Integrated approach in solving parallel machine scheduling and location (ScheLoc) problem

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1 Internatonal Journal of Industral Engneerng Computatons 7 (2016) Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: Integrated approach n solvng parallel machne schedulng and locaton (ScheLoc) problem Mohsen Raabzadeh a, Mohsen Zaee b and Al Bozorg-Amr c* a M.Sc., Industral Engneerng Group, College of Engneerng, Unversty of Bonord, Bonord, Iran b Assstant Professor, Industral Engneerng Group, College of Engneerng, Unversty of Bonord, Bonord, Iran c School of Industral Engneerng, College of Engneerng, Unversty of Tehran, Tehran, Iran C H R O N I C L E A B S T R A C T Artcle hstory: Receved November Receved n Revsed Format Aprl Accepted Aprl Avalable onlne Aprl Keywords: Schedulng Layout plannng Parallel machnes Schedulng and layout plannng are two mportant areas of operatons research, whch are used n the areas of producton plannng, logstcs and supply chan management. In many ndustres locatons of machnes are not specfed, prevously, therefore, t s necessary to consder both locaton and schedulng, smultaneously. Ths paper presents a mathematcal model to consder both schedulng and layout plannng for parallel machnes n dscrete and contnuous spaces, concurrently. The prelmnary results have ndcated that the ntegrated model s capable of handlng problems more effcently Growng Scence Ltd. All rghts reserved 1. Introducton Whle n many schedulng problems, the locatons of machnes are fxed t s possble to show how to consder locaton and schedulng problems smultaneously. Obvously, ths ntegrated method enhances the modelng power of schedulng for dfferent real-lfe problems (Heßler & Deghda, 2015). Hennes and Hamacher (2002) are beleved to be the frst who ntroduced the dea of schedulng and layout plannng. In ther study, schedules and locaton are examned on a graph consstng of n nodes and m edges where, each node s consdered as a storage locaton for a ob. The prmary obectve was to fnd the optmal locaton and schedule of a sngle machne by mnmzng the maxmum completon tme n two dfferent scenaros. The frst scenaro consders locaton of faclty could be consdered only on edges whle n the second scenaro, locatons can be consdered on other parts. Elvs et al. (2007, 2009) presented some polynomal soluton methods for the planar ScheLoc maespan problem, whch ncorporates an specal nd of a schedulng and a rather general, planar locaton problem, respectvely. * Correspondng author. Tel: E-mal: albozorg@ut.ac.r (A. Bozorg-Amr) 2016 Growng Scence Ltd. All rghts reserved. do: /.ec

2 574 Kalsch and Drezner (2010) ntegrated both the locaton of the machne and the schedulng of the obs executed by the machne by analyzng two dfferent obectves of the maespan and the total completon tme. They consdered some mportant propertes of the models and provded a lower bound for the obectve functons. The sngle machne ScheLoc problems wth Eucldean, rectlnear and general lq norms were solved by the bg trangle small trangle branch-and-bound technque (Lawler, 1973; Drezner & Suzu, 2004). Accordng to Scholz (2012a,b), geometrc branch-and-bound approaches wth mxed contnuous and combnatoral varables are more sutable soluton methods for ScheLoc problems. Table 1 shows dfferent sngle machne ScheLoc problems tacled by varous authors. Table 1 Dfferent studes n machne schedulng and locaton (ScheLoc) problem Researchers Hennes & Hamacher, 2002 Soluton space Obectve functon Producton envronment dscrete Networ Plane C max TC Other Sngle machne Parallel machne Other Elvs et al., Elvs et al., Kalsch & Drezner, Scholz, Present study The proposed study ScheLoc s assocated wth schedulng and layout plannng of a certan obs on some unque machnes wth determned processng tmes. All obs are avalable on machnes accordng to the followng, Dstance between storage and machne The tme obs are avalable for processng on machne = The tme obs are avalable at storage+ Speed of transportaton facltes The prmary obectve s to mnmze the maespan. The problem can be consdered n two forms of dscrete and contnues Dscrete ScheLoc In ths case, machnes can be located on specal places. For nstance, as shown n Fg. 1, fve alternatve locatons are consdered for two machnes and four obs are stored n nventory wth specfed processng tmes Job Warehouse 3 4 Canddate poston Fg. 1. The dscrete parallel machne ScheLoc problem

