Robust Multi-Objective Facility Location Model of Closed-Loop Supply Chain Network under Interval Uncertainty

Transcription

1 Internatona Journa of Operatons Research Internatona Journa of Operatons Research Vo. 14, No. 2, (2017) Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty Nea Makroon, Mazar Saah* Department of Appe Mathematcs, Facuty of Mathematca Scences Unversty of Guan, Rasht, Iran Receve October 2016; Revse January 2017; Accepte Apr 2017 Abstract: In ths paper, we conser a suppy chan network that ncues mutpe pants, coecton centers, eman markets, an proucts, where a mut-obectve mxe nteger programmng moe has been eveope to mnmze cost an maxmze some envronmenta ssues by Amn an Zhang (2013a). Due to the uncertanty of the emans an returns, the robust counterpart of the moe s scusse uner nterva uncertanty. Accorng to some numerca resuts, the percentage changes of robust an stochastc moes are compare reatve to etermnstc moes n fferent cases. The numerca resuts show that the robust moe n comparson wth the stochastc programmng moe gves a coser ft to the resuts of the etermnstc moe. Keywor Cose-oop suppy chan, Mxe-nteger near programmng, Mut-obectve programmng, Robust Optmzaton. 1. INTRODUCTION Cose oop suppy chan (CLSC) network ams at reucng the waste an generatng proft for enterprses through ntegratng forwar an reverse ogstcs. CLSC has been consere as one of the sustanabe practce to push conventona open-oop systems to cose-oop ones by Eah an Franchett (2014). Reusng returne proucts by recycng process, prevents epetng resources, reuces envronmenta pouton an optmzes utty by Gue Jr. an Van Wassenhove (2006). Therefore, conserng the process of hanng the prouct returns n suppy chan s of sgnfcant mportance by Zarean Jahrom et a. (2014). Due to ts varous aspect mportance such as envronmenta, mte resources, t s the focus of current research, Ayres et a (1997), Boemhof an Corbett (2010), De Govann (2014), Mohaer an Faah (2014), Paksoy an Ozceyan (2014), an severa mxe nteger programmng moes are eveope to optmze CLSC. However, ue to the confct between economc optmzaton an envronmenta protectons, severa research has consere mut-obectve an goa programmng approach for CLSC. For exampe, n Amn an Zhang (2013a), a mxe nteger near programmng (MILP) moe s propose that mnmzes the tota cost. Beses, the moe s extene to conser envronmenta factors by weghe sums an -constrant methos. Snce n a CLSC network, n partcuar, n reverse fow, some parameters mght be uncertan, many researchers have use fferent approaches to ea wth t. E-Saye et a. (2010) eveope an MILP moe for a CLSC network n the presence of uncertanty n eman. Pshvaee et a. (2011) formuate an MILP moe n the exstence of uncertanty n emans, returns an transportaton costs parameters base on Ben-Ta an Nemrovsk approach (1998), (2000). An enhance verson of the uncertan moe n Pshvaee et a. (2011) s gven n Makroon an Saah (2016) that has much ess constrants when emans, returns an transportaton costs between factes are uncertan. Saema et a. (2007) extene the reverse ogstcs moe of Feschmann et a. (2001) n orer to take nto account uncertanty n emans. Francas an Mnner (2009) propose a two-stage stochastc moe to esgn a cose-oop network uner uncertan emans an returns. Pshvaee et a. (2009) propose a etermnstc optmzaton moe for a reverse ogstcs network. However, envronmenta factors have not been taken nto account n the moe. Lee an Dong (2009) propose a two-stage stochastc programmng moe for CLSC network. They aso eveope a souton