Total tardiness minimisation in permutation flow shops: a simple approach based on a variable greedy algorithm

Transcription

1 Total tardness mnmsaton n permutaton flow shops: a smple approach based on a varable greedy algorthm Jose Manuel Framnan, Raner Lesten To cte ths verson: Jose Manuel Framnan, Raner Lesten. Total tardness mnmsaton n permutaton flow shops: a smple approach based on a varable greedy algorthm. Internatonal Journal of Producton Research, Taylor Francs, 00, (), pp.-. <0.00/000000>. <hal-00> HAL Id: hal-00 Submtted on Sep 00 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Internatonal Journal of Producton Research Total tardness mnmsaton n permutaton flow shops: a smple approach based on a varable greedy algorthm For Peer Revew Only Journal: Internatonal Journal of Producton Research Manuscrpt ID: TPRS-00-IJPR-0.R Manuscrpt Type: Orgnal Manuscrpt Date Submtted by the Author: 0-Feb-00 Complete Lst of Authors: Framnan, Jose; Unversty of Sevlle, Industral Management, School of Engneerng Lesten, Raner; Unversty Dusburg-Essen, Producton Management Keywords: SCHEDULING, FLOW SHOP SCHEDULING, META-HEURISTICS Keywords (user): Emal: jpr@lboro.ac.uk

3 Page of Internatonal Journal of Producton Research Total tardness mnmsaton n permutaton flow shops: a smple approach based on a varable greedy algorthm J.M. FRAMINAN and R. LEISTEN Word count (ncl. references):, Keywords: Schedulng, flow shop, due dates, greedy algorthm. For Peer Revew Only Industral Management, School of Engneerng, Unversty of Sevlle, Ave. Descubrmentos s/n, E0 Sevlle, Span; jose@es.us.es. Tel: +, Fax: +. Producton and Operatons Management, Department of Technology and Operatons Management, Mercator School of Management, Unversty Dusburg-Essen, Lotharstr., D- 0 Dusburg, Germany; raner.lesten@un-due.de. To whom correspondence should be addressed. Emal: jpr@lboro.ac.uk

4 Internatonal Journal of Producton Research Page of Total tardness mnmsaton n permutaton flow shops: a smple approach based on a varable greedy algorthm For Peer Revew Only J.M. FRAMINAN and R. LEISTEN In ths paper, we address the problem of schedulng jobs n a permutaton flowshop wth the objectve of mnmzng the total tardness of jobs. To tackle ths problem, we suggest employng a procedure based on a greedy algorthm that successvely terates over an ncreasng number of canddate solutons. The computatonal experments carred out show ths algorthm to outperform the best exstng one for the problem under consderaton. In addton, we carry out some tests to analyse the effcency of the adopted desgn.. Introducton A flowshop s a common manufacturng layout where the operaton sequence for all jobs s the same. Addtonally, t s usually assumed that jobs cannot overtake one another from machne to machne and therefore, once a sequence s fxed for all jobs on the frst machne, ths sequence s mantaned for all machnes (permutaton flowshop). Industral Management, School of Engneerng, Unversty of Sevlle, Ave. Descubrmentos s/n, E0 Sevlle, Span; jose@es.us.es. Producton and Operatons Management, Department of Technology and Operatons Management, Mercator School of Management,, Unversty Dusburg-Essen, Lotharstr., D- 0 Dusburg, Germany; raner.lesten@un-due.de. To whom correspondence should be addressed. Emal: jpr@lboro.ac.uk

5 Page of Internatonal Journal of Producton Research Several objectves may be consdered when schedulng n a flowshop. Wthout any doubt, the most wdely studed s the makespan objectve (see e.g. Reza Hejaz and Saghafan 00; or Ruz and Maroto 00 for recent revews on ths problem). In contrast, sgnfcantly less attenton has been pad untl recent years to other objectves, partcularly those nvolvng due dates. Among due-date related objectves, mnmsaton For Peer Revew Only of total tardness ams to fnd schedules that satsfy the due dates commtted to customers and t s therefore of sgnfcance and mportance to real lfe consderatons (Hasja and Rajendran 00). In ths paper, we propose an algorthm wth the objectve of mnmzng the total tardness of jobs. The algorthm s based on a varable greedy algorthm concept. A salent feature of the algorthm s that t employs a talored desgn for the problem, whch s based on the slack-concept,.e. jobs wth mnmum slacks are removed from a gven soluton n order to fnd for them a better poston n the schedule. Besdes, the algorthm s parameter-free (wth the excepton of the runnng tme), so t does not requre any parameter settng and parameter tunng. Ths smplcty and the lack of control parameters s a desred characterstc of any heurstc approach as makes t more user-frendly (Hansen and Mladenovć 00). The remander of the paper s organsed as follows: n the next secton we revew n more detal the problem under consderaton and the contrbutons up-to now. Next, we present the proposed algorthm for the problem. We then dscuss and set up an extensve computatonal experment n order to valdate the suggested approach and to compare wth the best-so-far ones. Fnally, we present the man conclusons and some future research lnes. Emal: jpr@lboro.ac.uk

