A FUZZY EXTENT ANALYSIS METHOD FOR VENDOR SELECTION IN SUPPLY CHAIN: A CASE STUDY FROM THE AUTOMOTIVE INDUSTRY

Kumar, M

ABSTRACT

The analytic hierarchy process (AHP) of Saaty (1980) is a popular method for solving multi-criteria decision problems involving qualitative data, and has been applied to the vendor selection process (Nydick and Hill, 1992; Barbarosoglu and Yazgac, 1997; Narasimhan, 1983; and Masella and Rangone, 2000). AHP is often criticized for its inability to handle the inherent uncertainty and imprecision of the pairwise comparison process. This paper focuses on addressing uncertainty issues in the process of vendor selection by using a Fuzzy Extent Analysis method. The applicability of the proposed Fuzzy Extent Analysis method for vendor selection is tested in Auto Exhaust System Ltd, Gurgaon, India. A study of the Fuzzy Extent Analysis Method is also provided in this paper by experimenting with sensitivity analysis. The result shows that the Fuzzy Extent Analysis Method for vendor selection is simple and comprehensive in concept, efficient in computation, and robust in modeling human evaluation processes for vendor selection.

OPSOMMING

Die analitiese hiërargieproses (AHP) van Saaty (1980) bied ‘n populêre metode vir die oplossing van veeldoelige kwalitatiewe besluitvormingsprobleme. Dit is reeds toegepas op gevalle van leweransierseleksie (Nydick en Hill, 1992; Barbarosoglu en Yazgac, 1997; Narasimhan, 1983; en Masella en Rangone, 2000). Die AHP-metode word dikwels gekritiseer omdat dit nie inherente onsekerhede en onakkuraatheid van paargewyse vergelykings kan hanteer nie. Die artikel omseil die kritiek deur gebruik te maak van ‘n Wasigheidsleermetode. Die metode word beproef vir ‘n leweransierseleksievraagstuk by Auto Exhaust Systems Bpk., Gurgaon, Indië. Daar word ook geëksperimenteer met sensitiwiteitsanalise wat bewys lewer van die doeltreffendheid van die metode van ontleding.

(ProQuest: … denotes formulae omitted.)

1. INTRODUCTION

Supply chain is a network of facilities and distribution options that perform the functions of procuring materials, transforming these materials into intermediate and finished products, and distributing these products to customers. One important factor in the supply chain is vendor selection in the purchasing process. This decision is even more important because strategic partnerships are being formed with vendors for competitive advantage. The main purchasing objective of an item is to obtain the lowest possible price by creating strong competition between vendors, and negotiating with them. However, in the modern business world, many firms prefer a strategy of using few vendors. ‘Few vendor’ strategies imply that a buyer wants to have a long-term relationship with, and the cooperation of, a few dedicated vendors. In view of the high percentage of the purchased material cost in the total cost, the key objective of the purchasing department ought to be purchasing the right quality of a product in the right quantity from the right vendors at the right time. The right vendors can provide the right quality of material on time at a reasonable price. Vendors can enhance the effectiveness of product and process design by ensuring reliability, reducing system wide inventories, reducing total system costs, and optimizing the flow of information throughout the system and the quality of the supply of materials, components, and services in the supply chain. Therefore, vendor selection has an important role in the supply chain process. The vendor selection process is a multi-criteria decision, including many tangible and intangible criteria with uncertainty inherent in each criterion (Kumar et al, 2006a). Kumar et al (2006a) provided several reasons for imprecision and fuzziness in the process of vendor selection. Fuzzy set theories are employed owing to vagueness and imprecision in the vendor selection process, and are used to transform the imprecise and vague information in the criteria into fuzzy criteria. Chang’s (1992, 1996) Fuzzy Extent Analysis method is able to handle tangible and intangible criteria with the inherent uncertainty involved in the vendor selection decision. Hence, in this paper the application of the Fuzzy Extent Analysis for the formulation and solution of the vendor selection process is developed to incorporate imprecision in estimating the decision parameters.

