INTEGRATIVE COOPERATIVE APPROACH FOR SOLVING PERMUTATION FLOWSHOP SCHEDULING PROBLEM WITH SEQUENCE DEPENDENT FAMILY SETUP TIMES

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1 8 th International Conference of Modeling and Siulation - MOSIM 10 - May 10-12, Haaet - Tunisia Evaluation and optiization of innovative production systes of goods and services INTEGRATIVE COOPERATIVE APPROACH FOR SOLVING PERMUTATION FLOWSHOP SCHEDULING PROBLEM WITH SEQUENCE DEPENDENT FAMILY SETUP TIMES R. Bouabda, M. Eddaly FSEGS Route de l aéroport Sfax -Tunisie radhouan.bouabda@gail.co, eddaly.ansour@gail.co B. Jarboui, A. Rebai FSEGS Route de l aéroport Sfax -Tunisie basse_jarboui@yahoo.fr, abdelwaheb.rebai@fsegs.rnu.tn ABSTRACT: This paper addresses to the perutation flowline anufacturing cell with sequence dependent faily setup ties proble. The objective is to iniize the aespan criterion. We propose a cooperative approach including a genetic algorith and a branch and bound procedure. The latter is probabilistically integrated in the genetic algorith in order to enhance the current solution. Moreover, the application of the branch and bound algorith is based upon the decoposition of the proble into subprobles. The perforance of the proposed approach is tested by nuerical experients on a large nuber of representative test probles. KEYWORDS: Branch and bound; Genetic algorith; Flowline anufacturing cell; Setup ties; Maespan 1 INTRODUCTION Since the early 1960 s, the developent of Cellular Manufacturing (CM), as one of Group Technology (GT) subsets, has received uch attention of practitioners and researchers in the anufacturing systes. The ajor issues are the reduction of the coplexity and the increase of the productivity of job shops (Greene and Sadowsi, 1984). Operationally, the CM consists in arranging a set of siilar jobs into part failies. These siilarities concern the jobs setups, achinery requireents and operations. In (Greene and Sadowsi, 1984) any industrial applications have been presented in CM environents such that chip producing, etal fabricating, and anual assebly industries. Hendizadeh et al. (2008) have presented a detailed literature review of scheduling in cellular anufacturing systes. In this paper we consider the scheduling of CM environent while processing the jobs with the sae setup faily together. In each faily, there is a set of n jobs that ust be processed on a set of achines in the sae order without pre-eption (no interruption is allowed). Such cells are nown as flowline cells or pure flowshop anufacturing cells (Schaller et al., 2000). This proble occurs when the tie required to shift fro one part (job) to another is negligible or included in the processing ties. However, the tie required to shift fro one faily to another is high and cannot be neglected. A coprehensive review can be found in (Cheng et al.,2000) for the flowshop scheduling probles with setup ties. They have included the proble under consideration in sequence dependent faily setup ties (SDFST) category. Schaller et al. (2000) have presented a real world application in the anufacturing printing circuit boards where the use of CM technology is highly required. Garey and Johnson (1979) have showed that this proble is strongly NP-hard with respect to the aespan criterion. Schaller et al. (2000) have developed lower bounds for solving this proble. The obtained results have showed that the branch and bound is efficient for the sall test probles. However, when the nuber of failies or the nuber of achines increases, the perforance of the proposed lower bounds decreases. Also, several constructive heuristics have been proposed in (Schaller et al., 2000). The coputational results have showed that, for scheduling the jobs within failies, the called "CDS" heuristic (Capbell et al., 1970) is the best one. Besides, the NEH heuristic (Nawaz et al., 1983) is the ost efficient regarding the job failies scheduling. França et al. (2005) have presented two evolutionary algoriths: a Genetic Algorith (GA) and a Meetic Algorith (MA). The coputational results showed that these two algoriths outperfor existing best heuristic. In addition MA presents a slight superiority according to the GA. Hendizadeh et al. (2008) have proposed a Tabu Search algorith (TS). They have eployed the Siulated Annealing algorith (SA) concepts for balancing between intensification and diversification of the search s direction. Their results have showed that the proposed algorith, in average, outperfors the constructive heuristic of Schaller et al., (2000) in ters of solution quality but not in ters of CPU ties. Moreover, the TS algorith and the MA of França et al. (2005) appear siilar in average. Lin et al. (2009) have proposed three different etaheureustics including SA, GA and TS for solving this proble. It s noted that the proposed SA and GA outperfor MA of and França et al. (2005) and TS of Hendizadeh et al.

