ORIGINAL ARTICLEMakespan minimization of a flowshop sequence-dependentgroup scheduling problemNasser Salmasi & Rasaratnam Logendran &Mohammad Reza SkandariReceived: 19 May 2009 / Accepted: 25 January 2011 / Published online: 26 February 2011# Springer-Verlag London Limited 2011Abstract Theflowshopsequencedependentgroupsched-uling problemwith minimization of makespan as theobjective(Fm|fmls, Splk, prmu|Cmax) is consideredinthispaper. It is assumed that several groups with differentnumber of jobs are assigned to a flow shop cell that has mmachines. The goal is to find the best sequence ofprocessing the jobs in each group and the groupsthemselves withminimizationof makespanas theobjec-tive. Amathematical model for the researchproblemisdeveloped inthispaper. Astheresearchproblemis showntobeNP-hard, ahybridant colonyoptimization(HACO)algorithmis developed to solve the problem. Alowerboundingtechniquebasedonrelaxingafewconstraints ofthe mathematical model developed for the original problemis proposed to evaluate the quality of the HACO algorithm.Three different problem structures, with two, three, and sixmachines, are used in the generation of the test problems totest the performance of the algorithmand the lowerboundingtechniquedeveloped. TheresultsobtainedfromtheHACOalgorithmandthosethat haveappearedinthepublished literature are also compared. The comparativeresults showthat the HACOalgorithmhas a superiorperformance compared to the best available algorithmbasedonmemetic algorithmwithanaverage percentagedeviation of around 1.0% fromthe lower bound.KeywordsSequence-dependent group scheduling.Integerprogramming.Meta-heuristics.Lower bound.Flow shopscheduling1 IntroductionIn thispaper, it is assumed that in a flow shopcell severalgroups withdifferent number of jobs areassignedtobeprocessed. All jobsthat belongtoagrouprequiresimilarsetuponmachines. Thus, amajorsetupisrequiredbeforeprocessing each group on every machine. It is assumed thatthe setup time of a group for each machine depends on theimmediatelyprecedinggroupthat was processedonthatmachine. The problem is classified as flow shop sequencedependent group scheduling (FSDGS) problem. Theimportanceof sequencedependent setuptimeschedulingproblemshasbeendiscussedinseveral studiesbyAllah-verdi et al. [1], Panwalker et al. [2], Wortman [3], andSchalleret al. [4]. Therearemanyreal-worldapplicationsofsequence-dependent schedulingproblems. Forinstance,paintingautomobiles withdifferent colors insmall batchsizesisanexampleofsequence-dependent setupschedul-ing problems.Allahverdi et al. [1], Cheng et al. [5], Zhu and Wilhelm[6], and Allahverdi et al. [7] reported a comprehensiveliterature reviewof scheduling problems by consideringseparatesetuptime. HejaziandSaghafian[8]performedacomprehensiveliteraturereviewonflowshopschedulingproblems. Jordan[9]discussedtheextensionofageneticN. Salmasi:M. R. SkandariDepartment of Industrial Engineering,Sharif University of Technology,Tehran, IranN. Salmasie-mail: nsalmasi@sharif.eduM. R. Skandarie-mail: r.skandari@ie.sharif.eduR. Logendran (*)School of Mechanical, Industrial, and Manufacturing Engineering,Oregon State University,Corvallis, OR 97331, USAe-mail: Logen.Logendran@oregonstate.eduInt J Adv Manuf Technol (2011) 56:699710DOI 10.1007/s00170-011-3206-9algorithm(GA) to solve the FSDGSproblemwith twomachines in order to minimize the weighted sumofearlinessandtardinesspenalties.Schalleretal.