Sequencing Heuristics for Flexible Flow Shop Scheduling Problems with Unrelated Parallel Machines and Setup Times

Transcription

1 IE NETWORK 26 Sequencing Heurisics for Flexible Flow Shop Scheduling s wih Unrelaed Parallel Machines and Seup Times Jii Jungwaanaki 1, Manop Reodecha 1, Paveena Chaovaliwongse 1, and Frank Werner 2 1 Deparmen of Indusrial Engineering, Faculy of Engineering, Chulalongkorn Universiy, Phayahai Rd., Paumwan, Bangkok 133 Tel: , , FAX: , jii.j@suden.chula.ac.h 2 Faculy of Mahemaics, Oo-von-Guericke-Universiy, P.O. Box 412, D-3916 Magdeburg, Germany Tel: Tel.: ; fax: , frank.werner@mahemaik.uni-magdeburg.de Keywords : sequencing heurisics, flexible flow shop, unrelaed parallel machines Absrac This paper presens an invesigaion of scheduling procedures o seek he minimum of a posiively weighed convex sum of makespan and he number of ardy jobs in a saic flexible flow shop scheduling environmen. The flexible flow shop problem is a scheduling of jobs in a flow shop environmen consising of series of producion sages, some of which may have only one machine, bu a leas one sage mus have muliple machines. In addiion, sequence - and machine - dependen seup imes are considered. No preempion of jobs is allowed. Some dispaching rules and flow shop makespan heurisics are adaped for sequencing in he flexible flow shop problem. The improvemen heurisic algorihms such as shif and pairwise inerchange algorihms have been proposed. Upon hese heurisics, he firs sage sequence is generaed by using hem. The firs sage sequence will now be used, in conjuncion wih eiher permuaion or FIFO rules, o consruc a schedule for he problem. The soluion is hen se equal o he bes funcion value obained by boh rules. The performance of he heurisics is compared relaive o he opimal soluion on a se of small-scale es problems. 1. Inroducion This paper is primarily concerned wih he scheduling problem occurring in he producion indusries such as he glass-conainer (Paul, [1]), rubber (Yanney and Kuo, [2]), phoographic film (Tsubone, Ohba, Takamuki, and Miyake, [3]), seel (Finke and Medeiros, [4]), exile, and food indusries. These indusries are esablished as mulisage producion flow shop faciliies where a producion sage may be made up of parallel machines known as flexible flow shop or hybrid flow shop environmen, ha is, i is a generalizaion of he classical flow shop model. There are k sages and some sages have only one machine, bu a leas one sage mus have muliple machines such ha all jobs have o pass hrough a number of sages in he same order. A machine can process a mos one job a a ime and a job can be processed by a mos one machine a a ime. Preempion of processing is no allowed. I consiss of assigning jobs o machines a each sage and sequencing he jobs assigned o he same machine so ha some cerain opimaliy crieria are minimized. Alhough he flexible flow shop problem has been widely sudied, mos of he sudies relaed o flexible flow shop problems are concenraed on problems wih idenical processors, see for insance, Gupa, Krüger, Lauff, Werner and Soskov [5], Alisanoso, Khoo, and Jiang [6], Lin and Liao [7] and Wang and Hunsucker [8]. However, in a real world siuaion, i is common o find newer or more modern machines running side by side wih older and less efficien machines. Even hough he older machines are less efficien, hey may be kep in he producion lines because of heir high replacemen coss. The older machines may perform he same operaions as he newer ones, bu would generally require a longer operaing ime for he same operaion. In his paper, he flexible flow shop problem wih unrelaed parallel machines is considered, i.e., here are differen parallel machines a every sage and speeds of he machines are dependen on he jobs. Moreover,

