Facility Location and Inventory Balancing in a Multi-period Multiechelon Multi-objective Supply Chain: An MOEA Approach

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1 Journal of Optmzaton n Industral Engneerng 16 (2014) Faclty Locaton and Inventory Balancng n a Mult-perod Multechelon Mult-objectve Supply Chan: An MOEA Approach Seyed Habb A. Rahmat a, Abbas Ahmad b,*, Behrooz Karm c a PhD Canddate, Industral Engneerng Department, Amrkabr Unversty of Technology, Tehran, Iran b Assstant Professor, Industral Engneerng Department, Amrkabr Unversty of Technology, Tehran, Iran c Assocate Professor, Industral Engneerng Department, Amrkabr Unversty of Technology, Tehran, Iran Receved 18 September, 2013; Revsed 20 November, 2013; Accepted 27 March, 2014 Abstract A comprehensve and ntegrated study of any supply chan (SC) envronment s a vtal requrement that can create varous advantages for the SC owners. Ths consderaton causes productve managng of the SC through ts whole wde components from upstream supplers to downstream retalers and customers. On ths ssue, despte many valuable studes reported n the current lterature, consderable gaps stll preval. These gaps nclude ntegraton and nserton of basc concepts, such as queung theory, faclty locaton, nventory management, or even fuzzy theory, as well as other new concepts such as strategc plannng, data mnng, busness ntellgence, and nformaton technology. Ths study seeks to address some of these gaps. To do so, t proposes an ntegrated four-echelon mult-perod mult-objectve SC model. To make the model closer to the real world problems, t s also composed of nventory and faclty locaton plannng, smultaneously. The proposed model has a mxed nteger lnear programmng (MILP) structure. The objectves of the model are reducng cost and mnmzng the non-fll rate of customer zones demand. The cost reducton part ncludes cost values of raw materal shppng from supplers to plants, plant locaton, nventory holdng costs n plants, dstrbuton cost from plants to warehouses or dstrbuton centers (DCs), and shppng costs from DCs to customer zones. Fnally, snce the lterature of SC lacks effcent Pareto-based mult-objectve evolutonary algorthms (MOEAs), a new mult-objectve verson of the bogeography-based optmzaton algorthm (MOBBO) s ntroduced to the lterature of the SC. The effcency of the algorthm s proved through ts comparson wth an exstng algorthm called mult-objectve harmony search (MOHS). Keywords: Integrated supply chan management, Producton-dstrbuton, Faclty locaton problem, nventory balancng, Plannng problem, MOBBO, MOHS. 1. Introducton Dfferent defntons have been proposed for supply chan (SC) n the lterature. Generally, t s defned as an ntegrated system of facltes and actvtes that synchronzes nter-related busness functons of materal procurement, materal transformaton to ntermedates and fnal products and dstrbuton of these products to customers (Smch-Lev, 2000). In other words, the goal of the supply chan management s to ntegrate supplers, manufacturers, warehouses, and stores, so that the producton of the merchandses and ther dstrbuton can be done at the rght quanttes, to the rght locatons, and at the rght tme. The total objectve of the system s to mnmze system-wde costs whle satsfyng servce level requrements of the customers. Snce SC covers a vast range of concepts and methods, varous ssues are evolved as the subjects of the ndustral and academc research n SC. Some common ssues nclude suppler selecton, producton-dstrbuton plannng, transportaton and dstrbuton, faclty locaton of the dstrbuton centers and logstc warehouse, queung ssues, nventory controllng and balancng. Some researchers have focused on only one of these tems whle others have consdered the combnaton of two or more of them. One of the mportant combnatons n the lterature s the combnaton of producton area of the SC wth the dstrbuton part of t, known as producton-dstrbuton problems. In ths class, Jayarman and Prkul (2001) consdered factory, dstrbuton center, and demand area wthn ther mult-product determnstc SC. Ylmaz and Cagatay (2006) ntroduced a three-stage strategc plannng for ther producton-dstrbuton network. In ths model, whch consders one product, mult-supplers, mult-manufacturers, and mult-dstrbuters, demand s determnstc and the objectve s to mnmze cost of producton, transportaton, and nventory. Some revew papers are also presented on ths subject (Erenguc et al.,1999, Chen, 2004). Some papers focus on nventory * Correspondng author Emal address: abbas.ahmad@aut.ac.r 83

