Available online at ScienceDirect. Procedia CIRP 17 (2014 )

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1 Avalable onlne at ScenceDrect Proceda CIRP 7 (4 ) 48 4 Varety Management n Manufacturng. Proceedngs of the 47th CIRP Conference on Manufacturng Systems Robust Metaheurstcs for Schedulng Cellular Flowshop wth Famly Sequence- Dependent Setup Tmes Al-mehd Ibrahem a*, Tarek Elmekkawy b and Qngjn Peng c a,c Department of Mechancal and Manufacturng Engneerng, Unversty of Mantoba, Wnnpeg, RT N, Canada b Department of Mechancal and Industral Engneerng, Qatar Unversty, Doha, Qatar * Correspondng author. Tel.: ; fax: E-mal address: umalmehd@cc.umantoba.ca Abstract In manufacturng systems, mnmzaton of the total flow tme has a great mpact on the producton tme, the productvty and the proftablty of a frm. Ths paper consders a cellular flowshop schedulng problem wth famly sequence setup tme to mnmze the total flow tme. Two metaheurstc algorthms based on Genetc algorthm () and partcle swarm optmzaton () are proposed to solve the proposed problem. As t s customarly accepted, the performance of the proposed algorthms s evaluated usng Desgn of Experments (DOE) to study the robustness of the proposed metaheurstcs based on the Relatve Percentage Devaton (RPD) from the lower bounds. The results of the DOE evaluaton of the proposed algorthms show that -based metaheurstc s better than for solvng schedulng problems n cellular flow shop, whch ams to mnmze the total flow tme. 4 Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense 4 The Authors. Publshed by Elsever B.V. ( Selecton and peer-revew under responsblty of the Internatonal Scentfc Commttee of The 47th CIRP Conference on Manufacturng Systems Selecton n and the peer-revew person of the under Conference responsblty Char Professor of the Internatonal Hoda ElMaraghy. Scentfc Commttee of The 47th CIRP Conference on Manufacturng Systems n the person of the Conference Char Professor Hoda ElMaraghy Keywords: Cellular manufacturng flowshop; Sequencng-dependent setup; Total flow tme; Genetc Algorthm; Partcle Swarm Optmzaton. Introducton The challenges faced by manufacturng companes have forced them to place a sgnfcant value on effcency, tmng, and cost to reman compettve []. Cellular manufacturng system (CMS) ams to optmze the effcency, tmng, and cost effectveness. In flowshop manufacturng, cells usually consst of a number of machnes that are dedcated to produce a specfc group of part famles that have smlar producton requrements. In a manufacturng cell, parts (jobs) havng smlar toolng or requred setup on machnes are assgned to be processed as one famly. Ths s done, n order to mprove the effcency of the producton cell. The requred setup tme on a machnes used for swtchng between jobs that belong to the same part famly s usually consdered as a part of processng tme of these jobs. Ths s because the setup requred for swtchng the process between part famles s qute sgnfcant. Further, the setup tme depends on the famly to be processed and the precedng part famly. Such a problem s called cellular flowshop schedulng problem wth sequence dependent setup tmes. Readers may refer to Allahverd et al [] for a comprehensve study of the schedulng problems wth setup tmes. Broadly, effcent job schedulng s a crucal aspect of any manufacturng envronment []. Schedulng a flowshop cellular manufacturng wth famly setup tmes has been the subject of several studes reported n the lterature, whch was ntated by Schaller et al [4]. They developed several heurstc algorthms wth mnmzaton of the makespan as the crtera. Further, they developed a lower boundng method to evaluate Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense ( Selecton and peer-revew under responsblty of the Internatonal Scentfc Commttee of The 47th CIRP Conference on Manufacturng Systems n the person of the Conference Char Professor Hoda ElMaraghy do:.6/j.procr.4..7

