A fuzzy goal programming approach for vendor selection problem … · A fuzzy goal programming...

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

Manoj Kumara, Prem Vratb, R. Shankarc,*

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

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

Accepted 19 September 2003

Abstract

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

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

1. Introduction

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

70 M. Kumar et al. / Computers & Industrial Engineering 46 (2004) 69—85

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

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

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

2. Literature review

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

M. Kumar et al. / Computers & Industrial Engineering 46 (2004) 69—85 71

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

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

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

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

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

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.


Index set

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

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

Decision variable

xi order quantity for the vendor i

Parameters

D aggregate demand of the item over a fixed planning period

n number of vendors competing for selectionpi price of a unit item of the ordered quantity xi from the vendor iqi percentage of the rejected units delivered by the vendor idi percentage of the late delivered units by the vendor iUi upper bound of the quantity available with vendor i

fi vendor quota flexibility for vendor iF lower bound of flexibility in supply quota that a vendor should haveri vendor rating value for vendor iP lower bound to total purchasing value that a vendor should haveBi budget allocated to each vendor

3.1. Model formulation

A classical multiple-objective mixed integer-programming problem can be written as follows:

Maximize Zjfo) = [Z1(xi),Z2(xi), ...,ZL(x;)], 1=1,2, ...,L

subject to

gjðxiÞ # aj; j= 1,2,...,J

(1)= bk, k=l,2,...,K

xi $ 0 and integer

In formulation (1), xi are n decision variables, Z1(xi),Z2(xi),...,ZL(xi), are L distinct objectivefunctions, gj are the inequality constraints and hk are the equality constraints. aj and bk are the right handside constants for inequality and equality relationships, respectively.


The mixed integer programming vendor selection problem (MIP_VSP) formulation for threeobjectives and a set of system and policy constraints can be formulated as follows:

n

Minimize Z1 — V/?;(x;) (2)i=\

n

Minimize Z2 = X #,(*,) (3)i=\

n

Minimize Z3 = X dt{x^) (4)!=1

subject to

i=\

xi#Ui; for all i= 1;2 nð 6Þ

<^) ^ F (?)i=\

piðxiÞ#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 theitems.

In real life situations, a lot of informational inputs required for selecting vendors are imprecise, vagueor uncertain. This may be true for objectives as well as parameters. Such imprecise information can bemodeled through representations.


3.2. Fuzzy mixed integer goal programming model

When vague information related to the objectives are present then the problem can be formulated as afuzzy goal-programming problem. A typical fuzzy mixed integer goal programming problem (f-MIGP)formulation can be stated as follows:

Find xi i = 1;2; ...,n

to satisfy ZlðxiÞ ø Z~ I = 1,2, ...,L

gjixd^aj, . / = 1 , 2 , . . . , / (11)hk(xt) = bk; k=l,2,...,K

xi $ 0 and integer; i— 1; 2 ; . . . , n

where

ZlðxiÞ is the lth goal constraint,gjðxiÞ is thejth inequality constraint,hkðxiÞ 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 linguisticterm 'about' and it means that ZlðxiÞ should be in the vicinity of the aspiration Z~l

The lth fuzzy goal ZlðxiÞ = % signifies that the decision maker would be satisfied even for valuesslightly greater than (or lesser than) % up to a stated deviations signified by tolerance limit. Thejth systemconstraint gjðxiÞ # aj and the kth system constraint hk(xt) = 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 aset. In classical set theory, each element of a universe X either belongs to a set A or not, whereas in fuzzyset 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,/xA(x))/xGX} (12)

where mAðxÞ : X !* 0;1] is called the membership function of A and mAðxÞ is the degree of membershipto which x belongs to A:

The fuzzy set A in X is thus uniquely characterized by its membership function mAðxÞ; whichassociates with each point in X; a nonnegative real number whose value is finite and usually finds a placein the interval [0,1], with the value of mAðxÞ at x representing the 'grade of membership' of x in A: Thusnearer the value of mAðxÞ 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 mAðxÞ and mBðxÞis defined as a fuzzy set C whose membership function is as follows:

/xc(x) = max[/xA(x), /JLB(X)], X G X (13)


