research article a hybrid genetic algorithm with a...

Research ArticleA Hybrid Genetic Algorithm with a Knowledge-Based Operatorfor Solving the Job Shop Scheduling Problems

Hamed Piroozfard Kuan Yew Wong and Adnan Hassan

Department of Manufacturing and Industrial Engineering Faculty of Mechanical Engineering Universiti Teknologi Malaysia (UTM)81310 Skudai Johor Malaysia

Correspondence should be addressed to Hamed Piroozfard phamed2liveutmmy

Received 4 December 2015 Accepted 7 March 2016

Academic Editor Jein-Shan Chen

Copyright copy 2016 Hamed Piroozfard et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Scheduling is considered as an important topic in productionmanagement and combinatorial optimization in which it ubiquitouslyexists in most of the real-world applications The attempts of finding optimal or near optimal solutions for the job shop schedulingproblems are deemed important because they are characterized as highly complex and119873119875-hard problems This paper describesthe development of a hybrid genetic algorithm for solving the nonpreemptive job shop scheduling problems with the objective ofminimizing makespan In order to solve the presented problem more effectively an operation-based representation was used toenable the construction of feasible schedules In addition a new knowledge-based operator was designed based on the problemrsquoscharacteristics in order to use machinesrsquo idle times to improve the solution quality and it was developed in the context of functionevaluation A machine based precedence preserving order-based crossover was proposed to generate the offspring Furthermorea simulated annealing based neighborhood search technique was used to improve the local exploitation ability of the algorithmand to increase its population diversity In order to prove the efficiency and effectiveness of the proposed algorithm numerousbenchmarked instances were collected from theOperations Research Library Computational results of the proposed hybrid geneticalgorithm demonstrate its effectiveness

1 Introduction

Scheduling is one of the most important topics that ubiq-uitously exist in many real-world applications Schedulingis assigning a set of tasks on resources to be performedduring a time period considering the time capability andcapacity constraints [1] The main focus is to improve theproduction efficiency and utilization of resources in order tomaximize profit In manufacturing many of the schedulingproblems are considered to be exceedingly complex in whichthey are difficult to be solved with exact methods andconventional algorithms Scheduling problems have been theresearch interest of many researchers and a great deal ofresearch efforts can be found in different fields of engineeringand science such as operations research computer scienceindustrial engineering mathematics and management sci-ence since 1950

The job shop scheduling problems (JSSPs) are well-known important and complex problems in productionmanagement and combinatorial optimization fields whichare characterized as 119873119875-hard Complexity of JSSPs can becalculated by (119899)119898 in terms of all possible schedules inwhichtheir complexity is exceedingly increasing as the problem sizegets bigger Garey et al [2] and Ullman [3] have proved thatthe JSSPs are amongst the119873119875-hard problems thus they can-not be solved with polynomial-time algorithms (unless 119875 =

119873119875) The JSSPs have tended to be treated with conventionaltechniques and exactmethods such as Lagrangian relaxationbranch and bound heuristic rules shifting bottleneck (egCarlier and Pinson [4] Adams et al [5] Vancheeswaran andTownsend [6] Brucker et al [7] and Lageweg et al [8])since job shop instances have been presented by Fisher andThompson [9] Exact methods like branch and bound couldguarantee global optima however computational time could

Hindawi Publishing CorporationJournal of OptimizationVolume 2016 Article ID 7319036 13 pageshttpdxdoiorg10115520167319036

2 Journal of Optimization

be significantly increased with growing problem size Overthe preceding decade many different methodologies havebeen inspired from nature biology and physical processingThese techniques have been successfully applied on numer-ous optimization problems especially the JSSPs Amongthese metaheuristics are genetic algorithm [10 11] ant colonyoptimization [12] imperialist competitive algorithm [13 14]tabu search [15] simulated annealing (SA) [16 17] particleswarm optimization [18] and immune system [19] Jain andMeeran [20] and Calis and Bulkan [21] have performedcomprehensive reviews on JSSP techniques which can bereferred to for more detail

Genetic algorithm (GA) as a powerful search techniqueimitates the biological evolution and natural selection pro-cess GA was first proposed by Holland [22] and furtherdeveloped by David Goldberg This metaheuristic approachis widely employed to find optimal or near optimal solutionsfor different optimization problems in comparison to otheralgorithms GA was first applied on JSSPs by Davis [23] andsince then many GA-based algorithms have been presentedfor solving JSSPs Croce et al [11] presented a GA for solvingJSSPs with an encoding scheme that was based on preferencerules Yamada and Nakano [24] proposed a GA with anew representation scheme that was based on operationcompletion time and its crossover was able to generate activeschedules Lee et al [25] presented a GA with an operation-based representation and a precedence preserving order-based crossover for the JSSPs Sun et al [26] developed amodified GA with a clonal selection and a life span strategyfor the JSSPs and the developed algorithmwas able to find 21best known solutions out of 23 benchmarked instances GA isa powerful search technique in which its global search abilityis conspicuous from the literature however this metaheuris-tic algorithm suffers in terms of premature convergence andlocal search ability

Hybridization strategy is mainly employed to overcomethe drawbacks of GA in terms of local search ability andpremature convergence in order to make the algorithmmoreefficient and powerful Wang and Zheng [27] proposed ahybrid optimization strategy that was a combination of GAand local search In this approach local search algorithmdecreased the probability of GA from getting trapped in localoptima and this hybrid framework relaxed the parameterdependency of both algorithms Zhou et al [28] developeda hybrid heuristic-GA for solving the JSSPs with an adaptivemutation operator for preventing premature convergence Inthis framework GA was applied on the first operations ofthe machines and heuristics were used to determine theremaining operations in the restricted solution space In addi-tion a neighborhood search technique was applied to furtherimprove the solution quality obtained from the hybridheuristic-GA Ombuki and Ventresca [29] presented a deadlock free local search GA with an operation-based represen-tation and UOX crossover that was able to generate feasiblesolutions In this algorithm GA was employed to performglobal search and a local search operator was applied for localexploitationThey have also developed another genetic-basedhybrid algorithm with tabu search and based on computa-tional results the hybrid GA with tabu search outperformed

the local search GA Goncalves et al [30] developed a hybridGA for the JSSPs by combining GA schedule builder and alocal search operator In this algorithm schedule builder wasapplied to generate schedules using a priority rule and GAwas used to determine prioritiesThen a local search operatorfurther improved the solution quality Lin and Yugeng [31]developed a hybrid algorithm with a new representationscheme called random keys encoding In this algorithm GAwas used to obtain an optimal schedule and then a neigh-borhood search was introduced to perform local exploitationand increase the solution quality obtained from GA Zhou etal [32] proposed a hybridGA tominimizeweighted tardinessin job shop scheduling In this algorithm GA and a heuristicwere employed for obtaining optimal solutions in whichGA was applied for determining the first operations and theheuristic was used for assigning the remaining operationsResults showed that the hybrid framework performed betterthan GA and heuristic alone Asadzadeh and Zamanifar [33]proposed aGA that was implemented in parallel using agentsand agents were also used to create initial populations Zhangand Wu [17] introduced a hybrid-SA immune system algo-rithm for minimizing the total weighted tardiness of job shopscheduling Yusof et al [34] developed a hybrid micro-GAthat was implemented in parallel for the JSSPsThis algorithmwas a combination of asynchronous colony GA that con-sisted of colonies with a small number of populations andautonomous immigration GA with subpopulations

In this paper an effective hybrid GA is presented for solv-ing the nonpreemptive JSSPs In order to solve the presentedproblem more effectively an operation-based representationis used In addition a new knowledge-based operator isdesigned and adopted within the context of function evalua-tionThis knowledge-based operator imitates the JSSPrsquos char-acteristics in order to use themachinesrsquo idle times in assigningthe operations to the machines In the reproduction phaseof GA a machine based precedence preserving order-basedcrossover and two types of mutation operators are developedin order to produce the offspring and mutants Moreover SAis employed to increase the solution quality of the schedulesobtained from GA and to increase the population diversityin GA to some extent Having highlighted the important fea-tures of the proposed algorithm it is developed to minimizethe makespan of schedules Finally computational results ofbenchmarked problems are used to prove the efficiency ofthe proposed algorithm