2 several indusries encouner seup imes which resul in even more difficul scheduling problems. In his paper, boh sequence- and machine-dependen seup ime resricions are aken ino accoun as well. Since mos scheduling problems are combinaorial opimizaion problems which are oo difficul o be solved opimally, heurisic mehods are used o obain good soluions in accepable imes. Consequenly, he purpose of his paper is o presen he consrucive heurisics based on boh he simple dispaching rules such as he Shorage Processing Time (SPT) and he Longes Processing Time (LPT) rules and he flow shop makespan heurisics such as he Palmer [9], Campbell, Dudek, and Smih [1], Gupa [11], and Dannenbring [12] algorihms as well as Nawaz, Enscore and Ham [13] algorihm adaped o solve he flexible flow shop problem wih unrelaed parallel machines and sequence-dependen seup imes. In addiion, he crieria of makespan and he number of ardy jobs have been receiving considerable aenion in his paper. One reason for his consideraion is he increasing pressure of high compeiion while cusomers expec ordered goods o be delivered on ime. The remainder of his paper is srucured as follows: in Secion 2, he problem under sudy is explained. Secion 3 shows he heurisic algorihms. The compuaional resuls are explained in Secion 4. The conclusions are shown in Secion Saemen of he The flexible flow shop sysem is defined by he se O = {1,,,, k} of k processing sages. A each sage, O, here is a se M = {1,, i,, m } of m unrelaed machines. The se J = {1,, j,, n} of n independen jobs has o be processed on a se M = {M 1,, M k }. Each job j, j J, has is release dae r j and a due dae d j. I has is fixed sandard processing ime for every sage, O. Owing o he unrelaed machines, he processing ime p of job j on machine i a sage is equal o ps v, where ps is he sandard processing ime of job j a sage, and v j j is he relaive speed of job j which is processed by he machine i a sage. There are processing resricions of jobs as follows: (1) jobs are processed wihou preempions on any machine; (2) a job canno be processed before is compleion of he previous operaion; (3) every machine can process only one operaion a a ime; (4) operaions have o be realized sequenially, wihou overlapping beween sages; (5) job spliing is no permied. Seup imes considered in his problem are classified ino wo ypes, namely machinedependen seup ime and sequence-dependen seup ime. A seup ime of a job is machinedependen if i depends on he machine o which he job is assigned. I is assumed o occur only when he job is he firs job assigned on he machine. ch denoes he lengh of he machinedependen seup ime, (or changeover ime), of job j if job j is he firs job assigned o machine i a sage. A sequence-dependen seup ime is considered beween successive jobs. A seup ime of a job on a machine is sequence-dependen if i depends on he job jus compleed on ha machine. s lj denoes he ime needed o changeover from job l o job j a sage, where job l is processed direcly before job j on he same machine. All seup imes are known and consan. The scheduling problem has dual objecives, namely minimizing he makespan and minimizing he number of ardy jobs. Therefore, he objecive funcion o be minimized is λc max + ( 1 λ)η T, where C max is he makespan, which is equivalen o he compleion ime of he las job o leave he sysem, η T is he oal number of ardy jobs in he schedule, and λ is he weigh (or relaive imporance) given o C max and η T, ( λ 1). 3. Heurisic Algorihms Heurisic algorihms have been developed o provide good and quick soluions. They obain soluions o large problems wih accepable compuaional imes, bu hey do no generae opimaliy and i may be difficul o judge heir effeciveness. They can be divided ino eiher consrucive or improvemen algorihms. The former algorihms build a feasible soluion from scrach. The laer algorihms ry o improve a previously generaed soluion by normally using

3 some form of specific problem knowledge. However, he ime required for compuaion is usually larger compared o he consrucive algorihms. 