2 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory... subject n ther supply chan (Muckstadt and Roundy, 1987; Chan et al., 2002; Lev et al., 2005). Inventory has consderable role n the studes of SCs as the man artery of any supply chan. The basc SC model that consders ths tem s generally known as sngle warehouse multretaler (SWMR) problem (Muckstadt and Roundy, 1987). Federgruen and Zpkn (1984) studed sngle perod, one warehouse, mult-retaler problem under uncertan demands. Roundy (1985) proposed a polcy wth 98% effectveness n O(nlogn) tme for analyzng a problem whch permts no shortage or backloggng. Chan and Kumar (2009 a, b) nvestgated a manufacturng envronment that ncluded warehouse-schedulng problem n a manufacturng envronment. Poon et al. (2009) studed order pckng operatons n warehouses. Another mportant ssue, whch s nserted nto SC models, s faclty locaton. Javd and Azad (2010) solved an ntegrated model of faclty locaton, capacty, nventory, and routng. Bdhand et al. (2009) developed a mxed lnear nteger programmng problem of mult-commodty supply chan. They solved ther problem usng decomposton methods. Rappold and Van Roo (2009) studed two-echelon supply chan, whch combned faclty locaton, nventory allocaton, and capacty nvestment. Among the presented studes, most of the researches are sngle objectve (Wllams 1981, Gen and Syarf, 2005; Tsaks and Papageorgou, 2008). Further, some studes focus on mult-objectve problems n the SC areas. Ths class s more realstc, because most of the real world problems, specfcally n the complex envronment of the SC problems, cope wth several goals. In ths class, Altparmak et al. (2006) developed a mult-objectve shortage forbdden model that nvestgated network structure of manufacturers and customer area. Ther model tred to mnmze costs, delver tme and balance of the capacty of the factores. Jola et al. (2011) proposed a lnear mult-objectve producton-dstrbuton model. Ther model consdered a SC wth mult-products, levels, and perods. However, they changed ther model nto a sngle objectve model n ther solvng approach. Sadegh et al. (2011) developed a sngle-vendor sngle-retaler n a mult-product supply chan. Alakbar and Sefbarghy (2011) ntroduced a socal responsble suppler selecton model. Songsong and Lazaros (2012) also studed a multobjectve producton-dstrbuton model. Ther model consdered a unversal SC wth three objectves of costs, response, and servce level. Taherkhan and Sefbarghy (2012) determned the materal flows n a mult-echelon assembly supply chan. Shankar et al. (2013), n ther mult-objectve producton-dstrbuton, proposed sngleproduct four-echelon supply chan archtecture. They also consdered faclty locaton plannng n ther problem. However, they dd not consder the nventory ssue n ther ntegrated model. They solved ther model va a mult-objectve hybrd partcle swarm optmzaton (MOHPSO) algorthm. Ther approach s a Pareto-based approach n whch the mult-objectve s not converted nto a sngle objectve model. These approaches are more popular these days (Deb et al., 2001). The number of these algorthms s not consderable n the lterature of the SC though. Vahdan and Sharf (2013) developed an nexact-fuzzy-stochastc optmzaton model for a closed loop supply chan network desgn problem. Accordng to the lterature, ths research proposes an ntegrated model whch flls some gaps of the lterature. To do so, snce, n the lterature of SC, specfcally for producton-dstrbuton plannng problem, fewer researches have studed the shortage permtted assumpton, ths tem s consdered n our model. Moreover, some other terms of the nventory ssue, as the man artery of any SC model, are ncluded n the developed ntegrated mult-echelon mult-perod nventory parts of the model. In addton, to make the model more realstc, t s also encompassed faclty locaton plannng. The fnal structure of ths model s as a mxed nteger lnear programmng (MILP) problem. Furthermore, snce the lterature lacks effcent Paretobased mult-objectve evolutonary algorthms (MOEAs), a new mult-objectve verson of the bogeography-based optmzaton algorthm (MOBBO) s ntroduced to the lterature of the SC. Fnally, ths algorthm s compared wth an exstng algorthm called MOHS (Rahmat et al. 2013). The results are also evaluated through dfferent statstcal and non-statstcal tests, tables, and fgures. The paper s organzed as follow. The developed ntegrated model s descrbed n secton 2. Ths secton ncludes problem defnton ncludng all parts of the model rangng from assumptons and ndces to objectve functons and constrants. Secton 3 presents the requred concepts and operators of the proposed MOEA. Secton 4, through dfferent computatonal experments, proves effcency of the proposed algorthms. Secton 5 concludes the paper and presents the future works. 2. Problem Defnton In ths secton, the ntegrated SC model s descrbed. The proposed SC model of ths research s a multechelon mult-perod model whch encompasses nventory and faclty locaton plannng smultaneously. The nventory part of the model ncludes four-echelon multperod nventory cost as the objectve functon and nventory balancng among dfferent echelon wthn dfferent perods. The structure of the proposed model s as MINLP. Fgure 1 llustrates a smple structure of ths model, schematcally. The rest of ths secton defnes the requred defntons and notatons and then formulates the man model n dfferent subsectons Notatons l: Number of supplers (h =1, 2,, l) n: Number of potental plant locatons ( = 1, 2,, n) 84