6 Internatonal Journal of Producton Research Page of Problem statement and lterature revew The problem under consderaton refers to the schedulng of n jobs n a permutaton flow shop consstng of m machnes. Each job has a processng tme on each machne j, whch s denoted by p j. Addtonally, a due date d for each job s gven. In a permutaton flowshop, the completon tme of the -th job n a gven For Peer Revew Only schedule σ on a machne j, C []j (σ), s gven by the followng recursve equatons: C []j (σ) = max{ C [-]j (σ) ; C []j- (σ) } + p j C [0]j (σ) :=0; C []0 (σ):=0; Wth these equatons, C (σ), the completon tme of a job (.e. the earlest tme when ths job can be can released to the customer), can be computed. T, the tardness of a job s then defned as the maxmum of the dfference between ts completon tme and ts due date, and zero,.e.: T = max{ C (σ) d ;0}. The total tardness TARD s the sum of the tardness across all jobs: TARD = Σ T. Clearly, tardness mnmsaton s related to mantanng hgh servce levels, as t ams to respect the commtted due dates. Research on the topc has been carred out by Gelders and Sambandam (), Km (), Parthasarathy and Rajendran (), Armentano and Roncon (), and Hasja and Rajendran (00). Gelders and Sambandam develop constructve heurstcs for the problem, whle Km suggests an algorthm based on the NEH heurstc (Nawaz et al., ) for makespan mnmsaton. Hs proposal s not compared to these from Gelders and Sambandam. Parthasarathy and Rajendran present two versons of Smulated Annealng (SA) for the problem and compare them wth those of Gelders and Sambandam, and that of Km. They found ther proposal to outperform all these heurstcs. Armentano Emal: jpr@lboro.ac.uk

7 Page of Internatonal Journal of Producton Research and Roncon desgn a tabu search procedure that s compared wth an adaptaton of the NEH heurstc for the problem under consderaton, but not wth other heurstcs avalable for the problem. Hasja and Rajendran propose a SA algorthm for the problem (HR n the followng) by ntroducng new perturbaton and mprovement schemes. They compare ther algorthm wth the most effcent verson of the heurstc For Peer Revew Only by Parthasarathy and Rajendran (PR n the followng) and the one by Armentano and Roncon usng an adaptaton of the benchmark problems by Tallard (). The results of ths computatonal experence show that (at least for the test-bed employed), the heurstc by Hasja and Rajendran clearly outperforms the exstng ones and henceforth t should be consdered the best heurstc for the problem up to date. In ther computatonal experence, the heurstc by Parthasarathy and Rajendran also outperforms the one by Armentano and Roncon. As a summary, t can be concluded that the most effcent heurstcs for the problem under consderaton are PR and HR. Both heurstcs use the same heurstc for generatng the ntal sequence,.e. the so-called Earlest Apportoned Due Date Method (EADD). In EADD, jobs are ordered for each machne n ascendng order of a value proportonal to the processng tmes of the job and ts due date. In ths manner, one order of jobs s obtaned for each machne. Among these, the orderng wth the least value of total tardness s chosen. PR employs a Random Inserton Perturbaton Scheme (RIPS) for generatng new solutons from a gven sequence n the followng manner: for each job n the seed sequence, two random postons are generated (one on ts rght and other on ts left) to nsert t, thus obtanng two new sequences. Obvously, the frst (last) job cannot be nserted on any poston on ts left (rght), hence only one new sequence can be obtaned from these extreme postons. In total, (n ) new sequences are obtaned. Among them, the one wth the least total tardness s selected. Emal: jpr@lboro.ac.uk

8 Internatonal Journal of Producton Research Page of HR employs sophstcated mechansms for the generaton of new solutons, namely Job Index Based Inserton Scheme (JIBIS), Job Shft-Based (JSB) scheme, and Probablstc Step Swap (PSS) perturbaton scheme. Usng JIBIS, each job (startng from job to job n) n a seed sequence s selected for nserton n all possble postons of the sequence. If the total tardness of the so-obtaned sequence s lower than the one For Peer Revew Only for the seed, t replaces the seed. Note that JIBIS obtans a soluton whch s not worse than the ntal seed. In the JSB scheme, a job n the seed sequence s chosen based on a probablstc functon and then nserted ether on the rght or on the left of ts orgnal poston (the same probablty s assgned to the nserton on the left and on the rght). The job n the seed sequence s chosen accordng to the relatve dsplacement of the job s poston n the current sequence from the job s poston n the best-so-far sequence. Fnally, PSS generates sequences from a seed sequence by probablstcally swappng jobs close to each other, and far from each other. Among them, the one wth the least total tardness s chosen. In Fgures and we present the detaled pseudocode of both heurstcs, keepng the same steps and the programme structure as n the orgnal descrptons. [INSERT FIGURE ABOUT HERE] [INSERT FIGURE ABOUT HERE]. Proposed heurstc In recent years, a number of general-purpose heurstc approaches (also known as meta-heurstcs) have been ntroduced to tackle combnatoral optmsaton problems. Two of the newest approaches are the Iterated Greedy (IG) algorthm and the Varable Emal: jpr@lboro.ac.uk