2 Al-mehd Ibrahem et al. / Proceda CIRP 7 ( 4 ) the soluton qualty of the proposed algorthms. Franca et al [5] developed the Genetc Algorthm () and Memetc algorthm (MA) wth local search for the makespan mnmzaton performance measure. The authors concluded that the soluton qualty of the MA outperformed the avalable algorthms. Hendzadeh et al [6] developed a meta-heurstc algorthm based on Tabu Search (TS) for the proposed problem to mnmze the makespan. They concluded that the soluton qualty s the same as that proposed by Franca et al[5]. Ln et al [7] studed the same problem and proposed a smulated annealng based meta-heurstc algorthm to mnmze the makespan as the crteron. Salmas et al [8] nvestgated the aforementoned problem for total flow tme mnmzaton for the frst tme; they then developed two metaheurstcs based on Tabu Search and Ant Colony Optmzaton algorthm to mnmze the total flow tme as the crteron for the frst tme. They clamed that the algorthm has a superor performance n comparson the TS algorthm. Ther algorthm s consdered as the best avalable metaheurstc. Moreover, tght lower bounds for makespan mnmzaton and total flow tme mnmzaton are developed by Salmas et al [8]and [9]. Later, Nader, and Salmas [] developed two dfferent mxed nteger lnear programmng models. These models were effectve n solvng even medumszed problems and provdng the optmum soluton n a reasonable tme. As t hghlghted n the prevous paragraphs, most of the research effort has focused on developng approxmaton algorthms to mnmze the makespan. However, the total flow tme mnmzaton s an mportant measure that needs to be consdered. Optmzng total flow tme reflects a stable utlzaton of resources, reducng the work-n-process nventory, and mnmzng the setup tme costs. In addton, the research reported n the lterature evaluates the qualty of the soluton based on the Percentage Devaton Error (PDE) from the lower bounds. In our vew, accordng to El-Ghazal (9) [], the analyss of metaheurstcs should be done n three steps: expermental desgn, measurement (e.g., qualty of solutons, computatonal effort, and robustness), and reportng (e.g., box plots, nteracton plots) []. In our research, two metaheurstcs based on Genetc Algorthm () and Partcle Swarm Optmzaton () are developed to mnmze the total flow tme n a cellular flowshop schedulng problem wth famly sequence setup tme. Moreover, the performance of the proposed algorthms s evaluated to fnd the most robust algorthm by means of the Desgn of Experments (DOE) approach. Ths paper ams to present the paradgm to dentfy the most robust metaheurstc n a cellular flowshop schedulng problem.. Problem descrpton The focus of ths research s to schedule a flowshop cellular manufacturng cell wth sequence dependent famly setup tmes wth the am of mnmzng the total flow tme whch s notated as: Fm fmls, Splk, prum C j. The objectve functon of the proposed problem s: Mnmze N C jm, j ( ) To descrbe the problem, we frst defne the followng notatons: F: Number of the famles, N : Number of jobs; N f : Number of the jobs n each famly, f ={,,..,F} M: Number of machnes; m: Index for machnes, ={,,.,m} j, k: Index for jobs, j, k ϵ {,,.,N } f,l : Index for famles, f,l ϵ {,,.,F} p j, : Processng tme for job j on machne Se l,f,m : Setup tme of famly f f t processed mmedately after the famly l on machne The major assumptons are nvoked n ths study are: The number of parts (jobs), ther processng tmes, the number of part famles, and ther setup tmes are known n advance. The sequence of parts n a part famly, and part famles are the same on all machnes (permutaton schedulng). Once a part starts to be processed on a machne, the process cannot be nterrupted before completon (preempton s dsallowed). Each machne can handle only one part (job)at a tme. The jobs belongng to each famly should be processed wthout any preempton by other jobs of other famles (group technology assumpton). The ready tme of all parts s zero, whch means that all parts n all part famles are avalable for processng at the start tme. Setup tme depends on both the part to be processed and ts precedng part. There s mnor