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A fuzzy goal programming approach for vendor selection problem in a supply chain
Computers & Industrial Engineering 46 (2004) 69–85 www.elsevier.com/locate/dsw
A fuzzy goal programming approach for vendor selection problem in a supply chain Manoj Kumara, Prem Vratb, R. Shankarc,* a
Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India b Indian Institute of Technology Roorkee, Roorkee 247 667, Uttaranchal, India c Department of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, 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. q 2003 Elsevier Ltd. All rights reserved. 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 * Corresponding author. Tel.: þ91-11-26596421/64371; fax: þ91-11-26862620. E-mail address: [email protected] (R. Shankar). 0360-8352/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2003.09.010
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quota allocation. The choice of the right vendor is a crucial decision with wide ranging implications in a supply chain. Vendors play an important role 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 5.
2. 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
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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 Buffa (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) (ii) (iii) (iv)
Only one item is purchased from one vendor. Quantity discounts are not taken into consideration. No shortage of the item is allowed for any vendor. Demand of the item is constant and known with certainty.
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Index set i j k l
index for vendor, for all i ¼ 1; 2; …; n index for inequality constraints, for all j ¼ 1; 2; …; J index for equality constraints, for all k ¼ 1; 2; …; K index for objectives, for all l ¼ 1; 2; …; L Decision variable
xi
order quantity for the vendor i Parameters
D n pi qi di Ui fi F ri P Bi
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 xi 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 Zl ðxi Þ ¼ ½Z1 ðxi Þ; Z2 ðxi Þ; …; ZL ðxi Þ;
l ¼ 1; 2; …; L
subject to gj ðxi Þ # aj ;
j ¼ 1; 2; …; J
hk ðxi Þ ¼ bk ;
k ¼ 1; 2; …; K
ð1Þ
xi $ 0 and integer In formulation (1), xi are n decision variables, Z1 ðxi Þ; Z2 ðxi Þ; …; ZL ðxi Þ; are L distinct objective functions, gj are the inequality constraints and hk are the equality constraints. aj and bk are the right hand side constants for inequality and equality relationships, respectively.
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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: Minimize Z1 ¼
n X
pi ðxi Þ
ð2Þ
qi ðxi Þ
ð3Þ
di ðxi Þ
ð4Þ
i¼1
Minimize Z2 ¼
n X i¼1
Minimize Z3 ¼
n X i¼1
subject to n X
xi ¼ D
ð5Þ
i¼1
xi # Ui ; n X
for all i ¼ 1; 2; …; n
ð6Þ
fi ðxi Þ $ F
ð7Þ
ri ðxi Þ $ P
ð8Þ
i¼1 n X i¼1
pi ðxi Þ # Bi ;
for all i ¼ 1; 2; …; n
xi $ 0 and integer
ð9Þ 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.
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3.2. 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: i ¼ 1; 2; …; n Find xi to satisfy Zl ðxi Þ ø Z~ l ; l ¼ 1; 2; …; L gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ;
j ¼ 1; 2; …; J k ¼ 1; 2; …; K
ð11Þ
xi $ 0 and integer; i ¼ 1; 2; …; n where Zl ðxi Þ is the lth goal constraint, gj ðxi Þ is the jth inequality constraint, hk ðxi 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, bk 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 Zl ðxi Þ should be in the vicinity of the aspiration Z~ l : The lth fuzzy goal Zl ðxi Þ ø Z~ l signifies that the decision maker would be satisfied even for values slightly greater than (or lesser than) Z~ l up to a stated deviations signified by tolerance limit. The jth system constraint gj ðxi Þ # aj and the kth system constraint hk ðxi Þ ¼ bk are assumed to be crisp. 3.3. 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 2 {ðx; mA ðxÞÞ=x [ X}
