Constructiveand SimulatedAnnealing Algorithms for Hybrid Flow ShopProblems with UnrelatedParallelMachines Jitti Jungwattanakit, Manop Reodecha,PaveenaChaovalitwongse Departmentof Industrial Engineering,Faculty of Engineering, ChulalongkornUniversity,Bangkok
[email protected] Frank Werner Facultyof Mathematics,Otto-von-Guericke-University, P.O. Box 4l 20, D-390I 6 Magdeburg,
[email protected]
Abstract Most schedulingproblemsare combinatorialoptimizationproblemswhich are too difficult to be solved optimally, and henceheuristicsare used to obtain good solutionsin a reasonabletime. The specificgoal of this paperis to investigateschedulingheuristics,to seekthe minimum of a positively weighted convex sum of makespan,and the number of tardy jobs, in a static hybrid flow shop environment,where at least one production stage is made up of unrelatedparallel machines. In addition,sequence-and machine-dependent setuptimes are considered.Somesimple dispatchingrules and flow shop makespan heuristics are adapted for the sequencing problem under consideration. Then, this solution may be improved by a fast polynomial reinsertion algorithm. Moreover, a simulated annealing algorithm is presentedin this paper. Three basic parameters(i.e., cooling schedules,neighborhoodstructures,and initial temperatures)of a simulatedannealingalgorithm are briefly discussedin this paper.The performanceof the heuristicsis comparedrelativeto eachother on a setof testproblemswith up to 50 jobs and20 stages. Keywords: Hybrid flow shop scheduling; Constructive algorithms; Improvement heuristics; SimulatedAnnealingalgorithms. 1. Introduction This paper is primarily concerned with industrial schedulingproblems,where one first has to assignjobs with limited resourcesand then to sequencethe assignedjobs on each resourceover time. It is mainly concernedwith processing industries that are establishedas multi-stage production facilities with multiple production units per stage (i.e., parallel machines),e.g. a textile company(Karacapilidis and Pappis,[]), an automobileassemblyplant (Agnetis et al., l2l), a printed circuit board manufacturer(Alisantosoet al.,[3]), and so on. In such industries,at some stagesthe facilities are duplicated in parallel to increasethe overall capacities,or to balance the capacitiesof the stages,or either to eliminate or to reduce the impact of bottleneck stages on the shop floor capacities.The mixed characterbetween flow
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shopand parallelmachinesis known as a hybrid or flexible flow shopenvironment. Although the hybrid flow shopproblemhas beenwidely studiedin the literature,most of the studiesrelatedto hybrid flow shopproblemsare concentrated on problems with identical processors, seefor instance,Guptaet al.,[4] and Wang and Hunsucker [5]. In a real world sifuation,it is common to find newer or more modern machines running side by side with older and less efficient machines.Even though the older machines are less efficient, they may be kept in the productionlines becauseof their high replacementcosts.The older machinesmay perform the same operationsas the newer ones, but would generallyrequire a longer operating time for the same operation.In this paper, the hybrid flow shop problem with unrelated parallel machinesis considered,i.e., there are
ThammasatInt. J. Sc. Tech.. Vol. 12, No. l, January-March2007
heuristics, and then it is followed by the applicationof the First-In-First-Out(FIFO) rule. To obtain a near-optimal solution, metaheuristic algorithms have also been proposed.For example,Nowicki and Smutnicki [17] propose a tabu search(TS) algorithm for the hybrid flow shop makespan problem. Gourgandet al.,llSl presentseveralsimulated annealing (SA)-based algorithms for the hybrid flow shop problem. A specific neighborhood is used and the authors apply the methods to a realistic industrial problem. Jin et al. |91 propose two approachesto generatethe initial job sequence and use an SA algorithm to improve it. It can be seenthat the SA algorithm has been successfully applied to various combinatorial optimization problems. For an extensive survey of the theory and applications of the SA algorithm,seeKoulamaset al.,1201. In this paper, a hybrid flow shop problem with unrelatedparallel machinesand setuptimes is studied. The goal is to seek a schedulewhich minimizes,a positively weightedconvex sum of makespan and the number of tardy jobs. The constructive heuristics based on dispatching rules and pure flow shop makespan heuristics are adaptedand SA-basedalgorithms as iterative algorithmsare proposed. The rest of this paper is organized as follows: The problem under consideration is describedin Section 2. Heuristic algorithms are sketchedin Section3. Section4 and Section5 present the variants of the SA algorithm. Computational results with the heuristics are briefly discussedin Section 6 and conclusions are given in Section7.