The rest of the paper is organized as follows. Section 2 presents a brief literature review of the existing quantitative approaches related to the vendor selection decision, and criticism of the Analytic Hierarchy Process (AHP). Section 3 describes the vendor selection process in a case company as well as the Fuzzy Extent Analysis method for vendor selection process, and demonstrates the usefulness of the proposed method with the help of a case illustration from an automobile manufacturer. A sensitivity analysis of the proposed model is also shown in this section. The main conclusions derived from this research are presented in Section 4.

2. LITERATURE REVIEW

There are many decision methods reported in the literature for supporting the vendor selection process. Different methods used for vendor selection are:

(i) Analytic Hierarchy Process (AHP) and Analytical Network Process (ANP),

(ii) Mathematical programming models

(iii) Data Envelopment Analysis (DEA)

(iv) Other methods

The analytic hierarchy process (AHP) of Saaty (1980) is a popular method for solving multi-criteria decision problems including tangible and intangible criteria, and has been applied to the vendor selection process. Nydick and Hill (1992), Barbarosoglu and Yazgac (1997), Narasimhan (1983), and Masella and Rangone (2000) proposed the use of the analytic hierarchy process (AHP) to deal with imprecision in vendor choice. Narasimhan (1983) used the analytical hierarchical process (AHP) to generate weights for the vendor selection decision. Partovi et al (1990) reviewed the published applications of AHP in vendor selection. Ghodsypour and O’Brien (1998) proposed an integration of an analytical hierarchy process and linear programming to consider both tangible and intangible factors in choosing the best vendors and placing the optimum order quantities among them such that the total value of purchasing is maximised. Sarkis and Talluri (2000) proposed the use of the analytical network process (ANP), a more sophisticated version of AHP, for vendor selection.

Mathematical programming approaches include only tangible criteria, and have been extensively used for vendor selection. They include fuzzy programming, utility theory and chance constrained programming, utility theory and interval programming, fuzzy goal programming, interval-programming, mixed integer nonlinear programming, and stochastic integer programming. Kumar et al (2006a) developed a fuzzy programming approach to deal with the effect of information uncertainty in the constraints of vendor selection process, and showed the results at increasing levels of uncertainty. Kumar et al (2006b) developed a solution procedure based on utility theory and chance constrained programming to solve a supplier quota allocation problem where a realistic constraint in budget allocation to individual vendors was a random variable, and showed the results at various degrees of uncertainty. Kumar (2006) proposed a solution procedure based on a hybrid approach of utility theory and interval programming to solve multi-objective vendor selection problems where some of the parameters were uncertain. Kumar et al (2004) developed a fuzzy goal programming approach to deal with the effect of information uncertainty in the objectives of vendor selection process, and showed how the quota allocation of vendors is varied with uncertainty. Kumar (2004) developed an interval-programming model that incorporates sales revenue as a maximization goal with realistic constraints such as number of rejections, service level, on-time delivery, buyers’ demand, and vendors’ budget amount, etc. Ghodsypour and O’Brien (2001) presented a mixed integer non-linear programming model to solve the multiple sourcing problem, which takes into account the total cost of logistics, including net price, storage, transportation, and ordering costs. Feng et al (2001) presented a stochastic integer programming approach for simultaneous selection of tolerances and vendors based on the quality loss function and process capability indices.

Data envelopment analysis (DEA) includes tangible criteria and is used to compare the efficiency of vendors. Weber et al (2000) presented the data envelopment analysis (DEA) method for the multiple objective vendor selection decision. Braglia and Petroni (2000) described a multi-attribute utility theory based on the use of DEA, aimed at helping purchasing managers to formulate viable sourcing strategies in the changing market place.

