Robust-fuzzy model for supplier selection under uncertainty: An application to the automobile industry

Transcription

1 Robust-fuzzy model for suppler selecton under uncertanty: An applcaton to the automoble ndustry Masood Rabehª,*, Mohammad Modarres b, Adel Azar c ª Department of Industral Management, hahd Behesht Unversty, Tehran, Iran. b Department of Industral Engneerng, harf Unversty of Technology, Tehran, Iran. c Department of Industral Management, Tarbat Modares Unversty, Tehran, Iran. Ths paper proposes an nnovatve robust-fuzzy method for mult-obectve, mult-perod suppler selecton problem under multple uncertantes. Ths approach ntegrates robust optmzaton and fuzzy programmng. Uncertan parameters are modeled as random varables that take value wthn a symmetrcal nterval. However, due to the complexty or ambguty of some real world problems and specally the nature of some of the avalable nput data, the length of nterval s also hghly uncertan. Ths ambguty motvated us to present a new approach, whch can be applcable to multple uncertantes condtons. Thus, n our approach the half-length of these ntervals s also represented by fuzzy membershp functon. We develop a model and a soluton approach to select supplers by consderng rsk. The proposed method s appled to a real case of suppler selecton n automoble ndustry under uncertanty and ambguty condtons. To verfy the proposed model, we evaluated the results by smulaton technque and compared values of obectve functon under dfferent scenaros. Keywords: uppler selecton; Uncertanty; Robust optmzaton; Fuzzy programmng; Robust-fuzzy model; Auto ndustry.. Introducton uppler selecton s an mportant strategc decson makng for managers n supply chan management []. One can fnd numerous studes and models n the suppler selecton lterature, see ecton 2. The decson makng models n the lterature are bascally developed to answer the followng questons: How many supplers are need? Whch supplers should be selected? What s * Correspondng author: M. Rabeh(m_rabeh@sbu.ac.r)

2 the optmal orderng polcy to each suppler? One can fnd many determnstc models developed n the lterature to answer these questons. The man dsadvantage of determnstc models s that they fal to be accountable when encounter the uncertan nature of real world systems. On the other hand, rsk and uncertanty management s an nherent part of suppler selecton. The mportance and the effects of uncertanty n suppler selecton have been wdely emphaszed n the supply chan lterature [-3]. To handle the effect of uncertanty of nput parameters, ether probablstc or fuzzy approaches can be appled. However, n many real cases stochastc programmng cannot be appled, due to the lack of enough hstorcal data to ft approprate probablty dstrbuton for uncertan parameters. Furthermore, computatonal complexty of stochastc programmng models s another reason to lmt ts applcaton. Therefore, robust optmzaton methods can be consdered as an approprate alternatve effcent method for handlng ths type of uncertanty [3]. Recently some researchers have developed robust optmzaton approach to handle uncertantes n suppler selecton [, 2]. In recent years, robust optmzaton has been emerged as a methodology for managng supply chan rsks. Robustness plays an mportant role n sustanablty of supply wth uncertan nput data. Robustness of supply chan has been the fronter of supply chan research [4]. Fuzzy programmng s another method for handlng a type of uncertan parameters whch are usually expressed n lngustc and vague terms and then transformed nto fuzzy numbers. In real cases, due to the vagueness of the nformaton about parameters, determnstc methods are not sutable for an effectve suppler selecton. Under these condtons, the theory of fuzzy sets s one of the best tools to handle the uncertanty [5]. Fuzzy programmng based on fuzzy sets theory s appled due to the presence of vagueness and mprecson of nformaton n suppler selecton. In ths research, we develop a new method of robust optmzaton to handle uncertantes. In robust optmzaton; the models are developed based on the assumpton that no knowledge about the dstrbuton of uncertan parameters exsts. In these models, uncertanty of parameters s modeled as symmetrc and bounded random varables. Every symmetrcal nterval characterzed by two numbers: mddle number of nterval (nomnal value) and nterval half-length. It s assumed that these numbers are gven. In real world applcatons however, forecastng ther exact 2

3 values seems to be dffcult or even mpossble. In fact, errors and ambguty n forecastng and measurng model parameters makes the decson makng model prone to many resources underutlzaton and/or overutlzaton problems. In our real case study, we faced the problem of determnng the half-length of nterval of suppler s capacty parameter. That case motvated us to provde an nnovatve robust-fuzzy model. In ths approach, the half-length of nterval s presented by a fuzzy number. In fact, the man nnovaton of ths research s combnng the robustness and fuzzness concepts n the presence of multple uncertantes. In order to llustrate our proposed robust-fuzzy(r-f) method, t s mplemented for a real case of suppler selecton n automoble ndustry under multple uncertantes. Under varous condtons, the results proved to be accurate enough, when they are compared wth the results by smulaton technques. In other words, the proposed model can properly manage a varety of uncertan condtons and provde useful and relable nformaton for flexble decson makng processes. The rest of ths paper s organzed as follows. In ecton 2 lterature revew s presented. In ecton 3, the case study s descrbed. teps of mplementng the model of case s presented n ecton 4. We present the methodology of robust-fuzzy (R-F) method n ecton 5. In ecton 6, the mathematcal model s formulated. Data collecton and parameters are descrbed n ecton 7. Computatonal result s presented n ecton 8 and fnally, some concludng remarks are gven n ecton Lterature revew Due to the mportance of suppler selecton decson, an extensve lterature revew, both determnstc and uncertan models, s presented to address ths knd of decson. In secton pertnent to uncertan models, three categores of mentoned models were stated n forms of stochastc, fuzzy and robust. It should be remnded that, relevant researches were revewed for each category. Fnally, consderng both the strengths and weaknesses of each category and current gap n lterature, artcle contrbuton was presented. 2.. Determnstc models In ths secton, a bref overvew of relevant studes about determnstc suppler selecton models s presented. Benton [6] used Lagrange relaxaton to develop a non-lnear program for suppler selecton under varous condtons ncludng multple supplers, multple tems, resource 3

