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A multiobjective approach to vendor selection
European Journal of Operational Research 68 (1993) 173- 184 North-Holland
173
Theory and Methodology
A multiobjective approach to vendor selection Charles A. Weber and John R. Current Faculty of Management Sciences, The Ohio State University, Columbus, OH 43210-1399, USA Received May 1991; revised September 1991 Abstract: Purchases from vendors involve significant costs for many firms. Decisions related to these purchases include the selection of vendors and the determination of order quantities to be placed with the selected vendors. Such decisions are frequently multiobjective in nature. That is, they are evaluated by more than one criterion. At least 23 criteria for various vendor selection problems have been identified. In this article, we present a multiobjective approach to systematically analyze the inherent tradeoffs involved in multicriteria vendor selection problems. The approach is motivated by, and demonstrated with, an actual purchasing problem facing a division of a Fortune 500 company. Keywords: Multiobjective programming; Purohasing; Vendor selection; Decision support system
Introduction
The cost of raw materials and component parts purchased from external vendors is significant for most manufacturing firms. For example, the cost of components and parts purchased from external sources by large automotive manufacturers may total more than 50% of revenues. Coal purchases for large utilities such as T V A approach $1 billion annually (Bender et al., 1985). For high-technology firms, purchased material and services represent up to 80% of total product costs (Burton, 1988). Total purchases of services and goods from all US industries was conservatively estimated at approximately $4.211 trillion in 1982. The manufacturing sector accounted for approxi-
Correspondence to: J. Current, Faculty of Management Sciences, The Ohio State University, Columbus, OH 43210-1399, USA.
mately $1.2 trillion of this total with materials and supplies comprising over 80% of these purchases (Heberling, 1990). There are two basic decisions that need to be made in the vendor selection process. The firm must decide which vendors it should contract and it must determine the appropriate order quantity for each vendor selected. We refer to these decisions as the vendor selection problem. The problem is complicated by the fact that vendor selection is often an inherently multiobjective one. That is, the firm's decision may be driven by more than one objective. For example, Dickson (1966) identified 23 different criteria evaluated in the vendor selection process. In that article, quality was seen as being of extreme importance while delivery, performance history, warranties and claim policies, production facilities and capacity, price, technical capability and financial position were viewed as being of considerable importance
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
174
C.A. Weber, J.R. Current , / A multiobjective approach to vendor selection
in the vendor selection process. Wind and Robinson (1968) demonstrate that tradeoffs may exist among the various criteria and that these tradeoffs may not be apparent when using single-objective models. A review of criteria used in the vendors selection process is presented in Weber et al. (1991). Frequently, the relevant objectives are in conflict. For example, the vendor with the lowest per unit price may not have the best quality or delivery rating of the various vendors under consideration. Consequently, the firm must analyze the tradeoffs among the relevant criteria when making its vendor decisions. The analysis of these tradeoffs is particularly important in modern manufacturing strategies. For example, in JustIn-Time (JIT) environments the tradeoffs among price, quality, and delivery reliability are particularly important (e.g., Ansari and Modarress, 1986; Chapman, 1989; Rao and Scheraga, 1988). Given the economic importance and inherent complexity of the vendor selection process, it is somewhat surprising that little research has been devoted to developing mathematical programming techniques to address the problem. A recent review of vendor selection criteria and methods (Weber et al., 1991) identifies only ten such approaches. Four articles (Moore and Fearon, 1973; Anthony and Buffa, 1977; Kingsman, 1986; Pan, 1989) proposed linear programming approaches to vendor selection and four articles (Gaballa, 1974; Bender et al., 1985; Narasimhan and Stoynoff, 1986; Turner, 1988) proposed mixed integer programming approaches. Given the inherently multiobjective nature of many vendor selection decisions, it is even more surprising that only two articles structure the problem in terms of multiobjective mathematical programming techniques. Buffa and Jackson (1983) formulated the problem as a goal program and included goals to address quality, price and delivery criteria. Sharma et al. (1989) address a somewhat different problem than the one addressed here. Their goal programming formulation was designed to analyze strategic issues related to vendor selection and does not yield specific vendor selection or order quantity options. In this paper we propose a multiobjective approach to generate various vendor selection options which demonstrate the efficient tradeoffs among the relevant criteria. The approach was
motivated by a vendor selection problem faced by a division of a Fortune 500 company. Multiobjective analysis has several advantages over singleobjective analysis. For example, it allows the various criteria to be evaluated in their natural units of measurement (e.g., cost per delivered unit; percent of nonconforming units; percent of units delivered on time) and therefore, eliminate the necessity of transforming them to a common unit of measurement such as dollars. In addition, such techniques present the decision maker with a set of noninferior (or non-dominated) solutions. A major criticism of optimization approaches is that they provide the decision maker with a single, 'optimal' solution rather than with a set of alternative courses of action. As a consequence, the distinction between the roles of decision maker and analyst are blurred (Hall, 1985). By providing a set of noninferior alternatives and the tradeoffs associated with them, multiobjective techniques permit the decision maker to incorporate personal experience and insight in making his or her final decision. Another major advantage of multiobjective techniques is that they provide a methodology to analyze the impacts of strategic policy decisions. Such decisions frequently entail a reordering of the priorities on a firm's objectives. For example, the adoption of a JIT manufacturing strategy increases the emphasis on the quality and timeliness of delivery of components purchased from external sources (Hahn et al., 1983; Manoochehri, 1984). In addition, firms employing JIT strategies often attempt to reduce the number of vendors which supply material inputs (Hahn et al., 1983; Manoochehri, 1984). Changes in emphasis such as these often affect the cost that firms must pay for items purchased from vendors. The multiobjective approach to vendor selection presented in this paper provides decision makers with a method to systematically analyze the effects of policy decisions on the relevant criteria in their vendor selection decisions. The remainder of this paper is organized as follows. Mathematical formulations of the multiobjective vendor selection problem are presented in the next section. The approach is demonstrated in the third section with real-world application using data provided by a Fortune 500 company. A summary and conclusions are presented in the final section.
C.A. Weber, J.R. Current / A multiobjectice approach to vendor selection
Formulations
Vendor selection problems are applicationspecific. That is, the appropriate constraints and the relative importance of the objectives vary with the problem setting (Dempsey, 1978). Therefore, it is not possible to specifically model a single functional form which is appropriate for all potential scenarios. Consequently, we first present and discuss a general formulation of the problem. Then we present a specific formulation designed for a single-item, single-plant problem faced by a Fortune 500 company which employs a JIT manufacturing strategy. In general, the multiobjective vendor selection problem may be formulated as follows: (P1) Min
Z=
[Z,(x, y), Z2(x, y) ..... Zp(x, y)] (1)
subject to
fi(x, y)>__bi forall i = 1 . . . . n, g j ( x , y ) > b j for a l l j = l , . . . m ,
(2)
x > 0,
(4)
y
(5)
(0, 1)
(3)
where Zl(x, y), Z2(x, y),...,Zv(x, y) are the objective to be optimized, fi(x, y)> bi is the set of system constraints, gj(x, y)> bj is the set of policy constraints, and x represents the vector of order quantities for the various vendors and y represents the vector o~ binary variables which indicate which vendor are selected. As stated earlier, the specific functional forms assumed by (1)-(3) depend upon the particular application under analysis. A detailed description of these potential forms appears in Weber 1990). Potential objectives include: (i) Minimize the total monetary cost. (ii) Minimize the number of non-conforming a n d / o r rejected items. (iii) Minimize the number of early a n d / o r late deliveries. (iv) Minimize the total number of vendors employed. (v) Minimize orders from unstable (financial or political) regions. (vi) Minimize delivery distance (or time).