ð12Þ
where mA ðxÞ : X ! ½0; 1 is called the membership function of A and mA 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 mA ðxÞ; which associates with each point in X; a nonnegative real number whose value is finite and usually finds a place in the interval ½0; 1; with the value of mA ðxÞ at x representing the ‘grade of membership’ of x in A: Thus nearer the value of mA 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 mA ðxÞ and mB ðxÞ is defined as a fuzzy set C whose membership function is as follows:
mC ðxÞ ¼ max½mA ðxÞ; mB ðxÞ;
x[X
ð13Þ
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Definition 3: Intersection of two fuzzy sets A and B with respective membership functions mA ðxÞ and mB ðxÞ is defined as a fuzzy set C whose membership function is as follows:
mC ðxÞ ¼ min½mA ðxÞ; mB ðxÞ;
x[X
ð14Þ
Using approach of Yang et al. (1991), the triangular membership function m½Zl ðxi Þ for 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 m½Zl ðxi Þ is shown in Fig. 1 and is defined as follows: 8" # > Zl ðxi Þ 2 Zlmin > > ; if Zlmin # Zl ðxi Þ # Z~ l > > ~ l 2 Zlmin Z > > <" # max ð15Þ m½Zl ðxi Þ ¼ Z 2 Z ðx Þ l i l > ~ l , Zl ðxi Þ # Zlmax > ; if Z > > Zlmax 2 Z~ l > > > : 0; otherwise In relationships (15) Z~ l is the aspiration level of the lth fuzzy goal, Zlmin is a minimum limit of deviation to Z~ l ; Zlmax is a maximum limit of deviation to Z~ l ; m½Zl ðxi Þ is a strictly monotonically decreasing (or increasing) continuous function. 3.4. 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
Fig. 1. Membership function of Zl ðxi Þ:
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fuzzy constraints. The membership function of the fuzzy decision is given by
mS ðxÞ ¼ mZ ðxÞ > mC ðxÞ ¼ min½mZ ðxÞ; mC ðxÞ
ð16Þ
where mZ ðxÞ and mC ðxÞ represent the membership functions of all the fuzzy objectives and fuzzy constraints, respectively. The fuzzy decision of all the fuzzy sets Zl for representing l ¼ 1; 2; …; L fuzzy goals and of all the fuzzy sets Cm for representing fuzzy constraints m ¼ 1; 2; …; M may be given as: ! ! L M \ \ mS ðxÞ ¼ mZl ðxÞ > mCm ðxÞ ¼ min min mZl ðxÞ; min mCm ðxÞ ð17Þ l¼1;2;…;L
m¼1
l¼1
m¼1;2;…;M
An optimum element means selecting that element ðxp Þ which has the highest degree of membership value to the fuzzy decision set: p mS ðx Þ ¼ max mS ðxÞ ¼ max min min mZl ðxÞ; min mCm ðxÞ ð18Þ x[S
x[S
l¼1;2;…;L
m¼1;2;…;M
If the goals are described by the membership function mZl ðxi Þ on X ¼ {gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ; xi $ 0} then the membership function of the optimal solution ðxp Þ is given by: p mS ðx Þ ¼ max mS ðxÞ ¼ max min min mZl ðxi Þ ; …; mZl ðxi Þ ð19Þ x[S
x[S
l¼1;2;…;L
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 mZl ðxi Þ 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: l # mZl ðxi Þ ;
for all l ¼ 1; 2; …; L
gj ðxi Þ # aj ;
j ¼ 1; 2; …; J
hk ðxi Þ ¼ bk ;
k ¼ 1; 2; …K
xi $ 0 and integer
ð20Þ
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3.6. Application of f-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: n X
pi ðxi Þ ø Z~ 1
ð21Þ
qi ðxi Þ ø Z~ 2
ð22Þ
di ðxi Þ ø Z~ 3
ð23Þ
xi ¼ D
ð24Þ
i¼1 n X i¼1
n X i¼1 n X i¼1
xi # Ui ; n X
for all i ¼ 1; 2; …; n
ð25Þ
fi ðxi Þ $ F
ð26Þ
ri ðxi Þ $ P
ð27Þ
i¼1 n X i¼1
pi ðxi Þ # Bi ;
for all i ¼ 1; 2; …; n
xi $ 0 and integer; for all i ¼ 1; 2; …; n
ð28Þ 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.