Definition 3: Intersection of two fuzzy sets A and B with respective membership functions mAðxÞ andmBðxÞ is defined as a fuzzy set C whose membership function is as follows:

fic(x) = mm[fiA(x), fiB(x)], X ð14Þ

Using approach of Yang et al. (1991), the triangular membership function /̂ [z,fe)] f° r the lth fuzzygoal in the f-MIGP formulation has been considered. This type of membership function is adopted due toease in defining the maximum and minimum limit of deviations of the each fuzzy goal from its centralvalue. The triangular membership function /̂ [z,fe)] is shown in Fig. 1 and is defined as follows:

if

0; otherwise

< Z~

ð15Þ

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

/Ji>[Zi(Xi)] 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 theintersection of the all the fuzzy sets representing the fuzzy objectives and all the fuzzy sets representing

0 Z,1™1 Zl

Fig. 1. Membership function of ZlðxiÞ:


fuzzy constraints. The membership function of the fuzzy decision is given by

= /xz(x) 0 /xc(x) = min[/xz(x); /xc(x)] (16)

where mZðxÞ and mCðxÞ represent the membership functions of all the fuzzy objectives and fuzzyconstraints, respectively.

The fuzzy decision of all the fuzzy sets Zl for representing I— 1 ;2 ; ...,L fuzzy goals and of all thefuzzy sets Cm for representing fuzzy constraints m— 1; 2; ...,M may be given as:

ixt(i) = I O u7 (x) I Pi I O ur (x) I = min min « 7 (x) , min ur (x) (17)\'=i / U=i / \j=\X-,L l ' m=\X-M m \

An optimum element means selecting that element ðxpÞ which has the highest degree of membershipvalue to the fuzzy decision set:

fis(x*) = max fis(x) = max min min /xZ;(x), min /xCm(x) ð18Þ

If the goals are described by the membership function /x^*;) on X = {g/^O ^ aj,hk(xt) — bk;xi $ 0}then the membership function of the optimal solution ðxpÞ is given by:

fis(x*) = max fis(x) = max min min /xZife),..., fiZl(Xi) ð19Þ

3.5. Crisp mathematical formulation

Using approach of Yang et al. (1991), which is based on a piecewise linear approximation with themin-operator for aggregating the fuzzy goals, the f-MIGP formulation may be solved to determine thedecision set, and then by maximization of the set. The fuzzy goals are defined by using a triangularmembership function as represented in Eq. (15). Once the membership functions of the fuzzy objectivesZlðxiÞ a r e known, the fuzzy optimization problem (f-MIGP) formulation is transformed into anequivalent crisp formulation (c-MIGP) for optimization. An equivalent crisp mathematical program-ming (c-MIGP) formulation is given by:

max l

s.t. A < Pzitxi)' f ° r aU I = 1>2, . . . ,£

ð20ÞgjðxiÞ # aj; j= 1,2, . . . , /

hk(xt) = bk, k=\,2,...K

xi $ 0 and integer


3.6. Application off-MIGP model for solving VSP

The ambiguity of the decision situation due to imprecise information concerning the minimization ofthree objectives related to the net cost, the net rejections and the net late deliveries is captured and af-MIGP_VSP formulation is presented as follows:

fe) = Zx (21)

$ ( * , ) = Z2 (22)

= Z, (23)i=\

i=\

i#Ui; for all i = 1;2 n ð25Þ

xiÞ $ F ð26Þi=\

riðxiÞ$P ð27Þi=\

piðxiÞ #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 adata set represented in Table 1. The data relates to a realistic situation of a manufacturing sector dealingwith auto parts. The adopted situation can easily be extended to any other industry. The f-MIGP_VSPformulation is developed from requirements espoused by the situation during the initial stages ofimplementing a formal program for better management of its supply chain which subsequently undertooka vendor certification plan for its purchased items. Those vendors who successfully passed the screeningprocesses were eligible for procurement. Four established vendors for a projected demand of the part hadbeen screened for supplying this purchased item. A f-MIGP_VSP model is developed for the selection andthe quota allocations of the vendors from a list of four potential vendors under uncertain environments.