setup among parts wthn a famly whereas there are major setup tmes among the part famles. In general, a soluton of the cellular flowshop schedulng problem s performed n two phases: sequencng of the part famles and sequencng parts (jobs) wthn each famly. In other words, the soluton representaton conssts of F + segments. Whle the frst segment F represents the sequence of part famles on each machne, the other segments correspond to the sequence of parts (jobs) wthn each part famly[]. For a feasble schedule, the sequence of the famles s: π = {π [], π [] π [f -], π [f ], π [f +],.., π [F] } Where π [f ] ={ δ [f ][], δ [f ][], δ [f ][ j],.., δ [f ][Nf] } s the sequence of the jobs wthn the famly. Two metaheurstcs whch are based on Genetc algorthm () and partcle swarm optmzaton () are developed to solve the proposed problem heurstcally.. Genetc Algorthm Genetc Algorthm () has successfully been appled to the schedulng problems. It conssts of makng a populaton of solutons. These ndvduals are evolved by mutaton and reproducton processes. The best ftted solutons of the populaton survve whle the worse ftted wll be replaced [5]. The basc steps of the proposed are as follows:

3 4 Al-mehd Ibrahem et al. / Proceda CIRP 7 ( 4 ) Selecton Step : Intalze the populaton by generatng ntal solutons. Step : Assess each soluton n the populaton based on the total flow tme and assgnng a ftness value. Step : Select ndvduals for recombnaton and go to step 4. Step 4: Recombne ndvduals generatng new ones and enter step 5. Step 5: Mutate the new ndvduals and go to step 6. Step 6: If the stoppng crteron s met, STOP, otherwse, replace old ndvduals wth the new ones, and return to step. The proposed algorthm starts wth an ntal soluton that s generated randomly from the populaton (Pop. sze). After that, the ftness functon s estmated based on the objectve functon whch s used to mnmze the total completon tme or total flow tme (TFT). The ftness functon s equal to the nverse of TFT. Hence each soluton, namely soluton π, the desrable values of the ftness functon s the hgher values, and the normal probablty can be determned by the followng formula []: PopSze TFT ( ) TFT ( X ) X Based on the ftness values, two solutons are randomly pcked for the reproducton process usng the crossover and mutaton operators... Crossover Operator The crossover operator s used to fnd better solutons by recombnng good genes from dfferent parent chromosomes. The crossover operator combnes the selected solutons to buld two new ones. For nstance, a random soluton of the aforementoned problem can be represented n two segments: the sequence of the famles, and the sequence of jobs n each famly. In ths paper, crossover has been performed smlar to the crossover presented by Franca et al [5] and Hendzadeh et al []. A new offsprng s constructed from two parents as follows: A segment of parent one s randomly selected and coped n the same poston n the new offsprng. The rest of the frst part of the offsprng s completed or flled from the second parent []... Mutaton Operator The mutaton operator generates offsprng from a sngle parent. Frst, two random postons n each part of a parent are chosen and then ther postons are swapped. The offsprng s compared wth ts correspondng parent. Ths dfferent from () the tradtonal, where ether computaton tme or the number of generatons s selected as termnaton crteron [4]. In ths paper, ntensve the parameters are tuned usng Desgn of Experments (DOE) to fnd the best combnaton of the algorthm parameters. We concluded that the populaton sze s, the maxmum CPU 6 sec, mutaton rate s two, and crossover rate s Partcle Swarm Optmzaton Partcle Swarm Optmzaton () s a novel teratve computatonal evoluton model that was developed by Kennedy and Eberhart [5]. They smulated brds swarm behavour n, and made every partcle n the swarm move accordng to ts experence and the best partcle s experence to fnd a new better poston. After the evoluton, the best partcle n the swarm was seen as the best soluton for the nput