4 lmtatons and quantty dscount. Godsypour and O'Bren [7] used ntegrated analytcal herarchy process (AHP) wth mxed nteger programmng to develop a decson support system (D), n order to reduce the number of supplers. Godsypour and O'Bren [8]also developed a model to take nto account both qualtatve and quanttatve factors, based on the ntegraton of AHP and lnear programmng model. In a further development Ghodsypour and O'Bren [9]. presented a mxed nteger non-lnear programmng model by consderng the total cost of logstcs for multple sourcng cases. Basnet and Leang [] nvestgated the problem of suppler selecton by consderng the lot-szng.ustun and demrtas [] used an ntegrated approach of analytc network process (ANP) and mult-obectve mxed nteger lnear programmng (MOMILP) for suppler selecton problem. Ther approach consders both tangble and ntangble factors n choosng the best supplers and defnes the optmum quanttes among selected supplers to maxmze the total value of purchasng (TVP), and to mnmze the total cost and total defect rate and to balance the total cost among perods. Fnally, Mendoza and Ventura [2] presented two mxed nteger nonlnear programmng models to select the best supplers and determne order quanttes, whle mnmzng the annual orderng, nventory holdng, and purchasng costs wth respect to supplers capacty and qualty constrants. Ther research ntegrated the ssues of nventory management and suppler selecton. Determnstc model lterature consdered the followngs as sgnfcant tems: ome of these models are mult obectve (see Basnet and Leang, 25). ome of these models consdered some mportant characterstcs such as multple supplers, multple tems smultaneously (see Benton, 99). These models are not applcable for uncertan envronments obvously Uncertan models In ths secton, a bref overvew of relevant studes about uncertan suppler selecton models (.e., fuzzy, stochastc and robust models) s presented Fuzzy models ome researchers developed fuzzy models. Kumar et al. [3] used a fuzzy mxed nteger goal programmng approach to solve the vendor selecton problem n case of multple obectves. 4

5 Chen et al. [4] presented a fuzzy decson makng approach to solve the suppler selecton problem. They used lngustc values to evaluate the ratngs for a number of quanttatve and qualtatve factors ncludng qualty, prce, flexblty, and delvery performance. Ther model s shown to be a good tool for suppler selecton decson makng stuaton. Klc [5] presented a new ntegrated approach, ncludng fuzzy TOPI method and a mxed nteger lnear programmng model, to select sutable suppler n a mult-tem/mult-suppler envronment. In the frst stage of ths approach, mportance value of each suppler for each product s obtaned va fuzzy TOPI. In the second stage, outputs of the frst stage are used as nputs to the proposed mathematcal model and select the best supplers and determne amount of order for each product (tem) to each suppler va solvng the model. Nazar-hrkoh et al. [6] proposed an nteractve fuzzy mult-obectve lnear programmng (FMOLP) model to solve the multobectve suppler selecton problem and order allocaton under multple prce levels and multple products. The model attempts to mnmze cost as well as numbers of defectve unts and late delvered unts ordered from supplers. Fnally, Roshandel et al. [7] presented a fuzzy herarchcal TOPI approach to select and evaluate supplers of mported raw materals for the detergent producton ndustry n Iran. The authors clam that ther approach overcomes dsadvantages of a pror well-known method n the area of fuzzy TOPI (Chen' method). ouf Neyestan et al. [8] presented an evolutonary algorthm for suppler order allocaton wth fuzzy parameters consderng lnear and volume dscount. The authors clam that some nformaton, such as buyer demand and suppler delvery rate, s uncertan, so, fuzzy sets are appled to handle uncertanty. Mahmoud et al. [9] were extended fuzzy VIKOR usng an effcent fuzzy dstance measure to solve applcable uppler electon Problem (P) under group decson-makng. The authors clam that n real-lfe stuatons, P parameters are often mprecse, vague, uncertan, or ncomplete and n ths respect, fuzzy sets theory s the best developed approach to handle these uncertantes. There are some general shortcomngs about fuzzy approaches. Fuzzy approaches are only applcable n condton of uncertanty n a fuzzy form. Determnng real Membershp functon of the fuzzy numbers s not easy. In all above mentoned studes the ssues of sngle uncertanty are nvestgated, but are not fully matched wth real world cases 5

6 tochastc models Kaslngam and Lee. [2] developed a chance-constraned nteger programmng formulaton for vendor selecton and order allocaton by mnmzng costs. Ther model also consders the lead tme requrements and vendors capactes and uses a chance constrant due to the stochastc nature of demand. Feng et al. [2] presented a stochastc nteger programmng (IP) method for smultaneous selecton of tolerances and supplers based on the qualty loss functon and process capablty ndces. In ths research, to acheve the requred qualty level and mnmum total cost process capablty ndces and qualty loss functon ncorporated also n a mathematcal model. Bonser and Wu [22] developed a stochastc program for solvng procurement problem n an electrcal utltes frm under uncertan demand and market prce. everal other works have also consdered ths type of uncertanty n suppler selecton problem. eshadret al. [23] developed a probablstc bddng model to represent the connecton between multple sourcng and ts consequences. Ther bddng model nvestgates the effect of multple sourcng on compettve behavor pror to suppler selecton. Ranbar Tezen et al. [24] presented an ntegrated model for suppler locaton-selecton and order allocaton under capacty constrants n an uncertan envronment. The authors clam that costs are uncertan, and were consdered to be stochastc. In many real cases, applcatons of stochastc programmng are lmted, due to: Enough hstorcal data about approprate probablty dstrbuton for uncertan parameters usually s not avalable. These models have some computatonal complextes [3]. The aforementoned shortcomngs motvated us to select an approprate alternatve method especally for our real case wth especal characterstcs. In such cases, robust optmzaton methods could be adopted because these methods (especally Bertsmas and m s approach) are more practcal. In fact, robust optmzaton methods tres to present solutons whch are less senstve versus changes of parameters [3] Robust models 6

