A fuzzy goal programming approach for vendor selection problem in a supply chain

Transcription

1 A fuzzy goal programming approach for vendor selection problem in a supply chain Manoj Kumar a, Prem Vrat b, R. Shankar c, * adepartment of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi , India b Indian Institute of Technology Roorkee, Roorkee , Uttaranchal, India c Department of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi , India Accepted 19 September 2003 Abstract A fuzzy goal programming approach is applied in this paper for solving the vendor selection problem with multiple objectives, in which some of the parameters are fuzzy in nature. A vendor selection problem has been formulated as a fuzzy mixed integer goal programming vendor selection problem that includes three primary goals: minimizing the net cost, minimizing the net rejections, and minimizing the net late deliveries subject to realistic constraints regarding buyer's demand, vendors' capacity, vendors' quota flexibility, purchase value of items, budget allocation to individual vendor, etc. An illustration with a data set from a realistic situation is included to demonstrate the effectiveness of the proposed model. The proposed approach has the capability to handle realistic situations in a fuzzy environment and provides a better decision tool for the vendor selection decision 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 the flow of materials from suppliers in order to strike a balance between what are often seen as conflicting goals of 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 a supply chain, a decision maker must consider decisions regarding the selection of the right vendors and their

2 70 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) quota allocation. The choice of the right vendor is a crucial decision with wide ranging implications in a supply chain. Vendors play an importantrole in achieving the objectives of the supply management. Vendors enhance customer satisfaction in a value chain. Hence, strategic partnership with better performing vendors should be integrated within the supply chain for improving the performance in many directions including reducing costs by eliminating wastages, continuously improving quality to achieve zero defects, improving flexibility 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 the potential vendors. Hence, vendor selection is a vital strategic issue for evolving an effective supply chain and the right vendors play a significant role in deciding the overall performance. The VSP is a complex problem due to several reasons. By nature, the VSP is a multi-criterion decision making problem. Individual vendor may perform differently on different criteria. A supply chain decision faces many constraints, some of these are 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 data set. 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 to the various decision-making processes in a fuzzy environment. Zimmerman (1976, 1978) presented a fuzzy 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 areas such as artificial intelligence, image processing, robotics, pattern recognition, etc. (Hannan, 1981; Yager, 1977). Narsimhan (1980) proposed a fuzzy goal programming (FGP) technique to specify imprecise aspiration levels of the fuzzy goals. Yang, Ignizio, and Kim (1991) formulated the FGP with nonlinear membership functions. The FGP technique has been applied to various other fields such as structural 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 generally uncertain and different types of fuzziness exist at the decision stages. In this paper, the fuzzy mixed integer goal programming vendor selection problem (f-migp_vsp) formulation is used to incorporate the imprecise aspiration levels of the goals. This paper is further organized as follows. Section 2 presents a brief literature review of the existing quantitative approaches related to the VSP. Section 3 describes the f-migp_vsp formulation by considering three important fuzzy goals, viz. minimizing the net cost, minimizing the net rejections, and minimizing the net late deliveries. The corresponding equivalent crisp transformation (c-migp_vsp) is also provided. In Section 4, an illustration with a data set from a case company is included to demonstrate the effectiveness of the proposed approach. Finally, we provide conclusions regarding the effectiveness of the proposed approach in Section Literature review Linear weighting method proposed by Wind and Robinson (1968) for vendor selection decision is the most 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 different vendors for their quota allocations. Monozka and Trecha (1988) proposed multiple criteria vendor service factor ratings and an overall supplier performance index. Mathematical programming models