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Table 1 Vendor source data of the illustrative example Vendor number
pi ($)
qi (%)
di (%)
Ui (units)
fi
ri
Bi ($)
1 2 3 4
5 7 6 2
0.05 0.03 0 0.02
0.04 0.02 0.08 0.01
5000 15,000 6000 3000
0.02 0.01 0.06 0.04
0.88 0.91 0.97 0.85
25,000 100,000 35,000 5500
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 (pi in $) the percentage rejections ðqi Þ; the percentage late deliveries ðdi Þ; vendors’ capacities ðUi Þ; vendors’ quota flexibility ðfi Þ on a scale of 0 – 1, vendor rating ðri Þ on a scale of 0 – 1, and the budget allocations for the vendors ðBi Þ: The least value offlexibility 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 ¼ fD and the least total purchase value of supplied items is given as P ¼ rD: If overall flexibility ðf Þ 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 ¼ {gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ; xi $ 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).
m½Z1 ðxi Þ
8 Z1 ðxi Þ 2 10; 5000 > > ; > > 20; 000 > < 145; 000 2 Z1 ðxi Þ ¼ > ; > > 20; 000 > > : 0;
if 105; 000 # Z1 ðxi Þ # 125; 000 if 125; 000 , Z1 ðxi Þ # 145; 000 otherwise
ð30Þ
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m½Z2 ðxi Þ
m½Z3 ðxi Þ
79
8 Z2 ðxi Þ 2 320 > > ; if 320 # Z2 ðxi Þ # 420 > > 100 > < 520 2 Z2 ðxi Þ ¼ > ; if 420 , Z2 ðxi Þ # 520 > > 100 > > : 0; otherwise
ð31Þ
8 Z3 ðxi Þ 2 550 > > ; if 550 # Z3 ðxi Þ # 700 > > 150 > < 850 2 Z3 ðxi Þ ¼ > ; if 700 , Z3 ðxi Þ # 850 > > 150 > > : 0; otherwise
ð32Þ
Analogous to the formulation (2)– (10), the mathematical formulation is developed. This MIP_VSP formulation is provided in Appendix A.
Fig. 2. Membership function of different goals.
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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. 4.1. 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
l # 2ð2:5X1 þ 3:5X2 þ 3X3 þ X4 Þ þ 7:25 l # ð2:5X1 þ 3:5X2 þ 3X3 þ X4 Þ 2 5:25 l # 2ð5X1 þ 3X2 þ 2X4 Þ þ 5:2 l # ð5X1 þ 3X2 þ 2X4 Þ 2 3:2 l # 2ð2:6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4 Þ þ 5:6 l # ð2:6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4 Þ 2 3:6 X1 þ X2 þ X3 þ X4 ¼ 20; 000 X1 # 5000 X2 # 15; 000 X3 # 6000 X4 # 3000
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0:88X1 þ 0:91X2 þ 0:97X3 þ 0:85X4 $ 18; 400 0:02X1 þ 0:01X2 þ 0:06X3 þ 0:04X4 $ 600 5X1 # 25; 000 7X2 # 100; 000 6X3 # 35; 000 2X4 # 5500 where Xi ¼ 1024 xi ; xi $ 0 and integer, i ¼ 1; 2; 3; 4: The optimal solution for the above formulation is x1 ¼ 0; x2 ¼ 12; 714; x3 ¼ 5336 and x4 ¼ 1950: In the optimal solution the degree of achievement of the fuzzy goals ðlmax ¼ 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 ðlmax ¼ 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.).
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Table 2 Nature of the quota allocations at the optimal solution Vendor number
Quota allocation
Performance criteria
1
0
2
12,714 (89% of capacity, 92% of his allocated budget is consumed)
3
5336 (85% of capacity, 89% of his allocated budget is consumed)
4
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
Fig. 3. Quota allocations to the vendors at different degree of achievement of fuzzy goals.
