Solving a dock assignment problem as a three-stage flexible flow-shop problem

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1 Solving a dock assignmen problem as a hree-sage flexible flow-shop problem L. Berghman, R. Leus ORSTAT, K.U.Leuven, Leuven, Belgium Absrac - This paper presens a model for a dock assignmen problem based on he siuaion encounered in a pracical case. Trailers are assigned o gaes during a specific period in ime for loading or unloading aciviies. The parking lo is used as a buffer zone. Transporaion beween he parking lo and he gaes is performed by addiional resources called erminal racors. The problem is modeled as a hree-sage flexible flow shop, where he firs and he hird sage share he same idenical parallel machines and besides ha, all sages share a differen se of idenical parallel machines. Differen mahemaical formulaions are given and a Lagrangian relaxaion approach is examined o solve his flexible flow-shop problem. Keywords - dock assignmen, flexible flow shop, addiional resources, Lagrangian relaxaion I. INTRODUCTION Toyoa is one of he world s larges auomobile manufacurers, selling over 7.5 million models (including Hino and Daihasu) annually on all five coninens, and generaing almos 130 billion euro in ne revenues. Since 1999, he oal warehouse space floor of he European Disribuion Cenre TPCE (Toyoa Pars Cenre Europe) locaed in Dies (Belgium) has been expanded o m 2. Toyoa s Disribuion Cenre delivers o 28 European disribuors on a daily basis. A TPCE, he warehouse has some fify gaes, each wih a capaciy of one railer, where goods can eiher be loaded on an empy railer or be unloaded from a loaded railer. Besides he warehouse wih he gaes, he sie also conains wo parking los, which can be seen as a buffer where railers can be emporarily parked, boh before and afer he loading and unloading. All ransporaion aciviies of railers beween hese parking los and he gaes are done by wo erminal racors, which are racors designed for use in pors, erminals and heavy indusry. Because of he numerous loading, unloading and ransporaion aciviies, TPCE needs a schedule specifying he saring ime and he assigned gae or erminal racor for each aciviy. The remainder of his paper is srucured as follows: he siuaion a TPCE will be modeled as a hree-sage flexible flow shop. This is described in deail in Secion II. A brief review of he relevan lieraure is given in Secion III. In Secion IV, some mahemaical formulaions are given. And finally, a Lagrangian relaxaion approach is discussed in Secion V. II. FLEXIBLE FLOW-SHOP SCHEDULING Each job is composed of hree asks, one for each sage. The fis sage consiss of one of he erminal racors moving he railer o he gae, while he second sage consiss of he loading and unloading aciviies. The differen gaes a TPCE are considered o be idenical parallel machines and every ask of sage wo has o be assigned o one of he parallel machines. The hird sage is one of he erminal racors ha moves he railer back o he parking lo. Two idenical parallel machines execue boh he firs and he hird sage. For safey reasons, we consider he gae on which he corresponding loading or unloading aciviy of sage wo is planned as being unavailable during he execuion of he ransporaion aciviy. In hese sages, he processing imes are no dependen on he disance because he acual driving ime of he erminal racors is small compared o he ime i akes he driver o follow he safey insrucions and aach he railer o he racor. Some fify idenical parallel machines execue he second sage, so he processing ime depends only on he job and is independen of he machine. For a given job, sage wo always sars immediaely afer sage one finishes; he same does no hold for sage hree. Afer loading or unloading, a railer canno be ranspored o he parking lo by a erminal racor if boh racors are busy. The railer mus remain on he gae unil a erminal racor becomes available, which makes i impossible for anoher railer o be loaded or unloaded here. The faciliies are disjuncive, in he sense ha each machine may process only one ask a a ime. Preempion of a ask is no allowed and