International Journal of Industrial Engineering Computations

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1 Internatonal Journal of Industral Engneerng Computatons 3 (2012) Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: A fuzzy mxed nteger lnear programmng model for ntegratng procurement-productondstrbuton plannng n supply chan Alreza Pourrousta a*, Saleh Dehbar a, Reza Tavakkol-Moghaddam b, Amr Imenpour c and Mahd Nader-Ben a a Department of Industral Engneerng, Islamc Azad Unversty, South Tehran Branch, Tehran, Iran b Department of Industral Engneerng, Colleague of Engneerng, Unversty of Tehran, Tehran, Iran c Department Graduate School of Management and Economc, Sharf Unversty of Technology, Tehran, Iran A R T I C L E I N F O A B S T R A C T Artcle hstory: Receved 15 October 2011 Receved n revsed form November, 26, 2011 Accepted 26 December 2011 Avalable onlne 14 December 2011 Keywords: Jmenez fuzzy technque Cadenas & Verdegay fuzzy technque VRP Tme wndow In ths paper, we study a supply chan problem where a whole seller/producer dstrbutes goods among dfferent retalers. Such problems are always faces wth uncertanty wth nput data and we have to use varous technques to handle the uncertanty. The proposed model of ths paper consders dfferent nput parameters such as demand, capacty and cost n trapezod fuzzy forms and usng two rankng methods, we handle the uncertanty. The results of the proposed model of ths paper have been compared wth the crsp and other exstng fuzzy technques usng some randomly generated data. The prelmnary results ndcate that the proposed models of ths paper provdes better values for the obectve functon and do not ncrease the complexty of the resulted problem Growng Scence Ltd. All rghts reserved 1. Introducton Uncertanty s one of common ssues among many ndustral engneerng problems and there have been tremendous efforts to address uncertanty usng dfferent mathematcal models such as fuzzy theory, robust optmzaton, etc. There are also another ssues assocated wth ntegrty of a supply chan and there have been substantal efforts to ntroduce ntegrated supply chan problems, whch ncludes all components of supplers, manufacturers, dstrbutors and retalers. The prmary obectve of an ntegrated supply chan (SC) s to optmze all cost components from convertng raw materals nto fnal products delvered to end users (Davs, 1993; McDonald & Karm, 1997; Smch-Lev et al., 2000). Sabr and Beamon (2000) developed a comprehensve mult-obectve SC to mplement n smultaneous strategc and operatonal SC plannng. The adapted mult-obectve decson analyss allows us to use a performance measurement system, whch ncludes cost, customer servce levels (fll rates), and flexblty. * Correspondng author. Tel.: E-mal: pourrousta@yahoo.com (A. Pouroosta) 2012 Growng Scence Ltd. All rghts reserved. do: /.ec

2 404 Jayaraman and Ross (2003) nvestgated a system of dstrbuton network desgn problems characterzed by multple product famles, a central manufacturng plant ste, multple dstrbuton center and cross-dockng stes, and retal outlets (customer zones) whch demand multple unts of varous commodtes. Syarf et al. (2002) presented a logstc chan network problem, whch s a 0 1 mxed nteger lnear programmng model. The desgn tasks of ths problem presented the choce of the facltes to be opened and the dstrbuton network desgn to satsfy the demand wth mnmum cost. They used the spannng tree-based genetc algorthm by mplementng Prüfer number representaton. The effcency of the proposed method was examned by comparng ts numercal experment results wth those of conventonal matrx-based genetc algorthm. Zhou et al. (2002) mplemented a genetc algorthm for a balanced allocaton of customers to multple dstrbuton centers n the supply chan