891rarr 119872푘 Let 120593(119888(119866)119872) be the actionset cor푖 997891rarr 119872푘 | 119894 = 1 2 119899119888 119872푘 isin 119872cor119894 whichcontainssum푛119888푖=1 119899cor119894 actions Now a hierarchical strategy [27] isadopted to calculate Δ119905 and Δ119888 which respectively represent


(1) iterlarr 0(2) 119866 larr ChromosomeDecodingMSOS(3) 119866best larr 119866(4) while 119866 = 0 and 119894119905119890119903 lt itermax do(5) 119866耠 larr GetNeighborhood(119866)(6) if 119866耠 = 0 then(7) if (F(119866耠) ≺ F(119866best)) or ((F(119866耠) F(119866best))amp(F(119866耠) = F(119866best))) then(8) 119866best larr 119866耠(9) end if(10) end if(11) 119866 larr 119866耠(12) iter = iter + 1(13) end while(14) 119864耠 larr ChromosomeEncoding(119866best)(15) return 119864耠Algorithm 3 LocalSearchForIndividual (MSOS)120582푖

the variation of total workload and critical workload Allactions in 120593(119888(119866)119872) are sorted by nondescending order ofΔ119905 If both actions have the same Δ119905 then the lowest Δ119888 ischosen as a second criterionThe value of Δ119905 Δ119888 is calculatedas follows

Δ119905 (cor푖 997891997888rarr 119872푘) = 119901cor119894 푘 minus 119901cor119894 푀(퐺cor119894)Δ119888 (cor푖 997891997888rarr 119872푘) = 119882푘 (119866) + 119901cor119894 푘 (6)

These actions in sorted 120593(119888(119866)119872) are considered in suc-cession until a neighborhood 119866耠 is established Whereafterwe compare 119866耠 with 119866best by adopting an acceptance ruledescribed as Step (7) in Algorithm 3 119866best will be replacedby 119866耠 providing that 119866耠 is not dominated by 119866 and they aredifferent from each other It should be noted that the lengthof once local search is controlled by two points in order tosave the computing cost First when getting a neighborhoodof 119866 once an effective action in 120593(119888(119866)119872) is found aneighbor schedule 119866耠 is formedThen the remaining actionsin 120593(119888(119866)119872)will no longer be consideredThe second effortis to set max iteration number itermax so that the search willbe terminated when the itermax is exhausted

543 Computational Complexity Analysis The major com-putational costs in MOMAD are involved in Step (16)simStep (18) and Step (21) Step (16) performs 119874(119872) com-parisons and assignments The objective normalization inStep (17) requires 119874(119872) operations Because the computing119892푡푒 for 119879 neighborhood solutions and119874(119872) basic operationsare required in one computation119874(119872119879) basic operations areneeded for Step (18) Therefore the total computational costin Step (16)simStep (18) is 119874(1198721198732) + 119874(1198721198732) + 119874(119872119873119879)since it computes 119873 times When comparing F(119866) andF(119866best) in local search it requires 119874(119872) basic operationsin one iteration and the worst case is 119874(119872itermax) if themaximal iterations are reached The computational cost ofStep (21) is119874(119872119873119875lsitermax) since it has119875lstimes119873 passesThusthe total computational complexity ofMOMAD is119874(1198721198732)+119874(119872119873119875lsitermax)

Table 2 Parameter settings of the proposed MOMAD algorithm

Parameters ValuesPopulation size (119873) 120Crossover probability (119875푐) 10Mutation probability (119875푚) 01Neighborhood size (119879) 01 times 119873Controls parameters (120575) 09Maximal replacing number (119899푟) 1Division (119867) 14Local search probability (119875ls) 01Number of weight vector cluster (120577) 119875ls times 119873Maximal iterations of local search (itermax) 50

6 Experiments and Results

With the aim of testing the performance of MOMAD 5 well-knownKacem instances (Ka 4times5 Ka 8times8 Ka 10times7 Ka 10times10and Ka 15 times 10) [23] and 10 BRdata instances (MK01simMK10)[49] are underinvestigated in the experiment The MOMADis implemented in C++ language and on an Intel Core i3-4170370GHz processor with 4GB of RAM

61 Parameter Settings With the purpose of eliminatingthe influence of random factors on the performance of thealgorithm it is necessary to independently run the proposedMOMAD 10 times on each test instance and the algorithmwill be terminated when reaching the maximal number ofobjective function evaluations in one run The parametersused in MOMAD algorithm are listed in Table 2 Moreoverbecause of the different complexity of each test instancethe predefined maximal numbers of objective evaluationscorresponding to different problems are listed in the secondcolumn in Table 3

62 Performance Metrics In order to quantitatively eval-uate the performance of the compared algorithms three


Table 3 Comparison of three different aggregation functions on average IGD and HV values over 10 independent runs for all Kacem andBRdata instances

Instance NFssect IGD HVMOMAD MOMAD-WS MOMAD-PBI MOMAD MOMAD-WS MOMAD-PBI

ka 4 times 5 40000 01054 02108 01845 05910 05843 05860ka 8 times 8 40000 01071 01252 01541 06574 06191 05917ka 10 times 7 40000 00298 00400 00651 10594 10481 09762ka 10 times 10 40000 01394 00767 01676 06802 09127Dagger 06873ka 15 times 10 40000 04002 04108 04405 03727 03474 02849MK01 50000 00304 00254 00619dagger 10316 10368 09975dagger

MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger

MK06 60000 00473 00529dagger 00993dagger 09700 09610 08906dagger

MK07 50000 00218 00313 01288dagger 09906 09734dagger 08484dagger

MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger


MK10 70000 00391 00765dagger 00597dagger 11480 10751dagger 10743dagger

sect means the number of function evaluations dagger means that the results are significantly outperformed by MOMAD Dagger means that the results are significantlybetter than MOMAD

quantitative indicators are employed tomake the comparisonmore convincing and they are described as follows

(1) Inverted Generational Distance (IGD) [50] Let 119875known bethe approximate PF obtained by the compared algorithm and119875lowast be the reference PF uniformly distributed in the object

space IGDmetric measures the distance between 119875known and119875lowast with smaller value representing better performance TheIGD metric can well reflect the convergence and diversitysimultaneously to some extent if 119875lowast is larger enough to wellrepresent the reference PF that is

IGD (119875lowast 119875known) = sumylowastisin푃lowast 119889 (ylowast 119875known)|119875lowast| 119889 (ylowast 119875known) = min

yisin푃known

radic푀sum푖=1

(119910lowast푖 minus 119910푖)2 ylowast = (119910lowast1 119910lowast2 119910lowast푀)푇 y = (1199101 1199102 119910푀)푇

(7)

In this experiment since all benchmark instances aretested without knowledge about their actual PFs 119875lowast usedin calculating IGD metric is obtained by two steps Firstwe merge all final nondominated solutions obtained byall compared algorithms during all the independent runsThen we select the nondominated solutions from the mixedsolution set as 119875lowast(2) Set Coverage (C) [51] 119862 metric can directly reflect thedominance relationship between two groups of approximatePFs Let 119860 and 119861 be two approximate PFs that are obtainedby two different algorithms then 119862(119860 119861) is defined as

119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)

Although 119862(119860 119861) represents the percentage of solutionsin 119861 that are dominated by at least one solution in 119860 ingeneral 119862(119860 119861) = 1 minus 119862(119861 119860) If 119862(119860 119861) is larger than

119862(119861 119860) algorithm 119860 is better than algorithm 119861 to someextent