different parallel machines at every stage and speeds of the machines are dependent on the jobs. Moreover, several industries encounter setup times which result in even more difficult schedulingproblems. A detailed survey for the hybrid flow shop problem has been given in Linn and Zhang 16l and Wang [7]. Most of the earlier literaturehas consideredthe simple case of only two stages. Arthanari and Ramamurthy [8] and Salvador [9] are among the first who define the hybrid flow shopproblem. They proposea branch and bound method to tackle the problem. However, it can only be applied to very small instances.Other exact approachesare proposedby many authors, e.g. Brah and Hunsucker[0] and Moursli and Pochet[1 l]. When an exact algorithm is applied to large problems, such an approach can take hours or days to derive a solution. On the other hand, a heuristic approach is much faster but does not guarantee an optimum solution. Gupta [12] proposesheuristic techniquesfor a simplified hybrid flow shop makespanproblem with two stagesand only one machine at stage two. The proposedheuristicsare basedon extensionsof Johnson's algorithm. Sriskandarajahand Sethi [13] developsimple heuristicalgorithmsfor the two-stage hybrid flow shop problem. They discussthe worst and averagecaseperformance of algorithms for finding minimum makespan schedules.Guinet et al., ll4] proposea heuristic for the makespanproblem in a two-stage hybrid flow shop. They comparethis heuristic with the ShortestProcessingTime (SPT) and the Longest ProcessingTime (LPT) dispatching rules. They concludethat the LPT rule gives good results for the two-stage makespan problem. Gupta and Tunc [5] consider the two-stage hybrid flow shop scheduling problem where there is one machine at stage one and the number of identical machinesin parallel at stagetwo is less than the total number of jobs. The setup and removal times of each job at each stage are separated from the processing times. They proposeheuristic algorithms that are empirically tested to determine the effectivenessin finding an optimal solution. Santos et al., tl6] investigateschedulingprocedureswhich seekto minimize the makespanin a static hybrid flow shop. Their method is to generate an initial permutation schedule based on the Palmer, CDS, Gupta and Dannenbring flow shop
2. Problem Statement The hybrid flow shop system is defined by a s e tO : { 1 , . . . , t , . . . , k } o f k p r o c e s s i nsgt a g e s . At each staget, t eO, thereis a set trtt = {1...., i,..., *'| of zt unrelatedmachines.The set-I: { 7 , . . . ,j , . . . , n } o f n i n d e p e n d e jnot b s h a s t o b e processed on machineof set M',..., M". Each job j, j eJ, has its release date 11> 0 and a due dated1 > 0. It has its fixed standardprocessing time for every stage t, t eO. Owing to the unrelated machines, the processing ttme p'i1 of jobT on machine i at stageI is equal to ps'i / v'4, where ps| is the standardprocessingtime of job / at stage t, and v';i is the relative speedof jobT which is processedby the machine I at stagel.
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There are processingrestrictionsofjobs as follows: (l) jobs are processed without preemptionson any machine; (2) every machine can processonly one operation at a time; (3) operations of a job have to be realized sequentially, without overlapping between stages;(a) job splittingis not permitted. Setuptimes consideredin this problem are classified into two types, namely machinedependent sefup time and sequence-dependent setup time. A setup time of a job is machinedependentif it dependson the machine to which the job is assigned. It is assumedto occur only when the job is the first job assignedon the machine. c&';; denotes the machine-dependent setuptime, (or changeovertime), ofjobT ifjobT is the first job assignedto machine I at staget. A sequence-dependentsetup time is considered jobs. A setup time of a job betweensuccessive on a machine is sequence-dependent if it depends on the job just completed on that machine. tttj denotes the time needed to changeoverfrom job I to job j at stage /, where job / is processeddirectly before job 7 on the same machine. All data are known and constant. The scheduling problem has dual objectives, namely minimizing the makespan and minimizing the number of tardy jobs. The objectivefunctionto be minimizedis: .7C-*+ (l - l)ry, where C.* is the makespan,which is equivalent to the completion time of the last job to leave the system, 77 is the total number of tardy jobs in the schedule,and 2 is the weight (or relative importance)given to C.* and ryr, (0 3 ).< 1). 3. Heuristic Algorithms Heuristic algorithms have been developec to provide good and quick solutions. They obtain solutions to large problems with acceptable computational times. They can be divided into either constructive or improvement algorithms. The former algorithms build a feasible solution from scratch. The latter algorithms try to improve a previously generated solution by normally using some forms of specific problem knowledge. However, the time required for computation is usually greater compared to the constructive algorithms. The drawback of heuristic algorithms is that they do not generateoptimality and it may be difficult to judge their effectiveness (Youssefel al.,l21l.