3.1 Heurisic Consrucion of a Schedule Since he hybrid flow shop scheduling problem is NP-hard, algorihms for finding an opimal soluion in polynomial ime are unlikely o exis. Thus, heurisic mehods are sudied o find approximae soluions. Mos researchers develop exising heurisics for he classical hybrid flow shop problem wih idenical machines by using a paricular sequencing rule for he firs sage. They follow he same scheme, see Sanos, Hunsucker, and Deal [14]. Firsly, a job sequence is deermined according o a paricular sequencing rule, and we will briefly discuss he modificaions for he problem under consideraion in he nex secion. Secondly, jobs are assigned as soon as possible o he machines a every sage using he job sequence deermined for he firs sage. There are basically wo approaches for his subproblem. The firs way is ha for he oher sages, i.e. from sage wo o sage k, jobs are ordered according o heir compleion imes a he previous sage. This means ha he FIFO (Firs in Firs ou) rule is used o find he job sequence for he nex sage by means of he job sequence of he previous sage. The second way is o sequence he jobs for he oher sages by using he same job sequence as for he firs sage, called he permuaion rule. Assume now ha a job sequence for he firs sage has already been deermined. Then we have o solve he problem of scheduling n jobs on unrelaed parallel machines wih sequence- and machine-dependen seup imes using his given job sequence for he firs sage. We apply a greedy algorihm which consrucs a schedule for he n jobs a a paricular sage provided ha a cerain job sequence for his sage is known (remind ha he job sequence for his paricular sage is derived eiher from he FIFO rule or from he permuaion rule), where he objecive is o minimize he flow ime and he idle ime of he machines. The idea is o balance evenly he workload in a heurisic way as much as possible. 3.2 Consrucive Heurisics In order o deermine he job sequence for he firs sage by some heurisics, we remind ha he processing and seup imes for every job are dependen on he machine and he previous job, respecively. This means ha hey are no fixed, unil an assignmen of jobs o machines for he corresponding sage has been done. Thus, for applying an algorihm for fixing he job sequence for sage one, an algorihm for finding he represenaives of he machine speeds and he seup imes is necessary. The represenaives of machine speed v / and seup ime s / for sage, =1, k, use he lj minimum, maximum and average values of he daa. Thus, he represenaive of he operaing ime / of job j a sage is he sum of he processing ime ps j v plus he represenaive of he seup ime s / lj. Nine combinaions of relaive speeds and seup imes will be used in our algorihms. The job sequence for he firs sage is hen fixed as he job sequence wih he bes funcion value obained by all combinaions of he nine differen relaive speeds and seup imes. For deermining he job sequence for he firs sage, we adap and develop several basic dispaching rules and consrucive algorihms for he flow shop makespan scheduling problem. Some of he dispaching rules are relaed o ardiness-based crieria, whereas ohers are used mainly for comparison purposes. The Shores Processing Time (SPT) rule is a simple dispaching rule, in which he jobs are sequenced in non-decreasing order of he processing imes, whereas he Longes Processing Time (LPT) rule orders he jobs in non-increasing order of heir processing imes. The Earlies Release Dae firs (ERD) rule is equivalen o he well-known firs-in-firs-ou (FIFO) rule. The Earlies Due Dae firs (EDD) rule schedules he jobs according o non-decreasing due daes of he jobs. The Minimum Slack Time firs (MST) rule is a variaion of he EDD rule. This rule concerns he remaining slack of each job, defined as is due dae minus he processing ime required o process i. The Slack ime per Processing ime (S/P) is similar o he MST rule, bu is slack ime is divided by he processing ime required as well (Baker [15], and Pinedo and Chao [16]). The hybrid SPT and EDD (HSE) rule is developed o combine boh SPT and EDD rules. Firsly, consider he processing imes of each job and deermine he relaive processing ime