[4]presentbranch-and-boundapproachesaswell asseveral heuristicalgorithms to solve the FSDGS problemwith multiplemachineswithminimizationofmakespanasthecriterion.They suggest applying the result of their best heuristicalgorithmas an initial solution for other heuristic algo-rithms such as tabu search (TS). They also showed that theFSDGSproblemwith minimization of makespan as thecriterion is NP-hard. Franca et al. [10] developed analgorithmbasedonGAanda memetic algorithm(MA)with local search to solve the FSDGS problemwithminimizationofmakespanasthecriterion. Theyusedthetest problems of Schaller et al. [4] to evaluate their heuristicalgorithm. Logendran et al. [11] developed a heuristicalgorithmbasedonTStosolvethetwo-machineFSDGSproblemsbyconsideringminimizationofmakespan. Theyalsodevelopedalower boundingmethodtoevaluatethequalityoftheirheuristicalgorithm. Hendizadehet al. [12]developedaheuristicalgorithmbasedonTSbyapplyingthe concept of elitism and the acceptance of worse movesfromsimulatedannealingtoimprove the intensificationand diversification moves to solve the same problem.Theirstudyalsoincludedacomparisonoftheiralgorith-micresultswiththoseobtainedbyFrancaetal.[10]. Theresults showed that the MA algorithmdeveloped byFrancaet al. [10] has asuperior performancecomparedtotheTSalgorithmdevelopedbyHendizadehatal.[12].Salmasi andLogendran[13] developedseveral heuristicalgorithmsbasedonTSfor solvingtheFSDGSproblemby considering minimization of makespan. They tested thesuggestionofSchalleretal. [4]aboutapplyingtheresultof their heuristic algorithmas an initial solution for ameta-heuristicalgorithm. Theresultsof their experimentshowedthatthere is no significantdifference between theperformance of using the results of Schaller et al. [4]algorithmas an initial solution and using a randomsequenceforgroupsaswell thejobsthat belongtoeachgroupasan initialsolution.Inthisresearch, amongtheavailablemeta-heuristics, aheuristic algorithmdeveloped based upon ant colonyoptimization(ACO)isusedtosolvetheproblem, andtheresultsobtainedfromit arecomparedwiththepreviouslyattempted heuristics. An efficient lower bounding method hasalso been developed to evaluate the performance of theheuristic algorithm by means of enhancing and generalizingthe lower bounding method, previously proposed byLogendran et al. [11] specifically for two-machine problems.Thegoal istofindthebest sequenceofprocessingthejobsineachgroupaswell asthegroupsthemselveswithminimization of makespan (Cmax) as the criterion. Based onPinedo[14],theproblemweinvestigatecanbenotatedasFm|fmls, Splk, prmu|Cmax with the following descriptions:& Fm denotes a flow shop environment that comprises of aseries of (distinct) machines.& fmls denotes that the jobs are assigned to differentgroups. Each group (p=1, 2, , g) includes bp jobs (thenumber of jobs in groups can be different).& Splkdenotes that the problembelongs to sequencedependent setuptimescheduling. Thesetuptimeof agroup(groupl)oneachmachine(machinek)dependson the immediately preceding group (group p) that wasprocessed on that machine. The setup time of eachgroup on each machine can be different.& prmu denotes permutation scheduling assumption that alljobs and groups are processed in the same sequence on allmachines. This is the only method used to facilitateproduction in some industries. For instance, if a conveyerisusedtomovejobsbetweenmachines, thenall jobsshould be processed in the same sequence on all machines.Notethat eachjobof agroupis assumedtohavenoseparatesetuptime. Ifitdoes, therequiredsetuptimecanbe included in its processing time. We make the well-known grouptechnologyassumption thatrequiresthejobsin a group to be scheduled contiguously. The setup processofamachineforagroupcanbestartedbeforeajobthatbelongs to the group is available, a feature commonlyknown as anticipatory setups in group scheduling.2 Mathematical modelWe introduce the concept of slots in the mixed integer linearprogramming modeling construct. A slot is a position which istobeoccupiedbyoneof thegroupsinorder tofindthesequence of groups. Thus, each group should be assigned toonly one slot and each slot should be dedicated to receivingonlyonegroup. Inreal-worldproblems, groups