3 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) approach the VSP in a more effective manner than the linear weighting model due to their ability to optimize the explicitly stated objective. The literature survey reveals that in mathematical programming models, LP, mixed integer programming (MIP) and goal programming (GP) are the commonly used techniques (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 the total purchasing and storage costs. Pan (1989) developed a single item LP model to minimize the aggregate price under constraints of quality, service level and lead-time. Bendor, Brown, Issac, and Shapiro (1985) proposed a MIP approach with the objective of minimizing purchasing, inventory and transportation related costs without any specific mathematical formulation and demonstrated it through selecting the vendors at IBM. Sharma et al. (1989) proposed a GP formulation for attaining goals pertaining 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 with multiple objectives. Handfield, Walton, Sroufe, and Melnyk (2002) and Narsimhan (1983) used the analytical hierarchical process to generate weights for VSP. Ghodsypour and O'Brien (1998) developed a 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 interval objective coefficients. Feng, Wang, and Wang (2001) presented a stochastic integer programming model for simultaneous selection of tolerances and suppliers based on the quality loss function and process capability index. The deterministic models proposed in literature suffer from the limitation in a real VSP due to the fact that a decision maker does not have sufficient information related to the different criteria. These data are typically fuzzy in nature. For a VSP, values of many criteria are expressed in imprecise terms like 'very poor in late deliveries', 'hardly any rejected items', etc. All the above-referred deterministic methods lack the capability to handle the linguistic vagueness of fuzzy type. The optimal results obtained from these deterministic formulations may not serve the real purpose of modeling the problem. A consideration to incorporate information vagueness in the VSP has not been found in the existing literature. This paper presents a f-migp_vsp formulation to capture the uncertainty related to the VSP. 3. Formulation of fuzzy mixed integer goal programming vendor selection problem The VSP is typically a multi-criterion decision making problem. The following assumptions, index set, decision variable and parameters are considered. Assumptions (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.

4 72 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Index set i index for vendor, for all i = 1 ; 2 ;. ;, n j 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 x i order quantity for the vendor i Parameters D n pi q i di U i fi F r i P B i aggregate demand of the item over a fixed planning period number of vendors competing for selection price of a unit item of the ordered quantity x i from the vendor i percentage of the rejected units delivered by the vendor i percentage of the late delivered units by the vendor i upper bound of the quantity available with vendor i vendor quota flexibility for vendor i lower bound of flexibility in supply quota that a vendor should have vendor rating value for vendor i lower bound to total purchasing value that a vendor should have budget allocated to each vendor 3.1. Model formulation A classical multiple-objective mixed integer-programming problem can be written as follows: Maximize Zjfo) = [Z 1 (x i ),Z 2 (x i ),...,Z L (x;)], 1=1,2,...,L subject to gjðxiþ # aj ; j= 1,2,...,J = b k, k=l,2,...,k (1) x i $ 0 and integer In formulation (1), x i are n decision variables, Z 1 (x i ),Z 2 (x i ),...,Z L (x i ), are L distinct objective functions, gj are the inequality constraints and h k are the equality constraints. a j and b k are the right hand side constants for inequality and equality relationships, respectively.

5 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) The mixed integer programming vendor selection problem (MIP_VSP) formulation for three objectives and a set of system and policy constraints can be formulated as follows: subject to n Minimize Z 1 V/?;(x;) (2) i=\ n Minimize Z 2 = X #,(*,) (3) i=\ n Minimize Z 3 = X d t {x^) (4)!=1 i=\ xi#ui ; for all i= 1 ; 2 nð 6Þ i=\ <^) ^ F (? ) pið xi Þ#Bi ; for all /= 1,2,...,n ð9þ xi $ 0 and integer ð10þ Objective function (2) minimizes the net cost for all the items. Objective function (3) minimizes the net number of rejected items from the vendors. Objective function (4) minimizes the net number of late delivered items from the vendors. Constraint (5) puts restrictions due to the overall demand of items. Constraint (6) puts restrictions due to the maximum capacity of the vendors. Constraint (7) incorporates flexibility needed with the vendors' quota. Constraint (8) incorporates total purchase value constraint for all the ordered quantities. Constraint (9) puts restrictions on budget amount allocated to the vendors for supplying the items. In real life situations, a lot of informational inputs required for selecting vendors are imprecise, vague or uncertain. This may be true for objectives as well as parameters. Such imprecise information can be modeled through representations.