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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: G1 : 5x1 þ 7x2 þ 6x3 þ 2x4 ø 125; 000 G2 : 0:05x1 þ 0:03x2 þ 0:02x4 ø 420 G3 : 0:04x1 þ 0:02x2 þ 0:08x3 þ 0:01x4 ø 700 subject to x1 þ x2 þ x3 þ x4 ¼ 20; 000 x1 # 5000 x2 # 15; 000 x3 # 6000 x4 # 3000 0:88x1 þ 0:91x2 þ 0:97x3 þ 0:85x4 $ 18; 400 0:02x1 þ 0:01x2 þ 0:06x3 þ 0:04x4 $ 600 5x1 # 25; 000 7x2 # 100; 000 6x3 # 35; 000
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2x4 # 5500 xi $ 0 and integer; i ¼ 1; 2; 3; 4 where G1, G2, and G3 indicate goals for minimizing the net cost, minimizing the net rejections, and minimizing the net late deliveries, respectively. References Anthony, T. F., & Buffa, F. P. (1977). Strategic purchase scheduling. Journal of Purchasing and Materials Management, Fall, 27 – 31. Bellman, R. E., & Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Sciences, 17, B141– B164. Bendor, P. S., Brown, R. W., Issac, M. H., & Shapiro, J. F. (1985). Improving purchasing productivity at IBM with a normative decision support system. Interfaces, 15, 106– 115. Buffa, F. P., & Jackson, W. M. (1983). A goal programming model for purchase planning. Journal of Purchasing and Materials Management, Fall, 27 – 34. Feng, C. X., Wang, J., & Wang, J. S. (2001). An optimization model for concurrent selection of tolerances and suppliers. Computers and Industrial Engineering, 40, 15 – 33. Ghodsypour, S. H., & O’Brien, C. (1998). A decision support system for supplier selection using an integrated analytic hierarchy process and linear programming. International Journal of Production Economics, 56 – 57, 199– 212. Gregory, R. E. (1986). Source selection: a matrix approach. Journal of Purchasing and Materials Management, Summer, 24 – 29. Handfield, R., Walton, S. V., Sroufe, R., & Melnyk, S. A. (2002). Applying environmental criteria to supplier assessment: a study in the application of the Analytical Hierarchy Process. European Journal of Operational Research, 141, 70 – 87. Hannan, E. L. (1981). Linear programming with multiple goals. Fuzzy Sets and Systems, 6, 235– 248. Kumar, M., Vrat, P., & Shankar, R. (2002). A multi-objective interval programming approach for vendor selection problem in a supply chain. In: Proceedings of the 2002 International Conference on e-manufacturing: an emerging need for 21st century world class enterprises. Bhopal, India: IEI, pp. 17 – 19. Liu, F., Ding, F. Y., & Lall, V. (2000). Using data envelopment analysis to compare suppliers for supplier selection and performance improvement. Supply Chain Management: An International Journal, 5(3), 143– 150. Monozka, R. M., & Trecha, S. J. (1988). Cost-based supplier performance evaluation. Journal of Purchasing and Materials Management, Spring, 2 – 7. Moore, D. L., & Fearon, H. E. (1972). Computer-assisted decision-making in purchasing. Journal of Purchasing, 9(4), 5 – 25. Narsimhan, R. (1980). Goal programming in a fuzzy environment. Decision Sciences, 11, 325– 336. Narsimhan, R. (1983). An analytical approach to supplier selection. Journal of Purchasing and Materials Management, Winter, 27 – 32. Oliveria, R. C., & Lourenco, J. C. (2002). A multi-criteria model for assigning new orders to service suppliers. European Journal of Operational Research, 139, 390– 399. Pan, A. C. (1989). Allocation of order quantity among suppliers. Journal of Purchasing and Materials Management, Fall, 36 – 39. Pickens, J. B., & Hof, J. G. (1991). Fuzzy goal programming in forestry: an application with special solution problems. Fuzzy Sets and Systems, 39(3), 239– 246. Rao, S. S., Sundaraju, K., Prakash, B. G., & Balakrishna, C. (1992). Fuzzy goal programming approach for structural optimization. AIAA Journal, 30(5), 1425 –1432. Ronen, B., & Trietsch, D. (1988). A decision support system for purchasing management of large projects. Operations Research, 36(6), 882– 890. Shankar, R., & Vrat, P. (1999). Some design issues in cellular manufacturing using fuzzy programming approach. International Journal of Production Research, 37(11), 2345– 2363. Sharma, D., Benton, W. C., & Srivastava, R. (1989). Competitive strategy and purchasing decisions. In: Proceedings of the 1989 Annual Conference of the Decision Sciences Institute, pp. 1088– 1090.
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Sinha, S. B., Rao, K. A., & Mangaraj, B. K. (1988). Fuzzy goal programming in multi-criteria decision systems: a case study in agricultural planning. Socio-Economic Planning Sciences, 22(2), 93– 101. Stevens, G. C. (1989). Integrating the supply chain. International Journal of Physical Distribution and Materials Management, 19(8), 3 – 8. Weber, C. A., Current, J. R., & Desai, A. (2000). An optimization approach to determining the number of vendors to employ. Supply Chain Management: An International Journal, 2(5), 90 – 98. Wind, Y., & Robinson, P. J. (1968). The determinants of vendor selection: the evaluation function approach. Journal of Purchasing and Materials Management, Fall, 29 – 41. Yager, R. R. (1977). Multiple objective decision-making using fuzzy sets. International Journal of Man-Machine Studies, 9, 375– 382. Yang, T., Ignizio, J. P., & Kim, H. J. (1991). Fuzzy programming with nonlinear membership functions: piecewise linear approximation. Fuzzy Sets and Systems, 11, 39 – 53. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338– 353. Zimmermann, H. J. (1976). Description and optimization of fuzzy systems. International Journal of General Systems, 2, 209– 215. Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45 –56.