approach by smuate anneang. Pshvaee an Torab (2010) eveope a possbstc mxe nteger programmng moe to ea wth uncertanty n CLSC confguraton. Sh et a. (2010) propose a moe to maxmze the proft of a remanufacturng system usng a Lagrangan reaxaton metho. Wang an Hsu (2010) propose an nterva programmng moe where the uncertanty s n the form of fuzzy numbers. Sh et a. (2011) stue a proucton pannng probem for a mut-prouct cose-oop system. The authors consere uncertan emans an returns by stochastc programmng. Amn an Zhang (2013b) eveope an optmzaton moe uner uncertan * Corresponng author s e-ma: saahm@guan.ac.r Te:

2 54 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) emans an ecson envronment for a CLSC. Vahan et a. (2012) appe fuzzy mut-obectve robust optmzaton to confgure a CLSC network. Amn an Zhang (2013a) have nvestgate the mpact of emans an returns uncertantes on the network confguraton by stochastc programmng (scenaro-base). In ths paper our focus s on the moe of Amn an Zhang (2013a). We conser nterva uncertanty n returns an emans an present the robust counterpart of the moe n Amn an Zhang (2013a). Moreover, on severa exampes, the robust approach s compare wth stochastc programmng approach of Amn an Zhang (2013a) showng that the robust approach performs better. 2. PROBLEM DEFINITION In ths secton, a genera CLSC network s escrbe that ncues pants, coecton centers, an eman markets (Fg. 1) Amn an Zhang (2013a). The pants can manufacture new proucts an remanufacture returne proucts. The proucts are sent to eman markets by pants. Then, the returne proucts are sent to coecton centers, where coectng use proucts from eman markets, etermnng the conton of them by nspecton an/or separaton to fn out whether they are recoverabe or not. Then senng recoverabe returns to the pants an unrecoverabe returns to the sposa center. The goa s to know how many an whch pants an coecton centers shou be open, an whch proucts an n whch quanttes shou be stock n them n orer mnmze the cost. Moreover, we assume that The network s for a snge pero. A returne proucts from eman markets are coecte n coecton centers. Locatons of eman markets are fxe. Locatons an capactes of pants an coecton centers are known n avance Pants 1,,,, Deman markets 1,,,, Dsposa Center Coecton centers 1,,,, Fgure 1. The cose oop suppy chan network Amn an Zhang (2013a) 3. MATHEMATICAL MODEL The network s formuate as an MILP probem, where sets, parameters an ecson varabes are efne as foows Amn an Zhang (2013a): Sets I = set of potenta manufacturng an remanufacturng pants ocatons ( 1,,,, I) J = set of proucts ( 1,,,, J) K = set of eman markets ocatons ( 1,, k,, K) L = set of potenta coecton centers ocatons ( 1,,,, L) Parameters A = proucton cost of prouct B = transportaton cost of prouct per km between pants an eman markets C = transportaton cost of prouct per km between eman markets an coecton centers D = transportaton cost of prouct per km between coecton centers an pants X Copyrght 2017 ORSTW

3 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) 55 O = transportaton cost of prouct per km between coecton centers an sposa centers E = fxe cost for openng pant F = fxe cost for openng coecton center G = cost savng of prouct (because of prouct recovery) H = sposa cost of prouct P = capacty of pant for prouct Q = capacty of coecton center for prouct t = the stance between ocaton an k generate base on the Eucean metho ( t k k an t are efne n the same way). t = s the stance between coecton center an sposa center = eman of customer