An Integrated Strategy for a Production Planning and Warehouse Layout Problem: Modeling and Solution Approaches

Transcription

1 Unversty of Wndsor Scholarshp at UWndsor Mechancal, Automotve & Materals Engneerng Publcatons Department of Mechancal, Automotve & Materals Engneerng Summer An Integrated Strategy for a Producton Plannng and Warehouse Layout Problem: Modelng and Soluton Approaches Guoqng Zhang Tatsush Nsh Sarna D. O. Turner Kesuke Oga Xndan L Follow ths and addtonal works at: Part of the Busness Intellgence Commons, Industral Engneerng Commons, Management Scences and Quanttatve Methods Commons, Operatonal Research Commons, and the Operatons and Supply Chan Management Commons Recommended Ctaton Zhang, Guoqng; Nsh, Tatsush; Turner, Sarna D. O.; Oga, Kesuke; and L, Xndan. (2016). An Integrated Strategy for a Producton Plannng and Warehouse Layout Problem: Modelng and Soluton Approaches. Omega, 68, Ths Artcle s brought to you for free and open access by the Department of Mechancal, Automotve & Materals Engneerng at Scholarshp at UWndsor. It has been accepted for ncluson n Mechancal, Automotve & Materals Engneerng Publcatons by an authorzed admnstrator of Scholarshp at UWndsor. For more nformaton, please contact scholarshp@uwndsor.ca.

2 An Integrated Strategy for a Producton Plannng and Warehouse Layout Problem: Modelng and Soluton Approaches Guoqng Zhang 1*, Tatsush Nsh 2, Sarna D. O. Turner 3, Kesuke Oga 2, Xndan L 4 1 Supply Chan and Logstcs Optmzaton Research Center, Department of Mechancal, Automotve & Materals Engneerng, Unversty of Wndsor, Canada 2 Graduate School of Engneerng Scence, Osaka Unversty, Japan 3 Department of Mechancal and Industral Engneerng, Unversty of Toronto, Canada 4 Research Center of Management Scence and Engneerng, Nanjng Unversty, Chna Abstract We study a real-world producton warehousng case, where the company always faces the challenge to fnd avaable space for ts products and to manage the tems n the warehouse. To resolve the problem, an ntegrated strategy that combnes warehouse layout wth the capactated lot-szng problem s presented, whch have been tradtonally treated separately n the exstng lterature. We develop a mxed nteger lnear programmng model to formulate the ntegrated optmzaton problem wth the objectve of mnmzng the total cost of producton and warehouse operatons. The problem wth real data s a large-scale nstance that s beyond the capabty of optmzaton solvers. A novel Lagrangan relax-and-fx heurstc approach and ts varants are proposed to solve the large-scale problem. The prelmnary numercal results from the heurstc approaches are reported. Keywords: warehouse layout, capactated lot-szng, nteger lnear programmng, Lagrangan heurstcs *Correspondng author: gzhang@uwndsor.ca, tel: ext 2637, fax:

3 1 Introducton The warehouse layout problem s one of the key ssues n warehouse management, whch nvolves all stages of a supply chan. An effectve warehouse layout can dstnctly reduce the operaton costs. The storage locaton assgnment problem addressed n ths paper s to determne the locatons for dfferent tems n a warehouse; thus we consder warehouse layout and storage locaton assgnment as nterchangeable n ths research. Producton plannng s the process of determnng how to use and allocate the resources effcently to satsfy the customer demands. The capactated lot-szng problem s a medum-term producton plannng, whch decdes how much to produce for each product so that the total cost of producton, setup, and nventory s mnmzed. Usually, warehouse layout problems and producton plannng are analyzed separately. However, whe warehouse space s one of the key resources for producton, enhancng the utzaton of the space can mprove the producton output and ncrease the manufacturer s proft. Our research s motvated by a real-world producton warehousng problem. We perform the study at a food company that processes and packages rce products. The company always faces the challenges of fndng avaable space for ther products and managng the tems n the warehouse. The lack of coordnaton between producton plannng and warehouse management leads to sgnfcant neffcency n both departments. Thus we nvestgate a producton plannng that consders the warehouse space resource and a warehouse layout wth new storage assgnment. In essence, we propose an ntegrated strategy to combne the capactated lot-szng problem wth warehouse layout problem so that producton process and warehousng can be coordnated effectvely. The warehouse layout problem and the capactated lot-szng problem have been studed extensvely n research. However, to our knowledge, these two problems have been addressed separately n the current lterature and they have not been consdered together n a mathematcal model. Jans and Degraeve (2008) revew the modelng for ndustral lot-szng problems and dscuss the four extensons of lot-szng models. It s ndcated n ther dscusson on new research drectons that the ntegraton of lot-szng, schedulng, and storage constrants affects the plannng and schedulng. Buschkühl et al. (2010) presents a revew of four decades of research on dynamc capactated lot-szng and focuses on modelng and algorthm for the problems. Although they hghlght that producton plannng s strongly related to the layout type and organzatonal structure of a producton system, the warehouse layout s not taken nto account n ther revewed lterature on the capactated lot-szng problem. Exstng lterature ndcates that most of the lot-szng problems are hard to solve. Zhang and La (2010) revew the mult-level warehouse layout problem wth a dedcated storage polcy wthout consderng the lot-szng problem. 2