175
In (P1), we have divided the constraints into two sets: system and policy constraints. System constraints are defined as those that are not directly under the control of the purchasing department (or firm) while policy constraints are those which the purchasing department can directly influence. This division has not mathematical significance. However, it proved useful in discussions with the purchasing manager (PM), First, it helped the PM to structure the problem. Secondly, it emphasized the usefullness of the approach in analyzing the impacts of policy decisions. Potential system constraints include: (i) vendor capacities, (ii) demand satisfaction, (iii) minimum order quantities established by the vendors, and (iv) total purchasing budget. Potential policy constraints include: (i) minimum a n d / o r maximum order quantities placed with particular vendors, (ii) minimum a n d / o r maximum number of vendors to employ, (iii) geographic preferences, and (iv) minority a n d / o r handicapped vendor selection. Some constraints may fall into either of the constraint categories depending upon the specific scenario being modeled. For example, minority vendor selection may be required by government regulation. In such a case, it would fall into the system constraint category as opposed to the policy category. In general, in multiobjective models such as (P1), the objectives are in conflict. That is, no one solution exists which is optimal for all of the objectives. In such problems, the notion of optimality is replaced by that of nondominance or noninferiority. A nondominated or noninferior solution is one in which an improvement in any one objective value will result in the degradation of at least one of the other objective's values. Therefore, multiobjective models are used to generate various noninferior solutions to the problem rather than to identify a single optimal solution. The two most commonly used methods for generating noninferior solutions to multiobjective problems are the weighting method and the constraint method (Cohon, 1978). The weighting
C.A. Weber,J.R. Current / A multiobjectiveapproach to vendor selection
176
method solves a single-objective problem where the objective is a convex combination of the original objective functions. This convex combination is determined by assigning relative weights to the original objectives and combining them. Parameterizing the weighting schemes results in various noninferior solutions to the original multiobjective problem. The constraint method identifies noninferior solutions by optimizing one of the original objectives subject to constraints on the values for the other objectives. Various noninferior solutions are generated by varying the bounds on the other objectives. One drawback to the weighting method in mixed-integer programs such as (P1) is that it will only identify solutions on the convex hull of the noninferior solution set. Because the noninferior solution set for such problems is not convex, the constraint method must be used to identify 'duality gap' solutions which do not lie on the convex hull of the noninferior solution set. For a more detailed description of multiobjective solution techniques and the problems associated with identifying 'duality gap' solutions, the interested reader is referred to Cohon (1978). To demonstrate the applicability of multiobjective programming to the vendor selection problem, we now formulate a specific model. This particular formulation was designed for the single-item, single-plant vendor selection problem faced by a Fortune 500 company which employs a JIT manufacturing strategy. (P2) Min
Z = (Z1, Z2, Z3)
(6)
subject to
~ Xj > d,
(7)
/=1 Xj < min(v~, w~)Yj for all j,
(8)
Xj > max( v], w) ) Yj for all j,
(9)
n
Y'. Yj = p ,
(10)
j=l
Xj>0
for all j,
Yj~(0,1)
for all j,
(11)
(12)
where Z1 =
pjXj,
(13)
Z2 = E AjXj,
(14)
j=l n
j=l
~jX+,
Z3=
(15)
j=l
Xj = Quantity ordered from vendor j. Yj = 1 if vendor j is selected; 0, otherwise. d = Aggregate demand for item over the planning horizon. v~ = Maximum amount of business to be given to vendor j. v) = Minimum amount of business to be given to vendor j if selected. w/u= Maximum order quantity available from vendor j. w] = Minimum order quantity vendor j will supply if selected. p = Number of vendors to be selected. & = Net per unit purchase price from vendor j. hj = Percentage of units delivered late by vendor
j. /~j = Percentage of rejected units delivered by vendor j. Functional forms (7)-(15) were developed in consultation with the purchasing manager of the firm involved in the study. Constraint (7) ensures that the quantity ordered for the item from all selected vendors meets the quantity demanded during the planning horizon. Constraint set (8) ensures that a vendor's capacity to supply the item (or the extent of the firm's willingness to do business with that vendor) is not exceeded. Constraint set (9) establishes minimum order requirements for selected vendors. These may be established by either the firm or by the individual vendors. Constraint (10) establishes the number of vendors to be selected. Constraint set (11) prohibits negative orders and constraint set (12) reflects the binary nature of the vendor selection decision. The three objectives (13), (14) and (15) are formulated to minimize total monetary cost of purchases, total number of late deliveries, and total number of rejected units, respectively. Formulations (P1) and (P2) are mixed integer (zero-one) problems. It should be noted how-
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
Computational
ever, that in general, the number of zero-one variables will not be very large. Consequently, most problem instances can be solved using standard branch-and-bound techniques. This assertion is based on the fact that the formulations require one zero-one variable for each potential vendor and, in general, the number of potential vendors for any specific item is not very large (i.e., < 30). The existence of a relatively small number of potential vendors is particularly common for high-cost/high-volume items such as raw material inputs and major component parts. Given the total cost and high demand for such items, they are the most likely items for which the time and cost of systematic analysis are justified. A case in point is the application presented here. The vendor selection options presented in the next section are for the purchase of a refined material input. Approximately $14 million are spent yearly by one division for purchases of this material with the plant studied in this application accounting for about 20% of this total. Due to the volume of these purchases and the quality standards of the division, only six vendors compete for the firm's business.
results
The results presented in this study were generated on an IBM XT using General Optimization Incorporated's integer programming software package, 'What's Best'. The cost, delivery and quality data were provided by the purchasing firm. Given the large minimum order quantities established by the firm, we did not need to consider quantity discounts in purchase price. The delivery and quality data were determined using vendor quotations and prior experience with the vendors. The data appear in Weber (1990). To demonstrate the utility of the approach to the purchasing manager, (P2) was initially solved as a two-objective problem by assigning a weight of zero to the quality objective, Z 3. It is easier to interpret the tradeoffs in two-objective problems than it is in problems with more than two objectives. Consequently, this reduction allowed us to more readily demonstrate the analytical capabilities of the multiobjective formulation, as well as more readily solicit input from the purchasing manager regarding the relevant objectives and constraints and their appropriate functional
Delivery (Iba. late) 325
K
300 K 275 K Ii I
250 K 225
K
200 K 175
K
I 2.22M
177
I
I
2.24M
I 2.26M
I
I
I
2.28M
Price
($ Cost) (~ [-I
P = 4 Model S o l u t i o n s P = 5 Model S o l u t i o n s P = 6 Model S o l u t i o n s
Figure 1. Noninferior solutions for 4, 5 and 6 vendors
>
178
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
forms. For example, discussion of two-objective results led to the inclusion of a lower bound on order quantities, with the addition of constraint set (9). Six vendors were considered as potential suppliers of the item. Due to the capacity constraints on individual vendors, the total number of vendors selected, constraint (10), was set equal to 4, 5, and 6. For each of these values of p, five sets of weights were generated using the NISE method (Cohon, 1978). The NISE method is a variation of the weighting method and was selected because it quickly generates a good approximation of the underlying noninferior set. The actual weight used and the resulting solution values are presented in Tables 1-3. Figure 1 depicts these solutions graphically. This figure readily demonstrates the inherent tradeoffs among price, delivery and the value of p. For example, in this particular problem, the five-vendor solutions dominate the sixvendor solutions regardless of the weighting scheme employed. Also, at a delivery level of less than 198000 pounds of late delivered items, the p = 4 policy dominates the p = 5 and p = 6 vendor policies. This would not be the case if there was not a policy constraint on the minimum purchase from any selected vendor (set here at 40000 pounds) because vendors then could be selected but given an order of zero units. The noninferior solutions presented in Tables 1-3 were all generated using the weighting method. As stated earlier, noninferior 'duality gap' solutions may exist which cannot be identified by weighting methods. To identify these duality gap solutions, we employed the constraint method for identifying noninferior solutions. Table 4 lists the duality gap solutions identified for the five-vendor scenario. The tradeoffs inherent among these solutions and the convex hull solutions (from Table 2) are depicted in Figure 2. As Table 4 demonstrates, there may exist a large number of duality gap solutions. Consequently, an interactive approach similar to that proposed in Current et al. (1990) may be desirable. In essence, this approach only examines 'gaps' which appear promising to the decision maker. Additional two-objective noninferior solutions were generated by solving (P2) with only the price and quality objectives (Z~ and Z 3) included (i.e., Z 2 was assigned a weight of zero.) After reviewing various two-objective results with the purchas-
Table 1 Price vs. delivery,4-vendor model. Convexhull solutions Sol.
Obj.