canhavedifferent number of jobs. Because each group can be assignedto any slot, to facilitate the development of the mathematicalmodel, it is assumed that every group has the same number ofjobs, comprised of real and dummy jobs. This number is equaltobmaxwhichis themaximumnumber of real jobs inagroup. If a group has fewer real jobs than bmax, the difference,i.e., bmaxnumber of real jobs, is assumed to be occupied bydummy jobs. The list of indices, parameters, decisionvariables, and the mathematical model are presented below:Indices and parameters:g Number of groupsm Number of machinesbpNumber of jobs in group p p=1,2,...,g700 Int J Adv Manuf Technol (2011) 56:699710bmaxmaximumnumber of jobs in groupstpjkFor real jobs; run time ofjob jin group p on machine kFor dummy jobs; 08>>>>:i 0; 1; 2; :::; g 1p 0; 1; 2; :::; gl 1; 2; :::; g p 6 lModel:MinimizeZ Cgm1Subject to:Xgi1Wip 1 p 1; 2; . . . g 2aXgp1Wip 1 i 1; 2; . . . ; g 2bXgp0Xgl6pl1Aipl 1 i 0; 1; 2; . . . ; g 1 3AiplWipi 0; 1; 2; :::; g 1 p 0; 1; 2; . . . ; g4aAiplWi1ll 1; 2; . . . ; gp 6 l 4bOik Xgp0Xgp6ll1Ai1pl Splki 1; 2; :::; g k 1; 2; :::m5Ci1 Ci11Oi1Xgp1Wip Tp1i 1; 2; 3; . . . ; g 6XijkCi1k Oik Xgp1Wip t0pjki 1; 2; :::; g j 1; 2; :::; bmaxk 1; 2; 3 . . . m7Xijk Xiqk M Yijq Pgp1Wip t0pjki 1; 2; . . . ; gj; q 1; 2; . . . bmaxj < qk 1; 2; 3 . . . m8Xiqk Xijk M 1 Yijq

Pgp1Wip t0pjkM: A large number9Xijk Xijk1 Pgp1Wip tpjki 1; 2; . . . ; g j 1; 2; . . . ; bmaxk 2; 3 . . . m10CikXijki 1; 2; . . . ; g j 1; 2; . . . ; bmaxk 2; 3 . . . mXijk; Cik; Oik0 Wip; Aipl 2 0; 1 f gYijq 2 0; 1 f g j < q 11Int J Adv Manuf Technol (2011) 56:699710 701Again, there are g slots and each group should beassignedtooneofthem. Constraints(2a)and(2b)ensurethat each slot should contain just one group and everygroup should be assigned to only one slot. Constraint (3) isincorporated in the model to ensure that the setup time of agrouponamachineis dependent onthat groupandthegroupprocessedimmediatelyprecedingthe group. Con-straints(4a)and(4b)ensurethatifgrouppisassignedtoslot i and group l is assigned to slot i +1, then Aipl must beequal to one. Likewise, if group p is not assigned to slot i orgroup l is not assigned to slot i +1, then Aipl must be equaltozero.Constraint (5) evaluatesthe requiredsetuptimeofgroups on machines. The required setup time for a group ona machine is evaluated based on the group assigned to a slotand the group assigned to the preceding slot. Thecompletion time of the group assigned to a slot on the firstmachine is evaluated in constraint (6). The completion timeofagroupassignedtoaslotisequaltothesummationofthe completion time of the group assigned to the precedingslot,therequiredsetuptimeforthegroupinthisslot,andthesumof theprocessingtimeof all jobsinthat group.Constraint (7) is incorporated in the model to find thecompletion time of jobs on machines. The completion timeofajobthatbelongs toa groupisgreaterthanorequaltothe summation of the completion time of the groupprocessed in the previous slot, the setup time for the group,andtheprocessingtimeofthejob.Constraints(8)and(9)are a kind of either/or constraints. They are incorporated inthe model to find the sequence of processing jobs thatbelong to a group. If job j in a group is processed after job qof the same group, then the difference between thecompletiontimeof jobj andjobq on allmachinesshouldbegreater thanor equal totheprocessingtimeof jobj.Constraint (10) is incorporated in the model to support thatamachinecanstartprocessingajobonlyifitsprocessinghas alreadybeencompletedonthe previous machine. Itmeans that the completion time of a job on a machineshouldbegreater thanor equal tothesummationof thecompletion time of that job on the preceding machine plusthe processing time of the job on the current