7 Recently some researchers have developed robust optmzaton approach to handle uncertantes n suppler selecton [, 2]. L and Zabnsky [] developed stochastc mathematcal programmng models to capture the rsk assocated wth uncertan custumer demand and suppler capacty. They used mult-parametrc programmng technques to analyze tradeoffs and determne a robust set of supplers wth balanced costs and rsks. In fact, the fnal result of ther research s a robust selecton of supplers. Nam et al. [2] developed a model for robust supply base management under both demand and supply uncertanty. In fact, the mportance of supply base management under condton of uncertanty, requers an understan dng of benefts assocated wth a contractor havng an optmal number of supplers. Brefly, the model determnes the robust optmal number of supplers that a contractor should mantan n ts supply base n condton of uncertanty. The lterature of robust modeles showes that: In mentoned studes only the ssue of sngle uncertanty s nvestgated. Lterature has not attenton to robust optmzaton methods especally Bertsmas and m approach as an effcent method. Wth consderng shortcommngs of each secton (determnstc and uncertan models), curremt lterature has pad lttle attenton the followng temes: multaneous attenton to characterstcs such as multple tems, multple supplers, multple obectves (goals), multple perods and multple products. Robust optmzaton methods especally Bertsmas and m approach as an effcent method. Multple uncertanty. Ths research ams to fll these gaps. For example, the proposed comprehensve model ncorporates assumptons and characterstcs such as multple tems (parts), multple supplers, multple obectves (goals), multple perods, multple products (automobles), multple uncertantes and two-echelon supply chan ncludng supplers and manufactures (plants). Also, n ths research was consdered mportant crteron such as prce, qualty and delvery. The contrbuton of ths research n vew pont of value added to theory s presented as follow: 7

8 In fact, the man contrbuton of ths research s combnng the robustness and fuzzness concepts n the presence of multple uncertantes. In other word, ths research proposes a novel approach about multple uncertantes. 3. Case applcaton Generatng hgh employment rate, hgh fnancal turnover and sgnfcant rate of dealng wth foregn automotve ndustry, the automotve ndustry s one of the mportant ndustres after ol and gas ndustry. Ths ndustry affect sgnfcantly on natonal economy. Therefore, ts supply chan s one of the most dynamc chans n ndustres. Ths chan provdes thousands of parts for each vehcle. Ignorng the parts suppled by sngle source, many others have multple sources. These parts nclude more than 7 percent of the total value of each vehcle. Consderng the number of supplers, whch can be from one to four. We mplement the proposed suppler selecton method n Iran Khodro, the largest automotve company n Iran. Due to the hgh volume of requred parts, choosng sutable supplers s of great mportance for ths ndustry. ome exstng data (suppler s capacty and transportaton cost) are uncertan by ther nature. Thus, due to the exstence of uncertan data, we adopt robust optmzaton for suppler selecton plannng. It motvated us to develop a new method named, Robust-Fuzzy method. Although the problem s formulated n ts general form, we collected data for two models of vehcles, Peugeot 45 and Pars and for Tehran plant only, whch s the largest and most mportant plant of the company. 4. Methodology of Robust-Fuzzy As was mentoned, one way of dealng wth uncertanty s by stochastc programmng modelng. The man dffcultes of stochastc programmng approach are: (a) lack of exstence of dstrbuton functon of data and (b) computatonal challenges [25].The concept and advantages of robust approach n general, and n supply chan plannng n partcular, can be llustrated n Fgure. Determnstc approaches obtan the soluton based on averagng or good guess. In contrast, robust approaches provde a soluton whch s near-optmal. Although the resultng 8

9 cost s more than that of determnstc approach, the soluton s more relable. In other words, takng nto account the varablty of parameters n a range of values, the soluton s stll relable wth hgh-confdence, [26]. oyster [27] took the very frst step toward usng robust optmzaton, proposng lnear programmng model. Ths model produces solutons that are too conservatve. The next steps s by Ben-Tal and Nemrovsk [28-3], El-Ghav et al. [3], El-Ghav and Lebert [32] to address over-conservatsm. They appled robust optmzaton to lnear programmng problem wth ellpsodal uncertanty sets, thus obtanng conc quadratc programs. [Please nsert Fgure about here.] Bertsmas and m [33] presented a new approach, specfcally talored for polyhedral uncertanty that leads to lnear robust counterparts,.e. retans advantage of lnear framework of oyster (973), whle controllng conservatveness level of the soluton. Thus, ther method s also approprate for solvng dscrete optmzaton problems and that s the reason we have adopted Bertsmas and m model. As explaned before, n robust optmzaton approach, uncertan data takes value wthn an nterval. Although no dstrbuton functon of ths random varable s avalable, the length of nterval s gven. However, n some real cases t s dffcult for decson maker to determne the exact nterval length (or half-length f t s symmetrcal). If the decson makers consder a hgh half-length of nterval, then the level of conservatsm and consequently hgher cost would be ncreased. Conversely, f the nterval half-length be consdered low, the rsk would be ncreased. Furthermore, n some stuatons the decson maker suggests ambguous nterval half-length. To overcome ths problem, we develop an nnovatve approach and represent the nterval half-length n term of fuzzy number. In fact, we develop a robust-fuzzy (R-F) method whch ntegrates the concept and technques of robust optmzaton )Bertsmas and m approach ( and fuzzy programmng, Werners approach [34, 35]. In addton to havng the applcable features of Bertsmas and m model, ths method s properly responsve to mental ambguty of decson maker about nterval half-length and also provdes necessary flexblty for decson makers. In other words, ths method s able to solve the problems n multple uncertanty condtons. 9

