A Novel Fuzzy Multi-Objective Method for Supplier Selection and Order Allocation Problem Using NSGA II

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1 A Novel Fuzzy Mult-Objectve Method for Suppler Selecton and Order Allocaton Problem Usng NSGA II Mohammad Al Sobhanolah a, Ahmad Mahmoodzadeh *, Bahman Nader b Department of Industral Engneerng, Faculty of Engneerng, Kharazm Unversty, Tehran, Iran. * Correspondng author. Tel.: ; ahmad.mahmoodzade1@gmal.com a sobhanallah@khu.ac.r b bahman.nader@aut.ac.r Ths paper ntroduces a suppler selecton and order allocaton problem n a sngle-buyer-multsuppler supply chan n whch approprate supplers are selected and orders allocated to them. Transportaton costs, quantty dscount, fuzzy type uncertanty and some practcal constrants are taken nto account n the problem. The problem s formulated as a b-objectve model to mnmze annual supply chan costs and to maxmze the annual purchasng value. The fuzzy weghts of supplers, whch are the output of one of the suppler evaluaton methods, are consdered n the second objectve functon. Then, we propose a novel fuzzy mult-objectve programmng method for obtanng Pareto solutons. The method s the extenson of a sngle-objectve method exst n the lterature. Ths method s based on the decson maker's degree of satsfacton from each fuzzy objectves consderng the fulfllment level of fuzzy constrants. In the proposed method, the problem remans mult-objectve and, unlke exstng methods, does not transformed nto a sngle-objectve model. At the last stage of proposed method, the fuzzy results are compared wth an ndex, and decson maker can dentfy the approprate or napproprate solutons. To solve the problem, non-domnated sortng genetc algorthm (NSGA II) s desgned and computatonal results are presented usng numercal examples. Keywords: Suppler selecton; Order allocaton; Fuzzy theory; Mult objectve programng; NSGA II. 1. Introducton In the current compettve envronment, suppler evaluaton and selecton s one of the most mportant processes n supply chan management of any organzaton. It s crtcal snce supplers have a major mpact on strategc and operatonal performance of organzatons. Also, ths process plays an mportant role n determnng the cost, the qualty, and other aspects of the fnshed product [1]. Hence, organzatons rely more on supplers to reduce ther costs, to mprove the qualty of ther products, or to focus on a specfc part of ther operatons []. Suppler selecton s complex snce organzatons must take nto account multple aspects ncludng both quanttatve and qualtatve crtera [3, 4]. In such cases, crtera are conflctng that requre trade-offs among them. Therefore, selecton of best supplers become a mult-crtera decson-makng problem. Furthermore, the process becomes more complcated f parameters are ncomplete or uncertan. On the other hand, hgher levels of nventory leads to ncreased supply chan responsveness, but decrease the cost effcency because of wth nventory holdng costs [5]. Snce, nventory costs can 1

2 represent anywhere between 0 and 40% of the total product value, Hence, nventory management s one of the sgnfcant parts of supply chan management [6]. Allocatng orders to the selected supplers, allows for some economes of scale through the rght choce of the quanttes to order from each suppler. Sometmes, supplers offer quantty dscounts as a powerful ncentve to motvate buyers to ncrease the amount of ther ordered quanttes [7]. Indeed, the unt prce pad by buyers for large orders s usually smaller than the unt prce of small orders [8]. Incremental quantty dscounts, busness volume quantty dscounts, and all-unt quantty dscounts are the three man types of quantty dscounts [9]. Ths paper ntroduces a suppler selecton and order allocaton problem n a sngle-buyer-multsuppler supply chan n whch approprate supplers are selected and orders allocated to them. Transportaton costs, quantty dscount, fuzzy type uncertanty and some practcal constrants are taken nto account n the problem. Frstly, the problem s formulated as a b-objectve model as mnmzng annual supply chan costs and maxmzng the annual purchasng value. The fuzzy weghts of supplers, whch are the output of one of the suppler evaluaton methods, are consdered n the second objectve functon. The assumptons of the model fts to the real word condton, so that the model can be appled n any type of supply chan where a buyer supples the demanded tems from some potental supplers. To overcome the complexty of the process, we propose a novel fuzzy mult-objectve programmng method for obtanng Pareto solutons. The method s the extenson of a sngle-objectve method proposed by Jménez et al. [10]. Ths method s based on the level of decson maker's satsfacton from each fuzzy objectves consderng the degree of realzaton of fuzzy constrants. In the proposed method, the problem remans mult-objectve and, unlke exstng methods, does not transformed nto a sngle-objectve model. At the last stage of proposed method, the fuzzy results are compared wth an ndex, and the decson maker can dentfy the approprate or napproprate solutons. To solve the problem, Non Domnated Sortng Genetc Algorthm (NSGA II) s desgned and computatonal results are presented usng numercal examples. The rest of ths paper proceeds as follows: In Secton, we brefly revew the related lterature. In Secton 3, we descrbe the problem, state the assumptons, and gve the parameters, varables and the formulaton of the model. In Secton 4, we explan our novel fuzzy mult-objectve programmng method. In Secton 5, we descrbe the soluton procedure ncludng changes on objectves and constrants, soluton encodng, repar algorthm and NSGA II procedure. In Secton 6, we llustrate the fndngs of the mplementaton of proposed methodology to some numercal examples. Fnally, Secton 7 concludes the study and presents future remarks.. Lterature revew.1 suppler selecton and order allocaton Here, we revew the researches whch have been conducted n the area of suppler selecton and order allocaton wth quantty dscount and consder ther objectves and soluton methods. Dahel [11] assume mult-tem volume dscounts and propose a mult-objectve mxed nteger programmng model. Ether a preference-orented approach or the generatng approach s utlzed to solve the model. Demrtas & Üstün [1] study a model to maxmze the purchasng value and to mnmze the budget and the defect rates. They use analytcal network process (ANP) and the mult-objectve mxed-nteger programmng and utlze epslon-constrant method and reservaton level by Tchebycheff procedure to solve the problem. Xa & Wu [13] Study the problem wth maxmzng the total weghted quantty of purchasng, mnmzng the total purchasng cost, mnmzng the number of defectve tems, and maxmzng the number of on-tme delvered tems. They propose a two-stage method to solve the problem. They use AHP mproved by rough set theory n the frst stage, and, a mult-objectve mxed nteger lnear