4 Varous researchers have combned the warehouse layout problem wth dfferent nventory models such as Economc Order Quantty (EOQ) models (e.g., Chu and Chu (2008), Wson (1997), and Malmborg and Deutsch (1988)), and the replenshment polcy models (e.g., Hassn (2008), and Kulturel et al. (1999)). It s noted that n exstng research where the warehouse layout problem and nventory models are consdered jonly, only statc nventory models are utzed. The model presented n ths paper not only combnes a storage locaton assgnment problem wth dynamc nventory, but also takes nto account producton plannng. To our knowledge, the dynamc jont mult-tem storage locaton assgnment combned wth producton plannng has not been consdered n the current lterature. The man contrbutons of the work nclude: 1) a new ntegrated strategy that combnes storage locaton assgnment and the capactated lot-szng problem, whch were solved separately n prevous research; 2) a new mxed nteger programmng (MIP) model for the jont problem; furthermore the model s dynamc, whe prevous models that have combned the storage locaton assgnment problem wth nventory models are statc; 3) a novel Lagrangan relax-and-fx heurstc approach and ts varants for solvng the ntegrated optmzaton of producton plannng and warehouse layout problem, 4) a real-world ndustral producton warehouse problem, whch s solved wth the proposed methods and the results obtaned could be appled to coordnate the producton and warehousng at other smar factes. The remander of the paper s organzed as follows. Secton 2 provdes lterature revew on the warehouse layout and the capactated lot-szng problem. A real-world problem and ts background are presented n Secton 3. A mxed nteger lnear programmng model s presented n Secton 4. Lagrangan decomposton and coordnaton s ntroduced n Secton 5. A Lagrangan relax-and-fx heurstc approach s proposed n Secton 6. The two varants of the Lagrangan relax-and-fx heurstc method are dscussed n Secton 7. The computatonal results for test problems and the prelmnary numercal results from the proposed Lagrangan methods for the ndustral problem are reported n Secton 8. Secton 9 concludes the paper wth a dscusson of future work. 2 Lterature revew The storage locaton assgnment/warehouse layout problem Extensve lterature has been conducted on warehouse desgn and operatons, whch examned the dfferent operatons, components and areas of mprovement n warehousng. Here we revew some exstng work and lterature related to warehouse layout problem and comparsons of dfferent storage assgnment polces. For a comprehensve revew of research on warehouse operatons, we refer readers to Gu et al. 3

5 (2007, 2010) and Francs et al. (1992). The warehouse layout problem s to determne the physcal locaton n storage departments/zones for ncomng tems whe takng nto consderaton the storage structure, the capacty and the storage/retreve process and requrement. An effcent warehouse layout can reduce materal handlng cost and mprove space utzaton (De Koster et al. (2007), and Gu et al. (2007)). Accordng to the dfferent rules used to assgn tems n a warehouse, a warehouse system can mplement dfferent layout polces,.e., storage assgnment polces. De Koster et al. (2007) descrbe the fve storage assgnment polces, ncludng dedcated, random, class-based, turnover-based, and closest open locaton storage polces. A consderable number of studes compared the storage assgnment polces under varous warehouse envronments. Malmborg and Altassan (1998) examned a less-than-unt load warehousng system and developed models to compare dedcated storage usng the cube per order ndex, and randomzed storage usng closest open locaton. Randomzed storage only acheved a 65% space utzaton compared wth 100% space utzaton acheved wth dedcated storage. Zeng et al. (2002) examned a short term-and long-term plan to show dfferent alternatves for reducng the tme spent flng customer orders. For the short-term plan they used an actvty-based storage phosophy. Petersen and Aase (2004) examned statc random storage, volume-based storage, and class-based storage polces, and used a smulaton model to compare the polces n a warehouse envronment. It was shown that when the number of order ncreased, the percentage of savngs for usng each polcy decreased. Muppan and Ad (2008) concluded that f classes were formed consderng only handlng costs, a dedcated storage assgnment polcy would produce the lowest costs, and f classes were formed based on space costs, then a completely randomzed storage assgnment polcy yelded the lowest cost soluton. However, they ponted out that when handlng costs and space costs were consdered together, a class-based storage assgnment polcy was optmal. Pohl et al. (2011) nvestgated the storage assgnment n unt-load warehouses wth non-tradtonal asles, and concluded that warehouse desgn parameters that perform best under random storage also perform well under turnover-based storage. Çelk and Süral (2014) studed order pckng performance under random and turnover-based storage polces n fshbone asle warehouses. They provded smple heurstcs for fshbone layouts and performed computatonal experments n order to compare the performances of fshbone and tradtonal layouts under optmal routng. Guo et al. (2016) nvestgated the mpact of requred storage space on the warehouse performance wth random, full turnover -based and class-based storage polces. 4

6 In the recent years, the class-based storage polcy has attracted consderable attenton. Larson et al. (1997) examned a class-based warehouse layout. The am was to acheve effectve use of floor space. The model developed by Muppan and Ad (2007) examned the mplementaton of a class-based storage assgnment polcy and ts effect on storage space and materal handlng costs, and the model was solved usng a branch and bound algorthm. Nsh and Konsh (2010) proposed beam search heurstcs for optmzaton of floor-storage warehousng systems. Pan et al. (2014) proposed a model to estmate a travel tme for a hgh-level pcker-to-part system wth class-based storage polces. Pan et al. (2015) studed the order batchng problem n a pck-and-pass warehousng system. Capactated Lot-szng Problem (CLSP) The capactated lot-szng problem (CLSP) s one of the most mportant and dffcult problems n producton plannng (Karm et al. (2003)). The capactated lot-szng problem was consdered to be NP hard by Floran and Klen (1971) and Btran and Yanesse (1982). Many exact and heurstc soluton methods have been developed to solve CLSP. Here we only provde a bref revew of exstng work on CLSP that takes nto account nventory and warehouse capacty. For the comprehensve revews of the models and algorthms on CLSP, the reader s referred to Karm et al. (2003), and Robnson et al. (2009). The CLSP has been consdered wth bounded nventores and drect applcaton to the warehouse envronment. Love (1972) was the frst to examne a bounded nventory problem. Page and Paul (1976) consdered the problem of mantanng nventory for multple products when there s a restrcton on the maxmum nventory nvestment or on the maxmum amount of warehouse space. Guterrez et al. (2002) examned a relevant class of producton nventory systems when the nventory levels were bounded. Lu and Tu (2008) examned a producton plannng problem where the nventory capacty was the lmtng factor. They formulated the model and developed an algorthm that was consdered to be O (T 2 ). The capactated lot-szng problem wth bounded nventory has also been extended to nclude multple products, e.g., Pochet and Wolsey (1991), Abs and Kedad-Sdhoum (2008), Nascmento et al. (2010). Hwang and Kang (2016) studed the lot-szng problem wth backloggng for stepwse transportaton cost. Warehouse capacty or layout can be a crtcal resource or ssue of CLSP (Zhang et al. 2012). Chu and Chu (2008) examned the sngle tem dynamc lot-szng model wth bounded nventory and outsourcng. The nventory was bounded by the storage capacty of the warehouse. Transchel and Mnner (2009) analyzed the replenshment of multple products to satsfy dynamc demand when the warehouse capacty or the avaable nventory budget was lmted. A savngs-based heurstc was suggested for the warehouse 5