9 Page of Internatonal Journal of Producton Research Neghbourhood Search (VNS). As our proposal combnes features of both metaheurstcs, we frst brefly ntroduce these. IG (see e.g. Marchor and Steenbeek 000, and Ruz and Stützle 00) generates a sequence of solutons by teratng over a constructve heurstc usng two phases, namely destructon and constructon. Durng the destructon phase, k components For Peer Revew Only (destructon level) are removed from a gven soluton π, thus obtanng a set of removed components Sk ( π ) and a partal soluton π R( k ) formed by the non-removed components of π. Durng the constructon phase, a greedy constructve heurstc s appled to the problem n order to obtan a (possbly new) soluton π ' by addng to π the components n S ( π ). Then, t s decded whether or not R( k ) k π ' substtutes π as the ncumbent soluton for the next teraton accordng to some pre-defned acceptance crteron (e.g. f π ' s better than π). In addton, a local search procedure can be mplemented before ths step n order to mprove π '. IG s closely related to another technque, Iterated Local Search (ILS), n whch the algorthm terates over a local search neghbourhood (see e.g. Lourenço et al. 00). To the best of our knowledge, IG has been only appled to few problems, ncludng crew schedulng (Marchor and Steenbeek 000) and flowshop schedulng wth makespan objectve (Ruz and Stützle 00). Salent features of IG are ts smplcty and the possblty to use a talored heurstc to obtan good solutons for the problem under consderaton. Clearly, an obvous drawback of the method s the stagnaton due to the greedy nature of the procedure. In order to overcome ths problem, Ruz and Stützle (00) adopt a smulated annealng-based acceptance crteron,.e.: acceptng worse solutons wth some probablty dependng on the number of teratons of the algorthm. A smlar approach s also adopted n Marchor and Steenbeek (000). Emal: jpr@lboro.ac.uk

10 Internatonal Journal of Producton Research Page of In ths paper, we propose a dfferent procedure for overcomng local optma, whch s based on the VNS concept (Mladenovć and Hansen ). VNS uses a fnte set of pre-selected neghbourhood structures that we denote by N k (k=,, k max ). Let us denote by N k (x) the set of solutons n the neghbourhood of x. The pseudo-code of the basc structure of a VNS algorthm can be descrbed as follows: For Peer Revew Only () Generate an ntal soluton x; () Whle termnaton condtons are not met, repeat: (.) Set k ; (.) Untl k = k max, repeat: (..) Fnd the best neghbour x of x ( x N k (x) ); (..) If x s better than x, then set x x and set k ; otherwse set k k + ; Sophstcated features can be added to ths most basc procedure. For these, we refer the reader to Hansen and Mladenovć (00). VNS has shown to be effcent for a number of NP-hard problems, ncludng some schedulng problems (see e.g. Gagné et al. 00, Schuster and Framnan 00, and Tasgetren et al. 00). In our proposal, we use the VNS concept of varyng the neghbourhood, but we apply t to the destructon and constructon phases of the IG algorthm. To do so, a greedy mechansm specfcally talored for the problem s descrbed n the next secton... Slack-based greedy algorthm For a schedule σ := ( σ,...), let us defne s ( σ ) the slack of job σ n the sequence σ n the followng manner: s ( σ ) : = d C, m. It s clear that s ( σ ) s sequence- σ σ Emal: jpr@lboro.ac.uk

11 Page of Internatonal Journal of Producton Research dependent and that t s possble to express the total tardness n terms of the slacks,.e.: TARD( σ ) = max[ s ( σ );0]. Consequently, f a job σ has a negatve value for s ( σ ), then ths job s contrbutng to ncrease the total tardness. Based on ths dea, t s possble to defne a greedy algorthm to obtan a sequence σ from another sequence σ n the followng manner: For Peer Revew Only () Remove from σ the k jobs wth lowest s ( σ ). Store them n remanng subsequence from σ after removng the k jobs. () For each job σ r Rk, repeat: R k. Let µ be the (.) Insert σ r n all possble slots of µ (startng from the frst poston). Let ν be the best subsequence obtaned, and b the poston of σ r n ν. (.) Perform an adjacent parwse exchange among all jobs n postons b+ to (n k + ). Let π be the best sequence obtaned by ths approach. (.) If π mproves ν, then set µ π ; otherwse set µ ν. () Set σ ' µ. Accordng to ths approach, t s possble to defne k dfferent types of movements, dependng on the number of jobs to be removed. Asde, ths constructon mechansm s related to the heurstc by Km, whch n turn s based on the NEH heurstc. Nevertheless, we added step (.) as we try to allow some small perturbaton of the soluton for those jobs after the one for whch the mnmum tardness has been found. In addton, t s possble to ntroduce a speed up n order to reduce the computatonal effort. When executng step (.), nstead of calculatng the completon tmes from scratch for every slot j (j =,, n k + ) n whch the job σ r s nserted, the completon tme n all machnes for jobs precedng job (j ) do not need to be Emal: jpr@lboro.ac.uk