2 MOSIM 10 - May 10-12, Haaet - Tunisia (2008) both regarding the solution quality and the CPU ties. In this paper, our purpose is to develop an efficient integrative cooperative approach based on a Genetic Algorith and a Branch and Bound procedure. The latter is eployed as an iproveent procedure for the evolutionary algorith. Our idea is to prove that, for this proble, the introduction of the exact algorith into the evolutionary are ore efficient than the cooperation with local search ethods. 2 PROBLEM FORMULATION The focus of this research is to schedule a perutation flowline anufacturing cell with sequence dependent faily setup ties where it is desired to iniize the aespan. The ajor assuptions of this study are: All jobs are available for processing at the start tie. Nuber of parts, their processing ties, the nuber of part failies, and their setup ties are nown in advance and are non-negative integers. Every part can be processed by at ost one achine at any given tie without preeption. Each achine can handle only one part at a tie and is persistently available to process all scheduled parts when required. Both the sequence of part failies and parts within each faily are the sae on all achines. All jobs in the current faily ust be finished before switching failies. The faily setup ties are sequence dependant. A changeover between jobs within a part faily requires inor setup tie and can be included in the processing ties of each job. Switching between part failies requires a ajor setup and hence requires an explicit treatent of setup ties. For a foral definition of the proble consider the following notation. M Nuber of achines. K Nuber of failies. Index for achines ( = 1, 2,..., M ). f Index for failies ( f = 1,2,..., K). N f Nuber of jobs in faily f. J Index for jobs ( j = 1,2,..., N f ). J f Set of jobs in faily f. s Setup tie for faily f processed iediately after rf 0 1 [ ] faily r at achine. S Setup tie for the first faily processed on achine. P Processing tie of job j in faily f on achine. fj π π π π π is = { [ ], 1 [ 2],..., [ ]} is a set of sequence where K the job sequence within the faily sequenced in position f of π. δ The job sequenced in position j of the job sequence f [ j] for faily f. C( δ, ) The copletion tie on achine of the [ j] job sequenced in position j of the job sequence for the faily sequenced in position f ofπ. C( δ [ ][ ],0) = 0 for j = 1,2,..., N f j and f = 1,2,..., K. C(0, ) = 0 for = 1,2,..., M. { δ } [ ][ ] C( δ, ) = ax C(, 1), S + P for [ ][ ] [ ][ ] [ ] = 1,2,..., M. C( δ[ ][ ], ) = ax C( δ 1 [ ][ 1 ], 1), C( δ[ 1 ], ) + S f f f N [ ] + P [ 1] 1 f f 1 f for f = 2,3,..., K. C( δ, ) = ax C( δ, 1), C( δ, ) + P [ j] [ j] [ j 1] f j for j = 2,3,..., N and f = 1,2,..., K. The objective of the proble is to find the π that results δ K N [ K ] in the iniu C( [ ], M ) 3 THE PROPOSED APPROACH Recently, a new wave of optiization tools has been arisen which attepts to cooperate between several algoriths of resolution. Two ways of cooperation or hybridization have been proposed on the literature based on the ind of the algoriths. We find the cooperation between approxiation algoriths, in such way two etaheuristics are coupled with each others and cooperation between etaheuristics and exact algoriths. Blu (2010) and Lozano and Martinez (2010) have addressed to the type of hybridization between different etaheuristics in a Special issue in Coputers and Operations Research (Blu, 2010). In the other hand, Jourdan et al., (2009) have presented the different approaches of cobining exact algoriths and etaheuristics to solve cobinatorial optiization probles. We propose a cooperative approach that cobines the Branch-and-Bound and genetic algorith ethods to solve the anufacturing cell scheduling proble considered here. First, the proble is tacled with a genetic algorith then a Branch-and-Bound (B&B) enhanceent procedure is proposed to iprove the genetic algorith perforance. 3.1 The Genetic algorith In what follows, we present the fraewor of our proposed algorith Solutions encoding and population initialization The used encoding schee can be represented erely by 1+ K vectors where K denotes the nuber of failies. The first vector corresponds to the perutation order of part failies processed on achines and the last K vectors represents the perutation order of parts