machine.Constraint(11)isincorporatedinthemodeltoensurethatthe completion time of a group on a machine is equal to thecompletion time of the last job of the group which isprocessed by the machine.As theresearchproblemis knowntobeNP-hard[4],heuristic algorithms are needed to solve industry-sizeproblems withinareasonabletime. Amongtheavailableheuristics, ACOalgorithmischosentobecomparedwiththe previously proposed meta-heuristics because of thewidespreadfavorable publicityACOalgorithmhas beenreceiving lately for its capability of solving difficultcombinatorial problems. The meta-heuristic algorithmdeveloped is described in Section 3.3 Hybrid ant colony optimization algorithmACOalgorithmhas been applied for flow shopschedulingproblems by Gajpal and Rajendran [15]. They applied ACOtosolvetheproblemof schedulinginpermutationflow-shops with the objective of minimizing the variance ofcompletiontimeof jobs. Gajpal et al. [16] appliedACOalgorithm for sequence-dependent flow shop job schedulingproblems.Inthissection, ahybridmeta-heuristicalgorithmbasedon ACO and the NEH algorithm [17] is proposed, which isdifferent fromthe one appliedbyGajpal andRajendran[15] and Gajpal et al. [16]. Asolution to the problemcomprises of two types of information: sequence of groups,and sequence of jobs in each group. For this type ofproblem, with makespan as the objective function, thesequence of groups is more important than the sequence ofjobsineachgroup,becausethesetuptimesaredependenton the sequence of groups, and the makespan of a solutionis not directlyaffected by the positionof an individual jobinagroup. Thisisnot validforotherobjectivefunctionssuch as due date-related objectives or total flowtime,wherein the sequence of jobs is as important as thesequence of groups.Inhybridantcolonyoptimizationalgorithm(HACO),ameta-heuristicthatisavariantofACO,namedantcolonysystem, is used for finding the sequence of groups(Section 3.1) and for sequence of jobs, two computationallyrapidheuristicalgorithms areimplementedasanintegralmodule of the proposed meta-heuristic(Section 3.2).3.1 Sequenceof groupsACOis a population-based construction meta-heuristic,inspiredbythe foragingbehavior of several ant species.Argentineantsdeposit pheromoneontheground, andusepheromone as an indirect medium of interchanging informa-tion to find the shortest path from colony to food. To find thefood, they initially wander, and upon finding food, return totheir colony while laying down pheromone trails. But if otherants find such a path, they are likely not to keep traveling byrandom, but insteadtheytendtofollowthetrail, returningcolony and reinforcing the path if they eventually find food.Over time, however, the pheromone trail evaporates, and itsattractiveness is reduced as a result. The more time it takes foran ant to travel from colony to food and return, the more theavailable time for pheromones to evaporate. Consequently, ashorter path, in comparison, gets used more and so thepheromone renews. Thus the pheromone density remainshigh. The more pheromone ona path, the higher is thelikelihoodof that pathbeingfollowedbyother ants. Soeventually all the ants follow a single path, i.e., the shortestpath fromcolony to food. The idea of the ant colony algorithm702 Int J Adv Manuf Technol (2011) 56:699710is to mimic this behavior by deploying artificial ants (agents)that walk around the graph of the combinatorial problem thatis to be solved.The first work on ACO was Ant System [18] proposed forsolving the traveling salesman problem. In Ant System,artificial ants areused to solve the problemby constructingsolutions via addingsolutioncomponents (cities), one byone, guided by the pheromone intensity and heuristicinformation(whichis basedonintercitydistances). Sincethe Ant Colony System(ACS) which was proposed byDorigo and Gambardella [19], has