Research Article A PSO-Based Hybrid Metaheuristic …downloads.hindawi.com/journals/tswj/2014/902950.pdfResearch Article A PSO-Based Hybrid Metaheuristic for Permutation Flowshop Scheduling

Research ArticleA PSO-Based Hybrid Metaheuristic for Permutation FlowshopScheduling Problems

Le Zhang12 and Jinnan Wu1

1 School of Information Engineering Shenyang University Shenyang 110044 China2 School of Information Science and Technology Tsinghua University Beijing 100084 China

Correspondence should be addressed to Le Zhang snowise126com

Received 4 August 2013 Accepted 5 November 2013 Published 29 January 2014

Academic Editors S Berres and W-C Lee

Copyright copy 2014 L Zhang and J WuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the permutation flowshop scheduling problem (PFSP) with the objectives of minimizing the makespan andthe total flowtime and proposes a hybridmetaheuristic based on the particle swarmoptimization (PSO) To enhance the explorationability of the hybrid metaheuristic a simulated annealing hybrid with a stochastic variable neighborhood search is incorporatedTo improve the search diversification of the hybrid metaheuristic a solution replacement strategy based on the pathrelinking ispresented to replace the particles that have been trapped in local optimum Computational results on benchmark instances showthat the proposed PSO-based hybrid metaheuristic is competitive with other powerful metaheuristics in the literature

1 Introduction

Due to the strong industrial background the permutationflowshop scheduling problem (PFSP) has attracted consid-erable attention from researchers all over the world In thisproblem a set of jobs 119869 = 1 2 119899 needs to be processedthrough a set of machines 119872 = 1 2 119898 Each job 119894 isin119869 should be processed through these 119898 machines with thesame machine order that is starting from machine 1 andfinishing on the last machine119898 The processing time of eachjob 119894 isin 119869 on machine 119895 isin 119872(119901

119894119895) is nonnegative and known

before scheduling It is assumed that all the jobs are availablebefore processing and once started the processing cannot beinterrupted It is required that each job can only be processedby only one machine at any time and at the same time eachmachine cannot processmore than one jobThe processing ofa job cannot start on the next machine 119895 + 1 until this job hasbeen completed on the current machine 119895 and machine 119895 + 1is idle The objective is to determine the sequence of these 119899jobs so that a certain performancemeasure can be optimizedThe most commonly studied performance measures are theminimization of makespan (119862max) and the minimization oftotal flowtime (TFT) Let 120587 = (120587(1) 120587(119899)) denote apermutation of the 119899 jobs in which 120587(119896) represents the job

arranged at the kth position and the completion time of eachjob 120587(119896) on each machine 119895 can be calculated as follows

119862120587(1)1

= 119901120587(1)1

119862120587(1)119895

= 119862120587(1)119895minus1

+ 119901120587(1)119895

119895 = 2 3 119898

119862120587(119896)1

= 119862120587(119896minus1)1

+ 119901120587(119896)1

119896 = 2 3 119899

119862120587(119896)119895

= max 119862120587(119896)119895minus1

119862120587(119896minus1)119895

+ 119901120587(119896)119895

119896 = 2 3 119899 119895 = 2 3 119898

(1)

Then the makespan can be defined as 119862max(120587) = 119862120587(119899)119898 andthe total flowtime can be defined as the sum of completiontimes of all jobs TFT(120587) = sum119899

119896=1119862120587(119896)119898

Since the first introduction of the PFSP [1] considerable

attention of researchers has been paid to this problemand many kinds of algorithms have been proposed in theliterature According to the comprehensive review of Ruizand Maroto [2] and Framinan et al [3] for PFSP thesesolutionmethods can be classified into three categories exactmethods heuristics methods and metaheuristic methods

Since it has been proven that the PFSP with makespanminimization is NP-complete in the strong sense when 119898 ge

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 902950 8 pageshttpdxdoiorg1011552014902950

2 The Scientific World Journal

3 and the PFSP with total flowtime minimization is NP-complete in the strong sense when 119898 ge 2 [4] few exactmethods have been proposed for PFSP in the literaturedue to their unacceptable computation time These exactmethods include the mixed integer linear programmingmethod [5] and the branch and bound algorithms ([6ndash8] forthe makespan minimization and [9ndash11] for the total flowtimeminimization) However these exact methods are feasible foronly small size problems because they cannot solve large sizeproblems in reasonable computation time

Heuristic methods can be classified into two categoriesconstructive heuristics and improvement heuristics Con-structive methods start from an empty solution and try tobuild a feasible solution in a short time Johnsonrsquos algorithm[1] is the earliest known heuristic for PFSP which canobtain optimal solutions for 119898 = 2 Campbell et al [12]proposed the CDS heuristic and Koulamas [13] proposed atwo-phase heuristic for PFSP which were both extensionsof Johnsonrsquos algorithm Palmer [14] proposed a slope indexheuristic for PFSP and then Gupta [15] and Hundal andRajgopal [16] extended Palmerrsquos heuristic and proposed twosimple heuristics Nawaz et al [17] proposed a so-calledNEH heuristic based on the idea that jobs with high totalprocessing times on all machines should be scheduled asearly as possible and this NEH heuristic is regarded as thebest heuristic for PFSP with makespan minimization (Ruizand Maroto [2] Taillard [18]) Recently complex heuristicshave been proposed for PFSP for example Liu and Reeves[19] and Framinan and Leisten [20] As far as the solutionquality is concerned the FL heuristic proposed by Framinanand Leisten [20] was the best one among simple heuristics(Framinan et al [3]) Contrary to constructive heuristicsimprovement heuristics start from an existing initial solutionand try to improve it by a given procedure for example a localsearch Fan and Winley [21] proposed a heuristic named theintelligent heuristic search algorithm for the PFSP Suliman[22] proposed a two-phase improvement heuristic in whichan initial solution is generated by theCDSheuristic in the firstphase and then improved by a pair exchange neighborhoodsearch in the second phase Framinan et al [23] proposed anew efficient heuristic for the PFSP with no-wait constraint