3 Journal of Optmzaton n Industral Engneerng 16 (2014) t: Number of warehouse (DC) locatons (e = 1, 2,, t) m: Number of customer zones (markets) or demand ponts (j = 1, 2,, m) p: Number of components (c = 1, 2,, p) s: Number of tme perods (k = 1, 2,, s) 2.2. Parameters D jk : Average demand from markets j at tme perod k K k : Potental capacty of plant at tme perod k K S ek : Potental capacty of warehouse e at tme perod k chk : Supply capacty by suppler h from component c at tme perod k F : Annual fxed cost of keepng open of plant F e : Annual fxed cost of keepng open of warehouse e C hck : Cost of makng and shppng a components c from supply source h to plant at tme perod k C ek : Cost of producng and shppng one unt from plant to warehouse e at tme perod k C ejk : Cost of throughput and shppng one unt from warehouse e to customer j at tme perod k IC k : Inventory holdng cost of one unt n plant at tme perod k IC ek : Inventory holdng cost of one unt n warehouse e at tme perod k IC ck : Inventory holdng cost of component c n plant at tme perod k c: Consume coeffcent of component c 2.3. Decson varables Y : 1, f plant s open, 0 otherwse Y e : 1, f warehouse e s open, 0 otherwse X hck : Quantty of component c shpped from suppler h to plant at tme perod k X ejk : Quantty shpped from warehouse e to customer zone j at tme perod k " X ek : Quantty shpped from plant to warehouse e at tme perod k I ck : Inventory quantty of component c n plant at tme perod k I ek : Inventory quantty n warehouse e at tme perod k " k I : Inventory quantty n plant at tme perod k 2.4. The man model Ths subsecton presents the man proposed model as follows. 1 n t n l p s n t s " e e hck hck ek ek 1 e1 1 h1 c1 k1 1 e1 k1 Mn Z f Y f Y C X C X t m s p n s n s t s " Cejk X ejk ICck Ick ICk I k ICek I ek e1 j1 k1 c1 1 k1 1 k1 e1 k1 Mn Z s.t. n 2 1 t m s e1 j1 k1 m s j1 k1 D X ejk jk Xhck Schk h 1, 2,..., l, c 1, 2,..., p, k 1, 2,..., s (3) 1 t X ejk D jk j 1, 2,..., m, k 1, 2,..., s (4) e1 t " X ek KkY 1, 2,..., n, k 1, 2,..., s (5) e1 (1) (2) 85

4 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory... m X ejk KekY e e 1, 2,..., t, k 1, 2,..., s (6) j1 l t " Xhck X ek 0 1, 2,..., n, c 1, 2,..., p, k 1, 2,..., s (7) h1 e1 n m " X ek X ejk 0 e 1, 2,..., t, k 1, 2,..., s (8) 1 j1 l t ck ck 1 hck c ek h1 e1 t " " " k k1 k ek e1 n " m ek ek1 ek ejk 1 j1 (9) I I X X c 1, 2,..., p, 1, 2,..., n, k 1, 2,..., s I I K X 1, 2,..., n, k 1, 2,..., s (10) (11) I I X X e 1, 2,..., t, k 1, 2,..., s " " hck ek ejk ck k ek X, X, X, I, I, I 0 e " c0 0 e0 Y, Y 0,1 I I, I 0 In ths model, Eq.1 models the frst objectve functon, whch mnmzes the total cost n supply chan. Total cost ncludes raw materal shppng from supplers to plants, plant locaton, nventory holdng costs n plants, dstrbuton cost from plants to warehouses or dstrbuton centers (DCs), throughput and shppng costs from DCs to customer zones. The objectve functon (2) s mnmzng the non-flled rate of customer zones demand. Equaton (3) ensures that the total quantty shpped from a suppler at each perod cannot exceed the supply capacty. Equaton (4) ndcates that the demand at customer zone should be satsfed to the maxmum extend. Equaton (5) shows that no plant can supply more than ts capacty f the plant s opened. The Equaton n (6) represents that no warehouse can supply more than ts capacty f the warehouse s opened. Equaton (7) ensures that the quantty shpped out of a plant cannot exceed the component quantty receved. Consequently, equaton (8) ensures that the quantty shpped out of a warehouse cannot exceed the quantty receved. Equatons (9) and (10) represent the nventory balance constrant n plant. Equaton (11) s the nventory balance equaton for DC. For example, n ths equaton, nventory of one unt n warehouse e and at tme perod k ( I ek ) s equal to the nventory at one tme unt of prevous perod ( I ek 1 ) plus amount of one unt shpped from plants to DC e at tme perod n 1 " ek (12) k( X ) mnus amount of one unt shpped from DC e to customer zones (CZs), at tme perod m j1 ejk k( X ). Fnally, constrant (12) shows postve and bnary varables. Fgure 1 llustrates the dagram of ths supply chan schematcally. 3. Solvng Methodology As mentoned earler, the developed model of ths research has MINLP structure. It s proved that smpler model than ths model are NP-Hard (Shankar et al., 2013). Therefore, a meta-heurstc approach s proposed to solved the problem. Ths approach ntroduces MOBBO algorthm to the SC area. Ths algorthm s a MOEA based on bogeography optmzaton (BBO) algorthm as the sngle objectve verson. BBO s a populaton-based optmzaton algorthm (Smon, 2008). Therefore, t has dfferent smlartes wth other exstng populaton-based algorthms lke genetc algorthm (GA) or partcle swarm optmzaton (PSO). Generally, n ths type of algorthm we have a set of ndvduals that s called populaton. The ndvdual n ths algorthm s called habtat or sland. Any feature of the ndvdual (lke gene n GA) here s known as a SIV. The ftness value of the ndvduals here s measured by hgh sutablty ndex (HSI). However, t has some dstnctve dfferences wth the exstng populatonbased algorthms. For nstance, n ths algorthm, nstead 86