Other methods on vendor selection decisions include the genetic algorithm, casebased reasoning, the multi-attribute selection model, and the fuzzy expert system. Ohdar and Ray (2004) developed a genetic algorithm-based methodology to evolve the optimal set of fuzzy rules base, and a fuzzy inference system of the MATLAB fuzzy logic toolbox to assess the suppliers’ performance. Choy and Lee (2002) proposed a case-based supplier management tool (CBSMT) using the case-based reasoning (CBR) technique in the areas of intelligent supplier selection and management that will enhance performance, compared with the traditional approach. Barla (2003) did a case study of vendor selection and evaluation for a manufacturing company under lean philosophy. In order to reduce the vendor base, the vendor selection and evaluation study is conducted using the multi-attribute selection model. Kwong et al (2002) developed an approach that combined the scoring method and the fuzzy expert system for vendor assessment.

Criticism of Analytic Hierarchy Process (AHP): AHP is not the panacea for realworld decision-making problems. The AHP is criticized by Dyer (1990) and Triantaphyllou (2001), despite its popularity. AHP is criticized for its inability to deal with uncertainty and imprecision of the decision maker’s perceptions (Deng, 1999). The major drawback of AHP is that it fails to address the uncertainty in expressing the preferences during pairwise comparison (PC). The inability of the AHP to address imprecision and uncertainty paved the way for the incorporation of fuzzy logic into the AHP (Deng, 1999). In order to overcome the shortcomings of the AHP, fuzzy set theory is used to integrate AHP to determine the best alternative (Cheng, 1996). The integration of fuzzy set and AHP gives a much better and more exact representation of the relationship between criteria and alternatives (Karsak and Tolga, 2001). Hence, in this paper the Fuzzy Extent Analysis method is used for vendor selection in supply chains.

3. FUZZY EXTENT ANALYSIS METHOD FOR VENDOR SELECTION

3.1 Vendor selection process:

The vendor selection decision is a complex process involving various criteria. These criteria may vary depending on the type of product being considered, and include many judgment factors (Sarkis and Talluri, 2002). The vendor selection process is a multi-criteria decision, encompassing many tangible and intangible factors in a hierarchical manner. Figure 1 shows the structuring of the vendor selection process hierarchy of three levels of an auto component manufacturing company.

The top level of the hierarchy represents the ultimate goal of the process: to select the best vendor of an automobile exhaust system that can meet customer requirements, bring profits to the firm, and compete strongly in the automobile exhaust system market. Four strategic criteria – cost, quality, delivery, and technical capability – are identified to achieve this goal. These factors were determined from reviewing literature and using a brainstorming tool among the members of the supply chain department. The second level of the hierarchy is grouped into these four strategic categories of criteria that may affect the choice of vendor. The cost factor is important because the lower the cost of the item, the higher the profit to the company. Quality is equally important as it focuses more on meeting customers’ satisfaction and becoming competitive in order to stay ahead in the marketplace. Delivery of the item on time, and the in-house technical capability of the vendor to manufacture the item, are also important, as they satisfy the customers’ requirements. Finally, the bottom level of the hierarchy consists of the alternatives – the potential vendors to be evaluated in order to select the best vendor. As shown in Figure 1, we used three potential vendors to represent arbitrarily those that the firm (Auto Exhaust System Ltd, Gurgaon, India) wishes to evaluate. The firm considered is not involved in research or design activities. Any new or improved muffler must be procured from qualified vendors in the exhaust system industry. Therefore, three potential vendors were short-listed for evaluation, and one of them would be selected to supply the muffler.

3.2 Fuzziness in the vendor selection process:

In the vendor selection decision, many criteria related to the various vendors are not known with certainty. For example, how the vendor will respond to a new design cannot be ascertained. At the time of selecting a vendor, the supply chain manager of Auto Exhaust System Ltd represented the values of many criteria using linguistic parameters such as ‘very poor in late deliveries’, ‘hardly any rejected quantities’, good technical capability of vendor’, etc. The real-life vendor selection decisions of the company are also influenced by many natural factors and processes that are difficult to measure and model precisely. The decision situations are surrounded by uncertainty. Thus, there is a need to develop a systematic vendor selection process in the company for identifying and prioritizing relevant criteria, evaluating the tradeoffs between quantitative and qualitative criteria (including fuzziness) in the comparisons of vendors with respect to each criterion. The approach should also increase accuracy, reduce time in vendor selection, and develop consensus decisionmaking. Hence, this paper uses the Fuzzy Extent Analysis method for vendor selection to improve the decision-making through a more systematic and logical approach, since uncertainty issues are present in the vendor selection process.