Metaheuristics are high-level strategies that combine andguide other heuristics in the hope of obtaining a moreefficient or more robust procedure so that better solutionscan be foundThemain procedure ofmetaheuristics generallystarts from an initial solution or a set of solutions generatedby heuristics and iterates to improve the initial solution orsolutions until a given stopping criterion is reached Themetaheuristics proposed for PFSP are mainly the geneticalgorithm (Chang et al [24] Ruiz et al [25]) the simulatedannealing (Hooda and Dhingra [26] Nouri et al [27])the tabu search (Gao et al [28]) the ant colony algorithm(Rajendran and Ziegler [29]) the iterated greedy algorithm(Ruiz and Stutzle [30]) and the particle swarm optimization(PSO) (Tasgetiren et al [31] Wang and Tang [32]) Thesemetaheuristics use the benchmark problems proposed byTaillard [33] to evaluate their performance The ant colonyalgorithms proposed by Rajendran and Ziegler [29] namedM-MMAS and PACO obtained much better solutions than

constructive heuristics of Framinan and Leisten [20] Theiterated greedy algorithm proposed by Ruiz and Stutzle[30] improved the best known results for some instancesof PFSP with makespan minimization The particle swarmoptimization (PSO) named PSOVNS which incorporatesvariable neighborhood search (VNS) into PSO proposed byTasgetiren et al [31] improved 57 out of 90 best known solu-tions reported by Framinan and Leisten [20] and Rajendranand Ziegler [29] for the total flowtime criterion

In this paper we propose an improved PSO for the PFSPTo enhance the exploration ability of PSO the path relinkingand the hybrid simulated annealing with stochastic VNSare incorporated To improve the search diversification ofPSO a population update method is applied to replace thenonpromising particles The rest of this paper is organized asfollows Section 2 is devoted to describe the proposed PSOalgorithm The computational results on benchmark prob-lems are presented in Section 3 Finally Section 4 concludesthe paper

2 PSO Algorithm for PFSP

21 Brief Introduction of PSO PSO algorithm is a populationbased metaheuristic method introduced by Kennedy andEberhart [34 35] based on the social behavior of bird flockingand fish schooling as well as the means of informationexchange between individuals to solve optimization prob-lems In the PSO a swarm consists of 119898 particles and theseparticles fly around in an 119899-dimensional search space Thesolution of a problem is represented by the position of aparticle that is the 119894th particle at the 119905th generation is denotedas 119883119905119894= [119909119905

1198941 119909119905

1198942 119909

119905

119894119899] At each generation the flight of

each particle is determined by three factors the inertia ofitself the best position found by itself (119901best) and the bestposition found by the whole swarm (119892best) Generally 119901bestand 119892best are represented as 119875119905

119894= [119901119905

1198941 119901119905

1198942 119901

119905

119894119899] and 119866119905 =

[119892119905

1 119892119905

2 119892

119905

119899] respectively Then the velocity of the particle

119881119905

119894= [V1199051198941 V1199051198942 V119905

119894119899] for the next generation can be obtained

from the following equation

V119905+1119894119895

= 119908 sdot V119905119894119895+ 11988811199031sdot (119901119905

119894119895minus 119909119905

119894119895) + 11988821199032sdot (119892119905

119895minus 119909119905

119894119895)

119909119905+1

119894119895= 119909119905

119894119895+ V119905+1119894119895

(2)

where 119908 is called the inertia parameter 1198881and 119888

2are

the cognitive and social parameters and 1199031 1199032are random

numbers between (0 1) Based on the above equations theparticle can fly through search space toward119901best and119892best in anavigatedwaywhile still exploring new areas by the stochasticmechanism to escape from local optima

22 Solution Representation Since the PSO operates in thecontinuous space a job is represented by a dimension of aparticle and then the 119899 jobs can be denoted as a particle119883119905

119894= [119909119905

1198941 119909119905

1198942 119909

119905

119894119899] in the continuous space Due to the

continuous characters of the position values of particles inthe PSO the smallest position value (SPV) rule proposed byTasgetiren et al [31] is adopted to transform a particle withcontinuous position values into a job permutation A simple

The Scientific World Journal 3

Table 1 Solution representation and the corresponding job permutation using SPV rule

Dimension 119895 1 2 3 4 5 6 7 8 9119909119905

119894119895035 minus175 minus002 minus021 002 120 103 067 minus121

Job 120587119905119894119895

6 1 4 3 5 9 8 7 2

Table 2 Interchange move on the job permutation and the corresponding position value adjustment

Dimension 119895 1 2 3 4 5 6 7 8Before interchange119909119905

119894119895054 minus075 minus102 minus041 092 minus120 023 012

Job 120587119905119894119895

6 3 2 4 8 7 1 5After interchange119909119905

119894119895054 minus075 023 minus041 092 ndash120 minus102 012

Job 120587119905119894119895

6 7 2 4 8 3 1 5The bold and italic values are used to show the interchange move applied to jobs 3 and 7

example is provided in Table 1 to show the mechanism ofthe SPV rule In this instance (119899 = 9) the smallest positionvalue is 119909119905

1198942= minus175 so job 2 is assigned to the first position

of the job permutation according to the SPV rule then job9 is assigned to the second position of the job permutationbecause it has the second smallest position value 119909

119905

1198949=

minus121 With the same way other jobs are assigned in theircorresponding position of the job permutation according totheir position values Thus based on the SPV rule the jobpermutation is obtained that is 120587119905

119894= (2 9 4 3 5 1 8 7 6)