3 programmng n the second stage. Burke et al. [14] consder three type of dscounts as the lnear quantty dscount, the ncremental unt prce, and the all-unt quantty dscount. They develop a heurstc to measure the effect of quantty dscounts n the problem. Kokangul & Susuz [15] nvestgate a b-objectve model wth mnmzng the total purchasng cost and maxmzng the purchasng value obtaned usng AHP. They propose a b-objectve non-lnear programng model usng goal programmng. Amd et al. [16] study a model whch mnmzes the total cost, the percentage of late delvery tems, and the percentage of rejected tems. They developed a fuzzy mult-objectve mxed nteger lnear programmng model to solve the problem. Wang & Yang [17] develop a model to mnmze total cost, defectve rate, and delvery lateness rate. They propose a two-stage procedure usng AHP and a mult-objectve mathematcal programmng. Ebrahm et al. [18] study a model to mnmze the cost, late delvered tems, and defectve tems. They consder three type of dscounts and propose a scatter search algorthm and exact method to solve the problem. Razm & Maghool [19] consder three type of dscounts n a fuzzy bobjectve model to mnmze the total purchasng cost, and to maxmze the total purchasng value. An augmented epslon-constrant and reservaton level by Tchebycheff models s adopted to solve the problem. Kamal et al. [0] nvestgate a model to mnmze the total annual cost, the total number of defectve tems, and the total number of late delvered tems and to maxmze the total purchasng value. They propose partcle swarm optmzaton and scatter search algorthm to solve the model. Zhang & Zhang [1] study a sngle objectve model to mnmze costs wth stochastc demand. They solve a mxednteger programmng (MIP) n whch all supplers met the qualtatve crtera level. Lee et al. [] study a sngle objectve model to mnmze the total purchasng cost under all-unt and ncremental quantty dscounts and desgnate genetc algorthm to solve the model. Pazhan et al. [5] study a sngle objectve model to mnmze total cost per unt tme consderng the purchasng, setup, holdng, and transportaton costs. A mxed nteger nonlnear programmng model s proposed to solve the problem usng exact methods. Moghadam [3] consder optmzng four objectves as maxmzng the total net proft, mnmzng the total defectve parts, the total late delveres and the total rsk factors of economc envronment assocated wth each suppler. Bohner & Mnner [4] consder both all-unts and ncremental quantty dscounts and assume falure rsk to mnmze total costs. They solve ther mxed-nteger lnear programmng model usng exact approach. Çeb & Otay [5] nvestgate a model to mnmze the total cost, total late delveres, and total defectve tems and to maxmze total utlty of the purchasng actvty. Hamdan & Cheatou [1] consder the problem wth green crtera to maxmze the total value of purchasng and to mnmze total costs. They propose a three stages method usng fuzzy TOPSIS n the frst stage to assgn two preference weghts to every potental suppler, AHP n the second stage to assgn an mportance weght, and b-objectve nteger lnear programmng n the thrd stage the model s solved usng the weghted comprehensve crteron method and the branch-and-cut algorthm. Hamdan & Cheatou [6] consder the problem wth green crtera, quantty dscounts and varyng suppler avalablty. They apply the smlar three stage method proposed by Hamdan & Cheatou [1] n order to maxmze total green and tradtonal value of purchasng and mnmze the total purchasng cost. Ranjbar Tezenj et al. [7] consder the suppler locaton-selecton & order allocaton under capacty constrants n a stochastc envronment. The objectve functon ncludes establshment, nventory, and transportaton costs. They develop a b-objectve model for optmzaton of the mean and varance of costs. The model s solved wth genetc algorthm and smulated annealng. In ths paper, we assume two common objectves as mnmzng total costs and maxmzng purchasng value.. Fuzzy Mult-Objectve Approaches The lterature presents some methods for aggregatng fuzzy goals and constrants. Jménez & Blbao [8] propose a method for mult-objectve programmng problem wth fuzzy objectves as 3