3.3 Fuzzy Extent Analysis Methodology

Graan (1980) first presented the Fuzzy AHP problem, and extracted the priority vector form in an AHP problem with fuzzy ratings. Within the AHP context, the decision-maker cannot provide deterministic preferences, only perception-based judgment intervals. This kind of uncertainty in preferences can be modeled using the fuzzy set theory. In the fuzzy set terminology, the ratio supplied by the decisionmaker is a fuzzy number described by a membership function. Because the preferences in AHP are essentially judgments of human beings based on perception (this is especially true for intangibles), we believe the fuzzy approach allows a more accurate description of the decision-making process (Van Laarhoven and Pedrycz, 1983). In the literature, various authors (such as Van Laarhoven and Pedrycz, 1983; Buckley, 1985; Chang, 1996) propose many types of Fuzzy AHP methods. Van Laarhoven and Pedrycz (1983) compared fuzzy ratios described by triangular membership functions. Buckley (1985) determined fuzzy priorities of comparison ratios whose membership functions are trapezoidal. Chang (1992, 1996) proposed a new approach to handling Fuzzy Extent Analysis, with the use of triangular fuzzy numbers for pairwise comparison scale, and the use of the extent analysis method for the systematic extent values of the pairwise comparisons.

From these methods, we prefer Chang’s (1992, 1996) Fuzzy Extent Analysis method, since the steps of this approach are relatively easier than other Fuzzy AHP approaches, and similar to the crisp AHP. Recently, Bozdag et al (2003) have used this approach in the evaluation of computer-integrated manufacturing alternatives. Kahraman et al (2003, 2004) also used this approach in the evaluation of catering firms in Turkey, and in the selection of the best location for a facility, respectively. Buyukozkan et al (2004) applied the same approach to select the best software development strategy. In the following, first the outlines of the fuzzy set and Fuzzy Extent Analysis method are given, and then the method is applied to a vendor selection process.

3.3.1 Fuzzy set

Zadeh (1965) defined a fuzzy set A in X as follows:

A={(x,µ^sub A^)/x ∈ X} (1)

Where µA (x)= 0, x is absolutely not in A, while µA (x)= 1 means x absolutely belongs in A. In other cases, µA (x)has a non-negative real number whose value is finite, and usually finds a place in the interval [0,1]. An assigned value to µA (x)gives the degree of x belonging to A where µ A (x): X [arrow right] [0,1] ? is called the membership function of A and µA (x) is the degree of membership to which x belongs to A.

It should be noted that precise membership values do not exist, and are usually subjectively assessed or assigned in each context. As a result, general fuzzy sets are seldom used in practice; instead the fuzzy number to be introduced below is widely used, especially the more specific triangular fuzzy number. For more on fuzzy set theory, please refer to Dubois and Prade (1978, 1980), Chen and Hwang (1992), etc.

3.3.2 Triangular fuzzy numbers

A fuzzy number M is a special fuzzy subset of real numbers R . Its membership function µM (x)is a continuous mapping from R to a closed interval [0,1]. It has the following characteristics:

exists x0 ∈ R such that µM (x0) = 1.