12 Internatonal Journal of Producton Research Page 0 of recomputed. By ths procedure, the computaton tmes are reduced roughly to the half of the tme... Inserton-based local search In order to mprove the soluton provded by the slack-based greedy algorthm, a For Peer Revew Only local search n the neghbourhood of the soluton s accomplshed. Usual neghbourhood defntons for flowshop schedulng problems nclude adjacent parwse nterchange (exchangng the postons of two adjacent jobs), general parwse nterchange (exchangng the postons of each two jobs), and nserton (removng one job from ts poston and nsertng t nto a dfferent one). Among these possbltes, the experences conducted produced the best result for the last one. Consequently, an nserton-based neghbourhood s adopted. The local search algorthm then obtans n postons at random (wthout repetton), and for each poston, the job occupyng ths poston s removed and nserted n all possble slots. If the best soluton out of the n nsertons s better than the current soluton, then t replaces the latter as the best ncumbent soluton. The pseudocode of ths procedure s as follows: () Let σ be the current soluton. () For = to n do: (.) Obtan k = random (, n) (wthout repetton). (.) Remove σ k, the job n poston k, n σ and nsert t n all possble slots. Let σ ' be the best soluton found by ths way. (.) If σ ' mproves σ, then set σ σ '. () Return σ as soluton. 0 Emal: jpr@lboro.ac.uk

13 Page of Internatonal Journal of Producton Research Pseudo-code of the algorthm The algorthm operates as follows: () Whle computaton tme does not exceed a gven tme lmt, repeat: For Peer Revew Only (.) Generate a soluton x at random. (.) Set k ; (.) Untl k = (n ), repeat: (..) Obtan x from x accordng to the procedure explaned n Secton...e. usng the slack-based greedy algorthm. (..) Obtan x from x accordng to the procedure explaned n Secton..,.e. usng the nserton-based local search. (..) If x s better than x, then set x x and set k ; otherwse set k k + ; The algorthm starts wth a random soluton. Then, ths soluton s mproved by a stepwse approach that systematcally enhances the capabltes of the greedy algorthm untl t shuffles all jobs. If a better soluton s found, then the process starts agan. Otherwse, f the greedy algorthm has been expanded wthout reachng a better soluton (.e., k = n ), then a new random soluton s tred. Ths allows a dversfcaton of the exploraton once all greedy structures have been tred. Note that ths algorthm s parameter-free, wth the excepton of the CPU tme. Wth respect to the stoppng crteron, t has been chosen as the decson nterval for schedulng problems s usually very short and hence the total runnng tme of a heurstc s a crtcal ssue. However, replacng the orgnal crteron by a dfferent one Emal: jpr@lboro.ac.uk

14 Internatonal Journal of Producton Research Page of based e.g. on the number of solutons evaluated, the number of teratons wthout mprovement, etc. s straghtforward.. Computatonal experments In ths secton we present the results of the computatonal experments carred out For Peer Revew Only n order to valdate our proposed procedure. We frst dscuss the expermental desgn and partcularly the settng of due dates n the problem nstances. We then compare our proposal wth the best-so-far algorthms for the problem under consderaton. Fnally, we analyse how the dfferent desgn decsons n the algorthm (namely the desgn of the destructon phase, and the concept of varable destructon ) affect ts performance... Expermental desgn It s clear that the desgn of the expermental test-bed may have a huge nfluence on the results. Partcularly, beng a due-date problem, several types of problems can be dstngushed, each one wth a dfferent degree of dffculty. Henceforth, we frst revew the dfferent settngs for due dates that have been adopted n the lterature. Rather dfferent procedures for establshng due dates n permutaton flowshops can be found n the papers by Gelders and Sambandam (), Armentano and Roncon (), Demrkol et al. (), and Hasja and Rajendan (00). In all these, the processng tmes of the problems are obtaned from a unform dstrbuton. In fact, the [, ] dstrbuton s employed for all test-beds except for the one by Demrkol et al., where processng tmes are drawn from a [, 00] unform dstrbuton. Therefore, n the followng we focus our dscusson on how due dates are establshed n these papers. Armentano and Roncon suggest usng the followng dstrbuton to generate due dates: Emal: jpr@lboro.ac.uk

15 Page of Internatonal Journal of Producton Research R R Unform[ P( T ), P( T + )] where T and R are the tardness factor of jobs and the dsperson range of due dates, respectvely, and P s a lower bound of the makespan, defned as follows: n j m m P= max max pj + mn pl + mn pl, max pj j m = l= l= j+ j= For Peer Revew Only Wth the above settng, they defne four scenaros dependng on the values of T and R. Ths way of generatng due dates s nearly dentcal to the one employed by Demrkol et al. who buld a test-bed for several due date problems, ncludng 0 problem nstances of flowshops wth due dates. They employ dfferent T and R factors that produce much tghter due dates as compared wth the paper by Armentano and Roncon. Addtonally, from a bgger subset of problems, the 0 hardest nstances were collected. Gelders and Sambandam propose a dfferent method for generatng due dates, consstng of employng the followng unform dstrbuton: m m [ pj j =, pj j = + 0. mean processng tme of a job on machne m]. Ths due-date settng would result n approxmately 0% of the jobs beng late, and t has been used also by Chakravarthy and Rajendran (), and Framnan and Lesten (00) due to ts ablty to produce tght due dates. Fnally, we descrbe the due date settng n the test-bed developed by Hasja and Rajendran, whch s based on Tallard s test-bed (Tallard ). Tallard s test-bed conssts of a collecton of 0 flowshop problem nstances chosen so as good makespan values are dffcult to obtan. In ths test-bed, (n,m) {(0,), (0,0), Emal: jpr@lboro.ac.uk