NIS-1
Weights
Value
Percentage
Vendors
Order quantity
Z1 a 1.0 Zz b 0.0001
2230322 306700
1.000 1.750
NIS-2
Z1 Z2
2.0 1.0
2259755 198210
1.013 1.131
NIS-3
Z1 Z2
2.0 3.0
2265499 191030
1.015 1.090
NIS-4
Zl Z2
0.0001 1.0
2292655 175210
1.028 1.000
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 0 0 3000000 2890000 2500000 2400000 0 2783000 30OOOO0 2607000 0 2041000 0 2783000 3000000 2966000 0 0 0 2783000 3000000 2507000 2500000
Z1 = $ purchase cost. b Z2 = Units delivered late.
ing manager, the three-objective formulation was determined to accurately model the problem at hand. Noninferior solutions were generated for three values of p ( p = 4, 5 and 6). For each value of p, 5 sets of relative weights on the objectives were used. Duality gap solutions were also generated for the problems but they are not presented here. The convex hull solutions for the five-vendor scenario are presented in Table 5. The tradeoffs among the criteria are demonstrated using value paths (Schilling et al., 1983) in Figure 3. Value paths were used because they have proven to be an effective way to present the tradeoffs in problems with more than two objectives. In that figure there is a vertical axis for each of the three objectives. The value assigned to each noninferior solution (NIS) on a particular axis is that solution's value for the appropriate objective divided by the best five-vendor solution possible for that objective. Consequently, the minimum value for each axis is 1.00. For example, noninferior solutions 1, 3 and 2 are the five-vendor solutions
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
which minimize Z t, Z 2 and Z3, respectively. Therefore, NIS-1 has a value of 1 on the Z t(price) axis; NIS-3 has a value of 1 on the Z z(delivery) axis; and NIS-2 has a value of 1 on the Z 3- (quality) axis. Alternately, the axes could have been calibrated by the objective function values for the solutions rather than these percentages. Value paths are an effective way to demonstrate the tradeoffs among criteria in multiobjective problems with more than two objectives. For example, a quick analysis of Figure 3 shows that NIS-1 minimizes price but it has 82% more late deliveries and 53% more nonconforming items than do the best solutions for those objectives. Analysis of Table 5 and Figure 3 indicates that the selection of NIS-5 over the cost minimizing solution, NIS-1, would reduce late deliveries by 108490 pounds (33.78%) and reduce nonconformTable 2 Price vs. delivery, 5-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
Percen- Ven- Order tage dors quantity
NIS-1
Z1 a
1.0 0.0001
1.000 1.823
Z2 b
NIS-2
2221790 321000
Z1 Z2
2.5 1.0
2251223 212610
1.013 1.208
Z1 Z2
1.35 1.0
2258807 199810
1.016 1.135
NIS-4 Z 1 Ze
1.0 1.0
2265191 191830
1.019 1.089
NIS-5
0.0001 1.0
2292015 176010
1.031 1.000
NIS-3
Z2 Z2
a Z I = $ purchase cost. b Z2 = Units delivered late.
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 360000 0
Table 3 Price vs. delivery, 6-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
Percen- Ven- Quantity tage dors
NIS-1
Z1 a
1.0 0.0001
2222134 319900
1.000 1.801
NIS-2 Z l Z2
2.5 1.0
2243204 246400
1.009 1.387
NIS-3 Z 1 Z2
1.9 1.0
2252667 225890
1.014 1.272
Z2 b
NIS-4 Z 1
Z2
NIS-5 Z I Z2
1.5 1.0
2253307 224810
1.014 1.266
0.0001 1.0
2291067 177610
1.031 1.000
3OO0OOO 2530000 2500000 2400000 360000 2783000 3000000 2247000 0 2400000 40000 2783000 3000000 2567000 0 2001000 40000 2783000 3000000 2966000 0 40000 0 2783000 3000000 2467000 2500000
179
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V4 V6 V1 V2 V3 V4 V5 V6 VI V2 V3 V4 V5 V6 VI V2 V3 V4 V5 V6
2400000 360000 40000 3000000 2490000 2500000 2400000 360000 2490000 3000000 40000 2500000 2400000 67000 2783000 3000000
4OO0O 2500000 2400000 40000 2783000 3000000 67000 2500000 40000 40000 2783000 3000000 2427000 2500000
a Zl = $ purchase cost. b Z2 = Units delivered late.
ing items by 36509 pounds (19.96%) and increase total cost by only $29433 (1.32%). Figure 3 also demonstrates a major difference between the approach proposed here and the goal programming approach proposed in Buffa and Jackson (1983). In the goal program of Buffa and Jackson, the goals were first est~iblished for each of the criteria and priorities were set for the attainment of these goals. Once this was done, a single solution was identified which minimized the sum of the weighted deviations from the goals. The single set of weights are based upon the prespecified priorities established by the purchasing manager. Consequently, they essentially address the problem as a single-objective one, although they do recognize the existence of various objectives.
180
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
The approach proposed here does not require the purchasing manager to prespecify goals or priorities on their attainment. Rather, it generates various noninferior options which demonstrate the inherent tradeoffs among the objectives. After analyzing these tradeoffs, the purchasing manager can then decide which option is preferred. Of course, once the purchasing manager selects the option to be implemented, an implicit statement of his or her preferences among the objectives is made. However, these preferences (or priorities) need not be specified until after the various tradeoffs and options are known. If goals are known in advance, they can be included on the axes of the value paths and the attainment levels for the various solutions can be readily evaluated. Another distinction between (P2) and the Buffa and Jackson (1983) model is that (P2) permits the purchasing manager to establish the number of vendors to be selected and to establish minimum order quantities to be placed with selected vendors. These minimum order quantites may reflect purchaser preferences or vendor stipulations.
with them are important decisions for many firms. Such decisions may greatly affect a firm's ability to compete in the market place as they frequently account for a large portion of a product's production cost and may involve long-term contracts. Vendor selection decisions also affect the ability of a firm to effectively implement production strategies. For example, although price is important, delivery reliability and product quality take on increased importance in JIT manufacturing systems. As Dickson (1966) demonstrated, there are many important criteria to be evaluated in the vendor selection process. That article identified 23 such criteria. A recent review by Weber et al. (1990) indicated that vendor selection decisions continue to require the analysis of multiple criteria. Given the financial importance and multiobjective nature of many vendor selection decisions, we have developed a multiobjective programming approach to assist the purchasing manager in making such decisions. The approach allows the purchasing manager to generate noninferior purchasing options and systematically analyze the inherent tradeoffs among the relevant criteria. This is a particularly important feature in that it allows firms to analyze potential impacts of strategic options. The problem specific nature of most vendor selection decisions makes it impossible to formu-
Summary and conclusions The selection of vendors and the determination of appropriate order quantities to be placed J
Delivery (Ibs. late) 325
K
300
K
276 K
250
K~
225
K
200
K~
175
K--
\
I 2.22
I M
I 2.24
I M
I 2.26
I M
I 2.28
I
I )
M 2.30
M
Price
(S Cost) D Indicates duality gap solution
Figure 2. Noninferior solutions for P = 5 model, D indicates gap solution
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection Table 4 Price vs. delivery, 5-vendor model. Duality gap solutions Sol.
Obj.
NIS-1
Z1 a 1.0 Z2 b 0.0
N|S-2
Z~ Z2
NIS-3
Z1 Z2
NIS-4
NIS-5
Z1 Z2
Z~ Z2
NIS-6 Z I Z2
NIS-7 Z 1 Z2
NIS-8 Z 1 Z2
NIS-9 Z 1 Z2
Weights Value
Percentage
Vendors
Order quantity
2222441 320000
1.000 1.818
1.0 0.0
2228366 310000
1.003 1.761
1.0 0.0
2232242 300000
1.004 1.704
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 332500 0 3000000 2557000 2500000 2400000 82500 0 3000000 2807500 2500000 2400000 0 2233333 3000000 2666667 2500000 2400000 0 556666 3000000 2333334 2500000 2400000 0 890000 3000000 2000000 2500000 2400000 0 1223333 3000000 1666667 2500000 2400000 360000 2530000 3000000 0 2500000 2400000 360000 2064000 3000000 2966000 0 2400000 360000 2203000 3000000 2826667 0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
2235109 290000
2237975 280000
2240842 270000
2243548 245200
2245040 234180
2246238 230000
1.005 1.648
1.007 1.591
1.008 1.534
1.009 1.393
1.010 1.330
1.011 1.306
181
Table 4 (continued) Sol.
Obj. Weights Value
Percenrage
Ven- Order dors quantity V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
NIS-10 Z l Z2
1.0 0.0
2249105 220000
1.012 1.250
NIS-11 Z 1 Z2
1.0 0.0
2286423 183000
1.029 1.039
2400000 360000 2536667 3000000 2493333 0 3895500 0 2783000 3000000 2117500 2500000
a ZI = $ purchase cost. b Z2 = Units delivered late.
