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  • A fuzzy goal programming approach for vendor selectionproblem in a supply chain

    Manoj Kumara, Prem Vratb, R. Shankarc,*

    aDepartment of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, IndiabIndian Institute of Technology Roorkee, Roorkee 247 667, Uttaranchal, India

    cDepartment of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

    Accepted 19 September 2003


    A fuzzy goal programming approach is applied in this paper for solving the vendor selection problem withmultiple objectives, in which some of the parameters are fuzzy in nature. A vendor selection problem has beenformulated as a fuzzy mixed integer goal programming vendor selection problem that includes three primarygoals: minimizing the net cost, minimizing the net rejections, and minimizing the net late deliveries subject torealistic constraints regarding buyer's demand, vendors' capacity, vendors' quota flexibility, purchase value ofitems, budget allocation to individual vendor, etc. An illustration with a data set from a realistic situation isincluded to demonstrate the effectiveness of the proposed model. The proposed approach has the capability tohandle realistic situations in a fuzzy environment and provides a better decision tool for the vendor selectiondecision in a supply chain.

    Keywords: Vendor selection problem; Supply chain; Multi-criterion decision; Fuzzy sets

    1. Introduction

    The objective of managing the supply chain is to synchronize the requirements of the customers with theflow of materials from suppliers in order to strike a balance between what are often seen as conflicting goalsof high customer service, low inventory, and low unit cost (Stevens, 1989). The vendor selection problem(VSP) deals with issues related to the selection of right vendors and their quota allocations. In designing asupply chain, a decision maker must consider decisions regarding the selection of the right vendors and their

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    quota allocation. The choice of the right vendor is a crucial decision with wide ranging implications in asupply chain. Vendors play an importantrole in achieving the objectives of the supply management. Vendorsenhance customer satisfaction in a value chain. Hence, strategic partnership with better performing vendorsshould be integrated within the supply chain for improving the performance in many directions includingreducing costs by eliminating wastages, continuously improving quality to achieve zero defects, improvingflexibility to meet the needs of the end-customers, reducing lead time at different stages of the supply chain,etc. In designing a supply chain, decision makers are attempting to involve strategic alliances with thepotential vendors. Hence, vendor selection is a vital strategic issue for evolving an effective supply chain andthe right vendors play a significant role in deciding the overall performance. The VSP is a complex problemdue to several reasons. By nature, the VSP is a multi-criterion decision making problem. Individual vendormay perform differently on different criteria. A supply chain decision faces many constraints, some of theseare related to vendors' internal policy and externally imposed system requirements.

    In such decision making situations, high degree of fuzziness and uncertainties are involved in the dataset. Fuzzy set theory provides a framework for handling the uncertainties of this type. Zadeh (1965)initiated the fuzzy set theory. Bellman and Zadeh (1970) presented some applications of fuzzy theories tothe various decision-making processes in a fuzzy environment. Zimmerman (1976, 1978) presented afuzzy optimization technique to linear programming (LP) problem with single and multiple objectives.Since then the fuzzy set theory has been applied to formulate and solve the problems in various areassuch as artificial intelligence, image processing, robotics, pattern recognition, etc. (Hannan, 1981;Yager, 1977). Narsimhan (1980) proposed a fuzzy goal programming (FGP) technique to specifyimprecise aspiration levels of the fuzzy goals. Yang, Ignizio, and Kim (1991) formulated the FGP withnonlinear membership functions. The FGP technique has been applied to various other fields such asstructural optimization (Rao, Sundaraju, Prakash, & Balakrishna, 1992), agricultural planning (Sinha,Rao, & Mangaraj, 1988), forestry (Pickens & Hof, 1991), cellular manufacturing system design(Shankar & Vrat, 1999), etc. In a vendor selection decision process the required information is generallyuncertain and different types of fuzziness exist at the decision stages. In this paper, the fuzzy mixedinteger goal programming vendor selection problem (f-MIGP_VSP) formulation is used to incorporatethe imprecise aspiration levels of the goals.