no machine has buffer sorage for work-inprocess. A. Definiion and deailed problem saemen T, he se of all asks wih T =3n, can be pariioned ino differen subses. A firs pariion of T is T 1,T 2,T 3. T 1 = (1) 1,(1) 2,...,(1) n } T conains all ransporaion aciviies from he parking lo o a gae, T 2 = (2) 1,(2) 2,...,(2) n } T conains all loading and unloading aciviies and T 3 = (3) 1,(3) 2,...,(3) n } T conains all ransporaion aciviies from a gae o he parking lo. Ti U,TL i is a pariion of T i for i =1, 2, 3. T U = T1 U T2 U T3 U is he se of asks relaed o an unloading aciviy, while T L = T1 L T2 L T3 L is he se of asks relaed o a loading aciviy. All asks T have a ready ime r. For he unloading asks T U, his ready ime equals he planned arrival ime of he railer. For he loading asks T L, r =0because we assume ha all he goods o be loaded on he railers are available in he warehouse and he empy railer is sanding a he parking lo. All asks T have a weigh w. All /09/$ IEEE 320

2 Proceedings of he 2009 IEEE IEEM unloading asks T U have a due dae d equal o he ready ime because he aim is o avoid ardiness. All loading asks T L have a deadline d based on he driving ime o he cusomer and he agreed arrival ime a he cusomer. All ransporaion aciviies beween he parking lo and he gaes have a consan duraion C, independen of he disance. There are wo machines available for boh he firs and he hird sage. G is he se of all idenical parallel machines g of sage wo, wih G = m<n. Each machine can process a mos one job a a ime. The processing ime p of a job T is he ime needed o load or unload he railer a he gae. Our problem consiss of scheduling he asks in such a way ha he oal weighed laeness (or ardiness, since d = r ) of he unloading asks is minimized and he oal weighed earliness of he ransporaion aciviies back o he gae ha are relaed o a loading ask is maximized. In Table I, daa for a problem wih en jobs, four gaes and one erminal racor are given. A feasible schedule for his problem is presened in Fig. 1. The colored blocks represen he ransporaion aciviies by he erminal racor and he whie blocks represen he loading or unloading aciviies a he gaes. Task w r p d d ype U U U U U L L L L L TABLE I A PROBLEM WITH TEN TASKS, FOUR GATES AND ONE TERMINAL TRACTOR Fig. 1. A feasible schedule III. LITERATURE REVIEW The goal of his aricle is o examine mahemaical programming formulaions for he pracical problem and o implemen a Lagrangian relaxaion approach o obain highqualiy soluions in reasonable compuaion ime. We briefly survey relaed work on mahemaical formulaions for machine scheduling problems. Subsequenly, we review he applicaion of Lagrangian relaxaion in machine scheduling problems. A. Mahemaical formulaions A review of parallel machine scheduling lieraure is given in Cheng and Sin [1] and a survey of mahemaical programming formulaions for machine scheduling, including parallel machine scheduling, can be found in Blazewicz e al. [2]. Differen examples of coninuous-ime formulaions can be found. For uniform parallel machine scheduling, Dessouky [3] examines an assignmen-based ineger non-linear programming formulaion where jobs are assigned o posiions on machines and Jain and Grossmann [4] propose an assignmenbased ineger linear programming formulaion where jobs are assigned o machines and addiional variables are used o deermine he sequence on each machine. Bard and Rojanasoonhon [5] presen a flow formulaion for idenical parallel machine scheduling while Mellouli e al. [6] sudy differen models including a flow formulaion, a sequence-based formulaion and an assignmen-based formulaion. Time-indexed formulaions have also received a grea deal of aenion because he linear programming relaxaions provide srong lower bounds. Luh e al. [7], Sousa and Wolsey [8] and Kedad- Sidhoum e al. [9] examine formulaions for a single machine problem ha can easily be exended o a parallel machine scheduling problem. The binary decision variables assign each job o a cerain saring period. A survey of flexible flow-shop lieraure can be found in Linn and Zhang [10]. Mos sudies deal wih wo-sage flow