10 Consder a nomnal mxed nteger programmng, as follows. Mnmze c ' x subect to Ax b l x u x Z,,, k Wthout loss of generalty, we assume only the elements of matrx A and C are uncertan. In practce, the mean value of coeffcents a and also ts nterval half-length, â,and smlarly, and () d are estmated. Model of data uncertanty are represented as follows: Bertsmas and m, c (23). (a) (Uncertanty for matrx A): let N=,2,...,n. Each entry a, N s modeled as ndependent, symmetrc and bounded random varable(but wth unknown dstrbuton) a ~, N that takes values n[ a aˆ ˆ ]. a a (b) (Uncertanty for cost vector c): Each entry c, N take values n[ c d represents the devaton from the nomnal cost coeffcent, Furthermore, we defne: c. J represents the number of uncertan components n th constrant.,,,...,m, that takes value n[ ]. c d ],where The role of parameter n constrants s to adust the robustness of the proposed method aganst the level of conservatsm of the soluton. mlarly, the level of robustness n obectve functon s controlled by parameter.therefore, the robust counterpart of problem() s as follows: Mnz cx z. t : z z p y z y l p p x x z a x x d aˆ y y y u z p p b (2)

11 Model 2 s Bertsmas and m robust counterpart 4.. Lnear programmng problem wth fuzzy resources: Werners approach Consder the followng general fuzzy lnear programmng model wth fuzzy resources (nonsymmetrcal model): MaxZ f ( x) CX ~ s. t : ( AX ) b,,2,..., m or X ~ where b, s n b, b p wth gven p. MaxZ f ( x) CX ~ s. t : ( AX ) b,,2,..., m X (3) To make ths model a symmetrcal one, Werners [34, 35] proposed ts obectve functon should be of fuzzy type. Therefore, fuzzy resource s tolerance p s assumed to be gven and certan. He ntally offers the followng defntons for Z nf f s. t : ( Ax) b X Max CX nf f and sup f : Z sup f Max CX s. t : ( Ax) b X Membershp functon for obectve functon and constrants s as follows see Fgure 2 and 3: p (4) [Please nsert Fgure2 about here.] [Please nsert Fgure3 about here.] To acheve optmal solutons, we use Max mn operator. Therefore, fuzzy lnear programmng can be converted to a symmetrcal model as follows: Max where Mn x), ( x),... ( ) (5) x, ( m x 4.2. Innovatve Robust-Fuzzy approach

12 2 As explaned before, our Robust-Fuzzy approach ntegrates technques of robust optmzaton (Bertsmas and m s approach ( and fuzzy programmng )Werners approach(. In ths secton, we summarze the model by consderng the decson makers ambguty n determnng nterval half-length of ths parameter as a fuzzy number ~ˆb, wth the followng membershp functon, see Fgure 4. [Please nsert Fgure4 about here.] The resultng model s a lnear programmng one wth fuzzy resources (nonsymmetrcal model) whch can be converted nto a symmetrc one and fnally nto the followng determnstc model: k z x u x l y x y z y p J b b b p z J y a p z J y d p z b q p z x a t Z Z Z p z x c z Max,...,,,, ˆ ) ˆ ˆ (, ˆ :. ) ( ; mn mn (6) In model 6, P and q are robustness varables. When convertng a nomnal model to ts robust counterpart model, robustness varables are added to the model. The proposed method s more flexble compared to base model (Bertsmas and m model) and also elmnates decson maker s ambguty about half-length ntervals. In other words, the decson maker s able to reach pessmstc, optmstc and more balanced results between them. If (nterval length s upper lmt) pessmstc results s obtaned by model 3 and f (nterval length s lower lmt) optmstc results s obtaned. In secton 8. Mont Carlo smulaton experments are conducted to ustfy robustness of solutons and also estmate real level of conservatsm n aganst determnate budget. In fact, by smulaton,

13 the probablty of constrant volaton or rsk rate wll be determned. Brefly, The Monte Carlo smulaton model was mplemented to test and valdate the robustness of the soluton obtaned va the robust model. 5. teps of our proposed approach Fgure 5 shows the steps of ths approach. [Please nsert Fgure5 about here.] Data has been collected va ntensve ntervews wth managers and experts and from the exstng documents of Iran Khodro Company, as well as from APCO Company, ts man subsdary whch s responsble for managng ts supplyng parts and component. Also profle of experts nclude: fve people wth at least years of experence and expertse ncludng logstcs manager, logstcs proect team leader, head of logstcs engneerng, logstcs engneerng expert, and plannng expert. 6. Formulatng the model In ths secton, we present mathematcal model of suppler selecton problem: 6.. Notaton ets I: et of products (automobles); J: et of plants; M: et of parts; N: et of supplers; T: et of plannng perods; R: et of obectve functons (goals). Certan Parameters P : Producton level of product n perod t for plant ; t C : Purchase cost of part m from suppler n; mn C : Capacty of suppler n to supply part m; mn Chmn : Inventory cost n manufacturng plant; PPM : Index of returned part m to suppler n; mn VC m : Consumpton coeffcent of part m for product ; 3

14 DP : On-tme delvery number of part m by suppler n; mn W : Importance weght of obectve r; r G : Value of goal r; r LT m : Lead tme of part m; : afety factor of determnng permssble levels of nventory; : Factor determnng the mnmum procurement from each suppler; : Robust protecton level. Uncertan parameters C mn t : Transportaton cost of part m from suppler n to plant ; C mn : uppler n capacty supplyng part m; Issue related to uncertan parameters C : The nomnal number of transportaton cost of part m from suppler n to plant ; t mn C : The Nomnal number of capacty of suppler n to supply part m; mn Cˆ t mn : Half-length of transportaton cost of part m from suppler n to plant ; ~ C : Half-length of capacty of suppler n to supply part m. mn Decson varables : upply rate of part m from suppler n n perod t for plant ; mnt I mnt : Inventory level of part m from suppler n n perod t for plant ; d, d : Devaton from goal, (postve, negatve); : Degree of constrant s satsfacton Mathematcal Model Obectve functons We consder three obectves for the model as follows: Mnmzng the cost of supplyng parts. Mnmzng the defectve parts suppled by supplers Maxmzng on-tme delvery The frst obectve functon conssts of purchasng cost, transportaton cost from suppler to manufacturng plant and holdng cost. The second obectve functon s related to PPM (part per 4