network. Petrovc et al. (1999) presented some soluton procedures for solvng supply chan problem usng the concept of fuzzy programmng. Lang (2008) presented a fuzzy mult-obectve producton/dstrbuton plannng decsons wth multproduct and mult-tme perod n a supply chan. Lang extended a fuzzy mult-obectve lnear programmng (FMOLP) system wth pecewse lnear membershp functon to handle ntegrated mult-product and mult-tme perod producton/dstrbuton plannng decsons (PDPD) problems where the obectves are formulated n fuzzy form. The work extends the orgnal mult-obectve lnear programmng to mnmze total costs and total delvery tme assocated wth nventory levels, avalable machne capacty and labor levels at each source, and predcts demand and avalable warehouse space at each destnaton and total budget. The proposed FMOLP model presents a systematc framework, whch facltates fuzzy decson-makng process to adust the search drecton durng the soluton procedure to obtan a DM s effcent soluton. In addton, the DM calculates the value n each cost category by studyng the tme value of money n the proposed model. Dubos et al. (2003) studed dfferent fuzzy set-based approaches for schedulng whch ncludes representng preference profles and modelng uncertanty dstrbutons. Chen and Chang (2006) presented a mathematcal programmng method for supply chan models wth fuzzy parameters. Lang (2008) presented an ntegrated producton-transportaton plannng decson wth fuzzy multple obectves n supply chans. Alev et al. (2007) used fuzzy-genetc approach to aggregate productondstrbuton plannng n supply chan management. Pedro et al. (2009) developed a fuzzy mathematcal programmng model for supply chan plannng, whch studes supply, demand and process uncertantes. The model was formulated as a fuzzy mxed-nteger lnear programmng model, where data were ll-known and modeled by trangular fuzzy numbers. Pedro et al. (2010), n another work, studed a fuzzy lnear programmng based method for tactcal supply chan plannng n an uncertanty envronment. Torab and Hassn (2008) presented an nteractve possblstc programmng method for multple obectve supply chan master plannng and ther computatonal results ndcated that the proposed fuzzy method relatvely performed better than other fuzzy technques. Blgen (2010) developed an ntegraton of producton and dstrbuton system nto a unfed model and addressed the producton and dstrbuton plannng problem n a supply chan system, whch ncludes the allocaton of producton volumes among varous producton lnes n the manufacturng plants, and the delvery of the goods to the dstrbuton centers. The proposed model was transformed nto fuzzy models, whch consders the fuzzness n the capacty constrants, and the aspraton level of costs based on varous aggregaton operators. Mula et al. (2010) examned the effectveness of a fuzzy mathematcal programmng system for supply chan producton plannng wth fuzzy demand. The work ncorporated a method of possblstc programmng, whch makes t possble to model the epstemc uncertanty n demand, whch could exst n the supply chan producton plannng problems as trangular fuzzy numbers.

3 A. Pourroosta et al./ Internatonal Journal of Industral Engneerng Computatons 3 (2012) 405 Dehbar et al. (2012) presented a supply chan problem where a whole seller/producer dstrbutes goods among varous retalers. The model was formulated as a more general and realstc form of tradtonal vehcle routng problem (VRP). The problem was solved usng a hybrd of partcle swarm optmzaton and smulated annealng (PSO-SA) and the results were compared wth other hybrd method, whch was a hybrd of Ant colony and Tabu search. They mplemented some well-known benchmark problems to compare the results of the proposed model wth other method. Lang and Cheng (2009) appled fuzzy sets for an ntegrated manufacturng/dstrbuton plannng decson (MDPD) problems wth mult-product and mult-tme perod n supply chans. The proposed model consdered tme value of money for each of the operatng cost categores and usng fuzzy mult-obectve lnear programmng model (FMOLP) mnmzes total costs and total delvery tme wth reference to nventory levels, avalable machne capacty and other ssues, smultaneously. They used an ndustral case to demonstrate the feasblty of the proposed model for a realstc MDPD problem. In ths paper, we present an ntegrated supply chan by consderng dfferent parameters wth uncertanty usng trapezod numbers. The proposed model of ths paper s solved usng two dfferent fuzzy programmng technques. The organzaton of ths paper frst presents the necessary notatons and problem formulatons n secton 2 and some numercal soluton s gven n secton 3, fnally, the paper concludes the results and suggests some future works. 2. Problem statement 2.1. Problem defnton The proposed model of ths paper conssts of four dfferent stages. In the frst stage, the SC consders supplers provdng raw materal and work n process for dfferent factores. The second stage consders factores, whch produce the fnal product. In the thrd stage, the network ncludes dstrbuton centers, whch are responsble for shppng fnal products to dfferent locatons. Fnally, the last stage ncludes sales zones. The followng assumptons hold for the proposed model of ths paper, The supply chan ncludes supplers, factores, dstrbuters and sales centers, There are four cost tems ncludng purchasng, producton, transportaton, setup and holdng, The nput data are nventory capacty, producton capacty, consumpton rate, demand and supply, The outputs are approprate program for purchasng, producton of each factory n each perod, optmal nventory level n factores and dstrbuton centers, the amount of raw materal shpped from suppler to factory and from factory to dstrbuton center and from dstrbuton center to end customer va sales' centers. We consder a medum term plannng and all parameters are n trapezod fuzzy numbers. Table 1 shows the necessary parameters and decson varables, Table 1 Necessary notatons and decson varables Set of ndces S set of supplers ( 1,2,, ) P set of plants ( 1,2,, ) W set of dstrbuton centers (DC) ( 1,2,, ) Z set of customer zones (CZ) ( 1,2,, ) T set of tme perods ( 1,2,, ) R set of raw materals ( 1,2,, ) G set of fnshed products ( 1,2,, )

4 406 Parameters rst SC PC F HR rpt HG HW gwt TR rspt TG gpwt TW gwzt gzt D cap β rst gwt V α rg Decson varables: k q rst x rspt RI rpt y GI WI gwt m gpwt n gwzt 1 = 0 fuzzy purchasng cost of raw materal r from suppler s at perod t fuzzy varable producton cost of fnshed product g n plant p at perod t fxed producton cost of fnshed product g n plant p at perod t holdng cost of raw materal r n plant p at perod t holdng cost of fnshed product g n plant p at perod t holdng cost of fnshed product g n dstrbuton center w at perod t transportaton cost of raw materal r from suppler s to plant p at perod t transportaton cost of fnshed product g from plant p to DC w at perod t transportaton cost of fnshed product g from DC w to CZ z at perod t fuzzy demand of fnshed product g at CZ z at perod t fuzzy producton capacty of plant p for fnshed product g at perod t maxmum supply raw materal r by suppler s at perod t fuzzy maxmum holdng capacty for fnshed product g n DC w at perod t quantty of raw materal r consumed n fnshed product g quantty of raw materal r suppled from suppler s at perod t quantty of raw materal r shpped from suppler s to plant p at perod t nventory level of raw materal r n plant p at perod t quantty of fnshed product g produced n plant p at perod t nventory level of fnshed product