4 maxmzng λ f (x) subject to μ Zf (x) λ, where λs the degree of satsfacton of decson maker wth regard to the achevement of the goals. A unque optmal soluton of the above problem s a fuzzy effcent soluton of the orgnal MOP problem [9]. There are some approaches such as the weghted addtve approach [30, 31], compromse approach [3], method wth achevement degrees [33], augmented max-mn model [34, 35], and two-phase approach [36] n the lterature, whch ensure exstence of a fuzzy effcent soluton. Moghaddam [3] appled Fuzzy goal programmng approach for suppler selecton and order allocaton problem n reverse logstcs systems. He utlzed a Monte Carlo smulaton ntegrated wth fuzzy goal programmng to determne the entre set of Pareto optmal solutons. By ncorporatng the lnear membershp functons for objectves and constrants wth other constrants, he formulated fuzzy goal programmng model. Ergnel & Gecer [37] study Fuzzy Mult- Objectve lnear programmng model n Suppler Selecton Problem. Weght, cost, and calbraton tme are handled as fuzzy numbers for modellng the mprecse data. A two-phase approach s used to solve the fuzzy mult-objectve decson model. They appled a two-phase approach method to obtan a Pareto optmal soluton. Çeb & Otay [5] studed suppler selecton and order allocaton problem consderng quantty dscounts and lead tme. They appled a two-stage fuzzy approach to solve the problem. They used augmented max-mn model guaranteeng non-domnated solutons, to transform the fuzzy multobjectve model nto a crsp sngle objectve functon model. To select supplers, they utlze fuzzy MULTIMOORA n the frst stage, then, fuzzy goal programmng was used to determne order amounts n the second stage. Govndan et al. [38] study Suppler Selecton Problem wth Transportaton Decsons n an Eco-effcent Closed Loop Supply Chan Network. They utlze weghted fuzzy mathematcal programmng approach for generatng a fuzzy, properly effcent soluton. Frstly, they defne lnear membershp functons for each fuzzy goal as ntroduced by Zadeh [39]. Then, the appled a weghted max-mn approach seeks for an optmal soluton so that the rato of the levels of achevement of the goals and the rato of the weghts of the goals come as close to each other as possble. As revewed n above, all fuzzy approaches transform the model nto sngle objectve model, however, n ths paper, we propose a method to solve the problem n ts orgnal mult objectve form. 3. Problem formulaton We assume a supply chan nclude a buyer and several supplers. The buyer ntends to evaluate supplers and allocate orders to them. Fgure 1 shows the supply chan. So, the buyer faces a suppler selecton and order allocaton problem. The man objectves of the buyer are to mnmze the annual supply chan costs and to maxmze the annual purchasng value so that he does not face any shortages. The man assumptons of the problem are as followng. Only a sngle product s consdered that the buyer can purchase ts requred amount from several supplers. The annual demand s known and s constant over tme. Each suppler has a specfc producton capacty. Inventory shortage s not allowed for any suppler or buyer. The nventory cannot be transferred from perod to perod. The transportaton cost from each suppler to the buyer depends on the dstance and the number of requred vehcles. The defectve rate of each suppler s known. 4

5 In each perod, the (+1)-th suppler s order cannot be entered unless all -th suppler order are consumed. All supplers use the prce dscount polcy to encourage the buyer to make large orders. Suppose that there are n supplers n the supply chan. Each suppler uses the quantty dscount polcy, whch ncludes K prce level, and k-th prce level s determned by prce c k and order range [u,k 1, u k ). The logcal assumpton s that for u 1 < u < < u,k, we have c 1 > c > > c,k. Please nsert Fgure 1 here The model studed n ths research has some smlartes to the one studed by Kamal et al. [0] and Alae and Khoshalhan [40]. Both papers consder the problem as mult objectve programmng model ncludng maxmzng annual purchasng value, and mnmzng annual supply chan costs, defectve tems, and, late delveres. To solve the model, Kamal et al. [0] utlze the partcle swarm optmzaton (PSO) and the scatter search (SS); and, Alae and Khoshalhan [40], n addton to PSO and SS, use the harmony search (HS) and the hybrd harmony-cultural algorthm (HS-CA). Both papers consder the problem n determnstc envronment. However, n ths study, the problem s consdered as a b-objectve optmzaton model consderng fuzzy parameters n objectves and constrants. Then a novel method s proposed for obtanng Pareto solutons for the fuzzy b-objectve problem. The method s the extenson of a sngle-objectve method proposed by Jmenez et al. [10]. We also utlze non-domnated sortng genetc algorthm (NSGA II) to solve the problem. 3.1 Notaton Parameters We defne the followng parameters and decson varables whch used n the model. D c k S A h h T b T c Buyer s annual demand rate Suppler 's fxed producton setup cost Fxed orderng cost for suppler Inventory holdng cost of suppler per unt per unt tme Inventory holdng cost of buyer per unt per unt tme Consumpton tme of an order quantty from suppler Buyer s length of perod Suppler 's varable cost per unt (Fuzzy) Dscounted unt prce of nterval k offered by suppler P Suppler 's producton rate u k Upper bound of Suppler 's dscount nterval r R ds Suppler 's defectve rate (Fuzzy) Maxmum acceptable defectve rate for all purchased quanttes Dstance between buyer and suppler 5

6 w cap C Vehcles capacty Fxed cost of transportaton per dstance unt (Fuzzy) Suppler 's mportance rate n suppler evaluaton methods (Fuzzy) Decson varables q k y k Q Q Z Purchased quantty per perod from suppler n dscount nterval k Bnary varable; equals to 1 f q k falls nto suppler 's dscount nterval k; otherwse s 0. Order quantty from suppler per perod Total order quantty per perod from all supplers Integer varable denotng number of requred vehcles for transportng Q 3. Model The mathematcal modelng of the problem s shown as a mxed nteger nonlnear programmng model as follows: n K n K n n D Q h h Mn Cost c c q A S y C ds Z Q D P 1 k 1 1 k b k k k (1) n D Max APV w Q () K k k 1 Q 1 Q q 1.. n n Q Q 1 D Q P 1.. n Q (5) cap Z 1 Q 1.. n (6) Q cap Z 1.. n (7) K k 1 yk n (8) u, k 1 yk qk 1.. n; k 1.. K (9) q u y 1.. n; k 1.. K (10) k k k 1 n Qr Q 1 R (3) (4) (11) 6