The rest of this paper is organized as follows In Section 2the problem formulation of the JSSP is presented Theproposed hybridGA is discussed in Section 3 Computationalresults of benchmarked instances are presented in Section 4which is followed by a discussion Finally conclusions andfuture work are provided in Section 5

2 Problem Formulation

The JSSP is a general form of classical scheduling prob-lems which can be defined as follows given 119899 jobs 119873 =

1198951 1198952 119895

119899 and each job consists of 119904 operations 119878 =

1198741198951 1198741198952 119874

119895119904 that must be processed on 119898 machines

in a given technological sequence The notation 119874119895119904

denotes

Journal of Optimization 3

the 119904th operation of job 119895 with a known processing time 119875119895119904

which has to be processed on one of the machines 119872 =

119872111987221198723 119872

119898 In this environment each machine

can process at most one operation at a time and an operationof a given job cannot be processed on two machines at thesame time Once operation119874

119895119904starts to be processed on one

of the predeterminedmachines itmust be completedwithoutany preemption In addition a job cannot visit the samemachine twice and there are no precedence requirements fordifferent operations of the jobs It is also assumed that themachines are continuously available and the traveling times

of operations are negligible Unlike flow shop schedulingin JSSPs each job has its unique predetermined route Thesequencing of all operations on all machines is scheduled tominimize 119862max that is the maximum completion time of thejobs (max 119862

1 1198622 119862

119895)

The mathematical model of a JSSP with the objective ofminimizing makespan is given in (1)ndash(9) [35 36] In thismodel 119861 is assumed as a big number 119905

119895119904denotes the start

time of operation 119874119895119904 119878119898119894119897is the start time of machine 119894

in the priority of 119897 and ℎ119894119895119904

is assigned 1 if operation 119874119895119904

isexecuted on machine 119894 and 0 otherwise

min 119885 = max 1198621 1198622 119862

119895 119895 = 1 2 119899 (1)

St 119905119895119904+ 119875119895119904le 119905119895119904+1

119895 = 1 2 119899 119904 = 1 2 119904119895minus 1 (2)

119878119898119894119897+ 119875119895119904119883119894119895119904119897

le 119878119898119894119897+1

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894minus 1 119904 = 1 2 119904

119895 (3)

119878119898119894119897le 119905119895119904+ (1 minus 119883

119894119895119904119897) 119861 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897

119894 119904 = 1 2 119904

119895(4)

119883119894119895119904119897

le ℎ119894119895119904

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (5)

sum

119895

sum

119904

119883119894119895119904119897

= 1 119894 = 1 2 119898 119897 = 1 2 119897119894 (6)

sum

119894

sum

119897

119883119894119895119904119897

= 1 119895 = 1 2 119899 119904 = 1 2 119904119895 (7)

119905119895119904ge 0 119895 = 1 2 119899 119904 = 1 2 119904

119895(8)

119883119894119895119904119897

isin 0 1 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (9)

In this model constraint (2) is concerned with the opera-tionsrsquo sequences inwhich they should follow a specified orderConstraint (3) prevents machines overlapping and enforceseach machine to process not more than one operation at thesame time Constraint (4) prevents operations overlappingso that an operation is assigned to a specified idle machine ina condition such that its previous operation is executed andfinished In addition for each of the operations a machineis determined in constraint (5) In constraint (6) operationsare assigned to the machines and they are sequenced on themachines Constraint (7) limits the number of operations tobe performed on one machine according to the machinersquospriority

3 The Proposed Hybrid Genetic Algorithm

The proposed hybrid algorithm is a combination of twoalgorithms namely GA and SA The advantages of bothalgorithms are employed in the proposed framework in orderto find the optimum solutions for the JSSPs In Figure 1 aflowchart of the proposed hybrid genetic algorithm (HGA)is presented It starts with the initialization of the populationat random The created population is evaluated based onthe fitness function and a new knowledge-based operator

is applied in this step to improve the solution quality of theindividuals In addition this knowledge-based operator ismerged with the function evaluation phase and it works withmachinesrsquo idle timesThis operator is presented in Section 32In the reproduction phase a selection operator is applied toselect the parents for the mating pool and then a crossoveroperator is performed to produce the offspring In additiona mutation operator is carried out on randomly selectedindividuals to create the mutants The created offspring andmutants are evaluated and then termination conditions areconsidered in order to guide the algorithm to be on the rightpathThe termination conditions are described in Section 37

In a situation when the algorithm continues through ter-mination condition 2 that is termination ofGArsquos generationsSA will be started with the best individual of GA In SA aneighborhood search structure is employed with three differ-ent operators namely swapping insertion and reversion asa proposal mechanism In addition beta percent of acceptedsolutions are kept in a pool of individuals as each coolingcondition of SA is reached The SA algorithm is presentedin depth in Section 36 In this phase of the algorithm threeconditions are set either to stop the algorithm or to continuewith new parameters If the latest condition or condition4 that is termination of SArsquos outer loop is reached the SA


Start

Initialize population

Evaluate fitness and apply newknowledge-based operator

Apply selection operator

Crossover

Mutation

Stop algorithm andreport best solution

Stopping conditions1 2 satisfied

Initialize SA with thebest solution of GA

GA from the created pool ofindividuals

Apply neighborhood search to

Evaluate fitness and apply new

beta and decrease the temperature

Stopping conditions1 3 4 satisfied

Apply new parameters ofGA and reset SA

parameters

No

Condition 1 is reached (BKS is obtained)

Condition 2 is reached (termination of GArsquos generations)

No

Condition 1 or 3is reached (BKSis obtained or

HGA isterminated)

Repr

oduc

tion

Condition 4 is reached (termination of SA)

t = 0

create S998400

knowledge-based operator for each S998400

Migrate zeta percent of S998400 to

Create a pool of S998400 with a rate of

Figure 1 Flowchart of the proposed hybrid genetic algorithm

parameters are reset and new parameters of GA are appliedMoreover zeta percent of unique individuals are migrated toGA from the migration pool of SA

31 Encoding and Decoding In any algorithm the first andthe most important step is to find appropriate encoding anddecoding procedures in order to represent the problem Inthis paper an operation-based representation is adopted torepresent the permutation of operations of different jobs[34 37] Based on this representation approach the schedulecould be constructed if the technological constraints aresatisfied In this approach a chromosome consists of 119899 times 119898genes in which each of the genes represents the sequenceof the operations that should be executed on the machinesEach of the operations is denoted with a positive integervalue that is starting from 1 to 119899 The number of occurrencesfor each of the integer values is equal to the number ofoperations In other words the 119896th occurrence of an integervalue in the chromosome represents the 119896th operation of thejob with respect to the technological sequences Consider aJSSP that is given in Table 1 In this table operation routingmachine and processing time of a small problem with 4 jobsand 3 machines are tabulated Suppose a randomly generated

Table 1 An example of a job shop problem with 4 jobs and 3machines

Job Processing time machine number1198951

1 1 3 2 2 31198952

8 2 5 1 10 31198953

5 1 4 3 8 21198954

4 3 10 1 6 2

chromosome is given as 3 2 4 1 3 1 2 3 2 4 1 4 Inthis chromosome each job consists of three operations anddue to this reason each job occurs three times within thelength of the chromosome For instance the sixth and ninthgenes of this chromosome represent the second operationof job one and third operation of job two respectivelyIn addition each chromosome is composed of additionalinformation such as machine number processing time starttime and finish time that are attached to the chromosome

In order to decode the chromosome and construct theschedule we start from the most left to the most rightof the chromosome that is the first gene in the left sideof the chromosome should be scheduled first followed by


0 5 10 15 20 25 30

3

2

4

1

3

1

2

3

2

4

1

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3

Figure 2 The schedule of 4 times 3 job shop scheduling problem

the second gene until the last gene of the chromosome BasedonTable 1 the firstmachine should process119874

11119874221198743111987442

the second machine should process 11987412 11987421 11987433 11987443 and

the last machine should process11987413119874231198743211987441 According

to the chromosome 3 2 4 1 3 1 2 3 2 4 1 4 andconsidering the process constraints to be met the operationsequences of the jobs to be performed on machines 1 2 and3 are as follows These sequences for the first second andthird machines are [119874