Table 5 Price vs. delivery vs. quality, 5-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
NIS-1
ZI a 1.0 Z2 b 0.0001 Z3 c 0.0001
Percentage
Vendors
Order quantity
2221790 321100 182870
1.00 1.82 1.53
1.01 1.39 1.00
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 360000 0 3000000 2530000 2500000 2400000 360000 2783000 2747000 0 2500000 40000 0 2783000 3000000 2467000 2500000 2001000 40000 2783000 3000000 2607000 0 2400000 360000 2783000 3000000 2247000 0
NIS-2 Z 1 Z2 Z3
0.0001 0.0001 1.0
2246659 245200 119367
NIS-3
0.0001 1.0 0.0001
2292015 1.03 176010 1.00 1 5 0 2 2 1 1.26
NIS-4 Z 1 Z2 Z3
1.25 1.0 1.0
2265191 191830 155550
1.02 1.09 1.30
NIS-5 Z 1 Z2 Z3
1.75 1.0 1.0
2251223 212610 160018
1.01 1.21 1.23
Z1
Z2 Z3
a Zx = $ purchase cost. b Z2 = Units delivered late. c Z3 = Units rejected.
CA. Weber, J.R. Current / A multiobjective approach to vendor selection
182
late a specific multiobjective program with general applicability. Consequently, we first formulated a general multiobjective programming approach to the problem and then developed a detailed model for a specific application. This model was developed in consultation with the purchasing manager of a Fortune 500 company to analyze the purchase of a single item for that firm. Purchases of this item cost the firm approximately 14 million dollars a year. Given that each of the firm's plants orders this item individually, the model was formulated as a single-plant problem. The plant used for the study spends over 2.5 million dollars a year on purchases of the item. The analysis was performed with the 'What's Best' integer programming software package on an IBM-XT personal computer using an 8086 chip (4mz). Execution times varied between two
~ 1.015
,. ~ L~
:\
and five minutes depending upon the availability of a starting basis from a previous solution. Faster times would be expected with the use of newer chip technology (e.g., 80286, 80386, 80486 chips) which were not available to us on this project. However, these times indicate that real-world problem instances can be efficiently analyzed using personal computers and readily available commercial software. Consequently, such analysis can be performed in-house either by the purchasing manager directly or by an analyst working interactively with the PM. To demonstrate the practicality of a multiobjective approach to vendor selection to the firm, we deliberately selected a single-plant,single-item application with high total cost. In addition, the large minimum order quantities established by the firm eliminated the need to consider quantity
/,
~:~~,""',/41'1 1.4~I.~ ,.,~
.
,.25:~
/ " ~'~'~::~:::
. j,,,"
./' "
/' nr
1.00
1.00 PRICE ($ COST)
I ......
- -
DELIVERY (Ibs. l a t e ) Solution Key:
Solution 1
2~
lOO
.----)
QUALITY (Ibs. rejected)
U ......
5 Figure 3. Percentage path analysis - price, quality, delivery
C.A. Weber, J.R. Current / A multiobjective approach to cendor selection
discounts on per unit cost. Clearly, future areas of research include extending the approach to multiple-plant, multiple-item problems and the inclusion of volume discounts. The number and specific form of the objectives were determined in consultation with the purchasing manager. The inclusion of more objectives would pose no theoretical problems; however, increased computational time (i.e., an increased number of noninferior solutions to be generated) and increased complexity of the output (i.e., more solutions and more tradeoffs to be analyzed) might create implementation problems. The practical implications of these potential problems are another area for future research. Finally, we have implicitly assumed that the firm is unable to influence vendor performance vis-fi-vis the objectives via negotiation or contractual inducements. The authors are currently examining methods to employ the results of the multiobjective approach with data envelopment analysis (Charnes et al., 1978) to develop benchmarks and strategies for the firm to incorporate in negotiations with the various vendors regarding their criteria performances.
Acknowledgement The authors wish to express their gratitude to the National Association of Purchasing Managers for partially funding this research under its Doctoral Grant program.
References Ansari, A., and Modarress, B. (1988), "JIT purchasing as a quality and productivity centre", International Journal of Production Research 26/1, 19-26. Anthony, T.F., and Buffa, F.P. (1977), "Strategic purchase scheduling", Journal of Purchasing and Materials Management fall, 27-31. Bender, P.S., Brown, R.W., Isaac, H., and J.F. Shapiro, (1985), "Improving purchasing productivity at IBM with a normative Decision Support System", Interfaces 15, 106-115. Buffa, F.P., and Jackson, W.M. Jackson, (1983), "A goal programming model for purchase planning", Journal of Purchasing and Materials Management fall, 27-34. Burton, T.T. (1988), "JIT repetitive sourcing strategies: "Tying the knot" with your suppliers", Production and Inventory Management Journal fourth quarter, 38-41.
183
Chapman, S.N. (1989), "Just-in-Time supplier inventory: An empirical implementation model", International Journal of Production Research 27, 1993-2007. Charnes, A., Cooper, W.W., and Rhodes, E. (1978), "Measuring the efficiency of decision-making units", European Journal of Operational Research 6, 429-444. Cohon, J.L. (1978), Multiobjectiue Programming and Planning, Academic Press, New York. Current, J.R., ReVelle, C.S., and Cohort, J.L. (1990), "An interactive approach to identify the best compromise solution for two objective shortest path problems", Computers and Operations Research 17, 187-198. Dempsey, W.A. (1979), "Vendor selection and the buying process", Industrial Marketing Management 7, 257-267. Dickson, G.W. (1966), "An analysis of vendor selection systems and decisions", Journal of Purchasing 2, 5 17. Gabbala, A.A. "Minimum cost allocation of tenders", Operations Research Quarterly 25, 389-98. Hahn, C.K., Pinto, P.A., and Bragg, D.J. (1983), "Just-in-Time production and purchasing", Journal of Purchasing and Materials Management fall, 2-10. Hall, R.W. (1985), "What's so scientific about M S / O R ? ' , Interfaces 15, 40-45. Heberling, M. (1990), "The dollar magnitude of purchases in American businesses and governments," in: Proceedings of the Fourth Purchasing and Materials Management Research Symposium, Lehigh University, October 26 27. Kingsman, B.G. (1986), "Purchasing raw materials with uncertain fluctuating prices", European Journal of Operational Research 25, 358-72. Manoochehri, G.H. (1984), "Suppliers and the Just-In-Time concept", Journal of Purchasing and Materials Management winter, 110-41. Moore, D.L., and Fearon, H.E. (1973), "Computer-assisted decision-making in purchasing", Journal of Purchasing 9, 5-25. Narasimhan, R., and Stoynoff, L.K. (1986), "Optimizing aggregate procurement allocation decisions", Journal of Purchasing and Materials Management spring, 23-30. Pan, A.C. (1989), "Allocation of order quantity among suppliers", Journal of Purchasing and Materials Management fall, 36-39. Rao, A., and Scheraga, D. (1988), "Moving from manufacturing resource planning to Just-In-Time manufacturing", Production and Inventory Management Journal first quarter, 44-49. Schilling, D.A., ReVelle, C., and Cohon, J. (1983), "An approach to the display and analysis of multiobjective problems", Socio-Economic Planning Science 17, 57-63. Sharma, D., Benton, W.C., and Srivastava, R. (1989), "Competitive strategy and purchasing decision", Proceedings of the 1989 Annual Conference of the Decision Sciences Institute, 1088-90. Turner, I. (1988), "An independent system for the evaluation of contract tenders", Journal of the Operations Research Society 39, 551-561. Weber, C.A. (1990), " A Decision Support System using multi-criteria techniques for vendor selection", Ph.D. Dissertation, College of Business, The Ohio State University, Columbus, OH.
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CA. Weber,J.R. Current / A multiobjectiveapproach to vendor selection
Weber, C.A., Current, J.R., and Benton, W.C. (1991), "Vendor selection criteria and methods", European Journal of Operational Research 50, 1-17. Wind, Y., and Robinson, P.J. (1968), "The determinants of
vendor selection: The evaluation function approach" Journal of Purchasing and Materials Management fall, 2941.