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3.1 Heuristic Construction of a Schedule Since the hybrid flow shop scheduling problem is NP-hard, algorithms for finding an optimal solutionin polynomialtime are unlikely to exist. Thus, heuristic methodsare studiedto hnd approximate solutions. Most researchers develop existing heuristics for the classical hybrid flow shop problem with identical machines by using a particular sequencingrule for the first stage.They follow the samescheme, seeSantoset al.,[16]. Firstly, a job sequence is determined according to a particular sequencing rule, and we will briefly discussthe modifications for the problem under considerationin the next section. Secondly,jobs are assignedas soon as possible to the machines at every stage using the job sequencedetermined for the first stage. There are basically two approaches for this subproblem. The first way is that for the other stages,i.e. from stagetwo to stageft, jobs are ordered according to their completion times at the previous stage. This means that the FIFO (First-In-First-Out)rule is used to find the job sequencefor the next stageby meansofthe job sequenceofthe previousstage.The secondway is to sequencethe jobs for the other stagesby using the same job sequenceas the first stage, called the permutationrule. Assume now that a job sequencefor the first stagehas alreadybeen determined.Then we have to solve the problem of schedulingr jobs on unrelated parallel machines with sequenceand machine-dependentsetup times using this given job sequencefor the first stage.We apply a greedy algorithm which constructsa schedule for the n jobs at a particular stageprovided that a certain job sequencefor this stage is known (the job sequence for this particular stage is derived either from the FIFO or from the permutation rule), where the objective is to minimize the flow time and the idle time of the machines. The idea is to balance evenly the workload in a heuristic way as much as possible. 3.2 Constructive Heuristics In order to determine the job sequencefor the first stageby some heuristics, it is noted that the processingand setup times for every job are dependenton the machine and the previous job, respectively.This means that they are not fixed, until an assignmentof jobs to machines for the corresponding stage has been done. Thus, for
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applying an algorithm for fixing the job sequencefor stageone, an algorithm for finding the representativesof the machine speeds and the setuptimes is necessary. of machinespeedv";i The representatives and setup time s"ry for stageI use the minimum, maximum and averagevalues of the data. Thus, the representativeof the operatingtime of jobT at stage I is the sum of the processingtime of the setuptime ps'i lv"11plusthe representative s"4. Nine combinationsof relative speedsand setuptimes will be used in our algorithms.The job sequencefor the first stageis then fixed as the job sequencewith the best function value obtained by all combinations of the nine differentrelativespeedsand setuptimes. For determiningthe job sequencefor the first stage, we adapt and develop several basic dispatchingrules and constructivealgorithms for the flow shop makespanscheduling problem. Some of the dispatching rules are related to criteria, while others are used tardiness-based mainly for comparisonpurposes. The ShortestProcessingTime (SPT) rule is a simple dispatchingrule, in which the jobs are sequenced in non-decreasing order of the processing times, whereas the Longest ProcessingTime (LPT) rule orders the jobs in non-increasingorder of their processingtimes. The Earliest ReleaseDate first (ERD) rule is equivalentto the First-ln-First-Out(FIFO) rule. The Earliest Due Date first (EDD) rule schedulesthe jobs accordingto non-decreasing due datesof thejobs. The Minimum Slack Time first (MST) rule concernsthe remainingslack of each job, defined as its due date minus its processingtime. The Slack time per Processing time (S/P) is the slack time divided by the processingtime required(Baker, [22]). Palmer's heuristic l23l is a makespan heuristicdenotedby PAL by proposinga slope order index to sequence the jobs on the machinesbased on the processingtimes. The idea is to give priority to jobs that have a tendencyof progressingfrom shorttimes to long times as they move through the stages. Campbell,Dudek, and Smith [24] develop one of the makespan heuristic methods known as CDS algorithm. Since Johnson'srule is a twostage algorithm, a ft-stage problem must be collapsedinto a two-stageproblem. In so doing, ft - I sub-problems are created and Johnson's rule is applied to each of the sub-problems.
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Then, a "best" sequenceis selected.Gupta [25] provides an algorithm denoted by GUP, in a similar manner as algorithm PAL by using a different slope index and schedulingthe jobs accordingto the slope order. Dannenbring[26] denotedby DAN developsa method by using foundation. Johnson's algorithm as a Furthermore,the CDS and PAL algorithmsare also exhibited. Dannenbringconstructsonly one two-stageproblem,but the processingtimes for the constructedjobs reflect the behavior of PAL's slopeindex. Nawaz, Enscore and Ham [27] develop a flow shop makespanheuristic,called the NEH algorithm.It is basedon the idea that a job with a high total operating time on the machines should be placed first at an appropriaterelative order in the sequence.Thus, jobs are sorted in non-increasingorder of their total operatingtime requirements.The final sequenceis built in a constructiveway, adding a new job at eachstep and finding the best partial solution. For example,the NEH algorithm insertsa third job into the previous partial solution that gives the under value objective function best consideration(the relative position of the two previous job sequenceremains fixed). The algorithm repeatsthe processfor the remaining jobs accordingto the initial orderingof the tota operatingtime requirements. To