31 11987411 11987422 11987442] [11987421 11987412 11987433 11987443]

and [11987441 11987432 11987423 11987413] respectively According to these

sequences the first operation of each machine should bescheduled by considering the process and time constraintsTherefore operations 119874

31 11987421 and 119874

41must be sched-

uled on machines 1 2 and 3 at the permissible timeone after another respectively Then the second operations[11987411 11987412 11987432] third operations [119874

22 11987433 11987423] and fourth

operations [11987442 11987443 11987413] of eachmachinemust be scheduled

by considering the process and time constraints In additioneach set of them must be scheduled separately one afteranother on machines 1 2 and 3 respectively at the permis-sible time The procedures that are applied in this encodingand decoding can guarantee feasible schedules In Figure 2the schedule of the considered chromosome is shown

32 Fitness Function and the Knowledge-Based OperatorThe fitness function in an optimization problem usuallydetermines the probability of a solution that can be passedto the next generation In other words the solution quality isdetermined by applying this operator and the chromosomeswith higher quality will have a higher chance of survivinghowever the less fitted chromosomesmust be discarded fromthe population In the JSSPs many different performanceevaluators exist for defining the fitness function In thisresearch we use makespan or 119862max as the fitness function inorder to evaluate each of the chromosomes

In this algorithmand in the context of the fitness functiona new knowledge-based operator is designed based on theproblem characteristics This operator is designed based onthe machinesrsquo idle times that exist in the job shop environ-ments In addition this knowledge-based operator is appliedduring the function evaluation phase in order to decreasethe computational time of the algorithm and to cover all

the chromosomes that need to be evaluated To design thisoperator more efficiently the following steps are applied

Step 1 The idle points of each machine must be foundThenfor each of these idle points the idle start time idle finishtime and length of idle time must be calculated

Step 2 Based on the position of the idle point on themachinesequence list a candidate operation is chosen from the rightside of the machine sequence list in order to be shifted to theidle position by considering the duration of idle time and theprocessing time of the candidate operation

Step 3 If the length of idle time is bigger than or equal tothe processing time of the candidate operation it will beaccepted conditionally Otherwise it will be rejected

Step 4 If the candidate operation is rejected for transferthe operator goes back to the second step and chooses thesubsequent operation

Step 5 For the conditionally accepted candidate operationall of the processing constraints must be considered in orderto reject the shift or accept it For instance the previousoperation of the candidate operation must be completed

Step 6 If all of the constraints are satisfied the candidateoperation will be transferred to its new position Otherwisethe candidate operation will remain in its own position

Step 7 For each of the machines Steps 2ndash6 should be contin-ued until the last operation on the machine sequence list

Consider the 4 jobs 3 machines job shop problem thatis given in Table 1 Suppose the chromosome is given as3 2 4 1 3 1 2 3 2 4 1 4 in which its schedule isdepicted in Figure 2 The new knowledge-based operatorwith the mentioned procedures 1 to 7 is applied to the thirdmachine of this chromosome as follows In the thirdmachinethere are two idle points in which the first one starts at 4 andfinishes at 5 and the second one starts at 9 and finishes at 13Based on the machine sequence list [119874

41 11987432 11987423 11987413] and

Figure 2 the candidate operations for the first and secondidle points can be [119874

32 11987423 11987413] and [119874

23 11987413] respectively

Consider the candidate operations for the first idle point withthe duration of 1 minute the processing time for any of thegiven candidate operations [119874

32= 4 119874

23= 10 119874

13= 2] is

more than 1 minuteTherefore the first idle point will remainuntouched However the candidate operations for the secondidle point with the processing time of [119874

23= 10 119874

13= 2]

and its idle duration of 4 minutes lead to a possibility of ashift The second candidate operation in the list 119874

13 has a

processing time less than the length of the second idle pointThen for this operation the fifth step (ie considering all ofthe constraints) must be executed in order to have a feasibleschedule It is clear that if operation 119874

13is transferred to

the second idle position none of the constraints is violatedTherefore operation119874

13is transferred to its new position and

the finish time of the thirdmachine is decreased to 23minutesas depicted in Figure 3


0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3

Figure 3The schedule of a 4 times 3 job shop scheduling problem afterapplying the new knowledge-based operator on machine 3

33 Selection Operator A good selection technique canincrease GArsquos performance in terms of reaching faster to theoptimal solutions In this paper the RouletteWheel selectiontechnique which is the most commonly used operator isemployed for the selection of parents [34] In addition theelitism approach is applied in this selection technique in orderto retain the fittest chromosomes for the next generation andto prevent the solution from getting worse from one genera-tion to another In the Roulette Wheel selection we used theBoltzmann Probability 119875(119909) = exp(minus1205731015840 times 119891(119909))119891

1015840

(119909) tocalculate the probability of each chromosome In this equa-tion 119875(119909) is the probability of each chromosome 1205731015840 is theselection pressure 119891(119909) is the fitness of each individual and1198911015840

(119909) is the fitness of the worst individual in the generation Itshould be noted that we added 1198911015840(119909) to the original equationof Boltzmann Probability in order to make the selectionpressure independent of the problem scale In addition thenormalized probability of each selected individual is given as119875(119909)norm = 119875(119909)sum

maxpop119909=1

119875(119909)

34 Crossover In GA crossover is the most importantoperator in comparison to other operators and in fact itis the backbone of the algorithm The crossover operator isperformed by combining the information of the first andsecond parents and the produced offspring with the featuresof both parents can be either better or worse than their par-ents In addition the main goal of this operator is to producebetter and feasible offspring from the parental informationIn this paper a machine based precedence preserving order-based crossover (POX) is proposed for generating the feasibleoffspring [25]The following detailed steps are taken in orderto implement the POX operator

Step 1 First two individuals are selected as parents (1198751 1198752)

by applying the Roulette Wheel selection

Step 2 Then two sets of subjobs are selected and called 1199041198951

and 1199041198952 One of the subjobs is selected from the bottleneck

machines and the other one is selected randomly amongst the119899 remaining jobs

Step 3 In this step the elements of the first subjob (1199041198951) are

copied from the first parent (1198751) to the exact positions in

3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2

Figure 4 The procedures of a POX crossover in generating theoffspring

the first child (1198741) and the same goes for the second subjob

that is the elements of the second subjob (1199041198952) are copied

from the second parent (1198752) to the exact alleles in the second

child (1198742)

Step 4 All of the alleles in the first subjob (1199041198951) are deleted

in the second parent (1198752) and the same goes for the second

subjob that is all of the alleles in the second subjob (1199041198952) are

deleted in the first parent (1198751)

Step 5 The remaining alleles in the first and second parents(1198751 1198752) are transferred to the second and first offspring

(1198742 1198741) from the most left to the most right respectively

The procedures that are used in implementing the pro-posed POX operator lead to the feasible solutions in whichit does not need a repairing mechanism Figure 4 showsthe procedures of producing offspring from the parentalinformation by applying the POX operator on a 4 times 3 JSSP

35 Mutation In the reproduction phase mutation is thesecond way of exploring the solution space Mutation oper-ator can prevent the algorithm from being trapped in thelocal optima and it makes the algorithm faster in achievingbetter solutions In addition it could make perturbation inthe chromosome in order to increase the population diversityIn this algorithm two types of mutation operators namelyswapping and insertion are employed Not only can thesemutation operators increase the diversity of the populationbut the insertion operator could carry out intensive search Itshould be noted that one of these mutation operators shouldbe selected randomly in order to create an offspring and theyare described as follows

(1) Swapping operator To apply the swapping operatorfirst two random numbers are generated which arethe positions of two alleles in the chromosome (eg119877 = 2 5) Then all of the parental informationis copied to the exact positions in the offspringexcept the randomly selected alleles which must be


exchanged or swapped in the offspring For instanceconsider the parent as 2 1 1 3 2 3 which is ran-domly selected from the population The offspring ofthis chromosome by applying the swapping operatorwould be 2 2 1 3 1 3