7 schedulng problem, and three smple approaches to the replenshment of multple products wth dynamc demand and lmted warehouse capacty were developed. Buschkühl et al. (2010) ndcated that producton plannng and partcularly lot-szng s strongly relevant to the layout type and organzatonal structure, but no warehouse layout has been consdered n the lot-szng problem n ther revewed lterature. Ths lterature revew shows that prevous research has been conducted for establshng the need to coordnate storage assgnment wth nventory control, and there are some studes combnng storage assgnment wth statc EOQ and replenshment polces. Extensve research has been devoted to the storage locaton assgnment problem and the capactated lot-szng problem separately. However, to our knowledge, no work has been reported that combnes the storage locaton assgnment problem wth the capactated lot-szng problem. The study on the ntegraton of the two problems s motvated by a real-world case where there s a lmted producton warehouse space. In the next secton, the real-world problem and background nformaton are ntroduced. 3 The Problem and Background In the real-world producton and warehousng problem, the company ams to reorganze ther warehouse and determne f they have enough space to store each product to meet customer demand, whe keepng up wth current company operatons and allowng for future growth. Fgure 1 depcts the flow of tems from the producton area to the output/loadng pont of the facty. The areas of nterest assocated wth each secton of the process have been hghlghted n Fgure 1. Fgure 1. Item flow from producton to warehousng to output and related costs 6

8 In the producton area, there are eght producton lnes. The producton planner decdes on the lot sze of all tems at each perod wth capacty lmts. The plannng s concerned wth producton costs, setup costs, and nventory costs durng the plannng horzon. Setup costs are related to the amount of tme an operator would spend preparng a machne for producton. The operator s pad per hour and the salary s determned by the amount of tme that the operator needs to perform the setup operaton. Ths data s taken from the tme study that was conducted, and the cost for each setup vares dependng on the producton lne and the type of tem that s gong to be produced. Inventory s determned by the number of tems that are n the warehouse at the end of the perod. Demand s forecasted and known. However, the current producton plannng does not consder the avaabty of the warehouse space, whch could lead to nfeasble solutons that would exceed the warehouse capacty. The transportaton cost s ncurred whenever a product s carred by a forklft truck from the producton area to the storage locatons. Each storage locaton has ts own ndvdual address that tems could be delvered to, and the cost s proportonal to the transportaton dstance n feet by the forklft operator from the producton area to that specfc storage locaton. A dedcated storage assgnment polcy s used to organze the warehouse space, by assgnng locatons to each tem. A dedcated storage assgnment polcy also has ndustral relevance to the warehouse as t allows order pckers to become famar wth tem locatons, and t typcally reduces the materal handlng costs. The optmal storage locatons are determned based on cost: ths cost s proportonal to the transportaton dstance n feet by the forklft operator durng the placement and retreval of tems n the warehouse. The cost s based on the wage of the operator, order processng, labelng, and shrnk wrappng of the tems. The travel costs assocated wth the movement of product from the storage locatons to the output pont and from the producton area to the storage locatons are calculated based on operator travel. The only dfference s the startng and endng ponts whch alter the cost of travel. Therefore, n most cases the dstance from the producton area to the storage locatons dffers from the dstance from the storage locatons to the loadng/output pont. At the output pont, the demand s the order from the customers for each tem and s consdered as the pull mechansm n the tem flow process. The forecasted demand, whch s known, determnes how many 7

9 tems would flow to the loadng pont at each perod to be shpped to customers. Ths s where the tem s flow ended n the model. The challenge faced by the company s the dffculty n fndng avaable space durng perods when the products are transferred from producton area to the warehouse. Occasonally, because of msplaced tems due to employee errors or lack of vsbty, the operator cannot fnd the tems to be pcked up even though they are avaable, whch leads to extra producton. Several soluton approaches are nvestgated, ncludng reorganzng the warehouse and ncreasng vsbty. More mportantly, a proposal to combne the producton plannng wth the storage locaton assgnment s presented and leads to ths research. It s noted that the storage locaton assgnment can be adjusted annually, whch s actually the practcal way n the company. Thus the tme horzon for both producton plannng and storage locaton assgnment s one year. Ths paper develops an ntegrated strategy that combnes a dedcated storage locaton assgnment wth the capactated lot-szng problem nto a sngle mathematcal model that mnmze the total cost of travel, reserved storage space, handlng, producton, nventory holdng, and setup costs. The formulaton of the problem s developed n the next secton. 4 Model 4.1. Model Assumptons The basc assumptons of ths model are as follows: Demand s forecasted and known, and shortages are not permtted. A dedcated warehouse layout polcy s utzed. The tme horzon for both producton plannng and storage locaton assgnment s the same. For the real-world problem, the horzon s one year. All tems are stored and moved on pallets. Pallets are consdered to be the same sze, weght, and geometrc confguraton and these factors had no effect on the storage and handlng costs. A dscrete number of warehouse storage locatons are used. Items are delvered and retreved usng a sngle-command forklft truck. One unt of tem accounted for a column of three pallets. There s one general producton area that products come from, and has one output pont. Costs assocated wth the placement and retreval are drectly proportonal to the transportaton dstance. 8