15 mllon) factor. Ths factor s one of the most mportant ndces n ths ndustry. In fact, ths factor calculates rato of number of defects n the volume of shpment. It s clear that any suppler that have lower PPM wll attract more orders. The thrd obectve s an mportant ndcator, hghly emphaszed by ndustry experts as well as n the lterature. Obvously, that any supplers that have more on-tme delvery wll attract more orders. To consder all three obectves, we formulate the problem by applyng goal programmng technque. Constrants The constrants n general can be classfed nto several categores, functonal constrants, goal constrants and robust constrants. Functonal (man) constrants of the model are: Parts demand: Demand n each perod depends on three factors: producton level per product, consumpton coeffcent of part per product, and nventory. Inventory level, maxmum and mnmum: Inventory maxmum and mnmum levels depend on four factors: producton level per product, consumpton coeffcent of part per product, lead tme of part, and safety factor of determnng permssble levels of nventory. upplers capacty. The mnmum quantty of a part whch can be purchased from each suppler. Due to the company's polcy on purchasng from all supplers of a part, the mnmum quantty of a part whch can be purchased from each suppler s constraned by a factor that determnes the mnmum procurement level from each suppler ( ) Nomnal goal model: The nomnal goal model s: 5

16 MnZ. t : I m n t I m n t n n n N m n t n M n n J J I I mnt M M mnt mnt mnt J N N mnt mnt, I r r T T T C mnt w ( d PPM DP r C mn VC mn mn VC VC, d P P P m, n, t n r m mn m m t n r,nteger J mnt mnt wd ) G mnt d t t mnt d n M n N d I d mnt m, n, t w2d G2 T m n t G Ct G n / 3 LT J m I m, n,, t w3d 3 G3 mnt / 3 ( ) LT 3 2 mn 2 mnt m M, m, t, m, t N m n t J, m, t T Ch mn I mnt d d G (7) 6.4. Robust-Fuzzy counterpart model In ths secton, we present the fnal model by consderng the decson makers ambguty n ~ determnng nterval half-length of parameter c as a fuzzy number c. Thus, wth consderng ths nnovatve Robust-Fuzzy approach presented n ecton 2-2, the nomnal goal model s converted nto the followng robust-fuzzy counterpart model: 6

17 MaxZ. t : w d G I I n n n ZZ N mnt w2d 2 G2 m n t N m n t N T m n t n PP n M n J J I I mnt mnt M M mnt mnt mnt mnt mnt ZZ ppp, I T T J C PPM DP w3d 3 G3 VC mnt mnt VC VC n P t n ZZ Ĉt mnt mn mn m 2 mn m m P P mnt ( C,nteger J mnt mnt mnt ( Z d t t mnt d d I C m, n, t m, n,, t med n M n ppp mnt N C Z ) Z m n t 3 2 d mnt mn T G m, n, t ) C Ct G n / 3 LT I m, n,, t mnt / 3 ( ) LT mn J m mn mn 2 m mnt m, n, t M, m, t, m, t N m n t J, m, t T Ch mn I mnt M N T m n t J PP mnt Z d G (8) 7. Model Parameters and data collecton Model parameters (data) can generally be dvded nto two general categores: Certan parameters: producton level of each product, purchase cost (prce), nventory cost, the number of on-tme delvery, values of the goal, and goal s mportance coeffcents. Uncertan parameters: capacty and transportaton cost. By adoptng ABC method and removng sngle source parts from the lst, 29 supplers were examned. As mentoned before, the model s mplemented for only two vehcle models and for one manufacturng plant (Tehran) only. The robust model was solved based on producton plan of three perodc (monthly). In order to determne the prorty of goals (obectves), we appled AHP technque. To do that, we collected the opnon of top managers and experts of the company through AHP 7

18 questonnare, and then analyzed them by usng Excel spreadsheets and Expert Choce software (see table ). The nconsstency rato s set equal to.4, although. s usually relable for ths ndex. [Please nsert Table about here.] 8. Computatonal Results After solvng the model, three followng solutons called Normal, Pessmstc and Optmstc were obtaned. Normal oluton: solvng robust-fuzzy model n normal mode, namely consderng nterval half-length of parameter of suppler capacty n fuzzy form. Pessmstc oluton: solvng robust-fuzzy model wth, consderng maxmum amount for nterval half-length of parameter of suppler capacty. Optmstc oluton: solvng robust-fuzzy model wth, consderng mnmum amount for nterval half-length of parameter of suppler capacty. Pessmstc and optmstc solutons can show the accuracy of robust-fuzzy model s results. Obvously, Normal soluton should be between soluton Pessmstc and Optmstc. We consder and 2 for dfferent values. Therefore, for each type of soluton (Normal, Pessmstc and Optmstc) models were solved. The role of protecton level parameter s usually set by model bulder, accordng to rsk preference of decson maker. Its value s equal the number of uncertan parameters that ther uncertantes should be consdered by the model. In other words, value of parameter s equal the number of uncertan parameters that are allowed to volate from ther nomnal value. For example, corresponds to an absolute conservatve manager. For more nformaton, refer to Bertsmas and m (24). 8.. Verfyng the model by usng Monte Carlo smulaton To measure the accuracy of model performance (qualty of robust solutons) and also to calculate the rsk, we ran a smulaton model by generatng uncertan parameters tmes randomly and wth dfferent determnstc parameters. In general, the total of 33 smulaton runs was 8