g n plant p at perod t nventory level of fnshed product g n DC w at perod t quantty of fnshed product g shpped form plant p to DC w at perod t quantty of fnshed product g shpped form DC w to CZ z at perod t f fnshed product g produced n plant p at perod t Otherwse 2.2 Problems formulaton The frst obectve functon of the proposed model gven n Eq. (1) mnmzes total cost of purchasng tems, setup of each product n each factory, producton, nventory cost tems ncludng the cost of raw materal, fnal product n factory and dstrbuton centers. The obectve functon of the proposed model also mnmzes transportaton cost of raw materal, fnal product n factory and dstrbuton centers. ( ) Z = SC q + F z + PC y + HG GI + TR mn rst rst rspt xrspt r s t g p t r s p t + TG m + TW n + HR RI + gpwt gpwt gwzt gwzt rpt rpt gwt gwt g p w t g w z t r p t g w t HW WI (1) subect to q rst x p rspt RI RI x y rpt = rp, t 1 + rspt αrg. s g GI = GI + y m gp, t 1 gpwt w rst,, r, p, t g, pt, (2) (3) (4)

5 WI = WI + m n gwt gw, t 1 gpwt gwzt p z D gzt n w gwzt y cap k q p w rst m gpwt β m gpwt rst V gwt M. k A. Pourroosta et al./ Internatonal Journal of Industral Engneerng Computatons 3 (2012) 407 q, x, RI, y, GI, WI, m, n, B 0, k {0,1} rst rspt rpt gwt gpwt gwzt gwt g, w, t g, z, t g, pt, g, w, t rst,, g, pt, rs,, ptgw,,,, z Constrants (2) ensures that the amount of suppled raw materal s, at least, equal to the amount of raw materal shpped to all factores. Eq. (3) shows that the amount of raw materal n each perod s equal to the amount of nventory n the prevous perod and the amount of raw materal shpped to factory n ths perod mnus the consumpton n ths perod. Eq. (4) and Eq. (5) do smlarly for producton and dstrbuton centers. Eq. (6) determnes the maxmum demand for each dstrbuton center. Eq. (7) and Eq. (8) show the maxmum producton capacty of each producton and dstrbuton centers, respectvely. Eq. (9) determnes the maxmum supply and Eq. (10) ensures that when product s about to be delvered, the producton must be setup and accomplshed. Fnally, Eq. (11) ensures the non-negatvty of varables. (5) (6) (7) (8) (9) (10) (11) 2.3. Fuzzy model one (Jmenez's model) In ths secton, we present a fuzzy approach to handle the uncertanty and mplement the rankng method ntroduced by Jmenez et al. (2007) to defuzzfy the fuzzy numbers. Consder a trapezod fuzzy number A = { a1, a2, a3, a4} as follows, 0 ; x (, a1 ] ( 12) f A ( x ) ; x [ a1, a2] μ ( x ) = 1 ; [ 2, 3] A x a a g A ( x ) ; x [ a3, a4] 0 ; x [ a4, ) we assume that f A ( x ) s a contnuous and non- In order to make sure that 1 1 f A ( x ) and g A ( x ) decreasng functon and g A ( x ) s a contnuous and non-ncreasng functon. Therefore, we have a2 a4 ( A A EI A) = E1, E 2 = xdf A( x), xdg A( x). a1 a 3 Usng some smplfcaton yelds the followng, ( 13) 1 1 ( A A 1 1 EI A) = E1, E 2 = f A ( α ) dα, g A ( α) dα. 0 0 (14)

6 408 When ( ) A f x and ( ) g x are lnear we can smply the equatons as follows, A ( 1 1 EI A) = ( a1+ a2), ( a3 + a4) 2 2 In addton, to compare two fuzzy numbers we use the followng, (15) a b 0 f E2 E1 < 0 A B ( E2 E1 a b a b μm AB, ) = = f 0 E1 E2, E A B A B 2 E1 E2 E2 ( E1 E, (16) 2 ) a b 1 f E1 E2 > 0 where, and, are expected values of and, respectvely. For more detals the nterested readers are suggested to read Jmenez et al. (2007). The mathematcal model gven n Eq. (1) to Eq. (11) n fuzzy form s as follows, mn Z = ( SC + SC + SC + SC ) q 4 r s t rst rst rst rst rst 1 ( ) + F z + PC + PC PC PC y + HG GI g p t 4 + TR x + TG m + TW n rspt rspt gpwt gpwt gwzt gwzt r s p t g p w t g w z t + HR RI + HW WI rpt rpt gwt gwt r p t g w t (17) subect to Dgzt + Dgzt Dgzt + Dgzt + α n 2 2 ( 1 α) w gwzt g, z, t (18) y cap + cap cap + cap ( 1 α) + α gwt gwt gwt gwt k g, pt, (19) p V + V V + V m gpwt ( 1 α) + α gwt gwt gwt gwt g, w, t (20) The other equatons are the same as the crsp model Fuzzy model two (Cadenas and Verdegay model) The second fuzzy mathematcal model used n ths paper uses the method developed by Cadenas and Verdegay (1997). To understand the detals consder the followng fuzzy lnear programmng model,