10 All costs of producton, set up, and holdng are known, and the value of varable unt space s set to 1. Setup costs are proportonal to the tme spent by the operator to prepare for each producton Notatons The parameters of the proposed model nclude the number of tems, storage locatons, the plannng horzon, forecasted demand, as well as the costs of producton, setup, storage, handlng, travel costs, and reservaton cost. The notatons for those parameters are defned as the followng. Indces: : the ndex of an tem. t : the ndex of a perod wthn the plannng horzon l : the ndex of a storage locaton wthn the warehouse. The parameters used n the model are: R l : Unt cost of reservng storage locaton l. O l : Unt cost of movng a column of any tem from storage locaton l to the output pont. P l : Unt cost of movng a column of any tem from the producton area to storage locaton l. h t : Unt nventory cost of holdng tem at perod t. c t : Varable unt producton cost of tem at perod t. u t : Unt setup cost of tem at perod t. d t : The amount of requred demand of tem n perod t. Ths parameter denotes the demand of each tem, whch vares from product to product and perod to perod. v t : The varable capacty of tem at perod t. f t : A key resource constrant on producton, for example, budget, producton, labor, etc. M : A bg number, used for logc constrants. The problem ncludes lot-szng decson and storage locaton assgnment. The followng are the decson varables of the model: x t : The quantty of tem produced durng perod t. s t : The nventory level for tem at the end of perod t. y t : 1, f tem s produced durng perod t; 0, otherwse. w t : 1, f tem s moved from the producton area and placed n storage locaton l durng perod t; 0, otherwse. q t : 1, f tem s requested (by demand) from locaton l durng perod t; 0, otherwse. 9

11 n t z : 1, f tem s nventored n locaton l durng perod t; 0, otherwse : 1, f locaton l s reserved for tem for the plannng horzon; 0, otherwse. Note: Each locaton can only hold one unt at a tme Formulaton The ntegrated problem s formulated as the followng mxed nteger programmng model Objectve Functon I L I L T Z = R l z + P l w t =1 l=1 =1 l=1 t=1 I L T I T + O l q t + (c t x t + u t y t + h t s t ) =1 l=1 t=1 =1 t=1 (1) The objectve functon s dvded nto four terms: the frst term s the cost of reservng locatons for tems, n a dedcated storage polcy. The second term s the cost assocated wth the travel of products from the producton area to assgned storage locatons. The thrd term s the cost assocated wth the transportaton of products from the storage locatons to the output pont, whch occurs once a product s demanded durng a gven tme perod. The fourth term s all the costs assocated wth producton plannng, producton, setup, and holdng nventory. The objectve s to fnd the optmal producton plannng and the best locatons of the products n order to mnmze the total costs Constrants I z 1, l (2) =1 Constrants (2) lmt the number of tems that could be assgned to a reserved locaton to one. They ensure that a locaton can only be reserved by one tem. I q t 1, l, t (3) =1 Constrants n (3) ensure that at most one tem can be requested at each storage locaton n each perod. I w t 1, l, t (4) =1 Constrants n (4) state at most one tem can be moved from producton area to a specfc storage locaton l n perod t. I n t 1, l, t (5) =1 Constrants n (5) state that at most one tem can be nventored at each storage locaton n each perod. 10

12 L q t = d t, t (6) l=1 Constrants (6) state that the number of products requested from storage locatons durng perod t s equal to the demand of that product durng perod t. L w t = x t, t (7) l=1 Constrants (7) ensure that then number of products moved from producton area to the warehouse n a perod s the same as what produced n the perod. L n t = s t, t (8) l=1 Constrants (8) state that all the locatons that tems reman n at the end of tme perod t are consstent wth the nventory at the end of tme perod t. q t w t + n t 1, l, t (9) Constrants n (9) state that tems could only be retreved from locatons whch they have prevously been moved n from producton area or nventored. w t + n t 1 z, l, t (10) Constrants n (10) state that when movng an tem from producton area to the warehouse, the tem s only can be placed at an avaable storage locaton. Therefore, no tem could be assgned to a locaton where an tem has already been nventored n the prevous perod. n t = w t q t + n t 1, l, t (11) Constrants (11) are the flow balance of an tem from the producton area to a storage locaton, and from the storage locaton to the output pont. q t z, l, t (12) Constrants (12) ensure that tems are only retreved from locatons that had been reserved for them. w t z, l, t (13) Constrants (13) ensure that tems are only placed n locatons that had been reserved for them. n t z, l, t (14) Constrants (14) ensure that tems are only nventored n locatons that have been reserved for them. 11

13 I L v t w t f t t =1 l=1 (15) Constrants (15) are producton capacty restrctons. The constrants ensure that the number of each tem produced multpled by the varable capacty of each tem dd not exceed any key resource durng the plannng horzon. L w t My t, t (16) l=1 Constrants (16) are set up constrants; the constrants put a lmt on producton durng each perod. The value of M s fxed n terms of both producton capacty and tem demands. z, w t, q t, n t, y t {0,1}, l, t (17) The model descrbed by (1) to (17) s a mxed nteger programmng model. Smar to most capactated lot- szng problems, the problem for the real-world data s very dffcult to solve. A smpler model can be establshed by substtutng q t = w t + n t 1 n t nto (9) leads n t 0, l, t whch s clearly satsfed. Then, constrants (9) are elmnated. Constrants (6) can be rewrtten as L w l=1 t + n t 1 n t = d t, t. Smarly, constrants (3) and (12) are satsfed by (2), (9) and (10). Constrants (17) are also satsfed snce w t and n t are bnary varables. Snce z {0,1}, l, constrants (13) and (14) are always satsfed by (10) snce n t 0, l, t. All together mples the followng smplfed model. Objectve functon: I L I L T I T Z = R l z + ((P l + c t )w t +O l (w t + n t 1 n t ) + h t n t ) + (u t y t ) =1 l=1 =1 l=1 t=1 =1 t=1 (18) Subject to constrants (2), (4), (5), (6), (10), (11), (15), (16), z, w t, q t, n t, y t {0,1}, l, t (19) The approaches for solvng the problem are dscussed n the next secton. 5 LAGRANGIAN DECOMPOSITION AND COORDINATION ALGORITHM The Lagrangan decomposton and coordnaton method s one of the wdely used approaches for capactated lot-szng problem (Buschkühl et al., 2010), ntegrated optmzaton of producton schedulng and dstrbuton plannng (Nsh et al., 2007), supply chan coordnaton (Nsh et al., 2008), mult-product newsvendor problem wth supply dscount (Zhang, 2010), and so on. The key dea of ths method s to relax the couplng constrants through 12

