Internatonal Journal of Industral Engneerng Computatons 2 (20) 93 202 Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: wwwgrowngscencecom/ec A mult-obectve possblstc programmng approach for locatng dstrbuton centers and allocatng customers demands n supply chans Seyed Ahmad Yazdan a and Kamran Shahanagh a* a Department of Industral Engneerng, Iran Unversty of Scence and Technology, Narmak, Tehran, Iran A R T I C E I N O A B S T R A C T Artcle hstory: Receved January 200 Receved n revsed form 5 May 200 Accepted 8 May 200 Avalable onlne 8 May 200 Keywords: aclty locaton Dstrbuton center Supply chan uzzy number Possblstc programmng In ths paper, we present a mult-obectve possblstc programmng model to locate dstrbuton centers (DCs) and allocate customers' demands n a supply chan network desgn (SCND) problem The SCND problem deals wth determnng locatons of facltes (DCs and/or plants), and also shpment quanttes between each two consecutve ter of the supply chan The prmary obectve of ths study s to consder dfferent rsk factors whch are nvolved n both locatng DCs and shppng products as an obectve functon The rsk conssts of varous components: the rsks related to each potental DC locaton, the rsk assocated wth each arc connectng a plant to a DC and the rsk of shpment from a DC to a customer The proposed method of ths paper consders the rsk phenomenon n fuzzy forms to handle the uncertantes nherent n these factors A possblstc programmng approach s proposed to solve the resulted mult-obectve problem and a numercal example for three levels of possblty s conducted to analyze the model 200 Growng Scence td All rghts reserved Introducton One of the prmary ssues n faclty locaton problem s to locate a set of new facltes such that the transportaton cost from varous facltes to customers s mnmzed Due to ncreasng mportance of effcent desgn of supply chan networks, faclty locaton problems n the context of supply chan management (SCM) have attracted attentons of many researchers or comprehensve revew on faclty locaton models and soluton approaches refer to Akens (985), Owen & Daskn (998), and Klose & Drexel (2005) Especally, Mello et al (2009) dedcated a revew paper to faclty locaton models n the supply chan management In the SCM context we generally seek the best stes for locatng dstrbuton centers (DCs) or warehouses n a dscrete soluton space such that total fxed cost of locatng DCs and varable transportaton costs for dstrbutng products (commodtes) from manufacturng plants to customers through opened DCs are mnmzed Ths type of problems s normally modeled as mxed nteger programmng (MIP) formulatons One functon of DCs s consoldaton such that they sort and combne products receved from plants for shpment to customers Moreover, provdng a frm wth flexblty n respondng to changes n the marketplace n a qucker manner, and takng advantage of economes of scale n transportaton costs are other benefts of usng DCs (Amr, 2006) Although the man concern s to select the best possble stes for DCs, but n the lterature, plant locaton s also consdered as decson varables (or example, Kaufman et al, 977; Ro & Tcha, 984, Prkul & Jayaraman, 998; Marín & Plegrn, 999; Amr, 2006; u & Bostel, 2007) There are cases where no lmt s assgned to plants and/or DCs (Kuehn & amburger, 963; Kaufman et al, 977; Ro & Tcha, 984; Brmberg et al, 2000) and therefore the resulted models are formulated as uncapactated faclty locaton problem (UP), whle there are also other cases where some realstc constrants such as producton power of plants and storage space * Correspondng author Tel/fax: +98-27724048 E-mal addresses: shahanagh@ustacr (K Shahanagh) 200 Growng Scence td All rghts reserved do: 05267/ec20003003