(3) Hypervolume (HV) [51] HV metric is employed tomeasure the volume of hypercube enclosed by PF 119860 andreference vector rref = (1199031 1199032 119903푀)푇 with lager valuesrepresenting better performance It can be obtained by

HV (119860) = ⋃푎isin퐴

vol (119886) (9)

where vol(119886) is the volume of hypercube enclosed by solution119886 in PF 119860 and reference vector rref = (1199031 1199032 119903푀)푇HV measure can reflect both convergence and diversity ofcorresponding PF to a certain degree

For the convenience of computation all objective vectorsof the Pareto solutions are normalized based on (10) beforecalculating all three metrics Thus the reference vectors of


Table 4 Comparison of three different aggregation functions on average C value over 10 independent runs for all 15 problems

Instance MOMAD(A) versus MOMAD-WS(B) MOMAD(A) versus MOMAD-PBI(C)119862(119860 119861) 119862(119861 119860) 119862(119860 119862) 119862(119862 119860)ka 4 times 5 00000 00000 00000 00000ka 8 times 8 00250 00000 00983 00333ka 10 times 7 01000 00000 01167 00333ka 10 times 10 00583 00667 00250 00333ka 15 times 10 03167 02000 02500 02500MK01 02070 02221 04386 00373MK02 02296 01261 06180 01143MK03 02034 00453 05069 00121MK04 01689 01868 03457 00704MK05 01727 01300 05658 00174MK06 05158 03765 03546 02746MK07 03294 01690 05408 01346MK08 00143 00000 03300 00000MK09 05997 02937 05915 01200MK10 05408 01637 02795 03700

Table 5 Performance evaluation of the effect of local search using IGD HV and C values over 10 independent runs for all 15 problems

Instance IGD HV MOMAD(A) versus MOEAD(B)MOMAD MOEAD MOMAD MOEAD 119862(119860 119861) 119862(119861 119860)

ka 4 times 5 01054 02635 05910 05810 00000 00000ka 8 times 8 01071 01398 06574 06188 00500 00000ka 10 times 7 00298 00687 10594 09828 01333 00333ka 10 times 10 01394 03617 06802 03178 01667 00333ka 15 times 10 04002 03666 03727 04776 03833 00000MK01 00304 00537 10316 10161 05218 00358MK02 00415 00740 09908 09263 04055 00386MK03 00233 00077 08296 08363 00258 00330MK04 00340 00441 11437 11297 02585 00800MK05 00072 00062 10576 10621 02326 00616MK06 00473 00540 09700 09223 03511 04992MK07 00218 00418 09906 09552 02602 01224MK08 00000 00000 05755 05755 00000 00000MK09 00365 00406 11893 11622 07050 02499MK10 00391 00737 11480 10525 05546 02328

all benchmark instances for calculating HV value are set as(11 11 11)푇119891푖 (x) = (119891푖 (x) minus 119891min

푖 )(119891max푖 minus 119891min

푖 ) (10)

where119891max푖 and119891min

푖 are the supremum and infimum of119891푖(x)over all nondominated solutions obtained by gathering allcompared algorithms

63 Performance Comparison with Several Variants ofMOMAD Because the implementation of algorithm frame-work is not unique and there are some different strategies

employed to instantiate it several variants of MOMAD willbe first studied MOEAD is a simplified algorithm designedby eliminating local search from MOMAD to investigate itseffectiveness In the interest of studying the effect of differentaggregation functions MOMAD-WS and MOMAD-PBI areformed by replacing the Tchebycheff approach with weightedsum (WS) [7] and penalty-based boundary intersection(PBI) approach [7] The Wilcoxon signed-rank test [52] isperformed on the sample data of three metrics obtained after10 independent runs with the significance of 005 and the onewhich is significantly better than others is marked in boldThree metric values are listed in Tables 3ndash5


MK10

MOMADMOEAD

times104

2 4 6 80Number of function evaluations

0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104

1 2 3 4 50Number of function evaluations

40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue

Figure 3 Convergence graphs in terms of average IGD value obtained by MOMAD and MOEAD for MK01 MK02 MK07 and MK10problems

In Tables 3 and 4 the results of performance comparisonbetween MOMAD MOMAD-WS and MOMAD-PBI arelisted MOMAD is significantly better than MOMAD-WSon MK10 for IGD HV and C values Besides MOMADperforms better than MOMAD-WS on MK07 and MK09 interms of HV and C value and is also better than MOMAD-WS onMK06 for IGDmetric In contrast MOMAD-WS onlyachieves a better HV value on Ka 10 times 10 When comparingwith MOMAD-PBI MOMAD is better than MOMAD-PBIfor all the BRdata instances in terms of IGD and HV valuesIn additionMOMADalso obtains betterCmetric values on 8out of 10 instances In summary the presented results indicatethat Tchebycheff aggregation function is more suitable to beutilized in MOEAD framework when solving MOFJSP

To understand the effectiveness of problem-specific localsearch the comparison between MOMAD and MOEAD isconducted and the three metric values over 10 independent

runs are shown in Table 5 It is easily observed that as for IGDvalue there are significant differences in the performancesof the two algorithms on 8 test problems and MOMADsignificantly outperforms MOEAD on all these instancesAs for the other two metrics the situations are similar toIGD metric MOMAD outperforms MOEAD on 9 out oftotal 15 instances in terms of HV metric and 7 out of 15instances for C metric while MOEAD only obtains a betterC value onMK06 Based on the above comparison results andanalyses MOMAD is much powerful than MOEAD whichwell verifies the effectiveness of local search

With the aim of intuitively comparing the convergenceperformance of the two algorithms the evolutions of averageIGD values in MOMAD and MOEAD on four selectedBRdata instances are illustrated in Figure 3 As can beclearly seen from this figure with the increasing number offunction evaluations the IGD values in all four instances


Table 6 Comparison of the Kacem instance by listing the nondominated solutions

Instance F PSO + SA hGA PSO + TS Xing HTSA EDA MOMAD1 2 1 1 2 1 2 1 2 3 1 2 3 4 1 2 3 4

ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10

Table 7 Comparison between MOMAD and other algorithms using IGD metric for Kacem and BR data instances

Instance AlgorithmMOMAD MOGA PLS HSFLA HMOEA SEA P-EDA hDPSO PRMOTS + IS

ka 4 times 5 00000 01954 00000 mdash 00000 00000 00000 mdash 00000ka 8 times 8 01414 02532 01414 01414 02532 01414 01414 01414 01414ka 10 times 7 00000 mdash 00000 mdash 00000 00000 00000 mdash 00000ka 10 times 10 00000 01250 00000 00000 00000 00000 00000 00000 00000ka 15 times 10 00000 03571 00000 00000 00000 00000 00000 01429 00000MK01 00037 01525 00525 01909 00300 00078 00307 00984 00042MK02 00000 00680 00662 01493 00119 00357 00119 00000 00287MK03 00643 03933 02119 02119 00838 00643 01911 00949 00643MK04 00242 01470 mdash 01508 mdash 00271 00617 00540 00340MK05 00245 02486 mdash 00185 mdash 00245 00245 00596 00223MK06 00243 01457 mdash 01283 mdash 00296 00774 01377 00245MK07 00924 00243 mdash 01174 mdash 01029 00924 00976 00841MK08 00440 01709 01519 01519 00440 00567 00440 00540 00440MK09 00115 02491 mdash 01083 mdash 00056 00648 01520 00305MK10 00186 01111 mdash 00902 mdash 00419 00762 01653 00517For each instance the minimal IGD values obtained by the compared algorithms are marked in bold

gradually decrease and tend to be stable which indicatesthat both algorithms have good convergence SinceMOMADachieves lower IGD convergence curves it is easily observedthat MOMAD achieves better convergence property andconvergence efficiency than MOEAD