23 Population Initialization The population with 119899pop solu-tions is initialized with random solutions according to 1199090

119894119895=

119909min + randtimes (119909max minus119909min) where rand is a uniform randomnumber in [0 1] 119909min = minus40 and 119909max = 40 Also wegenerate the corresponding velocity of each particle by asimilar way V0

119894119895= Vmin+randtimes(VmaxminusVmin) where Vmin = minus10

and Vmax = 10 In addition another solution generated bythe NEH heuristic [18] is added to the initial population andreplaces a random selected solution so as to ensure the qualityof initial population

24 Hybrid Method of Simulated Annealing and StochasticVNS In the PSOVNS proposed by Tasgetiren et al [31] astochastic VNS which itself is a variant of VNS (Hansen andMladenovic [36]) is developed as the local search For a givendiscrete job permutation 120587119905 let 119908 and 119911 denote two differentrandom integer numbers generated in [1 119899] and then thetwo stochastic neighborhoods moves used in the stochasticVNS to generate a neighbor solution 120587

1199051015840

are (1) 1205871199051015840

=

insert(120587119905 119908 119911) remove the job at the wth position and insertit in the zth position and (2) 120587

1199051015840

= interchange(120587119905 119908 119911)interchange two jobs arranged at the wth position and thezth position After a job permutation is changed accordingto a local search operator such as insert or interchange theposition value of each dimension is adjusted correspondinglyto guarantee that the permutation that resulted by the SPVrule for new position values is the same as the permutation

that resulted by the local search operator For example Table 2shows the interchange move applied to two jobs 3 and 7and the corresponding position value changes It is clearthat the interchange of jobs 3 and 7 is corresponding to theinterchange of position values minus102 and 023 The positionvalue adjustment for the insert move is similar

To further improve the exploration ability of the localsearch we incorporate the solution acceptance scheme ofsimulated annealing into the stochastic VNS and thus obtaina hybrid method of simulated annealing and stochastic VNS(denoted as SA VNS) To reduce the computation time andmake the search process focus on the intensification phasewe use a decreasing acceptance threshold to act as the coolingprocedure of simulated annealing The procedure of theproposed SA VNS algorithm is illustrated in Algorithm 1

In the PSOVNS proposed by Tasgetiren et al [31] thestochastic VNS is applied on the global best particle foundat each iteration However a drawback of such applicationis that the starting point of the stochastic VNS may be thesame solution if the global best particle cannot be improvedfor a number of consecutive iterations and consequently theexploration ability of the PSO may be decreased Thereforefor a given population at iteration 119905 we propose to use thefollowing strategy

Step 1 Apply the SA VNS on the promising particles satisfy-ing (119891(120587119905) minus 119891(119892best))119891(119892best) le 002 in which 119891(120587119905) is theobjective value of particle 120587119905

Step 2 Update the global best particle If a new global bestparticle is found then further improve it using the SA VNS

25 Population Update Method It is well known that theadvantage of PSO is that it has a high convergence speedHowever this advantage may become the disadvantage forcomplex scheduling problems because the scheduling prob-lems generally have many local optimal regions in the searchspace That is for the PSO applied to PFSP some particlesmay always fly around a local region and thus are trapped inlocal optimumTherefore we propose a solution replacement


BeginInitialization

Let 1205870be the input initial solution Set 120587 = 120587

0 the acceptance threshold 119879 = 005 and

119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 = 0while (119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 lt 1198995) do

(1) Generate a random number 119903 in [0 1] and two random integer numbers 119908 and 119911If 119903 gt 05 generate 1205871015840 = insert (120587119908 119911) otherwise generate 1205871015840 = interchange (120587119908 119911)

(2) Set 119894119899119899119890119903 119888119900119906119899119905119890119903 = 0(3) while (119894119899119899119890119903 119888119900119906119899119905119890119903 lt 119899 times (119899 minus 1)) do

Set 119896 = 1while (119896 le 2) do

(1) Generate two random integer numbers 119908 and 119911(2) If 119896 = 1 generate 12058710158401015840 = 119894119899119904119890119903119905 (1205871015840 119908 119911)(3) If 119896 = 2 generate 12058710158401015840 = 119894119899119905119890119903119888ℎ119886119899119892119890 (1205871015840 119908 119911)(4) If 119891(12058710158401015840) lt 119891(1205871015840) then set 1205871015840 = 12058710158401015840 and 119896 = 1 otherwise set 119896 = 119896 + 1

end whileSet 119894119899119899119890119903 119888119900119906119899119905119890119903 = 119894119899119899119890119903 119888119900119906119899119905119890119903 + 1

end while(4) If 119891(1205871015840) lt 119891(120587

0) set 120587

0= 1205871015840and 120587 = 120587

1015840 and then go to step 6 otherwise go to step 5(5) If 119891(1205871015840) ge 119891(120587

0) but (119891(1205871015840) minus 119891(120587

0))119891(120587

0) le 119879 set 120587 = 1205871015840 otherwise generate a random

number r in [0 1] and then set 120587 = 1205870if 119903 gt 05 and 120587 = 1205871015840 if 119903 le 05

(6) Set 119879 = 119879 times 095 and 119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 = 119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 + 1end whileReport the improved solution 1205870

End

Algorithm 1 The main procedure of the SA VNS

strategy based on the pathrelinking [37] to remove thesesolutions with new solutions with good quality

In our algorithm a particle is viewed as being trapped inlocal optimum if its personal best solution 119901best has not beenimproved for a number of consecutive generations (ie 20)For these particles we give them the last chance to stay inthe population by applying the path relinking algorithm on itto check if its personal best 119901best can be improved If so thisparticle can remain in the population otherwise we replacethis particle with a new random particle