14 Lagrangan multplers and to decompose the orgnal problem nto some smple subproblems. Gven the Lagrangan multplers, the relaxed problem provdes a lower bound for the optmal prmal objectve value n a mnmzaton problem. Generally, by means of updatng the Lagrangan multplers effectvely, the lower bound can be mproved gradually Lagrangan decomposton and coordnaton method Applyng a Lagrangan decomposton to the model, we choose to relax couplng constrants (2), (4), (5), and (15). Ths leads to a decomposton wth as many ndependent subproblems as tems. Usng non-negatve Lagrange multplers α l, β t l, π t l and μ t, we can generate the followng relaxed problem (LRP). Let the set of tems, the set of perods and the set of locatons be I, T, L, respectvely. (LRP) Mnmze L( z, w t, q t, n t, y t ), wth L(z, w t, q t, n t, y t ) = R l z I L =1 l=1 I L T + ((P l + c t )w t +O l (w t + n t 1 n t ) + h t n t ) =1 l=1 t=1 I T + (u t y t ) =1 t=1 L I L T I L T I + α l (z 1) + β l t (w t 1) + π l t (n t 1) l=1 =1 l=1 t=1 =1 l=1 t=1 =1 T I L + μ t ( v t w t f t ) t=1 =1 l=1 (20) subject to constrants (6), (10), (11), (16), (19). The problem (LRP) can be decomposed nto ndependent subproblems, SP 1, SP 2,, SP I, as follows. (SP ) Mnmze L ( z, w t, q t, n t, y t ), wth L L T T L = R l z + ((P l + c t )w t +O l (w t + n t 1 n t ) + h t n t ) + (u t y t ) l=1 l=1 t=1 t=1 L L T L T T + α l z + β l t w t + π l t n t + μ t v t x t l=1 l=1 t=1 l=1 t=1 t=1 (21) subject to constrants (6), (10), (11), (16), (19). 13

15 Problem SP s equvalent to the sngle tem producton plannng and layout problem. Ths problem s a mxed nteger lnear programmng problem and can be solved optmally usng standard optmzaton software Subgradent optmzaton To mprove the lower bound, we update the Lagrangan multplers by solvng the followng Lagrangan dual problem. (LDP) max α l,β l t,π l t,μ t mn L(z, w t, q t, n t, y t ) subject to the constrants n problem LRP, where mn L( z, w t, q t, n t, y t ) s concave and pecewse lnear contnuous to α l, β l l t, π t and μ t. We use the subgradent method to optmze the dual problem by updatng the multplers at the kth teraton as follows. For example, the Lagrange multpler α l,(k) s updated by α l,(k+1) = α l,(k) + φ(ub Heu L (k) I )( z 1 L I ( z 1 l=1 =1 ) =1 ) 2 (22) where α l,(k) s the Lagrange multplers at kth teraton. UB Heu, L (k) are the value of upper bound and Lagrange functon value at kth teraton respectvely. φ s the step sze whch s determned by prelmnary expermental results Constructon of a feasble soluton The soluton to the relaxed problem s usually nfeasble to the orgnal problem due to the relaxaton of the couplng constrants. A feasble soluton,.e., upper bound, s constructed by the followng heurstc algorthm. The heurstc algorthm successvely checks the volaton of constrants (2), (4), (5) and (15) and modfes the soluton of the LRP problem. We also propose a new algorthm to obtan better upper bound n Secton 6. Heurstc algorthm Step 1: Intalzaton. Set the parameter I plan I, I error. Step 2: Remove those tems wth reserved space larger than needed:.e., f there exsts tem that satsfes the condton l L > max t T d t, tem s removed from the plannng. I plan I plan {}. z Step 3: Check f there are any tems that do not satsfy the constrants (2). If tem does not satsfy (2), then I error I error {}. Step 4: For each tem I error, f there exsts an feasble storage locaton l 2,.e., z 2 I reassgned to the locaton l 2 ; otherwse, the plannng of tem s removed. 14 = 0, then the tem s

16 Update I plan I plan {}. I error I error {}. If I error = φ, go to Step5. Step 5: Generate x t, s t, y t for I plan wth fxed reserved locaton such that the demand for each tem s satsfed. Generate w t, n t for I plan wth fxed reserved locaton such that the constrants (4) - (14) are satsfed. Step 6: Check f there s any perod t that does not satsfy the constrants (15). If the constrant s volated at perod t, then select the tem wth x t > 0 and set w t+1 w t, n t+1 n t for I plan, l L. Update w t+1 If (15) s satsfed for all tme horzon, go to Step 7., w t, n t+1, n t, y t+1, y t for I plan. Step 7: Calculaton of upper bound. Obtan an upper bound (UB) for a feasble soluton. The volatons of the constrants (2), (4) and (5) are checked at Step 3 and modfed at Step 4. Constrant (15) s checked and modfed at Step Constrants for reducng the computaton tme To mprove the performance of the algorthm, the constrants for reducng the computaton tme are ntroduced nto the subproblem SP. From the flow conservaton law, we derve the followng constrants. n t 1 l L d t + l L n t ( I, t T) (23) d t l L + l L n t 1 ( I, t T) (24) w t The maxmum number of avaable locatons for tem s calculated as L (max t T d I {} t ). From the flow conservaton law and the locaton capacty, we derve the followng constrants. l L + l L n t 1 L I {} (max t T d t ) ( I, t T) (25) w t n t 1 l L L (max t T d I {} t ) 5.5. Overall optmzaton algorthm d t 1 ( I, t T) (26) The algorthm of the Lagrangan decomposton and coordnaton s denoted by LDC Algorthm. LDC algorthm Step 1: Intalzaton. Set the parameters and Lagrange multplers α l,(0), β t l,(0), πt l,(0), μt (0) 0. Step 2: Solvng subproblem. The subproblem SP s solved for each tem I. If the Lagrange multplers are updated once at Step 5, the Lagrange multplers are updated by subgradent method (22) after solvng 15