For any a ∈[0, 1],

A^sub a^ = [x,µ^sub Aa^ (x) = is a closed interval. (2)

In this paper triangular fuzzy numbers are used to quantify the uncertainty owing to vagueness regarding fuzzy numbers M . They are used because of the ease in defining fuzzy numbers M for the approximate reasoning for the quantitative and qualitative values of vague linguistic variables. A fuzzy number M on R to be a triangular fuzzy number if its membership function µM (x): R [arrow right] [0, 1 ] is equal to

… (3)

where l ≤ m ≤ u, l and u stand for the lower and upper value of the support of M respectively, and m for the modal value. One of the most basic concepts of fuzzy set theory that can be used to generalize crisp mathematical concepts to fuzzy sets is the extension principle. Let X be a Cartesian product of universes X = x1, x2, . . . , xr, and A1, A2, . . . , Ar be r fuzzy sets in X1, X2, . . . , Xr, respectively. f is a mapping from X to a universe Y, y = f(x1, x2, . . . , xr). Then the extension principle allows us to define a fuzzy set B in Y by Zimmerman (1994):

… (4)

where …

otherwise 0

Where f -1 is the inverse of f.

In what follows we briefly summarize the basic arithmetic operations for triangular fuzzy numbers based on Dubosis and Prade (1980) and Chen and Hwang (1992). With this notation, and by use of the extension principle, some of the extended algebraic operations of triangular fuzzy numbers are expressed in Appendix 1.

3.3.3 Fuzzy Extent Analysis method

The first task of the Fuzzy Extent Analysis method is to decide on the relative importance of each pair of factors in the same hierarchy. In this paper, this kind of expression is used to compare two considered criteria in a fuzzy environment on a nine-level scale. Decision-makers were asked to specify the relative importance of vendor selection criteria. The Fuzzy Extent Analysis obtains the necessary information in the form of a pairwise comparisons matrix (PCM). After completing all of these, a pairwise comparisons matrix (PCM) is constructed. Inputs for the Fuzzy Extent Analysis approach are the crisp PCMs. The crisp PCMs are fuzzified using the triangular fuzzy numbers. The fuzzy PCMs for each criterion are the inputs for Fuzzy Extent Analysis to result in fuzzy performances per criteria. In the same way, the PCMs constructed by the comparison among criteria in a group are the hierarchies that are fuzzified to obtain fuzzy performances per criteria. By using triangular fuzzy numbers, via pairwise comparison, the fuzzy evaluation matrix A = (aij)n × m is constructed. For example, where essential or strong importance of element i over element j under a certain criterion: then (aij) = (l, 5, u i) where l and u represent a fuzzy degree of judgment. The greater (u – l), the fuzzier the degree, when (u – l) = 0, the judgment is a nonfuzzy number. This stays the same to scale 5 under general meaning. If strong importance of element j over element i holds, then the pairwise comparison scale can be represented by the fuzzy number ….

Let X = {x1, x2, . . . , xn} be an object set, and U = {u1, u2, . . . , un} be a goal set. According to the method of Chang’s (1992) Fuzzy Extent Analysis, each object is taken and extent analysis for each goal is performed, respectively. Therefore, m extent analysis values of each object, given as:

… (5)

where all the … are triangular fuzzy numbers representing the performance of the object xi with regard to each goal uj . The value of fuzzy synthetic extent with respect to the i – th object xi (i = 1, 2, . . . , n) that represents the overall performance of the object across all goals involved can be determined by (Chang, 1992):

… (6)

where i nj m = 1, 2, . . . , = 1, 2, . . . , ; .

The degree of possibility of M1 ≥ M2 is defined as:

… (7)

When a pair (x, y) exists such that x ≥ and µM1 (x) = µM2 (y) = 1, then we have

V(M1 ≥ M2) = Since M1 and M2 and are convex fuzzy numbers we have that:

V(M1 ≥ M2) = 1. iff m1 ≥ m2,

V(M2 ≥ M1) = hgt (M1 ∩ M2) = µM1(d), (8)

where d is the ordinate of the highest intersection point D between µM1 and µM2 (see Figure 2).

When M1 = (l1, m1, u1) and M2 = (l2, m2, u2), the ordinate of D is given by the following equation (9):

… (9)

To compare M1 and M2, we need both the values of V(M1 ≥ M2) and V(M2 ≥ M1).