6 74 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Fuzzy mixed integer goal programming model When vague information related to the objectives are present then the problem can be formulated as a fuzzy goal-programming problem. A typical fuzzy mixed integer goal programming problem (f-migp) formulation can be stated as follows: Find x i i = 1 ; 2 ;...,n where to satisfy ZlðxiÞ ø Z ~ I = 1,2,...,L gjixd^aj,./=1,2,...,/ (11) h k (x t ) = b k; k=l,2,...,k x i $ 0 and integer ; i 1 ; 2 ;..., n Zlðx i Þ is the lth goal constraint, gjðx i Þ is thejth inequality constraint, h k ðx i Þ is the kth equality constraint, Z~l is the target value of the lth goal, aj is the available resource of inequality constraint j, b k is the available resource of the equality constraint k : In the formulation (11), the symbol' ø ' indicates the fuzziness of the goal. It represents the linguistic term 'about' and it means that Zlðx i Þ should be in the vicinity of the aspiration Z ~l The lth fuzzy goal Zlðx i Þ = % signifies that the decision maker would be satisfied even for values slightly greater than (or lesser than) % up to a stated deviations signified by tolerance limit. Thejth system constraint gjðx i Þ # a j and the kth system constraint h k (x t ) = b k are assumed to be crisp Membership function According to Zadeh (1965), fuzzy set theory is based on the extension of the classical definition of a set. In classical set theory, each element of a universe X either belongs to a set A or not, whereas in fuzzy set theory an element belongs to a set A with a certain degree of membership. Definition 1: A fuzzy set A in X is defined by: A-{(x,/x A (x))/xgx} (12) where m A ðxþ : X!* 0 ; 1] is called the membership function of A and m A ðxþ is the degree of membership to which x belongs to A : The fuzzy set A in X is thus uniquely characterized by its membership function m A ðxþ ; which associates with each point in X ; a nonnegative real number whose value is finite and usually finds a place in the interval [0,1], with the value of m A ðxþ at x representing the 'grade of membership' of x in A : Thus nearer the value of m A ðxþ to 1, higher the grade of 'belongingness' of x in A : Definition 2: Union of two fuzzy sets A and B with respective membership functions m A ðxþ and m B ðxþ is defined as a fuzzy set C whose membership function is as follows: /x c (x) = max[/x A (x), /JL B (X)], X G X (13)

7 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Definition 3: Intersection of two fuzzy sets A and B with respective membership functions m A ðxþ and m B ð x Þ is defined as a fuzzy set C whose membership function is as follows: fi c (x) = mm[fi A (x), fi B (x)], X ð14þ Using approach of Yang et al. (1991), the triangular membership function /^[z,fe)] f r the lth fuzzy goal in the f-migp formulation has been considered. This type of membership function is adopted due to ease in defining the maximum and minimum limit of deviations of the each fuzzy goal from its central value. The triangular membership function /^[z,fe)] is shown in Fig. 1 and is defined as follows: if < Z ~ ð15þ In relationships (15) 0 ; otherwise Z~ l is the aspiration level of the lth fuzzy goal, Zlm in is a minimum limit of deviation to Z ~l Zl max is a maximum limit of deviation to Z ~l /Ji>[Zi( Xi )] is a strictly monotonically decreasing (or increasing) continuous function Solution approach Using approach of Bellman and Zadeh (1970), a fuzzy decision is a fuzzy set and is obtained by the intersection of the all the fuzzy sets representing the fuzzy objectives and all the fuzzy sets representing 0 Z, 1 1 Z l Fig. 1. Membership function of Z l ðx i Þ :

8 76 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) fuzzy constraints. The membership function of the fuzzy decision is given by = /x z (x) 0 /x c (x) = min[/x z (x); /x c (x)] (16) where m Z ðxþ and m C ðxþ represent the membership functions of all the fuzzy objectives and fuzzy constraints, respectively. The fuzzy decision of all the fuzzy sets Z l for representing I 1 ; 2 ;...,L fuzzy goals and of all the fuzzy sets C m for representing fuzzy constraints m 1 ; 2 ;...,M may be given as: ixt(i) = I O u 7 (x) I Pi I O u r (x) I = min min «7 (x), min u r (x) (17) l \'=i / U=i / \j=\x-,l ' m=\x-m m \ An optimum element means selecting that element ðxpþ which has the highest degree of membership value to the fuzzy decision set: fi s (x*) = max fi s (x) = max min min /x Z; (x), min /x Cm (x) ð18þ If the goals are described by the membership function /x^*;) on X = {g/^o ^ aj,h k (x t ) b k; x i $ 0} then the membership function of the optimal solution ðxpþ is given by: fi s (x*) = max fi s (x) = max min min /x Zife),..., fi Zl(Xi) ð19þ 3.5. Crisp mathematical formulation Using approach of Yang et al. (1991), which is based on a piecewise linear approximation with the min-operator for aggregating the fuzzy goals, the f-migp formulation may be solved to determine the decision set, and then by maximization of the set. The fuzzy goals are defined by using a triangular membership function as represented in Eq. (15). Once the membership functions of the fuzzy objectives ZlðxiÞ are known, the fuzzy optimization problem (f-migp) formulation is transformed into an equivalent crisp formulation (c-migp) for optimization. An equivalent crisp mathematical programming (c-migp) formulation is given by: max l s.t. A < Pzitxi)' f r a U I = 1>2,..., gjðxiþ # aj ; j= 1,2,...,/ ð20þ h k (x t ) = b k, x i $ 0 and integer k=\,2,...k