The path relinking is originally proposed by Glover et al[37] to generate new solutions by exploring a path thatconnects an initial solution and a guiding solution In this pathmoves are selected that introduce attributes contained in theguiding solution To present the path relinking algorithmwe define the distance between two particles 119883

119894and 119883

119895as

119889(119883119894 119883119895) = 119889(120587

119894 120587119895) = |ℎ(120587

119894) minus ℎ(120587

119895)| where 120587 is the

corresponding job permutation (obtained by the SPV rule)of particle 119883 and ℎ(120587) is the Hash function value of 120587 =

(120587(1) 120587(119899)) that is calculated as ℎ(120587) = sum119899

119896=1119896 times 120587(119896) times

120587(119896) Then the path relinking algorithm can be described asfollows

Step 1 Set 120587119894= (120587119894(1) 120587119894(2) 120587

119894(119899)) as the initial solution

Find the particle that has the largest distance to particle119883119894in the current population (denoted as 119883

119895) and set its

corresponding job permutation 120587119895= (120587119895(1) 120587119895(2) 120587

119895(119899))

as the guiding solution Set 119896 = 1 and the local optimum120587opt = 120587119894

Step 2 If 120587119894(119896) = 120587

119895(119896) find the job with index 120587

119895(119896) in 120587

119894and

swap it with 120587119894(119896) to generate a new job permutation 1205871015840

119894 If 1205871015840119894

is better than 120587opt then set 120587opt = 1205871015840

119894

Step 3 Set 119896 = 119896 + 1 If 119896 gt 119899 minus 10 stop otherwise go toStep 2

It should be noted that the above path relinking stopswhen 119896 is larger than 119899 minus 10 because if particle 119883

119895is better

than 119883119894 then 119883

119894will be replaced by 119883

119895if the path relinking

stops when 119896 = 119899 which may result in duplicated particlesand thus decrease the search diversification

26 Complete Procedure of the Proposed PSO

Step 1 Initialization

Step 11 Set initial values for the population size 119899pop theinertia weight and the cognitive and social parameters Set119905 = 0 and 119873119874

119894= 0 (119894 = 1 2 119899pop) for each particle in

the population Create the initial population and the initialvelocities for each particle using the method described inSection 23

Step 12 Generate the job permutation for each particle inthe population using the SPV rule and calculate the objectivevalue of each particle


Step 13 Set the personal best of each particle to be theparticle itself and the global best to be the best one amongthe population

Step 2 Update Particle Positions

Step 21 Update iteration counter 119905 = 119905 + 1

Step 22 Update inertia weight by 119908 = 119908 times 120573

Step 23 For each particle update the velocity and positionvalues according to (2)

Step 24 Generate the job permutation for each particle inthe current population using the SPV rule and calculate theobjective value of each particle

Step 3 Local Search Phase

Step 31 Use the SA VNS algorithm to improve the promisingparticles in the current population and then the global bestparticle found so far according to the adoption strategy ofSA VNS described in Section 24

Step 32 For each particle in the current population updateits personal best 119901best If 119901best of particle 119894 is improved thenset119873119874

119894= 0 otherwise set119873119874

119894= 119873119874

119894+ 1

Step 4 Population Update For each particle in the currentpopulation use the population update method described inSection 25 to update the current population

Step 5 Stopping Criterion If 119905 gt 119879max (the maximumiteration number) or the runtime has reached the limit thenstop otherwise go to Step 2

3 Computational Experiments

To test the performance of our PSO algorithm (denoted asPSOlowast) computational experiments were carried out on thewell-known standard benchmark set of Taillard [33] thatis composed of 110 instances ranging from 20 jobs and 5machines to 200 jobs and 20 machines This benchmark setcontains some instances proven to be very difficult to solveIn this benchmark set there are 10 instances for each problemsize Our PSO algorithm was implemented using C++ andtested on a personal PC with Pentium IV 30GHz CPU and512MBmemory Tomake a fair comparisonwith the PSOVNSwe use the same parameter setting proposed by Tasgetiren etal [31] That is the population size is taken as 119899pop = 2119899 theinitial inertia weight is set to 119908 = 09 and never less than 04the decrement factor120573 for119908 is taken as 0975 the accelerationcoefficients are set to 119888

1= 1198882= 2 the maximum iteration

number 119879max is taken as 500

31 Results for PFSP with Makespan Minimization For themakespan minimization objective our PSO algorithm wascompared with other powerful methods for example theant colony algorithm named PACO of Rajendran and Ziegler

[29] the genetic algorithm named HGA RMA of Ruiz et al[25] the iterated greedy algorithm named IG RSLS of Ruizand Stutzle [30] and the PSOVNS algorithmofTasgetiren et al[31]The solution qualitywasmeasured by the average relativepercent deviation (denoted as ARPD) over 119877 replicated runsfor each instance in makespan with respect to the bestknown upper bounds More specifically ARPD is calculatedas ARPD = sum

119877

119894=1(((119867119894minus 119880119894) times 100)119880

119894)119877 in which 119867

119894is

the makespan obtained by a certain algorithm whereas 119880119894is

the best known upper bound value for Taillardrsquos instances asof April 2004 for the makespan criterion As done by manyresearchers 119877 is set to 119877 = 10 in our experiments

The comparison results for these algorithms are given inTable 3 in which the values are the average performance ofthe 10 instances for each problem size As seen in Table 3 ourPSOlowast algorithm achieves the best average performance and itobtains the best results for instances of 20 times 5 20 times 10 50 times 550 times 10 100 times 5 100 times 20 and 200 times 20 The IG RSLS methodalso performs well with the HGA RMA method being closeMore specifically the PACOmethod cannot obtain the lowestARPD for any group of problem size compared to other rivalmethods The HGA RMA method has the lowest ARPD forinstances of 20times10 50times5 and 100times10The IG RSLS methoddemonstrates the best results for instances of 20 times 20 50 times 550 times 10 50 times 20 and 200 times 10 For instances of 20 times 10both the HGA RMA method and the PSOVNS method givethe best performance For instances of 50 times 5 all the fourmethods except for PACO can obtain the lowest ARPD Forinstances of 100 times 5 only the two PSO algorithms give thebest performance Our PSOlowast algorithm performs better thanits rivals in 100 times 20 and 200 times 20 instances which have beenproven more difficult to solve Therefore it can be concludedthat our PSOlowast algorithm is competitive with other powerfulmethods in the literature