94 of DCs are taken nto account and the resulted formulaton s called capactated faclty locaton problem (CP) (ee, 99; nd & Basta, 994, Prkul & Jayaraman, 996, 998; Jayaraman & Prkul, 200; Amr 2006; Keskn & Uster, 2007, among others) Tragantalerngsak et al (2000) studed a two-echelon faclty locaton problem where the facltes n the frst echelon are consdered as uncapactated whle the facltes n the second echelon are capactated There are many nput parameters nvolved n confguraton of a supply network whch are ether determnstc or stochastc such as dstance or customers demands Although some of the parameters, such as dstance, could be determnstc but there are other mportant factors, lke demand, whch are subect to uncertanty and have been dealt wth through stochastc programmng models (ogendran & Terrell, 988; Sheral & Rzzo, 99; wang, 2002, Mranda & Garrdo, 2008) A normal case to handle the uncertanty s to use hstorcal data to fnd an estmate of the uncertan data owever, n many cases, when the hstorcal data are unrelable or the perod of the study s short, the probablty dstrbutons of customers demands cannot be obtaned easly Instead, we could use fuzzy decson makng methods where we are able to use expert's opnon n a form of lngustc terms such as lttle, moderate, large, etc to provde estmatons for the uncertan parameters Therefore, the fuzzy set theory (Zadeh, 965) can be used to deal wth ths stuaton There are many cases where faclty locaton problems are analyzed usng the concept of fuzzy programmng (Darzentas, 987; Rao & Saraswat, 988) Bhattacharya et al (992) developed a fuzzy goal programmng approach to deal wth ths problem Zhou and u (2007) consdered a capactated faclty locaton-allocaton problem, and used a fuzzy programmng method to solve t In ths paper we develop a fuzzy mult-obectve mxed nteger lnear programmng (MOMIP) model for capactated DC locaton and dstrbuton decsons n supply chans where demands of customers and capactes of the DCs are assumed to have some possblty dstrbuton, and rsks assocated wth each potental DC locaton as well as each arc of the network are consdered as fuzzy numbers The possblty programmng approach s used for transformng the resultng fuzzy model to ts crsp equvalent, and the compromse programmng method s adopted to solve ths mult-obectve model Rsk of each potental DC locaton s expressed usng lngustc varables assocated wth natural dsasters such as fre hstory, earthquake possblty, tornados, hurrcanes, etc for evaluatng each locaton n terms of dsrupton rsks Rsks related to the arcs mply nherent rsks n the transportaton of products from each plant to each DC and from each DC to each customer These rsks may be vewed as rsks related to dfferent transportaton modes or nstance, we assume the shpments of the products from DC A to customer C are less rsky f the transportaton faclty s tran and they are more rsky f regular trucks are used, nstead We also consder other condtons affectng the qualty of shpment between each two nodes n each two echelons of the network Zhou and u (2007) studed locatng of facltes n the contnuous soluton space and consdered a sngle-stage dstrbuton problem The problem formulaton of ths study, however, deals wth faclty locaton decsons n dscrete space n a twostage dstrbuton problem urthermore, we consder a b-obectve model whereas they dealt wth the classcal sngle-obectve capactated locaton-allocaton problem The paper s organzed as follows In Secton 2, we brefly revew possblty programmng for multobectve lnear programmng models Mathematcal formulaton of the proposed network desgn problem s developed n Secton 3 In Secton 4, we provde a numercal example for the problem under nvestgaton and dscuss the results nally, n Secton 5, conclusons are gven to summarze the contrbuton of the work 2 Possblty programmng for mult-obectve lnear programmng models Neg and ee (993) proposed a fuzzy mult-obectve lnear programmng model as follows, Maxmze n = c r x, r =,2, p, ()