k for prouct k r = return of customer k for prouct k α = mnmum sposa fracton of prouct Varabes X = quantty of prouct prouce by pant for eman market k k Y = quantty of returne prouct from eman market k to coecton center k S = quantty of returne prouct from coecton center to pant T = quantty of returne prouct from coecton center to sposa center Z = 1, f a pant s ocate an set up at potenta ste, 0, otherwse W = 1, f a coecton center s ocate an set up at potenta ste, 0, otherwse ( H + ) ( ) ( ) mnz = E Z + FW + A + B t X + C t Y + G + D t S + 1 k k k k k k O t T st.. X, k, (1) k k S + X Z P, (2) k k Y k X, k, (3) k α Y T,, (4) k k Y W Q, (5) k k Y = S + T,, (6) k k Y = r, k, (7) k k { 1} Z, W 0,,, (8) X, Y, S, T 0,, k,, (9) k k X Copyrght 2017 ORSTW

4 56 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) The obectve functon s mnmzaton of the tota cost. The frst an secon terms are the fxe costs of openng pants an coecton centers, respectvey. The thr term s the proucton an transportaton costs of the new proucts. The forth term s reate to prouct recovery an transportaton costs of returne proucts. The ffth term represents the tota recovery an transportaton costs of returne proucts from coecton centers to pants. Fnay, the sxth term s the sposa an transportaton costs. The constrant (1) guarantees that the tota number of each manufacture prouct for each eman market s equa or greater than the eman. Capacty constrant of pants s gven by (2). Constrant (3) ncates that forwar fow s greater than reverse fow. Constrant (4) enforces a mnmum sposa fracton for each prouct. Constrant (5) gves capacty constrant of coecton centers. Constrant (6) tes that the quantty of returne proucts from eman market s equa to the quantty of returne proucts to pants an quantty of proucts n sposa center for each coecton center an each prouct. Constrant (7) represents the returne proucts. Constrants (8) an (9) are the bnary an non-negatve ecson varabes. Bese the tota costs whch s mnmze, the author n Amn an Zhang (2013a) consere envronmenta ssues n the moe. To o so, new parameters are efne. M s parameter of usng envronmenta freny materas by pant to prouce prouct an N s parameter of usng cean technoogy by coecton center to process prouct. Thus the secon obectve functon can be wrtten as foows: Z M X S N Y S T max = k k k k To sove ths mut-obectve optmzaton probem ε -constrant metho s use n Amn an Zhang (2013a) as foow. ( ) ( ) mnz = E Z + FW + A + B t X + C t Y + G + D t S + ( H + ) k k k k k k O t T st M X S N.. Y S T ε (10) k k k k X, k, (11) k k S + X Z P, (12) k k Y k X, k, (13) k α Y T,, (14) k k Y W Q, (15) k k Y = S + T,, (16) k k Y = r, k, (17) k k { 1} Z, W 0,,, (18) X, Y, S, T 0,, k,, (19) k k 4. ROBUST MODEL UNDER INTERVAL UNCERTAINTY Uncertanty n emans an returns are maor ssues n a suppy chan network. Thus t s benefca to take them X Copyrght 2017 ORSTW

5 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) 57 nto account n the optmzaton moe. In Amn an Zhang (2013a) the authors have use stochastc programmng n orer to ncue uncertanty n the presente moe. However, here we scuss the robust counterpart of the presente moe uner nterva uncertanty for emans an returns. Conser the foowng etermnstc near optmzaton probem: mn cx + st.. Ax b Base on Ben-Ta an Nemrovsk (1998, 2000), the reate uncertan near optmzaton probem that conssts of a coecton of near optmzaton probems can be efne as foows: mn cx + st.. Ax b (20) c,, Ab, U where U s the uncertanty set for uncertan ata. A vector x s a robust