17 each subproblem. Step 3: Constructon of a feasble soluton. Generate a feasble soluton from the soluton of subproblems derved at Step 2 by heurstc algorthm. Step 4: Evaluaton of convergence. If the dfference between the lower bound and upper bound are not updated for predefned tme, end the algorthm. Step5: Update of Lagrange multplers. The Lagrange multplers are updated by subgradent method (22) and return to Step 2. 6 LAGRANGIAN RELAX-AND-FIX HEURISTICS The LDC algorthm presented n the last secton does not generate a good feasble soluton and upper bound. We mprove the algorthm by proposng a fx heurstcs based on COI rule, whch s ntroduced later n ths secton. The dea of the fx heurstcs s that the solutons of the subproblem of the tem descrbed n Secton 5.1 are fxed one by one (Ohga et al. 2013). The smar approach s appled to other capactated lot szng problems (Chen 2015). The man pont s to add the constrants to reduce the feasble area so that better feasble solutons can be found Constrants to derve a feasble soluton We add the followng constrants to SP. Let I rest denote the set of tems whose solutons are not fxed and z (fx), w t,(fx), n t,(fx), y t,(fx) denote fxed decson varables z, w t, n t, y t. L z + z (fx) 1, I rest, l (27) I I rest w t + w t,(fx) 1, I rest, l, t (28) I I rest n t + n t,(fx) 1, I rest, l, t (29) I I rest v t w t + v t w t l=1 f t, I rest, t (30) I I rest l L We also add the followng constrants to SP n order to make the solutons feasble regardless of the fxng order. The constrants whch lmt the avaable locaton for the rest tem have to be consdered when unfxed tems nfluence ts producton plannng and warehouse layout. 16

18 z + (max t T d t ) l L I rest {} 6.2. Cube per-order Index (COI) rule + z (fx) L, I rest (31) I I rest l L The cube-per-order ndex (COI) rule s one of the classcal and wdely used dedcated layout polces (e.g., Hodgson and Lowe (1982), and Malmborg and Deutsch (1988)). Snce the rule s used n our heurstc soluton, a bref ntroducton and dscusson are gven as follows. The COI rule of Heskett (1963) s defned as the rato of the tem's total space requrement to number of trps requred based on the tems demanded. The orgnal heurstcs conssted of locatng the tems wth the lowest COI value closest to the nput/output ponts, and puttng tems that combned a hgh turnover frequency wth a low space requrement n the best storage locatons n the warehouse. Items were then assgned to locatons progressvely farther away from the I/O pont by ncreasng COI. Both Francs (1967) and Harmatuck (1976) proved the optmalty of the COI rule for the assgnment of products to storage locatons to mnmze travel dstance durng storage/retreval. COI rule and varants are stl wdely employed for varous warehouse layout problems, mostly combnng wth other approaches, whch can be found n the recent lterature on the topc, e.g, Malmborg and Krshnakumar (1989), Muppan and Ad (2008), and Zhang and La (2010). We defne COI value for tem as COI = l L z d t T t. (32) The tem wth the lowest value of COI s selected and fxed n the Lagrangan relax-and-fx heurstcs. It s expected that allocaton based on the prorty of COI rule leads to reduced total costs Overall optmzaton algorthm The algorthm of the fx heurstcs based on COI rule n combnaton wth LDC algorthm s denoted by LDC-LF algorthm. The soluton of the subproblem derved by the LDC algorthm s fxed accordng to the mnmum value of COI value for tem. LDC-LF algorthm Step 1: Intalzaton. Step 2: LDC. Set the parameter, I rest I. Obtan the lower bound and the upper bound wth LDC algorthm (Algorthm 1). 17

19 Step 3: Fxng the soluton. Select the tem mn that has the mnmum COI of I rest. Fx the soluton of the SP mn wth (27) -(31). I rest I rest { mn }. Step 4: Check f all tems are fxed. If I rest, go to Step 3. Step 5: Evaluaton of the upper bound. The upper bound derved by the fxed solutons s compared wth the one derved from LDC algorthm at Step 2 and obtan a better upper bound. 7 A large scale neghborhood search based on decomposton structure In Secton 6, the better feasble solutons and the upper bound are obtaned n a short tme. In ths secton, we further mprove the LDC algorthm by usng the neghbourhood search. The dea of the search s to locally optmze subproblem, once a soluton of the problem s avaable. Ths s based on the POPMUSIC (Talard and Voß (2002), Alvm et al. (2009), Ostertag et al. (2009)). These local optmzatons are repeated unt no mprovements are found. The man pont s the defnton of the subproblem. The defnton of the subproblem s presented n the next secton The defnton of the subproblem The orgnal problem s decomposed nto I subproblems by removng the couplng constrants (2), (4), (5) and (15). Several subproblems are chosen randomly. They are aggregated to bud a subproblem by addng extended (2), (4), (5) and (15). The formulaton of the subproblem s the followng when r parts are chosen. Let J I (r = J ) denote the set of the selected tems. Let z (pop), w t,(pop), n t,(pop), y t,(pop) denote the varables of the SP ( J ) and z (fxpop), w t,(fxpop), n t,(fxpop) n t,(fxpop), y t,(fxpop) denote the varables of the SP ( J ). The varables of the SP ( J ) are decson varables and the varables of the SP ( J ) are constant n the followng formulaton. (ASP r ) L L T T mn { R l z + ((P l + c t )w t +O l (w t + n t 1 n t ) + h t n t ) + (u t y t )} J l=1 l=1 t=1 t=1 (33) s. t (6), (10), (11), (16), (19) z (pop) J w t,(pop) J + z (fxpop) I J + w t,(fxpop) I J 1, l (34) 1, l, t (36) 18