32 Results for PFSP with Total Flowtime Minimization Forthe total flowtimeminimization objective our PSO algorithmwas compared with other powerful methods for examplethe constructive heuristics of Framinan and Leisten [20]the ant colony algorithm of Rajendran and Ziegler [29] andthe PSOVNS of Tasgetiren et al [31] using the benchmarkproblems of Taillard [33] The solution quality was measuredby the relative percent deviation (denoted as RPD) of thebest solution found among 119877 (119877 = 5) replicated runs foreach instance in the total flowtime criterion with respect tothe best known results That is RPD is calculated as RPD =

min((119867119894minus 119880119894) times 100)119880

119894 119894 isin 119877 in which 119867

119894is the total

flowtime value obtained by a certain algorithm whereas119880119894is

the best result obtained among the algorithms of Framinanand Leisten [20] and Rajendran and Ziegler [29] (this bestresult is denoted as LR and RZ)

For the minimization of the total flowtime criterionthe PSOVNS algorithm [31] is demonstrated to be a verypowerful PSO algorithmbecause it improved 57 out of 90 bestknown solutions reported in [20 29]The comparison resultsbetween our PSOlowast and the PSOVNS are given in Table 4 Fromthis table we can see that the PSOVNS algorithm can obtainthe best results for instances of 20times5 20times10 50times5 100times5 and


Table 3 ARPD comparison of different methods for makespan criterion

Problem PACO HGA RMA IG RSLS PSOVNS PSOlowast

20 times 5 018 004 004 003 00020 times 10 024 002 006 002 00220 times 20 018 005 003 005 00550 times 5 005 000 000 000 00050 times 10 081 072 056 057 05650 times 20 141 099 094 136 099100 times 5 002 001 001 000 000100 times 10 029 016 020 018 020100 times 20 193 130 130 145 118200 times 10 023 014 012 018 018200 times 20 182 126 126 135 116Average 065 043 041 047 039The bold font is used to highlight the better solutionslowastused to denote our algorithm

Table 4 Performance comparison of PSOVNS and PSOlowast for total flowtime criterion

Problem PSOVNS PSOlowast

RPD CPU (s) RPD CPU (s)20 times 5 minus0175 318 minus0168 81120 times 10 minus0037 721 minus0035 48620 times 20 2758 1193 minus0068 249750 times 5 minus0603 4171 minus0531 402850 times 10 minus0819 7449 minus0892 443250 times 20 0857 14332 minus0543 5067100 times 5 minus0570 22228 minus0546 40975100 times 10 minus0692 40788 minus0636 41414100 times 20 minus0104 82441 minus0801 44237Average 0068 19293 minus0469 159941The bold font is used to highlight the better solutionslowastused to denote our algorithm

100 times 10 while our PSOlowast algorithm obtains the best resultsfor the other large size instances of 20 times 20 50 times 10 50 times 20and 100 times 20 On average our PSOlowast shows a much betterperformance in the solution quality and robustness than thePSOVNS algorithm

4 Conclusions

This paper presents a PSO-based hybridmetaheuristic for thepermutation flowshop problems to minimize the makespanand the total flowtime In this algorithm a hybrid methodof simulated annealing and stochastic variable neighborhoodsearch is incorporated to improve the exploitation ability anda solution replacement strategy based on the path relinkingmethod is developed to improve the exploration abilityComputational experiments are carried out to test the perfor-mance of the proposed PSO-based hybrid metaheuristic andthe results show that the proposed algorithm is competitive or

superior to some other powerful algorithms in the literaturefor this problem Future research may lie in the application ofthis algorithm in practical production scheduling problems

Conflict of Interests

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

Acknowledgments

This work is supported by National Nature Science Foun-dation under Grant 61004039 by the Science Foundation ofEducational Department of Liaoning Province under GrantL2010374 and also by the Key Laboratory for ManufacturingIndustrial Integrated Automation of Liaoning Province Theauthors would like to thank the Associate Professor HongYang for her valuable comments


References

[1] S M Johnson ldquoOptimal two- and three-stage productionschedules with setup times includedrdquo Naval Research LogisticsQuarterly vol 1 no 1 pp 61ndash68 1954

[2] R Ruiz and C Maroto ldquoA comprehensive review and evalua-tion of permutation flowshop heuristicsrdquo European Journal ofOperational Research vol 165 no 2 pp 479ndash494 2005

[3] J M Framinan R Leisten and R Ruiz-Usano ldquoComparisonof heuristics for flowtime minimisation in permutation flow-shopsrdquo Computers and Operations Research vol 32 no 5 pp1237ndash1254 2005

[4] 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

[5] E F Stafford ldquoOn the development of a mixed integer linearprogramming model for the flowshop sequencing problemrdquoJournal of the Operational Research Society vol 39 pp 1163ndash1174 1988

[6] Z A Lomnicki ldquoA branch and bound algorithm for the exactsolution of the threemachine scheduling problemrdquoOperationalResearch Quarterly vol 16 pp 89ndash100 1965

[7] A P G Brown and Z A Lomnicki ldquoSome applications ofthe branch and bound algorithm to the machine schedulingproblemrdquo Operational Research Quarterly vol 17 pp 173ndash1861966

[8] G B McMahon and P G Burton ldquoFlowshop scheduling withthe branch and boundmethodrdquoOperations Research vol 15 no3 pp 473ndash481 1967