3 MOSIM 10 - May 10-12, Haaet - Tunisia within each part faily. We choose to generate each chroosoe in the initial population of size P randoly in our experients Selection In our algorith, the selected individuals (parents) are chosen randoly fro the initial population Crossover In our case we use two-point crossover operator which can be described as follows: the parents are divided at a rando into two points that is nevertheless identical for both parents. The content outside the selected two points is constantly transferred fro the first parent to the children whereas the contents inside is taen fro the second parent in the sae order as they are showed fro left to right. For the faily sequence, this operator is used at each iteration of our genetic algorith where the nuber of faily is greater than 3, i.e. the probability crossover of the faily sequences is equal to 1. However, within each faily, the sequences of jobs are also recobined following this fraewor: a rando nuber in the range [0.1] is drawn. If this nuber is greater than or equal to a job given probability pcross and the nuber of jobs exceeds 3 then we apply the crossover operator described above. If the rando nuber is greater than p and the nuber job cross of jobs in the sequence is less than 3 then the sequence of the jobs of the children tae the sae sequence of the first parent. Otherwise, a rando sequence of jobs is generated Mutation The second operator used by genetic algorith is the utation operator. The ai of utation is to escape fro the trap of local optiu by introducing faint odification in the sequence. In this way, utation helps genetic algorith to explore the whole search space and thus preserving population diversity. In our case, we use swap utation which consists of choosing two rando points in the sequence of faily and perforing all possible swap oves in the section between these two points. Also, the siilar operator is used for each sequence of jobs within faily. 3.2 The B&B procedure The solution π of our proble obtained by genetic algorith tae the following structure { [ ], [ ],..., [ ], [ ], [ 1],..., f f f + [ K] } π = π π π π π π where {,,..., } π = δ δ δ is the sequence of job [ 1] [ 2] N within the faily sequenced in the position f of π. For each solution generated by the genetic algorith, we define a probability α which indicates if the enhanceent procedure will be applied. This probability depends on the quality of the solution where the individuals with good quality will be favored. So, in our paper, this probability is deterined as follows: incubent best < α best where "incubent" designates the current solution provided by the genetic algorith and "best" denotes the best solution obtained so far. This restriction is considered as profitable in ters of coputational tie of the algorith when only a part of individuals will be subjected to the application of the branch and bound algorith. In the literature of hybrid evolutionary algoriths, this ind of probability is used by Jarboui et al. (2008) as a threshold acceptance and in Ruiz et al. (2006) is called enhanceent probability. In order to iprove the solution obtained by the genetic algorith we re-optiize the partial schedule within each faily f, π, by the branch and bound algorith. Therefore, the latter is based upon decoposition of the whole proble into subprobles i.e. by considering one proble into each faily. In such way, in each faily, we consider the proble as perutation flowshop scheduling proble with achine availability constraint. In our study, the branch and bound algorith can be considered as an intensification operator. We start by sequencing the jobs within the faily sequenced in the first position then to the jobs within the faily sequenced in the second position and so on. The branch and bound algorith is used only for the sequence of jobs of only one faily and the reaining K-1 sequences of jobs are already fixed by the genetic algorith. It should be noted that the probability α leads to guarantee a good partial schedule when the fixed part of the solution is provided following to a certain deviation according to the best solution. It is possible to perfor branch and bound algorith several ties to optiise the schedule of jobs within the sae faily, until no iproveent can be attained. The proposed branch and bound algorith is used for finding optial job sequences within a faily sequenced in position f of π, f = 1,2,..., K. The procedure was terinated when the nuber of nodes generated exceeded η. ( ) ax By considering the search tree for our algorith, the root node φ corresponds to the null schedule of jobs within the faily sequenced in position f. Each of the other nodes corresponds to a partial schedule δ where {,,..., } δ = δ δ δ is a subset of jobs to be [ 1] [ 2] [ e] placed at the beginning of the sequence of jobs within the faily sequenced in the position f. The job δ [ j] occupies the 1 f th j position in the schedule of faily sequenced in position f, e N[ ], N is the nuber of jobs in J. Let δ = J[ ] / δ f, by placing any unscheduled job q, q δ in position e + 1 of π, we