16 Internatonal Journal of Producton Research Page of (0,0), (0,), (0,0), (0,0), (00,), (00,0), (00,0), (00,0), (00,0), (00,0)}. For each problem sze, 0 nstances are avalable. In order to provde due dates d for the problem under consderaton, Hasja and Rajendran propose the followng method to generate them: m d = [+ u] p For Peer Revew Only j= j where u s a real random number unformly dstrbuted n the nterval [0,]. In vew of the dfferent contrbutons, we select three dfferent test-beds for the comparson of the heurstcs. For these, we use the followng acronyms: DMU: The test-bed by Demrkol et al. As mentoned before, ths test-bed conssts of 0 nstances, where (n,m) {(0,), (0,0), (0,), (0,0), (0,), (0,0), (0,), (0,0)}. For each problem sze, there are 0 nstances. T-GS: The test-bed obtaned usng the processng tmes of the test-bed of Tallard, and due dates generated accordng to the method proposed by Gelders and Sambandam. In vew of the prohbtve computaton tmes requred to solve the 0 bggest nstances of Tallard's test-bed (nstances ta0-ta0, n = 00) by usng the exstng algorthms for the problem, we exclude them from T-GS. Therefore, the test-bed under consderaton conssts of 0 nstances. T-HR: The test-bed obtaned usng the processng tmes of the test-bed of Tallard, and due dates generated accordng to the method proposed by Hasja and Rajendran. Analogously to the prevous test-bed, the 0 bggest nstances by Tallard are removed... Comparson wth exstng heurstcs As mentoned n the ntroducton, the procedure proposed by Hasja and Rajendran (HR n the followng) s the best algorthm so far for the problem under consderaton, Emal: jpr@lboro.ac.uk

17 Page of Internatonal Journal of Producton Research followed by the procedure of Parthasarathy and Rajendran (PR n the followng). In order to compare them wth our algorthm, we employ the three test-beds descrbed n the prevous secton. Note that nether HR nor PR have a smple termnaton crteron (such as the computaton tme, or the number of teratons wthout mprovement), but use a rather sophstcated combnaton of parameters dependng on the temperature For Peer Revew Only (number of teratons, number of accepted moves, and number of sequences generated). Moreover, t s not possble to establsh a far comparson based on the number of solutons evaluated by each algorthm, as our proposal constructs (complete) solutons by exhaustvely selectng and explorng partal sequences, therefore the number of fnal solutons evaluated s much lower than n PR and HR (less than % n the DMU testbed, as we wll show later). Consequently, the farest comparson wth HR s runnng ths algorthm exactly wth the same parameter settng descrbed by the authors, and then compare ts results to those obtaned by the new heurstc proposal, usng as termnaton crteron the same CPU tme taken by the HR algorthm for ths problem. Each nstance s solved by runs of each algorthm. The average and the best value of the total tardness for each algorthm and nstance are obtaned. Then, the qualty of the solutons obtaned for each nstance and each heurstc s measured n terms of RPD (Relatve Percentage Devaton) wth respect to the average and the best soluton, n the followng manner: ( T T ) = 00 T ( T T ) = 00 T where T s the average tardness obtaned over the runs of algorthm, best tardness among the runs, and T s the T s the best known soluton for ths nstance. For all nstances of a gven problem sze, the average and the standard devaton of both Emal: jpr@lboro.ac.uk

18 Internatonal Journal of Producton Research Page of and are obtaned, and shown n Tables,, and for the DMU, T-GS, and T-HR test-beds, respectvely. The tables show the qualty of the solutons obtaned n terms of the (Average Relatve Percentage Devaton) of heurstc for each problem sze. For Peer Revew Only ARPD ARPD values are obtaned by averagng across a number of nstances the Relatve Percentage Devaton ( RPD ), measured as follows: where RPD nstance, and ( RPD RPD ) RPD mn = mn 00 RPD s the tardness obtaned by the applcaton of the heurstc to the problem RPD mn s the mnmum tardness found by any heurstc for ths nstance. [INSERT TABLE ABOUT HERE] [INSERT TABLE ABOUT HERE] [INSERT TABLE ABOUT HERE] As can be seen from Tables,, and, the algorthm proposed exhbts a very good performance for the three test-beds. Both the best soluton (out of runs) and the average obtan better results than those of HR and PR. In order to test the performance of our proposal for dfferent CPU tmes, we change the orgnal desgn of PR and HR n order to ntroduce CPU tme as stoppng crteron. Then, we run all three algorthms wth fractons of the CPU tme nvested for the orgnal desgn. More specfcally, let TIME be the CPU tme requred (average over runs) for a gven nstance n the experments presented n Tables,, and. Then we solve the DMU test-bed wth all three algorthms for a CPU tme of TIME/, and Emal: jpr@lboro.ac.uk