A fuzzy goal programming approach for vendor selection problem in a supply chain Manoj Kumara, Prem Vratb, R. Shankarc,* a
Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India b Indian Institute of Technology Roorkee, Roorkee 247 667, Uttaranchal, India c Department of Management Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, 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. q 2003 Elsevier Ltd. All rights reserved. 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 * Corresponding author. Tel.: þ91-11-26596421/64371; fax: þ91-11-26862620. E-mail address: [email protected] (R. Shankar). 0360-8352/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2003.09.010
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quota allocation. The choice of the right vendor is a crucial decision with wide ranging implications in a supply chain. Vendors play an important role 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 5.
2. 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
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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 Buffa (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) (ii) (iii) (iv)
Only one item is purchased from one vendor. Quantity discounts are not taken into consideration. No shortage of the item is allowed for any vendor. Demand of the item is constant and known with certainty.
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Index set i j k l
index for vendor, for all i ¼ 1; 2; …; n index for inequality constraints, for all j ¼ 1; 2; …; J index for equality constraints, for all k ¼ 1; 2; …; K index for objectives, for all l ¼ 1; 2; …; L Decision variable
xi
order quantity for the vendor i Parameters
D n pi qi di Ui fi F ri P Bi
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 xi 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 Zl ðxi Þ ¼ ½Z1 ðxi Þ; Z2 ðxi Þ; …; ZL ðxi Þ;
l ¼ 1; 2; …; L
subject to gj ðxi Þ # aj ;
j ¼ 1; 2; …; J
hk ðxi Þ ¼ bk ;
k ¼ 1; 2; …; K
ð1Þ
xi $ 0 and integer In formulation (1), xi are n decision variables, Z1 ðxi Þ; Z2 ðxi Þ; …; ZL ðxi Þ; are L distinct objective functions, gj are the inequality constraints and hk are the equality constraints. aj and bk are the right hand side constants for inequality and equality relationships, respectively.
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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: Minimize Z1 ¼
n X
pi ðxi Þ
ð2Þ
qi ðxi Þ
ð3Þ
di ðxi Þ
ð4Þ
i¼1
Minimize Z2 ¼
n X i¼1
Minimize Z3 ¼
n X i¼1
subject to n X
xi ¼ D
ð5Þ
i¼1
xi # Ui ; n X
for all i ¼ 1; 2; …; n
ð6Þ
fi ðxi Þ $ F
ð7Þ
ri ðxi Þ $ P
ð8Þ
i¼1 n X i¼1
pi ðxi Þ # Bi ;
for all i ¼ 1; 2; …; n
xi $ 0 and integer
ð9Þ 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.