20 L n t,(pop) J v t w t,(fxpop) J l=1 + n t,(fxpop) I J L + v t w t,(fxpop) I J l=1 If the soluton of ASP r s updated, the current tentatve soluton s also updated. 1, l, t (37) f t, t (38) 7.2. Overall optmzaton algorthm The algorthm of the large-scale neghborhood search based on POPMUSIC n combnaton wth LDC algorthm s descrbed as LDC-POP algorthm. LDC-POP algorthm Step 1: LDC. Obtan the lower bound and the upper bound wth LDC algorthm. Step 2: Constructng the subproblem. Select the r parts randomly and construct the subproblem ASP r. Step 3: Optmzng the subproblem and updatng the solutons. Optmze the subproblem ASP r. If the solutons mprove the objectve of the orgnal problem, the solutons are updated. Step 4: Evaluaton of convergence. If the number of the search teratons meets a predefned repeat count, gve the upper bound as output. Otherwse go to Step 2. 8 Numercal results In our experment, fve test nstances are solved wth proposed model and approaches. As shown n Table 1, among those nstances, four nstances are developed to verfy the proposed model and soluton: two are of small sze and another two are of medum sze, whch are smar to four of eght producton lnes. Ffth nstance s from the orgnal real-world case, whch s a large-scale problem. Table 1. Test nstances CASE Items No Storages No Perod No. Small_ Small_ Medum_ Medum_ Large (ndustry case)

21 The sze parameters for the small sze nstance 1 are I =5, L =10, and T =5. The man data of small-szed nstance 1 s provded n Table 2. The other data are avaable upon request. Due to the space lmtaton of the paper, we omt the detaed data of medum and large sze nstance. The data for all nstances are avaable upon request. Table 2. Data for the small nstance 1 R l : 4.24 u t : 4.50~15.0 O l : 3.51~5.51 d t : 1~3 P l : 8.14~9.06 v t : 1 h t : 2.94 f t : 10 c t : 5.82~6.73 M: 5 We frst attempt to use GAMS/CPLEX to solve the fve nstances. Snce the proposed model s a mxed nteger programmng, the branch and bound method by CPLEX s utzed to solve the problems. Table 3 shows computatonal results of the branch and bound method wth CPLEX for small and medum nstances, where S-, and M- represent the small and medum szes of nstances respectvely. BB ndcates the branch and bound method by CPLEX. CPLEX cannot obtan any feasble soluton n 375 seconds for the second medum sze nstance but obtans a feasble soluton wth the gap of 0.60% after almost 1 hour. However, for the large scale-nstance of the real-world case, the solver fas to gan any feasble soluton due to the complexty and sze of the model. Table 3. Performance of a branch and bound method CASE LB [-] UB [-] GAP[%] Tme [s] S-BB S-BB M-BB M-BB Large (ndustry case) - - * *Process s termnated wth memory over We use the small and medum nstances to verfy the proposal of the model and solutons wth practcal stuatons. The solutons provde what to produce for each perod, how many locatons to assgn to for each tem, where the tems are to be located and retreved from, and how much of each tem should be nventored over a specfed tme horzon. Table 4 and Table 5 report the producton plannng and storage locatons for small nstance 1 respectvely. From Table 5, we can see that tem 1 uses locaton slots 1, 2, and 5 n all fve months; tem 3 uses locaton 10 n February and Apr. Due to space lmt, the other solutons are not reported, but are avaable upon request. 20

22 Table 4. Producton x(,t): Results for producton plannng Jan Feb Mar Apr May Item Item Item Item Item Table 5. Item Placement w(,l,t): The storage locatons durng each perod Jan Feb Mar Apr May Item1.L Item1.L Item1.L Item2.L Item2.L Item3.L Item3.L Item4.L Item4.L Item5.L To solve the real-world case problem, the related data on demands and warehouse s collected. The parameter sets for ths large nstance are 153 tems, 12 perods, and 813 storage locatons. An attempt s made to solve the problem drectly usng GAMS/CPLEX but t fas to gan any feasble soluton. The computatonal experments were executed to nvestgate the effectveness of the proposed methods. The program code for the proposed method was coded by the Mcrosoft Vsual C++ Express 2010 Express. SP for each tem was solved by the CPLEX12.6 (IBM ILOG). Although the Lagrangan heurstcs developed ams at solvng the large-scale nstance, the fve nstance problems are solved wth the heurstcs to verfy the algorthms. Our computatonal experences show that the smplfed model can save the computatonal tmes for both CPLEX solver and our heurstcs. The followng computatonal results are based on the smplfed model. Table 6 shows the computatonal results of the conventonal method (LDC algorthm ntroduced n Secton 5.5) and the proposed method (LDC-LF algorthm and LDC-POP algorthm proposed n Secton 6.3 and