19 performed n order to determne the number of constrants whch are volated. In other words, for each protecton level the rato of the total number of volated constrants to total constrants was determned. The smulaton model was mplemented to test and valdate the result obtaned va the robust model. The outputs of smulaton model s the probablty of constrant volaton or rsk rate. In ths research, the role of smulaton and the relatonshp t wth mathematcal model s shown n ths form: Certan parameters Uncertan parameters Robust-Fuzzy Mathematcal Model The role: producng robust solutons olutons or value of varables mulaton The role: Generatng uncertan parameters Rsk rate or Probablty of constrants volaton In fact, managers could determne value of decson varables after selectng a desred rsk rate Results and analyss From Table 2and Fgure 6 t can be nferred that ncreasng the protecton level causes the level of constrant s satsfacton ( ) to drop. [Please nsert Table2 about here.] [Please nsert Fgure6 about here.] Fgure 7 shows that the robust-fuzzy model values (Normal soluton) s n balance compared wth two other solutons (though three of them begn from the same pont). On the other hand, t can be nferred that when the protecton level ncreases, graphs (Fgure7) are further away from each other. o, n these ponts t s clear that nterval s half-length are fuzzy. [Please nsert Fgure7 about here.] Table 2 and Fgure 7 show that the levels of rsk or level conservaton hghly affect the slope of obectve functon s value lne. Thus, the robust model, compared wth determnstc model,has more sgnfcant affects on rsk reducton. 9

20 In Table 3, columns a-c show the percentage of devaton from goal to 3 for 3 types of solutons, respectvely. [Please nsert Table3 about here.] In Table 4 columns a-c show the probablty of frst goal constrants volaton, probablty of capacty constrants volaton and the sum of probablty of volaton, respectvely. Table 4 ndcates the rsk rate (probablty of constrants volaton) based on ndcator. [Please nsert Table4 about here.] From Table 2, t can be seen that ncreasng the protecton levels worsens the obectve functon ( ) decreases the probablty of volaton. In fact, as the level of protecton ncreases, the model varable values wll be chosen more strngently n allowed nterval so that lessen the probablty of lmtaton volaton and fnally worsen the obectve functon s value. Ths could be a reason why robust modelng and model s performance s accurate. The numbers, derved from Table 4, have been resulted from smulaton and show that ncreasng protecton level decreases the probablty of volaton. Ths process ndcates the accuracy of robust model s performance and smulaton. ee Fgure 8 and 9: [Please nsert Fgure 8 about here.] [Please nsert Fgure9about here.] Frst goal constrant has 387 uncertan parameters and ncludes protecton level of. The other constrants ncludng uncertan parameters consst of 387 capacty constrants wth protecton level of 2. [Please nsert Fgure about here.] 2

21 [Please nsert Fgureabout here.] mulaton results ndcated that only some of the capacty constrants (up to maxmumof73) are possble to be volated. The reason s that some supplers capacty are more than needed or the model has consdered a share whch s less than exsted capacty so the consdered uncertanty for some parameters of suppler s cannot affect them. Accordng to ths explanaton, two ndcators were consdered for rsk calculaton. Fgures and demonstrate the trade-off between sum of devatons and probablty of constrants volaton based on ndcator values and 2, respectvely. Indcator.To dvde the total number of volated states on the total number of states. Indcator2.To dvde the total number of volated states on the total number of possble states whch are dependent on the maxmum number of volable states. 9. Concluson In ths study, an nnovatve robust-fuzzy methodology was developed and appled to a real case study of suppler selecton problem wthn automoble ndustry wth multple uncertantes. Ths approach s based on an ntegraton of technques of robust optmzaton developed by Bertsmas and ms (24) approach and fuzzy programmng (Werners (978b) approach ( to handle two modes of uncertanty. In robust optmzaton models, uncertanty of parameters s modeled as random varables that take value wthn a symmetrcal nterval. In fact, our nnovatve robustfuzzy method s an mproved verson of robust optmzaton model. Ths approach s properly responsve to mental ambguty of decson maker about nterval half-length and also provdes necessary flexblty for decson makers. In other words, ths method s able to be responsve to both types (modes) of uncertanty. The performance of the proposed model results n comparson wth the results obtaned va smulaton technques are proved to be relable under varous condtons. In other words, the proposed model can properly manage a varety of uncertan condtons and provde useful and relable nformaton for flexble decson makng processes. The flexblty of the decson makng process s shown n Fgures 7, 8 and 9. 2

22 Although ths study s the frst attempt by ths approach for suppler selecton decson makng problem, the results suggest that the approach also can be extended from both theoretcal and practcal ponts of vew. In addton, the followng manageral mplcatons are of great mportance: As stated n ecton, 2, 3, the ssue of uncertanty s very mportant n suppler selecton problem because of nherent uncertanty assocated wth varous factors such as capacty and transportaton cost. Due to the hgh volume of requred parts n ths case study and consderng multple crtera and also the condton of uncertanty, choosng sutable supplers s of great mportance. In general, robust optmzaton concept and especally nnovatve robust-fuzzy method are easy to understand and mplement, especally by automotve ndustry managers. Robust optmzaton approach s especally more practcal and applcable n comparson wth stochastc programmng models. As explaned before, n robust optmzaton approach, uncertan data takes value wthn an nterval. In some real cases such as our case study, t s dffcult for managers to determne the exact nterval length. Furthermore, n these stuatons the decson maker suggests ambguous nterval half-length. In fact, to overcome ths problem, we developed an nnovatve approach and proposed the nterval half-length n term of fuzzy number. Moreover, the new method can help managers to acheve flexblty n decson makng n case of ambguous nterval half-length. Consequently, the nnovatve method can be of great nterest to managers as well as academcs. In addton, the followng mplcatons can be nferred from the mathematcal model and computatonal results:. The mathematcal model consders multple crtera n form of three obectves (goals). Ths pont s more matched wth real world. Ths model also helps automotve ndustry supply chan managers to make ntegrated and comprehensve decsons consderng varous ssues such as multple products, multple plants, multple supplers and multple decson-makng perods. 2. Computatonal results demonstrate that managers wth a desred rsk level can make a sutable decson. In the other word, as Table 2, 3 and 4, ndcate there are optons to select. An optmal combnaton of supplers can exst for each selecton opton. 22