(2) Insertion operator To apply the insertion operatorfirst two random numbers are generated to be thepositions of two alleles in the chromosome (eg119877 = 6 4) Then all of the parental informationis copied to the exact positions in the offspringexcept the randomly selected alleles The randomlyselected allele with a smaller job-value is positionedon the left of the other random allele with a biggerjob-value For instance given the randomly selectedparent as 3 1 2 3 1 2 the offspring of this chromo-some by applying the insertion operator would be3 1 2 2 3 1

36 Simulated Annealing The SA approach was inspired bythe physical annealing process of solid materials and it ischaracterized as a stochastic local search approach [38] Thesearch operator in SA is occasionally permitted to go throughany unfavorable direction and it enables the algorithm toescape from the local solutions and get towards the globalsolutions In SA this characteristic could be attained throughprobabilistically accepting the worse solutions

In the proposed HGA SA starts with the best solutionof GA Then a proposal mechanism that consists of threeoperators namely swapping insertion and reversion isapplied in order to generate a newneighborhood solution (1198781015840)based on the current solution (119878) The new knowledge-basedoperator is applied on the newly generated solution (1198781015840) andthen it is evaluated based on the objective function If thenewly evaluated neighborhood solution is equal to or betterthan the current solution the new neighborhood solutionwill be accepted (119891(1198781015840) le 119891(119878)) Otherwise the algorithmwill continue the search process with a solution (1198781015840 or 119878) thatis decided through a probabilistic acceptance function Inaddition acceptance of a solution is based on the individualrsquosobjective function value and the current temperature (119879) ofthe algorithm (119890[(119891(119878)minus119891(119878

1015840))119879]) In each inner loop of SA an

accepted solution is kept in a new pool of individuals As theinner loops of the algorithm are terminated the initial valueof the temperature should bemodified based on the annealingschedule In addition beta percent of unique individuals arekept and the remaining individuals are discarded from thenew pool of individuals in each outer loop of SA Moreoverwith the termination of the outer loop zeta percent ofunique individuals are kept for the migration purpose andthe remainders are discarded It should be noted that themigration rate must be low and it is used to increase thepopulation diversity of GA to some extent

In the proposal mechanism of SA three operatorsnamely swapping insertion and reversion are used inwhich one of them is randomly selected and applied in orderto create the neighborhood solution Among these operatorsswapping and insertionwere explained in Section 35 and thereversion operator is described as follows First two random

positions are selected within the length of the individual(eg 119877 = 1 4) and then the substrings between themare inverted For instance given the current solution as2 1 3 1 2 3 by applying the reversion operator the newneighborhood solution would be 1 3 1 2 2 3

37 Ending Conditions In the proposed HGA four differentconditions are provided in order to terminate the algorithmentirely or partially The first termination condition is theachievement of the best known solution and the third termi-nation condition is set as the maximum number of genera-tions in the main loop of HGA If the algorithm reacheseither the first or third condition the whole algorithmwill beterminated at onceThe second and fourth partial conditionsare defined as the maximum number of generations in GAand the maximum number of outer loops in SA respectively

4 Computational Experiments and Discussion

41 Basic Data Themain aim of this section is to evaluate theperformance and validate the efficiency of the proposedHGAbased on the well-studied job shop instances For this reasontwo classes of the job shop instances were used in which thefirst class was presented by Lawrence [39] LA01 to LA40 andthe second class was introduced by Fisher andThompson [9]FT06 and FT20 Different dimensions of instances in termsof jobs and machines were used including 6 times 6 10 times 5 15 times5 20 times 5 10 times 10 15 times 10 20 times 10 30 times 10 and 15 times 15which were collected from the Operations Research LibraryIn addition some of the reported algorithms in the literature[16 19 27 29ndash31 33 40ndash45] were used in order to comparewith the proposed HGA

42 Computational Results To develop the proposed HGAMATLAB R2010a was used and the algorithm was run ona computer which was functioning with Windows XP IntelCoreDuoCPUT2450 at 2GHz and 249GB of RAM Eachof the benchmarked instances was tested 10 times indepen-dently with the following tuned parameters population size119873pop = 200 crossover probability on a population 119875

119888=

05 mutation probability on a population 119875mu = 08 initialtemperature of SA 119879

0= 30 cooling rate of temperature in

SA 120572 = 09 keeping rate of individuals 120573 = 005 migrationrate of individuals 120577 = 0002 and updated value of 119875mu =

09 In addition the selection pressure of the Roulette Wheeloperator is set at 1205731015840 = 7 for all benchmarked instances

The computational experiments of the 42 well-studiedjob shop instances with the above tuned parameters and10 replications per each benchmarked instance have beencarried out and the results are depicted in Table 2 Table 2consists of the instance name dimension of the problem(jobs timesmachines) best known solution (BKS) and results ofour proposed HGA which consist of four columns includingthe best solution of the replications relative deviation (RD)and average solution andworst solution of the algorithm in 10runsThese are themost important indexes in the comparisonprocess of the algorithm in order to check its effectivenessand consistency The remaining columns of Table 2 arethe obtained results of the compared algorithms including


Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA


Table 3 Comparison of the results

Reference CA NBPS NBKSO ARD () Improvement in the proposed HGACA Proposed HGA CA Proposed HGA ()

Rakkiannan and Palanisamy [16] HGAPSA 11 2 6 07 07 0Asadzadeh and Zamanifar [33] PGA 34 14 26 34 02 32Luh and Chueh [40] MMIA 18 16 17 032 002 03Yang et al [41] MA 22 20 20 004 003 001Hasan et al [42] HGA 14 11 14 049 000 049Lin and Yugeng [31] HGA 42 26 29 065 035 03Goncalves et al [30] Param 42 30 29 041 035 006

Ombuki and Ventresca [29] LSGA 27 3 14 541 054 487HGA 27 6 14 407 054 353

Coello et al [19] AIS 23 12 14 161 044 117Binato et al [43] Grasp 42 23 29 18 035 145Wang and Zheng [27] HGA 9 7 6 038 028 01Sabuncuoglu and Bayiz [44] BS 40 16 27 423 036 387Nuijten and Aarts [45] RCS 42 30 29 063 035 028NBPS number of benchmarked problems solvedNBKSO number of best known solutions obtainedARD average relative deviation in percentageCA compared algorithms

HGAPSA PGA MMIA MA HGA HGA Param LSGAHGA AIS Grasp HGA BS and RCS which were presentedby Rakkiannan and Palanisamy [16] Asadzadeh and Zaman-ifar [33] Luh and Chueh [40] Yang et al [41] Hasan et al[42] Lin and Yugeng [31] Goncalves et al [30] Ombukiand Ventresca [29] Coello et al [19] Binato et al [43]Wang and Zheng [27] Sabuncuoglu and Bayiz [44] andNuijten and Aarts [45] respectively In order to calculatethe relative deviation for each of the benchmarked instancesthe formula RD = 100 times (BFM minus BKS)BKS was used Inthis formula BFM is the best found makespan and BKS isthe best known solution In addition the average executiontimes of 10 replications for the biggest benchmarked instancesLA31 to LA35 were recorded as 1522 2521 1436 3738 and2484 seconds respectively The average execution times forthe other smaller benchmarked instances were shorter thanthese Overall it can be said that the computational time ofour proposed HGA was reasonable

43 Discussion It is obvious that our proposed HGA wasable to find optimal or near optimal solutions for thebenchmarked problems Table 2 presents that our HGA hasobtained the best known solutions for 6905 percent of theinstances that is 29 out of 42 instances have reached the bestknown solutions In the small sized benchmarked instancessuch as LA01 to LA15 FT06 and FT20 most of the reportedalgorithms and the proposed HGA were able to achieve thebest known solutions However in the bigger sized bench-marked instances LA16ndashLA40 the proposed algorithm wasable to achieve equal or better solutions in comparison tomost of the reported algorithms Moreover the results of theproposed HGA which were obtained without applying thenew knowledge-based operator on the big sized instances

(LA36ndashLA40) were 1300 1428 1232 1251 and 1242 respec-tively It is clear that the obtained results using the newknowledge-based operator in the proposedHGA are better incomparison to those without the knowledge-based operatorThis implies that the new knowledge-based operator canincrease the solution quality of the proposed HGA