23 respectvely). The convergence condton s set that the lower bound (LB) and the upper bound (UB) are not updated 3 tmes n small and medum cases. In large case, the LDC algorthm s stopped when the teraton s repeated 5 tmes. The GAP column reports the relatve gap between upper bound and lower bound. Table 6. Performance comparson between the conventonal and the proposed method CASE LB[-] UB[-] GAP[%] Iteraton[-] Total tme[s] S-LDC S-LDC-LF S-LDC-POP S-LDC S-LDC-LF S-LDC-POP M-LDC M-LDC-LF M-LDC-POP M-LDC M-LDC-LF M-LDC-POP L-LDC L-LDC-LF L-LDC-POP For the frst small sze nstance, all developed algorthms can fnd an optmal soluton, whch s the same as the soluton from CPLEX solver, whe algorthms LDC-LF and LDC-POP obtan good solutons wth the small gaps from 0.12% to 0.20% for the second small sze nstance. For the frst medum sze nstance, the computaton tmes for LDC-LF and LDC-POP are longer than CPLEX. The GAPs of the solutons for the two algorthms are 0.43% and 0.26%, respectvely, whch are very small. The second medum sze nstance s much larger than the frst nstance, and the computaton tmes for LDC-LF and LDC-POP are much shorter than CPLEX (CPLEX cannot fnd any feasble soluton n 375 seconds). The GAPs of the solutons for the two algorthms are 1.14% and 0.72%, respectvely, whch are small. For the large sze case, the proposed algorthms can gan good feasble solutons whe CPLEX fas to fnd any feasble soluton due to lmted memory. GAPs of the solutons from the three algorthms are from 6.29% to 8.06%, whe the runnng tmes are around 30 mnutes. Among those algorthms, the performance LDC-LF and LDC-POP s better than that of LDC n all nstances, whch shows the effectveness of COI rule for fxng the subproblem. Between LDC-LF and LDC-POP, the computaton tme for LDC-LF s shorter than that of LDC-POP except for the large sze case, whe the performance 22

24 of LDC-POP s better than LDC-LF, whch lustrates the neghborhood search n subproblems can mprove soluton performance. The above computatonal results lustrate that the proposed model and soluton approaches are effectve, especally for medum and large sze nstances. Among all algorthms developed, the mproved Lagrangan algorthm LDC-POP wth COI rule and neghborhood search s the best n terms of soluton qualty. the gap of 6.29% for the real-world large sze problem shows the soluton s very close to optmal. Consderng that n practce the plannng s done only one or two tmes for the whole year, the runnng tme of 30 mnutes s qute acceptable. It s notable to menton that no feasble soluton can be found f the warehouse layout problem and capactated lot-szng problem for the ndustry case are solved separately due to the space lmt of the warehouse. The proposed ntegrated strategy provdes an effectve way to coordnate the producton plannng and warehouse management. 9 Conclusons In ths paper, we propose an ntegrated strategy to combne a storage locaton assgnment problem wth a capactated lot-szng plannng, whch s motvated by a real-world case. The jont mult-tem storage locaton assgnment capactated lot-szng problem determnes when and how much to produce for each product, and smultaneously determnes where to place the products n the warehouse based on a dedcated storage polcy. A mxed nteger lnear programmng model s developed wth the objectve of mnmzng the total costs of producton, setup, storage, handlng, and reservng space. It also ensures the avaabty of warehouse space n the plannng horzon. The model and soluton approaches are verfed though dfferent sze of nstances. The model wth the data from the real-world case s large scale and the problem complexty s beyond the capacty of the current optmzaton solvers. To solve the large-scale real-world case, three Lagrangan heurstc approaches are developed. Dfferent from the conventonal Lagrangan heurstcs, the Lagrangan relax-and-fx heurstc approach presents a new way to construct a feasble soluton by usng COI rule to fx the subproblem one by one. We test the proposed approach wth small, mddle and large-scale nstances, where the largest one corresponds the real-world case. Computatonal results show that the performance of the Lagrangan relax-and- fx heurstcs wth COI rule outperforms that of the conventonal Lagrangan relaxaton method. Further, an mproved varant of the novel Lagrangan relax-and-fx heurstcs s developed by ntroducng a neghborhood search method, and the computaton results show that the heurstcs can obtan a near optmal soluton wth a reasonable runnng tme. It would be worthwhe to nvestgate the way to reduce the convergence tme of the proposed method or 23

25 develop other soluton approaches for the problems. The current model s developed under the stuaton where the demands are gven and determnstc. An extenson to ths research s to take nto account uncertan demands. Wth uncertan demands, a stochastc model and new soluton approaches can be studed. Acknowledgments Ths work s partally supported by the Natural Scences and Engneerng Research Counc of Canada dscovery grant (RGPIN ), MITACS Internshp, JSPS Grant-n-Ad for Scentfc Research (15H02971), and Japan Socety for the Promoton of Scence Fellowshp. References Abs, N., and Kedad-Sdhoum, S. (2008) The mult-tem capactated lot-szng problem wth setup tmes and shortage costs. European Journal of Operatonal Research, 185, Alvm, A. C. F., Talard, E. D. (2009) Popmusc for the pont feature label placement problem. European Journal of Operatonal Research, 192, Btran, G.R., and Yanasse, H.H. (1982) Computatonal complexty of the capactated lot sze problem. Operatons Research, 28, Buschkühl, L., Sahlng, F., Helber, S., and Tempelmeer, H. (2010) Dynamc capactated lot-szng problems: a classfcaton and revew of soluton approaches. OR Spectrum, 32, Çelk, M., and Süral, H. (2014) Order pckng under random and turnover-based storage polces n fshbone asle warehouses. IIE transactons, 46(3), Chen, H. (2015) Fx-and-optmze and varable neghborhood search approaches for mult-level capactated lot szng problems. Omega, 56, Chu, F., and Chu, C. (2008) Sngle-tem dynamc lot-szng models wth bounded nventory and outsourcng. IEEE Transactons on Systems, Man and Cybernetcs-Part A: Systems and Humans, 38, De Koster, R., Le-Duc, T., and Roodbergen, K.J. (2007) Desgn and control of warehouse order pckng: a lterature revew. European Journal of Operatonal Research, 182, Floran, M., and Klen, M. (1971) Determnstc Producton Plannng wth Concave Costs and Capacty Constrants. Management Scence, 18, Francs, R. L. (1967) On Some Problems of Rectangular Warehouse Desgn and Layout. The Journal of Industral Engneerng, 18,