64 Performance Comparison with Other Algorithms In thissubsection comparison betweenMOMAD and several state-of-the-art algorithms are made First to compare MOMADwith algorithms solving MOFJSP by using a priori approachMOMAD is compared on five Kacem instances with PSO+ SA [17] hGA [18] PSO + TS [19] Xing et alrsquos algorithm[20] HTSA [21] and EDA [22] All Pareto solutions markedin bold are listed in Table 6 Next in Tables 7ndash10 MOMAD

is compared with eight algorithms recently proposed forsolving MOFJSP by using Pareto approach that are MOGA[25] PLS [28] HSFLA [37] HMOEA [30] SEA [29] P-EDA [31] hDPSO [34] and PRMOTS + IS [32] It should bepointed out that these compared algorithms list the resultsafter predefined runs rather than each run in their originalliteratures Therefore the statistical comparisons as madebefore no longer apply In this subsection the three metricsare computed for the set of PFs collected over predefined runsof each algorithm

As shown in Table 6 first the MOMAD can obtain morePareto solutions than all other algorithms for solving the fiveinstances except EDA for Ka 10times10 and Ka 15times10 and Xingfor Ka 15times10 Second as for Ka 10times10 the solution (7 44 6)


Table 8 Comparison between MOMAD and other algorithms using HV metric for Kacem and BR data instances


ka 4 times 5 05977 05777 05977 mdash 05977 05977 05977 mdash 05977ka 8 times 8 05185 03385 05185 05185 03385 05185 05185 05185 05185ka 10 times 7 04560 mdash 04560 mdash 04560 04560 04560 mdash 04560ka 10 times 10 09060 06560 09060 09060 09060 09060 09060 09060 09060ka 15 times 10 10167 02739 10167 10167 10167 10167 10167 07310 10167MK01 12130 10833 11820 07579 12003 12110 12022 11680 12132MK02 09691 09152 09173 08032 09538 09402 09586 09691 09370MK03 06659 06296 05443 05443 06574 06659 06405 06331 06659MK04 11118 10882 mdash 09804 mdash 11084 10893 10990 11044MK05 08274 07805 mdash 08234 mdash 08274 08290 07912 08131MK06 09018 08163 mdash 07816 mdash 08766 08121 08307 09454MK07 01926 07863 mdash 01805 mdash 01804 01926 01900 01894MK08 05521 04469 04819 04819 05521 05451 05521 05375 05521MK09 13115 12575 mdash 12742 mdash 12858 12939 13013 12897MK10 10365 08106 mdash 09372 mdash 09496 09369 07976 09652For each instance the greater HV values obtained by the compared algorithms are marked in bold

Table 9 Comparison between MOMAD and other algorithms using Cmetric for Kacem and BR data instances

InstanceMOMAD(A) versus

MOGA(B) MOMAD(A) versus PLS(C) MOMAD(A) versusHSFLA(D)

MOMAD(A) versusHMOEA(E)119862(119860 119861) 119862(119861 119860) 119862(119860 119862) 119862(119862 119860) 119862(119860119863) 119862(119863119860) 119862(119860 119864) 119862(119864 119860)

ka 4 times 5 00000 00000 00000 00000 mdash mdash 00000 00000ka 8 times 8 00000 00000 00000 00000 00000 00000 00000 00000ka 10 times 7 mdash mdash 00000 00000 mdash mdash 00000 00000ka 10 times 10 02500 00000 00000 00000 00000 00000 00000 00000ka 15 times 10 06667 00000 00000 00000 00000 00000 00000 00000MK01 07500 00000 07143 00000 09091 00000 05833 00000MK02 05000 00000 07500 00000 10000 00000 02500 00000MK03 01000 00000 08571 00000 08571 00000 00000 00000MK04 04000 01034 mdash mdash 08000 01724 mdash mdashMK05 00000 00000 mdash mdash 03571 00000 mdash mdashMK06 03000 01387 mdash mdash 08667 00438 mdash mdashMK07 00000 05000 mdash mdash 06667 00000 mdash mdashMK08 00000 01111 06250 00000 06250 00000 00000 00000MK09 06667 00000 mdash mdash 10000 00000 mdash mdashMK10 10000 00000 mdash mdash 06000 01747 mdash mdashFor each instance the greater set coverage values obtained by the compared algorithms are marked in bold

obtained by PSO + SA and solution (7 43 6) obtained by PSO+TS are both dominated by (7 42 6) and (7 43 5) obtained byMOMAD Besides when comparing with Xing one solution(15 76 12) of Ka 8 times 8 obtained by Xing is dominated by(15 75 12) got by MOMAD By analyzing the results of Ka15 times 10 two solutions (ie (12 91 11) and (11 93 11)) whichare respectively obtained by PSO + SA and PSO + TS arerespectively dominated by (11 91 11) and (11 93 10) achievedbyMOMAD According to the above comparison results andanalyses when comparing with the algorithms based on a

priori approach MOMAD can obtain more nondominatedsolutions with higher quality

Tables 7 and 8 show the comparison results of IGDand HV values between MOMAD and eight Pareto-basedalgorithms First it can be observed that except MOGA andhDPSO MOMAD and other 6 algorithms can find all thenondominated solutions of Ka 4times5 Ka 10times7 Ka 10times10 andKa 15 times 10 Although there exist no algorithms that can findall nondominated solutions of Ka 8 times 8 MOMAD and other6 algorithms are better than MOGA and HMOEA Next we




SEA(B)MOMAD(A) versus

P-EDA(C)MOMAD(A) versus

hDPSO(D)MOMAD(A) versusPRMOTS + IS(E)119862(119860 119861) 119862(119861 119860) 119862(119860 119862) 119862(119862 119860) 119862(119860119863) 119862(119863119860) 119862(119860 119864) 119862(119864 119860)

ka 4 times 5 00000 00000 00000 00000 00000 00000 00000 00000ka 8 times 8 00000 00000 00000 00000 00000 00000 00000 00000ka 10 times 7 00000 00000 00000 00000 00000 00000 00000 00000ka 10 times 10 00000 00000 00000 00000 00000 00000 00000 00000ka 15 times 10 00000 00000 00000 00000 06667 00000 00000 00000MK01 01818 00000 04545 00000 00000 00000 01818 03000MK02 01429 00000 02500 00000 00000 00000 04444 00000MK03 00000 00000 01111 00000 00909 00000 00000 00000MK04 00000 04138 01875 00345 00000 03448 00870 02414MK05 00000 00000 00000 00000 00000 00000 04000 00000MK06 01165 06350 07869 01241 01250 01241 02619 05912MK07 00000 00000 00000 00000 01000 00000 03478 00000MK08 00000 00000 00000 00000 01111 00000 00000 00000MK09 03281 05714 09481 00286 01429 01286 08817 01143MK10 05072 02651 07250 01024 02857 00653 05602 02349For each instance the greater set coverage values obtained by the compared algorithms are marked in bold

focus on the BRdata test set When considering IGD metricMOMAD obtains the best values on 6 out of 10 instances andthe second-best IGD values on three instances The situationof HV metric is much similar to IGD metric MOMADachieves the best HV values on 7 out of 10 instances andyields the second-best HV values on all the other instancesMOGA obtains the best IGD value on MK07 but MOMADis better than MOGA on other 9 instances The situation ismuch similar to SEA P-EDA and PRMOTS + IS Althoughthey exhibit superior performance over MOMAD on severalinstances for IGD andHV values inmost cases they performrelatively worse than MOMAD