Table 3 lists the compared algorithms (CA) numberof benchmarked problems solved (NBPS) number of bestknown solutions obtained (NBKSO) and average relativedeviation (ARD) for the compared algorithms and the pro-posed HGA In addition the last column shows the improve-ment made in our HGA with respect to the other algorithmsin which it is the subtraction between the average relativedeviation of the compared algorithms and the proposedHGA Based on Table 3 the average relative deviation of theproposedHGA is only 035 percent for the 42well-studied jobshop instances (average deviation of the best found solutionsby the proposed HGA from the best known solutions) Itis clear that the proposed HGA has made a considerableimprovement in the solution quality of the benchmarkedinstances in comparison to those of the other algorithmsFor illustration purposes the optimum schedule of LA22 andnear optimal schedule of LA40 that were obtained by theproposed HGA are presented in Figures 5 and 6 respectively

5 Conclusions

This paper has proposed a hybrid genetic algorithm that isa combination of GA and SA for solving the nonpreemptiveJSSPs in order to minimize the makespan of schedules In theproposed algorithm GA was applied for global explorationamong the chromosomes of a population and SA was usedto carry out local exploitation around the individuals An


0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3

Figure 5 The optimal schedule of LA22 obtained by our HGA

0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3

Figure 6 The near optimal schedule of LA40 obtained by our HGA

operation-based representation was used for the solutionencoding of the algorithm In addition a new knowledge-based operator based on the problem characteristics wasdesigned and it was able to increase the solution qualityof the schedules To produce the offspring and mutants amachine based precedence preserving order-based crossoverand two types of mutation operators namely swapping andinsertion were used in order to increase the populationdiversity and intensify the searchMoreover the SA approachwith its neighborhood search ability was applied to furtherimprove the solution quality which was obtained from GAThe proposed HGA was tested on some of the well-studiedbenchmarked instances which were collected from the Oper-ations Research Library and the results were compared withother algorithms Computational results show that generallythe proposed algorithm has an average relative deviation less

than those of the compared algorithms and this proves theeffectiveness and efficiency of the proposed approach

For futurework we suggest taking into account greennessissue in scheduling problems as it is a new frontier that isbeing extended in the manufacturing areas In addition newdeveloped algorithms like imperialist competitive algorithmcan be implemented in the proposed framework with thesuggested knowledge-based operator to see its performanceMoreover one could consider developing new operators tofurther increase the population diversity of the algorithmand even developing an operator to measure the populationdiversity

Competing Interests

The authors declare that they have no competing interests


References

[1] M L Pinedo ldquoIntroductionrdquo in SchedulingTheory Algorithmsand Systems pp 1ndash10 Springer New York NY USA 2012

[2] 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

[3] J D Ullman ldquoNP-complete scheduling problemsrdquo Journal ofComputer and System Sciences vol 10 pp 384ndash393 1975

[4] J Carlier and E Pinson ldquoAn algorithm for solving the job-shopproblemrdquoManagement Science vol 35 no 2 pp 164ndash176 1989

[5] J Adams E Balas and D Zawack ldquoThe shifting bottleneckprocedure for job shop schedulingrdquo Management Science vol34 no 3 pp 391ndash401 1988

[6] R Vancheeswaran and M A Townsend ldquoTwo-stage heuristicprocedure for scheduling job shopsrdquo Journal of ManufacturingSystems vol 12 no 4 pp 315ndash325 1993

[7] P Brucker B Jurisch and B Sievers ldquoA branch and bound algo-rithm for the job-shop scheduling problemrdquo Discrete AppliedMathematics vol 49 no 1ndash3 pp 107ndash127 1994

[8] B J Lageweg J K Lenstra and A H Rinnooy Kan ldquoJob-shopscheduling by implicit enumerationrdquoManagement Science vol24 no 4 pp 441ndash450 197778

[9] H Fisher and G L Thompson Probabilistic Learning Com-binations of Local Job-Shop Scheduling Rules Prentice-HallEnglewood Cliffs NJ USA 1963

[10] R Qing-Dao-Er-Ji Y Wang and X Wang ldquoInventory basedtwo-objective job shop scheduling model and its hybrid geneticalgorithmrdquoApplied SoftComputing vol 13 no 3 pp 1400ndash14062013

[11] F D Croce R Tadei and G Volta ldquoA genetic algorithm for thejob shop problemrdquo Computers amp Operations Research vol 22no 1 pp 15ndash24 1995

[12] J Zhang X Hu X Tan J H Zhong and Q Huang ldquoImple-mentation of an ant colony optimization technique for jobshop scheduling problemrdquo Transactions of the Institute ofMeasurement and Control vol 28 no 1 pp 93ndash108 2006

[13] J Zhang P Zhang J Yang and Y Huang ldquoSolving the JobShop Scheduling Problem using the imperialist competitivealgorithmrdquo Advanced Materials Research vol 845 pp 737ndash7402012

[14] H Piroozfard and K Y Wong ldquoAn imperialist competitivealgorithm for the job shop scheduling problemsrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringand Engineering Management (IEEM rsquo14) pp 69ndash73 IEEESelangor Malaysia December 2014

[15] V A Armentano and C R Scrich ldquoTabu search for minimizingtotal tardiness in a job shoprdquo International Journal of ProductionEconomics vol 63 no 2 pp 131ndash140 2000

[16] T Rakkiannan and B Palanisamy ldquoHybridization of geneticalgorithm with parallel implementation of simulated annealingfor job shop schedulingrdquo American Journal of Applied Sciencesvol 9 no 10 pp 1694ndash1705 2012

[17] R Zhang and C Wu ldquoA hybrid immune simulated annealingalgorithm for the job shop scheduling problemrdquo Applied SoftComputing Journal vol 10 no 1 pp 79ndash89 2010

[18] D Lei ldquoA Pareto archive particle swarm optimization formulti-objective job shop schedulingrdquo Computers and IndustrialEngineering vol 54 no 4 pp 960ndash971 2008

[19] C A C Coello D C Rivera and N Cortes ldquoUse of an artificialimmune system for job shop schedulingrdquo in Artificial Immune

Systems J Timmis P Bentley and E Hart Eds pp 1ndash10Springer Berlin Germany 2003

[20] A S Jain and S Meeran ldquoDeterministic job-shop schedulingpast present and futurerdquo European Journal of OperationalResearch vol 113 no 2 pp 390ndash434 1999

[21] B Calis and S Bulkan ldquoA research survey review of AI solutionstrategies of job shop scheduling problemrdquo Journal of IntelligentManufacturing vol 26 no 5 pp 961ndash973 2015

[22] J H Holland ldquoGenetic algorithmsrdquo Scientific American vol267 no 1 pp 66ndash72 1992

[23] L Davis ldquoJob shop scheduling with genetic algorithmsrdquo inProceedings of the 1st International Conference on Genetic Algo-rithms pp 136ndash140 Hillsdale NJ USA 1985

[24] T Yamada and R Nakano ldquoA genetic algorithm applicable tolarge-scale job-shop problemsrdquo in Proceedings of the ParallelProblem Solving from Nature (PPSN-II rsquo92) pp 281ndash290 Else-vier Science Brussels Belgium 1992

[25] K-M Lee T Yamakawa and K-M Lee ldquoGenetic algorithmfor general machine scheduling problemsrdquo in Proceedings ofthe 2nd International Conference on knowledge-Based IntelligentElectronic Systems pp 60ndash66 IEEE Adelaide Australia April1998

[26] L Sun X Cheng and Y Liang ldquoSolving job shop schedulingproblem using genetic algorithm with penalty functionrdquo Inter-national Journal of Intelligent Information Processing vol 1 no2 pp 65ndash77 2010

[27] L Wang and D-Z Zheng ldquoAn effective hybrid optimizationstrategy for job-shop scheduling problemsrdquoComputers ampOper-ations Research vol 28 no 6 pp 585ndash596 2001

[28] H Zhou Y Feng and L Han ldquoThe hybrid heuristic geneticalgorithm for job shop schedulingrdquo Computers amp IndustrialEngineering vol 40 no 3 pp 191ndash200 2001

[29] B M Ombuki and M Ventresca ldquoLocal search genetic algo-rithms for the job shop scheduling problemrdquo Applied Intelli-gence vol 21 no 1 pp 99ndash109 2004