7 y 0,1 k 1: K, 1.. n (1) k Z Integer 1.. n (13) qk 0 k 1: K, 1.. n (14) Q n (15) Q 0 (16) Equaton 1 mnmzes the cost objectve functon, whch conssts of four parts. The frst part nvolves varable and purchases costs; the second part ncludes orderng and setup costs; the thrd part s the nventory holdng cost of the buyer and supplers; and the fourth part calculates the transportaton costs. Equaton maxmzes the total annual purchasng value (APV), whch s the weghted quanttes of orders. These weghts are the supplers mportance and can be the output of mult-crtera decsonmakng methods [5]. Maxmzng ths relaton leads to allocate more orders to more mportant supplers. Equaton 3 states that the sum of purchases from a suppler s dscount ntervals s equal to the purchases amount from that suppler. Gven the Equaton 4, the total purchase of the buyer n each cycle s equal to the sum of purchases from all supplers. Supplers capacty constrants are expressed by Equaton 5. The number of requred vehcles to carry the products of each suppler s calculated usng Equaton 6 and 7. Equaton 8 ensures that the order quantty to each suppler only be at one of ts dscount ntervals. Usng the Equatons 9 and 10, the order quantty from each suppler, falls nto one of the dscount ntervals offered by ths suppler. Equaton 11 ensure that the rato of defectve products should not exceed the predetermned value. Also, relatons 1-16 defne the type of varables. 4. Fuzzy mult-objectve approach Varous methods have been presented n lterature for fuzzy mult-objectve optmzaton, however, we propose a novel method to the problem. The method s based on the decson-maker's satsfacton degree of fuzzy objectves and the level of fulflment of the constrants. Ths method s the extenson of sngle-objectve fuzzy programmng proposed by Jmenez et al. [10] to the mult-objectve envronment. The method ams to dentfy the Pareto solutons whch s best match wth the desres of the decson maker. Fuzzy objectve functons are replaced by the degree of satsfacton defned n the algorthm and fuzzy constrants are also rewrtten as a functon of α, the level of fulflment. Step 1. Accordng to Jemenez et al. [10], fuzzy constrants must be rewrtten as a functon of alpha. Wth these modfcatons, the problem space s beng wder for α = 0, and more lmted for α = 1. For any constrant type, any fuzzy coeffcents a a1, a, a3 and fuzzy rght-hand sde b b1, b, b3, must be replaced by the followng expressons, respectvely. a1 a a a3 a 1 b1 b b b3 b 1 (17) (18) Step. The decson maker (DM) s asked to specfy the nterval GG, for each objectve. For a mnmzaton objectve, f z G, he wll fnd t totally satsfactory, but f z G, hs degree of satsfacton wll be null. Then the goal s expressed by means of a fuzzy set G whose membershp functon s as follows: 7

8 G G 1 f z G z decreasng on G z G 0 f z G 0,1 Smlarly, we defne G for maxmzaton objectves as follows. 1 f z G z ncreasng on G z G 0 f z G 0,1 The DM ams to obtan a maxmum satsfacton degree. But n order to get a better objectve value, a lower level of fulflment of constrants s acheved. Gven these crcumstances the DM mght want a lower satsfacton degree of objectves n exchange for a better constrants fulflment level [10]. Step 3. For each objectve, we need to compute the satsfacton degree of the fuzzy goal G by 0 each α-acceptable Pareto soluton, that s to say the membershp degree of each fuzzy number z to the fuzzy set G. There are several methods to do ths (see.e. Dubos et al. [41]). We suggest usng an ndex proposed by Yager [4] and used by Jmenez et al. [10]. K G z 0 z 0 z G 0 z z dz z dz where the denomnator s the area under occurrence z 0 (see Fgure ). (19) (0) k (1) 0 z and, n the numerator, the possblty of z of each crsp value z s weghted by ts satsfacton degree G z Please nsert Fgure here of the goal G Step 4. Here, n order to have a balance between satsfacton degree and constrants fulflment level, we use the condton smlar to Jmenez et al. [10] as followng: : * 0 k k k G k k x argmax K z () The greater the value of α, the more lmtted the problem space and the lower satsfacton degree; however, the smaller the value of α, the wder the problem space and the hgher satsfacton degree. The product of α and satsfacton degree creates a certan balance between them. Step 5. The solutons for each fuzzy objectve functon can be compared, and, the DM can dentfy the approprate or napproprate solutons among Pareto solutons. Here, are two defntons as followng: Defnton 1: for any par of fuzzy numbers a and b, the degree n whch a s greater than b, calculated as followng [10]: a b 0 f E E1 0 a b a b a, b f E 1 E, E E 1 0 E E1 E E 1 a b 1 f E1 E 0 a b E E1 M a a b b (3) 8

9 Where a a E1, E and b b E1, E ndfferent f M ab, 0.5. are the expected ntervals of a and b. We say that a and b are Defnton : For a trangular fuzzy number a a, a, a 1 3, ts expected nterval s easly calculated as follows [43]: 3 a 1, a a 1 a, a EI a E E a (4) 4.1 The case of trapezodal fuzzy numbers The proposed method can also be appled for trapezodal fuzzy numbers. In the step 1, for any fuzzy coeffcents a a, a, a, a and fuzzy rght-hand sde b b, b, b, b, the relatons can be rewrtten as [10]: a1 a a3 a4 a 1 b1 b b3 b4 b 1 In the Defnton of step 5, For a fuzzy number a a, a, a, a 3 4 a 1, a a 1 a, a EI a E E a The other steps reman unchanged. 5. Soluton procedure , ts expected nterval wll be [43]: The defned problem n the prevous secton s nonlnear, and the reason s that the varable Q s the denomnator of Equatons 1,, 5 and 11. Of course, Equaton 5 can be converted to lnear form, DQ PQ, and Equaton 11 can also be rewrtten as lnear form, (5) (6) (7) n Qr RQ. However, relatons 1 and 1 are totally nonlnear. Consderng that nonlnear problems cannot be solved wth exact methods, the nondomnated sortng genetc algorthm (NSGA II) s desgned for solvng the problem. Frstly, the fuzzy objectve functons and the fuzzy constrants must be modfed n accordance wth the proposed method. 5.1 Modfcaton of fuzzy objectves and fuzzy constrants 1 3 Gven the frst step of the proposed method, assumng that r r, r, r be rewrtten as follows:, then Equaton 11 can n 1 3 r r r r Q 1 QR (8) 1 It s also assumed that the fuzzy values of the objectve functons n a soluton are denoted by Cost,, APV,, GG, of each objectve can be determned by and The nterval 9