[9] E Ignall and L Schrage ldquoApplication of the branch and boundtechnique to some flow-shop scheduling problemsrdquo OperationsResearch vol 13 no 3 pp 400ndash412 1965

[10] S P Bansal ldquoMinimizing the sum of completion times of njobs over m machines in a flowshopmdasha branch and boundapproachrdquo AIIE Transactions vol 9 no 3 pp 306ndash311 1977

[11] C-S Chung J Flynn and O Kirca ldquoA branch and boundalgorithm to minimize the total flow time for m-machinepermutation flowshop problemsrdquo International Journal of Pro-duction Economics vol 79 no 3 pp 185ndash196 2002

[12] H G Campbell R A Dudek and M L Smith ldquoA heuristicalgorithm for the n job m machine sequencing problemrdquoManagement Science vol 16 no 10 pp 630ndash637 1970

[13] C Koulamas ldquoA new constructive heuristic for the flowshopscheduling problemrdquo European Journal of Operational Researchvol 105 no 1 pp 66ndash71 1998

[14] D Palmer ldquoSequencing jobs through a multi-stage process inthe minimum total timemdasha quick method of obtaining a nearoptimumrdquoOperational ResearchQuarterly vol 16 no 1 pp 101ndash107 1965

[15] J N D Gupta ldquoHeuristic algorithms for multistage flowshopscheduling problemrdquo AIIE Transactions vol 4 no 1 pp 11ndash181972

[16] T S Hundal and J Rajgopal ldquoAn extension of Palmerrsquos heuristicfor the flow shop scheduling problemrdquo International Journal ofProduction Research vol 26 no 6 pp 1119ndash1124 1988

[17] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983

[18] E Taillard ldquoSome efficient heuristic methods for the flowshop sequencing problemrdquo European Journal of OperationalResearch vol 47 no 1 pp 65ndash74 1990

[19] J Liu and C R Reeves ldquoConstructive and composite heuristicsolutions to the 119875sum119862

119894scheduling problemrdquo European Jour-

nal of Operational Research vol 132 no 2 pp 439ndash452 2001[20] J M Framinan and R Leisten ldquoAn efficient constructive

heuristic for flowtimeminimisation in permutation flow shopsrdquoOmega vol 31 no 4 pp 311ndash317 2003

[21] J P-O Fan and G K Winley ldquoA heuristic search algorithm forflow-shop schedulingrdquo Informatica vol 32 no 4 pp 453ndash4642008

[22] SM A Suliman ldquoTwo-phase heuristic approach to the permu-tation flow-shop scheduling problemrdquo International Journal ofProduction Economics vol 64 no 1 pp 143ndash152 2000

[23] J M Framinan M S Nagano and J V Moccellin ldquoAn efficientheuristic for total flowtimeminimisation in no-wait flowshopsrdquoInternational Journal of Advanced Manufacturing Technologyvol 46 no 9ndash12 pp 1049ndash1057 2010

[24] P C ChangWHHuang and J LWu ldquoA blockmining and re-combination enhanced genetic algorithm for the permutationflowshop scheduling problemrdquo International Journal of Produc-tion Economics vol 141 no 1 pp 45ndash55 2013

[25] R Ruiz C Maroto and J Alcaraz ldquoTwo new robust geneticalgorithms for the flowshop scheduling problemrdquo Omega vol34 no 5 pp 461ndash476 2006

[26] N Hooda and A K Dhingra ldquoFlow shop scheduling usingsimulated annealing a reviewrdquo International Journal of AppliedEngineering Research vol 2 no 1 pp 234ndash249 2011

[27] B V Nouri P Fattahi and R Ramezanian ldquoHybrid firefly-simulated annealing algorithm for the flow shop problem withlearning effects and flexible maintenance activitiesrdquo Interna-tional Journal of Production Research vol 51 no 12 pp 3501ndash3515 2013

[28] J Gao R Chen and W Deng ldquoAn efficient tabu searchalgorithm for the distributed permutation flowshop schedulingproblemrdquo International Journal of Production Research vol 51no 3 pp 641ndash651 2013

[29] C Rajendran and H Ziegler ldquoAnt-colony algorithms for per-mutation flowshop scheduling to minimize makespantotalflowtime of jobsrdquo European Journal of Operational Research vol155 no 2 pp 426ndash438 2004

[30] R Ruiz and T Stutzle ldquoA simple and effective iterated greedyalgorithm for the permutation flowshop scheduling problemrdquoEuropean Journal of Operational Research vol 177 no 3 pp2033ndash2049 2007

[31] M F Tasgetiren Y-C Liang M Sevkli and G Gencyilmaz ldquoAparticle swarm optimization algorithm for makespan and totalflowtime minimization in the permutation flowshop sequenc-ing problemrdquoEuropean Journal of Operational Research vol 177no 3 pp 1930ndash1947 2007

[32] X Wang and L Tang ldquoA discrete particle swarm optimizationalgorithm with self-adaptive diversity control for the permuta-tion flowshop problem with blockingrdquo Applied Soft ComputingJournal vol 12 no 2 pp 652ndash662 2012

[33] E Taillard ldquoBenchmarks for basic scheduling problemsrdquo Euro-pean Journal of Operational Research vol 64 no 2 pp 278ndash2851993

[34] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[35] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 October1995


[36] P Hansen and N Mladenovic ldquoVariable neighborhood searchprinciples and applicationsrdquo European Journal of OperationalResearch vol 130 no 3 pp 449ndash467 2001

[37] F Glover M Laguna and R Martı ldquoFundamentals of scattersearch and path relinkingrdquo Control and Cybernetics vol 29 no3 pp 652ndash684 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