MOMAD is compared with eight algorithms in Tables9 and 10 by using set coverage value It is clearly indicatedthat MOMAD is much superior to five algorithms exceptfor MOGA SEA and PRMOTS + IS When comparingwith MOGA MOMAD is worse than MOGA on MK07 andMK08 but MOMAD is better than MOGA on 7 BRdatainstances Compared with SEA MOMAD is generally betteron MK01 MK02 and MK10 instances but on MK04 MK06and MK09 SEA generally shows higher performance As forPRMOTS + IS MOMAD is superior to it on MK02 MK05MK07 MK09 and MK10 whereas PRMOTS + IS only winsin MK01 MK04 and MK06

Table 11 shows the comparison between MOMAD andMOGA on 18 DPdata instances which are designed in [53]The predefinedmaximal function evaluation of each instanceis shown in second column and MOMAD is independentlyrun 5 times at a time From the IGD and HV value it is easilyobserved that the performance of MOMAD is much betterthan MOGA since MOGA only obtains a better IGD valueon 07a and a better HV value on 02a The comparison ofC metric is similar to the IGD and HV MOMAD achieves

16 significantly better results and there is no significantdifference between MOMAD andMOGA on 02a and 09a Inaddition it can also be observed that C (MOMAD MOGA)equals one on 12 test instances which means as for these 12instances every solution obtained byMOGA is dominated byat least one solutions by MOMAD

The objective ranges of MOMAD and PRMOTS + IS forDPdata instances are given in Table 12 The MOMAD findsa wider spread of nondominated solutions than PRMOTS +IS especially in terms of makespan and total workload Thusit can be concluded that MOMAD is more effective thanPRMOTS + IS in terms of exploring a search space

According to the above extensive IGD and HV valuesthe average performance scores of 10 BRdata instances arefurther recorded to rank the compared algorithm whichmake it easier to quantify the overall performance of thesealgorithms by intuitive observation For each specific BRdatainstance suppose Alg1Alg2 Alg푙 respectively denotethe 119897 algorithms employed in comparison Let 120599푖푗 be 1 iff Alg푗obtains smaller IGD and biggerHV value thanAlg푖Then theperformance119875(Alg푖) of each algorithmAlg푖 can be calculatedas [54]

119875 (Alg푖) = 푙sum푗=1푗 =푖

120599푖푗 (11)

It should be noted that the smaller the score the betterthe algorithm Here PLS and HMOEA only consider part ofinstances so we first rank all algorithms onMK01simMK03 andMK08 instances and then we rank these algorithms exceptPLS and HMOEA on the remaining 6 instances Figure 4shows the average performance score of IGD and HV values


Table 11 Comparison between MOMAD and MOGA using IGD HV and Cmetric for DP data

Instance NFssect IGD HV MOMAD versus MOGAMOMAD MOGA MOMAD MOGA C(MOMAD MOGA) C(MOGA MOMAD)

01a 50000 00000 02121 13310 10743 10000 0000002a 50000 02705 03002 06117 08576 00000 0000003a 50000 01852 05899 09950 02615 05000 0000004a 50000 00000 04022 11059 06042 10000 0000005a 60000 00000 04845 12857 06156 10000 0000006a 60000 00000 06834 13126 04432 10000 0000007a 50000 05034 02517 11886 06220 06667 0000008a 50000 00755 06998 11682 01780 05000 0000009a 50000 03162 03802 11149 04647 00000 0000010a 70000 00028 03334 10840 05980 07500 0000011a 70000 00000 07135 12853 03112 10000 0000012a 70000 00000 07982 13137 02533 10000 0000013a 60000 00000 11040 12866 01062 10000 0000014a 60000 00000 09852 12799 01414 10000 0000015a 70000 00000 06547 12388 02967 10000 0000016a 70000 00000 04685 11568 03833 10000 0000017a 70000 00000 08188 12872 02075 10000 0000018a 70000 00000 10231 13079 00975 10000 00000sect means the number of function evaluations

Table 12 Objective ranges for the DPdata

Instance 1198911(min max) 1198912(min max) 1198913(min max)MOMAD PRMOTS + IS MOMAD PRMOTS + IS MOMAD PRMOTS + IS

01a (2561 2561) (2592 2596) (11137 11137) (11137 11137) (2505 2505) (2508 2549)02a (2340 2378) (2345 2441) (11137 11137) (11137 11137) (2232 2236) (2228 2238)03a (2267 2334) (2317 2351) (11137 11137) (11137 11137) (2230 2236) (2228 2245)04a (2533 2766) (2647 2786) (11064 11081) (11064 11087) (2503 2727) (2503 2727)05a (2263 2792) (2426 2852) (10941 10981) (10941 10988) (2203 2465) (2194 2497)06a (2202 2767) (2328 2769) (10809 10866) (10809 10848) (2166 2347) (2164 2347)07a (2454 2454) (2522 2522) (16485 16485) (16485 16485) (2187 2187) (2187 2187)08a (2186 2426) (2276 2465) (16485 16485) (16485 16485) (2064 2083) (2062 2093)09a (2168 2297) (2254 2458) (16485 16485) (16485 16485) (2064 2077) (2062 2068)10a (2443 2986) (2656 2912) (16464 16518) (16464 16512) (2178 2607) (2178 2629)11a (2137 2918) (2416 2954) (16135 16231) (16135 16194) (2031 2375) (2021 2328)12a (2036 2319) (2339 2745) (15748 15828) (15748 15809) (1974 2082) (1974 2115)13a (2392 2511) (2643 2708) (21610 21610) (21610 21610) (2216 2223) (2203 2215)14a (2297 2405) (2493 2522) (21610 21610) (21610 21610) (2165 2174) (2165 2168)15a (2235 2337) (2549 2595) (21610 21610) (21610 21610) (2165 2178) (2164 2173)16a (2409 3040) (2735 3236) (21478 21556) (21478 21562) (2237 2549) (2206 2478)17a (2163 2783) (2528 3002) (20875 20942) (20878 20972) (2105 2347) (2093 2268)18a (2109 2442) (2416 2901) (20562 20633) (20566 20621) (2068 2171) (2061 2198)

over 10 BRdata instances for 9 selected algorithms and therank accordance with the score of each algorithm is listed inthe corresponding bracket It is easily observed thatMOMADworks well nearly on all the instances in terms of IGD andHVmetrics since it achieves good performance on almost all thetest problems

Figures 5 and 6 show the final PFs of MK01simMK10instances obtained byMOMAD and the reference PFs gener-ated by selecting nondominated solutions from the mixtureof eight compared algorithm As can be seen MOMADfinds all Pareto solutions for MK02 As for MK01 MK03MK05 MK07 and MK08 MOMAD almost finds all Pareto


00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS

MOMAD P-EDA SEA hDPSO PLS HSFLA MOGAHMOEA0

2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS

P-EDA HMOEA SEA hDPSO PLS HSFLA MOGAMOMAD0

2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS

P-EDA SEA hDPSO MOGA HSFLAMOMAD

08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)