10 decson maker. Here, we assume that the deal value of each objectve s determned by optmzng that functon and gnorng the other one; and, the non-deal value s determned by the decson maker. In optmzng each objectve, the problem s solved for the smallest possble amount of α whch ensures that the problem space s n the broadest state. Then, the degree of satsfacton of objectves can be calculated as follows. K K G G Cost APV 0 0 z 1 z G 3 z G dz dz G G G G 3 z z 1 3 dz dz 3 z z 1 z G 3 z G dz dz G G G G 3 z z 1 3 dz dz 3 z It should be noted that the objectve functons n Equaton 1 and are mnmzaton and maxmzaton, respectvely; so, the membershp functon of G s descendng for the frst objectve, and ascendng for the second one. The ntegrals of the above relatons can be easly calculated. 5. Soluton representaton Each chromosome or answer vector can be expressed as Q Q Q Q vector s equal to Q ; : 1,,, n (9) (30), n whch the sum of the Q represents the order quantty assgned to suppler and s equal to Q : q1, q,, q, K. The procedure for generatng ntal solutons s shown n Fgure 3. Please nsert Fgure 3 here 5.3 Repar algorthm Due to the exstence of constrants n the problem, t s possble to face the nfeasble solutons n the ntal soluton generaton algorthm and repetton of the man algorthm. These nfeasble solutons must be controlled usng the constrants handlng methods. In ths secton, we propose a repar algorthm whch transform nfeasble solutons to feasble ones. Equatons 3 and 4 are beng establshed by soluton representaton. By calculatng the number of vehcles, Equatons 6 and 7 are enforced. Equatons 8, 9 and 10 are also beng establshed accordng to the order quantty to each suppler must be fall nto one of the dscount ntervals. The only constrants that may lead to nfeasble solutons, are Equatons 5 and 5. In the followng, we propose a repar algorthm. Repar algorthm 1: In an nfeasble soluton, accordng to Equaton 5, for each suppler, we defne a P DQ / Q. If a > 0, for suppler, that s, the suppler stll has some capacty for order assgnment; f a < 0, the amount of annual order assgned to ths suppler exceeds ts annual producton capacty. So we defne two sets as S : a 0 and S : a 0. The set S + represents the supplers wth an addtonal capacty for assgnment, and the set S - represents the supplers whose capacty 10

11 constrant are volated. The followng changes ensure the feasblty of a soluton f we have: a S a S ; otherwse, the soluton s rejected. Q Q P S (31) D a Q Q a a S (3) S S Accordng to Equaton 31, the annual order quantty to supplers wth volated capacty constrant, s set equal to ther annual producton rate. Equaton 3 also shares the addtonal order quantty ( a ) among other unvolated supplers, n proporton to ther avalable capacty. S Repar algorthm : For Equaton 8, we defne A Q r ' QR n whch ' 1 3 r 1 r r r r /. Also, due to changes n the repar algorthm 1, a s re-calculated for all supplers, and the sets S + and S - are formed. If we have A > 0, the constrant s volated, so, two supplers and j are randomly chosen, provded that we have r' r' and j S, then modfcatons Q Q ' and Q Q ' must be appled. In ths case, t s guaranteed that the total order quantty j j (Q) s not changed. Reducng the suppler s order quantty does not lead to any volaton; however, ncreasng the suppler j s order quantty may lead to exceedng the producton capacty constrant. So consder the followng changes: A r' r' j ' mn, a, Q (34) j Equaton 33 determnes the amount whch must be reduced from the suppler s and added to the suppler j s order quantty. Ths change should be consdered n accordance wth repar algorthm (a); therefore, accordng to Equaton 34, the mnmum value between the amount determned n Equaton 33 and the remanng capacty of the suppler j, a j, must be selected. Also, to prevent negatve values for Q, the mnmum of Q and determned value should be chosen. It should be noted that the procedure should be repeated untl the value of A s postve. If repar s not performed wthn the predefned replcatons, the soluton modfcatons of repar algorthm s gnored, and, the repar algorthm 1 and are repeated agan. j 1 (33) 5.4 Non-domnated sortng genetc algorthm (NSGA II) Ths algorthm s presented by Deb et al. [44] as one of the best algorthms for obtanng Pareto fronters. In ths algorthm, the populaton P 1 s generated wth respect to populaton sze N p. At repetton t, after selectng parent chromosomes from populaton P t, the offsprng (populaton O t ) are generated accordng to the crossover rate p c and the mutaton rate p m. Then, P t s merged wth O t and chromosomes are sorted nto non-domnated fronters based on ther rank and crowdng dstance; Then, N p best solutons form populaton P t+1. The algorthm contnue untl the best Pareto solutons are obtaned n accordance wth the stop condton. For more nformaton, see Deb et al. [44] and Deb [45]. Please nsert Fgure 4 here 11

12 The graphcal vew of the algorthm s shown n Fgure 4. After non-domnated sortng, the solutons are sorted nto F fronters, where F 1 s the best Pareto fronter. The chromosomes wthn each fronter are also sorted accordng to the crowdng dstance. After mergng parents and offsprng populatons, the best fronters are transferred to the new populaton. As shown n Fgure 4, the fronters F 1 and F and the chromosomes wth hgher crowdng dstance on fronter F 3 are transferred to the new populaton and the other fronters are elmnated. In the proposed algorthm, we utlze a unform crossover. Assume that chromosomes m and n to be selected, after choosng a random number τ n the nterval (0,1), the order quanttes of each suppler 1 m n m n n offsprng 1 and s determned as Q Q 1 Q and Q 1Q Q, respectvely. In addton, f a mutaton s appled to the chromosome, one suppler s randomly selected and ts order quantty s determned exactly n accordance wth the ntal soluton generaton procedure. 6. Computatonal results An example s presented n ths secton, all requred data except for fuzzy data and data related to vehcles s taken from Kamal et al. [0]..A buyer wth an annual demand of 100,000 unts plans to purchase the requred amount from 4 supplers. The buyer s nventory holdng cost s $.6 and the capacty of each vehcle s 5,000 unts. Supplers nformaton s shown n Table 1. The fxed cost of each vehcle per unt of dstance s assumed as a fuzzy number C = (400,530,640). Other fuzzy parameters are also gven n Table. Also, the dscount prce offered by supplers s shown n Table 3. Please nsert Table 1 here Please nsert Table here Let s denote fuzzy objectves as Cost 1,, 3 and APV 1,, 3. Accordng to the proposed method, we need to determne the nterval GG, for each objectve. For ths purpose, two sngle-objectve optmzaton models are solved for 0. By mnmzng 1, the fuzzy optmal soluton s obtaned as Cost = (158400, , ); also, by maxmzng 3, the fuzzy optmal soluton s APV = (67000, 70000, 75000). Gven the optmal solutons of two optmzaton models, t s determned Cost APV that G = and G = The decson maker also set the non-deal values for both objectve functons as Cost G = and be (158400, ) and (55000, 75000), respectvely. APV G = So, the nterval, GG for cost and APV wll Table 4 shows the Pareto solutons obtaned from solvng the problem for dfferent values of α. The problem has been solved for α =0.1, 0.,..., 1 and the decson-maker's satsfacton level has been reported. Also, normalzed satsfacton levels are shown n the ffth and sxth columns. Please nsert Table 3 here Please nsert Table 4 here Regardng the normalzed satsfacton levels, f the domnated solutons are elmnated, four solutons 17, 18, 19 and 0 are dentfed as the fnal Pareto solutons. Fgure 5 shows the remanng Pareto solutons. Please nsert Fgure 5 here 1