shops wih parallel machines eiher in he firs or in he second sage, bu no in boh. To he bes of our knowledge, very few examples of igh coninuous-ime formulaions can be found in lieraure. Riane e al. [11] propose a sequence-based formulaion, while Paernina-Arboleda e al. [12] presen an assignmen-based formulaion and Jungwaanaki e al. [13] describe a flow formulaion. Concerning he ime-indexed formulaions, Chen and Luh [14] consider a job shop scheduling problem consising of scheduling differen pars on differen ypes of machines where each machine ype has differen idenical machines and he compleion of each par requires a series of differen operaions. Tang e al. [15] and Tang and Xuan [16] sudy mahemaical formulaions for he scheduling problem wih muliple idenical parallel machines available for processing jobs a each sage. B. Lagrangian relaxaion for machine scheduling problems Mos aricles in he machine-scheduling lieraure ha make use of Lagrangian relaxaion, relax he capaciy consrains on machines of heir ime-indexed ineger programming formulaion by using Lagrange mulipliers and decompose he relaxed problem ino independen job level subproblems (parallel machine scheduling: Luh e al. [7]. Kedad-Sidhoum e al. [9]; job-shop or flow-shop scheduling: Chen e al. [17] and Tang and Xuan [16]), alhough in flow-shop or job-shop scheduling i migh be ha he precedence consrains are relaxed (Chen and Luh [14], Tang e al. [15]). Kedad-Sidhoum e al. [9] also relax he number of occurrences, such ha a job can be processed several imes. The Lagrange mulipliers are updaed by a (surrogae) subgradien mehod. When he dual soluion 321

3 Proceedings of he 2009 IEEE IEEM a he end of he algorihm is infeasible, a feasible schedule is mos of he ime consruced by a heurisic lis-scheduling approach. IV. MATHEMATICAL FORMULATIONS Since he available lieraure does no conain much informaion dedicaed o mahemaical formulaions for flexible flowshop problems, we sar by esing formulaions for only par of our problem. When we only look a he gae assignmen par of he problem and do no ake he erminal racors ino accoun, he problem a hand is close o P m r j,d j w j T j in he sandard hree-field noaion, alhough some asks have deadlines raher han due daes. Based on preliminary experimens, we have found ha ou of differen formulaions, he ime-indexed formulaion described in Subsecion C gives he bes resuls. Therefore, his formulaion is exended o he flexible flow-shop scheduling problem in Subsecion D. A. Assignmen-based formulaion for he parallel machine scheduling problem The decision variables of his firs formulaion are he following. For all asks T and for every gae g G, 1 if ask is scheduled a gae g, g = (1) 0 oherwise. For all asks, u T, 1 if ask is scheduled before ask u when boh z u = asks are execued a he same gae, 0 oherwise. (2) For every ask T we also have a saring ime s. A mahemaical formulaion of our problem is he following: min w (s + p d )+ w (s + p d ) (3) T U T L subjec o g G g =1 T (4) g + ug z u z u 1, u} T ; g G (5) s + p (1 z u )M s u, u} T (6) r s T (7) s + p d T L (8) g 0, 1} T ; g G (9) z u 0, 1}, u T (10) s 0 T (11) The objecive funcion (3) minimizes for all unloading asks T2 U he oal ime beween he finishing ime s + p of he unloading ask and he due dae d and for all loading asks T2 L he oal ime beween he he finishing ime s + p of he loading ask and he deadline d. Consrain (4) limis each loading or unloading ask o be assigned o exacly one gae, while consrain (5) ensures ha if ask and ask u are assigned o he same gae g, hen one mus be processed before he oher. Consrain (6) ensures ha he ending ime of a cerain ask is smaller han he saring ime of he following ask on ha same gae, where M sands for a large value. A possible value for M, which will be used in he implemenaions, is max r + T 2 p. Consrains (7) and (8) enforce he ime windows: no asks can sar before is ready ime and all loading asks have o be finished before heir deadline. Finally, consrains (9) and (10) sae and z u are binary variables and consrain (11) saes ha all saring imes have o be posiive. For a