The degree of possibility for a convex fuzzy number to be greater than k convex fuzzy numbers Mi (i = 1, 2, . . . , k) can be defined by:

V(M ≥ M1, M2, . . . , Mk) = V[(M ≥ M1) and (M ≥ M2) and . . . and (M ≥ Mk)]

= minV(M ≥ Mi), i = 1, 2, . . . , k. (10)

Assume that d (Ai) = min V(Si ≥ Sk), for k = 1, 2, . . . , n; k ≠ i. (11)

Then the weight vector is given by:

W = (d (A1), d (A2), . . . , d (An))T, (12)

where Ai (i = 1, 2, . . . , n) are n elements.

Via normalization, we get the normalized weight vectors as follows:

W – (d(A1), d(A2), . . . , d(An)T, (13)

Where W is a nonfuzzy number.

3.3.4 Application of Fuzzy Extent Analysis model to vendor selection

In the vendor selection problem of the selected item for the automobile exhaust system firm (Auto Exhaust System Ltd, Gurgaon, India), three serious vendors remain. We shall call them V1 , V2 and V3. The purchasing manager has convened a committee (made up of supply chain members, marketing members, quality inspectors, etc.) to decide which vendor is best qualified for the given purchasing scenario. The committee has twelve members, and they have identified from reviewing the literature and using a brainstorming tool among themselves the following decision criteria:

(i) price (C1);

(ii) quality (C2);

(iii) delivery (C3);

(iv) technical capability (C4).

A questionnaire consisting of all criteria of the three levels of the Fuzzy Extent Analysis model is designed and is used to collect the pairwise comparison judgments from all evaluation team members. This approach is found to be very useful in collecting data. The pairwise comparison judgments are made with respect to the attributes of one level of hierarchy given the attribute of the next higher level of hierarchy, starting from the first level. The results collected from the questionnaire are used to form the corresponding pairwise comparison matrix for determining the normalized weights. With respect to the above four criteria, a vendor is compared against the others by a pairwise comparison procedure of the Fuzzy Extent Analysis model. In our case study, the purchasing manager of the company acted as the evaluator, and assigned the ratings to each vendor with respect to each criterion as shown in the respective pairwise comparison matrix.

First step. Via pairwise comparison of the performance criteria of vendors, the fuzzy evaluation matrix that is relevant to the objective is constructed (Table 1).

Then by applying the formula of fuzzy synthetic extent, we have obtained the weight vectors with respect to the decision criteria C1, C2, C3 and C4, as follows:

W = (0.146, 0.339, 0.157, 0.357)T

Second step: At the second level of the decision procedure, the committee compares vendors V1, V2 and V3, under each of the criteria separately. The pairwise comparison matrices of vendors with respect to criterion C 1 are shown in Table 2.

Similarly, the pairwise comparison matrices of vendors with respect to the other criteria C2, C3 and C4, are calculated. As before, these matrices are used to estimate weights – in this case, the weights of each vendor under each criterion separately. The results are given in Table 3.

Finally, adding the weights per vendor multiplied by the weights of the corresponding criteria, a final score is obtained for each vendor. The final scores of all three vendors are provided in Table 4.

The final score of vendor 1, vendor 2, and vendor 3 are 0.357, 0.309, and 0.334 respectively. According to the final scores, it is clear that vendor V1 is the preferred vendor. The results of Table 4 show that, in the Fuzzy Extent Analysis approach, vendor 1 is selected as the best vendor. The results also show that vendor 3 is ranked second, and vendor 2 is the inferior vendor.

3.3.5 Sensitivity analysis:

Sensitivity analysis is a way to address the uncertainty in estimating the parameters (Malczewski, 1999). After the problem is evaluated for optimum conditions, sensitivity analysis assesses different conditions near the optimum values to check for the sensitivity of the criteria. Sensitivity analysis also aids in understanding the interaction between the criteria, the dominant criterion and its effect, e.g., the variation in the final results when the weight of that criterion is varied. Sensitivity analysis provides information on how changes to the components of a mathematical model impact the optimal solution. The information provided by sensitivity analysis enhances the value of the modeling process. Sensitivity analysis is extremely important since the real world is dynamic, and the impact of these changes must be evaluated. Therefore, sensitivity analysis is an important aspect of any decisionmaking process. Hence, in this paper sensitivity analysis is done in the results provided by Fuzzy Extent Analysis for vendor selection of the case company.