19 Page of Internatonal Journal of Producton Research TIME/. The qualty of the solutons obtaned s shown n Table n terms of the best soluton and the average over runs of each algorthm. [INSERT TABLE ABOUT HERE] For Peer Revew Only As can be seen from Table, the algorthm outperforms HR and PR for the dfferent CPU tmes. It s nterestng to note that, for all three algorthms, the results for dfferent CPU tmes are smlar wth respect to the best soluton. Regardng the average soluton, there s a steady mprovement for our proposal, but not for HR and PR, whch may ndcate that both algorthms could stagnate for longer CPU tmes. Fnally, n Table we gve the average number of solutons explored by HR and our proposal n the DMU test bed for each problem sze (the CPU allowed s that of the orgnal HR descrpton, see Table ). It has to be noted that ths value s not an ndcator of the computatonal effort of each algorthm, as n the case of our proposal, most of the effort s devoted to nsert the jobs removed n the destructon phase n the dfferent slots of the remanng schedules. Therefore, the number of (complete) solutons explored by our proposal s very low as compared to HR, whch use most of the CPU tme to explore and evaluate new (complete) solutons. [INSERT TABLE ABOUT HERE].. Effectveness of the varable greedy desgn The proposed algorthm s based on a varable neghbourhood dependng on k, whch determnes the number of elements n the sequence to be destroyed and later reconstructed n a range from to (n ). Therefore, t s of nterest to assess the Emal: jpr@lboro.ac.uk

20 Internatonal Journal of Producton Research Page of contrbuton of ths desgn to the performance of the algorthm. More specfcally, we are nterested n answerng the followng questons: a) Does the varable desgn contrbute to mprove the soluton wth respect e.g. fxed value of k? b) Are there alternatve upper values of k (other than n ) for whch the For Peer Revew Only algorthm yelds a hgher performance? In order to answer the frst queston, we analyse the contrbuton of each value of k to mprove the current soluton of the algorthm. More specfcally, for a gven problem nstance, we record the value of k for whch a new best-so-far soluton s obtaned. We then calculate the frequency of each value of k and plot the results. In Fgure, we show the plot for dfferent problems n the T-HR test-bed. Each lne shows the frequency of the mprovements of k for 0 runs of 0 nstances of a gven (n,m). [INSERT FIGURE ABOUT HERE] In Fgure, t s shown that all values of k n the range contrbute to the mprovement of the soluton. The frequency decreases wth k, whch s not surprsng, snce n the algorthm, every tme a new soluton s found, the value of k s set to and starts ncreasng once the solutons found wth ths destructon level are worse than the best-so-far sequence. The only excepton s k =, ndcatng the relatvely low ablty of ths destructon level to fnd better solutons as compared to k =. Nevertheless, the second hghest frequences are obtaned for k =. Fnally, t has to be noted that the pattern of the frequences s the same for dfferent number of machnes. Emal: jpr@lboro.ac.uk

21 Page of Internatonal Journal of Producton Research In a dfferent experment also amed to the frst queston, we compare the results obtaned by ths algorthm n the DMU test-bed wth those obtaned usng dfferent (fxed) numbers of components n the destructon phase. For all runs, the stoppng crteron s the same as n the prevous sub-secton,.e. the CPU tme requred for each problem by the HR heurstc. Consequently, the CPU tme s agan the same for all For Peer Revew Only algorthms. The results are shown n Table n terms of the A wth respect to the best soluton found by the dfferent settngs. [INSERT TABLE ABOUT HERE] In vew of the results n Table, the best overall results are obtaned by the varable destructon desgn. Whle some results wth a fxed k are very good, these are strongly dependent on the partcular problem sze. Once the convenence of usng a varable k as compared to employng fxed values has been establshed, we check whether there exst other upper values of k yeldng better results than the orgnal proposal (n ). To do so, we run dfferent versons of our heurstc wth several upper levels of k for the DMU test-bed. More specfcally, we test n/0, n/ and n/. All versons are allowed to use the same CPU tme (that employed for the comparson of the dfferent heurstcs n Secton.., see Table ). The results are shown n Table n terms of the RPD. [INSERT TABLE ABOUT HERE] As can be seen from Table, employng other values as upper lmt of the level of destructon s less effcent than the orgnal proposal. Partcularly, ncorporatng large Emal: jpr@lboro.ac.uk

22 Internatonal Journal of Producton Research Page 0 of destructon levels seems sutable to obtan better results for a gven CPU tme, confrmng the results n Fgure showng that also large k contrbute to mprove current best solutons... Effectveness of the slack-based greedy algorthm For Peer Revew Only In ths subsecton we nvestgate whether the mechansm adopted for selectng the k components to be destroyed (.e. those k wth least slack) s more effcent than, for nstance, pckng these k at random, or pckng them n reverse order (descendng order of ther slacks). The results are shown n Table n terms of the ARPD wth respect to the best results obtaned for the three approaches. In ths table, NEW denotes the approach proposed, whle random and nverse ndcates pckng the components n the destructon phase at random, or wth descendng order of the slacks, respectvely. [INSERT TABLE ABOUT HERE] As can be seen from Table, the slack-based defnton of the greedy algorthm s, on the overall results, more effcent than pckng the solutons at random or n descendng order of ther slack. It s to note that, for the problem sze n = 0 and m =, the nverse orderng obtans slghtly better results. However, ths small dfference s not supported for the rest of the problem szes.. Conclusons We have presented a new algorthm for the problem of mnmsng total tardness n permutaton flowshops. A salent feature of the proposed method s that t does not requre any parameter, wth the excepton of the computaton tme. Ths means that 0 Emal: jpr@lboro.ac.uk