apply the algorithmsto this problem,the total operating times for calculating the job sequencefor the first stageare calculatedfor the nine combinations of relative speeds of machinesand setuptimes. The best solution is selectedfrom them. 3.3 Improvement Heuristics Improvement heuristics start with an already built scheduleand attempt to improve it by somegiven procedure.Their use is necessary since the constructive algorithms (especially some algorithms that are adapted from pure makespanheuristicsand some dispatchingrules such as SPT. LPT) do not considerdue dates. We will improve the overall function value concerningthe due date criterion. In order to find a satisfactorysolutionofthe given problem involving due dates,we use a fast polynomial heuristic by applying the shift move (SM) neighborhoodas an improvement mechanism. The SM neighborhoodrepositionsa chosen job. An arbitraryjob a, at positionr is shiftedto
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position l, while leaving all other relative job ordersunchanged.If 1< r < i < n, it is called a right shift andyields ft': (trt,..., nr_1, /++1,..., tri, 8,,..., ft,). If l< I < r < n, rt is calleda left shift a n d y i e l d s f r ' : ( t h , . . . f r , ,n t i , . . . , 8 - r , 7 T ,.+. .r,, t h ) . For example, assumethat one solution in the currentgenerationis selected,say [5 9 8 1 3 | 6 2 41, and then a couple of job positions for performing the shift is selected,e.g. positions2 and 7 (in this case,it is a right shift). The new solutionwillbe [5 8 7 3 1 6 9 24]. However,if positions 7 and 2 are selected(i.e. it is a left shift), the new solutionwill be [5 6 9 8 ] 3 | 2 4]. In the SM neighborhood,the currentsolution (S.",)has(n l)' neighbors. We apply the SM neighborhood by consideringonly jobs that are tardy in a left-toright scanand move eachof them left and right to all n-l possible positions (all shift-move algorithm). In each step, the best scheduleis selectedif it improves the objective function value. Since every job is consideredat most once (ifall jobs under considerationare late), at most O(n2)job sequencesare examinedby the improvementheuristics). 4. Simulated Annealing Heuristic A simulated annealing(SA) heuristic has been introducedby Kirkpatrick et al.,1281.lt is an enhancedversion of local optimization, in which an initial solution is repeatedlyimproved by making small local alterations,but an SA procedureoften acceptsa poor solutionto avoid being trappedin a poor local optimum. A basic SA algorithm startsfrom an initial solution s e S, and it generatesa new solution s'e ,Sin the neighborhoodof the initial solution s by using a suitableoperator.This new point's objective function value,(s') is then comparedto the initial point's value l(s) (the objective function value of the full schedule generated from the job sequence for the first stage is taken). The change in the objective function value, 6 : fls')-fls), is calculated. If the objectivefunction value decreases(d < 0), it is automaticallyacceptedand it becomesthe point from which the search will continue. If the objective function value increases(d > 0), then higher valuesof the objectivefunction may also be accepted with a probability, usually determinedby a function, exp (-ET), where 7 e fr is a control parameter of an SA algorithm
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called the temperature. The role of the temperature 7 is significant in the operation of an SA algorithm. This temperature,which is simply a positive number, is periodically reduced every 1y'2iterations, where NZ denotes the epoch length, so that it moves gradually from a relatively high value to near zero as the method progresses according to a function referred to as the cooling schedule.In our tests, we investigatedin particularthe influenceof the chosen neighborhoodand the cooling scheme for controlling the temperature.We used a geometric(i.e., 7,".: axT,,u)and a Lundy-Mees reduction l29l (i.e., T,n, - T"Hl(l+|I.d) scheme and tested the parameters of these schemes (initial temperatures, temperature reductionsand neighborhoodstructures). Concerning the neighborhood, we considered both an SM neighborhood (see Section 3.3) and a pairwise interchange(PI) neighborhood.The idea for the PI neighborhood is to exchangea pair ofarbitrary jobs, a and t4, where I I i, r < n and i *r. Such an operation swapsthe job at position r and one at position i, which yields fr'-- (tr,,..., E-t, /ti,/t,+t,..., Ei-t,t, lTi+t,..., n ). For instance. assume that the current solution is [5 9 8 ] 3 1 6 2 41, and then randomly the couple of job positions to be exchangedis selected,e.g. positions I and 3. Thus, the new solutionwill be 18 9 5 7 3 | 6 2 4]. For the selectionof a neighbor, one of all possiblenx(n-1)12PI neighborsis checkedand then comparedto the starting one. 5. Choice of an initial solution An SA algorithm has been shown to be effective for many combinatorial optimization problems (see Koulamas et al., [20]), and it seems easy to apply such an approach to schedulingproblems.To improve the quality of the solution finally obtained, we also investigatedthe influence of the choice of an appropriate initial solution by using particular constructive and fast improvement algorithms. We used one constructivealgorithm of: SPT, LPT, ERD, EDD, MST, S/P, PAL, CDS, GUP, DAN and NEH; we also used another fast improvementheuristicsas an initial solution.
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constructive heuristics. We used problems with l0jobs x 5 stages,30jobs x l0 stages,and 50 jobs t 20 stages.For all problem sizes, we testedinstanceswith l" e {0, 0.05, 0.1, 0.5, and l) in the objective function. Ten different instancesfor eachproblem size have been run. The results for the algorithms from Section 3 are given in Table 1. We give the average (absolutefor l" : 0 and percentagefor 1,> 0) deviation ofa particular algorithm from the best solution in thesetests for all problem sizesnx k (the overall best variant is given in bold face).