23 3. The mplementaton of the proposed method does not requre extensve and complex hstorcal data but ust determnng an nterval for uncertan parameters by experts s enough. 4. In ths context, the mproved decson makng s related to robustness of solutons (see fg). In uncertan envronment, a proper decson s equvalent to a robust decson. In the proposed approach, the man decson s the value of supply rate of part m from suppler n n perod t for plant (value of mnt ). A robust value for ths varable s equvalent to robust decson. Managers could determne ths value after selectng a desred rsk rate. 5. Due to the nature of auto ndustry, the logstc manger has to manage thousands of parts and components. Obvously, the lack of even one part can stop the producton lne and mpose an ncredble cost as well some ntangble costs, such as customer dssatsfacton. From ths pont of vew, ths ndustry s much more senstve to approprate suppler selecton n comparson wth others. On the other hand, nherent uncertanty of suppler relatonshp s a crucal factor, especally n the developng countres. Therefore, ths ndustry needs a practcal, effcent and relable methods to manage the logstc of parts and components. The contrbuton of ths research n vew pont of value added to companes s presented as follow: The nnovatve model s applcable and effcent for manufacturng companes that the ssue of uncertanty s hghly sgnfcant for them. In other word, due to the exstence of many mportant sources of uncertantes n busness envronment, relable data cannot be found, the classcal supply selecton models are not applcable any more. Thus, our proposed method s applcable for the companes wth uncertan nput data. Although our study was motvated for a large auto ndustry, ths approach can be mplemented for other ndustres such as fnancal, bankng, nsurance and smlar cases. The maor lmtaton of the research s how to determne the membershp functon of fuzzy numbers. Furthermore, the model s not qute applcable for "make to order" manufacturng systems. Future research can extend ths approach to cover other systems. Other ssue n future research s usng artfcal ntellgence technques nstead of robust optmzaton methods. References 23

24 [] L, L. and Zabnsky, Z.B. Incorporatng uncertanty n to a suppler selecton problem, Int. J. Prod. Econ., 34(2), pp (2). [2] Nam, -H., Vtton, J. and Kurata, H. Robust supply base management: Determnng the optmal number of supplers utlzed by contractors, Int. J. Prod. Econ., 34(2), pp (2). [3] Hasan, A. Zegord,.H. and Nkbakhsh, E. Robust closed-loop supply chan network desgn for pershable goods n agle manufacturng under uncertanty, Int. J. Prod. Res., 5(6), pp. 2 (2). [4] Lu, Y. and Peng, B. Achevng robustness obectves wthn a supply chan by means of relablty allocaton, n: Proceedngs of the 29 Internatonal ymposum on Web Informaton ystems and Applcatons (WIA 9) Nanchang, P. R. Chna, pp. 2-23, (29). [5] Amd, A., Ghodsypour,.H., and O Bren, C. Fuzzy multobectve lnear model for suppler selecton n a supply chan, Int. J. Prod. Econ., 4 (2), pp (26). [6] Benton, W.C., Quantty dscount decson under condtons of multple tems, multple supplers and resource lmtaton, Int. J. Prod. Econ., 27 (99), pp (26). [7] Ghodsypour,.H. and O Bren, C.. A decson support system for reducng the number of supplers and managng the suppler partnershp n A JIT/ TQM envronment, The proceedng of 3rd nternatonal symposum on logstcs, Unversty of Padua, Italy, (997). [8] Ghodsypour,.H. and O Bren, C. A decson support system for suppler selecton usng an ntegrated analytc herarchy process and lnear programmng, Int. J. Prod. Econ., (-3), pp (998). [9] Ghodsypour,.H. and O Bren, C. The total cost of logstcs n suppler selecton, under condtons of multple sourcng, multple crtera and capacty constrant, Int. J. Prod. Econ., 73 (), pp (2). [] Basnet, C. and Leung, J.M.Y. Inventory lot-szng wth suppler selecton, Comput. Oper. Res., 32 () (25), pp. 4. [] Ustun, O. and AktarDemrtas, E. An ntegrated mult-obectve decson-makng process for mult-perod lotszng wth suppler selecton, Omega., 36, pp (28). [2] Mendoza, A. and Ventura, J.A. Analytcal models for suppler selecton and order quantty allocaton, Appl. Math. Model., 36, pp (22). [3] Kumar, M., Vrat, P. and hankar, R. A fuzzy goal programmng approach for vendor selecton problem n a supply chan, Comput. Ind. Eng., 46 (), pp (24). [4] Chen, C.-T., Ln, C.-T. and Huang,.-F. A fuzzy approach for suppler evaluaton and selecton n supply chan management, Int. J. Prod. Econ., 2 (2), pp (26). [5] Klc, H.. An ntegrated approach for suppler selecton n mult-tem/mult-suppler envronment, Appl. Math. Model., 37, pp (23). [6] Nazar-hrkoh,., hakour, H., Javad, B. and Keramat, A. uppler selecton and order allocaton problem usng a two-phase fuzzy mult-obectve lnear programmng, Appl. Math. Model., 37, pp (23). 24