S A Yazdan and K Shahanagh/ Internatonal Journal of Industral Engneerng Computatons 2 (20) 95 subect to n = a x b, =,2, m, (2) x 0, =,2, n, (3) where x, =, K, n are crsp decson varables c r s the fuzzy coeffcent of the th decson varable n the rth, r =, K, p obectve functon a s the fuzzy coeffcent of the th decson varable n the th constrant ( =, K, n, =, K, m ), and b s the fuzzy rght-hand sde n the th, =, K, m constrant c r, a and b can be expressed as ether trapezodal or trangular fuzzy numbers ere, we represent ther trapezodal form as c r = ( cr, cr, cr2, cr ), a = ( a, a, a 2, a ), b = ( b, b, b 2, b ), and ther trangular form as c r = ( cr, cr0, cr ), a = ( a, a0, a ), b = ( b, b0, b ) Applyng the possblty programmng approach to fuzzy mult-obectve lnear programmng model () (3) under exceedance as well as strct exceedance possblty n the case of trapezodal fuzzy numbers s gven below 2 Case of exceedance possblty (5) (6) where α s a pre-determned value whch s the mnmum requred possblty, and falls n the nterval of (0,] 22 Case of strct exceedance possblty In ths case, only constrant set (5) n the above Eq (4) (6) s replaced by the followng constrant set, In the case of trangular fuzzy numbers, c r 2 s replaced wth c r0, a and a 2 are replaced wth a 0, and b 2 s replaced wth b 0 n the models Note that, n a gven fuzzy model, t s possble to use trapezodal numbers for some nput parameters and trangular numbers for other parameters 3 Model formulaton The network desgn problem consdered n ths secton conssts of three echelons of plants, DCs, and customers and the locaton decsons are made n the DC level urthermore, dstrbuton decsons are made n two stages In the frst stage, products are shpped from capactated plants to capactated DCs, and n the second stage, shpments from capactated DCs to customers (retalers) are consdered n order to satsfy customers demands Demand of each customer and capacty of each DC are assumed to have some possblty dstrbutons whch are expressed usng trapezodal fuzzy numbers The goal s to fnd the best locatons for DCs to be opened and to determne shpment quantty on each arc of the network such that the total cost and the total rsk n the network are mnmzed There
96 s also an upper bound (p) on the number of DCs to be opened The SCM network under study s depcted n g We use the followng notaton for the formulaton of the model I J K D W P k f c e k R S k b set of customers, =,,m set of potental DCs, =,,n set of plants, k=,,k fuzzy demand of customer fuzzy capacty of DC capacty lmt at plant k fxed cost of openng a DC at ste unt transportaton cost from DC to customer unt transportaton cost from plant k to DC fuzzy rsk assocated wth the arc connectng DC to customer fuzzy rsk assocated wth the arc connectng plant k to DC fuzzy rsk assocated wth DC at ste The decson varables are also as follows, z x f DC at ste s opened, 0 otherwse, amount of product shpped from DC to customer y amount of product shpped from plant k to DC k The fuzzy supply chan network desgn (SCND) problem can be formulated as follows, Mn Z = f z + J I J J k K I J I c x J + e k y J k K k Mn Z2 = bx + Rx + S k y k (2) st x = D, I, (3) J x W z, J, (4) I ()