feasbe souton to probem (20) f t satsfes a reazatons of the constrants from the uncertanty set U. Ben-Ta an Nemrovsk (1999) efne the robust counterpart of probem (20) as foows mn cˆ ( x) = sup cx Ax b c Ab U + :,,,, (21) c,? Ab, U An optma souton to probem (21) s the optma robust souton of probem (20). Such a souton satsfes the constrants for a possbe reazatons of the ata, an guarantees an optma obectve functon vaue not worse than cˆ ( x * ). Probem (21) s a sem-nfnte near optmzaton probem an seems to be computatonay ntractabe. Nevertheess, t turns out that for a we varety of compact, convex uncertanty sets, the robust counterpart moe s a tractabe (poynomas sovabe) convex optmzaton probem, usuay a near optmzaton or a conc quaratc probem (see Ben-Ta et a. (2009), Ben-Ta an Nemrovsk (2000, 2002)). Uner box uncertanty, = { } s unknown but boune n a box of the form ξ ξ = 1,.., m, = 1,, n { ξ : ξ ξ ρ, 1,,, 1,, } u = R G = m = n Box where ξ s the nomna vaue of the ξ an the postve numbers G represent uncertanty scae an ρ >0 s the uncertanty eve. A partcuar case of nterest s G = ξ, whch correspons to a smpe case where box contans ξ whose reatve evaton from the nomna ata s of sze up to ρ. To eveop the robust counterpart of moe (1), emans an returns are consere as uncertan parameters an t s assume they beong to certan ntervas. Thus constrant (11) n the uncertan case s as foows: { ρ, },,, = : = 1,,, = 1,, k k k Box k k k k X k such that u R G k K J To have ths nequaty feasbe for any n the gven uncertanty set k t s suffcent to have u so as to mmunze aganst nfeasbty, Box max,, k k u k Box X k Ths s further equvaent to X Copyrght 2017 ORSTW

6 58 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) + ρ,, k k k X G k However, for the returns constrants snce t s n equaty form n (17), thus frst we reax t to the nequaty one wthout osng anythng. Snce the moe ams to mnmze the returne proucts as t s a part of the obectve functon. Then anaogous to the emans constrants, the uncertan verson of returns constrants (17) wth nterva uncertanty s as foows: r + ρ,, k k k Y r G k Therefore, the robust counterpart of moe (10)-(19) uner nterva uncertanty for emans an returns s the foowng MILP probem: ( ) ( ) mnz = E Z + FW + A + B t X + C t Y + G + D t S + ( H + ) k k k k k k O t T st M X S N.. Y S T ε k k k k + ρ,, k k k X G k, S + X Z P k k k,, k Y X k α Y T,, k k, Y W Q k k,, Y = S + T k k r + ρ,, k k k Y r G k { 1} Z, W 0,,, X, Y, S, T 0,, k,, k k 5. COMPUTATIONAL EXPERIMENTS In ths secton, on severa ranomy generate exampes, we compare the robust optmzaton moe wth the stochastc programmng approach use n Amn an Zhang (2013a). A optmzaton probems are sove usng CPLEX In Amn an Zhang (2013a) scenaro anayss has been use to observe the effects of uncertanty. In ths paper, at frst ke Amn an Zhang (2013a), etermnstc moe s beng sove by usng nomna ata an the exstng scenaros n Tabe 2. Nomna ata are ranomy generate usng the ranom strbutons specfe n Tabe 1. In the etermnstc moe, ε s a very mportant parameter. In Amn an Zhang (2013a), the vaue of the obectve functon s evauate for fferent ε vaues an t s observe that f the vaue of ε ncreases then the vaue of the obectve functon w ncrease too. Accorng to Fgure 5 n Amn an Zhang (2013a), for ε from 40,000 to 500,000, the changes of the obectve functon s not sgnfcant but for ε greater than 500,000 the obectve functon ncreases magnfcenty. Thus n ths paper, we conser ε s 500,000. By conserng the probabty of accuracy of each scenaro, the resut of the stochastc moe s obtane an ste n Tabe 2. In the ast coumn of Tabe 2, the percentage changes of resuts from both etermnstc an stochastc moes reatve to X Copyrght 2017 ORSTW