9 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Application off-migp model for solving VSP The ambiguity of the decision situation due to imprecise information concerning the minimization of three objectives related to the net cost, the net rejections and the net late deliveries is captured and a f-migp_vsp formulation is presented as follows: fe) = Zx (21) $(*,) = Z 2 (22) i=\ = Z, (23) i=\ i#ui ; for all i = 1 ; 2 n ð25þ i=\ xiþ $ F ð26þ i=\ riðxiþ$p ð27þ pið xi Þ #Bi ; for all i = 1 ; 2 ;. ;, n ð28þ xi $ 0 and integer ; for all i = 1 ; 2 ;. ;, n ð29þ 4. An illustration The effectiveness of the FGP technique for the VSP, presented in this paper is demonstrated through a data set represented in Table 1. The data relates to a realistic situation of a manufacturing sector dealing with auto parts. The adopted situation can easily be extended to any other industry. The f-migp_vsp formulation is developed from requirements espoused by the situation during the initial stages of implementing a formal program for better management of its supply chain which subsequently undertook a vendor certification plan for its purchased items. Those vendors who successfully passed the screening processes were eligible for procurement. Four established vendors for a projected demand of the part had been screened for supplying this purchased item. A f-migp_vsp model is developed for the selection and the quota allocations of the vendors from a list of four potential vendors under uncertain environments.

10 78 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Table 1 Vendor source data of the illustrative example Vendor number pi ($) * (%) 4 (%) Ui (units) fi Bi ($) ,000 15, , ,000 35, The objective functions and constraint sets reflect the procurement requirements for a purchased item in the supply chain. The three objectives, viz. minimizing the net cost, the net rejections, and the net late deliveries have been considered subject to few practical constraints regarding demand of the item, vendors' capacity limitations, vendors' budget allocations, etc. We have considered a typical situation faced by a firm. The vendor profiles shown in Table 1 represent the data set for the price quoted (p i in $) the percentage rejections ðq i Þ ; the percentage late deliveries ðd i Þ ; vendors' capacities ðuiþ ; vendors' quota flexibility ðfiþ on a scale of 0-1, vendor rating ðr i Þ on a scale of 0-1, and the budget allocations for the vendors ðb i Þ : The least value of flexibility in vendors' quota and least total purchase value of supplied items are policy decisions and depend on the demand. The least value of flexibility in suppliers' quota is given as F =fd and the least total purchase value of supplied items is given as P = rd : If overall flexibility ðfþ is 0.03 on the scale of 0-1, the overall vendor rating ðrþ is 0.92 on the scale of 0-1 and the aggregate demand ðdþ is 20,000, then the least value of flexibility in suppliers' quota ðfþ and the least total purchase value of supplied items ðpþ are 600 and 18,400, respectively. The aspiration level of a fuzzy goal is obtained by using a simple heuristic. The individual function of that particular goal is first treated as the objective function of a MIP problem having constraint set same as the one defined in MIGP_VSP formulation. For example, with reference to formulation (11), the aspiration-level of the, first goal, i.e. minimizing the net-cost is obtained by solving a corresponding (MIP) having objective function and constraint set as X = {gjðx i Þ # aj,h k (x t ) = b k; x i $ 0} and integer. The aspiration levels of the three fuzzy goals, viz. the minimization of the net cost, the net rejections and the net late deliveries have thus been obtained as 125,000, 420 and 700, respectively. These values also indicate the best possible solution as these values have been obtained after ignoring other goals. The maximum and minimum limit for the deviation of each fuzzy goal is the same on both sides and are set as 20,000, 100 and 150, respectively, for the three fuzzy goals (Fig. 2). In this problem, triangular membership functions as given in Eq. (15) are used. The triangular membership functions of the three fuzzy goals, viz. minimizing the net cost, minimizing the net rejections and minimizing the net deliveries are constructed as given in Eqs. (30)-(32). 145 ; 000 -Z^J lfl25,000<z l( x^ 145 ; 000 L 20 ; ; otherwise ð30þ

11 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Lo, ioo *}' if 420, Z2ÐXIÞ # 520 otherwise [ g ] if 550^,), f Z 3 ðx i L i ;.0, Þ if 700, Zjfo) < 850 otherwise ð31þ ð32þ Analogous to the formulation (2)-(10), the mathematical formulation is developed. This MIP_VSP formulation is provided in Appendix A (a) Membership function of Z,(x,) (b) Membership function of Z 2 (x f ) (c) Membership function of Z 3 (x,.) z ' <r ' ) Fig. 2. Membership function of different goals.