S A Yazdan and K Shahanagh/ Internatonal Journal of Industral Engneerng Computatons 2 (20) 97 J I J x z p, x y k = y k K k, J, P, k K, k 0, I, J, (5) (6) (7) (8) y k 0, J, k K, (9) z { 0,}, J (0) The obectve functon () mnmzes the total costs of the openng and the operatng the DCs, and the total varable transportaton costs n the network The obectve functon (2) mnmzes the total rsk n the network whch has three components: The rsks assocated wth locatng DCs, the rsks of shppng products from plants to DCs, and the rsks of dstrbutng products from DCs to customers Constrant set (3) ensures that the demand of each customer s satsfed Constrant sets (4) and (7) ensure that the capacty restrctons at the DCs and the plants are not volated, respectvely Constrant (5) lmts the number of DCs to be opened to the pre-specfed value p Constrant set (6) s the flow conservaton constrant at each DC nally, constrants (8) (0) are non-negatvty and ntegralty constrants, respectvely To solve SCND model, the compromse programmng (CP) method s used Compromse programmng tres to fnd a soluton that comes as close as possble to the deal (optmal) values of each obectve functon (Zeleny, 982) ere Closeness s defned by the p dstance metrc as follows: p / p p k * f f = γ for p =, 2,,, * = f * n whch f, f 2,, f k are dfferent and conflctng obectve functons f = mn( f ), gnorng all other obectves, s called the deal value for the th obectve and γ s the weght of obectve The x s called a compromse soluton, f mnmzes p by consderngγ > 0, γ =, and p Dfferent effcent solutons can be obtaned by consderng dfferent values for parameters p and γ As p ncreases, larger devatons get more weght, such that for p=, the largest devaton completely domnates the dstance determnaton owever, the most common values are p =,2,and 4 Numercal Example In ths secton, a numercal example s studed to demonstrate the mplementaton of the proposed method and dscuss the advantage of usng the developed model The example conssts of two plants, sx potental DC locatons, and ten customers The decson maker uses the lngustc varables shown n Table to assess the rsks assocated wth potental locatons for DCs and rsks assocated wth each arc of the network for transportng the product
98 Table ngustc varables for assessng rsk of each potental DC locaton and each arc of the network Very ow (V) (0,0,,2) ow () (,2,2,3) Medum ow (M) (2,3,4,5) ar () (4,5,5,6) Medum gh (M) (5,6,7,8) gh () (7,8,8,9) Very gh (V) (8,9,0,0) Plants DCs Customer 2 2 2 K n m g Supply chan network under study 4 Model parameters Model parameters are as follows: = ((80,82,86,95),(75,80,83,86),(90,93,93,95),(87,90,92,97),(85,87,88,98),(78,82,83,85), D (76,8,82,89),(72,76,78,80),(80,86,89,90),(77,78,84,84)), b = (,, V,, M, ), f = (5600,7000,5200,6000,7200,8000), P = (620,570), = ((260,270,280,30),(270,280,290,300),(290,295,320,325),(300,30,320,340),(260,270,290,340), W (270,30,30,320)),
S A Yazdan and K Shahanagh/ Internatonal Journal of Industral Engneerng Computatons 2 (20) 99 T c 4 5 = 7 6 9 7 6 5 2 7 20 6 0 0 6 5 8 8 20 5 6 8 3 2 7 6 2 7 4 9 8 2 9 8 9 20 9 7 8 4 6 8 6 9 3 8 0 8 7 20, 2 7 5 R T = V M M M M V M V V V M V M V V M, V T 25 9 22 0 20 8 T M V e = S = 2 20 20 24 20 5 M Snce the coeffcents of varables n the constrants are all crsp numbers, therefore, there s no dfference between the case of exceedance possblty, and the case of strct exceedance possblty The parametrc crsp equvalent of vectors D and b T, and matrces R T and S are as follows: D ˆ = (95 9α, 86 3α, 95 2α, 97 5α, 98 0α, 85 2α, 89 7α, 80 2α, 90 α, 84 0α ), W ˆ = (30 30α, 300 0α, 325 5α, 340 20α, 340 50α, 320 0α ), R ˆ T = 2 α 5 α 8 α 5 α 5 α 2 α 8 α 0 0 0 5 α 2 α 8 α 2 α 2 α 8 α, 2 α b ˆ = (,, 2 α,, 5 α, ), S ˆ T 6 α = 3 α 5 α 9 α 3 α 8 α 9 α 6 α 6 α 3 α 2 α 3 α 42 Soluton results and analyss The proposed model of ths paper has been solved for α = 0, 05, whch represent very low, moderate and very hgh possbltes The optmal values of the obectve functons for each value of α, are gven n Table 2 Results of the compromse programmng (CP) model for γ =05 (=,2), and p= are represented n Table 3 Table 4 summarzes the soluton obtaned for dfferent values of α n terms of assgned DCs to plants, assgned customers to DCs, plant load and DC load ratos Plant and DC load ratos are also depcted n gures 2 and 3, respectvely Table 2 Optmal values of obectve functons when solved ndvdually α Ob fun 0 05 0 Z 68459 6798850 6768 Z 2 909 747750 6058