5 Journal of Optmzaton n Industral Engneerng 16 (2014) of ftness value mgraton rates are used to gude the algorthm. Actually, n bogeography scence mgraton s dvded nto two dfferent performances of the speces, ncludng emgraton and mmgraton. For each of these performances, a specfc rate s also consdered called emgraton rate ( ) and mmgraton rate ( ). j Emgraton rate determnes how lkely a speces (emgratng speces) shares ts features wth other speces (mmgratng speces). Lkewse, mmgraton rate determnes how lkely a speces (mmgratng speces) accepts features from other speces (emgratng speces). In a relaton wth HSI, t can be expected that features mgrate from hgh-hsi habtats (emgratng habtat) to low-hsi habtats (mmgratng habtat). Therefore, by usng mgraton rates, the am of the BBO s to gude the optmzaton process n a way that the HSI s maxmzed (Rahmat and Zandeh, 2012). Now, before explanng the operators of ths algorthm, snce ths paper s gong to ntroduce mult-objectve verson of the BBO, the fundamental prncples and defntons of MOAs are ntroduced Mult-objectve prncples In a mult-objectve problem lke f ( x) f ( x),..., f ( x) subject to 1 m g ( x) 0, 1,2,..., c, x X, soluton a can domnate soluton b ( a, b X ) f followng two condtons are held smultaneously: 1) f ( a) f ( b), 1, 2,..., m 2) {1, 2,..., m}: f ( a) f ( b) Now, n dfferent teratons of the MOEA, a set of solutons that cannot domnate each other s known as Pareto solutons set or Pareto front. Improvng ths Pareto front, durng dfferent teratons to acheve Pareto optmal Fg. 1. A four layers dagram of the proposed supply chan front, s the goal of MOEA. Improvement n multobjectve envronment has two sgns, ncludng (1) mprovng the convergence to the optmal front, or (2) mprovng the dversty of the exstng solutons of a Pareto front. Therefore, t s expected that the fnal obtaned Pareto front of an MOEA has an approprate convergence and dversty. To evaluate these two types of the mprovement n a MOEA, dfferent types of measures can be used, some of whch wll be ntroduced and used n the next secton Representaton, Intalzaton and decodng scheme of the habtats In MOBBO, lke any other populaton-based algorthm, the optmzaton process starts wth ntalzng the ntal populaton. The proposed habtat s composed of three rows by whch some constrants are satsfed based on the values of decson varables. The frst row represents that facltes such as plant and warehouse are open or close n bnary representaton. The second row ncludes the quantty shpped through supplers, plants, DCs and CZs. The thrd row ndcates the amount of nventory level n plants and warehouses. The general form of a habtat structure s represented n Fgure 2. In the frst row of ths habtat, potental locaton of plants and warehouses are coded as bnary varables. The amount of shpped components and one unt of product are calculated accordng to supply capacty of supplers, plants, DCs and demand of markets at each tme perod n second row of soluton. For example, quantty shpped from a plant to DCs at tme perod k s less than plant capacty. In addton, ths amount should be less equal than the component amount shpped from supplers to plant. 87

6 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory... Fg. 2. The habtat structure of the MOBBO In the thrd row of habtat, nventory levels n plants and warehouses are decoded consderng the second row of the habtat. For example, the amount of one unt that s delvered to DC must be equal to the amount of one unt that leaves from and s stored n ths DC. In order to prevent the negatve and nfeasble solutons n ths part, untl the nventory quantty s negatve, a random number between zero and one s generated. If the value of ths number s less than a predetermned value, a plant s selected randomly, and the amount of shpped tems s equal to the mnmum total of ths value wth nventory defct and plant capacty to shp. Otherwse, a customer zone (CZ) s selected randomly and the amount of shpped from DC s equal to maxmum dfference of ths amount wth nventory defct and zero value to generate feasble solutons of both states Sortng strategy Ths operator s the frst man factor that dstngushes MOBBO from ts sngle verson. In ths part, after decodng the habtats and calculatng ther HSIs, nstead of sortng the populaton accordng to the HSI s values, a mult-objectve strategy s used. Ths strategy s proposed by Deb et al. (2000). In ths strategy, an operator called FNDS s used for assgnng ranks to ndvduals of the populaton due to domnaton concept. Then, another operator called CD s used to estmate densty of solutons whch are lad surroundng a partcular soluton n the same rank. Now, accordng to these two operators the populaton s sorted. To do so, n the case of the dfferent ranks (or the ndvduals from dfferent fronts), the one wth lower rank s better. However, f the ranks are the same, the one wth hgher CD s preferred Selecton strategy and mgraton operator Ths operator s the second man factor that dstngushes MOBBO from ts sngle objectve verson. In ths selecton strategy, a bnary tournament selecton s used for selectng the emgratng habtat. To do so, after calculatng of the CDs and FNDSs, f a specfc habtat needs to be mmgrated, two habtats are selected randomly. Then, f they are from dfferent ranks (or dfferent front), the one wth lower rank s selected; otherwse, the one wth hgher CD s selected as the emgratng habtat. A scheme of ths selecton strategy can be seen n Fgure 7. Now, to mplement mgraton operator, t s necessary to calculate and j. The method of ths calculaton s the thrd (and the last) man factor that dstngushes MOBBO from ts sngle verson. After sortng the populaton, mmgraton rate and emgraton rate can be evaluated as Eq.13 and Eq. 14, respectvely. In these equatons, k represents rank of th habtat after sortng all habtats accordng to mult-objectve strategy and n represents sze of the populaton. Of course, t should be mentoned that, k rangng from 1 to n and the hgher values are of more nterest. k I(1 ) (13) n k E( ) (14) n The nverse relaton between these two rates s shown n Fgure 3. In ths fgure, E and I, represents maxmum number of the mgraton rates and are usually set at zero, and S denotes the state of the speces amount. Accordng to what was mentoned and ths fgure, by ncreasng of the number of speces (or gong for a more sutable habtat), the mmgraton rate s decreasng and the emgraton rate s ncreasng. It means that the features n a more sutable habtat wth mscellaneous speces are more lkely to be emgrated rather than to be mmgrated. Ths s the man concept of the mgraton. j 88