25 [7] Roshandel, J. Mr-Narges,.., and Hatam-hrkouh, L. Evaluatng and selectng the suppler n detergent producton ndustry usng herarchcal fuzzy TOPI, Appl. Math. Model., 37, pp. 7 8(23). [8] ouf Neyestan, M., Jola, F. and Golmakan, H.R. An evolutonary algorthm for suppler order allocaton wth fuzzy parameters consderng lnear and volume dscount, centa Iranca E, 22(3), pp. 3-4 (25). [9] Mahmoud, A., ad-nezhad and Maku, A. An extended fuzzy VIKOR for group decson-makng based on fuzzy dstance to suppler selecton, centa Iranca E, 23(4), pp (26). [2] Kaslngam, R.G. and Lee, C.P. electon of vendors a mxed nteger programmng approach, Comput. Ind. Eng., 3, pp (996). [2] Feng, C-X., Wang, J. and Wang, J.. An optmzaton model for concurrent selecton of tolerances and supplers, Comput. Ind. Eng., 4(-2), pp (2). [22] Bonser, J.. and Wu,.D. Procurement Plannng to Mantan Both hort-term Adaptveness and Long-Term Perspectve, Manage. c., 47, pp (2). [23] eshadr,., Chatteree, K. and Llen, GL. Multple source procurement compettons, Market. c.,, pp (99). [24] Bertsmas, D. and m, M. Robust dscrete optmzaton and network flows, Math. Programmng. er. B. 98, pp (23). [25] Van Landeghem, H. and Vanmaele, H. Robust plannng: a new paradgm for demand chan plannng. J. Oper. Manage., 2, pp (22). [26] Ranbar Tezen, F., Mohammad, M., Pasanddeh,.H.R. and Nour Koupae, M. An ntegrated model for suppler locaton-selecton and order allocaton under capacty constrants n an uncertan envronment, centa Iranca E, 23(6), pp (26). [27] oyster, A.L. Convex programmng wth set-nclusve constrants and applcatons to nexact lnear programmng, Oper. Res., 2, pp (973). [28] Ben-Tal, A. and Nemrovsk, A. Robust convex optmzaton, Math, Oper. Res., 23, pp (998). [29] Ben-Tal, A. and Nemrovsk, A. Robust solutons of uncertan lnear programs, Oper. Res. Lett., 25, pp. -3 (999). [3] Ben-Tal, A. and Nemrovsk, A. Robust solutons of lnear programmng problems contamnated wth uncertan data, Math. Programmng, er. A., 88, pp (2). [3] El Ghaou, L., Oustry, F. and Lebret, H. Robust solutons to uncertan semdefnte programs, IAM. J. Optm., 9, pp (998). [32] El-Ghaou, L. and Lebret, H. Robust solutons to least-square problems to uncertan data matrces, IAM. J. Matrx Anal. Appl., 8 (4), pp )997(. [33] Bertsmas, D. and m, M. The Prce of Robustness. Oper Rese, 52(), pp (24). [34] Werners, B. Interactve multple obectve programmng subect to flexble constrants, Euro. J. Oper Rese., 3, pp (987). [35] Werners, B. An nteractve Fuzzy programmng system, Fuzzy ets and ystems., 23, pp (987). 25

26 Masood Rabeh receved hs PhD from Tarbat Modaress Unversty, Iran. He s an assstant professor n Department of Industral Management at hahd Behesht Unversty, Iran. Hs research nterests nclude Operatons Research, ystem Dynamcs & ystem Thnkng, upply Chan Management and Proect Management. Mohammad Modarres receved hs PhD from Unversty of Calforna, Los Angles (UCLA), UA. He s professor n Department of Industral Engneerng at harf Unversty of Technology, Iran. Hs research nterests nclude Operatons research, Mathematcal Modelng and tochastc Modelng. Adel Azar receved hs PhD from Unversty of Tehran, Iran. He s professor n Department of Industral Management at Tarbat Modaress Unversty, Iran. Hs research nterests nclude Fuzzy Management cence, Manageral and Economc ystems Modelng, Busness Process Reengneerng, ystem Engneerng, Performance Management and Performance Based Budgetng. 26

27 Fgures: upply Chan Cost Determnstc plannng Robust plannng best value optmal Uncertanty range Parameter (e.g. demand) value Fgure.The effect of robust plannng on total supply chan cost [23]. CX Z ( x) Z Z f f f Z CX Z CX Z Z CX Fgure 2. Membershp functon of the obectve functon. b p ( AX ) ( x) p f f f ( AX ) b b ( AX ) ( AX ) b p b p 27

28 Fgure3. Membershp functon of a constrant. 28

29 29 mn mn mn mn ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) ( med p z b f b p z b f b b b p z b p z f x Fgure 4. Membershp functon of ~ˆb

30 Problem Defnton Expert Opnons Mathematcal Modelng for problem Defnng varables Defnng parameters Modelng obectve functons Modelng constrants Avalable Documents Data Collecton Determne uncertan parameters Determne type of uncertan parameters Convert the model to Robust-Fuzzy model Robust-Fuzzy approach olvng Robust-Fuzzy model mulatng results Evaluate qualty of result Compare wth pessmstc and optmstc mode Present results Fgure 5. teps of mplementng the model of case 3

31 Model's obectve functon protecton level Fgure 6. Behavor of obectve functon wth respect to ( ) n Normal oluton. um of devatons from goals Optmstc oluton Normal soluton Pessmstc soluton Fgure 7. Behavor of sum of devatons of goals 3

32 probablty of constrants volaton Pessmstc oluton Normal soluton Optmstc oluton Fgure 8. The probablty of constrants volaton s trend n dfferent modes, based on ndcator probablty of constrants volaton Pessmstc oluton Normal soluton Optmstc oluton Fgure 9. The probablty of constrants volaton s trend n dfferent modes, based on ndcator 2. 32

33 sum of devatons of goals rsk Fgure. Trade-off between sum of devatons and probablty of constrants volaton based on ndcator sum of devatons of goals rsk Fgure. Trade-off between sum of devatons and probablty of constrants volaton based on ndcator2. 33

34 Tables: Table : mportance coeffcents of goals obectves (goals) Frst econd thrd mportance coeffcents of goals Table 2: the obectve functon values for Normal, Pessmstc and Optmstc solutons Normal oluton Pessmstc oluton Optmstc oluton protecton levels sum of devatons 2 of goals sum of devatons of goals sum of devatons of goals

35 Table3: the percentage of devaton from each goal. protecton levels Normal soluton Pessmstc soluton Optmstc soluton a b c a b c a b c Table 4: Probablty of constrants volaton (rsk rate) n dfferent modes and based on ndcator. protecton levels Normal soluton Pessmstc soluton Optmstc soluton a b c a b c a b c