4 compared o he maximum processing ime required. Secondly, deermine he relaive due dae compared o he maximum due dae. Nex, calculae he prioriy value of each job by using he weigh (or relaive imporance) given o C max and η T for he relaive processing ime and relaive due dae. Palmer s heurisic [9] is a makespan heurisic denoed by PAL in an effor o use Johnson s rule by proposing a slope order index o sequence he jobs on he machines based on he processing imes. The idea is o give prioriy o jobs ha have a endency of progressing from shor imes o long imes as hey move hrough he sages. Campbell, Dudek, and Smih [1] develop one of he mos significan heurisic mehods for he makespan problem known as CDS algorihm. Is srengh lies in wo properies: (1) i uses Johnson s rule in a heurisic fashion, and (2) i generally creaes several schedules from which a bes schedule can be chosen. In so doing, k 1 subproblems are creaed and Johnson s rule is applied o each of he sub-problems. Thus, k 1 sequences are generaed. Since Johnson s algorihm is a wo-sage algorihm, a k-sage problem mus be collapsed ino a wo-sage problem. Gupa [11] provides an algorihm denoed by GUP, in a similar manner as algorihm PAL by using a differen slope index and schedules he jobs according o he slope order. Dannenbring [12] denoed by DAN develops a mehod by using Johnson s algorihm as a foundaion. Furhermore, he CDS and PAL algorihms are also exhibied. Dannenbring consrucs only one wo-sage problem, bu he processing imes for he consruced jobs reflec he behavior of PAL s slope index. Is purpose is o provide good and quick soluions. Nawaz, Enscore and Ham [13] develop he probably bes consrucive heurisic mehod for he permuaion flow shop makespan problem, called he NEH algorihm. I is based on he idea ha a job wih a high oal operaing ime on he machines should be placed firs a an appropriae relaive order in he sequence. Thus, jobs are sored in non-increasing order of heir oal operaing ime requiremens. The final sequence is buil in a consrucive way, adding a new job a each sep and finding he bes parial soluion. For example, he NEH algorihm insers a hird job ino he previous parial soluion ha gives he bes objecive funcion value under consideraion. However, he relaive posiion of he wo previous job sequence remains fixed. The algorihm repeas he process for he remaining jobs according o he iniial ordering of he oal operaing ime requiremens. Again, o apply hese algorihms o he hybrid flow shop problem wih unrelaed parallel machines, he oal operaing imes for calculaing he job sequence for he firs sage are calculaed for he nine combinaions of relaive speeds of machines and seup imes. 3.3 Improvemen Heurisics Unlike consrucive algorihms, improvemen heurisics sar wih an already buil schedule and ry o improve i by some given procedures. Their uses are necessary since he consrucive algorihms (especially some algorihms ha are adaped from pure makespan heurisics and some dispaching rules such as he SPT, and LPT rules) wihou due dae consideraions. In his secion, some fas improvemen heurisics will be invesigaed o improve he overall funcion value by concerning mainly he due dae crierion. In order o find a saisfacory soluion of he due dae problem, he polynomial heurisics by applying eiher he shif move (SM) algorihm as an improvemen mechanism based on he idea ha we will consider he jobs ha are ardy and move hem lef and righ or he pairwise inerchange (PI) algorihm ha he ardy jobs are swapped o he differen job posiions lef and righ in boh random wo posiions (denoed by leer 2 ) and all oher posiions (denoed by leer A ). The bes schedule among he generaed neighbors is hen aken as he resul. 4. Compuaional Resuls Firsly, he overall consrucive algorihms and differen improvemen heurisics are sudied. The consrucive algorihms (denoed by leer CA ) are he simple dispaching rules such as he SPT, LPT, ERD, EDD, MST, S/P, and HSE rules, and he flow shop makespan heurisics adaped such as he PAL, CDS, GUP, DAN, and NEH rules. Then, we applied he polynomial improvemen heurisics based on four cases saed above in Secion 3.3. They are denoed by 2- SM, A-SM, 2-PI, and A-PI, respecively. We used problems wih beween hree o seven jobs. For