7 Journal of Optmzaton n Industral Engneerng 16 (2014) perform mgraton operator, the structure ntroduced for the sngle objectve BBO s used. The Unform neghborhood structure s used for conductng the mgraton as Fgure 4. In ths fgure, H represents habtat and n denotes number of the SIVs of each habtat. To show how Unform mgraton works, a scheme s plotted n Fgure 5. In ths structure, the mmgratng habtat accept features from sharng or emgratng habtat for those cells that ther random vector numbers (Rand s 0 or 1) s Mutaton operators Fg. 3. The varaton of mgraton rates toward number of the speces (Smon, 2008) Fgure 3 summarzes the concepts that were proposed n ths sub secton to do a qute mgraton process. Accordng to ths fgure, t s clear that a good habtat (wth low ) s less lkely to be mmgrated, whle as, a poor habtat (wth hgh ) s so lkely to be. Fnally, to In ths research, for mutaton structure Mask operator s mplemented. The scheme of ths operator s llustrated n Fgure 5. Fg. 4. The mgraton operator and the selecton strategy Fg. 5. Unform mgraton operator scheme Fg. 6. Mask mutaton operator scheme In ths fgure, a Mask vector s generated randomly wth the number from the nterval [0,1]. Now, for those cells of the Mask vector, whch have the values less than 0.5, the habtat s mutated. For conductng ths mutaton, the mentoned cell s regenerated and ts value s assgned randomly. 89

8 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory The MOBBO s optmzaton process The evoluton process of the MOBBO s llustrated schematcally n Fgure 7. Ths process s started by ntalzng the ntal populaton of the habtat R t. Then, BBO s operators, ncludng mgraton and mutaton, are mplemented on R t to create the new populaton Q t. The blendng of habtats of t P and t Q t creates R t. In ths step, R are sorted n several fronts by means of the explaned strategy n sub secton 3.3. Now, to create populaton of the next teraton Pt 1, whle the capacty of P s not exceeded, the fronts are added to Pt 1, accordng to ncreasng order of ther ranks. But, when wthout a front, Pt 1 has fewer members than populaton sze and wth t, Pt 1 has more members than populaton sze, the habtats must be selected partally to reach the predetermned populaton sze. In ths stuaton, the habtats of the front are sorted n decreasng order of ther CDs, and the habtats of next teraton are chosen from top of the front. In fact, the most dfferentaton of the MOBBO wth the NSGAII (Deb et al., 2000) s ther evoluton operators. In other words, the searchng heart of the NSGAII s GA, but the searchng heart of MOBBO s BBO. In other terms, nstead of some smple dfferences, they gude the multobjectve process smlarty. The pseudo code of the MOBBO s also presented n Fgure 8. In ths fgure, the searchng heart of the MOBBO, whch s BBO, s separated n the mddle of the pseudo code. 3.7 The MOHS As mentoned above, MOBBO s compared wth the MOHS from the lterature (Rahmat, 2013). MOHS s a Pareto-based mult-objectve verson of the sngle objectve harmony search (HS) algorthm, whch s renforced by the same operators as the ones mplemented n ths study to get a mult-objectve. Ths algorthm mmcs the mprovsng process of muscans. In HS, three dfferent operators are used, ncludng harmony memory operator, ptchng operator, and random operator. The more elaborate descrpton of ths algorthm s found n Rahmat et al. (2013). However, t s requred to menton that for the ptchng operator of the MOHS n ths study, a structure just lke what explaned for the mutaton of the MOBBO s used. 4. Computatonal Results In ths secton, to assess the developed model and the proposed algorthm, frst some test problems are generated. Then, the algorthms are compared based on the whole generated test problems. Ths comparson s made through utlzng dfferent types of the statstcal and non-statstcal tests and varous explanatory llustratons Test problem generatng In the model study, the parameters are generated as Table 1. In ths table, U(1000,1500) represents a random number generated n the nterval (1000,1500) from unform dstrbuton. Fg. 7. Evoluton process of the MOBBO 90

9 Journal of Optmzaton n Industral Engneerng 16 (2014) Fg. 8. The pseudo code of MOBBO Table1 Input data of the model Parameter Dstrbuton functon Parameter Dstrbuton functon D jk U(1000,1500) C ejk U(30,50) F F U(2000,8000) e U(2000,8000) C hck U(30,50) C ek U(40,70) IC ck U(5,10) IC k U(8,12) IC ek U(10,15) c U(0,1) Then, to test the generated model, 14 test problems were created va the nformaton presented n Table 2. In ths table, the factors that make a dstnctve problem are number of supplers (l), plant locaton (n), DCs (T), and CZs (m). Moreover, 5 raw materal types and at 3 tme perods are consdered. 91