Figure 4 Ranking in the average performance score over MK01 MK02 MK03 and MK08 problem instances for the compared ninealgorithms and over other six BRdata instances for the compared seven algorithms The smaller the score the better the overall performancein terms of HV and IGD metrics

solutions Besides MOMAD can find the vast majority of theoptimal solutions for MK04 MK06 MK09 andMK10ThusMOMAD is capable of finding a wide spread of PF for eachinstance and showing the tradeoffs among three differentobjectives

Average CPU time consumed by MOMAD and othercompared algorithms are listed in Table 13 However thedifferent experimental platform programming language andprogramming skills make this comparison not entirely rea-sonable Hence we enumerate the computational CPU time

combined with original experimental platform and program-ming language for each corresponding algorithm whichhelp us distinguish the efficiency of the referred algorithmsThe values show that the computational time consumed byMOMAD is much smaller than other algorithms except forKa 4 times 5

In summary from the above-mentioned experimentalresults and comparisons it can be concluded that MOMADoutperforms or at least has comparative performance to allthe other typical algorithms when solving MOFJSP


MK01

4042

44 46

150160

17036

38

40

42

MakespanTotal work load

Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF

Figure 5 Final nondominated solutions of MK01simMk06 instance found by MOMAD

65 Impacts of Parameters Settings

(1) Impacts of Population Size 119873 Since population size 119873 isan important parameter for MOMAD we test the sensitivityof MOMAD to different settings of 119873 for MK04 All otherparameters except 119873 are the same as shown in Table 2 Thealgorithm independently runs 10 times with each different119873 independently As clearly shown in Table 14 the HVvalue changes weakly with the increasing of 119873 The IGDvalue decreases at first then it will have a growing tendencyTotally speaking MOMAD is not significantly sensitive topopulation size 119873 and properly increasing value of 119873 canimprove the algorithm performance to some extent

(2) Impacts of Neighborhood Size 119879 119879 is another key param-eter in MOMAD With the aim of studying the sensitivityof 119879 MOMAD is implemented by setting different valuesof 119879 and keeps other parameter settings unchanged We

run the MOMAD with each 119879 10 times independently forMK06 Table 14 shows how the IGD and HV values changealong with the increasing of 119879 from where we can obtain thesame observations Both values are enhanced first with theincreasing of 119879 then the algorithm performance is degradedwith the continuous increasing So the influence of 119879 onMOMAD is similar to119873

7 Conclusions and Future Work

This paper solves the MOFJSP in a decomposition mannerfor the purpose of simultaneously minimizing makespantotal workload and critical workload To propose an effec-tive MOMAD algorithm the framework of MOEAD isadopted First a mixture of different machine assignmentand operation sequencing rules is utilized to generate aninitial population Then objective normalization technique


MK07

100150

200250

0

500

1000100

150

200

250

MakespanTotal workload

Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF


Table 13 The average CPU time (in seconds) consumed by different algorithms on Kacem and BRdata instances

Instance AlgorithmMOMADa MOGAb HSFLAc HMOEAd hDPSOe PRMOTS + ISf

ka 4 times 5 196 58 126 309 mdash mdashka 8 times 8 269 95 mdash 969 82 mdashka 10 times 7 378 mdash 1014 1452 mdash mdashka 10 times 10 550 142 mdash 1441 117 mdashka 15 times 10 1528 875 2113 3246 816 mdashMK01 702 294 17218 4727 236 583MK02 1373 450 22956 5126 258 617MK03 2228 2850 13987 31813 703 1858MK04 1684 1056 42612 mdash 663 861MK05 1693 1404 15312 mdash 1278 911MK06 5905 1158 57780 mdash 1225 2182MK07 3002 2952 18523 mdash 2312 1013MK08 3293 7224 16548 167414 1471 5606MK09 4659 11688 56570 mdash 7234 7941MK10 9758 10722 107220 mdash 8836 8228aThe CPU time on an Intel Core i3-4170 CPU 37GHz processor in C++ bThe CPU time on a 2GHz processor in C++ cThe CPU time on a Pentium IV18 GHz processor in C++ dThe CPU time on an Intel Core(TM)2 Duo CPU 266GHz processor in C♯ eThe CPU time on a 2GHz processor in C++ fTheCPU time on an Intel Core TM2 T8100 CPU 21 GHz processor in C♯


Table 14 IGD and HV metric corresponding to different119873 and 119879MK04 MK06119873(119867) IGD HV 119879 IGD HV55(8) 00465 11332 6 00448 09397120(14) 00361 11380 12 00417 09547153(16) 00437 11296 24 00402 09533210(19) 00364 11338 48 00426 09463253(21) 00326 11383 96 00487 09348300(23) 00417 11336 120 00502 09360

is used in Tchebycheff approach and the MOP is convertedinto many single-objective optimization subproblems Byclustering the weight vectors into different groups a localexploitation based on moving critical operations is incorpo-rated in MOEAD and applied on candidate solutions withbest aggregation function compared with solutions whoseweight vectors belong to the same groupAfter embedding thelocal search into MOEAD our MOMAD is established Insimulation experiments the Tchebycheff approach is studiedto be more suitable for using in MOEAD framework thanWS and PBI approaches and the effectiveness of local searchis also verified Moreover MOMAD is compared with eightcompetitive algorithms in terms of three quantitativemetricsFinally the effect of two key parameters population size andneighborhood size are analysed Extensive computationalresults and comparisons indicate that the proposedMOMADoutperforms or at least has a comparative performance toother representative approaches and MOMAD is fit to solveMOFJSP

In the future we want to study the MOFJSP with morethan three objectives at first Second it would be significant tointroduce a novel local search method which considers mov-ing more than one critical operation Finally it would alsobe interesting to apply the MOMAD to dynamic schedulingproblem

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National ScienceFoundation of China (no 61572238) and the ProvincialOutstanding Youth Foundation of Jiangsu Province (noBK20160001)

References

[1] M R Garey D S Johnson and R Sethi ldquoThe complexity offlowshop and jobshop schedulingrdquo Mathematics of OperationsResearch vol 1 no 2 pp 117ndash129 1976

[2] M Mastrolilli and L M Gambardella ldquoEffective neighbour-hood functions for the flexible job shop problemrdquo Journal ofScheduling vol 3 no 1 pp 3ndash20 2000

[3] G Zhang L Gao and Y Shi ldquoAn effective genetic algorithm forthe flexible job-shop scheduling problemrdquo Expert Systems withApplications vol 38 no 4 pp 3563ndash3573 2011

[4] LWang S Wang Y Xu G Zhou andM Liu ldquoA bi-populationbased estimation of distribution algorithm for the flexible job-shop scheduling problemrdquoComputers amp Industrial Engineeringvol 62 no 4 pp 917ndash926 2012

[5] Y Yuan and H Xu ldquoAn integrated search heuristic for large-scale flexible job shop scheduling problemsrdquo Computers ampOperations Research vol 40 no 12 pp 2864ndash2877 2013

[6] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[7] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[8] H Li and Q Zhang ldquoMulti-objective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[9] Q Zhang W Liu E Tsang and B Virginas ldquoExpensivemultiobjective optimization byMOEADwith gaussian processmodelrdquo IEEE Transactions on Evolutionary Computation vol14 no 3 pp 456ndash474 2010

[10] Y-Y Tan Y-C Jiao H Li and X-K Wang ldquoA modificationtoMOEAD-DE formultiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[11] Y Yuan H Xu B Wang B Zhang and X Yao ldquoA newdominance relation-based evolutionary algorithm for many-objective optimizationrdquo IEEE Transactons on EvolutionaryComputation vol 20 no 1 pp 16ndash37 2016