shown better performancethanother availableversionsofACOindifferent typesofproblems, it isusedasapart ofHACOinthispaper. Thecommon mechanics behind every ACO algorithm are:1. Acolonyof (anumber of) antswhichconstructssolutions independently2. Thesolutionconstructionphasethat worksbasedon pheromone values and heuristic information3. Local search that is applied around the best solution(s) found to improve the solution(s)4. Updating pheromone values phase that updates thepheromone values based on the quality of thesolutions found by ants; so as to guide thealgorithmtoward promising solutionsThe following are the steps of ACO as pseudo code:Set parameters, initialize pheromone trailswhile termination conditionnot met doConstruct Ant SolutionsApply Local Search (optional)Update Pheromonesend while3.1.1 Solution constructionAgroup-sequence solution is a permutation of groups,denoted by 1=(G1, G2,, Gg), in which the absoluteposition of the groups is important. Asolution to theproblem can be constructed in two different ways:1. Adding groups one by one to thesequence initiallyincluding the reference group. For instance, sequence1 is constructed as follows: G1 immediately followsthe reference group, G2 follows G1, and so on.2. Defininguslots for ugroups, andassigningthegroups tothe slots. For instance, sequence1isconstructedasfollows: G1isassignedtothefirstslot, G2 is assigned to the second slot, and so on.It is worthmentioningthat whilethetwoways seemdifferent in the first glance, there is a one-to-one accordancebetween themin interpretinga solution.3.1.2 Pheromone definitionTwodifferentdefinitionsofpheromonecanbedevelopedaccording to the aforementioned ways of solutionconstruction:1. pq: desirability of processing group q immediatelyafter group p, p=0,1,,g q=1,2,..,g2. pq: desirabilityof processinggroupqinthepthpositionof the sequence, p=1,2,,g q=1,2,,gThe first definition can be justified because of theimportanceoftheabsolutepositionsofgroupsandcanbefound in Gajpal and Rajendran [15]. The second definitioncanbe justifiedbecause of the sequence dependence ofsetuptimes.Sincethesetuptimesaresequencedependentand hold a large share of the solutions makespan, thesecond one has an advantage. Suppose that processinggroup p immediately after group q led to a favorablesolution, so it is promising to change the absolute positionsofgroupswhilemaintainingthisrelativeorder,inordertoachieve a better solution. Although we noted the one-to-oneaccordance of the construction ways in the previoussection, it isvital toacknowledgethedifferencebetweenutilizing the values of the two pheromone definitions duringtheconstructionphase. Thehighintensityofpqsuggestsprocessinggroupqimmediatelyafter groupp, whilethehigh intensity of pq suggests processing group q in the pthposition of the sequence. It is easily possible for pq and pqto be used in different steps of construction phase.3.1.3 Heuristic informationTo guide the construction algorithmtoward promisingareas, inadditiontothepheromonemechanism, heuristicinformation is deployed in ACO. In ACO literature,heuristic information indicates an intuitive guess thatmeasureshowaddingasolutioncomponenttothepartialsolution will affect the final solutions cost (solutionscost is its objective function value that is to beminimizedinmost ant colonyoptimizationalgorithms).Tocoordinate the directionof pheromone andheuristicinformation, usuallythereverseof theguessedcost (orother relatedfunctions) isusedinstead. Inthisproblem,the heuristic information for each group that is acandidatetobeaddedtothepartial sequence(thepartialsolutionconstructedsofar)canbedefinedasthereverseof differential makespan (makespanpl) produced be-cause of adding the candidate group (group l) to the partialsequence inthe pthstepof solutionconstruction, as inEq. 12. In order