6. Computational results Firstly, we studied the algorithms from Section 3 (determination of an initial solution for SA) which are separated into four main goups. The first heuristic group includes the simple dispatching rules such as SPT, LPT, ERD, EDD, MST, and S/P.The secondheuristic group contains the flow shop makespan heuristicsadaptationsuch as PAL, CDS, GUP, DAN, and NEH. The third and fourth heuristic groups are generated from the first two heuristics by applying additionally an all-shiftmove algorithm (see Section3.3), and they are "I" before the letters for the denotedby the letter
Table I Average perfonnanceof constructiveand fast improvementalgorithms ?,
DAN NEH GUP PAL CDS S/P MST EDD ERD 2.O 0.5 1.8 1.7 1.9 3.0 32 2.8 3.1 1.7 2,4 7.7 6.4 7.8 8.0 t2.2 12.4 t2.3 8.9 8.3 30xl0 9.3 2.3 7 7.3 t4.3 9.7 t6.2 16.2 7.7 8.6 50x t t t'7.5 15.4 19.6 31.7 18.8 31.7 t9.2 Sum 2'52 11.66 10.03 14.12 11.47 12.81 24.23 24.09 22.21 22.10 I0x5 3 0 ^ 1 0 | 7 . 7 8 | 4 . 7 2 1 9 . 6 1 2 0 . 9 | 1 8 . 2 1 | 7 . 6 | 1 6 ' 8 0 | 2 . 3 5 | 4 . 1 1 | 4 ' 7 7 0 . 5 9 8.43 0.30 8.03 6.99 7.90 9.87 10.14 11.75 10.96 8.28 s0,20 8.53
Problem size
u u)
ut
SPT
LPT
10x5
2.3^ 8.0 7.4
30\10 50"20
16'61 8.13
|3.|2 7.75
18.46 9]2
19.|4 10.4i
16.38 9.65
|5'7| 8.77
15.59 7.27
||.|7 6.45
13.30 7.5'7
13.61 7'79
0.40 0'08
30x10 50x20 Sum
16.12 8.11 qt:tt n.qg 16.34 8.12 41.94
12.29 7.59 tru t].1l 12.46 7.58 31.24
18.01 9.68 qs.za 22.ti 18.24 9.69 50.10
18.13 9.1r 49.78 2r.94 18.29 9.63 49.85
15.30 8.84 42.94 18.81 15.43 8.75 42'98
14.46 8.12 4t.79 19.21 14.57 8.05 41.82
15.03 704 32.41 10.33 15.24 7.03 32.59
10.50 6.28 24.65 7.87 10.68 27 24'82
12.60 7.50 32.s5 12.44 12.77 7.50
13.04 156 30.34 9.74 1324 7.54 30.52
0.26 0'09 3.33 2.98 0.38 0.08 3.44
ISPT
ILPT
IERD
IEDD
IMST
IPAL
ICDS
IGUP
IDAN
INEH
0.7 4.5 4.3
1.4 4.3 3.5
0.7 3.2 3.5
1.0 2.4 3.9
0.6 4.4 4.6
0.5 4.0 4.5
0.9 4.1 4.8
l0 4.4 5.3
0.5 2.4 2.3
5.06 6.03 4.46 15.55 4.63 6.10 4.27 t4.99 4.31 6.13 4.37 14.81
3.60 6.66 5.22 t5.49 3.95 6.t4 5.22 t5.31 2.84 6.10
6.84 8.00 6.03 20.88 6.40 7.94 5.56 19.89 6.01 7.85 5.28 1 9l.4
2.64 7.07 4.22 13.93 1.80 6.23 3.78 I l.8l t.1t 5.19 4.06 10.96 1.71 5.29 4.05 I 1.05
3.63 6.84 4.83 15.30 3.20 5.57 4.88 13.66 2.55 4.88 4.79 12.22
3.46 6.24 5.38 15.08 |.94 5.51 4.73 t2 . l 8 0.80 5.95 4.61 I 1.36
2.84 6 . 18 5.01 t4
5.32 I l.l5 s.66 22.14 5.78 9.93 5.66 21.36 5.80 9.18 5.44 20.42 5.80 9.33 5.45 20.58
3.39 4.78 3.53 I 1.70 2.0\ 2.78 3.15 7 2.24 1.65 2.92 6.82
4.31 6.33 4.39 15.04
3.82 7.75 5.10 t6.67 4.60 8.39 4.78 t7.78 4.91 8.94 4.81 18.66 4.91 9.17 4.91 18.98
2.24 r.69 2.93 6.86
2.55 4.98 4.77 12.31
0.80 5.95 4.59 11.34
2.52 0.59 0.30 3.41 2.86 0.40 0.08 3.33 2.98 0.26 0.09 3.33 2.98 0.38 0.08 3.44
- - - _ | 0.5 -ro's 1.0
l,
30x10 50x20 Sum
Problem size
t0'5 30xl0
50x20
0.05
0.1
0.5
l0x5 30x10 50x20 Sum l0x5 30x10 50x20 Sum l0x5 30x10 50x20 Sum l0x5
1.0
30x10 50x20 Sum
t.0 4.2 3.6
13.66
6.01 7.72 5.17 18.91
0.9 5.7 7.8 5.43 8.29 6.28 19.99 4.48 7. 8 3 4.61 16.93 3.98 7.20 4.73 15.91 3.98 7.40 4.09 t5.47
o n averageabsolutedeviation for )" = 0, and averagepercentagedeviation for l"> O
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Thammasat Int. J. Sc. Tech., Vol. 12, No. l, January-MarchzOOj
From these results it is obvious that the algorithms in the fourth heuristic group (i.e., IPAL, ICDS, IGUP, IDAN, and INEH) can improve the pure makespanheuristics from the secondheuristic group (i.e., PAL, CDS, GUP, DAN, and NEH), and they are better than the dispatchingrules in the first heuristic group (i.e., SPT, LPT, EDD, MST, and S/P) as well as the third heuristic group improved from them. Among the simple dispatching rules (heuristic Group I), the SPT rule outperforms the other dispatching rules for I : 0, and the LPT rule is better than the other rules for l" > 0. Among the adapted flow shop makespan heuristics in the heuristic Group II, the NEH algorithm is clearly the best algorithm among all The CDS studied constructive heuristics. algorithm is certainly the secondrank algorithm, whereasthe remaining algorithms differ slightly from each other. When we apply a fast (re-)insertion algorithm (denoted by the letter "I" first) to the dispatching rules and adapted makespan heuristics, we have found that the quality of the solution can be improved by about 50-70 percent except for the NEH rule. It is noted that the NEH rule is not improved by using the improvement heuristics in algorithm INEH because the NEH algorithm is embedded by such an (re-) insertion algorithm itself. However, the improvement of the heuristics from the adapted pure makespan heuristics in the heuristic Group IV is better than the