[12] Y Yuan H Xu B Wang B Zhang and X Yao ldquoBalancingConvergence and Diversity in Decomposition-Based Many-ObjectiveOptimizersrdquo IEEETransactions on EvolutionaryCom-putation vol 20 no 2 pp 180ndash198 2016

[13] P C Chang S H Chen Q Zhang and J L Lin ldquoMOEADfor flowshop scheduling problemsrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo08) pp 1433ndash1438 Hong Kong June 2008

[14] A Alhindi and Q Zhang ldquoMOEAD with Tabu Search formultiobjective permutation flow shop scheduling problemsrdquoin Proceedings of the 2014 IEEE Congress on EvolutionaryComputation CEC 2014 pp 1155ndash1164 China July 2014

[15] X-N Shen Y Han and J-Z Fu ldquoRobustness measures androbust scheduling for multi-objective stochastic flexible jobshop scheduling problemsrdquo Soft Computing pp 1ndash24 2016

[16] F Zhao Z Chen J Wang and C Zhang ldquoAn improvedMOEAD for multi-objective job shop scheduling problemrdquoInternational Journal of Computer Integrated Manufacturingvol 30 no 6 pp 616ndash640 2017

[17] W Xia and Z Wu ldquoAn effective hybrid optimization approachfor multi-objective flexible job-shop scheduling problemsrdquoComputers amp Industrial Engineering vol 48 no 2 pp 409ndash4252005

[18] J Gao M Gen L Sun and X Zhao ldquoA hybrid of geneticalgorithm and bottleneck shifting for multiobjective flexible jobshop scheduling problemsrdquo Computers amp Industrial Engineer-ing vol 53 no 1 pp 149ndash162 2007


[19] G H Zhang X Y Shao P G Li and L Gao ldquoAn effec-tive hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problemrdquo Computers ampIndustrial Engineering vol 56 no 4 pp 1309ndash1318 2009

[20] L-N Xing Y-W Chen and K-W Yang ldquoAn efficient searchmethod for multi-objective flexible job shop scheduling prob-lemsrdquo Journal of Intelligent Manufacturing vol 20 no 3 pp283ndash293 2009

[21] J-Q Li Q-K Pan and Y-C Liang ldquoAn effective hybridtabu search algorithm for multi-objective flexible job-shopscheduling problemsrdquo Computers amp Industrial Engineering vol59 no 4 pp 647ndash662 2010

[22] S Wang L Wang M Liu and Y Xu ldquoAn estimation ofdistribution algorithm for the multi-objective flexible job-shopscheduling problemrdquo inProceedings of the 2013 IEEE Symposiumon Computational Intelligence in Scheduling CISched 2013 -2013 IEEE SymposiumSeries onComputational Intelligence SSCI2013 pp 1ndash8 Singapore April 2013

[23] I Kacem S Hammadi and P Borne ldquoPareto-optimalityapproach for flexible job-shop scheduling problems hybridiza-tion of evolutionary algorithms and fuzzy logicrdquo MathematicsandComputers in Simulation vol 60 no 3-5 pp 245ndash276 2002

[24] N B Ho and J C Tay ldquoSolving multiple-objective flexible jobshop problems by evolution and local searchrdquo IEEE Transac-tions on SystemsMan and Cybernetics Part C Applications andReviews vol 38 no 5 pp 674ndash685 2008

[25] X Wang L Gao C Zhang and X Shao ldquoA multi-objectivegenetic algorithm based on immune and entropy principlefor flexible job-shop scheduling problemrdquo The InternationalJournal of Advanced Manufacturing Technology vol 51 no 5-8pp 757ndash767 2010

[26] M Frutos A C Olivera and F Tohm ldquoA memetic algorithmbased on a NSGAII scheme for the flexible job-shop schedulingproblemrdquo Annals of Operations Research vol 181 pp 745ndash7652010

[27] Y Yuan and H Xu ldquoMultiobjective flexible job shop schedulingusing memetic algorithmsrdquo IEEE Transactions on AutomationScience and Engineering vol 12 no 1 pp 336ndash353 2015

[28] J-Q Li Q-K Pan and J Chen ldquoA hybrid Pareto-based localsearch algorithm for multi-objective flexible job shop schedul-ing problemsrdquo International Journal of Production Research vol50 no 4 pp 1063ndash1078 2012

[29] T-C Chiang andH-J Lin ldquoA simple and effective evolutionaryalgorithm for multiobjective flexible job shop schedulingrdquoInternational Journal of Production Economics vol 141 no 1 pp87ndash98 2013

[30] J Xiong X Tan K Yang L Xing and Y Chen ldquoA HybridMultiobjective Evolutionary Approach for Flexible Job-ShopScheduling Problemsrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash27 2012

[31] L Wang S Y Wang and M Liu ldquoA Pareto-based estimationof distribution algorithm for the multi-objective flexible job-shop scheduling problemrdquo International Journal of ProductionResearch vol 51 no 12 pp 3574ndash3592 2013

[32] S Jia and Z-H Hu ldquoPath-relinking Tabu search for the multi-objective flexible job shop scheduling problemrdquo Computers ampOperations Research vol 47 pp 11ndash26 2014

[33] G Moslehi and M Mahnam ldquoA Pareto approach to multi-objective flexible job-shop scheduling problem using particleswarm optimization and local searchrdquo International Journal ofProduction Economics vol 129 no 1 pp 14ndash22 2011

[34] X Shao W Liu Q Liu and C Zhang ldquoHybrid discrete par-ticle swarm optimization for multi-objective flexible job-shopscheduling problemrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 9ndash12 pp 2885ndash29012013

[35] L C F Carvalho andMA Fernandes ldquoMulti-objective FlexibleJob-Shop scheduling problem with DIPSO More diversitygreater efficiencyrdquo in Proceedings of the 2014 IEEE Congress onEvolutionary Computation CEC 2014 pp 282ndash289 China July2014

[36] N Tian and Z Ji ldquoPareto-ranking based quantum-behavedparticle swarm optimization for multiobjective optimizationrdquoMathematical Problems in Engineering vol 2015 Article ID940592 10 pages 2015

[37] J Li Q Pan and S Xie ldquoAn effective shuffled frog-leapingalgorithm for multi-objective flexible job shop schedulingproblemsrdquo Applied Mathematics and Computation vol 218 no18 pp 9353ndash9371 2012

[38] L Wang G Zhou Y Xu and M Liu ldquoAn enhanced Pareto-based artificial bee colony algorithm for the multi-objectiveflexible job-shop schedulingrdquo The International Journal ofAdvanced Manufacturing Technology vol 60 no 9ndash12 pp 1111ndash1123 2012

[39] Q Zhang H Li D Maringer and E Tsang ldquoMOEAD withNBI-style Tchebycheff approach for portfolio managementrdquoin Proceedings of the 2010 6th IEEE World Congress on Com-putational Intelligence WCCI 2010 - 2010 IEEE Congress onEvolutionary Computation CEC 2010 Spain July 2010

[40] Y Mei K Tang and X Yao ldquoDecomposition-based memeticalgorithm for multiobjective capacitated arc routing problemrdquoIEEE Transactions on Evolutionary Computation vol 15 no 2pp 151ndash165 2011

[41] V A Shim K C Tan and C Y Cheong ldquoA hybrid estimationof distribution algorithm with decomposition for solving themultiobjective multiple traveling salesman problemrdquo IEEETransactions on Systems Man and Cybernetics Part C Appli-cations and Reviews vol 42 no 5 pp 682ndash691 2012