3 Sets I Set of all obs J Set of alternatve locatons for machnes Indces, Jobs wth ( 1,2,..., I ) Locatons of dfferent machnes ( 1, 2,..., J ) Parameters ma Number of smlar machnes p Processng tme of ob l dst r M Tme of avalablty of ob n storage M. Raabzadeh et al. / Internatonal Journal of Industral Engneerng Computatons 7 (2016) 575 elocty of movng ob from storage to machnes Dstance between ob to canddate locaton Tme of readness of ob on machne r A bg number calculated by M c Completon tme of ob arables c Maespan max dst l (1) I, p 1 max (r ) y Bnary varable, whch s one f one of machnes s located on alternatve, zero, otherwse x Bnary varable, whch s one f ob s assgned to machne, zero, otherwse z Bnary varable, whch s one f ob ' s processed before ob, zero, otherwse The mathematcal problem s stated as follows, mn Z=c max subect to J 1 J 1 y x ma 1 c p r M(1 x ), c p c M(1 z ) M(2 x x ), ( ), c p c M( z ) M(2 x x ), ( ), x y, c c max c 0, c 0; y, x,z 0,1 max As we can observe from the proposed model, the obectve functon gven n Eq. (2) mnmzes the maespan. Eq. (3) nsures that all machnes are assgned whle Eq. (4) determnes that each ob has to be executed only on sngle machne. Eq. (5) specfes that ob can only be started when t reaches to machne. Eq. (6) and Eq. (7) specfy that ob can be started when the process of the prevous ob was already fnshed. Accordng to Eq. (8), a ob can be processed only on a partcular machne. Eq. (9) computes the maespan and fnally Eq. (10) demonstrates the type of varables. The problem can be examned under two scenaros. For the frst scenaro, layout plannng s accomplshed by mnmzng sum of the tmes for all obs assgned to machnes and then schedulng of obs on machnes are determned. Ths problem s frst formulated as follows, (2) (3) (4) (5) (6) (7) (8) (9) (10)

4 576 dst mn Z= ( ) y J I l 1 1 subect to constrant 3 y 0,1. When all y are determned, the optmal values are denoted as mn Z=c max subect to constrants (4)(5)(6)(7)(9) x y, c 0, c 0; x, z 0,1 max For the second scenaro, we frst solve the followng problem J I dst mn Z= ( l ) x 1 1 subect to constrants (3)(4)(8) y, x 0,1 y and the followng problem s solved, (11) (12) (13) (14) (15) (16) (17) Now, the obs are sorted accordng to non-decreasng order of avalablty tmes. In case, there are two obs wth the same avalablty tmes, the one wth less processng tme s consdered and then the obs are assgned to machnes Contnous ScheLoc In ths case, machnes can be assgned anywhere n ob floor as shown n Fg. 2 as follows, Job Warehouse Fg. 2. Contnuos parallel machne ScheLoc problem For ths problem, we defne the followng assumptons Sets I K Indces Set of all obs Set of all machnes

5 M. Raabzadeh et al. / Internatonal Journal of Industral Engneerng Computatons 7 (2016) 577 Jobs wth ( 1,2,..., I ) Locatons of dfferent machnes ( 1, 2,..., J ) Parameters n Number of obs ma p l ( a, b ) M arables dst r c max Number of smlar machnes Processng tme of ob Tme of avalablty of ob n storage elocty of movng ob from storage to machnes Locaton of storage for ob A bg number Dstance between locaton of machn and storage of ob Tme of readness of ob on machne r Maespan ( x, y ) Locaton of machne w If ob s assgned to machne, zero, otherwse z dst l Bnary varable, whch s one f ob ' s processed before ob, zero, otherwse The proposed study s formulated as follows, mn Z=c max subect to K 1 w 1 dst c p l M(1 w), c p c M(1 z ) M(2 w w ), ( ), c p c M( z ) M(2 w w ), ( ), c c max c 0, c 0; w, z 0,1 max (20) (21) (22) (23) (24) (25) (26) The obectve functon mnmzes the maespan, Eq. (21) ensures that each ob s processed only on one machne, Eq. (22) specfes that ob can only be started when t reaches to machne. Eq. (23) and Eq. (24) specfy that ob can be started when the process of the prevous ob was already fnshed. Eq. (25) computes the maespan and fnally Eq. (26) demonstrates the type of varables. In ths study, dstances are computed as follows, dst x a y b, (27) and Eq. (27) s lnearalzed as follows, x a u v, (28)