13 Table 5 shows the order quanttes assgned for supplers n Pareto solutons. Also, the fuzzy membershp functon for cost and APV objectves are shown n Fgures 6 and 7, respectvely. Accordng to them, the decson maker can choose the approprate soluton. Please nsert Table 5 here Please nsert Fgure 6 here Gven the fuzzy values of the cost n Fgure 6, we compare solutons. Accordng to the calculatons of fuzzy cost values: (a) The degree n whch soluton #0 s bgger than soluton #17, #18, and #19 are 1, 0.97 and 1, respectvely; (b) The degree n whch soluton #18 s bgger than soluton #17 and #19 are 0.85 and 0.8, respectvely; (c) The degree n whch soluton #19 s bgger than soluton #17 s Gven the mnmzaton of the cost objectve functon, the soluton #0 s the worst among other solutons. Please nsert Fgure 7 here Smlarly, we can compare the solutons gven the fuzzy values of APV objectve n Fgure 7: (a) The solutons #17 and #19 are ndfferent; (b) The degree n whch both soluton #18 and #0 are bgger than ether soluton #17 or #19 s equal to 1; (c) The solutons #18 and #0 are ndfferent. For ndfferent solutons n APV, one can choose a soluton that s more cost-effectve. Accordng to the analyss, the solutons #18 and #0 are ndfferent n APV; and the degree n whch soluton #0 s bgger than #18, s 0.97; therefore, soluton #18 s preferred to soluton #0. Smlarly, the solutons #17 and #19 are ndfferent n APV; and the degree n whch soluton #19 s bgger than #17, s 0.55; therefore, soluton #17 s preferred to soluton #19. In the example above, the maxmum acceptable defectve rate (R) s assumed to be 0.0. In the followng, the problem s also solved for R=0.0, 0.04, 0.030, and the same procedure s followed to dentfy the fnal Pareto solutons. Fgure 8 shows the results. Wth regard to maxmzaton of satsfacton levels, the more the Pareto fronter s farther away the orgn, the better the qualty of ts solutons. It s expected that by ncreasng R, the soluton space gets wder and better results are obtaned; Fgure 8 also confrms ths. 7. Concluson Please nsert Fgure 8 here In the current compettve envronment, suppler selecton s one of the major processes n supply chan management of any organzaton. On the other hand, allocatng orders to the selected supplers, allows for some economes of scale through the rght choce of the quanttes to order from each suppler. In ths paper, the suppler selecton and order allocaton problem has studed n a sngle-buyer-multsuppler supply chan. Supplers offer quantty dscounts as an ncentve to motvate buyers to ncrease the amount of ther ordered quanttes. Also, transportaton costs, fuzzy type uncertanty and some practcal constrants are taken nto account n the problem. The problem s formulated as a b-objectve model to mnmze annual supply chan costs and to maxmze the annual purchasng value. We proposed a novel fuzzy mult-objectve programmng method based on the decson maker's degree of satsfacton and the fulfllment level of fuzzy constrants. After solvng the model and determnng Pareto solutons, the fuzzy results are compared wth an ndex, and the decson maker can dentfy the approprate or napproprate solutons. We utlzed NSGA II to solve the model and the results are presented usng numercal examples. The nterested researchers can also nvestgate demand uncertanty, the effect of dfferent potental supplers n each perod and other quantty dscount schemes such as Incremental quantty dscounts and busness volume quantty dscounts n the problem. Also, other algorthms such as mult 13