7 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) 59 scenaro 5 (base case) are obtane. As state n Amn an Zhang (2013a), the resuts show that the stochastc programmng moe can gan fexbe optma CLSC confguraton wth the obectve functon near to the scenaro 5 (base case). Tabe 1. The sources of ranom generaton of the nomna ata Amn an Zhang (2013a) Parameter Corresponng ranom strbuton I J K L U A ~unform (13.5, 16.5) B ~unform (0.0131, ) C ~unform (0.0045, ) D ~unform (0.0027, ) O ~unform (0.0014, ) G ~unform (6.3, 7.7) H ~unform (2.25, 2.75) α ~unform (0.27, 0.33) E ~unform (4,500,000, 5,500,000) F ~unform (450,000, 550,000) t, t, t, t k k ~unform (0,100) M, N ~unform (0,1) P ~unform (75,600, 92,400) Q ~unform (30,600, 37,400) k r k Tabe 2. Scenaro anayss. Scenaros Deman Return Probabty Obectve Vaue Change % ,906, ,731, ,859, Determnstc ,773, (Base Case) ,815, ,948, ,723, ,992, ,720, Stochastc 10 Combnaton of nne scenaros 17,826, In orer to evauate the robust moe we conser two cases. In the frst case, the number of emans an returne proucts are constant but n the secon case these quanttes are chosen ranomy n an nterva n orer to conser the entre amounts scenaro n ths nterva. We conser 5 uncertanty eves (ρ=0.2,0.5,1,2,3) an we et G =1 an k r G =1. k Frst case Determnstc an robust moes are sove usng nomna ata prove n Tabe 1, an the resuts are ste n the thr coumn of Tabe 3. Then uner each uncertanty eve, three ranom reazatons are unformy generate n X Copyrght 2017 ORSTW

8 60 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) the corresponng uncertanty set (.e. nomna vaue ρg, nomna vaue+ ρg ) to anayze the performance of the soutons obtane by the propose robust an etermnstc moes. The resuts are gven n the forth coumns of Tabes 3. The ffth coumn of Tabe 3 shows the percentage changes of the vaues of the obectve functon for etermnstc an robust (fourth coumn) reatve to the vaue of the obectve functon of the etermnstc moe uner the nomna ata. Secon case In ths case, t s suppose that nomna ata for parameters wth uncertanty an r are ranomy k k generate usng the ranom strbutons specfe n the ntervas (27000, 33000) an (9000, 11000), respectvey an the nomna ata for other parameters generate from Tabe 1. Smar to the metho expane n the frst case, etermnstc an robust moes are nvestgate uner nomna ata an reazaton an percentage changes are cacuate. The resuts are obtane n Tabe 4. The resuts obtane n Tabes 3 an 4 show that ncreasng uncertanty eve causes an ncrease n the vaue of the obectve functon for the robust moe an resuts n an ncrease n the percentage of ts changes reatve to etermnstc moe uner nomna ata. For ower uncertanty eves, percentage changes of robust moe n many cases s ess than the stochastc moe but wth ncreasng the uncertanty eve, the percentage of changes of robust moe s more than stochastc moe, because n stochastc moe, movng far away from nomna ata brngs own the probabty of accuracy of scenaros whe n robust moe, ata are strbute unformy aroun the nomna ata. Tabe 3. Summary of Frst case resuts uner uncertan return an emans., r k k =30000 k r =10000 k ρ Obectve Functon Vaue Obectve functon vaues Percentage change over uner nomna ata uner reazatons the fna obectve functon vaue uner nomna ata (%) Determnstc Robust Determnstc Robust Determnstc Robust ,815,494 17,818,939 17,818,818 17,822, ,813,953 17,817, ,813,147 