[42] L Ke and Q Zhang ldquoMultiobjective combinatorial optimiza-tion by using decomposition and ant colonyrdquo IEEE Transactionson Cybernetics vol 43 no 6 pp 1845ndash1859 2013

[43] L Ke andQ Zhang ldquoHybridzation ofDecomposition and LocalSearch for Multiobjective Optimizationrdquo IEEE Transactions onCybernetics vol 44 no 44 pp 1808ndash1819 2014

[44] I Kacem S Hammadi and P Borne ldquoApproach by localizationand multiobjective evolutionary optimization for flexible job-shop scheduling problemsrdquo IEEETransactions on SystemsManand Cybernetics Part C Applications and Reviews vol 32 no 1pp 1ndash13 2002

[45] K Deb and H Jain ldquoAn evolutionary many-objective optimiza-tion algorithm using reference-point- based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[46] S-K Zhao S-L Fang and X-J Gu ldquoMachine selection andFJSP solution based on limit scheduling completion time min-imizationrdquo Jisuanji Jicheng Zhizao XitongComputer IntegratedManufacturing Systems CIMS vol 20 no 4 pp 854ndash865 2014

[47] F Pezzella GMorganti and G Ciaschetti ldquoA genetic algorithmfor the flexible job-shop scheduling problemrdquo Computers ampOperations Research vol 35 no 10 pp 3202ndash3212 2008

[48] K Z Gao P N Suganthan Q K Pan T J Chua T X Cai andC S Chong ldquoPareto-based grouping discrete harmony search


algorithm for multi-objective flexible job shop schedulingrdquoInformation Sciences vol 289 pp 76ndash90 2014

[49] P Brandimarte ldquoRouting and scheduling in a flexible job shopby tabu searchrdquoAnnals of Operations Research vol 41 no 3 pp157ndash183 1993

[50] E Zitzler L Thiele M Laumanns C M Fonseca and VG Da Fonseca ldquoPerformance assessment of multiobjectiveoptimizers an analysis and reviewrdquo IEEE Transactions onEvolutionary Computation vol 7 no 2 pp 117ndash132 2003

[51] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms A comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[52] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006

[53] S Dauzere-Peres and J Paulli ldquoAn integrated approach formodeling and solving the general multiprocessor job-shopscheduling problem using tabu searchrdquo Annals of OperationsResearch vol 70 pp 281ndash306 1997

[54] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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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


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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

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


xlowast All the Pareto optimal solutions constitute the PS andPF is the mapped vector of PS in the objective space SincePareto approach can achieve a set of Pareto solutions ratherthan a specific one it has receivedmuchmore attention than apriori approach and is recognized to bemore suitable to solveMOFJSP

Because the three objectives makespan total workloadand critical workload are conflicted with each other it isbetter to handle this model with knowledge about their PFMultiobjective evolutionary algorithm (MOEA) is a kind ofmature global optimization method with high robustnessand wide applicability Due to the fact that MOEAs havelow requirements on the optimization problem itself andhigh ability to obtain multiple Pareto solutions during eachrun they are suitable for solving multiobjective optimizationproblems (MOPs) The multiobjective evolutionary algo-rithm based on decomposition (MOEAD) that integratesmathematical programming with evolutionary algorithm(EA) can obtain a set of Pareto solutions by aggregat-ing multiple objectives into different single-objectives withmany predefined weight vectors [7] MOEAD has showngreat superiority on continuous optimization problems [8ndash12] thus it is necessary to investigate its performance onmultiobjective combinatorial optimization problems (MO-COPs) such as MOFJSP To the best of our knowledge in theliterature reported although MOEAD has been applied todifferent kinds of multiobjective scheduling problems such asmultiobjective flow shop scheduling problem (MOFSP) [13]multiobjective permutation flow shop scheduling problem(MOPFSP) [14] multiobjective stochastic flexible job shopscheduling problem (MOSFJSP) [15] and multiobjective jobshop scheduling problem (MOJSP) [16] there is seldomcorresponding application on MOFJSP

The primary aim of this paper is to solve MOFJSPin a decomposition manner by proposing a multiobjectivememetic algorithm based on decomposition (MOMAD)hybridizing MOEAD with local search With the purpose ofmaking the proposed algorithmmore applicable four aspectsare studied (1) integration of different machine assignmentand operation sequencing strategies are presented to con-struct the initial population (2) objective normalization isincorporated into Tchebycheff approach to convert an MOPinto a number of single-objective optimization subproblems(3) all weight vectors are divided into a few groups based onK-means clustering some superior individuals correspond-ing to different weight vector groups are selected by usinga selection mechanism (4) local search based on movingcritical operations is applied on selected individuals Toevaluate the effectiveness of the proposed algorithm somebenchmark instances are tested with three purposes (1)investigating the effects of different aggregation functions andvalidating the effectiveness of local search (2) analyzing theinfluence of the key parameters on the performance of thealgorithm (3) comparing the performance of MOMAD withother state-of-the-art algorithms for solving MOFJSP

The rest of the paper is organized as follows Next sectionpresents a short overview of the existing related work InSection 3 the background knowledge of MOFJSP is intro-duced Section 4 introduces the framework of MOMADThe

implementation details of the proposed MOMAD includinggenetic global search and problem specific local search aredescribed in Section 5 Afterwards experimental studies areprovided in Section 6 Finally Section 7 concludes this paperand outlines some avenues for future research

2 Related Work

As mentioned above there are two main methods to solveMOFJSP a priori approach and Pareto approach As fora priori approach Xia and Wu [17] discussed a hybridalgorithm where particle swarm optimization (PSO) andsimulated annealing (SA)were employed in global search andlocal search respectively A bottleneck shifting-based localsearch was incorporated into genetic global search by Gaoet al [18] Zhang et al [19] introduced an effective hybridPSO algorithm combined with tabu search (TS) Xing etal [20] used ten different weight vectors to collect effectivesolution sets A hybrid TS (HTS) algorithm was structuredby combining adaptive rules with two neighborhoods Inthis algorithm three weight coefficients 1205821 1205822 and 1205823with different settings were given to test different problems[21] An effective estimation of distribution algorithm wasproposed by Wang et al [22] in which the new individualswere generated by sampling a probability model

Contrary to the a priori approach a PS can be obtainedby using Pareto approach and the tradeoffs among differentobjectives can be presented The integration of fuzzy logicand EA was proposed by Kacem et al [23] A guide localsearch was incorporated into EA to enhance the convergence[24] With the aim of keeping population diversity immuneand entropy principle were adopted in multiobjective geneticalgorithm (MOGA) [25] Two memetic algorithms (MAs)were respectively proposed both of which integrate non-dominated sorting genetic algorithm II (NSGA-II) [6] witheffective local search techniques [26 27] Several effectiveneighborhood approaches were used in variable neighbor-hood search to enhance the convergence ability in a hybridPareto-based local search (PLS) [28] Chiang and Lin [29]proposed a simple and effective evolutionary algorithmwhich only requires two parameters Both the neighborhoodsof machine assignment and operation sequence are consid-ered in Xiong et al [30] An effective Pareto-based EDAwas proposed by Wang et al [31] A novel path-relinkingmultiobjective TS algorithm was proposed in [32] in whicha routing solution is identified by problem-specific neighbor-hood search and is then further refined by the TS with back-jump tracking for a sequencing decision In addition to thesuccessful use of EA several swarm intelligence algorithmshave also been widely used for global search PSOs were usedas global search algorithms in [33ndash36] Besides shuffled frogleaping [37] and artificial bee colony [38]were integratedwithlocal search in related hybrid algorithms