173
Theory and Methodology
A multiobjective approach to vendor selection Charles A. Weber and John R. Current Faculty of Management Sciences, The Ohio State University, Columbus, OH 43210-1399, USA Received May 1991; revised September 1991 Abstract: Purchases from vendors involve significant costs for many firms. Decisions related to these purchases include the selection of vendors and the determination of order quantities to be placed with the selected vendors. Such decisions are frequently multiobjective in nature. That is, they are evaluated by more than one criterion. At least 23 criteria for various vendor selection problems have been identified. In this article, we present a multiobjective approach to systematically analyze the inherent tradeoffs involved in multicriteria vendor selection problems. The approach is motivated by, and demonstrated with, an actual purchasing problem facing a division of a Fortune 500 company. Keywords: Multiobjective programming; Purohasing; Vendor selection; Decision support system
Introduction
The cost of raw materials and component parts purchased from external vendors is significant for most manufacturing firms. For example, the cost of components and parts purchased from external sources by large automotive manufacturers may total more than 50% of revenues. Coal purchases for large utilities such as T V A approach $1 billion annually (Bender et al., 1985). For high-technology firms, purchased material and services represent up to 80% of total product costs (Burton, 1988). Total purchases of services and goods from all US industries was conservatively estimated at approximately $4.211 trillion in 1982. The manufacturing sector accounted for approxi-
Correspondence to: J. Current, Faculty of Management Sciences, The Ohio State University, Columbus, OH 43210-1399, USA.
mately $1.2 trillion of this total with materials and supplies comprising over 80% of these purchases (Heberling, 1990). There are two basic decisions that need to be made in the vendor selection process. The firm must decide which vendors it should contract and it must determine the appropriate order quantity for each vendor selected. We refer to these decisions as the vendor selection problem. The problem is complicated by the fact that vendor selection is often an inherently multiobjective one. That is, the firm's decision may be driven by more than one objective. For example, Dickson (1966) identified 23 different criteria evaluated in the vendor selection process. In that article, quality was seen as being of extreme importance while delivery, performance history, warranties and claim policies, production facilities and capacity, price, technical capability and financial position were viewed as being of considerable importance
0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
174
C.A. Weber, J.R. Current , / A multiobjective approach to vendor selection
in the vendor selection process. Wind and Robinson (1968) demonstrate that tradeoffs may exist among the various criteria and that these tradeoffs may not be apparent when using single-objective models. A review of criteria used in the vendors selection process is presented in Weber et al. (1991). Frequently, the relevant objectives are in conflict. For example, the vendor with the lowest per unit price may not have the best quality or delivery rating of the various vendors under consideration. Consequently, the firm must analyze the tradeoffs among the relevant criteria when making its vendor decisions. The analysis of these tradeoffs is particularly important in modern manufacturing strategies. For example, in JustIn-Time (JIT) environments the tradeoffs among price, quality, and delivery reliability are particularly important (e.g., Ansari and Modarress, 1986; Chapman, 1989; Rao and Scheraga, 1988). Given the economic importance and inherent complexity of the vendor selection process, it is somewhat surprising that little research has been devoted to developing mathematical programming techniques to address the problem. A recent review of vendor selection criteria and methods (Weber et al., 1991) identifies only ten such approaches. Four articles (Moore and Fearon, 1973; Anthony and Buffa, 1977; Kingsman, 1986; Pan, 1989) proposed linear programming approaches to vendor selection and four articles (Gaballa, 1974; Bender et al., 1985; Narasimhan and Stoynoff, 1986; Turner, 1988) proposed mixed integer programming approaches. Given the inherently multiobjective nature of many vendor selection decisions, it is even more surprising that only two articles structure the problem in terms of multiobjective mathematical programming techniques. Buffa and Jackson (1983) formulated the problem as a goal program and included goals to address quality, price and delivery criteria. Sharma et al. (1989) address a somewhat different problem than the one addressed here. Their goal programming formulation was designed to analyze strategic issues related to vendor selection and does not yield specific vendor selection or order quantity options. In this paper we propose a multiobjective approach to generate various vendor selection options which demonstrate the efficient tradeoffs among the relevant criteria. The approach was
motivated by a vendor selection problem faced by a division of a Fortune 500 company. Multiobjective analysis has several advantages over singleobjective analysis. For example, it allows the various criteria to be evaluated in their natural units of measurement (e.g., cost per delivered unit; percent of nonconforming units; percent of units delivered on time) and therefore, eliminate the necessity of transforming them to a common unit of measurement such as dollars. In addition, such techniques present the decision maker with a set of noninferior (or non-dominated) solutions. A major criticism of optimization approaches is that they provide the decision maker with a single, 'optimal' solution rather than with a set of alternative courses of action. As a consequence, the distinction between the roles of decision maker and analyst are blurred (Hall, 1985). By providing a set of noninferior alternatives and the tradeoffs associated with them, multiobjective techniques permit the decision maker to incorporate personal experience and insight in making his or her final decision. Another major advantage of multiobjective techniques is that they provide a methodology to analyze the impacts of strategic policy decisions. Such decisions frequently entail a reordering of the priorities on a firm's objectives. For example, the adoption of a JIT manufacturing strategy increases the emphasis on the quality and timeliness of delivery of components purchased from external sources (Hahn et al., 1983; Manoochehri, 1984). In addition, firms employing JIT strategies often attempt to reduce the number of vendors which supply material inputs (Hahn et al., 1983; Manoochehri, 1984). Changes in emphasis such as these often affect the cost that firms must pay for items purchased from vendors. The multiobjective approach to vendor selection presented in this paper provides decision makers with a method to systematically analyze the effects of policy decisions on the relevant criteria in their vendor selection decisions. The remainder of this paper is organized as follows. Mathematical formulations of the multiobjective vendor selection problem are presented in the next section. The approach is demonstrated in the third section with real-world application using data provided by a Fortune 500 company. A summary and conclusions are presented in the final section.
C.A. Weber, J.R. Current / A multiobjectice approach to vendor selection
Formulations
Vendor selection problems are applicationspecific. That is, the appropriate constraints and the relative importance of the objectives vary with the problem setting (Dempsey, 1978). Therefore, it is not possible to specifically model a single functional form which is appropriate for all potential scenarios. Consequently, we first present and discuss a general formulation of the problem. Then we present a specific formulation designed for a single-item, single-plant problem faced by a Fortune 500 company which employs a JIT manufacturing strategy. In general, the multiobjective vendor selection problem may be formulated as follows: (P1) Min
Z=
[Z,(x, y), Z2(x, y) ..... Zp(x, y)] (1)
subject to
fi(x, y)>__bi forall i = 1 . . . . n, g j ( x , y ) > b j for a l l j = l , . . . m ,
(2)
x > 0,
(4)
y
(5)
(0, 1)
(3)
where Zl(x, y), Z2(x, y),...,Zv(x, y) are the objective to be optimized, fi(x, y)> bi is the set of system constraints, gj(x, y)> bj is the set of policy constraints, and x represents the vector of order quantities for the various vendors and y represents the vector o~ binary variables which indicate which vendor are selected. As stated earlier, the specific functional forms assumed by (1)-(3) depend upon the particular application under analysis. A detailed description of these potential forms appears in Weber 1990). Potential objectives include: (i) Minimize the total monetary cost. (ii) Minimize the number of non-conforming a n d / o r rejected items. (iii) Minimize the number of early a n d / o r late deliveries. (iv) Minimize the total number of vendors employed. (v) Minimize orders from unstable (financial or political) regions. (vi) Minimize delivery distance (or time).