7 A. Pourroosta et al./ Internatonal Journal of Industral Engneerng Computatons 3 (2012) 409 max Z n = c x = 1 n a x f b = 1 x 0, M, N where the cost s defned as follows, R such thatμ : R [0,1] N. μ F( ) The rght hand sdes are also defned as follows, μ F( R) such that μ : R [0,1] M (21) (22) (23) In addton, for each constrant, we have, μ F(F( R) such that M μ :F( R ) [0,1] (24) One general approach to solve the fuzzy lnear programmng gven by Eq. (25) s to use convex fuzzy sets as follows, t ; a x g b ψ a x, b = t a x b ; b g a x g b t 0 ; a x g b t (25) and the fuzzy lnear programmng model s summarzed n the followng form, n max c = 1 subect to x n a x g b + t ( 1 α ) M = 1 x 0, α [0,1], N For more detals, please see Cadenas and Verdegay (1997). (26) 3. Numercal soluton In ths secton we present some results for the mplementaton of the proposed fuzzy models and compare our results wth crsp model. The fuzzy parameters of the proposed models are unformly dstrbuted and they are gven n Table 2. Table 2 Input parameters Parameter Unform(a,b) (12,18) (18,25) (18,20) (5,15) (5,10) (6,12) (750,1500) (1,3)

8 410 In ths model, demand s gven n fuzzy form as 60,80,100,120, the capacty for each product n each perod s 670, the producton capacty of each factory s 340,360,400,420 and fnally the capacty of each dstrbuton center s 390,400,490,520. We consder quarterly or sem-annual plannng horzon. The resulted problem formulatons for two fuzzy models as well as crsp one have been solved usng Lngo software and the results are summarzed n Table 3. Table 3 The results of the mplementaton of the propose fuzzy models Jmenez Method Problem S P W Z Crsp Cadenas α=0.2 α=0.7 α= As we can observe from the results of Table 3, both fuzzy models provde better obectve values compared wth the crsp model and Fg. 1 demonstrates the results of the mplementaton of the frst fuzzy model versus crsp model. obectve functon menez (α=0.2) menez (α=0.7) menez (α=1) crsp Test problem Fg. 1. The performance of cadenzas and crsp model As we can observe from the results of Fg. 1, when α = 0.2 the proposed fuzzy model outperforms the crsp model and for other cases, there are not much dfferences between two methods. In addton, Fg. 2 shows the relatve performance of the second fuzzy model versus crsp model.

9 A. Pourroosta et al./ Internatonal Journal of Industral Engneerng Computatons 3 (2012) 411 obectve functon cadenas crsp Test problem Fg. 2. The performance of the crsp model versus the second fuzzy model It s clear from the results that the proposed model mostly beats the crsp model. 4. Concluson In ths paper, we have nvestgated a supply chan problem where a whole seller/producer dstrbutes goods among dfferent retalers. The resulted problem were often faced wth uncertanty and the nput data are perturbed wth some noses. Therefore, we need to mplement varous technques to handle the uncertanty. The proposed model of ths paper nvestgated dfferent nput parameters such as demand, capacty and cost n trapezod fuzzy forms and usng two rankng methods, we handled the uncertanty. The results of the proposed model of ths paper have been compared wth the crsp and other exstng fuzzy technques usng some randomly generated data and our comparson ndcated that the proposed model of ths paper could provde somewhat lower values for the obectve functon and do not ncrease the complexty of the resulted problem. Acknowledgment The authors would lke to thank the anonymous referees for the comments on earler verson of ths work, whch helped us mprove the qualty of the paper. References Alev, R.A., Fazlollah, B., Gurmov, B.G., & Alev, R.R.