12 80 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Now we summarize the complete solution procedure for the f-migp_vsp formulation through the following steps: Step 1: Construct a vendor source data similar to one given in Table 1. Step 2: Transform the vendor source data into a vector-minimization form of the MIP_VSP formulation. Step 3: Construct the f-migp_vsp formulation, considering some of the parameters of the goals as fuzzy in nature. Step 4: Set the aspiration level of the fuzzy goals. 4.1: Select the first objective along with the constraint set of MIP_VSP formulation. Solve this problem as a single objective MIP problem and get the optimum value of the objective function. Treat this value as the aspiration level of first goal. 4.2: Repeat Step 4.1 for all the other objectives of MIP_VSP formulation. Step 5: Determine the maximum and minimum limits of deviations from the aspiration level. Step 6: Define the membership function of each fuzzy goal in the f-migp_vsp formulation. Step 7: Construct the equivalent crisp (c-migp) formulation of the f-migp_vsp. Step 8: Solve the equivalent crisp (c-migp) formulation and obtain the decision regarding quota allocations to the vendors Application of c-migp model to the illustration For the illustrative application, optimal quota allocations among different vendors have been obtained using f-migp_vsp formulation. Once the membership functions of the three fuzzy goals of the case in illustration are defined, then the fuzzy problem (f-migp_vsp) formulation can be converted into an equivalent crisp (c-migp_vsp) formulation. The equivalent crisp (c-migp_vsp) formulation is given as: Maximize l subject to A < ð2 : 5X 1 þ 3 : 5X 2 þ 3X 3 þ X 4 Þ þ 7 : 25 A # ð2 : 5X 1 þ 3 : 5X 2 þ 3X 3 þ X 4 Þ 2 5 : 25 A < ð5x 1 þ 3X 2 þ 2X 4 Þ þ 5 : 2 A # ð5x 1 þ 3X 2 þ 2X4Þ 2 3 : 2 A < ð2 : 6X 1 þ 1 : 3X 2 þ 5 : 3X3 þ 0 : 6X4Þ þ 5 : 6 A # ð2 : 6X 1 þ 1 : 3X 2 þ 5 : 3X3 þ 0 : 6X4Þ 2 3 : 6 X 1 þ X 2 þ X 3 þ X 4 = 20 ; 000 X 1 # 5000 X 2 # 15 ; 000 X 3 # 6000 X 4 # 3000

13 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) : 88X 1 þ 0 : 91X 2 þ 0 : 97X 3 þ 0 : 85X 4 $ 18 ; : 02X 1 þ 0 : 01X 2 þ 0 : 06X 3 þ 0 : 04X4 $ 600 5X 1 # 25 ; 000 7X 2 # 100 ; 000 6X3 # 35 ; 000 2X4 # 5500 where X i = xi; x i $ 0 and integer, i = 1 ; 2 ; 3 ; 4 : The optimal solution for the above formulation is x 1 =0, x 2 = 12 ; 714 ; x 3 = 5336 and x 4 = 1950 : In the optimal solution the degree of achievement of the fuzzy goals ðl max = 0 : 996Þ is significantly high. This provides the best value solution for an aggregate demand of 20,000, yields the net cost as $124,914, the net rejections as 420 and the net late deliveries as 700. Vendor 1 lost the entire quota due to the most inferior performance on quality criterion and substantially poor performance on other criteria. However, vendor 3 has received more quota allocation as he performed the best on quality criterion. The remaining quota allocations in the optimal solution are with vendors 2 and 4. Vendors 2 and 4 have the ability to supply the remaining items to fulfill the demand requirement. Vendor 4 is the best on low cost and timely delivery criteria but does not have sufficient capacity to supply. The natures of quota allocations on the different performance criteria of different vendors are described in Table 2. The c-migp_vsp formulation is solved for different degree of achievements for the fuzzy goals for comparing the results. These results are shown in Fig. 3. The results indicate that the quota allocations to the vendors are quite different when fuzziness in some of the parameters of the problem is captured. As the values for the degree of the achievement of the fuzzy goals are increased, the quota allocations to the different vendors also change. The overall demand for the item is assumed to be constant. The quota allocations to the vendors depend on the performance criteria and the degree of the achievement of the fuzzy goals. The quota allocations would increase for those vendors who have superior performance on different criteria. This is expected even intuitively. In some cases, few vendors such as the vendor 1 loses the entire quota to the other vendors when fuzziness is considered in the c-migp_vsp formulation. Initially, when the goals are not achieved vendor 1 has significant allocation. When the value for the degree of achievement of the fuzzy goals is increased then the quota allocation to the vendor 1 starts decreasing in a consistent manner. The decrease in quota allocation to the vendor 1 is due to the poor performance (Table 2). In some cases, some vendors such as vendor 4 gains considerable amount of quota at the optimal solution ðl max = 0 : 996Þ : Initially, when the degrees of achievement of the fuzzy goals are zero, vendor 4 does not get any quota allocation. As the value for the degrees of achievement of the fuzzy goals increases, the quota allocation to the vendor 4 starts increasing in a consistent manner. This increase in quota allocation of vendor 4 is due to the superior performance (Table 2). Vendor 2 has the maximum capacity to supply the part. Vendor 2 has also got the maximum budget allocation. The performance indicators for the vendor 2 are quite superior (viz. comparatively lower percentage rejections, lower late delivery percentage, comparatively better vendor rating, etc.).