175
In (P1), we have divided the constraints into two sets: system and policy constraints. System constraints are defined as those that are not directly under the control of the purchasing department (or firm) while policy constraints are those which the purchasing department can directly influence. This division has not mathematical significance. However, it proved useful in discussions with the purchasing manager (PM), First, it helped the PM to structure the problem. Secondly, it emphasized the usefullness of the approach in analyzing the impacts of policy decisions. Potential system constraints include: (i) vendor capacities, (ii) demand satisfaction, (iii) minimum order quantities established by the vendors, and (iv) total purchasing budget. Potential policy constraints include: (i) minimum a n d / o r maximum order quantities placed with particular vendors, (ii) minimum a n d / o r maximum number of vendors to employ, (iii) geographic preferences, and (iv) minority a n d / o r handicapped vendor selection. Some constraints may fall into either of the constraint categories depending upon the specific scenario being modeled. For example, minority vendor selection may be required by government regulation. In such a case, it would fall into the system constraint category as opposed to the policy category. In general, in multiobjective models such as (P1), the objectives are in conflict. That is, no one solution exists which is optimal for all of the objectives. In such problems, the notion of optimality is replaced by that of nondominance or noninferiority. A nondominated or noninferior solution is one in which an improvement in any one objective value will result in the degradation of at least one of the other objective's values. Therefore, multiobjective models are used to generate various noninferior solutions to the problem rather than to identify a single optimal solution. The two most commonly used methods for generating noninferior solutions to multiobjective problems are the weighting method and the constraint method (Cohon, 1978). The weighting
C.A. Weber,J.R. Current / A multiobjectiveapproach to vendor selection
176
method solves a single-objective problem where the objective is a convex combination of the original objective functions. This convex combination is determined by assigning relative weights to the original objectives and combining them. Parameterizing the weighting schemes results in various noninferior solutions to the original multiobjective problem. The constraint method identifies noninferior solutions by optimizing one of the original objectives subject to constraints on the values for the other objectives. Various noninferior solutions are generated by varying the bounds on the other objectives. One drawback to the weighting method in mixed-integer programs such as (P1) is that it will only identify solutions on the convex hull of the noninferior solution set. Because the noninferior solution set for such problems is not convex, the constraint method must be used to identify 'duality gap' solutions which do not lie on the convex hull of the noninferior solution set. For a more detailed description of multiobjective solution techniques and the problems associated with identifying 'duality gap' solutions, the interested reader is referred to Cohon (1978). To demonstrate the applicability of multiobjective programming to the vendor selection problem, we now formulate a specific model. This particular formulation was designed for the single-item, single-plant vendor selection problem faced by a Fortune 500 company which employs a JIT manufacturing strategy. (P2) Min
Z = (Z1, Z2, Z3)
(6)
subject to
~ Xj > d,
(7)
/=1 Xj < min(v~, w~)Yj for all j,
(8)
Xj > max( v], w) ) Yj for all j,
(9)
n
Y'. Yj = p ,
(10)
j=l
Xj>0
for all j,
Yj~(0,1)
for all j,
(11)
(12)
where Z1 =
pjXj,
(13)
Z2 = E AjXj,
(14)
j=l n
j=l
~jX+,
Z3=
(15)
j=l
Xj = Quantity ordered from vendor j. Yj = 1 if vendor j is selected; 0, otherwise. d = Aggregate demand for item over the planning horizon. v~ = Maximum amount of business to be given to vendor j. v) = Minimum amount of business to be given to vendor j if selected. w/u= Maximum order quantity available from vendor j. w] = Minimum order quantity vendor j will supply if selected. p = Number of vendors to be selected. & = Net per unit purchase price from vendor j. hj = Percentage of units delivered late by vendor
j. /~j = Percentage of rejected units delivered by vendor j. Functional forms (7)-(15) were developed in consultation with the purchasing manager of the firm involved in the study. Constraint (7) ensures that the quantity ordered for the item from all selected vendors meets the quantity demanded during the planning horizon. Constraint set (8) ensures that a vendor's capacity to supply the item (or the extent of the firm's willingness to do business with that vendor) is not exceeded. Constraint set (9) establishes minimum order requirements for selected vendors. These may be established by either the firm or by the individual vendors. Constraint (10) establishes the number of vendors to be selected. Constraint set (11) prohibits negative orders and constraint set (12) reflects the binary nature of the vendor selection decision. The three objectives (13), (14) and (15) are formulated to minimize total monetary cost of purchases, total number of late deliveries, and total number of rejected units, respectively. Formulations (P1) and (P2) are mixed integer (zero-one) problems. It should be noted how-
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
Computational
ever, that in general, the number of zero-one variables will not be very large. Consequently, most problem instances can be solved using standard branch-and-bound techniques. This assertion is based on the fact that the formulations require one zero-one variable for each potential vendor and, in general, the number of potential vendors for any specific item is not very large (i.e., < 30). The existence of a relatively small number of potential vendors is particularly common for high-cost/high-volume items such as raw material inputs and major component parts. Given the total cost and high demand for such items, they are the most likely items for which the time and cost of systematic analysis are justified. A case in point is the application presented here. The vendor selection options presented in the next section are for the purchase of a refined material input. Approximately $14 million are spent yearly by one division for purchases of this material with the plant studied in this application accounting for about 20% of this total. Due to the volume of these purchases and the quality standards of the division, only six vendors compete for the firm's business.
results
The results presented in this study were generated on an IBM XT using General Optimization Incorporated's integer programming software package, 'What's Best'. The cost, delivery and quality data were provided by the purchasing firm. Given the large minimum order quantities established by the firm, we did not need to consider quantity discounts in purchase price. The delivery and quality data were determined using vendor quotations and prior experience with the vendors. The data appear in Weber (1990). To demonstrate the utility of the approach to the purchasing manager, (P2) was initially solved as a two-objective problem by assigning a weight of zero to the quality objective, Z 3. It is easier to interpret the tradeoffs in two-objective problems than it is in problems with more than two objectives. Consequently, this reduction allowed us to more readily demonstrate the analytical capabilities of the multiobjective formulation, as well as more readily solicit input from the purchasing manager regarding the relevant objectives and constraints and their appropriate functional
Delivery (Iba. late) 325
K
300 K 275 K Ii I
250 K 225
K
200 K 175
K
I 2.22M
177
I
I
2.24M
I 2.26M
I
I
I
2.28M
Price
($ Cost) (~ [-I
P = 4 Model S o l u t i o n s P = 5 Model S o l u t i o n s P = 6 Model S o l u t i o n s
Figure 1. Noninferior solutions for 4, 5 and 6 vendors
>
178
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
forms. For example, discussion of two-objective results led to the inclusion of a lower bound on order quantities, with the addition of constraint set (9). Six vendors were considered as potential suppliers of the item. Due to the capacity constraints on individual vendors, the total number of vendors selected, constraint (10), was set equal to 4, 5, and 6. For each of these values of p, five sets of weights were generated using the NISE method (Cohon, 1978). The NISE method is a variation of the weighting method and was selected because it quickly generates a good approximation of the underlying noninferior set. The actual weight used and the resulting solution values are presented in Tables 1-3. Figure 1 depicts these solutions graphically. This figure readily demonstrates the inherent tradeoffs among price, delivery and the value of p. For example, in this particular problem, the five-vendor solutions dominate the sixvendor solutions regardless of the weighting scheme employed. Also, at a delivery level of less than 198000 pounds of late delivered items, the p = 4 policy dominates the p = 5 and p = 6 vendor policies. This would not be the case if there was not a policy constraint on the minimum purchase from any selected vendor (set here at 40000 pounds) because vendors then could be selected but given an order of zero units. The noninferior solutions presented in Tables 1-3 were all generated using the weighting method. As stated earlier, noninferior 'duality gap' solutions may exist which cannot be identified by weighting methods. To identify these duality gap solutions, we employed the constraint method for identifying noninferior solutions. Table 4 lists the duality gap solutions identified for the five-vendor scenario. The tradeoffs inherent among these solutions and the convex hull solutions (from Table 2) are depicted in Figure 2. As Table 4 demonstrates, there may exist a large number of duality gap solutions. Consequently, an interactive approach similar to that proposed in Current et al. (1990) may be desirable. In essence, this approach only examines 'gaps' which appear promising to the decision maker. Additional two-objective noninferior solutions were generated by solving (P2) with only the price and quality objectives (Z~ and Z 3) included (i.e., Z 2 was assigned a weight of zero.) After reviewing various two-objective results with the purchas-
Table 1 Price vs. delivery,4-vendor model. Convexhull solutions Sol.
Obj.