23 Page of Internatonal Journal of Producton Research there s no need to adjust or tune t for dfferent problems. The computatonal experence carred out shows the proposal to outperform the best-so-far avalable algorthms. The algorthm s shown to be effcent for dfferent computatonal efforts, and ts desgn s checked to be more sutable as compared to dfferent optons regardng a varable/fxed destructon level, dfferent upper lmts for k, and other mechansms for For Peer Revew Only selectng the components to be destroyed. Further sophstcated features may be added to the suggested algorthm, such employng a more complex crteron for acceptng new solutons. However, these enhancements must be contrasted wth the smplcty and the speed of the current desgn. Acknowledgements The authors wsh to thank the referees for the nsghtful comments on earler versons of the paper. Ths research was accomplshed whle the frst author was wth the Unversty of Dusburg-Essen (Germany) as recpent of an Alexander Von Humboldt Fellowshp. References Armentano, V.A., Roncon, D.,, Tabu search for total tardness mnmzaton n flowshop schedulng problems, Computers & Operatons Research,, pp. -. Chakravarthy, K. C. Rajendran,, A heurstc for schedulng n a flowshop wth the bcrtera of makespan and maxmum tardness mnmzaton, Producton Plannng and Control, 0, pp. 0-. Demrkol, E., Mehta, S., Uzsoy, R.,, Benchmarks for shop schedulng problems, European Journal of Operatonal Research, 0, pp Emal: jpr@lboro.ac.uk

24 Internatonal Journal of Producton Research Page of Framnan, J.M., Lesten, R., 00, A heurstc for schedulng a permutaton flow shop wth makespan objectve subject to maxmum tardness, Internatonal Journal of Producton Economcs,, pp. -0. Gagné C, Gravel M, Prce WL., 00, A new hybrd Tabu-VNS metaheurstc for solvng multple objectve schedulng problems. Proceedngs of the Ffth For Peer Revew Only Metaheurstcs Internatonal Conference. Kyoto, Japan. Gelders, L.F., Sambandam, N.,, Four smple heurstcs for schedulng a flowshop, Internatonal Journal of Producton Research,, pp. -. Hansen, P., Mladenovć, N., 00, Varable neghbourhood search: Prncples and applcatons, European Journal of Operatonal Research, 00, pp. -. Hasja, S., Rajendran, C., 00, Schedulng n flowshops to mnmze total tardness of jobs, Internatonal Journal of Producton Research,, pp. -0. Km, Y.D.,, Heurstcs for flow shop schedulng problems mnmzng mean tardness, Journal of the Operatonal Research Socety,, pp. -. Lourenço, H.R., Martn, O., Stützle, T., 00, Iterated Local Search. In: Glover, F., and Kochenberger, G., edtors, Handbook of Metaheurstcs, pp. -. Kluwer Academc Publshers. Marchor, E., Steenbeek, A., 000, An evolutonary algorthm for large set coverng problems wth applcatons to arlne crew schedulng. In: Cagnon, S., et al., edtor, Real Applcatons of Evolutonary Computng, EvoWorkshops 000, Lecture Notes n Computer Scence, 0, pp. -. Sprnger Verlag, Berln. Mladenovć, N., Hansen, P.,, Varable Neghborhood Search, Computers Operatons Research,, pp Nawaz, M., Enscore, E. E., Ham, I.,, A heurstc algorthm for the m-machne, n- job flow-shop sequencng problem, OMEGA,, pp Emal: jpr@lboro.ac.uk

25 Page of Internatonal Journal of Producton Research Parthasarthy, S., Rajendran, C., Schedulng to mnmze mean tardness and weghted mean tardness n flowshop and flowlne-based manufacturng cell, Computers & Industral Engneerng,, pp. -. Schuster, C., Framnan, J.M., 00, Approxmatve procedures for no-wat job shop schedulng, Operatons Research Letters,, pp. 0-. For Peer Revew Only Tallard, E.,, Benchmark for basc schedulng problems, European Journal of Operatonal Research,, pp. -. Tasgetren MF, Sevkl M, Lang YC, Gencylmaz G, 00, Partcle swarm optmzaton algorthm for sngle machne total weghted tardness problem. In: Proceedngs of the IEEE congress on evolutonary computaton, Oregon, Portland, pp.. Reza Hejaz, S., and Saghafan, S., 00, Flowshop-schedulng problems wth makespan crteron: a revew, Internatonal Journal of Producton Research,, pp. -. Ruz, R., Maroto, C., 00, A comprehensve revew and evaluaton of flowshop heurstcs, European Journal of Operatonal Research,, pp. -. Ruz, R., Stützle, T., 00, A smple and effectve terated greedy algorthm for the permutaton flowshop schedulng problem, European Journal of Operatonal Research (n press, avalable onlne). Uzsoy, R., Benchmark schedulng problems [Last consulted.0.00], Emal: jpr@lboro.ac.uk

26 Internatonal Journal of Producton Research Page of n m PR HR NEW Average Standard dev. Average Standard dev. Average Standard dev. For Peer Revew Only Avg. CPU (secs) n m Table. ARPD obtaned n DMU test-bed (all heurstcs use the same CPU tme) PR HR NEW Average Standard dev. Average Standard dev. Average Standard dev Table. ARPD obtaned n the T-GS test-bed (all heurstcs use the same CPU tme) n m PR HR NEW Average Standard dev. Average Standard dev. Average Standard dev Table. ARPD obtaned n the T-HR test-bed (all heurstcs use the same CPU tme) Avg. CPU (secs) Avg. CPU (secs) Emal: jpr@lboro.ac.uk