5 all problem s, we esed insances wih λ {,,,,,, and 1} in he objecive funcion. Ten differen insances for each problem have been run. An experimen was conduced o es wih daa such as he sandard processing imes, relaive machine speeds, seup imes, release daes and due daes. The sandard processing imes are generaed uniformly from he inerval [1,1]. Due o he unrelaed machine problem, he relaive speeds are disribued uniformly in he inerval [.7,1.3]. The seup imes, boh sequence- and machine-dependen seup imes, are generaed uniformly from he inerval [,5], whereas he release daes are generaed uniformly from he inerval beween and half of heir oal sandard processing ime mean. The due dae of a job is se in a way ha i is similar o he approach presened by Rajendaran and Ziegler [17] and is as follows: k d j = oal of mean seup ime of a job on all sages + ps j + (n 1) (mean processing ime of a job on one machine) U(,1) + r j The resuls for he consrucive algorihms and improvemen heurisics are given in Table 1. We give he average (absolue for λ = resp. percenage for λ > ) deviaion of a paricular consrucive algorihm from he opimal soluion obained by formulaing he -1 mixed linear ineger programming and running a commercial mahemaical programming sofware, CPLEX 8.. and AMPL, wih an Inel Penium 4 2.GHz CPU; however, he CPU ime is limied a mos 2 hours. Table 1 average overall performance of consrucive algorihms and improvemen heurisics. λ CA 2-SM A-SM 2-PI A-PI.34 a Sum b Sum Sum Sum λ = 1 CA 2-SM A-SM 2-PI A-PI Sum Sum Sum a average absolue deviaion for λ =, b average percenage deviaion for λ > Sum From hese resuls i is obvious ha he improvemen heurisics can improve he qualiy of consrucive algorihms by abou 6 7 percen. In addiion, we have found ha he all pairwise inerchange -based improvemen heurisics are beer han he ohers in general. Consequenly, in his paper we have shown only he all pairwise inerchange-based improvemen heurisics. However, compared o beween he 2-SM and 2-PI algorihms whose CPU ime is smaller han boh he A-SM and A-PI algorihms, we have found ha he 2-SM algorihm became beer han he 2-PI algorihm. Nex, we presen he consrucive algorihms ha are separaed ino four main groups. The firs heurisic group is he simple dispaching rules such as SPT, LPT, ERD, EDD, MST, S/P, and HSE. The second heurisic group is he flow shop makespan heurisics adaped such as PAL, CDS, GUP, DAN, and NEH. The hird and fourh heurisic groups are generaed from he firs wo groups of heurisics where he soluions are improved by he seleced polynomial improvemen

6 algorihm based on all pairwise inerchange-based improvemen heurisics, and hey are denoed by he firs leer I in fron of he leers describing he heurisics of he firs wo groups. The resuls for he consrucive algorihms are given in Table 2. From hese resuls i can be seen ha he consrucive algorihms in he fourh heurisic group improved from flow shop makespan heurisics from he second heurisic group (i.e., PAL, CDS, GUP, DAN, and NEH) are beer han he dispaching rules in he firs heurisic group (i.e., SPT, LPT, EDD, MST, S/P, and HSE) as well as he hird heurisic group improved from hem. Among he simple dispaching rules (heurisic Group I), he SPT, LPT, ERD, and HSE rules are good dispaching rules. However, in general he HSE rule slighly ouperforms he oher dispaching rules for λ, and he LPT rule is beer han he oher rules for oherwise. Among he adaped flow shop makespan heurisics in heurisic Group II, he NEH algorihm is clearly he bes algorihm among all sudied consrucive heurisics (bu in fac, his algorihm akes he convex combinaion of boh crieria ino accoun when selecing parial sequences). The CDS algorihm is cerainly he algorihm on he second rank (bu i is subsanially worse han he NEH algorihm even if he makespan porion in he objecive funcion value is dominan, i.e. for large λ values). Table 2 Average performance of consrucive algorihms. λ SPT LPT ERD EDD MST S/P HSE λ a Sum b Sum Sum Sum Sum Sum Sum Sum Table 3 Average performance of improvemen algorihms. PAL CDS GUP DAN NEH Sum Sum Sum Sum Sum Sum Sum Sum