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3.2. 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: i ¼ 1; 2; …; n Find xi to satisfy Zl ðxi Þ ø Z~ l ; l ¼ 1; 2; …; L gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ;
j ¼ 1; 2; …; J k ¼ 1; 2; …; K
ð11Þ
xi $ 0 and integer; i ¼ 1; 2; …; n where Zl ðxi Þ is the lth goal constraint, gj ðxi Þ is the jth inequality constraint, hk ðxi 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, bk 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 Zl ðxi Þ should be in the vicinity of the aspiration Z~ l : The lth fuzzy goal Zl ðxi Þ ø Z~ l signifies that the decision maker would be satisfied even for values slightly greater than (or lesser than) Z~ l up to a stated deviations signified by tolerance limit. The jth system constraint gj ðxi Þ # aj and the kth system constraint hk ðxi Þ ¼ bk are assumed to be crisp. 3.3. 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 2 {ðx; mA ðxÞÞ=x [ X}
ð12Þ
where mA ðxÞ : X ! ½0; 1 is called the membership function of A and mA 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 mA ðxÞ; which associates with each point in X; a nonnegative real number whose value is finite and usually finds a place in the interval ½0; 1; with the value of mA ðxÞ at x representing the ‘grade of membership’ of x in A: Thus nearer the value of mA 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 mA ðxÞ and mB ðxÞ is defined as a fuzzy set C whose membership function is as follows:
mC ðxÞ ¼ max½mA ðxÞ; mB ðxÞ;
x[X
ð13Þ
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Definition 3: Intersection of two fuzzy sets A and B with respective membership functions mA ðxÞ and mB ðxÞ is defined as a fuzzy set C whose membership function is as follows:
mC ðxÞ ¼ min½mA ðxÞ; mB ðxÞ;
x[X
ð14Þ
Using approach of Yang et al. (1991), the triangular membership function m½Zl ðxi Þ for 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 m½Zl ðxi Þ is shown in Fig. 1 and is defined as follows: 8" # > Zl ðxi Þ 2 Zlmin > > ; if Zlmin # Zl ðxi Þ # Z~ l > > ~ l 2 Zlmin Z > > <" # max ð15Þ m½Zl ðxi Þ ¼ Z 2 Z ðx Þ l i l > ~ l , Zl ðxi Þ # Zlmax > ; if Z > > Zlmax 2 Z~ l > > > : 0; otherwise In relationships (15) Z~ l is the aspiration level of the lth fuzzy goal, Zlmin is a minimum limit of deviation to Z~ l ; Zlmax is a maximum limit of deviation to Z~ l ; m½Zl ðxi Þ is a strictly monotonically decreasing (or increasing) continuous function. 3.4. 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
Fig. 1. Membership function of Zl ðxi Þ:
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fuzzy constraints. The membership function of the fuzzy decision is given by
mS ðxÞ ¼ mZ ðxÞ > mC ðxÞ ¼ min½mZ ðxÞ; mC ðxÞ
ð16Þ
where mZ ðxÞ and mC ðxÞ represent the membership functions of all the fuzzy objectives and fuzzy constraints, respectively. The fuzzy decision of all the fuzzy sets Zl for representing l ¼ 1; 2; …; L fuzzy goals and of all the fuzzy sets Cm for representing fuzzy constraints m ¼ 1; 2; …; M may be given as: ! ! L M \ \ mS ðxÞ ¼ mZl ðxÞ > mCm ðxÞ ¼ min min mZl ðxÞ; min mCm ðxÞ ð17Þ l¼1;2;…;L
m¼1
l¼1
m¼1;2;…;M
An optimum element means selecting that element ðxp Þ which has the highest degree of membership value to the fuzzy decision set: p mS ðx Þ ¼ max mS ðxÞ ¼ max min min mZl ðxÞ; min mCm ðxÞ ð18Þ x[S
x[S
l¼1;2;…;L
m¼1;2;…;M
If the goals are described by the membership function mZl ðxi Þ on X ¼ {gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ; xi $ 0} then the membership function of the optimal solution ðxp Þ is given by: p mS ðx Þ ¼ max mS ðxÞ ¼ max min min mZl ðxi Þ ; …; mZl ðxi Þ ð19Þ x[S
x[S
l¼1;2;…;L
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 mZl ðxi Þ 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: l # mZl ðxi Þ ;
for all l ¼ 1; 2; …; L
gj ðxi Þ # aj ;
j ¼ 1; 2; …; J
hk ðxi Þ ¼ bk ;
k ¼ 1; 2; …K
xi $ 0 and integer
ð20Þ
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3.6. Application of f-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: n X
pi ðxi Þ ø Z~ 1
ð21Þ
qi ðxi Þ ø Z~ 2
ð22Þ
di ðxi Þ ø Z~ 3
ð23Þ
xi ¼ D
ð24Þ
i¼1 n X i¼1
n X i¼1 n X i¼1
xi # Ui ; n X
for all i ¼ 1; 2; …; n
ð25Þ
fi ðxi Þ $ F
ð26Þ
ri ðxi Þ $ P
ð27Þ
i¼1 n X i¼1
pi ðxi Þ # Bi ;
for all i ¼ 1; 2; …; n
xi $ 0 and integer; for all i ¼ 1; 2; …; n
ð28Þ 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.