17,816, ,824,106 17,815,305 17,823, ,823,466 17,832, ,820,653 17,829, ,832,718 17,802,966 17,820, ,812,948 17,830, ,825,892 17,843, ,849,941 17,844,760 17,879, ,825,951 17,860, ,840,238 17,874, ,867,164 17,863,793 17,915, ,834,543 17,886, ,840,799 17,892, Tabe 4. Summary of Secon case resuts uner uncertan return an emans., r k k ρ Obectve Functon Vaue uner Nomna ata Obectve functon vaues uner reazatons Percentage change over the fna obectve functon vaue uner nomna ata (%) (27000,33000) k r (9000,11000) k Determnstc Robust Determnstc Robust Determnstc Robust ,666,427 17,669,874 17,668,658 17,672, ,667,825 17,671, ,664,831 17,668, ,675,044 17,660,008 17,668, ,659,467 17,668, ,669,787 17,678, X Copyrght 2017 ORSTW

9 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) ,683,660 17,670,873 17,688, ,681,938 17,699, ,664,239 17,681, ,700,893 17,651,009 17,685, ,685,209 17,719, ,644,566 17,679, ,718,126 17,714,916 17,766, ,660,919 17,712, ,688,670 17,740, Fnay, to show the effect of mnmum sposa fracton of prouct ( α ), whch s an mportant parameter reate to reverse suppy on the obectve functon, senstvty anayss s performe. The three moes, etermnstc, robust an stochastc are evauate for fferent vaues of α. Fg. 2 an 3 show the resuts for a three etermnstc (uner nomna ata), robust an stochastc moes. These two fgures show that by ncreasng parameter ( α ), the vaue of the obectve functon for a the three moes ncreases. Aso t can be observe that the vaue of the obectve functon for etermnstc an robust moes are cose to each other Robust Determnstc Stochastc Fgure 2. Senstvty anayss of α for Frst case n etermnstc (uner nomna ata), robust (wth ρ=0.5) an stochastc scenaros Robust Determnstc Stochastc Fgure 3. Senstvty anayss of α for Secon case n etermnstc (uner nomna ata), robust (wth ρ=0.5) an stochastc scenaros X Copyrght 2017 ORSTW

10 62 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) 6. CONCLUSION In ths paper, a mxe nteger programmng moe for a cose oop suppy chan network s consere n accorance wth Amn an Zhang (2013a). Due to the exstence of uncertanty n eman an return parameters, the robust counterpart of the moe was presente. Then for some exampes, the senstveness of the moe reatve to fferent uncertanty eves an parameter was assesse an the obtane numerca resuts showe that the percentage of changes of resuts n robust moes has been ess than stochastc moe n most cases whe by ncreasng the eve of uncertanty, ths rate for stochastc moe has been ess. Aso the numerca resuts showe that the vaue of the obectve functon for a three moes wth parameter have a rect reaton, t means that by ncreasng the quantty of the parameter the vaue of the obectve functon for a the three moes ncreases. REFERENCES 1. Amn, S.H., Zhang, G. (2013a). A mut-obectve facty ocaton moe for cose-oop suppy chan network uner uncertan eman an return. Appe Mathematca Moeng, 37 (6): Amn, S.H., Zhang, G. (2013b). A three-stage moe for cose-oop suppy chan confguraton uner uncertanty. Internatona Journa of Proucton Research, 51(5): Ayres, R., Ferrer, G., V a n Leynseee, T. (1997). Eco-effcency, asset recovery an remanufacturng. European Management Journa, 15(5): Ben-Ta, A., E-Ghaou, L., Nemrovsk, A. (2009). Robust Optmzaton. Prnceton Unversty Press. 