    This paper is further organized as follows. Section 2 presents a brief literature review of the existingquantitative approaches related to the VSP. Section 3 describes the f-MIGP_VSP formulation byconsidering three important fuzzy goals, viz. minimizing the net cost, minimizing the net rejections, andminimizing the net late deliveries. The corresponding equivalent crisp transformation (c-MIGP_VSP) isalso provided. In Section 4, an illustration with a data set from a case company is included todemonstrate the effectiveness of the proposed approach. Finally, we provide conclusions regarding theeffectiveness of the proposed approach in Section 5.

    2. Literature review

    Linear weighting method proposed by Wind and Robinson (1968) for vendor selection decision is themost common way of rating different vendors on the performance criteria for their quota allocations.Gregory (1986) linked this approach to a matrix representation of data and then rated the differentvendors for their quota allocations. Monozka and Trecha (1988) proposed multiple criteria vendorservice factor ratings and an overall supplier performance index. Mathematical programming models

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    approach the VSP in a more effective manner than the linear weighting model due to their ability tooptimize the explicitly stated objective. The literature survey reveals that in mathematical programmingmodels, LP, mixed integer programming (MIP) and goal programming (GP) are the commonly usedtechniques (Moore & Fearon, 1972; Oliveria & Lourenco, 2002; Sharma, Benton, & Srivastava, 1989).Anthony and Buff a (1977) formulated the vendor selection decision as a LP problem to minimize thetotal purchasing and storage costs. Pan (1989) developed a single item LP model to minimize theaggregate price under constraints of quality, service level and lead-time. Bendor, Brown, Issac, andShapiro (1985) proposed a MIP approach with the objective of minimizing purchasing, inventory andtransportation related costs without any specific mathematical formulation and demonstrated it throughselecting the vendors at IBM. Sharma et al. (1989) proposed a GP formulation for attaining goalspertaining to price, quality and lead-time under demand and budget constraints. Buffa and Jackson(1983) also proposed the use of GP for price, quality and delivery objectives. Liu, Ding, and Lall (2000)and Weber, Current, & Desai (2000) presented a data envelopment analysis method for a VSP withmultiple objectives. Handfield, Walton, Sroufe, and Melnyk (2002) and Narsimhan (1983) used theanalytical hierarchical process to generate weights for VSP. Ghodsypour and O'Brien (1998) developeda decision support system by integrating the analytical hierarchy process with linear programming.Ronen and Trietsch (1988) incorporated uncertainty and proposed a statistical model for VSP. Kumar,Vrat, and Shankar (2002) analyzed the effect of information uncertainty in the VSP with intervalobjective coefficients. Feng, Wang, and Wang (2001) presented a stochastic integer programming modelfor simultaneous selection of tolerances and suppliers based on the quality loss function and processcapability index.

    The deterministic models proposed in literature suffer from the limitation in a real VSP due to thefact that a decision maker does not have sufficient information related to the different criteria. Thesedata are typically fuzzy in nature. For a VSP, values of many criteria are expressed in imprecise termslike 'very poor in late deliveries', 'hardly any rejected items', etc. All the above-referred deterministicmethods lack the capability to handle the linguistic vagueness of fuzzy type. The optimal resultsobtained from these deterministic formulations may not serve the real purpose of modeling theproblem.

    A consideration to incorporate information vagueness in the VSP has not been found in the existingliterature. This paper presents a f-MIGP_VSP formulation to capture the uncertainty related to theVSP.

    3. Formulation of fuzzy mixed integer goal programming vendor selection problem

    The VSP is typically a multi-criterion decision making problem. The following assumptions, indexset, decision variable and parameters are considered.


    (i) Only one item is purchased from one vendor.(ii) Quantity discounts are not taken into consideration.(iii) No shortage of the item is allowed for any vendor.(iv) Demand of the item is constant and known with certainty.

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    Index set

    i index for vendor, for all i = 1; 2 ; . ; , nj index for inequality constraints, for all j = 1,2,. . . , /k index for equality constraints, for all k = 1 ; 2 ; ...,K

    I index for objectives, for all l = 1; 2 ; . ; , L

    Decision variable

    xi order quantity for the vendor i


    D aggregate demand of the