improvement of the heuristics derived from the dispatchingrules in the heuristic Group IIL Secondly, we studied the SA algorithm with a random initial solution. The purpose of this study is to determine the favorable SA parameters, i.e., initial temperatures (100 through 1000, in steps of 100), neighborhood structures (PI and SM), and cooling schedules (CSl - CS3 refer to a geometric reduction schedulewith a e {0.85, 0.90, and 0.95}, and CS4{S6 are the schedulesby Lundy and Mees with p e {0.0005,0.001,and0.002}). Given the above three different problem sizes, the SA parameter values were tested. From our preliminary tests,we set the time limit equal to one second for the problems with ten jobs, ten secondsfor the problemswith 30 jobs, and 30 secondsfor the problems with 50 jobs. Table2 throughTable 4 presentthe effect ofthe initial temperatures, neighborhood structures and cooling schedules by using the average (absolute resp. relative) deviation from the best value as the performancemeasure. From the full factorial experiment, we analyzedour results by means of a multi-factor Analysis of Variance (ANOVA) technique using a 5% significance level. We have found that for neighborhood structuresand cooling schedules, there are statistically significant differences, whereas there are not statistically significant differences in the initial temperatures. A low initial temperatureis however slightly preferable (we recommend 100). It can be observedthat PI
Table 2 The effect of variousinitial temperatures on the performanceof the SA aleorithm l.
Problem size l0x5 30x10 50x20 l0x5 30xl0 50x20
I0x5 30x10 50x20 I 0x5 30xl0 50x20
t.0
l0x5 3 0 x1 0 50x20
100 0.019" 2.747 2.461 5.227 1.954' 7.662 3.901 13.517 | .707 6.126 3.440 r.273 0.850 3.723 2.t25 6-698 0.s13 3.337 1.761 s.511
200 0.022 2.781 2.531 5.334 2.140 7.727 4.010 13.877 1.647 6.13't 3.446 11.230 0.884 3.781 2.240 6.905 0.641 3.392 1.837 5.870
300 0.019 2.767 2.561 5.347 2.171 '7 .979 4.100 14.250 L840 6.218 3.535 l1.593 0.873 3.814 2.285 6.972 0.633 3.470 1.847 5.950
400 0.011 2.781 2.561 5.353 2.079 7.8t6 4.L61 14.056 1.917 6.237 3.596 l1.750 0.947 3.898 2.3t2 7.157 0.653 3.452 1.915 6.020
500 0.025 2.792 2.539 5.356 2.010 7.880 4.117 14.007 1.922 6.218 3.608 11.748 0.962 3.931 2.360 7.253 0.690 3.504 1.924 6.118
600 0.019 2.822 2.603 5.444 2.195 7.809 4.t4s t4.t49 L864 6.254 3.652 tl.770 0.959 3.909 2.404 7.272 0.726 3.497 1.987 6.2t0
700 0.019 2.811 2.628 5.458 2.195 7.925 4.151 14.27t I .839 6.304 3.626 11.769 0.968 3.947 2.381 7.296 0.705 3.520 1.942 6.167
u averageabsolutedeviation for 2,= 0, and b averagepercentagedeviation for 2 > 0
800 0.017 2.839 2.608 5.4& 2.192 7.877 4.232 14.301 | .969 6.291 3.658 l1.918 I .048 3.926 2.377 7.15t 0.756 3.554 t.963 6.273
900 0.022 2.864 2.653 5.539 2.26t 7.770 4.t97 14.228 1.895 6.386 3.750 12.031 |.025 3.915 2.414 7.354 0.721 3.546 2.007 6.274
1000 0.025 2.883 2.681 5.589 2.251 7.838 4.277 t4.366 1.893 6.361 3.675 11.929 1.030 3.974 2.338 7.342 0.684 3.534 1.953 6.171
ThammasatInt. J. Sc. Tech., Vol. 12, No. l, January-March2OO7
Table 3 The effect of various neighborhood Table 4 The effect ofvarious cooling scheduleson the structures on the performance of the SA performanceof the SA algorithm algorithm ^
Problem sze
I0x5 3 0 xl 0 50x20
0.05
0.1
0.5
1.0 _
u
I 0x5 3 0 xl 0 50x20 I 0x5 3 0 xl 0 50x20
I0x5 30x10 50x20 I0x5 30x10 50x20 Sum
?' 0.016' 2.794 2.522 5.332 2.270' 8.098 4.192 I4 . 5 6 0 1.973 6.522 3.646 1.1 36 4.249 2.425 7.810 0.865 3.891 2.049 6 . 8 1I
average absolute deviation
Problem csl sze
cs2