scheduling problem on idenical parallel machines, permuing he machine indices does no affec he srucure of he problem. For example, swiching he assignmen of all jobs assigned o machine 1 and hose assigned o machine 2 will resul in essenially he same soluion. Due o his srucure, branching becomes ineffecive because many assignmens of values o variables represen he same soluion. For his reason, differen symmery-breaking consrains have been implemened and compared. Moreover, differen valid inequaliies have been esed o ighen and speed-up he formulaion. Preliminary experimens show ha adding he following consrains resuls in less compuaional effor. ha our decision variables g g=1 g =1 1,..., m} (12) z u + z u 1, u} T (13) g + uh + z u + z u 2, u} T ; g, h} G (14) Consrain (12) is a symmery-breaking consrain ha saes ha for m, ask is scheduled on one of he machines 1,...,. Consrains (13) and (14) are valid inequaliies. The firs one saes ha for every pair of asks, eiher hey are no scheduled on he same gae, or one ask comes before he oher or he oher way around. The second one guaranees ha he sequencing variables z u and z u are boh zero if asks and u are assigned o differen gaes. This formulaion has much in common wih he ones presened in Jain and Grossman [4] alhough in hose formulaions he parallel machines are no idenical such ha he processing imes depend on he machine and he objecive is o minimize he sum of he processing coss c m j of assigning ask j o machine m. Consrain (14) sems from Zhu and Heady [18]. B. Flow-based formulaion for he parallel machine scheduling problem The second mahemaical formulaion is based on Bard and Rojanasoonhon [5]. The main differences wih our seing are he non-idenical parallel machines, he wo ime windows per ask, he seup imes, he prioriy classes conaining he asks and he corresponding conribuions. The decision variables of 322

4 Proceedings of he 2009 IEEE IEEM his formulaion are he following. For all asks, u T, 1 if ask is scheduled immediaely before ask u, u = 0 oherwise. (15) A mahemaical formulaion of our problem is he following: min w (s + p d )+ w (s + p d ) (16) T U T L subjec o u T 0 \} u T 0 \} u T u =1 T (17) 0 = m (18) u T 0 \} u =0 T 0 (19) s + p (1 u )M s u, u} T (20) r s T (21) s + p d T L (22) u 0, 1}, u T 0 (23) s 0 T (24) Consrain (17) limis each ask o be processed exacly once. I also ensures ha, if a ask is scheduled, i has no more han one successor, which migh be he dummy ask 0. Consrain (18) limis he number of iniial asks. These consrains (17) and (18) indirecly specify ha each machine can process a mos one ask a a ime. Consrain (19) ensures he conservaion of flow. If ask is assigned o gae g, boh is predecessor and successor mus be processed by gae g. Consrain (20) ensures ha for a given gae, he ending ime of a cerain ask is no longer han he saring ime of he following ask. As before, M sands for a large value, and a possible value for M equals max r + T p. This is he value we will use in he implemenaion. Consrains (21) and (22) enforce he ime windows. Finally, consrain (23) are binary variables and consrain (24) saes ha all saring imes have o be posiive. The symmery problem is solved by producing a se of m schedules insead of assigning paricular schedules o paricular gaes. saes ha our decision variables u C. Time-indexed formulaion for he parallel machine scheduling problem This ype of formulaion is based on ime-discreizaion where ime is divided ino periods. Le H denoe he scheduling horizon, hus we consider he ime periods 0,...,H max wih H max = max r + T p. This mahemaical formulaion is based on Sousa and Wolsey [8] and Kedad-Sidhoum e al. [9]. For all asks T and for all periods u H (2) u = 1 if ask sars in ime period u, 0 oherwise., (25) wih H (2) = r,...,h max if T U and H (2) = r,...,d p if T L. A mahemaical formulaion of our problem is he following: min w ( (u u )+p d ) T U u H + w ( (u u )+p d ) T L u H subjec o u [r,h max p ] T 2 v [maxr,u p },u] (26) u =1 T (27) v m u H (28) u 0, 1} T ; u H (29) The objecive funcion (26) minimizes for all unloading asks T U he oal ime beween he