200 Table 3 Results of the compromse programmng (CP) model p α 0 05 0 p 0063 0062 0060 Z 770 7645 75773 Z 2 909 74775 6058 Table 4 Summary of the soluton obtaned of the CP model for dfferent values of α p= α 0 05 0 Assgned Assgned customers oad rato * customers oad rato Assgned customers oad rato DC load rato 4,5,6,0 00 4,5,6,0 00 4,5,6,0 00 DC 2 load rato - - - - - - DC 3 load rato 2,3,5,9 00 2,3,5,9 098 2,3,5,9 00 DC 4 load rato - - - - - - DC 5 load rato,7,8 078,7,8 08,5,7,8 089 DC 6 load rato - - - - - - Assgned DCs oad rato Assgned DCs oad rato Assgned DCs oad rato Plant load rato 3,5 053 3 05 3 052 Plant 2 load rato,5 00,5 098,5 094 * oad rato = amount of product shpped from DC(plant)/capacty of the DC(plant) rom gure 2 t s seen that whle load rato for plant has lttle fluctuatons, the load rato for plant 2 s consstently droppng Ths mples that under dfferent possbltes the amount shpped from each plant to each DC wll vary Smlarly, gure 3 shows that whle DC s always fully loaded, the load rato for DC 3 when possblty ncreases from α=0 to α=05 decreases, and smultaneously ths value for DC 5 ncreases Also, note that whle for α=0,05 all of the demand of customer 5 can be satsfed by DCs and 3, for α=, DC 5 should also satsfy some porton of ths customer s demand
S A Yazdan and K Shahanagh/ Internatonal Journal of Industral Engneerng Computatons 2 (20) 20 5 Concluson In ths paper, we have proposed a possblstc programmng approach for supply chan network desgn problem under fuzzy envronment (SCND) Specfcally, we dealt wth locaton and dstrbuton decsons n a supply chan system The proposed model of ths paper has consdered two dfferent obectves n order to ncorporate dfferent rsk factors assocated wth each locaton and each arc of the network such as openng a DC, connectng a plant to a DC, and a DC to a customer nto the model The fuzzy mult-obectve mxed nteger lnear program (MOMIP) also assumes possblty dstrbutons for customers' demands and the capactes of DCs through fuzzy sets theory The proposed model of ths paper, n addton to mnmzng the locaton and the transportaton costs, also mnmzes the total rsk of locaton and dstrbuton n the network through an ntegrated and comprehensve model or the possblstc programmng model we have consdered three levels of possbltes of very low, medum, and very hgh, and for each level of possblty we have analyzed the results through a numercal example Acknowledgements Authors would lke to thank the Edtor and anonymous revewers for ther helpful comments and suggestons References Akens, C (985) aclty locaton models for dstrbuton plannng European Journal of Operatonal Research, 22, 263 279 Amr, A (2006) Desgnng a dstrbuton network n a supply chan system: formulaton and soluton procedure European Journal of Operatonal Research, 7, 567 576 Bhattacharya, U, Rao, J R & RN Twar (992) uzzy mult-crtera faclty locaton problem uzzy Sets and Systems, 5, 277 287 Brmberg, J, ansen, P, Mladenovc, N & Tallard, E D, (2000) Improvements and comparson of heurstcs for solvng the uncapactated multsource Weber problem Operatons Research, 48(3), 444 460 Darzentas, J (987) A dscrete locaton model wth fuzzy accessblty measures uzzy Sets and Systems, 23, 49 54 nd, KS & Basta, T (994) Computatonally effcent soluton of a mult-product, two-stage dstrbuton-locaton problem Journal of Operatons Research Socety, 45, 36 323
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