Table 1Vendor source data of the illustrative example

Vendor number

1234

pi ($)

5762

* (%)

0.050.0300.02

4 (%)

0.040.020.080.01

Ui (units)

5,00015,000

3600.03000

fi

0.020.010.060.04

0.88

0.910.970.85

Bi ($)

25,000100,00035,000

5500

The objective functions and constraint sets reflect the procurement requirements for a purchased item inthe supply chain. The three objectives, viz. minimizing the net cost, the net rejections, and the net latedeliveries 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 situationfaced by a firm. The vendor profiles shown in Table 1 represent the data set for the price quoted (pi in $) thepercentage rejections ðqiÞ; the percentage late deliveries ðdiÞ; vendors' capacities ðUiÞ; vendors' quotaflexibility ðfiÞ on a scale of 0 - 1 , vendor rating ðriÞ on a scale of 0 - 1 , and the budget allocations for thevendors ðBiÞ: The least value of flexibility in vendors' quota and least total purchase value of supplied itemsare policy decisions and depend on the demand. The least value of flexibility in suppliers' quota is given asF =fD and the least total purchase value of supplied items is given as P = rD: If overall flexibility ðfÞ is0.03 on the scale of 0 - 1 , the overall vendor rating ðrÞ is 0.92 on the scale of 0-1 and the aggregate demandðDÞ is 20,000, then the least value of flexibility in suppliers' quota ðFÞ and the least total purchase value ofsupplied 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 ofthat particular goal is first treated as the objective function of a MIP problem having constraint set same asthe one defined in MIGP_VSP formulation. For example, with reference to formulation (11), theaspiration-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 = {gjðxiÞ # aj,hk(xt) = bk;xi $ 0} and integer.The aspiration levels of the three fuzzy goals, viz. the minimization of the net cost, the net rejections andthe net late deliveries have thus been obtained as 125,000, 420 and 700, respectively. These values alsoindicate 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 setas 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 triangularmembership functions of the three fuzzy goals, viz. minimizing the net cost, minimizing the netrejections and minimizing the net deliveries are constructed as given in Eqs. (30)-(32).

145;000 - Z ^ J l f l 2 5 , 0 0 0 < Z l ( x ^ 145;000 ð30Þ

L 20;000

0; otherwise


520 2

ioo * } 'Lo,

if 420 , Z2ÐXIÞ # 520

otherwise

ð31Þ

[™g™] if 550^,) ,700150

850 2f 850 2 Z 3 ð x i Þ

L 150 3 i ;

.0,

if 700 , Zjfo) < 850

otherwise

ð32Þ

Analogous to the formulation (2)-(10), the mathematical formulation is developed. This MIP_VSPformulation is provided in Appendix A.

0 105000 125000 145000

(a) Membership functionof Z,(x,)

0 320 420 520

(b) Membership functionof Z2(xf)

0 550 700 850 z'<r')

(c) Membership functionof Z3(x,.)

Fig. 2. Membership function of different goals.


Now we summarize the complete solution procedure for the f-MIGP_VSP formulation through thefollowing 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_VSPformulation.Step 3: Construct the f-MIGP_VSP formulation, considering some of the parameters of the goals asfuzzy 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 thisproblem 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 quotaallocations 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 obtainedusing f-MIGP_VSP formulation. Once the membership functions of the three fuzzy goals of the case inillustration are defined, then the fuzzy problem (f-MIGP_VSP) formulation can be converted into anequivalent crisp (c-MIGP_VSP) formulation. The equivalent crisp (c-MIGP_VSP) formulation is givenas:

Maximize lsubject to

A < ð2 :5X1 þ 3:5X2 þ 3X3 þ X4Þ þ 7:25

A # ð2:5X1 þ 3:5X2 þ 3X3 þ X4Þ 2 5:25

A < ð5X1 þ 3X2 þ 2X4Þ þ 5:2

A # ð5X1 þ 3X2 þ 2X4Þ 2 3:2

A < ð2 :6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4Þ þ 5:6

A # ð2:6X1 þ 1:3X2 þ 5:3X3 þ 0:6X4Þ 2 3:6

X1 þ X2 þ X3 þ X4 = 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

2X4 # 5500

where Xi = 1024xi; xi $ 0 and integer, i = 1; 2; 3; 4:

The optimal solution for the above formulation is x1 =0, x2 = 12;714; x3 = 5336 and x4 = 1950:

In the optimal solution the degree of achievement of the fuzzy goals ðlmax = 0:996Þ is significantlyhigh. 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 quotadue to the most inferior performance on quality criterion and substantially poor performance onother criteria. However, vendor 3 has received more quota allocation as he performed the best onquality 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 tosupply. The natures of quota allocations on the different performance criteria of different vendorsare described in Table 2.

The c-MIGP_VSP formulation is solved for different degree of achievements for the fuzzy goals forcomparing the results. These results are shown in Fig. 3.

The results indicate that the quota allocations to the vendors are quite different when fuzzinessin some of the parameters of the problem is captured. As the values for the degree of theachievement of the fuzzy goals are increased, the quota allocations to the different vendors alsochange. The overall demand for the item is assumed to be constant. The quota allocations to thevendors 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 differentcriteria. This is expected even intuitively.