problem. The populaton of s called swarm, and each ndvdual or partcle whch s a potental soluton s known wth ts current poston and current velocty. The new poston of each ndvdual partcle s obtaned by assgnng a new poston as well as a new velocty to the partcle. Each partcle gans a dfferent poston, and the value of each poston s evaluated based on the objectve functon value of the poston. The man advantage of ths approach s that every partcle always remembers ts best poston n the experence. When a partcle moves to another poston, t must refer to ts best experence and the best experence of all partcles n the swarm. The best poston of each partcle that has been ganed so far durng the prevous steps s named the best partcle (p-best). The best poston ganed by all partcle so far s named global best (g-best) poston of the search. The new poston as well as the new velocty of each partcle s obtaned based on the prevous postons, the p-best, and the g-best. Consderng an n-dmenson search space; there are S partcles (swarm sze) cooperatng to fnd the global optmum n the search space. A warm of S partcles, the th partcle s assocated wth the x x, x,.., x, and the velocty poston vector n v v, v,.., v n the best soluton s obtaned by the th partcle (p-best), and the best so-far soluton s obtaned by whole swarm (global best) are updated each teraton. Each partcle uses ts own search experence and the global experence by the swarm to update the velocty and fles to a new poston based on the followng equatons: * j j v t wv t C r y t x t j j j j C r Y t x t x t v t x t (4) j j j Where s the nerta weght that controls the nfluence of the prevous velocty of partcles, C, C are called acceleraton coeffcent to weght the socal nfluence. The parameters r, and r are unformly dstrbuted random varables n (, ), and. Partcles fly n the search space based on Eqs. () and (4). Every partcle always remembers ts best ()

4 Al-mehd Ibrahem et al. / Proceda CIRP 7 ( 4 ) poston n the experence, where,,., S, and j,,., n. When a partcle moves to another poston, new velocty s calculated accordng to the prevous velocty and the dstance of ts poston both p-best and g-best. However, the new velocty s lmted to the range [ Vmn, V max ] to control the extreme travelng of partcles outsde the search space. Partcles gan ther new postons accordng to the new velocty and the prevous poston (Eq.4). These contnuous postons of the partcles wll be converted nto dscrete values usng the Ranked Order Value (ROV). It converts the contnuous poston value of the partcles to job and famly sequences. For more detals, the reader may refer to Lu et al [6]. The steps of the algorthm are executed as follows: Step: Intalze parameters. Step: Iteraton =;. Generate ntal poston, ntal velocty: { X,,,, n} and { V,,,, n}. Apply the (ROV) to fnd the permutaton;. Evaluate each partcle based on the objectve functon{ f,,,, n} ;.4 For each partcle n the swarm usng the objectve functon f for,,..., n for each partcle set: P X, wherep p x, p x,., pn x n pb together wth ts best ftness value f for,,... n.5 Fnd the best ftness value among the whole swarm such as f mn {f } for,,, n.6 Set the global best to l G g xl,, g xl,,., gn xl, n G X l such that wth ts gb ftness value f f l Step: Repeat untl Iteraton =Maxmum Iteraton Step 4: Update velocty v t wv t C r y t x t j j j j * j j C r Y t x t Step 5: Update poston x tv t x t j j j Step6: Apply the (ROV) to fnd the permutaton Step7: Update personal best t t Step8: Update global best to G Xl Step9: If the stoppng crteron s satsfed, STOP; otherwse, return to step. The algorthm starts wth generatng the ntal velocty and poston accordng to the followng equatons: kj k X X rand X X mn max mn V V rand V V mn max mn Where X mn and X max represent the mnmum and maxmum poston values. V mn And V max represent the mnmum and (5) (6) maxmum veloctes. Also Rand s a random varable between (, ). These values are presented n Table. Table : The Mn and Max. values of the parameters. Parameter Max. Values Mn. Value