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Table 1 Vendor source data of the illustrative example Vendor number
pi ($)
qi (%)
di (%)
Ui (units)
fi
ri
Bi ($)
1 2 3 4
5 7 6 2
0.05 0.03 0 0.02
0.04 0.02 0.08 0.01
5000 15,000 6000 3000
0.02 0.01 0.06 0.04
0.88 0.91 0.97 0.85
25,000 100,000 35,000 5500
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 (pi in $) the percentage rejections ðqi Þ; the percentage late deliveries ðdi Þ; vendors’ capacities ðUi Þ; vendors’ quota flexibility ðfi Þ on a scale of 0 – 1, vendor rating ðri Þ on a scale of 0 – 1, and the budget allocations for the vendors ðBi Þ: The least value offlexibility 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 ¼ fD and the least total purchase value of supplied items is given as P ¼ rD: If overall flexibility ðf Þ 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 ¼ {gj ðxi Þ # aj ; hk ðxi Þ ¼ bk ; xi $ 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).
m½Z1 ðxi Þ
8 Z1 ðxi Þ 2 10; 5000 > > ; > > 20; 000 > < 145; 000 2 Z1 ðxi Þ ¼ > ; > > 20; 000 > > : 0;
if 105; 000 # Z1 ðxi Þ # 125; 000 if 125; 000 , Z1 ðxi Þ # 145; 000 otherwise
ð30Þ
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m½Z2 ðxi Þ
m½Z3 ðxi Þ
79
8 Z2 ðxi Þ 2 320 > > ; if 320 # Z2 ðxi Þ # 420 > > 100 > < 520 2 Z2 ðxi Þ ¼ > ; if 420 , Z2 ðxi Þ # 520 > > 100 > > : 0; otherwise
ð31Þ
8 Z3 ðxi Þ 2 550 > > ; if 550 # Z3 ðxi Þ # 700 > > 150 > < 850 2 Z3 ðxi Þ ¼ > ; if 700 , Z3 ðxi Þ # 850 > > 150 > > : 0; otherwise
ð32Þ
Analogous to the formulation (2)– (10), the mathematical formulation is developed. This MIP_VSP formulation is provided in Appendix A.
Fig. 2. Membership function of different goals.
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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. 4.1. 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
l # 2ð2:5X1 þ 3:5X2 þ 3X3 þ X4 Þ þ 7:25 l # ð2:5X1 þ 3:5X2 þ 3X3 þ X4 Þ 2 5:25 l # 2ð5X1 þ 3X2 þ 2X4 Þ þ 5:2 l # ð5X1 þ 3X2 þ 2X4 Þ 2 3:2 l # 2ð2:6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4 Þ þ 5:6 l # ð2:6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4 Þ 2 3:6 X1 þ X2 þ X3 þ X4 ¼ 20; 000 X1 # 5000 X2 # 15; 000 X3 # 6000 X4 # 3000
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0:88X1 þ 0:91X2 þ 0:97X3 þ 0:85X4 $ 18; 400 0:02X1 þ 0:01X2 þ 0:06X3 þ 0:04X4 $ 600 5X1 # 25; 000 7X2 # 100; 000 6X3 # 35; 000 2X4 # 5500 where Xi ¼ 1024 xi ; xi $ 0 and integer, i ¼ 1; 2; 3; 4: The optimal solution for the above formulation is x1 ¼ 0; x2 ¼ 12; 714; x3 ¼ 5336 and x4 ¼ 1950: In the optimal solution the degree of achievement of the fuzzy goals ðlmax ¼ 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 ðlmax ¼ 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.).
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Table 2 Nature of the quota allocations at the optimal solution Vendor number
Quota allocation
Performance criteria
1
0
2
12,714 (89% of capacity, 92% of his allocated budget is consumed)
3
5336 (85% of capacity, 89% of his allocated budget is consumed)
4
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
Fig. 3. Quota allocations to the vendors at different degree of achievement of fuzzy goals.
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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: G1 : 5x1 þ 7x2 þ 6x3 þ 2x4 ø 125; 000 G2 : 0:05x1 þ 0:03x2 þ 0:02x4 ø 420 G3 : 0:04x1 þ 0:02x2 þ 0:08x3 þ 0:01x4 ø 700 subject to x1 þ x2 þ x3 þ x4 ¼ 20; 000 x1 # 5000 x2 # 15; 000 x3 # 6000 x4 # 3000 0:88x1 þ 0:91x2 þ 0:97x3 þ 0:85x4 $ 18; 400 0:02x1 þ 0:01x2 þ 0:06x3 þ 0:04x4 $ 600 5x1 # 25; 000 7x2 # 100; 000 6x3 # 35; 000
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