0.002 0.024 0.915 2.823 0.638 2.643 1.555 5.490 1.052 2.020 3.443 7.522 uu) 4.411 4.067 8.906 13.609 0.959 1.725 2.839 5.985 ul r.2q5 3.551 5.093 0.636 0.773 2.560 3.515 n< ) ))1 1.168 4.364 6.510 0.546 0.479 2.700 3.06s Ll51 1.778 4.403 5.322 _ b for ) : 0 and average percentage deviation for )' > 0 10x5 30x10 50x20 Sum l0x5 l0- l0 50.20 Sum l0x5 30.I0 50.20 Sum l0x5 10.l0 s0.20 Sum l0x5 10.r0 50. 20 Sum
moves are better than SM neighborhoodsfor )" : 0, whereasSM neighborhoodsare better than PI moves for the other values. Consequently,the neighborhoodstrucfuresshould be basedon PI moves for )" - 0 and on SM neighborhood otherwise.For the cooling schedules,we have observed that a geometric cooling scheme outperforms the other cooling schedules.In particular,the reductionschemeTnn:0.85xTo14, where 7,n,,, and Tuladenote the new and old temperatures, can be recommended. Finally, we used the recommended SA parametersto test the choice of an appropriate initial solution.The lettersbefore SA denotethe heuristic rule for finding an initial solution for the SA algorithm. For example,SPTSA means that the SPT rule is usedas an initial solutionfor the SA algorithm. From these results in Table 5, we have found that there are no statistically significant differences when using different initial solutions. We have however found that the IEDDSA rule is a good algorithm for problems with ),: 0, and the NEIISA and INEHSA rules are slightly better than the others for problems with )" > 0. Consequently,in general the NEHSA aru INEHSA algorithms are good , r,oicesfor the SA algorithmwith using a biased initial solution.
38
0.000 0.653 0.320 0.973 0.936 J.388 r.0s9 5.383 0.855 2.741 r.000 4.596 0.658 2.450 0.9s9 4.067 0.590 2.754 0.963 4.306
cs3 0.028 1.663 1.880 3.571 1.930 4.6\4 3.101 9.645 1.561 3.725 2.620 7.906 0.920 2.940 1.969 5.829 0.182 3.102 1.827 5111
cs4 0.033 4.625 4.237 8.895 3.142 t2.314 6.559 22.015 2.7'70 10.487 6.008 19.265 1.'755 6.833 4.429 13.017 1.122 5.051 3.167 9.340
cs5 0.022 4.562 4.283 8.867 3.043 12.034 6.472 21.549 2.696 9.717 5.758 l8.l7l Ll49 4.816 3.103 9.068 0.628 3.885 2.400 6 913
0.035 4.433 4.137 8.605 2.767 I 1.068 6.t'71 20.006 2.255 8.01I 4.9t1 15177 0.610 3.693 2.313 6.616 0.364 3.391 1.968 5.726
7. Conclusions In this paper, we have investigatedboth (SA-based) iterative constructive and convex minimizing a approaches lor combination of makespanand the number of tardyjobs for the hybrid flow shopproblemwith unrelated parallel machines and setup times, which often occurs in the textile industry. A11 algorithms are based on the list scheduling principle by developingjob sequencesfor the first stage and assigning and sequencingthe remaining stages by both the permutation and FIFO approaches.The constructivealgorithms are comparedto each other. lt is shown that the NEH and CDS algorithms outperform the others, respectively. In particular, the NEH algorithm is most superior to the other regardless of constructive algorithms improvementheuristics.After the applicationof the fast improvement heuristics, the INEH algorithm basedon the NEH rule is still better thanthe otheralgorithms. In addition, we have used SA-based algorithmsas improvementalgorithms. Before we studiedthe influence of the initial solution on the performance of the SA algorithm, we tested the SA parameters, i.e., initial temperatures, neighborhood structures, and cooling schedules. We have found that a low initial temperatureis slightly preferable (we recommend100). The neighborhoodstructures should be basedon PI moves for )" - 0 and on
SM neighborhoodsotherwise. The geometric cooling scheme7,"-:0.85x 7,7,,is recommended. For the recommended SA parameters, we investigated the selection of a starting solution by using several constructive algorithms. The variants NEHSA and INEHSA can both be in general. recommended Further research can be done to use other iterative algorithms such as tabu search,genetic
algorithm,or ant colony algorithms. The choice of good parametersfor them should be tested. In addition, the influence of the starting solution should be investigated. Moreover, hybrid algorithms should be developed by using simulated annealing as a local searchalgorithm within a genetic algorithm or the other algorithms.