In some cases, few vendors such as the vendor 1 loses the entire quota to the other vendorswhen fuzziness is considered in the c-MIGP_VSP formulation. Initially, when the goals are notachieved vendor 1 has significant allocation. When the value for the degree of achievement of thefuzzy goals is increased then the quota allocation to the vendor 1 starts decreasing in aconsistent 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 theoptimal solution ðlmax = 0:996Þ: Initially, when the degrees of achievement of the fuzzy goals arezero, vendor 4 does not get any quota allocation. As the value for the degrees of achievement ofthe fuzzy goals increases, the quota allocation to the vendor 4 starts increasing in a consistentmanner. 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 maximumbudget allocation. The performance indicators for the vendor 2 are quite superior (viz. comparativelylower percentage rejections, lower late delivery percentage, comparatively better vendor rating, etc.).


Table 2Nature of the quota allocations at the optimal solution

Vendor number Quota allocation Performance criteria

12,714 (89% of capacity, 92% of hisallocated budget is consumed)

5336 (85% of capacity, 89% of hisallocated budget is consumed)

1950 (65% of capacity, 71% of hisallocated budget is consumed)

Inferior on the performance criteria (viz. highestpercentage rejections, high percentage latedeliveries, less vendor rating value, lessquota flexibility value, etc).Highest supplying capacity, highest budgetallocation, better on the performance criteria (viz. lesspercentage rejections, less percentage of late deliveries,comparatively better vendor rating value, etc.)Better on the performance criteria (viz. no rejectionsat all, best vendor rating value, high quota flexibility,highest vendor rating value, etc.)Less capacity to supply, less budget allocation, lessvendor rating value, less per unit cost, lesspercentage rejections, and less percentage of late deliveries

Vendor 2 receives more quota allocation for the higher values of the degree of achievement of the fuzzygoals. As the value of the degree of the achievement of the fuzzy goals is increased, the quota allocationto the vendor 2 also increases.

Vendor 3 is consistently superior on some performance criteria (viz. no rejections at all, highestvendor rating value and highest quota flexibility). The quota allocation to the vendor 3 has slightly

14000 -\

12000 -

10000 -

ore 8000 "o

s 6000

4000 "

2000 "

— • — Vendor 1- - - * - - - vendor 2— •— Vendor 3— *•••• Vendor4

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Degree of achievement of the fuzzy goals

0.9 1.0

Fig. 3. Quota allocations to the vendors at different degree of achievement of fuzzy goals.


decreased, yet it is quite stable and consistent over a range of values for the degree of the achievement ofthe fuzzy goals (Fig. 3).

5. Conclusion

The f-MIGP_VSP formulated in this paper is extremely useful for solving the VSP in a supply chainwhen the goals are not clearly stated. The formulation can effectively handle the vagueness andimprecision in the statement of the objectives. It presents a rational approach to decision-making processfor the VSP in a supply chain in the sense that there is no longer any distinction between the goals and theconstraints. Further, the f-MIGP_VSP formulation gives the optimal trade-offs among the values fordifferent goals for a VSP. Due to conflicting nature of the multiple objectives and vagueness in theinformation related to the parameters of the decision variables, the deterministic techniques areunsuitable to obtain an effective solution. The proposed f-MIGP_VSP formulation has the advantagesthat 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 realsituations, as very precise and deterministic information is generally not available for designing andmanaging a supply chain.

Appendix A

The MIP_VSP model containing three fuzzy goals, eleven crisp constraints and four decisionvariables can be formulated as follows:

G 1 :

G2 :

G 3 :

subject to

x1 þ

x1 #

x2 #

x3 #

x 4 #

5x1 þ 7x2 þ 6x3 þ 2x4 ø 125 ;000

0:05x1 þ 0:03x2 þ 0:02x4 ø 420

0:04x1 þ 0:02x2 þ 0:08x3 þ 0:01x4 ø 700

• x2 þ x3 þ x4 = 20 ;000

= 5000

• 15 ;000

• 6000

• 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

7x2

6x3

< 25;000

< 100;000

< 35;000


2x4 # 5500

xi $ 0 and integer; i — 1; 2; 3 ;4

where G1, G2, and G3 indicate goals for minimizing the net cost, minimizing the net rejections, andminimizing the net late deliveries, respectively.

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