[30] J F Goncalves J J D M Mendes and M G C Resende ldquoAhybrid genetic algorithm for the job shop scheduling problemrdquoEuropean Journal of Operational Research vol 167 no 1 pp 77ndash95 2005

[31] L Lin and X Yugeng ldquoA hybrid genetic algorithm for job shopscheduling problem to minimize makespanrdquo in Proceedings ofthe 6th World Congress on Intelligent Control and Automation(WCICA rsquo06) pp 3709ndash3713 IEEE Dalian China June 2006

[32] H Zhou W Cheung and L C Leung ldquoMinimizing weightedtardiness of job-shop scheduling using a hybrid genetic algo-rithmrdquo European Journal of Operational Research vol 194 no3 pp 637ndash649 2009

[33] L Asadzadeh and K Zamanifar ldquoAn agent-based parallelapproach for the job shop scheduling problem with geneticalgorithmsrdquoMathematical and Computer Modelling vol 52 no11-12 pp 1957ndash1965 2010

[34] R Yusof M Khalid G T Hui S Md Yusof and M F OthmanldquoSolving job shop scheduling problem using a hybrid parallelmicro genetic algorithmrdquo Applied Soft Computing Journal vol11 no 8 pp 5782ndash5792 2011

[35] M Seda ldquoMathematical models of flow shop and job shopscheduling problemsrdquo International Journal of Applied Mathe-matics amp Computer Sciences vol 4 no 4 pp 122ndash127 2008

[36] K-H Kim and P J Egbelu ldquoAmathematical model for job shopscheduling with multiple process plan consideration per jobrdquoProduction Planning andControl vol 9 no 3 pp 250ndash259 1998


[37] W Cheung and H Zhou ldquoUsing genetic algorithms andheuristics for job shop scheduling with sequence-dependentsetup timesrdquoAnnals of Operations Research vol 107 no 1ndash4 pp65ndash81 2001

[38] S Kirkpatrick C D Gelatt Jr andM P Vecchi ldquoOptimizationby simulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[39] S Lawrence ldquoResource constrained project scheduling anexperimental investigation of heuristic scheduling techniquesrdquoTech Rep GSIA Carnegie Mellon University 1984

[40] G-C Luh andC-HChueh ldquoAmulti-modal immune algorithmfor the job-shop scheduling problemrdquo Information Sciences vol179 no 10 pp 1516ndash1532 2009

[41] J-H Yang L Sun H P Lee Y Qian and Y-C Liang ldquoClonalselection based memetic algorithm for job shop schedulingproblemsrdquo Journal of Bionic Engineering vol 5 no 2 pp 111ndash119 2008

[42] S M K Hasan R Sarker and D Cornforth ldquoHybrid geneticalgorithm for solving job-shop scheduling problemrdquo in Pro-ceedings of the 6th IEEEACIS International Conference onComputer and Information Science (ICIS rsquo07) pp 519ndash524 IEEEMelbourne Australia July 2007

[43] S Binato W J Hery D M Loewenstern and M G ResendeldquoA GRASP for job shop schedulingrdquo in Essays and Surveys onMetaheuristics pp 59ndash79 2002

[44] I Sabuncuoglu and M Bayiz ldquoJob shop scheduling with beamsearchrdquo European Journal of Operational Research vol 118 no2 pp 390ndash412 1999

[45] W P M Nuijten and E H L Aarts ldquoA computational studyof constraint satisfaction for multiple capacitated job shopschedulingrdquo European Journal of Operational Research vol 90no 2 pp 269ndash284 1996

Submit your manuscripts athttpwwwhindawicom

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

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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

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Stochastic AnalysisInternational Journal of


be significantly increased with growing problem size Overthe preceding decade many different methodologies havebeen inspired from nature biology and physical processingThese techniques have been successfully applied on numer-ous optimization problems especially the JSSPs Amongthese metaheuristics are genetic algorithm [10 11] ant colonyoptimization [12] imperialist competitive algorithm [13 14]tabu search [15] simulated annealing (SA) [16 17] particleswarm optimization [18] and immune system [19] Jain andMeeran [20] and Calis and Bulkan [21] have performedcomprehensive reviews on JSSP techniques which can bereferred to for more detail

Genetic algorithm (GA) as a powerful search techniqueimitates the biological evolution and natural selection pro-cess GA was first proposed by Holland [22] and furtherdeveloped by David Goldberg This metaheuristic approachis widely employed to find optimal or near optimal solutionsfor different optimization problems in comparison to otheralgorithms GA was first applied on JSSPs by Davis [23] andsince then many GA-based algorithms have been presentedfor solving JSSPs Croce et al [11] presented a GA for solvingJSSPs with an encoding scheme that was based on preferencerules Yamada and Nakano [24] proposed a GA with anew representation scheme that was based on operationcompletion time and its crossover was able to generate activeschedules Lee et al [25] presented a GA with an operation-based representation and a precedence preserving order-based crossover for the JSSPs Sun et al [26] developed amodified GA with a clonal selection and a life span strategyfor the JSSPs and the developed algorithmwas able to find 21best known solutions out of 23 benchmarked instances GA isa powerful search technique in which its global search abilityis conspicuous from the literature however this metaheuris-tic algorithm suffers in terms of premature convergence andlocal search ability

Hybridization strategy is mainly employed to overcomethe drawbacks of GA in terms of local search ability andpremature convergence in order to make the algorithmmoreefficient and powerful Wang and Zheng [27] proposed ahybrid optimization strategy that was a combination of GAand local search In this approach local search algorithmdecreased the probability of GA from getting trapped in localoptima and this hybrid framework relaxed the parameterdependency of both algorithms Zhou et al [28] developeda hybrid heuristic-GA for solving the JSSPs with an adaptivemutation operator for preventing premature convergence Inthis framework GA was applied on the first operations ofthe machines and heuristics were used to determine theremaining operations in the restricted solution space In addi-tion a neighborhood search technique was applied to furtherimprove the solution quality obtained from the hybridheuristic-GA Ombuki and Ventresca [29] presented a deadlock free local search GA with an operation-based represen-tation and UOX crossover that was able to generate feasiblesolutions In this algorithm GA was employed to performglobal search and a local search operator was applied for localexploitationThey have also developed another genetic-basedhybrid algorithm with tabu search and based on computa-tional results the hybrid GA with tabu search outperformed

the local search GA Goncalves et al [30] developed a hybridGA for the JSSPs by combining GA schedule builder and alocal search operator In this algorithm schedule builder wasapplied to generate schedules using a priority rule and GAwas used to determine prioritiesThen a local search operatorfurther improved the solution quality Lin and Yugeng [31]developed a hybrid algorithm with a new representationscheme called random keys encoding In this algorithm GAwas used to obtain an optimal schedule and then a neigh-borhood search was introduced to perform local exploitationand increase the solution quality obtained from GA Zhou etal [32] proposed a hybridGA tominimizeweighted tardinessin job shop scheduling In this algorithm GA and a heuristicwere employed for obtaining optimal solutions in whichGA was applied for determining the first operations and theheuristic was used for assigning the remaining operationsResults showed that the hybrid framework performed betterthan GA and heuristic alone Asadzadeh and Zamanifar [33]proposed aGA that was implemented in parallel using agentsand agents were also used to create initial populations Zhangand Wu [17] introduced a hybrid-SA immune system algo-rithm for minimizing the total weighted tardiness of job shopscheduling Yusof et al [34] developed a hybrid micro-GAthat was implemented in parallel for the JSSPsThis algorithmwas a combination of asynchronous colony GA that con-sisted of colonies with a small number of populations andautonomous immigration GA with subpopulations

In this paper an effective hybrid GA is presented for solv-ing the nonpreemptive JSSPs In order to solve the presentedproblem more effectively an operation-based representationis used In addition a new knowledge-based operator isdesigned and adopted within the context of function evalua-tionThis knowledge-based operator imitates the JSSPrsquos char-acteristics in order to use themachinesrsquo idle times in assigningthe operations to the machines In the reproduction phaseof GA a machine based precedence preserving order-basedcrossover and two types of mutation operators are developedin order to produce the offspring and mutants Moreover SAis employed to increase the solution quality of the schedulesobtained from GA and to increase the population diversityin GA to some extent Having highlighted the important fea-tures of the proposed algorithm it is developed to minimizethe makespan of schedules Finally computational results ofbenchmarked problems are used to prove the efficiency ofthe proposed algorithm