3 and the PFSP with total flowtime minimization is NP-complete in the strong sense when 119898 ge 2 [4] few exactmethods have been proposed for PFSP in the literaturedue to their unacceptable computation time These exactmethods include the mixed integer linear programmingmethod [5] and the branch and bound algorithms ([6ndash8] forthe makespan minimization and [9ndash11] for the total flowtimeminimization) However these exact methods are feasible foronly small size problems because they cannot solve large sizeproblems in reasonable computation time

Heuristic methods can be classified into two categoriesconstructive heuristics and improvement heuristics Con-structive methods start from an empty solution and try tobuild a feasible solution in a short time Johnsonrsquos algorithm[1] is the earliest known heuristic for PFSP which canobtain optimal solutions for 119898 = 2 Campbell et al [12]proposed the CDS heuristic and Koulamas [13] proposed atwo-phase heuristic for PFSP which were both extensionsof Johnsonrsquos algorithm Palmer [14] proposed a slope indexheuristic for PFSP and then Gupta [15] and Hundal andRajgopal [16] extended Palmerrsquos heuristic and proposed twosimple heuristics Nawaz et al [17] proposed a so-calledNEH heuristic based on the idea that jobs with high totalprocessing times on all machines should be scheduled asearly as possible and this NEH heuristic is regarded as thebest heuristic for PFSP with makespan minimization (Ruizand Maroto [2] Taillard [18]) Recently complex heuristicshave been proposed for PFSP for example Liu and Reeves[19] and Framinan and Leisten [20] As far as the solutionquality is concerned the FL heuristic proposed by Framinanand Leisten [20] was the best one among simple heuristics(Framinan et al [3]) Contrary to constructive heuristicsimprovement heuristics start from an existing initial solutionand try to improve it by a given procedure for example a localsearch Fan and Winley [21] proposed a heuristic named theintelligent heuristic search algorithm for the PFSP Suliman[22] proposed a two-phase improvement heuristic in whichan initial solution is generated by theCDSheuristic in the firstphase and then improved by a pair exchange neighborhoodsearch in the second phase Framinan et al [23] proposed anew efficient heuristic for the PFSP with no-wait constraint

Metaheuristics are high-level strategies that combine andguide other heuristics in the hope of obtaining a moreefficient or more robust procedure so that better solutionscan be foundThemain procedure ofmetaheuristics generallystarts from an initial solution or a set of solutions generatedby heuristics and iterates to improve the initial solution orsolutions until a given stopping criterion is reached Themetaheuristics proposed for PFSP are mainly the geneticalgorithm (Chang et al [24] Ruiz et al [25]) the simulatedannealing (Hooda and Dhingra [26] Nouri et al [27])the tabu search (Gao et al [28]) the ant colony algorithm(Rajendran and Ziegler [29]) the iterated greedy algorithm(Ruiz and Stutzle [30]) and the particle swarm optimization(PSO) (Tasgetiren et al [31] Wang and Tang [32]) Thesemetaheuristics use the benchmark problems proposed byTaillard [33] to evaluate their performance The ant colonyalgorithms proposed by Rajendran and Ziegler [29] namedM-MMAS and PACO obtained much better solutions than

constructive heuristics of Framinan and Leisten [20] Theiterated greedy algorithm proposed by Ruiz and Stutzle[30] improved the best known results for some instancesof PFSP with makespan minimization The particle swarmoptimization (PSO) named PSOVNS which incorporatesvariable neighborhood search (VNS) into PSO proposed byTasgetiren et al [31] improved 57 out of 90 best known solu-tions reported by Framinan and Leisten [20] and Rajendranand Ziegler [29] for the total flowtime criterion

In this paper we propose an improved PSO for the PFSPTo enhance the exploration ability of PSO the path relinkingand the hybrid simulated annealing with stochastic VNSare incorporated To improve the search diversification ofPSO a population update method is applied to replace thenonpromising particles The rest of this paper is organized asfollows Section 2 is devoted to describe the proposed PSOalgorithm The computational results on benchmark prob-lems are presented in Section 3 Finally Section 4 concludesthe paper

2 PSO Algorithm for PFSP

21 Brief Introduction of PSO PSO algorithm is a populationbased metaheuristic method introduced by Kennedy andEberhart [34 35] based on the social behavior of bird flockingand fish schooling as well as the means of informationexchange between individuals to solve optimization prob-lems In the PSO a swarm consists of 119898 particles and theseparticles fly around in an 119899-dimensional search space Thesolution of a problem is represented by the position of aparticle that is the 119894th particle at the 119905th generation is denotedas 119883119905119894= [119909119905

1198941 119909119905

1198942 119909

119905

119894119899] At each generation the flight of

each particle is determined by three factors the inertia ofitself the best position found by itself (119901best) and the bestposition found by the whole swarm (119892best) Generally 119901bestand 119892best are represented as 119875119905

119894= [119901119905

1198941 119901119905

1198942 119901

119905

119894119899] and 119866119905 =

[119892119905

1 119892119905

2 119892

119905

119899] respectively Then the velocity of the particle

119881119905

119894= [V1199051198941 V1199051198942 V119905

119894119899] for the next generation can be obtained

from the following equation

V119905+1119894119895

= 119908 sdot V119905119894119895+ 11988811199031sdot (119901119905

119894119895minus 119909119905

119894119895) + 11988821199032sdot (119892119905

119895minus 119909119905

119894119895)

119909119905+1

119894119895= 119909119905

119894119895+ V119905+1119894119895

(2)

where 119908 is called the inertia parameter 1198881and 119888

2are

the cognitive and social parameters and 1199031 1199032are random

numbers between (0 1) Based on the above equations theparticle can fly through search space toward119901best and119892best in anavigatedwaywhile still exploring new areas by the stochasticmechanism to escape from local optima