NIS-1
Weights
Value
Percentage
Vendors
Order quantity
Z1 a 1.0 Zz b 0.0001
2230322 306700
1.000 1.750
NIS-2
Z1 Z2
2.0 1.0
2259755 198210
1.013 1.131
NIS-3
Z1 Z2
2.0 3.0
2265499 191030
1.015 1.090
NIS-4
Zl Z2
0.0001 1.0
2292655 175210
1.028 1.000
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 0 0 3000000 2890000 2500000 2400000 0 2783000 30OOOO0 2607000 0 2041000 0 2783000 3000000 2966000 0 0 0 2783000 3000000 2507000 2500000
Z1 = $ purchase cost. b Z2 = Units delivered late.
ing manager, the three-objective formulation was determined to accurately model the problem at hand. Noninferior solutions were generated for three values of p ( p = 4, 5 and 6). For each value of p, 5 sets of relative weights on the objectives were used. Duality gap solutions were also generated for the problems but they are not presented here. The convex hull solutions for the five-vendor scenario are presented in Table 5. The tradeoffs among the criteria are demonstrated using value paths (Schilling et al., 1983) in Figure 3. Value paths were used because they have proven to be an effective way to present the tradeoffs in problems with more than two objectives. In that figure there is a vertical axis for each of the three objectives. The value assigned to each noninferior solution (NIS) on a particular axis is that solution's value for the appropriate objective divided by the best five-vendor solution possible for that objective. Consequently, the minimum value for each axis is 1.00. For example, noninferior solutions 1, 3 and 2 are the five-vendor solutions
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
which minimize Z t, Z 2 and Z3, respectively. Therefore, NIS-1 has a value of 1 on the Z t(price) axis; NIS-3 has a value of 1 on the Z z(delivery) axis; and NIS-2 has a value of 1 on the Z 3- (quality) axis. Alternately, the axes could have been calibrated by the objective function values for the solutions rather than these percentages. Value paths are an effective way to demonstrate the tradeoffs among criteria in multiobjective problems with more than two objectives. For example, a quick analysis of Figure 3 shows that NIS-1 minimizes price but it has 82% more late deliveries and 53% more nonconforming items than do the best solutions for those objectives. Analysis of Table 5 and Figure 3 indicates that the selection of NIS-5 over the cost minimizing solution, NIS-1, would reduce late deliveries by 108490 pounds (33.78%) and reduce nonconformTable 2 Price vs. delivery, 5-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
Percen- Ven- Order tage dors quantity
NIS-1
Z1 a
1.0 0.0001
1.000 1.823
Z2 b
NIS-2
2221790 321000
Z1 Z2
2.5 1.0
2251223 212610
1.013 1.208
Z1 Z2
1.35 1.0
2258807 199810
1.016 1.135
NIS-4 Z 1 Ze
1.0 1.0
2265191 191830
1.019 1.089
NIS-5
0.0001 1.0
2292015 176010
1.031 1.000
NIS-3
Z2 Z2
a Z I = $ purchase cost. b Z2 = Units delivered late.
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 360000 0
Table 3 Price vs. delivery, 6-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
Percen- Ven- Quantity tage dors
NIS-1
Z1 a
1.0 0.0001
2222134 319900
1.000 1.801
NIS-2 Z l Z2
2.5 1.0
2243204 246400
1.009 1.387
NIS-3 Z 1 Z2
1.9 1.0
2252667 225890
1.014 1.272
Z2 b
NIS-4 Z 1
Z2
NIS-5 Z I Z2
1.5 1.0
2253307 224810
1.014 1.266
0.0001 1.0
2291067 177610
1.031 1.000
3OO0OOO 2530000 2500000 2400000 360000 2783000 3000000 2247000 0 2400000 40000 2783000 3000000 2567000 0 2001000 40000 2783000 3000000 2966000 0 40000 0 2783000 3000000 2467000 2500000
179
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V4 V6 V1 V2 V3 V4 V5 V6 VI V2 V3 V4 V5 V6 VI V2 V3 V4 V5 V6
2400000 360000 40000 3000000 2490000 2500000 2400000 360000 2490000 3000000 40000 2500000 2400000 67000 2783000 3000000
4OO0O 2500000 2400000 40000 2783000 3000000 67000 2500000 40000 40000 2783000 3000000 2427000 2500000
a Zl = $ purchase cost. b Z2 = Units delivered late.
ing items by 36509 pounds (19.96%) and increase total cost by only $29433 (1.32%). Figure 3 also demonstrates a major difference between the approach proposed here and the goal programming approach proposed in Buffa and Jackson (1983). In the goal program of Buffa and Jackson, the goals were first est~iblished for each of the criteria and priorities were set for the attainment of these goals. Once this was done, a single solution was identified which minimized the sum of the weighted deviations from the goals. The single set of weights are based upon the prespecified priorities established by the purchasing manager. Consequently, they essentially address the problem as a single-objective one, although they do recognize the existence of various objectives.
180
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection
The approach proposed here does not require the purchasing manager to prespecify goals or priorities on their attainment. Rather, it generates various noninferior options which demonstrate the inherent tradeoffs among the objectives. After analyzing these tradeoffs, the purchasing manager can then decide which option is preferred. Of course, once the purchasing manager selects the option to be implemented, an implicit statement of his or her preferences among the objectives is made. However, these preferences (or priorities) need not be specified until after the various tradeoffs and options are known. If goals are known in advance, they can be included on the axes of the value paths and the attainment levels for the various solutions can be readily evaluated. Another distinction between (P2) and the Buffa and Jackson (1983) model is that (P2) permits the purchasing manager to establish the number of vendors to be selected and to establish minimum order quantities to be placed with selected vendors. These minimum order quantites may reflect purchaser preferences or vendor stipulations.
with them are important decisions for many firms. Such decisions may greatly affect a firm's ability to compete in the market place as they frequently account for a large portion of a product's production cost and may involve long-term contracts. Vendor selection decisions also affect the ability of a firm to effectively implement production strategies. For example, although price is important, delivery reliability and product quality take on increased importance in JIT manufacturing systems. As Dickson (1966) demonstrated, there are many important criteria to be evaluated in the vendor selection process. That article identified 23 such criteria. A recent review by Weber et al. (1990) indicated that vendor selection decisions continue to require the analysis of multiple criteria. Given the financial importance and multiobjective nature of many vendor selection decisions, we have developed a multiobjective programming approach to assist the purchasing manager in making such decisions. The approach allows the purchasing manager to generate noninferior purchasing options and systematically analyze the inherent tradeoffs among the relevant criteria. This is a particularly important feature in that it allows firms to analyze potential impacts of strategic options. The problem specific nature of most vendor selection decisions makes it impossible to formu-
Summary and conclusions The selection of vendors and the determination of appropriate order quantities to be placed J
Delivery (Ibs. late) 325
K
300
K
276 K
250
K~
225
K
200
K~
175
K--
\
I 2.22
I M
I 2.24
I M
I 2.26
I M
I 2.28
I
I )
M 2.30
M
Price
(S Cost) D Indicates duality gap solution
Figure 2. Noninferior solutions for P = 5 model, D indicates gap solution
C.A. Weber, J.R. Current / A multiobjective approach to vendor selection Table 4 Price vs. delivery, 5-vendor model. Duality gap solutions Sol.
Obj.
NIS-1
Z1 a 1.0 Z2 b 0.0
N|S-2
Z~ Z2
NIS-3
Z1 Z2
NIS-4
NIS-5
Z1 Z2
Z~ Z2
NIS-6 Z I Z2
NIS-7 Z 1 Z2
NIS-8 Z 1 Z2
NIS-9 Z 1 Z2
Weights Value
Percentage
Vendors
Order quantity
2222441 320000
1.000 1.818
1.0 0.0
2228366 310000
1.003 1.761
1.0 0.0
2232242 300000
1.004 1.704
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 332500 0 3000000 2557000 2500000 2400000 82500 0 3000000 2807500 2500000 2400000 0 2233333 3000000 2666667 2500000 2400000 0 556666 3000000 2333334 2500000 2400000 0 890000 3000000 2000000 2500000 2400000 0 1223333 3000000 1666667 2500000 2400000 360000 2530000 3000000 0 2500000 2400000 360000 2064000 3000000 2966000 0 2400000 360000 2203000 3000000 2826667 0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
2235109 290000
2237975 280000
2240842 270000
2243548 245200
2245040 234180
2246238 230000
1.005 1.648
1.007 1.591
1.008 1.534
1.009 1.393
1.010 1.330
1.011 1.306
181
Table 4 (continued) Sol.
Obj. Weights Value
Percenrage
Ven- Order dors quantity V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
NIS-10 Z l Z2
1.0 0.0
2249105 220000
1.012 1.250
NIS-11 Z 1 Z2
1.0 0.0
2286423 183000
1.029 1.039
2400000 360000 2536667 3000000 2493333 0 3895500 0 2783000 3000000 2117500 2500000
a ZI = $ purchase cost. b Z2 = Units delivered late.