Besides the successful using in many scheduling prob-lems MOEADs have also been widely dedicated to otherMO-COPs A novel NBI-style Tchebycheff approach wasused in MOEAD to solve portfolio management MOP withdisparately scaled objectives [39] Mei et al [40] developedan effective MA by combining MOEAD with extended












푘=1

푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘



1198692 11987421 3 3 411987422 - 2 511987423 - 3 2

1198693 11987431 4 3 211987432 4 - 211987433 3 4 3


푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘 119906푖푗푘=









0

24 6

323

223

O O12

O21

O31 O32 O33

O22 O23

O13

M3

M3

M3

M1

M1

M1

M2

M2

M2

lowast









233 2 21 212 1 31 3 2 2 1 1 3


Job 1 Job 2 Job 3

M1 M3 M3J1 J1 J1J3 J3 J3J2 J2 J2M2M1


O22 O32 O12 O13 O23 O33O21O31O11





100381610038161003816100381610038161003816100381610038161003816 (3)
























(4)




(5)


















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















푘=1

푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘



1198692 11987421 3 3 411987422 - 2 511987423 - 3 2

1198693 11987431 4 3 211987432 4 - 211987433 3 4 3


푛sum푖=1

푛119894sum푗=1

119901푖푗푘119906푖푗푘 119906푖푗푘=









0

24 6

323

223

O O12

O21

O31 O32 O33

O22 O23

O13

M3

M3

M3

M1

M1

M1

M2

M2

M2

lowast









233 2 21 212 1 31 3 2 2 1 1 3


Job 1 Job 2 Job 3

M1 M3 M3J1 J1 J1J3 J3 J3J2 J2 J2M2M1


O22 O32 O12 O13 O23 O33O21O31O11





100381610038161003816100381610038161003816100381610038161003816 (3)
























(4)




(5)


















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








0

24 6

323

223

O O12

O21

O31 O32 O33

O22 O23

O13

M3

M3

M3

M1

M1

M1

M2

M2

M2

lowast









233 2 21 212 1 31 3 2 2 1 1 3


Job 1 Job 2 Job 3

M1 M3 M3J1 J1 J1J3 J3 J3J2 J2 J2M2M1


O22 O32 O12 O13 O23 O33O21O31O11





100381610038161003816100381610038161003816100381610038161003816 (3)
























(4)




(5)


















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





233 2 21 212 1 31 3 2 2 1 1 3


Job 1 Job 2 Job 3

M1 M3 M3J1 J1 J1J3 J3 J3J2 J2 J2M2M1


O22 O32 O12 O13 O23 O33O21O31O11





100381610038161003816100381610038161003816100381610038161003816 (3)
























(4)




(5)


















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of

















(4)




(5)


















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




















MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








MK02 50000 00415 00495 00972dagger 09908 09772 09374dagger

MK03 50000 00233 01251 02837dagger 08296 07027 04929dagger

MK04 50000 00340 00282 01128dagger 11437 11455 10758dagger

MK05 50000 00072 00057 01181dagger 10576 10636 09120dagger



MK08 50000 00000 00075 01943dagger 05755 05705 03832dagger








yisin푃known

radic푀sum푖=1


(7)


119862 (119860 119861) = |119887 isin 119861 | exist119886 isin 119860 119886 ≺ 119887||119861| (8)




HV (119860) = ⋃푎isin퐴

vol (119886) (9)











푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of












푖 ) (10)






MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





MK10

MOMADMOEAD

times104


0

20

40

60

80

100

120IG

D-m

etric

val

ue

MK07

MOMADMOEAD

times104


40

50

60

70

80

90

IGD

-met

ric v

alue

MK01

MOMADMOEAD

times104


0

2

4

6

8IG

D-m

etric

val

ueMK02

MOMADMOEAD

times104


0

2

4

6

8

IGD

-met

ric v

alue









ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







ka 4 times 51198911 11 12 11 12 11 11 12 11 11 12 131198912 32 32 32 32 34 32 32 34 32 32 331198913 10 8 10 8 9 10 8 9 10 8 7

ka 8 times 81198911 15 16 14 14 15 14 15 14 15 14 15 14 15 16 161198912 75 73 77 77 75 77 76 77 75 77 75 77 75 73 771198913 12 13 12 12 12 12 12 12 12 12 12 12 12 13 11

ka 10 times 71198911 11 11 11 11 11 11 11 11 121198912 61 62 61 62 61 62 61 62 601198913 11 10 11 10 11 10 11 10 12

ka 10 times 101198911 7 7 7 7 8 7 7 8 7 7 8 8 7 7 8 81198912 44 43 43 42 42 43 42 42 43 42 41 42 43 42 41 421198913 6 5 6 6 5 5 6 5 5 6 7 5 5 6 7 5

ka 15 times 101198911 12 11 11 11 11 11 11 11 11 111198912 91 91 93 91 93 93 91 93 91 931198913 11 11 11 11 10 10 11 10 11 10


































120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






























120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

erfo

rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

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orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


















120599푖푗 (11)












00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

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D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

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rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

erfo

rman

cesc

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PRMOTS +IS


08333(1)18333(3)15000(2)

28333(4)

40000(5)46667(6) 46667(6)

0

2

4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


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MK02

2530

35

140145

15025

30

35


Criti

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ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

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ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

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MK08

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500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

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cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

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cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

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age I

GD

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form

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e

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(b)







MK01

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44 46

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40

42


Criti

cal w

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load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

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500

1000100

150

200

250


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MK08

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500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





00000(1)12500(2)

20000(3)27500(4) 30000(5) 35000(6)

57500(7)70000(8) 75000(9)

PRMOTS +IS


2

4

6

8

10Av

erag

e IG

D p

erfo

rman

cesc

ore

02500(1)12500(2)

22500(3) 25000(4) 25000(4)40000(6)

60000(7)70000(8) 72500(9)

PRMOTS +IS


2

4

6

8

10

Aver

age H

V p

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rman

cesc

ore

(a)

05000(1)

23333(2) 26667(3) 26667(3)35000(5)

43333(6) 46667(7)

0

2

4

6

Aver

age H

V p

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rman

cesc

ore

PRMOTS +IS


08333(1)18333(3)15000(2)

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40000(5)46667(6) 46667(6)

0

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4

6

Aver

age I

GD

per

form

ance

scor

e

PRMOTS +IS


(b)







MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





MK01

4042

44 46

150160

17036

38

40

42


Criti

cal w

ork

load

MK02

2530

35

140145

15025

30

35


Criti

cal w

ork

load

MK03

200 250300

350

800850

900100

200

300

400


Criti

cal w

ork

load

MK04

50100

150

300350

40050

100

150


Criti

cal w

ork

load

MK05

170 180 190 200 210

670680

690160

180

200

220


Criti

cal w

ork

load

MOMADPF

MK06

40 6080 100 120

300400

50040

60

80

100


Criti

cal w

ork

load

MOMADPF









MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





MK07

100150

200250

0

500

1000100

150

200

250


Criti

cal w

orkl

oad

MK08

Criti

cal w

orkl

oad

500550

600

2450

2500

2550450

500

550

600


MK09

300400

500

2200

2300

2400200

300

400

500


Criti

cal w

orkl

oad

MOMADPF

MK10

Criti

cal w

orkl

oad

150200

250300

1800

2000

2200150

200

250

300


MOMADPF











Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of










Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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