22 Solution Representation Since the PSO operates in thecontinuous space a job is represented by a dimension of aparticle and then the 119899 jobs can be denoted as a particle119883119905

119894= [119909119905

1198941 119909119905

1198942 119909

119905

119894119899] in the continuous space Due to the

continuous characters of the position values of particles inthe PSO the smallest position value (SPV) rule proposed byTasgetiren et al [31] is adopted to transform a particle withcontinuous position values into a job permutation A simple



Dimension 119895 1 2 3 4 5 6 7 8 9119909119905


Job 120587119905119894119895

6 1 4 3 5 9 8 7 2




Job 120587119905119894119895



Job 120587119905119894119895





119905

1198949=


119894= (2 9 4 3 5 1 8 7 6)


119894119895=





1199051015840

are (1) 1205871199051015840

=


1199051015840









BeginInitialization



119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 = 0while (119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 lt 1198995) do



Set 119896 = 1while (119896 le 2) do


end whileSet 119894119899119899119890119903 119888119900119906119899119905119890119903 = 119894119899119899119890119903 119888119900119906119899119905119890119903 + 1

end while(4) If 119891(1205871015840) lt 119891(120587

0) set 120587

0= 1205871015840and 120587 = 120587


0) but (119891(1205871015840) minus 119891(120587

0))119891(120587




End





119894and 119883

119895as

119889(119883119894 119883119895) = 119889(120587

119894 120587119895) = |ℎ(120587

119894) minus ℎ(120587

119895)| where 120587 is the



119896=1119896 times 120587(119896) times


Step 1 Set 120587119894= (120587119894(1) 120587119894(2) 120587



119895) and set its


119895(119899))


Step 2 If 120587119894(119896) = 120587


119895(119896) in 120587

119894and


119894 If 1205871015840119894


119894



119895is better

than 119883119894 then 119883




















119894= 0 otherwise set119873119874

119894= 119873119874

119894+ 1









119877

119894=1(((119867119894minus 119880119894) times 100)119880

119894)119877 in which 119867

119894is





min((119867119894minus 119880119894) times 100)119880

119894 119894 isin 119877 in which 119867

119894is the total












4 Conclusions





Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Dimension 119895 1 2 3 4 5 6 7 8 9119909119905


Job 120587119905119894119895

6 1 4 3 5 9 8 7 2




Job 120587119905119894119895



Job 120587119905119894119895





119905

1198949=


119894= (2 9 4 3 5 1 8 7 6)


119894119895=





1199051015840

are (1) 1205871199051015840

=


1199051015840









BeginInitialization



119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 = 0while (119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 lt 1198995) do



Set 119896 = 1while (119896 le 2) do


end whileSet 119894119899119899119890119903 119888119900119906119899119905119890119903 = 119894119899119899119890119903 119888119900119906119899119905119890119903 + 1

end while(4) If 119891(1205871015840) lt 119891(120587

0) set 120587

0= 1205871015840and 120587 = 120587


0) but (119891(1205871015840) minus 119891(120587

0))119891(120587




End





119894and 119883

119895as

119889(119883119894 119883119895) = 119889(120587

119894 120587119895) = |ℎ(120587

119894) minus ℎ(120587

119895)| where 120587 is the



119896=1119896 times 120587(119896) times


Step 1 Set 120587119894= (120587119894(1) 120587119894(2) 120587



119895) and set its


119895(119899))


Step 2 If 120587119894(119896) = 120587


119895(119896) in 120587

119894and


119894 If 1205871015840119894


119894



119895is better

than 119883119894 then 119883




















119894= 0 otherwise set119873119874

119894= 119873119874

119894+ 1









119877

119894=1(((119867119894minus 119880119894) times 100)119880

119894)119877 in which 119867

119894is





min((119867119894minus 119880119894) times 100)119880

119894 119894 isin 119877 in which 119867

119894is the total












4 Conclusions





Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BeginInitialization



119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 = 0while (119900119906119905119890119903119897119900119900119901 119888119900119906119899119905119890119903 lt 1198995) do



Set 119896 = 1while (119896 le 2) do


end whileSet 119894119899119899119890119903 119888119900119906119899119905119890119903 = 119894119899119899119890119903 119888119900119906119899119905119890119903 + 1

end while(4) If 119891(1205871015840) lt 119891(120587

0) set 120587

0= 1205871015840and 120587 = 120587


0) but (119891(1205871015840) minus 119891(120587

0))119891(120587




End





119894and 119883

119895as

119889(119883119894 119883119895) = 119889(120587

119894 120587119895) = |ℎ(120587

119894) minus ℎ(120587

119895)| where 120587 is the



119896=1119896 times 120587(119896) times


Step 1 Set 120587119894= (120587119894(1) 120587119894(2) 120587



119895) and set its


119895(119899))


Step 2 If 120587119894(119896) = 120587


119895(119896) in 120587

119894and


119894 If 1205871015840119894


119894



119895is better

than 119883119894 then 119883




















119894= 0 otherwise set119873119874

119894= 119873119874

119894+ 1









119877

119894=1(((119867119894minus 119880119894) times 100)119880

119894)119877 in which 119867

119894is





min((119867119894minus 119880119894) times 100)119880

119894 119894 isin 119877 in which 119867

119894is the total












4 Conclusions





Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119894= 0 otherwise set119873119874

119894= 119873119874

119894+ 1









119877

119894=1(((119867119894minus 119880119894) times 100)119880

119894)119877 in which 119867

119894is





min((119867119894minus 119880119894) times 100)119880

119894 119894 isin 119877 in which 119867

119894is the total












4 Conclusions





Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















4 Conclusions





Acknowledgments



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
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A PSO-Based Hybrid Metaheuristic …downloads.hindawi.com/journals/tswj/2014/902950.pdfResearch Article A PSO-Based Hybrid Metaheuristic for Permutation Flowshop Scheduling