W.4 C.4 C.4 Poston value 4 Velocty -4 4 To evaluate the performance of the proposed algorthms, test problems developed by Salmas [9] have been used n ths research.. Three dfferent parameters defne a problem structure. These nclude: number of machnes, number of famles, and maxmum job n famly. The test problems are classfed nto small, medum, and large sze problems wth the number of famles s a random nteger from DU [, 5], DU [6, ], and DU [, 6] respectvely. The number of jobs n a famly s also a random nteger from DU [, 4], DU [5, 7], and DU [8, ] for small, medum, and large problems respectvely. Also, the test problems are classfed nto two and sx machne problems. 5. Robustness A schedule s consdered robust f ts qualty s only slghtly senstve to data uncertantes and to unexpected events [7]. Furthermore, robustness means the evaluaton process that s able to determne the best or the most effcent algorthm that gves a hgh qualty soluton. Samarghand et al [8] reported that the most mportant feature of Desgn of Experments (DOE) s ts ablty to study the nteracton effects between the consdered factors.. The levels correspondng to each parameter are presented n Table. Table Factor levels. Factor Level Level Level Max. Job. n Famly N. Famly Machne Results and dscusson In ths secton, we wll compare the performance of two metaheurstcs based on and. The comparson s conducted to answer the followng questons: ) whch algorthm s more robust? ) What are the factors affectng the response? To answer the questons, a statstcal analyss based on (DOE) s conducted. The response s based on the Relatve Percentage Devaton (RPD) that measures the devaton of the total flow tme from the correspondng lower bound developed by Salmas [8]. The RPD s calculated usng the followng formula: TFT LB RPD (7) LB

5 4 Al-mehd Ibrahem et al. / Proceda CIRP 7 ( 4 ) 48 4 Ths study showed that the number of famles has a more sgnfcant mpact than the other factors. Nevertheless, maxmum number of jobs n a famly sgnfcantly affected the RPD values when the numbers of famles s large as shown n Fgures, and. Source Table : ANOVA results Degrees of freedom Seq. Sums of Squares Adj. Sum of Squares Adj. Mean Square F- value P- value Max. Job. n Famly N. Famly Machne Algorthm Error Total.8744 Furthermore, based on the P-values, analyss of varance (ANOVA) showed that the most sgnfcant factor s the number of famles snce ts P-value s close to zero. Yet, as shown n Table, most of other factors are nsgnfcant snce ther values are greater than 5. Relatve Percentage Devaton Man Effects Plot for RPD N.Famly Max.Jops In Fam. Machne 6 Algorthm Fgure : The man effect plot Moreover, studyng the nteracton among the factors gves a better understandng about the effect of several smultaneous factors. As shown n Fgure, s better than f the number of famles s small and medum. Yet, large famles are more practcal n manufacturng. Even though the maxmum jobs n famles have greater effect on the RPD n both metaheurstcs as shown n Fgure 4, shows better performance than. It should also be noted that metaheurstc algorthm based on s more effcent than that based on. For nstant, for the two-machne problems, the varaton n RPD for s more than for as shown n Fgure 5. Yet, gves some solutons that are better than the ones obtaned by. However, n terms of the varaton, shows less varaton than. RPD Num.Famles Max.Jops In Famly Fgure : Surface Plot of RPD vs Max.Jops In Famly, and No. of Famles Relatve Percentage Devaton Algorthm L(-5) L(6-) Number of Famles L(-6) Fgure : Interacton plots of maxmum jobs n famly vs. the proposed algorthms Relatve Percentage Devaton Algorthm L(-4) L(5-7) Max.Jobs In Famly L(8-) Fgure 4: Interacton plots of maxmum jobs n famly vs. the proposed algorthms. In addton, there s a consderable dfference n the varaton n RPD between the two metaheurstcs for the sxmachne problems as shown n Fgure 6, ndcatng that s performed better than. Therefore, s more effcent than n solvng the cellular flowshop to mnmze the total flow tme.