5. Ben-Ta, A., Nemrovsk, A. (1998). Robust convex optmzaton. Mathematcs of Operatons Research, 2: Ben-Ta, A., Nemrovsk, A. (1999). Robust soutons to uncertan near programs. Operatons Research Letters, 25: Ben-Ta, A., Nemrovsk, A. (2000). Robust soutons of near programmng probems contamnate wth uncertan ata. Mathematca Programmng, 88: Ben-Ta, A., Nemrovsk, A. (2002). A robust optmzaton methooogy an appcatons. Mathematca Programmng, (Ser. B), 92: Boemhof, J.M., Corbett, C.J. (2010). Cose-oop suppy chans: envronmenta mpact. Wey Encycopea of Operatons Research an Management Scence. 10. De Govann, P. (2014). Envronmenta coaboraton n a cose-oop suppy chan wth a reverse revenue sharng contract. Annas of Operatons Research, 220(1): Eah, B., Franchett, M. (2014). A new optmzaton moe for cose-oop suppy chan networks. Internatona Technoogy Management Conference (ITMC), IEEE Internatona, E-Saye, M., Afa, N., E-Kharboty, A. (2010). A stochastc moe for forwar reverse ogstcs network esgn uner rsk. Computers & Inustra Engneerng, 58: Feschmann, M., Beuens, P., Boemhof-Ruwaar, J.M., Van Wassenhove, L.N. (2001). The mpact of prouct recovery on ogstcs network esgn. Proucton an Operatons Management, 10 (2): Francas, D., Mnner, S. (2009). Manufacturng network confguraton n suppy chans wth prouct recovery. Omega, 37 (4): Gue Jr., V.D.R., an Van Wassenhove, L.N. (2006). Cose-Loop Suppy Chans: An Introucton to the Feature Issue (Part 1). Proucton an Operaton Management, 15(3): Lee, D., Dong, M. (2009). Dynamc network esgn for reverse ogstcs operatons uner uncertanty. Transportaton Research, 45 (1): Makroon, N., Saah, M. (2016). On the robust optmzaton approach to cose-oop suppy chan network esgn uner uncertanty. Internatona Journa of Operatons Research, 13(2): Mohaer, A., an Faah, M. (2014). Cose-oop suppy chan moes wth conserng the envronmenta mpact. The Scentfc Wor Journa, Artce ID , 23 pages. 19. Paksoy, T., Ozceyan, E. (2014). Envronmentay conscous optmzaton of suppy chan networks. Journa of the Operatona Research Socety, 65(6): Pshvaee, M.S., Rabban, M., Torab, S.A. (2011). A robust optmzaton approach to cose-oop suppy chan network esgn uner uncertanty. Appe Mathematca Moeng, 35: Pshvaee, M.S., Joa, F., Razm, J. (2009). A stochastc optmzaton moe for ntegrate forwar/reverse ogstcs network esgn. Journa of Manufacturng Systems, 28 (4): Pshvaee, M.S., Torab, S.A. (2010). A possbstc programmng approach for cose-oop suppy chan network esgn uner uncertanty. Fuzzy Sets an Systems, 161 (20): X Copyrght 2017 ORSTW

11 Makroon an Saah: Robust Mut-Obectve Facty Locaton Moe of Cose-Loop Suppy Chan Network uner Interva Uncertanty IJOR Vo. 14, No. 2, (2017) Saema, M.I.G., Barbosa-Povoa, A.P., Novas, A.Q. (2007). An optmzaton moe for the esgn for a capactate mut-prouct reverse ogstcs network wth uncertanty. European Journa of Operatona Research, 179 (3): Sh, J., Zhang, G., Sha, J. (2011). Optma proucton pannng for a mut-prouct cose oop system wth uncertan eman an return. Computers & Operatons Research, 38 (3): Sh, J., Zhang, G., Sha, J., Amn, S.H. (2010). Coornatng proucton an recycng ecson wth stochastc eman an return. Journa of Systems Scence an Systems Engneerng, 19 (4): Vahan, B., Tavakko-Moghaam, R., Moarres, M., Babo, A. (2012). Reabe esgn of a forwar/reverse ogstcs network uner uncertanty: A Robust-M/M/c queung moe. Transportaton Research Part E, 48 (6): Wang, H., Hsu, H. (2010). Resouton of an uncertan cose-oop ogstcs moe: an appcaton to fuzzy near programs wth rsk anayss. Journa of Envronmenta Management, 91 (11): Zarean Jahrom, H., Faahnezha, M.S., Saegheh, A., Ahma Yaz, A. (2014). A Robust mut obectve optmzaton moe for sustanabe cose-oop suppy chan network esgn. Journa of Inustra Engneerng Research n Proucton Systems, 2(3): X Copyrght 2017 ORSTW