3.3.5.1 Weight of price criterion

An important use of sensitivity analysis is to determine how much a price criterion weight must change before there is change in the ranking of the alternatives. This type of break-even analysis is shown in Figure 3. In the real life vendor selection, the data available is vague – which is why Fuzzy Extent Analysis is the only practical solution for vendor selection. In case of Fuzzy Extent Analysis methodology for vendor selection, initially at zero priority of price Vendor 1 and Vendor 3 are on the same rank and encompass 35% alternative weight, while Vendor 2 is ranked least and encompasses 30% alternative weight. As the priority in price increases, the percentages of alternative weight change. The percentage of alternative weight of Vendor 3 decreases as the priority in price increases, and the ranking of Vendor 3 declines. The percentage of alternative weight of Vendor 2 increases as the priority in price increases, and the ranking of Vendor 2 improves. Typically, the percentage of alternative weight of Vendor 1 remains constant, owing to uncertainty in the decision parameters of vendor selection. Vendor 2 is ranked second on 0.58 or more priority weight of price. At 0.58 priority weight of price, Vendor 2 and Vendor 3 are ranked the same, lower than Vendor 1, and encompass approximately 32.5% alternative weight, whereas Vendor 1 is still ranked first and encompasses approximately 35% alternative weight. At 1.0 (or full priority weight of price) the ranking of the vendors changes: Vendor 2 occupies second position and encompasses 34% alternative weight, Vendor 3 is third by encompassing 31% alternative weight, and Vendor 1 remains first by encompassing 35% alternative weight.

3.3.5.2 Weight of quality criterion

The result of experimenting with the sensitivity of the quality criterion by changing the weight is shown in Figure 4.

In the case of the Fuzzy Extent Analysis methodology for vendor selection, initially at zero priority of quality, Vendor 1 is ranked first and encompasses 36% alternative weight, Vendor 3 is second and encompasses 35% alternative weight, and Vendor 2 is third and encompasses 29% alternative weight. As the priority of quality increases, the percentages of alternative weights of vendors change. The percentage of alternative weight for Vendor 2 increases as the priority of price increases, and Vendor 2’s ranking improves. The percentage of alternative weight for Vendor 3 decreases as the priority of price increases, and Vendor 3’s ranking declines. Typically, the percentage of alternative weight for Vendor 1 remains constant, and Vendor 1 occupies first position through all the changes in the priority of quality. This is owing to uncertainty in the decision parameters of vendor selection. Another observation to be made about Fuzzy Extent Analysis is that the trend of change in the priority of vendors in Figure 3 and Figure 4 is almost the same; and in both cases the percentage of alternative weight for Vendor 1 remains constant. So the results in sensitivity analysis of price and quality criteria in the case of Fuzzy Extent Analysis do not differ significantly. At 0.56 priority weight of quality, Vendor 2 and Vendor 3 are ranked the same, lower then Vendor 1, and encompass approximately 32% alternative weight, whereas Vendor 1 remains first and encompasses approximately 36% alternative weight. At 1.0 (or full priority weight) for price, the ranking of the vendors changes: Vendor 2 occupies second rank and encompasses 34% alternative weight, Vendor 3 is third, encompassing 30% alternative weight, and Vendor 1 remains first, encompassing 36% alternative weight

3.3.5.3 Weight of delivery criterion

Figure 5 shows the sensitivity analysis by changing the weight of the delivery criterion in Fuzzy Extent Analysis methodology for vendor selection.