6 Al-mehd Ibrahem et al. / Proceda CIRP 7 ( 4 ) Relatve Percentage Error Relatve Percentage Error Interval Plot of RPE for Machne problems 95% CI for the Mean Algorthm Fgure 5:Interval plot for two-machne problems. Interval Plot of RPE for 6Machne problems 95% CI for the Mean Algorthm Fgure 6: Interval plot for sx-machne problems 7. Concluson In ths paper, we examned two metaheurstcs to fnd robust schedules that mnmzng total flow tme n cellular flowshop schedulng problems wth sequence dependent setup tmes. Metaheurstc algorthms based on and s selected due to ther successful mplementaton n schedulng. The Robustness of these metaheurstcs are compared Desgn of Experments (DOE) technque based on the Relatve Percentage Devaton from the correspondng lower bounds proposed by Salmas et al [8]. It was found the proposed algorthm s more robust than even though hgh qualty solutons can be obtaned by based on the problems examned n ths study. Future research on ths study s to study the robustness of any other metaheurstc algorthms or other performance measures lke makespan. Addtonally, the proposed approach can be mplemented n mult-objectve optmzaton problems. [4] Schaller J, Gupta JND, Vakhara AJ. Schedulng a flowlne manufacturng cell wth sequence dependent famly setup tmes. Eur J Oper Res ;5:4 9. [5] Franca P, Gupta J, Mendes A, Moscato P, Veltnk K. Evolutonary algorthms for schedulng a flowshop manufacturng cell wth sequence dependent famly setups. Comput Ind Eng 5;48: [6] Hamed Hendzadeh S, Faramarz H, Mansour SA, Gupta JND and, Elmakkawy T. Meta-heurstcs for schedulng a flowlne manufacturng cell wth sequence dependent famly setup tmes. Int J Prod Econ 8;: [7] Ln S-W, Gupta JND, Yng K-C, Lee Z-J. Usng smulated annealng to schedule a flowshop manufacturng cell wth sequence-dependent famly setup tmes. Int J Prod Res 9;47:5 7. [8] Salmas N, Logendran R, Skandar MR. Total flow tme mnmzaton n a flowshop sequence-dependent group schedulng problem. Comput Oper Res ;7:99. [9] Salmas N, Logendran R, Skandar MR. Makespan mnmzaton of a flowshop sequence-dependent group schedulng problem. Int J Adv Manuf Technol. [] Nader B, Salmas N. Permutaton flowshops n group schedulng wth sequence- dependent setup tmes. Eur J Ind Eng ;6: [] El-Ghazal T. Metaheurstcs: from desgn to mplementaton. Jonh Wley Sons Inc, Chchester 9. [] Bouabda R, Jarbou B, Reba A. A nested terated local search algorthm for schedulng a flowlne manufacturng cell wth sequence dependent famly setup tmes. (LOGISTIQUA), 4th :56. [] Hendzadeh H, Elmekkawy T, Wang G. B-crtera schedulng of a flowshop manufacturng cell wth sequence dependent setup tmes. Eur J Ind Eng 7;: [4] Zhang Y, L X, Wang Q. Hybrd genetc algorthm for permutaton flowshop schedulng problems wth total flowtme mnmzaton. Eur J Oper Res 9;96: [5] Kennedy J, Eberhart R. Partcle swarm optmzaton. Proc ICNN 95 - Int Conf Neural Networks 995;4:94 8. [6] Lu B, Wang L, Jn Y-H. An effectve hybrd -based algorthm for flowshopschedulng wth lmted buffers. Comput Oper Res 8;5: [7] Wang L, Ng AHC, Deb K, edtors. Mult-objectve Evolutonary Optmsaton for Product Desgn and Manufacturng. London: Sprnger London;. [8] Samarghand H, Elmekkawy TY, Ibrahem AM. Studyng the effect of dfferent combnatons of tmetablng wth sequencng algorthms to solve the no-wat job shop schedulng problem. Int J Prod Res ;5-6: References [] Denne JS. Effcent job schedulng for a cellular manufacturng envronment. Rochester Insttute of Technology, 6. [] Allahverd A, Ng C, Cheng T, Kovalyov M. A survey of schedulng problems wth setup tmes or costs. Eur J Oper Res 8;87:985. [] Shyas CR, Madhusudanan Plla V. Cellular manufacturng system desgn usng groupng effcacy-based genetc algorthm. Int J Prod Res 4: 4.