10 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory Outputs of the algorthms on the generated test problems In ths subsecton, after defnng some requred defntons, the outputs of the algorthms are evaluated. The defntons nclude the mplemented metrcs and the tests used Mult-objectve metrc descrpton Generally, two man features are consdered to evaluate the performance of a MOEA. 1) The frst feature assesses whether the fnal Pareto front of the algorthm s converged to the Pareto optmal front or not. 2) The second feature evaluates the dversfcaton of the set of solutons of the Pareto front. Table 2 Generated test problems Test Problem L N T M Table 3 Utlzed performance measures Metrc Metrc calculaton Metrc bref descrpton Dversty (Ztzler, 1999) ( ) Spacng (Schott, 1995) ( ) Mean deal dstance (MID) (Rahmat et al., 2012) ( ) Smultaneous metrc (SM) (Rahmat et al., 2013) ( ) Number of the non-domnated solutons n fnal Pareto ( NOS ) (Rahmat et al., 2012) ( ) m D (m ax f m n f ) j1 1:n j 2 1:n 1 * ( ) n 2 S d d d n 1 1 k n Λ k m n k j j j 1 1 n mn f f d j d m c MID where c ( f ) j NOS SM In the lterature of MOEAs, dfferent metrcs are suggested to conduct the evaluaton of these two features. In ths paper, four metrcs are mplemented that are summarzed n Table 3. Followng notatons are used n ths table. Further nformaton about the metrcs can be found n the mentoned references n the Table3. MID D - j 1 2 It s used to evaluate the spread of the front. It s used to measure the unformty of the solutons wthn a front. It s used to measure the closeness of solutons n a Pareto front wth an deal pont whch s usually consdered as (0, 0). Ths metrc consders the two features of the MOEAs smultaneously. Measures number of the Pareto solutons. d : denotes the space between two neghbor solutons n : denotes the number of the exstng solutons n the Pareto fronts m: denotes the number of the objectve functons f : denotes the j th objectve functon of the th soluton j The notaton n ths table and the rest of the secton ndcates that hgher values are superor, whereas ndcates superorty of smaller values. 92

11 Journal of Optmzaton n Industral Engneerng 16 (2014) Mult-objectve metrc outputs In ths subsecton, by calculatng the mentoned metrcs of the prevous subsecton, for each metrc, the algorthms are compared. To do so, dfferent types of tests and evaluatons are mplemented. Intally, the outputs of the metrcs are calculated and summarzed n Tables 4 and 5. Table 4 presents the obtaned outputs for the three metrcs MID, Dversty (D), and Smultaneous metrc (SM). Ths classfcaton of the metrcs nto two dfferent tables has two man reasons. Frst, the capacty of the page and the requred explctness of the table restrct us to brng all of the outputs n a sngle table. Second, snce SM s calculated accordng to the MID and D, these three metrcs are summarzed n the same table. Table 4 Outputs of the algorthms for D, MID, and SM MOBBO MOHS # D MID SM D MID SM Av Table 5 Outputs of the algorthms for the S and NOS MOBBO MOHS # S NOS S NOS Av In these two tables, the last row, named average (Av.), calculates the average of each metrc on all test problems for a specfc algorthm. In ths row, the superor algorthm s bolded. For example, for NOS n Table 5, MOBBO s superor and s bolded. Accordng to these two tables, followng results can be obtaned: 1. For NOS, MOBBO s superor. Fgure 9 supports ths superorty. Ths type of fgure s also plotted for other metrcs n Fgures 9-13 to present the manner of the outputs for dfferent algorthms on the test problems more explctly. In fact, these fgures present the same concepts and evaluatons, as presented n Tables 5 and 6, graphcally. These fgures are called stacked column charts. Each column of these fgures compares the obtaned value of each metrc by the algorthms. The columns are dvded nto two parts; the upper part s the value obtaned from MOBBO whereas the beneath part represents the value obtaned from the MOHS. Clearly, n each fgure 15 columns that are related to the outputs of the algorthms on the 14 test problems, on that specfc metrc, exst. 2. For S, MOHS s superor. Fgure 9 supports ths superorty. It s clear that for most of the columns MOHS part has less value than MOBBO part. 93

12 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory For MID, MOBBO s superor. Fgure 11 supports ths superorty. It s clear that for most of the columns, MOBBO part has less value than MOHS part. 4. For D, MOBBO s superor. Fgure 12 supports ths superorty. In ths metrc, for most of the columns MOHS part has less value than MOBBO part consderably. In ths metrc, snce the hgher value shows superorty, MOBBO s superor. 5. For SM, MOBBO s superor. Fgure 13 supports ths superorty. Agan, n ths metrc, MOBBO has consderable superorty. MOBBO NOS MOHS NOS Test Problems Fg. 9. The outputs of the algorthms for NOS () MOBBO S MOHS S Test Problems Fg. 10. The outputs of the algorthms for S () MOBBO MID MOHS MID 3.5E+09 3E E+09 2E E+09 1E Test Problems Fg. 11. The outputs of the algorthms for MID () 94