Table 5 Price vs. delivery vs. quality, 5-vendor model. Convex hull solutions Sol.
Obj.
Weights Value
NIS-1
ZI a 1.0 Z2 b 0.0001 Z3 c 0.0001
Percentage
Vendors
Order quantity
2221790 321100 182870
1.00 1.82 1.53
1.01 1.39 1.00
V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6 V1 V2 V3 V4 V5 V6
2400000 360000 0 3000000 2530000 2500000 2400000 360000 2783000 2747000 0 2500000 40000 0 2783000 3000000 2467000 2500000 2001000 40000 2783000 3000000 2607000 0 2400000 360000 2783000 3000000 2247000 0
NIS-2 Z 1 Z2 Z3
0.0001 0.0001 1.0
2246659 245200 119367
NIS-3
0.0001 1.0 0.0001
2292015 1.03 176010 1.00 1 5 0 2 2 1 1.26
NIS-4 Z 1 Z2 Z3
1.25 1.0 1.0
2265191 191830 155550
1.02 1.09 1.30
NIS-5 Z 1 Z2 Z3
1.75 1.0 1.0
2251223 212610 160018
1.01 1.21 1.23
Z1
Z2 Z3
a Zx = $ purchase cost. b Z2 = Units delivered late. c Z3 = Units rejected.
CA. Weber, J.R. Current / A multiobjective approach to vendor selection
182
late a specific multiobjective program with general applicability. Consequently, we first formulated a general multiobjective programming approach to the problem and then developed a detailed model for a specific application. This model was developed in consultation with the purchasing manager of a Fortune 500 company to analyze the purchase of a single item for that firm. Purchases of this item cost the firm approximately 14 million dollars a year. Given that each of the firm's plants orders this item individually, the model was formulated as a single-plant problem. The plant used for the study spends over 2.5 million dollars a year on purchases of the item. The analysis was performed with the 'What's Best' integer programming software package on an IBM-XT personal computer using an 8086 chip (4mz). Execution times varied between two
~ 1.015
,. ~ L~
:\
and five minutes depending upon the availability of a starting basis from a previous solution. Faster times would be expected with the use of newer chip technology (e.g., 80286, 80386, 80486 chips) which were not available to us on this project. However, these times indicate that real-world problem instances can be efficiently analyzed using personal computers and readily available commercial software. Consequently, such analysis can be performed in-house either by the purchasing manager directly or by an analyst working interactively with the PM. To demonstrate the practicality of a multiobjective approach to vendor selection to the firm, we deliberately selected a single-plant,single-item application with high total cost. In addition, the large minimum order quantities established by the firm eliminated the need to consider quantity
/,
~:~~,""',/41'1 1.4~I.~ ,.,~
.
,.25:~
/ " ~'~'~::~:::
. j,,,"
./' "
/' nr
1.00
1.00 PRICE ($ COST)
I ......
- -
DELIVERY (Ibs. l a t e ) Solution Key:
Solution 1
2~
lOO
.----)
QUALITY (Ibs. rejected)
U ......
5 Figure 3. Percentage path analysis - price, quality, delivery
C.A. Weber, J.R. Current / A multiobjective approach to cendor selection
discounts on per unit cost. Clearly, future areas of research include extending the approach to multiple-plant, multiple-item problems and the inclusion of volume discounts. The number and specific form of the objectives were determined in consultation with the purchasing manager. The inclusion of more objectives would pose no theoretical problems; however, increased computational time (i.e., an increased number of noninferior solutions to be generated) and increased complexity of the output (i.e., more solutions and more tradeoffs to be analyzed) might create implementation problems. The practical implications of these potential problems are another area for future research. Finally, we have implicitly assumed that the firm is unable to influence vendor performance vis-fi-vis the objectives via negotiation or contractual inducements. The authors are currently examining methods to employ the results of the multiobjective approach with data envelopment analysis (Charnes et al., 1978) to develop benchmarks and strategies for the firm to incorporate in negotiations with the various vendors regarding their criteria performances.
Acknowledgement The authors wish to express their gratitude to the National Association of Purchasing Managers for partially funding this research under its Doctoral Grant program.
References Ansari, A., and Modarress, B. (1988), "JIT purchasing as a quality and productivity centre", International Journal of Production Research 26/1, 19-26. Anthony, T.F., and Buffa, F.P. (1977), "Strategic purchase scheduling", Journal of Purchasing and Materials Management fall, 27-31. Bender, P.S., Brown, R.W., Isaac, H., and J.F. Shapiro, (1985), "Improving purchasing productivity at IBM with a normative Decision Support System", Interfaces 15, 106-115. Buffa, F.P., and Jackson, W.M. Jackson, (1983), "A goal programming model for purchase planning", Journal of Purchasing and Materials Management fall, 27-34. Burton, T.T. (1988), "JIT repetitive sourcing strategies: "Tying the knot" with your suppliers", Production and Inventory Management Journal fourth quarter, 38-41.
183
Chapman, S.N. (1989), "Just-in-Time supplier inventory: An empirical implementation model", International Journal of Production Research 27, 1993-2007. Charnes, A., Cooper, W.W., and Rhodes, E. (1978), "Measuring the efficiency of decision-making units", European Journal of Operational Research 6, 429-444. Cohon, J.L. (1978), Multiobjectiue Programming and Planning, Academic Press, New York. Current, J.R., ReVelle, C.S., and Cohort, J.L. (1990), "An interactive approach to identify the best compromise solution for two objective shortest path problems", Computers and Operations Research 17, 187-198. Dempsey, W.A. (1979), "Vendor selection and the buying process", Industrial Marketing Management 7, 257-267. Dickson, G.W. (1966), "An analysis of vendor selection systems and decisions", Journal of Purchasing 2, 5 17. Gabbala, A.A. "Minimum cost allocation of tenders", Operations Research Quarterly 25, 389-98. Hahn, C.K., Pinto, P.A., and Bragg, D.J. (1983), "Just-in-Time production and purchasing", Journal of Purchasing and Materials Management fall, 2-10. Hall, R.W. (1985), "What's so scientific about M S / O R ? ' , Interfaces 15, 40-45. Heberling, M. (1990), "The dollar magnitude of purchases in American businesses and governments," in: Proceedings of the Fourth Purchasing and Materials Management Research Symposium, Lehigh University, October 26 27. Kingsman, B.G. (1986), "Purchasing raw materials with uncertain fluctuating prices", European Journal of Operational Research 25, 358-72. Manoochehri, G.H. (1984), "Suppliers and the Just-In-Time concept", Journal of Purchasing and Materials Management winter, 110-41. Moore, D.L., and Fearon, H.E. (1973), "Computer-assisted decision-making in purchasing", Journal of Purchasing 9, 5-25. Narasimhan, R., and Stoynoff, L.K. (1986), "Optimizing aggregate procurement allocation decisions", Journal of Purchasing and Materials Management spring, 23-30. Pan, A.C. (1989), "Allocation of order quantity among suppliers", Journal of Purchasing and Materials Management fall, 36-39. Rao, A., and Scheraga, D. (1988), "Moving from manufacturing resource planning to Just-In-Time manufacturing", Production and Inventory Management Journal first quarter, 44-49. Schilling, D.A., ReVelle, C., and Cohon, J. (1983), "An approach to the display and analysis of multiobjective problems", Socio-Economic Planning Science 17, 57-63. Sharma, D., Benton, W.C., and Srivastava, R. (1989), "Competitive strategy and purchasing decision", Proceedings of the 1989 Annual Conference of the Decision Sciences Institute, 1088-90. Turner, I. (1988), "An independent system for the evaluation of contract tenders", Journal of the Operations Research Society 39, 551-561. Weber, C.A. (1990), " A Decision Support System using multi-criteria techniques for vendor selection", Ph.D. Dissertation, College of Business, The Ohio State University, Columbus, OH.
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CA. Weber,J.R. Current / A multiobjectiveapproach to vendor selection
Weber, C.A., Current, J.R., and Benton, W.C. (1991), "Vendor selection criteria and methods", European Journal of Operational Research 50, 1-17. Wind, Y., and Robinson, P.J. (1968), "The determinants of
vendor selection: The evaluation function approach" Journal of Purchasing and Materials Management fall, 2941.