27 Page of Internatonal Journal of Producton Research n m PR HR NEW T/ T/ T/ T/ T/ T/ For Peer Revew Only Table. ARPD obtaned n for dfferent CPU tmes (average on runs) n m HR NEW 0 0,, 0 0 0,, 0,, 0 0, 0, 0,, 0 0 0,, 0,0, 0 0,0, Table. Number of solutons explored by HR and NEW n the DMU-testbed n m k= k= k=0 k= k=0 k= k=0 k= k=0 k= NEW Average : Table. ARPD for dfferent greedy algorthms for the nstances n the DMU test-bed. Emal: jpr@lboro.ac.uk

28 Internatonal Journal of Producton Research Page of Upper lmt of k n m n/0 n/ n/ n- (NEW) For Peer Revew Only Table. ARPD for several upper lmts of k for the nstances n the DMU test-bed (n- corresponds to the orgnal proposal). n m NEW nverse random Average: Table. ARPD for dfferent mechansm for pckng the components to be removed (DMU test-bed). Emal: jpr@lboro.ac.uk

29 Page of Internatonal Journal of Producton Research Step : Obtan one ntal sequence usng heurstc EADD and assgn t to S and B. Step : Intalze SA parameters,.e. T:=; ACCEPT := 0; TOTAL:=0; FR_CNT:=0. Step : If (FR_CNT = ) or (T 0) then go to Step. Step : Invoke perturbaton routne RIPS to obtan S. TARDINESS ( S ') TARDINESS ( S) = 00 TARDINESS( S) For Peer Revew Only If > 0 then go to Step. Step : S:= S ; ACCEPT := ACCEPT + TARDINESS ( S ') TARDINESS ( B) Step : If 00 > 0 then go to Step 0. TARDINESS ( B) Step : B := S ; FR_CNT := 0. Go to Step 0. Step : If exp( ) < Random[0;], then go to Step 0. T Step : S:= S, and ACCEPT := ACCEPT +. Step 0: TOTAL:= TOTAL +. Step : If (TOTAL > n) or (ACCEPT > n/) go to Step ; else go to Step. Step : If (ACCEPT 00/TOTAL ) then FR_CNT := FR_CNT + Step : Set T := T 0.; ACCEPT := 0; TOTAL := 0. Go to Step. Step : The algorthm s frozen. B contans the fnal soluton. Fgure. Pseudocode of PR Emal: jpr@lboro.ac.uk

30 Internatonal Journal of Producton Research Page of Step : Obtan one ntal sequence usng EADD. Assgn t to S and B. Step : Apply JIBIS to S and B. Step : Z = TARDINESS(S), Z B := TARDINESS(B) Step : Intalze SA parameters,.e. T :=0, COUNT := 0; REPLACE :=0; FREEZE :=0. Step : Generate and evaluate nne random sequences and store them (together wth B) n an archve of the best ten sequences. Step : If (FREEZE = ) or (T 0) then go to Step. For Peer Revew Only Step : Set Z a large number. Step : Do the followng steps twce: Do the followng steps n tmes: Invoke JSB to obtan S. Z := TARDINESS(S ). If Z s less than the maxmum value of the objectve functon of a sequence amongst all sequences n the archve, then replace that sequence and ts objectve functon by S and Z. If Z < Z then S :=S and Z :=Z. Step : Do the followng steps twce: Invoke perturbaton scheme PSS to obtan S ; Z :=TARDINESS(S ) If Z s less than the maxmum value of the objectve functon of a sequence amongst all sequences n the archve, then replace that sequence and ts objectve functon by S and Z. If Z < Z then S :=S and Z :=Z. TARDINESS ( S ') TARDINESS ( S) Step 0: = 00 TARDINESS( S) Step : If > 0 then go to Step. Step : S:= S ; Z:=Z ; REPLACE = REPLACE +. Step : If TARDINESS ( S '') TARDINESS ( B) 00 > 0, then go to Step. TARDINESS( B) Step : B:=S ; Z B := Z ; FREEZE = 0. Go to Step. Step : If exp( ) < Random[0;], then go to Step. T Step : S:= S ; Z:=Z ; REPLACE := REPLACE +. Step : COUNT := COUNT +. Step : If (COUNT n) go to Step. Step : If REPLACE 00/COUNT, then FREEZE := FREEZE +. Step 0: T = T 0.; COUNT := 0; REPLACE := 0. Go to Step. Step : Implement JIBIS on all sequences stored n the archve. The best amongst them s returned as soluton. Fgure. Pseudocode of HR Emal: jpr@lboro.ac.uk

31 Page of Internatonal Journal of Producton Research Frequency 0, 0, 0, ta00-ta00 (n=0, m=) ta0-ta00 (n=0, m=0) 0, ta0-ta00 (n=0, m=0) 0, 0, 0, 0, 0,0 0 Level of destructon For Peer Revew Only Fgure. Frequences correspondng to the values of k for whch a better soluton s found Emal: jpr@lboro.ac.uk