finishing ime u u H (2) u + p of he unloading ask and he due dae d and for all loading asks T L he oal ime beween he finishing ime u u H (2) u + p of he loading ask and he deadline d. Consrain (27) limis each ask o be processed exacly once. Consrain (28) ensures ha for a given ime inerval u, only m unloading or loading aciviies can be execued. Finally, consrain (29) saes ha our decision variables u are binary variables. As in Secion IV-B, he symmery problem is solved by producing a se of m schedules. D. Time-indexed formulaion for he flexible flow-shop problem When exending he above formulaion o our flexible flowshop problem, H max = max r + T p +2nC. The decision variables are he following. For all ask T i (for i =1, 2, 3) and for all momens in ime u H (i) x (i) u =, 1 if ask sars in ime period u, 0 oherwise. (30) wih H (1) = r,...,t p 2C} (wih T = max r + T 2 p +2nC), H (2) = r +C,...,T p C} and H (3) = r + C + p,...,t C} if T U and H (1) = r,...,d p 2C}, H (2) = r + C,...,d p C} and H (3) = r + C + p,...,d C} if T L. A mahemaical formulaion of our problem is he following: 323

5 Proceedings of he 2009 IEEE IEEM subjec o T min T U w + T L w u H (k) x (1) u + x (3) u + v u u H (2) u H (3) u u ux (3) u + p d + C d (31) x (k) u =1 T k ; k =1, 2, 3 (32) ( ) v x (3) v m u H (33) ( ) x (1) u + x (3) u 2 u H (34) T x (1) u =,u+c ; u H (35) T u v=u+p x (3),v ; u H (36) x (i) u 0, 1} for i =1, 2, 3; T i ; u H (i) (37) The objecive funcion (31) minimizes for all unloading asks T U he oal ime beween he finishing ime u H u u + p of he unloading ask and he due dae d and for all loading asks T L he oal ime beween he he finishing ime u H ux (3) u + C of he ransporaion ask back o he parking lo and he deadline d. In his way, we achieve ha all impor goods are as early as possible in he warehouse and all expor railers are ready as early as possible. Consrain (32) limis each ask o be processed exacly once eiher on a gae or by a erminal racor. Consrain (33) ensures ha for a given ime inerval, only m unloading, loading or ransporaion aciviies can be execued. Consrain (34) ensures he capaciy of he erminal racors. Consrains (35) and (36) ensure ha for each job he firs ask has o be finished before he second ask can sar and he second ask has o be finished before he hird ask can sar. Finally, consrain (37) saes ha our decision variables x (i) u (for i =1, 2, 3) are binary variables. V. LAGRANGIAN RELAXATION For some insances wih 34 asks, nine gaes and one erminal racor, he compuaion imes for he ime-indexed formulaion exceed wo hours. Noe ha a higher number of erminal racors (even wo) significanly decreases he required compuaional effor. In order o produce near-opimal soluions wihin reasonable running ime for pracical-size insances, we resor o Lagrangian relaxaion. Boh he resource consrains of he ime-indexed formulaion (consrains (33) and (34)) are relaxed using Lagrange mulipliers, such ha easily solvable, independen job-level subproblems are obained. The Lagrange mulipliers ac as prices o regulae he use of he machines. For each ask, he saring ime o srike he bes balance beween hese machine prices and he ardiness / earliness of he job is seleced. The Lagrange mulipliers are updaed according o he demand of he machines, by a subgradien mehod. When he dual soluion a he end of he algorihm is infeasible, a feasible schedule is consruced by a wo-phase heurisic approach based on lis scheduling, which uses he dual soluion, and a heurisic o ake he deadlines ino accoun (see [16], for example). REFERENCES [1] T. Cheng and C. Sin, A sae-of-he-ar review of parallel-machine scheduling research, European Journal of Operaional Research, vol. 47, pp , [2] J. Blazewicz, M. Dror, and J. Weglarz, Mahemaical programming formulaions for machinery scheduling: A survey, European Journal of Operaional Research, vol. 51, pp , [3] M. Dessouky, Scheduling idenical jobs wih unequal ready imes on uniform parallel machines o minimize he maximum laeness, Compuers and Indusrial Engineering, vol. 34 (4), pp , [4] V. Jain and I. 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