4 MOSIM 10 - May 10-12, Haaet - Tunisia create a descendant node {,,...,, q} δ q = δ δ δ. [ 1] [ 2] [ e] Our branch and bound algorith eploys an adaptive depth-first plus bactracing search strategy and uses a lower bound to fatho nodes. We calculate the lower bound for each created nodes such that the nuber of created nodes are always less than or equal to N. The node to be branched is selected aong the recently created nodes which has the iniu lower bound. The nodes with lower bounds that are saller than the current incubent solution are interesting for further consideration such that the initial incubent solution is the solution obtained by our genetic algorith. We develop a useful achine-based lower bound. To copute a lower bound of the aespan for each node of the search tree, we suppose that only one achine; ( = 1,..., M ) can process at ost one job at a tie and the reainder achines can handle at the sae tie an infinite nuber of jobs. By this relaxation, the proble is transfored to sequence a set of jobs on a one achine with release dates (heads) and delivery ties (tails). Thus, we calculate a lower bound on the aespan by solving a single-achine scheduling proble. In order to calculate a lower bound the ain idea is that a job within a faily f can be processed at a achine only when both the job and the achine are ready. Let: C( q, ) The copletion tie of job q on achine. With q is the last scheduled job within the faily sequenced in position f. And in P ji Z j / q l, = δ i= l for 1 l M 0 otherwise Then lb = C( q, ) + Z + P for = 1,2,...,. [ ], 1 f j j δ / q Hence, a valid lower bound on the copletion tie on achine of the last sequenced jobs within the faily sequenced in position f of π. lb = ax lb 1 We define by C ' ( δ [ ][ ], ) the lower bound value of the f j copletion tie of the job in position j within the faily sequenced in position f on achine. Then C ' ( δ,0) = 0 for = f + 1, f + 2,..., K. [ ][ j] C = lb + S + P. ' 1 ( δ +,1) [ f 1][ 1] [ f 1] [ f 1][ 1] ' ' C ( δ[ ][ ], ) = ax lb S[ ][ 1], C ( δ[ 1][ 1], 1) + P f + f f + f + f for = 2,3,..., M. ' ' ' C ( δ[ ][ ], ) = ax C ( δ 1 [ 1], ) + S[ 1][ ], C ( δ[ ][ 1], 1) + P N 1 [ 1] for = f + 2, f + 3,..., K and = 1,2,..., M. C ( δ, ) = ax C ( δ, 1), C ( δ, ) + P ' ' ' [ ][ j] [ ][ j] [ ][ j 1] j for j = 2,3,..., N[ ], = f + 1, f + 2,..., K and = 1,2,..., M. So, the lower bound value of the copletion tie of the last job in the last faily on achine M is obtained by C ' (, M ) δ[ K ] N [ K ] 4 COMPUTATIONAL RESULTS Several tests were carried out to evaluate the effectiveness and the efficiency of our cooperative approach. In our ipleentation, we use the test probles tested by Schaller et al. (2000), França et al. (2005) and Lin et al. (2009) for the sae proble. In total, there are 900 instance probles classified into three classes according to the range of setup ties naed "Sall Setups" (SSU), "Mediu Setups" (MSU) and "Large Setups" (LSU). A Detailed description of these proble tests is given in (Schaller et al., 2000) and França et al. (2005). The algorith was coded in C++ and run on a PC with Pentiu 4, 3 GHz processor, with 512 Mb of RAM. The paraeter settings were deterined by preliinary job tests as follows: P = 40, p cross =0.5, the probability α varies according to the class of instances. Thus, α is equal to 0.3 4K, and respectively for LSU, MSU 2K K and SSU classes. The paraeter ηax is set to N f 50. For each class of instances, we define a perforance easure of each algorith by the average relative percentage of iproveents 30 M ij LBi i= 1 LBi avg = 100 of 30 the aespan for the instance i obtained by the heuristic algorith j ( ) ij M with respect to the lower bound values of Schaller et al. (2000). in and ax denote, respectively, the iniu and the axiu relative percentage of iproveents over each class of instances. t avg reports the average CPU ties of different algoriths in seconds. We perfor a coparison of the proposed cooperative approach (GA-B&B) with the other proposed algoriths in the literature and the results are displayed in Table 1 and Table2. Table1 provides results fro evolutionary algoriths including our proposed algorith; the first colun illustrated the results of the Meetic Algorith (MA) of França et al. (2005), the second and the third colun represents respectively the results of the genetic algorith of Lin et al. (2009) and our proposed algorith. Table 2 illustrates the results obtained fro local search algorith that is results of Tabu search algorith (TS(2)) of Hendizadeh et al. (2008) into the first colun, results of Siulated Annealing (SA) and Tabu search algorith (TS(1)) developed both by Lin et al. (2009) respectively into the second and third colun.