(2007). Fuzzy-genetc approach to aggregate producton-dstrbuton plannng n supply chan management. Informaton Scences, 177, Blgen, B. (2010). Applcaton of fuzzy mathematcal programmng approach to the producton allocaton and dstrbuton supply chan network problem. Expert system wth applcatons, 37, Cadenas, J.M., & Verdegay J.L. (1997). Usng fuzzy numbers n lnear programmng. IEEE Transactons on Systems. Man and Cybernetcs Part B-Cybernetcs, 27, Chen, S.P., & Chang, P.C.(2006). A mathematcal programmng approach to supply chan models wth fuzzy parameters. Engneerng Optmzaton, 38, Davs, T. (1993). Effectve supply chan management. Sloan Management Revew, 34, Dehbar, S., Pourrousta, A., Ebrahm Neghad, S., Tavakkol-Moghaddam, R., & Javanshr, H.(2012). A new supply chan management method wth one-way tme wndow: A hybrd PSO-SA approach. Internatonal Journal of Industral Engneerng Computatons,3(2),

10 412 Dubos, D., Farger, H., & Fortemps, P. (2003). Fuzzy schedulng: modellng flexble constrants vs. copng wth ncomplete knowledge. European Journal of Operatonal Research, 147, Jayaraman, V., & Ross, A. (2003). A smulated annealng methodology to dstrbuton network desgn and management. European Journal of Operatonal Research, 144, Jmenez, M., Arenas, M., Blbao, A., & Guez, M.V. (2007). Lnear programmng wth fuzzy parameters: an nteractve method resoluton. European Journal of Operatonal Research, 177, Lang, T.F. (2008). Fuzzy mult-obectve producton/dstrbuton plannng decsons wth multproduct and mult-tme perod n a supply chan. Computers & Industral Engneerng, 55(3), Lang, T.F.(2008). Integratng producton-transportaton plannng decson wth fuzzy multple goals n supply chans. Internatonal Journal of Producton Research, 46, Lang, T.F., Cheng, & H.W. (2009). Applcaton of fuzzy sets to manufacturng/dstrbuton plannng decsons wth mult-product and mult-tme perod n supply chans. Expert Systems wth Applcatons, 36, McDonald, C.M., & Karm, I.A.(1997). Plannng and schedulng of parallel sem-contnuous processes. Industral & Engneerng Chemcal Research, 36, Mula, J., Pedro, D., & Poler, R. (2010). The effectveness of a fuzzy mathematcal programmng approach for supply chan producton plannng wth fuzzy demand. Internatonal Journal of Producton Economcs, 128, Pedro, D., Mula, J., Poler, R., & Verdegay, J.L. (2009). Fuzzy optmzaton for supply chan plannng under supply, demand, and process uncertantes. Fuzzy Sets and Systems, 160, Pedro, D., Mula, J., Jmenez, M., & Botela, M.D.M. (2010). A fuzzy lnear programmng based approach for tactcal supply chan plannng n an uncertanty envronment. European Journal of Operatonal Research, 205, Petrovc, D. Roy, R., & Petrovc, R.(1999). Supply chan modellng usng fuzzy sets. Internatonal Journal of Producton Economcs, 59, Smch-Lev, D., Kamnsky, P., & Smch-Lev, E. (2000). Desgnng and Managng the Supply Chan: Concepts, Strateges, and Case Studes. McGraw-Hll, New York. Sabr, E.H., & Beamon, B.N. (2000). A mult-obectve approach to smultaneous strategc and operatonal plannng n supply chan desgn. Omega, 28, Syarf, N., Yun, Y., & Gen, M. (2002). Study on mult-stage logstc chan network: a spannng tree based genetc algorthm approach. Computers & Industral Engneerng, 43, Torab, S.A., & Hassn, E. (2008). An nteractve possblstc programmng approach for multple obectve supply chan master plannng. Fuzzy Sets and Systems, 159, Zhou, G., Mn, H., & Gen, M. (2002). The balanced allocaton of customers to multple dstrbuton centers n the supply chan network: a genetc algorthm approach. Computers & Industral Engneerng, 43,