The rest of this paper is organized as follows In Section 2the problem formulation of the JSSP is presented Theproposed hybridGA is discussed in Section 3 Computationalresults of benchmarked instances are presented in Section 4which is followed by a discussion Finally conclusions andfuture work are provided in Section 5

2 Problem Formulation

The JSSP is a general form of classical scheduling prob-lems which can be defined as follows given 119899 jobs 119873 =

1198951 1198952 119895

119899 and each job consists of 119904 operations 119878 =

1198741198951 1198741198952 119874

119895119904 that must be processed on 119898 machines

in a given technological sequence The notation 119874119895119904

denotes




119872111987221198723 119872






1 1198622 119862

119895)







min 119885 = max 1198621 1198622 119862

119895 119895 = 1 2 119899 (1)

St 119905119895119904+ 119875119895119904le 119905119895119904+1

119895 = 1 2 119899 119904 = 1 2 119904119895minus 1 (2)

119878119898119894119897+ 119875119895119904119883119894119895119904119897

le 119878119898119894119897+1

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894minus 1 119904 = 1 2 119904

119895 (3)

119878119898119894119897le 119905119895119904+ (1 minus 119883

119894119895119904119897) 119861 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897

119894 119904 = 1 2 119904

119895(4)

119883119894119895119904119897

le ℎ119894119895119904

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (5)

sum

119895

sum

119904

119883119894119895119904119897

= 1 119894 = 1 2 119898 119897 = 1 2 119897119894 (6)

sum

119894

sum

119897

119883119894119895119904119897

= 1 119895 = 1 2 119899 119904 = 1 2 119904119895 (7)

119905119895119904ge 0 119895 = 1 2 119899 119904 = 1 2 119904

119895(8)

119883119894119895119904119897

isin 0 1 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (9)







Start




Crossover

Mutation










parameters

No



No


HGA isterminated)

Repr

oduc

tion


t = 0

create S998400









1 1 3 2 2 31198952

8 2 5 1 10 31198953

5 1 4 3 8 21198954

4 3 10 1 6 2




0 5 10 15 20 25 30

3

2

4

1

3

1

2

3

2

4

1

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



11119874221198743111987442




31 11987411 11987422 11987442] [11987421 11987412 11987433 11987443]



31 11987421 and 119874

41must be sched-


22 11987433 11987423] and fourth














41 11987432 11987423 11987413] and


32 11987423 11987413] and [119874



32= 4 119874

23= 10 119874

13= 2] is


23= 10 119874

13= 2]


13 has a


13is transferred to





0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



1015840



maxpop119909=1

119875(119909)









3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2





child (1198742)























119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872111987221198723 119872






1 1198622 119862

119895)







min 119885 = max 1198621 1198622 119862

119895 119895 = 1 2 119899 (1)

St 119905119895119904+ 119875119895119904le 119905119895119904+1

119895 = 1 2 119899 119904 = 1 2 119904119895minus 1 (2)

119878119898119894119897+ 119875119895119904119883119894119895119904119897

le 119878119898119894119897+1

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894minus 1 119904 = 1 2 119904

119895 (3)

119878119898119894119897le 119905119895119904+ (1 minus 119883

119894119895119904119897) 119861 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897

119894 119904 = 1 2 119904

119895(4)

119883119894119895119904119897

le ℎ119894119895119904

119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (5)

sum

119895

sum

119904

119883119894119895119904119897

= 1 119894 = 1 2 119898 119897 = 1 2 119897119894 (6)

sum

119894

sum

119897

119883119894119895119904119897

= 1 119895 = 1 2 119899 119904 = 1 2 119904119895 (7)

119905119895119904ge 0 119895 = 1 2 119899 119904 = 1 2 119904

119895(8)

119883119894119895119904119897

isin 0 1 119894 = 1 2 119898 119895 = 1 2 119899 119897 = 1 2 119897119894 119904 = 1 2 119904

119895 (9)







Start




Crossover

Mutation










parameters

No



No


HGA isterminated)

Repr

oduc

tion


t = 0

create S998400









1 1 3 2 2 31198952

8 2 5 1 10 31198953

5 1 4 3 8 21198954

4 3 10 1 6 2




0 5 10 15 20 25 30

3

2

4

1

3

1

2

3

2

4

1

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



11119874221198743111987442




31 11987411 11987422 11987442] [11987421 11987412 11987433 11987443]



31 11987421 and 119874

41must be sched-


22 11987433 11987423] and fourth














41 11987432 11987423 11987413] and


32 11987423 11987413] and [119874



32= 4 119874

23= 10 119874

13= 2] is


23= 10 119874

13= 2]


13 has a


13is transferred to





0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



1015840



maxpop119909=1

119875(119909)









3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2





child (1198742)























119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start




Crossover

Mutation










parameters

No



No


HGA isterminated)

Repr

oduc

tion


t = 0

create S998400









1 1 3 2 2 31198952

8 2 5 1 10 31198953

5 1 4 3 8 21198954

4 3 10 1 6 2




0 5 10 15 20 25 30

3

2

4

1

3

1

2

3

2

4

1

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



11119874221198743111987442




31 11987411 11987422 11987442] [11987421 11987412 11987433 11987443]



31 11987421 and 119874

41must be sched-


22 11987433 11987423] and fourth














41 11987432 11987423 11987413] and


32 11987423 11987413] and [119874



32= 4 119874

23= 10 119874

13= 2] is


23= 10 119874

13= 2]


13 has a


13is transferred to





0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



1015840



maxpop119909=1

119875(119909)









3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2





child (1198742)























119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30

3

2

4

1

3

1

2

3

2

4

1

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



11119874221198743111987442




31 11987411 11987422 11987442] [11987421 11987412 11987433 11987443]



31 11987421 and 119874

41must be sched-


22 11987433 11987423] and fourth














41 11987432 11987423 11987413] and


32 11987423 11987413] and [119874



32= 4 119874

23= 10 119874

13= 2] is


23= 10 119874

13= 2]


13 has a


13is transferred to





0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



1015840



maxpop119909=1

119875(119909)









3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2





child (1198742)























119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30

4

3

2

3

1

1

2

3

1 2

4

4

Mac

hine

Time (min)

Cmax = 29

M1

M2

M3



1015840



maxpop119909=1

119875(119909)









3 2 34 2 123 4 4 11

3 2 44 2 123 3 4 11

1 4 13 2 434 2 1 23

1 4 13 2 434 2 1 23

3 2 34 2 123 4 4 11

1 4 23 2 434 2 1 13

P1

P1

O1

P2

P2

O2

(sj1) = [1 2]

(sj2) = [3 4]

Subjob 1

Subjob 2





child (1198742)