14 82 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) Table 2 Nature of the quota allocations at the optimal solution Vendor number Quota allocation Performance criteria 12,714 (89% of capacity, 92% of his allocated budget is consumed) 5336 (85% of capacity, 89% of his allocated budget is consumed) 1950 (65% of capacity, 71% of his allocated budget is consumed) Inferior on the performance criteria (viz. highest percentage rejections, high percentage late deliveries, less vendor rating value, less quota flexibility value, etc). Highest supplying capacity, highest budget allocation, better on the performance criteria (viz. less percentage rejections, less percentage of late deliveries, comparatively better vendor rating value, etc.) Better on the performance criteria (viz. no rejections at all, best vendor rating value, high quota flexibility, highest vendor rating value, etc.) Less capacity to supply, less budget allocation, less vendor rating value, less per unit cost, less percentage rejections, and less percentage of late deliveries Vendor 2 receives more quota allocation for the higher values of the degree of achievement of the fuzzy goals. As the value of the degree of the achievement of the fuzzy goals is increased, the quota allocation to the vendor 2 also increases. Vendor 3 is consistently superior on some performance criteria (viz. no rejections at all, highest vendor rating value and highest quota flexibility). The quota allocation to the vendor 3 has slightly \ Vendor 1 ---*--- vendor 2 Vendor 3 * Vendor o re 8000 " o s " 2000 " Degree of achievement of the fuzzy goals Fig. 3. Quota allocations to the vendors at different degree of achievement of fuzzy goals.

15 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) decreased, yet it is quite stable and consistent over a range of values for the degree of the achievement of the fuzzy goals (Fig. 3). 5. Conclusion The f-migp_vsp formulated in this paper is extremely useful for solving the VSP in a supply chain when the goals are not clearly stated. The formulation can effectively handle the vagueness and imprecision in the statement of the objectives. It presents a rational approach to decision-making process for the VSP in a supply chain in the sense that there is no longer any distinction between the goals and the constraints. Further, the f-migp_vsp formulation gives the optimal trade-offs among the values for different goals for a VSP. Due to conflicting nature of the multiple objectives and vagueness in the information related to the parameters of the decision variables, the deterministic techniques are unsuitable to obtain an effective solution. The proposed f-migp_vsp formulation has the advantages that any commercially available software such as LINDO/LINGO may be used for solving it. The proposed formulation is more effective than the deterministic methods for handling the real situations, as very precise and deterministic information is generally not available for designing and managing a supply chain. Appendix A The MIP_VSP model containing three fuzzy goals, eleven crisp constraints and four decision variables can be formulated as follows: G 1 : G 2 : G 3 : 5x 1 þ 7x 2 þ 6x 3 þ 2x 4 ø 125 ; : 05x 1 þ 0 : 03x 2 þ 0 : 02x 4 ø : 04x 1 þ 0 : 02x 2 þ 0 : 08x 3 þ 0 : 01x 4 ø 700 subject to x 1 þ x 2 þ x 3 þ x 4 = 20 ; 000 x 1 # = 5000 x 2 # 15 ; 000 x 3 # 6000 x 4 # : 88x 1 þ 0 : 91x 2 þ 0 : 97x 3 þ 0 : 85x 4 $ 18 ; : 02x 1 þ 0 : 01x 2 þ 0 : 06x 3 þ 0 : 04x 4 $ 600 5x 1 < 25 ; 000 7x 2 < 100 ; 000 6x 3 < 35 ; 000

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