6 578 y b g h. Therefore, the followng equatons are added to the problem statement, x a u v, y b g h, u g v 0, h 0, u, v 0, g, h 0. (29) (30) (31) (32) (33) (34) (35) The resulted problem formulaton s stll nonlnear because of nonlnear terms gven n Eq. (32) and Eq. (33). However, snce only u or v appears nonzero n the fnal soluton and g or h appears n nonzero form n the obectve functon, we may dsregard them n the problem formulaton (Lawler, 1973) and the problem s formulated as follows, mn Z=c max (36) subect to constrants (23)(24)(25) K 1 w 1 ( u v ) ( g h ) c p l M(1 w), x a u v, y b g h, x 0, y 0, u 0, v 0, g 0, h 0, c 0, c 0; w, z 0,1 max To solve the resulted model n contnuous form, we consder two scenaros. For the frst scenaro, the followng problem formulaton has to be solved. (37) (38) (39) (40) (41) dst x a yb mn Z= ( ) ( l ) ( x,y ) I K I K l R. Usng Eqs. (28-35) yelds the followng problem statement, I K ( u v ) ( g h) mn Z= ( l ) 1 1 subect to constrants (39)(40) x 0, y 0, u 0, v 0, g 0, h 0 (42) (43) (44) (45) Once the problem s solved for optmalty, u v g و h are optmal values of u v g و h, respectvely. Therefore we have, mn Z=c max (46) subect to constrants (21)(23)(24)(25) ( u v ) ( g h ) c p l M(1 w), c 0, c 0; w, z 0,1 max (47) (48)

7 M. Raabzadeh et al. / Internatonal Journal of Industral Engneerng Computatons 7 (2016) 579 For the second scenaro, we frst solve the followng mn Z= I K 1 1 AT subect to constrants (21)(39)(40) ( u v ) ( g h ) AT l M (1 w ), AT 0, u 0, v 0, g 0, h 0, w {0,1} (49) (50) (51) Here f AT receves a value one, t means that ob s assgned to machne and the assgnment of obs to machnes are the same as dscrete form. 3. The results In ths secton, we present the results of the mplementaton of the proposed study usng some randomly generated numbers. All problems are coded n GAMS usng personal computer wth 2.2GHz core 7 CPU and 6 GB RAM. Table 2 shows the nput parameters. Table 2 Input data used for testng dfferent nstances Sample problem r Coordnaton of storage Coordnaton of canddate locatons p u [0,25] u [0,25] [15, 5] r 0.1p u [0,15] u [0,15] [25,5] r 10 p u [0,200] u [0,200] [10, 3] Processng tme elocty of vehcles Tme of avalablty of obs normrnd 1.5,2,2.5,3 normrnd u 3,3.5,...,6 normrnd u 0.5,1,1.5 u normrnd [8, 2] normrnd [4,1] normrnd [25,5] 3.1. Dscrete ScheLoc In ths case, we consder addtonal nput data for the proposed study as gven n Table 3 as follows, Table 3 Input data for dscrete ScheLoc Job number Processn tme (S) Tme of avalablty n storage elocty of vehcle (m/s) Table 4 shows the results of the mplementaton of the proposed study. In addton, Fg. 3 shows the results. Table 4 The summary of the results of the optmal soluton Problem Obectve functon Locatons of machnes CPU tme Integrated , Scenaro one 21 4, Scenaro two 21 1,

8 580 =1 =2 =1 =2 J=2 J=3 J=5 J=1 J=4 J=2 J=1 J=3 J=5 J=4 =3 =4 =3 =4 a) b) =1 =2 J=2 J=1 J=3 J=5 J=4 =3 =4 c) Job Warehouse Canddate poston Machne poston Job allocaton Fg. 3. The locatons of the machnes and ob schedule for dscrete case In order to have a better understandng on the performance of the proposed method we have generated 10 sample test problems, solved wth GAMS software pacage and n case GAMS software could not reach optmal soluton, we have reported the best soluton after 1000 seconds shown n (*). Table 5 shows the results of the problem. For all problems relatve gaps are computed as follows, sola solb Gap. a& b ( ) 100. solb (52) Table 5 The results of the optmal soluton for 10 sample test problems Problem Input data Integrated model Scenaro one Scenaro two # Canddate # of # of obs locatons machnes Cmax Tme Cmax Tme Gap Cmax Tme Gap * Mean