119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















119888=







Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Ex

perim

entalresults

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

FT06

6times6

5555lowast

000

5555

mdash55

5555

mdash55

5555

55mdash

5555

mdash55

FT20

20times5

1165

1165lowast

000

1174

1178

mdash119

6mdash

1165

mdash117

5116

51209

1192

mdash116

9116

5mdash

1165

LA01

10times5

666

666lowast

000

666

666

mdash66

666

666

666

666

666

6mdash

mdash66

666

666

666

666

6LA

0210times5

655

655lowast

000

6575

666

mdash655

655

655

655

655

655

mdashmdash

655

655

mdash704

655

LA03

10times5

597

597lowast

000

5977

604

mdash617

597

597

617

603

597

mdashmdash

597

604

mdash650

597

LA04

10times5

590

590lowast

000

590

590

mdash607

590

590

606

590

590

mdashmdash

590

590

mdash620

590

LA05

10times5

593

593lowast

000

593

593

mdash593

593

593

593

593

593

mdashmdash

593

593

mdash593

593

LA06

15times5

926

926lowast

000

926

926

mdash926

926

926

926

926

926

mdashmdash

926

926

926

926

926

LA07

15times5

890

890lowast

000

890

890

mdash890

890

890

897

890

890

mdashmdash

890

890

mdash890

890

LA08

15times5

863

863lowast

000

863

863

mdash863

863

863

863

863

863

mdashmdash

863

863

mdash863

863

LA09

15times5

951

951lowast

000

951

951

mdash951

951

951

951

951

951

mdashmdash

951

951

mdash951

951

LA10

15times5

958

958lowast

000

958

958

mdash958

958

958

958

958

958

mdashmdash

958

958

mdash958

958

LA11

20times5

1222

1222lowast

000

1222

1222

mdash1222

1222

1222

1222

1222

1222

mdashmdash

mdash1222

1222

1222

1222

LA12

20times5

1039

1039lowast

000

1039

1039

mdash1039

mdash1039

1039

1039

1039

mdashmdash

mdash1039

mdash1039

1039

LA13

20times5

1150

1150lowast

000

1150

1150

mdash1150

1150

1150

1150

1150

1150

mdashmdash

mdash1150

mdash1150

1150

LA14

20times5

1292

1292lowast

000

1292

1292

mdash1292

mdash1292

1292

1292

1292

mdashmdash

mdash1292

mdash1292

1292

LA15

20times5

1207

1207lowast

000

1207

1207

mdash1207

1207

1207

mdash1207

1207

mdashmdash

mdash1207

mdash1207

1207

LA16

10times10

945

946

011

9462

947

mdash994

mdash945

mdash945

945

959

959

945

946

945

988

945

LA17

10times10

784

784lowast

000

7864

787

mdash793

784

784

mdash784

784

792

792

785

784

mdash827

784

LA18

10times10

848

848lowast

000

8488

852

mdash860

mdash848

mdash848

848

857

857

848

848

mdash881

848

LA19

10times10

842

842lowast

000

8444

850

mdash873

857

844

mdash851

842

860

860

848

842

mdash882

848

LA20

10times10

902

907

055

907

907

mdash912

mdash907

mdash907

907

907

907

907

907

mdash948

907

LA21

15times10

1046

1050

038

1055

1061

mdash114

61088

mdashmdash

1067

1046

1114

1097

mdash1091

1058

1154

1069

LA22

15times10

927

927lowast

000

9371

943

mdash1007

mdashmdash

mdash941

935

989

980

mdash960

mdash985

937

LA23

15times10

1032

1032lowast

000

1032

1032

mdash1033

mdashmdash

mdash1032

1032

1035

1032

mdash1032

mdash1051

1032

LA24

15times10

935

943

086

9505

956

mdash1012

mdashmdash

mdash957

953

1032

1001

mdash978

mdash992

942

LA25

15times10

977

984

072

9923

1000

mdash1067

mdashmdash

mdash993

986

1047

1031

1022

1028

mdash1073

981

LA26

20times10

1218

1218lowast

000

1218

1218

mdash1323

mdashmdash

mdash1218

1218

1307

1295

mdash1271

mdash1269

1218

LA27

20times10

1235

1255

162

1257

1261

mdash1359

mdashmdash

mdash1265

1256

1350

1306

mdash1320

mdash1316

1285

LA28

20times10

1216

1217

008

1221

1225

mdash1369

mdashmdash

mdash1233

1232

1312

1302

1277

1293

mdash1373

1216

LA29

20times10

1152

1179

234

1189

1204

mdash1322

mdashmdash

mdash118

2119

61311

1280

1248

1293

mdash1252

1208

LA30

20times10

1355

1355lowast

000

1355

1355

1355

1437

mdashmdash

mdash1355

1355

1451

1406

mdash1368

mdash1435

1355

LA31

30times10

1784

1784lowast

000

1784

1784

1790

1844

1784

mdashmdash

1784

1784

1784

1784

mdash1784

1784

1784

1784

LA32

30times10

1850

1850lowast

000

1850

1850

1860

1907

mdashmdash

mdash1850

1850

1850

1850

mdash1850

mdash1850

1850

LA33

30times10

1719

1719lowast

000

1719

1719

1719

mdashmdash

mdashmdash

1719

1719

1745

1719

mdash1719

mdash1719

1719

LA34

30times10

1721

1721lowast

000

1721

1721

1725

mdashmdash

mdashmdash

1721

1721

1784

1758

mdash1753

mdash1780

1721

LA35

30times10

1888

1888lowast

000

1888

1888

1895

mdashmdash

mdashmdash

1888

1888

1958

1888

1903

1888

mdash1888

1888

LA36

15times15

1268

1294

205

1300

1307

1279

mdashmdash

mdashmdash

1292

1279

1358

1357

1323

1334

1292

1401

1292

LA37

15times15

1397

1418

150

1431

1436

1408

mdashmdash

mdashmdash

1418

1408

1517

1494

mdash1457

mdash1503

1411


Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Con

tinued

Instance

name

Size

BKS

Prop

osed

HGA

HGAPS

APG

AMMIA

MA

HGA

HGA

Param

LSGA

HGA

AIS

Grasp

HGA

BSRC

SBe

stRD

()

Ave

Worst

2012

2010

2009

2008

2007

2006

2005

2004

2003

2000

2001

1999

1996

LA38

15times15

1196

1222

217

1230

1238

1219

mdashmdash

mdashmdash

1231

1219

1362

1338

1274

1267

mdash1297

1278

LA39

15times15

1233

1249

130

1250

1251

1246

mdashmdash

mdashmdash

1251

1246

1391

1343

1270

1290

mdash1369

1233

LA40

15times15

1222

1233

090

1238

1243

1241

mdashmdash

mdashmdash

1246

1241

1323

1311

1258

1259

mdash1347

1247

Averager

elatived

eviatio

n(

)035

07

34

032

004

049

065

041

541

407

161

18038

423

063

BKSbestkn

ownsolutio

nRD

relatived

eviatio

nin

percentage

Ave

averageo

f10independ

entrun

slowast

Optim

alsolutio

nachieved

byou

rHGA











5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















5 Conclusions



0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 1000

10

14

3

6

9

8

13

14

1

10

4

8

5

2

10

9

8

3

13

6

14

3

1

12

4

3

9

5

10

12

11

8

13

6

9

3

10

2

14

5

6

9

12

4

8

11

13

7

3

15

6

9 5

10

8

13

9

11

8

14

7

12

4

15

5

15

6

12

7

2

15

5

4

11

14

2

1

2

6

3

7

13

4

9

8

6

15

11

11

14

10

1

13

2

9

15

3

4

13

5

3

8

10

6

1

9

4

11

13

1

2

10

7

3

14

12

11

8

15

5

14

12

2

4

5

1

7

12

11

10

4

1

2

15

12

7

14

6

13

7

15

11

15

12

1

1

2

7

5

7

Mac

hine

Time (min)

Cmax = 927

M7

M8

M9

M10

M4

M5

M6

M1

M2

M3


0 200 400 600 800 1000 1200 1400

7

9

3

12

5

11

4

13

9

15

8

4

7

1

13

10

3

6

15

12

8

7

5

5

1

15

8

4

7

8

12

10

2

14

10

3

4 1

8

3

7

4

12

11

2

9

8

6

9

3

14

5

12

4

2

8

10

14

15

6

13

12

5

7

14

3

11

8

3

10

4

15

14

12

8

9

13

6

2

5

6

14

15

9

3

14

15

4

5

2

6

2

11

1

12

15

8

10

1

15

12

4

3

9

13

11

6

5

9

14

15

1

7

12

3

14

8

11

10

6

11

1

13

5

14

11

9

8

4

7

3

12

5

7

14

1

15

10

9

4

1

14

7

6

11

1

10

13

3

5

7

4

12

14

5

12

15

6

10

4

9

13

2

9

6

15

3

11

1

7

8

1

5

13

2

14

3

7

4

2

11

7

10

6

12

9

3

13

6

12

8

7

11

10

13

9

5

15

13

1

2

4

2

6

10

8

6

13

2

11

2

5

15

10

1

11

2

13

9

10

11

14

2

13

1

Mac

hine

Time (min)

Cmax = 1233

M7

M8

M9

M10

M11

M12

M13

M14

M15

M4

M5

M6

M1

M2

M3





Competing Interests



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a hybrid genetic algorithm with a...

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