In real life vendor selection, the percentage of late deliveries is vague, so Fuzzy Extent Analysis is the only practicable solution. In the case of Fuzzy Extent Analysis methodology for vendor selection, with an initial zero priority for quality, Vendor 1 is ranked first and encompasses 35% alternative weight, Vendor 2 is second and encompasses 34% alternative weight, and Vendor 3 is third and encompasses 31% alternative weight. As the priority of delivery increases, percentages of alternative weights for vendors change. The percentage of alternative weight for Vendor 1 increases as the priority of delivery increases, and Vendor 1’s ranking improves. The percentage of alternative weight for Vendor 2 decreases as the priority in delivery increases, and Vendor 2’s ranking declines. Typically, the percentage of alternative weight for Vendor 3 remains constant, while Vendor 3 remains third through all the changes in priority for delivery. At 1.0 (or full priority weight) for delivery the ranking of the vendors does not change: all the vendors maintain their position. But,at 1.0 priority weight the percentage of alternative weight for vendors changes: Vendor 1 encompasses 38% alternative weight, Vendor 2 encompasses 31% alternative weight.

3.3.5.4 Weight of technical capability criterion

Changes in the ranking of vendors with changes in the delivery criterion in Fuzzy Extent Analysis methodology is shown in Figure 6.

In the case of Fuzzy Extent Analysis methodology for vendor selection, with an initial zero priority for quality, Vendor 1 is ranked first and encompasses 36% alternative weight; Vendor 3 is second and encompasses 34% alternative weight, and Vendor 2 is third and encompasses 30% alternative weight. As the priority of technical capability increases, the percentages of alternative weights for vendors change. The percentage of alternative weight of Vendor 2 increases as the priority of technical capability increases, and Vendor 2’s ranking improves. The percentage of alternative weight for Vendor 3 decreases as the priority of technical capability increases, and Vendor 3’s ranking declines. Typically, the percentage of alternative weight for Vendor 1 remains constant, and Vendor 1 remains ranked first through all the changes in priority for technical capability. At 0.82 priority weight for technical capability, Vendor 2 and Vendor 3 are ranked the same, lower than Vendor 1, and encompass approximately 32% alternative weight, whereas Vendor 1 is still first and encompasses approximately 36% alternative weight. At 1.0 (or full priority weight) for price, the ranking of the vendors changes: Vendor 2 occupies second position and encompasses 33% alternative weight, while Vendor 3 is third, encompassing 31% alternative weight; Vendor 1 remains first, encompassing 36% alternative weight.

Using sensitivity analysis in Fuzzy Extent Analysis, we find that by changing the priorities of the various criteria – price, quality, delivery, and technical capability – the decision for vendor selection is affected. Decision-makers must therefore observe the results of trade-offs in making better purchasing decisions, thus optimizing the supply chain objectives.

4. CONCLUSIONS

Many authors have pointed out the need for a ‘fuzzy’ evaluation system for the vendor selection process, and for the development of operationally practical processes to allow purchasing managers to accommodate quantitative and qualitative criteria and vagueness more effectively, in order to obtain supply chain goals. In this paper, the Fuzzy Extent Analysis model was used, criteria for supplier selection were clearly identified, and the problem was systematically structured. In this paper a Fuzzy Extent Analysis process is proposed as an effective fuzzy method to derive the weight of considered criteria, and then to rank the vendors. Moreover, the quantitative data used in Auto Exhaust System Ltd are accepted as valid parameters because of historical data gathered from the company records. Also, sensitivity analysis is explained as an example of the validity of parameters. The process is clear, transparent, and understandable to the average person. Each of the major subprocesses – e.g. problem hierarchy development, alternative criteria and subcriteria adaptation, and alternative choice – are relatively simple to understand and communicate. All the members of the team who assigned pairwise comparison judgments appear to be satisfied with the final selection of the vendor system. To overcome the problems of assessing pairwise comparison judgments, the evaluators were first trained in the principles of Fuzzy Extent Analysis and in assessment techniques. In this paper, triangular membership functions are used. Further research can be carried out by using other types of membership function such as trapezoidal, exponential, and S-type.

5. ACKNOWLEDGMENTS

The author wishes to acknowledge the managers of Auto Exhaust System Ltd, Gurgaon, India for supporting this research and providing all the necessary facilities for getting the required data.

6. REFERENCES

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