13 Journal of Optmzaton n Industral Engneerng 16 (2014) E MOBBO D MOHS D Test Problems Fg. 12. The outputs of the algorthms for D () MOBBO sm MOHS SM Test Problems Fg. 13. The outputs of the algorthms for SM () Up to ths part, the superorty of the proposed algorthms s recognzed on dfferent metrcs. However, the superorty recognton requres statstcal approval. Hence, two types of statstcal tests are mplemented whch are called 2-sample t test and Mann Whtney test. These two tests are alternatve parametrc and nonparametrc tests that are used for comparng two populatons of data statstcally (Chambar et al., 2012). The outputs of the statstcal tests are summarzed n Table 6. Ths table presents the outputs of the two consdered types of tests. In ths table, P-values of the tests are reported. Theoretcally, f the P-value s less than our consdered sgnfcant level, whch s 0.05, the null hypothess (H 0 ) s rejected. The outputs of ths table are consstent wth the prevous obtaned results and confrm them. It means both of the statstcal and non-statstcal tests approve that for NOS, D, and SM, MOBBO s superor whereas for the S, MOHS wns. For MID, non-statstcal tests and evaluatons ndcate the superorty of MOBBO. However, statstcal tests show that ths dfference s not sgnfcant. Moreover, the accuracy of results s also more renforced by checkng the same outputs of the parametrc and non-parametrc tests. For llustratng the results of the statstcal tests more explctly, the box-plots are also plotted for each metrc n Fgure 14. Accordng to ths fgure, t s clear that for the cases that null hypothess s rejected, whch algorthm s superor and shows why n MID there s no sgnfcant dfference graphcally. Fnally, to have a better sense of the pattern of Pareto solutons n the fnal Pareto optmal front of each algorthm and comparng the front pattern of the algorthms, Fgure 15 s plotted. In ths fgure, a sample fnal Pareto front s plotted for all test problems. Each secton of the fgure plots the front of the two algorthms for that specfc test problem. 95

14 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory... Table 6 The summarzed statstcal test results Mann Whtney test t-test P-value Result P-value Result NOS 0.00 MOBBO outperforms MOHS 0.00 MOBBO outperforms MOHS S MOHS outperforms MOBBO MOHS outperforms MOBBO MID 0.59 H 0 s not rejected H 0 s not rejected D 0.00 MOBBO outperforms MOHS 0.00 MOBBO outperforms MOHS SM 0.00 MOBBO outperforms MOHS 0.00 MOBBO outperforms MOHS Fg. 14. The box plots of the statstcal tests In ths way, t s easy to assess and understand why the mentoned results are obtaned n each part of the evaluatons. Besdes, t shows clearly that how much MOBBO s superor to reach to the mentoned two man features of the MOEAs n comparson wth MOHS. 96