9 M. Raabzadeh et al. / Internatonal Journal of Industral Engneerng Computatons 7 (2016) 581 Table 6 The results of the optmal soluton for 10 sample test problems Input data Integrated model Scenaro one Scenaro two Problem # # of Canddate # of Cmax Tme Cmax Tme Gap Cmax Tme Gap obs locatons machnes * * * * Mean Table 7 The results of the optmal soluton for 10 sample test problems Problem # Input data Integrated model Scenaro one Scenaro two # of obs Canddate # of locatons machnes Cmax Tme Cmax Tme Gap Cmax Tme Gap Mean As we can observe from the results of Tables (5-7), the proposed ntegrated model has relatvely performed better than alternatve methods Contnous model For the case of contnuous model, we have solved the problem and Table 8 and Fg. 4 show the results of the proposed study. Table 8 The summary of the results of the optmal soluton Problem Obectve functon Locatons of machnes CPU tme Integrated 18 (10, 6), (1, 6) Scenaro one 21.5 (10, 1), (10, 1) Scenaro two 18 (10, 6), (1, 6) Agan, we have generated 10 addonal test problems and Table 9, Table 10 and Table 11 present detals of our fndngs. Table 9 The results of the optmal soluton for 10 sample test problems Prob. Input data Integrated model Scenaro one Scenaro two # of obs locatons Cmax tme Cmax tme Gap Cmax tme Gap Mean

10 582 (1,6) (10,6) (1,6) (10,6) (1,1) (10,1) a) (1,1) b) (10,1) (1,6) (10,6) (1,1) c) (10,1) Job Warehouse Canddate poston Machne poston Job allocaton Fg. 4. The locatons of the machnes and ob schedule for contnous case Table 10 The results of the optmal soluton for 10 sample test problems Problem Input data Integrated model Scenaro one Scenaro two # # of Canddate obs locatons Cmax tme Cmax tme Gap Cmax tme Gap Mean Agan, as we can observe from the results of Table 9, Table 10 and Table 11, the proposed ntegrated model has been able to solve the randomly generated problems n less amount of tme. In most cases, the proposed model has provded better obectve functon values. In summary, n both cases, there have been some mprovement on the performance of the ScheLoc problem usng both dscrete and contnuous spaces.

11 M. Raabzadeh et al. / Internatonal Journal of Industral Engneerng Computatons 7 (2016) 583 Table 11 The results of the optmal soluton for 10 sample test problems Problem # # of obs Input data Integrated model Scenaro one Scenaro two Canddate locatons cmax tme cmax Tme Gap cmax Tme Gap Mean Concluson In ths paper, we have presented a mathematcal model for ScheLoc problem n dscrete and contnuous spaces. The proposed study has been formulated under dfferent condtons and the mplementatons were examned usng varous randomly generated numbers. The prelmnary results have ndcated that the proposed study of ths paper could provde promsng results. As future study, one may consder the problem under uncertan condtons usng fuzzy numbers and we leave t as a future study for nterested researchers. Anowledgement The authors would le to than the anonymous referees for constructve comments on earler verson of ths paper. References Drezner, Z., & Suzu, A. (2004). The bg trangle small trangle method for the soluton of nonconvex faclty locaton problems. Operatons Research,52(1), Elvs, D., Hamacher, H. W., & Kalsch, M. T. (2009). Smultaneous schedulng and locaton (ScheLoc): the planar ScheLoc maespan problem.journal of Schedulng, 12(4), Elvs, D., Hamacher, H. W., & Kalsch, M. T. (2007). Schedulng and locaton (ScheLoc): maespan problem wth varable release dates. Technsche Unverstät Kaserslautern, Fachberech Mathemat. Hennes, H., & Hamacher, H. W. (2002). Integrated schedulng and locaton models: sngle machne maespan problems. Technsche Unverstät Kaserslautern, Fachberech Mathemat. Heßler, C., & Deghda, K. (2015). Dscrete Parallel Machne Maespan ScheLoc Problem. Technsche Unverstät Kaserslautern, Fachberech Mathemat. Kalsch, M. T., & Drezner, Z. (2010). Solvng schedulng and locaton problems n the plane smultaneously. Computers & Operatons Research,37(2), Lawler, E. L. (1973). Optmal sequencng of a sngle machne subect to precedence constrants. Management scence, 19(5), Ncel, S., & Puerto, J. (1999). A unfed approach to networ locaton problems. Networs, 34(4), Pnedo, M. L. (2012). Schedulng: theory, algorthms, and systems. Sprnger Scence & Busness Meda. Scholz, D. (2011). Determnstc global optmzaton: geometrc branch-and-bound methods and ther applcatons (ol. 63). Sprnger Scence & Busness Meda.

12 584 Scholz, D. (2012a). Integrated schedulng and locaton problems. InDetermnstc Global Optmzaton (pp ). Sprnger New or. Scholz, D. (2012b). Summary and dscusson. In Determnstc Global Optmzaton (pp ). Sprnger New or.