14 objectve partcle swarm optmzaton (MOPSO) and mult objectve smulated annealng (MOSA) can be appled to the problem n order to nvestgate the effcency of the proposed NSGA II. References [1] Hamdan, S., & Cheatou, A. Suppler selecton and order allocaton wth green crtera: An MCDM and mult-objectve optmzaton approach, Comput. Oper. Res., 81, pp (017). [] Govndan, K., Rajendran, S., Sarks, J., et al. Mult crtera decson makng approaches for green suppler evaluaton and selecton: a lterature revew, J. Clean. Prod., 98, pp (015). [3] Guo, C., & L, X. A mult-echelon nventory system wth suppler selecton and order allocaton under stochastc demand, Int. J. Prod. Econ., 151, pp (014). [4] Kannan, D., Khodaverd, R., Olfat, et al. Integrated fuzzy mult crtera decson makng method and multobjectve programmng approach for suppler selecton and order allocaton n a green supply chan, J. Clean. Prod., 47, pp (013). [5] Pazhan, S., Ventura, J. A., & Mendoza, A. A seral nventory system wth suppler selecton and order quantty allocaton consderng transportaton costs, Appl. Math. Model., 40(1), pp (016). [6] Ballou, R. H. Busness Logstcs Management. Upper Saddle Rver, NY Prentce Hall (199). [7] Mansn, R., Savelsbergh, M. W. P., & Tocchella, B. The suppler selecton problem wth quantty dscounts and truckload shppng, Omega, 40(4), pp (01). [8] Talezadeh, A. A., Stojkovska, I., & Pentco, D. W. An economc order quantty model wth partal backorderng and ncremental dscount, Comput. Indust. Eng., 8, pp. 1 3 (015). [9] Ayhan, M. B., & Klc, H. S. A two stage approach for suppler selecton problem n mult-tem/multsuppler envronment wth quantty dscounts, Comput. Indust. Eng., 85, pp. 1 1 (015). [10] Jménez, M., Arenas, M., Blbao, A., et al. Lnear programmng wth fuzzy parameters: an nteractve method resoluton, Eur. J. Oper. Res., 177(3), pp (007). [11] Dahel, N.-E. Vendor selecton and order quantty allocaton n volume dscount envronments, Suppl. Chan Manag.: Int. J., 8(4), pp (003). [1] Demrtas E.A., & Üstün, Ö. An ntegrated multobjectve decson makng process for suppler selecton and order allocaton, Omega, 36, pp (008). [13] Xa, W., & Wu, Z. Suppler selecton wth multple crtera n volume dscount envronments, Omega, 35(5), pp (007). [14] Burke, G. J., Carrllo, J., & Vakhara, A. J. Heurstcs for sourcng from multple supplers wth alternatve quantty dscounts, Eur. J. Oper. Res., 186(1), pp (008). [15] Kokangul, A., & Susuz, Z. Integrated analytcal herarch process and mathematcal programmng to suppler selecton problem wth quantty dscount, Appl. Math. Model., 33(3), pp (009). [16] Amd, A., Ghodsypour, S. H., & O Bren, C. A weghted addtve fuzzy multobjectve model for the suppler selecton problem under prce breaks n a supply chan, Int. J. Prod. Econ., 11(), pp (009). 14

15 [17] Wang, T.-Y., & Yang, Y.-H. A fuzzy model for suppler selecton n quantty dscount envronments, Expert Syst. Appl., 36(10), pp (009). [18] Ebrahm, R. M., Razm, J., & Haleh, H. Scatter search algorthm for suppler selecton and order lot szng under multple prce dscount envronment, Adv. Eng. Soft., 40(9), pp (009). [19] Razm, J., & Maghool, E. Mult-tem suppler selecton and lot-szng plannng under multple prce dscounts usng augmented e-constrant and Tchebycheff method, Int. J. Adv. Manuf. Technol., 49(1 4), pp (010). [0] Kamal, A., Fatem-Ghom, S., & Jola, F. A mult-objectve quantty dscount and jont optmzaton model for coordnaton of a sngle-buyer mult-vendor supply chan, Comput. Math. Appl., 6(8), pp (011). [1] Zhang, J. L., & Zhang, M. Y. Suppler selecton and purchase problem wth fxed cost and constraned order quanttes under stochastc demand, Int. J. Prod. Econ., 19(1), pp. 1-7 (011). [] Lee, A. H. I., Kang, H.-Y., La, C.-M., et al. An ntegrated model for lot szng wth suppler selecton and quantty dscounts, Appl. Math. Model., 37(7), pp (013). [3] Moghaddam, K.S., Fuzzy mult-objectve model for suppler selecton and order allocaton n reverse logstcs systems under supply and demand uncertanty, Expert Syst. Appl.. 4, pp (015). [4] Bohner, C., & Mnner, S. Suppler selecton under falure rsk, quantty and busness volume dscounts, Comput. Indust. Eng., 104, pp (017). [5] Çeb, F., & Otay, İ. A two-stage fuzzy approach for suppler evaluaton and order allocaton problem wth quantty dscounts and lead tme, Inf. Sc., 339, pp (016). [6] Hamdan, S., & Cheatou. A. Dynamc green suppler selecton and order allocaton wth quantty dscounts and varyng suppler avalablty, Comput. Indust. Eng., 110, pp (017). [7] Tezenj, F. R., Mohammad, M., Pasanddeh, S. H. R., et al. An ntegrated model for suppler locatonselecton and order allocaton under capacty constrants n an uncertan envronment. Sc. Iran. E, 3(6), (016). [8] Jménez, M., Blbao, A., Pareto-optmal solutons n fuzzy mult-objectve lnear programmng, Fuzzy Set. Syst., 160(18), pp (009). [9] Sakawa, M. Fuzzy sets and nteractve multobjectve optmzaton. Sprnger Scence & Busness Meda (013). [30] Twar, R. N., Dharmar, S., & Rao, J. R. Fuzzy goal programmng an addtve model, Fuzzy Set. Syst., 4(1), pp (1987). [31] Chen, L. H., & Tsa, F. C. Fuzzy goal programmng wth dfferent mportance and prortes, Eur. J. Oper. Res., 133(3), pp (001). [3] Wu, Y. K., & Guu, S. M. A compromse model for solvng fuzzy multple objectve lnear programmng problems, J. Chnese Ins. Indust. Eng., 18(5), pp (001). [33] Aköz, O., & Petrovc, D. A fuzzy goal programmng method wth mprecse goal herarchy, Eur. J. Oper. Res., 181(3), pp (007). [34] Lee, E. S., & L, R. J. Fuzzy multple objectve programmng and compromse programmng wth Pareto optmum, Fuzzy Set. Syst., 53(3), pp (1993). 15