7 λ ISPT ILPT IERD IEDD IMST IS/P IHSE Sum Sum Sum Sum Sum Sum Sum Sum a average absolue deviaion for λ =, b average percenage deviaion for λ > λ IPAL ICDS IGUP IDAN INEH Sum Sum Sum Sum Sum Sum Sum Sum When we apply he polynomial improvemen ( inerchange ) algorihm (denoed by he leer I firs) o he soluions obained by he dispaching rules and adaped flow shop makespan heurisics, we have found ha he qualiy of he soluion in erms of he deviaion from he opimal soluion can be improved by abou 7 percen excep for he NEH rule. However, he improvemen of heurisics from he adaped flow shop makespan heurisics in he heurisic Group IV is beer han he improvemen of heurisics from he dispaching rules in he heurisic Group III (since for mos of he problems, here is an subsanial porion in he objecive funcion value resuling from he makespan). In Group III, he IS/P rule ouperforms he ohers, whereas in Group IV, he algorihms can slighly improve he soluions obained by he NEH algorihm. 5. Conclusions In his paper, some consrucive algorihms have been invesigaed for minimizing a convex combinaion of makespan and he number of ardy jobs for he flexible flow shop problem wih unrelaed parallel machines and seup imes, which is ofen occurring in he real world problems. All algorihms are based on he lis scheduling principle by developing job sequences for he firs

8 sage and assigning and sequencing he remaining sages by boh he permuaion and FIFO approaches. The consrucive algorihms are compared o he opimal soluions. I is shown ha in paricular, for he simple dispaching rules he SPT, LPT, ERD, and HSE rules are good algorihms whereas for he flow shop makespan heurisics, he NEH algorihm is mos superior o he oher consrucive algorihms. When we have applied he polynomial improvemen algorihm, we have found ha he all-pairwise inerchange algorihm is he good improvemen algorihm. However, here are some deviaions from he opimal value, so in he furher research, he ieraive heurisics such as simulaed annealing, abu search and geneic algorihms will be able o sudy in order o find he bes soluion. References [1] R.J.A. Paul. Producion scheduling problem in he glass-conainer indusry. Operaions Research. 27(2): page 29 32, [2] J.D. Yanney, and W. Kuo. A pracical approach o scheduling a mulisage, muliprocessor flow-shop problem. Inernaional Journal of Producion Research. 27(1), page , [3] H. Tsubone, M. Ohba, H. Takamuki, and Y. Miyake. Producion scheduling sysem in he hybrid flow shop. Omega. 21(2): page , [4] D.A. Finke and D.J. Medeiros. Shop scheduling using abu search and simulaion. Proceedings of he 22 Winer Simulaion Conference. page , 22. [5] J.N.D. Gupa, K. Krüger, V. Lauff, F. Werner, and Y.N. Soskov. Heurisics for hybrid flow shops wih conrollable processing imes. Compuers and Operaions Research. 29(1): page , 22. [6] D. Alisanoso, L.P. Khoo, and P.Y. Jiang. An immune algorihm approach o he scheduling of a PCB flexible flow shop. The Inernaional Journal of Advanced Manufacuring Technology. 22(11 12): page , 23. [7] H.T. Lin, and C.J. Liao. A case sudy in a wo-sage hybrid flow shop wih seup ime and dedicaed machines. Inernaional Journal of Producion Economics. 86(2): page , 23. [8] W. Wang, and J.L. Hunsucker. An evaluaion of he CDS heurisic in flow shops wih muliple processors. Journal of he Chinese Insiue of Indusrial Engineers. 2(3): page , 23. [9] D.S. Palmer. Sequencing jobs hrough a muli-sage process in he minimum oal ime--a quick mehod of obaining a near opimum. Operaional Research Quarerly. 16(1): page 11-17, [1] H.G. Campbell, R.A. Dudek and M.L. Smih. A Heurisic algorihm for he n-job m-machine sequencing problem. Managemen Science. 16(1): page , 197. [11] J.N.D. Gupa. A funcional heurisic algorihm for he flow-shop scheduling problem. Operaions Research Quarerly. 22(1): page 39-47, [12] D.G. Dannenbring. An evaluaion of flow shop sequencing heurisics. Managemen Science. 23(11): page , [13] M. Nawaz, E. Enscore, Jr., and I. Ham. A heurisic algorihm for he m-machine, n-job flowshop sequencing problem. Omega Inernaional Journal of Managemen Science. 11(1): page 91 95, [14] D.L. Sanos, J.L. Hunsucker and D.E. Deal. An evaluaion of sequencing heurisics in flow shops wih muliple processors. Compuers & Indusrial Engineering. 3(4): page , [15] K.R. Baker. Inroducion o Sequencing and Scheduling. John Wiley & Sons, New York, [16] M. Pinedo, and X. Chao. Operaions scheduling wih applicaions in manufacuring and services. Irwin/McGraw-Hill, New York, [17] C. Rajendran, and H. Ziegler. Scheduling o minimize he sum of weighed flowime and weighed ardiness of jobs in a flowshop wih sequence-dependen seup imes. European Journal of Operaional Research. 149(3): page , 23.