Table 5 Comparisonof the SA aleorithmwith differentinitial solutions Problem
30x10 50x20 Sum
0.05
0.1
0.5
t.0
l0x5 30x10 50x20 Sum l0x5 30x10 50x20 Sum l0x5 30x10 50x20 Sum l0x5 30x10 50x20 Sum
Problem size
0.05
0.1
t'.J
1.0 _
I0x5 30x10 50x20 Sum l0i5 30x10 50x20 Sum l0x5 30x10 50x20 Sum l0x5 3 0 xl 0 50t20 Sum l0x5 3 0 xl 0 50x20 Sum
SPTSA
LPTSA
0.84 0.34
0.76 0.3
ER,DSA
EDDSA
MSTSA
S/PSA
0.82 0.32
0.76 0.30
0.76 0.34
PALSA CDSSA GUPSA DANSA NEHSA
l. 14
1.06
1.10
0.70b 2.43 1.09 4.22 0.64 2.10 0.86 3.59 0.48 1.99 0.97 3.44 0.40 2.38 0.83 3.61
0.38 2.69 1.07 4.14 0.51 2.37 0.90 3.78 0.39 2.05 0.94 7 0.33 2.02 0.92 3.27
0.72 2.83 l.l2 4.67 0.39 2.41 0.92 3.72 0.30 1.82 0.83 2.9s 0.34 2.29 0.80 3.43
0.45 2.53 l.2l 4.19 0.43 2.22 0.96 3.60 0.39 1.98 0.79 3.16 0.24 2.08 0.81 3.13
0.59 2.70 1.06 4.35 0.65 2.20 1.03 3.88 0.33 1.88 0.75 2.96 0.23 2.34 0.90 3.48
0.60 2.63 1.25 4.48 0.55 2.0'l 0.96 3.58 0.34 2.06 0.90 3.30 0.38 2.10 0.85 3.33
0.56 2.79 l.l5 4.49 0.50 2.36 1.08 3.94 0.27 2.10 0.88 3.25 0.35 1.94 1.02 3.30
0 0.78 0.28 1.06 0.63 2.63 1.16 4.42 0.44 1.96 0.90 3.31 0.36 1.79 0.80 2.96 0.30 2.08 0.86 3.24
ISPTSA
ILPTSA
IERDSA
IEDDSA
IMSTSA
IS/PSA
IPALSA
ICDSSA
0 0.82 0.36 l.t8 0 . 5I 2.51 1.17 4.24 0.59 2.14 0.94 3.67 0.28 1.85 0.83 2.96 0.30 2.15 0.93 3.38
0 0.78 0.32
0 0.88 0.34 1.22 0.59 2.64 1.30 4.52 0.55 2.12 l.t9 3.86 0.33 2.06 0.78 3.17 0.16 2.25 0.83 3.25
0 0.68 0.12 o.8o 0.49 2.81 l.l5 4.45 0.42 2.45 1.02 3.90 0.32 2.24 0.81 3.38 0.24 2.05 0.90 3.18
0 0.78 0.42 l.2o 0.38 2.36 l.l7 3.91 0.64 2.16 1.02 3.82 0.32 2.07 0.83 3.22 0.27 2.15 0.82 3.24
0 0.86 0.46
0 0.86 0.50
0 0.12 0.34
0 0.14 0.32
|.32 0.61 2.38 l.l6
1.36 0.54 2.54 l.l3 4.22 0.70 2.04 0.99
L06 0.45 2.82 l.lt 4.38 0.48 2.22 1.06
1.06 0.62 2.39 Llr 4.tl 0.44 2.03 1.04 l.5l 0.37 L87 0.89 3.13 0.21 2.15 0.88 3.24
l. I8
1.06
l.lo 0.54 2.60 l.l2 4.26 0.65 2.21 0.91 3.77 0.38 1.70 0.83 2.91 0.28 2.09 0.81 3.19
0.86 0.44 L3
^L^^l--.^ l^-,:-1:^r^^ . ^ : bo ^ - , ^ - ^ ^ ^ -^-.--. a.uir'ti* r* ,lJ -- n0. und **ge-p"r."ntu-g. u*.ug. ubrotr,.
a--,-,^^^
39
4.14
0.36 2.41 1.05 3.82 0.25 2.09 0.80 0.26 2.30 0.89 3.45
0 0.76 0.36
).t)
J./o
0.40 | .93 0.90 3.23
0.42 L93 0.96 3 . 3I
1.88 0.84 3.11
2.07 0.90 3.29
d.uiuti* fo,.I t 0
0.84 0.38 t.22 0.66 2.39 1.06 4.1I 0.59 2.33 l.l0 4.02 0.34 1.94 0.86 3.14 0.27 2.ll 0.96 3.34
0.78 0.44 1.22 0.52 2.51 l.0l 4.10 0.55 2.10 1.13 3.78 0.43 2.10 0.77 3.29 0.28 2.31 0.73 3.33
0.82 0.38 1.20 0.52 2.33 l.l I 3.96 0.48 2.00 0.80 3.28 0.43 1.70
IGUPSA
IDANSA
INEHSA
0 0.72 0.42 l.l4 0.46 2.76 t.2t 4.43 0.66 2.37 1.04 4.07 0.28 1.90 0.89 3.07 0.22 |.94 0.83 2.98
0 0.'74 0.40
2.76 0.36 2.05 0.66 3.07
l.l4 0.50 2.32 1.09 3.91 0.37 2.04 0.82 3.23 0.37 1.51 0.63 2.52 0.37 2.03 0.67 3.07
ThammasatInt. J. Sc. Tech.. Vol. 12. No. 1. January-March2OOj
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