16 [35] Arkan, F. A fuzzy soluton approach for mult objectve suppler selecton, Expert Syst. Appl., 40(3), pp (013). [36] L, X. Q., Zhang, B., L, H., Computng effcent solutons to fuzzy multple objectve lnear programmng problems, Fuzzy Set. Syst., 157(10), pp (006). [37] Ergnel, N., & Gecer, A. Fuzzy mult-objectve decson model for calbraton suppler selecton problem, Comput. Indust. Eng., 10, pp (016). [38] Govndan, K., Darbar, J. D., Agarwal, V., et al. Fuzzy mult-objectve approach for optmal selecton of supplers and transportaton decsons n an eco-effcent closed loop supply chan network, J. Clean. Prod., 165, pp (017). [39] Zadeh, Lotf A., Fuzzy sets, Inf. Control, 8(3), pp (1965). [40] Alae, S., & Khoshalhan, F. A hybrd cultural-harmony algorthm for mult-objectve supply chan coordnaton, Sc. Iran. E, (3), pp (015). [41] Dubos, D., Kerre, E., Mesar, R., et al. Fuzzy nterval analyss. In: Dubos, D., Prade, H. (Eds.), Fundamentals of Fuzzy Sets. Kluwer, Massachusetts, pp (000). [4] Yager, R.R., Rankng fuzzy subsets over the unt nterval, In: Proc. of 17th IEEE Int. Conf. on Dec. Control, San Dego, CA, pp (1979). [43] Helpern, S., The expected valued of a fuzzy number, Fuzzy Set. Syst., 47, pp (199). [44] Deb, K., Agrawal, S., Pratap, A., et al. A fast eltst non-domnated sortng genetc algorthm for multobjectve optmzaton: NSGA-II, In Int. Conf. on Parallel Prob. Solv. From Nature (pp ). Sprnger, Berln, Hedelberg (000). [45] Deb, K. Mult-objectve optmzaton usng evolutonary algorthms (Vol. 16). John Wley & Sons (001). Mohammad Al Sobhanolah obtaned BSc and MSc n Mathematc from Tabrz Unversty, Tabrz, Iran, n 1973 and 1975, respectvely. He also receved MS n Publc Management from Tabrz Unversty, n 1991 and PhD n Industral Engneerng from Swnburne Unversty, Australa, n Currently, he s a Professor n the Faculty of Industral Engneerng at Kharazm Unversty. He has authored numerous papers presented at conferences and publshed n natonal and nternatonal journals. Ahmad Mahmoodzadeh receved BSc degree n Manufacturng Engneerng from Shahd Rajaee Teacher Tranng Unversty, Tehran, Iran, n 000, and MSc degree n Industral Engneerng from Mazandaran Unversty of Scence and Technology, Iran, n 003. Currently, He s a PhD canddate n Industral Engneerng at Kharazm Unversty. Bahman Nader completed hs BSc degree n Industral Engneerng from Mazandaran Unversty of Scence and Technology (Iran), MSc and PhD form Amrkabr Unversty of Technology Tehran, Iran, and post-doctoral program at Unversty of Wndsor, Canada. Currently, he s an Assstant Professor at Kharazm Unversty. Hs research nterests are appled operatons research (mathematcal modelng and soluton method). 16

17 Fgures capton: Fgure 1. The schematc vew of supply chan Fgure. Occurrence possblty of a crsp objectve value z and ts goal satsfacton degree. Fgure 3. Procedure of ntal soluton generaton Fgure 4. Graphcal vew of NSGA II Fgure 5. Fnal Pareto solutons wth respect to the normalzed satsfacton levels Fgure 6. Fuzzy membershp functon of cost objectve n Pareto solutons Fgure 7. Fuzzy membershp functon of APV objectve n Pareto solutons Fgure 8. Pareto fronters wth regard to dfferent values of R Tables capton: Table 1. Supplers Informaton n the example Table. Fuzzy parameters n the example Table 3. Quantty dscount offered by supplers Table 4. Pareto solutons of problem wth respect to the dfferent values of Table 5. Order quanttes n Pareto solutons 17

18 Fgure 1. The schematc vew of supply chan Fgure. Occurrence possblty of a crsp objectve value z and ts goal satsfacton degree Fgure 3. Procedure of ntal soluton generaton 18

19 Fgure 4. Graphcal vew of NSGA II Fgure 5. Fnal Pareto solutons wth respect to the normalzed satsfacton levels 19

20 Fgure 6. Fuzzy membershp functon of cost objectve n Pareto solutons Fgure 7. Fuzzy membershp functon of APV objectve n Pareto solutons 0

21 Fgure 8. Pareto fronters wth regard to dfferent values of R Table 1. Supplers Informaton n the example Parameter Suppler S P A h ds Table. Fuzzy parameters n the example Suppler c w r 1 (3, 4.04, 4.9) (0.4, 0.44, 0.48) (0.0307, , ) (6, 6.48, 7.1) (0.55, 0.64, 0.67) (0.0498, , ) 3 (7, 7.17, 7.8) (0.71, 0.7, 0.78) (0.0116, 0.011, ) 4 (5, 5.87, 6.3) (0.55, 0.57, 0.6) (0.005, 0.015, 0.065) Table 3. Quantty dscount offered by supplers Suppler Unt prce Order nterval 1

22 (0,5000) 8.9 [5000,10000) 8.8 [10000,15000) 8.7 [15000,0000) 8.6 [0000,5000) 8.5 [5000,30000) 8.4 [30000,35108) 9.1 [0,000) 9 [000,4000) 8.9 [4000,6000) 8.8 [6000,8000) 8.7 [8000,10000) 8.6 [10000,0000) 8.7 [0,3000) 8.6 [3000,6000) 8.5 [6000,9000) 8.4 [9000,1000) 8.3 [1000,15000) 8. [15000,18000) 8.1 [18000,1000) 8 [1000,30000) 10.5 [0,4000) 10.4 [4000,8000) 10.3 [8000,1000) 10. [1000,16000) 10.1 [16000,68777) Table 4. Pareto solutons of problem wth respect to the dfferent values of # K 1 K K 1 K

23 Table 5. Order quanttes n Pareto solutons Suppler #