to calculate the differential makespan, weneedtoknowthesequenceofjobsinthegroups. Sothegroup-sequencingalgorithmneeds jobsequencingalgo-Int J Adv Manuf Technol (2011) 56:699710 703rithm to perform and they go hand in hand. Therefore, thejobsequencingmoduleisimplementedhereasapart ofthe ACOalgorithm.hpl 1makespanpl123.1.4 Probabilistic model of solution constructionACOconstructsasolutionbyaddingsolutioncomponentsto a null sequence, one after the other guided by aprobabilisticmechanism. InHACO, weusethefollowingpseudorandomproportional rule, derived fromACS, toselect the next groupq tobe added tothe partial sequenceSPartfromthelist of not yet sequencedgroupslN(SPart).The rule is a balanced mixture of two different approaches:greedtowardthe best candidate, andbias towardmorefavorable candidates, both in terms of pheromone andheuristic values. For the pth group of the sequence, with 0probability the best group (in terms of pheromone intensityand heuristic information) is selected and with (1-0)probabilitytherandommodelidentifiesthenextgroup.Inotherwords, with0probabilitythegreedtowardthebestcandidatewill determinethenext group(thefirst lineinEq. 13) andwith1-0probability, thebias towardmorefavorable candidates will determine the next group (thesecond line in Eq. 13).q arg Max8l2NSPartta1p0l:t0pla2:hbpln o: if dd0;Q : : otherwise8:14In this model, 1, 2, and indicate the relativeimportance of sequencing group q immediately after groupp0(ortheimportanceofthefirst definitionofpheromone),importance of sequencing group q in the pth positionof thesequence(theimportanceof theseconddefinitionof pheromone), and heuristic information importance,respectively.3.1.5 Local searchBefore updating the pheromone valuesand after construct-ingthesolution, alocal searchcanbeappliedtothebestsolution. The integral part of every local search algorithm isthedefinitionof neighborhood. InHACO, threedifferentneighborhoods are used to improve the quality of solutionsand the computationalefficiency of each was tested:1. Swapping all possible pairs of groups (jobs)2. Removing a group (a job) and inserting in allpossible positions3. Permutations of all possible triplet of groups (jobs)After selecting the neighborhood definition, it is time toapply the local search. An innovative recursive local searchalgorithmisusedinHACOthatworksasfollows:first,ittries toimprove the givensolutionbyimprovingall itsgroup-wise neighboring solutions that have better objectivefunction value than the given solutionto their neighboringlocal optimumsolution (by means of calling the localsearchalgorithmrecursivelyforthesesolutions),andthenreplacesthegivensolutionwiththebest of thesegroup-wiseneighboringlocal optimumsolutions; second, it usesthe same approach, but this time byimprovingjob-wiseneighboring solutions of the updated solution to theirneighboringlocal optimumsolution. Ultimately, the bestsolution among the job-wise neighboring local optimumsolutions is returned as the output of the local searchalgorithm. The pseudo code of the recursive algorithm usedfor local searchin HACO is presented in Appendix 1.3.1.6 Pheromone updateIn ACS, two strategies are used to manipulate the pheromonevalues, inorder toguide the algorithm: local andofflinepheromone update. The former diversifies the search bydecreasing the pheromone of sequenced groups by each antduringtheconstructionphase(diversificationstrategy) andthe latter intensifies the search toward promising areas at theend of each iteration by the best ant (intensification strategy).Model (15) is used to perform local pheromone update:tpq1 :tpq :t0; 2 0; 1 15In(15), isthelocalpheromonedecaycoefficientand0 is the initial value of the pheromone. Model (16) is usedto perform offline pheromone update:tpq 1 r:tpq 1CBestAnt: ifbest ant uses edgei;jtpq : otherwise8