15 Journal of Optmzaton n Industral Engneerng 16 (2014) Fg. 15. A sample of the fnal Pareto front of the algorthms 97

16 Seyed Habb A. Rahmat et al./ Faclty Locaton and Inventory Concluson Ths study nvestgated an ntegrated four-echelon multperod mult-objectve SC model, whch ncludes nventory and faclty locaton plannng smultaneously. The model has two objectve functons: the mnmzaton of the total cost wth mscellaneous cost terms and mnmzaton of non-fll rate of customer zones demand. Then, snce the proposed MINLP model belongs to NPhard class of the optmzaton problem, a new MOEA, called MOBBO was developed for solvng the problem. MOBBO was valdated through a comparson wth an algorthm of the lterature. Ths comparson was conducted va both statstcal and non-statstcal tests on varous generated test problems by dfferent multobjectve metrcs. Furthermore, dfferent types of statstcal and non-statstcal fgures were mplemented. The results of these evaluatons prove the hgh superorty of the MOBBO for solvng the model. Future work can nclude other practcal ntegratons n terms of queung theory, faclty locaton, nventory management, strategc plannng, data mnng, busness ntellgence, and nformaton technology. Future research can also consder other terms such as queung consderatons, value chan optmzaton, green desgnng of the SC for makng the model more realstc. References [1] Alakbar A., Sefbarghy M., A Suppler Selecton Model for Socal Responsble Supply Chan, Journal of Optmzaton n Industral Engneerng 8, [2] Altparmak F., Gen M., Ln L., Karaoglan I., A steady-state genetc algorthm for mult-product supply chan network desgn. Computers and Industral Engneerng 56(2), [3] Bdhand H.M., Yusuff R.M., Ahmad M.M.H.M, Bakar M.R.A, Development of a new approach for determnstc supply chan, network desgn. European Journal of Operaton Research (198), [4] Chambar A., Rahmat S.H.R., Najaf A.A., Karm A., A b-objectve model to optmze relablty and cost of system wth a choce of redundancy strateges, Computers and Industral Engneerng (63), [5] Chan L.M.A., Murel A., Shen Z.J.M., Effectve zero-nventory-orderng polces for the sngle-warehouse mult retaler problem wth pecewse lnear cost structures. Management Scence 48(11), [6] Chan F.T.S., Kumar V., 2009a.Performance optmzaton of a leaglty nspred supply chan model: A CFGTSA algorthm based approach. Internatonal Journal of Producton Research 47(3), [7] Chan, F. T. S., Kumar, V., 2009b. Hybrd TSSA algorthm-based approach to solve warehouse-schedulng problems. Internatonal Journal of Producton Research 47(4), [8] Chen Z. Integrated producton and dstrbuton operatons: taxonomy, models and revew. In: D. Smch-Lev, SD. [9] Deb, K., Agrawal, S., Pratap, A., Meyarvan, T., A fast eltst non-domnated sortng genetc algorthm for mult-objectve optmzaton: NSGA-II. In: Proceedngs of the parallel problem solvng from nature VI (PPSN-VI) conference, [10] Erenguc S.S., Smpson N.C., Vakhara A.J., Integrated producton/dstrbuton plannng n supply chans: an nvted revew. European Journal of Operatonal Research 115 (2): [11] Federgruen, A., & Zpkn, P., Computatonal ssues n an nfnte horzon, mult-echelon nventory model. Operatons Research 32(4), [12] Gen M, Syarf A., Hybrd genetc algorthm for mult-tme perod producton/dstrbuton plannng. Computers and Industral Engneerng 48(4), [13] Jayarman, V., Prkul H., Plannng and coordnaton of producton and dstrbuton facltes for multple commodtes. European Journal of Operatonal Research (133), [14] Javd A. A., Azad N., Incorporatng locaton, routng and nventory decsons n supply chan network desgn. Transportaton Research Part E. do: /j.tre [15] Jola F, Razm J, Rostam NKM., A fuzzy goal programmng and meta heurstc algorthms for solvng ntegrated producton:dstrbuton plannng problem. Central European Journal of Operaton Research 19 (4): [16] Lev, R., Roundy R., Shmoys D.B., A constant approxmaton algorthm for the one-warehouse-multretaler problem, In Symposum on Dscrete Algorthms, Proceedngs of the Sxteenth Annual ACM-SIAM Symposum on Dscrete Algorthms [17] Muckstadt J.A., Roundy R.O., Mult-tem, onewarehouse, mult-retaler dstrbuton systems. Management Scence 33(12), [18] Poon T.C., Choy K.L., Chow H.K.H., Lau H.C.W., Chan F.T.S., Ho K.C., A RFID case-based logstcs resource management system for managng orderpckng operatons n warehouses. Expert Systems wth Applcaton 36, [19] Rahmat, S.H.A., Zandeh, M., A new bogeography-based optmzaton (BBO) algorthm for the flexble job shop schedulng problem, Internatonal Journal of Advanced Manufacturng Technology DOI /s [20] Rahmat S.H.A., Zandeh M., Yazdan M., Developng two mult-objectve evolutonary algorthms for the mult-objectve flexble job shop schedulng problem, Internatonal Journal of Advanced Manufacturng Technology (64) [21] Rahmat S.H.A., Hajpour V., Nak S. T. A., A softcomputng Pareto-based meta-heurstc algorthm for a mult-objectve mult-server faclty locaton problem, Appled Soft Computng (13) [22] Rappold J. A., Van Roo B. D., Desgnng multechelon servce parts networks wth fnte repar capacty. European Journal of Operaton Research (199), [23] Roundy, R., %-effectve nteger-rato lot-szng for one-warehouse multretaler systems. Management Scence 31(11), [24] Sadegh J., Sadegh A., Sad-Mehrabad M., A parameter-tuned Genetc Algorthm for Vendor Managed Inventory Model for a Case Sngle-vendor Sngle-retaler 98

17 Journal of Optmzaton n Industral Engneerng 16 (2014) wth Mult-product and Mult-constrant, Journal of Optmzaton n Industral Engneerng 9, [25] Schott, J. R., Fault tolerant desgn usng sngle and multcrtera genetc algorthms optmzaton. Master s thess, Department of Aeronautcs and Astronautcs, Massachusetts Insttute of Technology, Cambrdge, MA. [26] Smon, D., Bogeography-based optmzaton. IEEE Transactons on Evolutonary Computaton 12, [27] Shankar B. L., Basavarajappa S., Chen J. C. H., Kadadevaramath R. S. Locaton and allocaton decsons for mult-echelon supply chan network A mult-objectve evolutonary approach, Expert Systems wth Applcatons (40) [28] Smch-Lev D, Kamnsky P, Smch-Lev E.,2000. Desgnng and Managng the Supply Chan Concepts, Strateges and Case Studes, McGraw-Hll. Boston. [29] Songsong L., Lazaros GP., Mult objectve optmzaton of producton, dstrbuton and capacty plannng of global supply chans n the process ndustry. Omega 41 (2), [30] Taherkhan M., Sefbarghy M., Determnaton of Materal Flows n a Mult-echelon Assembly Supply Chan, Journal of Optmzaton n Industral Engneerng 11, [31] Tsaks P, Papaeorgou LG., Optmal producton allocaton and dstrbuton supply chan networks. Internatonal Journal of Producton Economcs 111(2), [32] Wllams J.F., Heurstc technques for smultaneous schedulng of producton and dstrbuton n mult-echelon structures: theory and emprcal comparsons. Management Scence 27 (3), [33] Wu, Z.J Shen (eds), Chapter 17 of the book handbook of quanttatve supply chan analyss: modelng n the e-busness era, Kluwar, Dordrecht. [34] Vahdan B., Sharf M., An Inexact-Fuzzy- Stochastc Optmzaton Model for a Closed Loop Supply Chan Network Desgn Problem, Journal of Optmzaton n Industral Engneerng 12, [35] Ztzler E., Evolutonary Algorthms for Multobjectve Optmzaton: Methods and Applcatons. PhD. Thess, Dssertaton ETH No , Swss Federal Insttute of Technology (ETH), Zürch, Swtzerland. 99

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