5 MOSIM 10 - May 10-12, Haaet - Tunisia Table 1 iplies that, on average for all tested instances, GA-B&B obtains the best results both on in, avg and ax. In other words, the Branch and Bound algorith can iprove the perforance of the Genetic Algorith. So, by coparing our approach against the Meetic algorith of França et al. (2005), where a siple local search procedure is added to the GA, it sees, on average that our algorith GA&BB is better than MA. Therefore, it s ore suitable to ebed an exact algorith into the GA. Moreover, in Table 2, it s shown that the results provided by GA&BB are better than the results of tabu search algoriths (TS(1) and TS(2) in ters of solution quality according to the eployed perforance easure.. Besides, when regarding the results of the SA of Lin et al. (2009), it s seen that the perforances of our approach and SA are very close. Although, in ters of in and avg these two algoriths are siilar, SA is better than GA&BB in ters of ax. About the CPU ties, we cannot conclude uch inforation because of the PC configurations used for the different approaches. However, the coputational effort of our algorith sees to be short. TS(2) SA TS(1) in avg ax t avg in avg ax t avg in avg ax t avg LSU LSU LSU LSU LSU LSU LSU LSU LSU LSU MSU MSU MSU MSU MSU MSU MSU MSU MSU MSU SSU SSU SSU SSU SSU SSU SSU SSU SSU SSU all Table 1: Coputational results of Tabu Search and Siulated Annealing algoriths

6 MOSIM 10 - May 10-12, Haaet - Tunisia instances MA GA GA&BB in avg ax t avg in avg ax t avg in avg ax t avg LSU LSU LSU LSU LSU LSU LSU LSU LSU LSU MSU MSU MSU MSU MSU MSU MSU MSU MSU MSU SSU SSU SSU SSU SSU SSU SSU SSU SSU SSU all Table 2: Coputational results of genetic and eetic algoriths

7 MOSIM 10 - May 10-12, Haaet - Tunisia 5 CONCLUSION In this paper a cooperative approach which integrates a Branch and Bound algorith, as an iproveent procedure, into a Genetic Algorith is presented. The addressed proble is the perutation flowline anufacturing cell with sequence dependent faily setup ties while iniizing the aespan criterion. The Branch and Bound algorith decoposes the whole proble by considering, in each faily, the probleas perutation flowshop scheduling proble with achine availability constraint. The experiental results, based on the used proble tests, show that our algorith provided good results and our approach is better than other approaches which cooperated between evolutionary algoriths and local search procedure. REFERENCES setup ties. Coputers and Operations Research ;36; Lozanoa M, Garcia-Martinez C., Hybrid etaheuristics with evolutionary algoriths specializing in intensification and diversification: Overview and progress report. Coputers and Operations Research; 37: Nawaz M., Enscore E., Ha I., A heuristic algorith for the -achine, n-job flow shop sequencing proble. OMEGA, International Journal of Manageent Science; 11: Ruiz R, Maroto C, Alcaraz J., Two new robust genetic algoriths for the flowshop scheduling proble. OMEGA, International Journal of Manageent Science ;34: Schaller JE, Gupta JND, Vaharia AJ., Scheduling a flowline anufacturing cell with sequence dependent faily setup ties. European Journal of Operational Research;125: Blu C., Hybrid Metaheuristics. Coputers and Operations Research; 37: Capbell, HG, Dude RA, Sith ML, A heuristic algorith for the n-job, -achine sequencing proble. Manageent Science; 16 : Cheng TCE, Gupta JND, Wang G., A review of flowshop scheduling research with setup ties. Production and Operations Manageent ;9: França PM, Gupta JND, Mendes AS, Moscato P, Veltin K., Evolutionary algoriths for scheduling a flowshop anufacturing cell with sequence dependent faily setups. Coputers and Industrial Engineering;48: Garey MR, Johnson DS., Coputers and Intractability: A Guide to the Theory of NP- Copleteness, Freean, San Francisco, CA. Greene TJ, Sadowsi RP., A review of cellular anufacturing assuptions, advantages and design techniques. Journal of Operations Manageent; 4: Hendizadeh SH, Faraarzi H, Mansouri SA, Gupta JND, ElMeawy TY., Meta-heuristics for scheduling a flowline anufacturing cell with sequence dependent faily setup ties. International Journal of Production Econoics;111: Jarboui B, Ibrahi S, Siarry P, Rebai A., A cobinatorial particle swar optiisation for solving perutation flowshop probles. Coputers and Industrial Engineering;54: Jourdan L, Basseur M, Talbi EG., Hybridizing exact ethods and etaheuristics: A taxonoy. European Journal of Operational Research; 199: Lin S-W, Ying K-C, Lee Z-J., Metaheuristics for scheduling a non-perutation flowline anufacturing cell with sequence dependent faily