Fuzzy Similarity in Multicriteria Decision-Making Problem Applied to Supplier Evaluation and Selection in Supply Chain Management

Abstract

It is proposed to use fuzzy similarity in fuzzy decision-making approach to deal with the supplier selection problem in supply chain system. According to the concept of fuzzy TOPSIS earlier methods use closeness coefficient which is defined to determine the ranking order of all suppliers by calculating the distances to both fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS) simultaneously. In this paper we propose a new method by doing the ranking using similarity. New proposed method can do ranking with less computations than original fuzzy TOPSIS. We also propose three different cases for selection of FPIS and FNIS and compare closeness coefficient criteria and fuzzy similarity criteria. Numerical example is used to demonstrate the process. Results show that the proposed model is well suited for multiple criteria decision-making for supplier selection. In this paper we also show that the evaluation of the supplier using traditional fuzzy TOPSIS depends highly on FPIS and FNIS, and one needs to select suitable fuzzy ideal solution to get reasonable evaluation.

1. Introduction

The overall objective of supplier selection process is to reduce purchase risk, maximize overall value to the purchaser, and build the closeness and long-term relationships between buyers and suppliers [1]. With the globalization of the economic markets and the development of information technology, many companies consider that a well-designed and implemented supply chain management (SCM) system is an important tool for increasing competitive advantage [2].

Previously, many methods have been proposed to solve the supplier selection problem, some of the popular ones being the linear weighting methods (LW) [3, 4], the analytic hierarchy process (AHP) [5, 6], the analytic network process [7], total cost approaches [8, 9], and mathematical programming (MP) techniques [10, 11]. However, several influence factors are often not taken into account in the decision-making process, such as incomplete information [12, 13], additional qualitative criteria, and imprecision preferences. A strategic approach towards supplier selection often involves multiple criteria [14] and several decision-makers [15, 16], and decision-making is often influenced by uncertainty in practice. Supplier selection may involve several and different types of criteria, combination of different decision models, group decision-making, and various forms of uncertainty. Technique for order performance by similarity to ideal solution (TOPSIS) [17], which is one of the known classical MCDM methods, may provide the basis for developing supplier selection models that can effectively deal with these properties. It bases upon the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). TOPSIS was further extended to fuzzy environment where linguistic assessments could be used instead of numerical values by Chen [18]. This method allows to consider the fuzziness in the decision data and group decision-making process, where linguistic variables are used to assess the weights of all criteria and the ratings of each alternative with respect to each criterion. In a paper by Chen et al. [19] this approach was further applied to supplier evaluation and selection in supply chain management. In this paper, this fuzzy TOPSIS described in [19] is extended to use fuzzy similarity [20]. In fuzzy TOPSIS a vertex method is applied to calculate the distance between two fuzzy ratings. Using this vertex method, they calculate the distance of each alternative from fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (NPIS), respectively. After this a closeness coefficient of each alternative is defined to determine the ranking order of all alternatives. The higher value of closeness coefficient indicates that an alternative is closer to FPIS and farther from FNIS. By using fuzzy similarity we are replacing this vertex method for calculating distance by using fuzzy similarity to calculate similarity between two fuzzy ratings. By doing this we can use similarity value gained using proposed method and we do not need to calculate closeness coefficient to get to the result. Second we also study three different alternatives for selecting FPIS and also FNIS in fuzzy TOPSIS and compare how assessments of suppliers are changing depending on the chosen alternatives.

Rest of the paper is organized as follows. Section 2 introduces the basic notations of fuzzy numbers and linguistic variables as well as used fuzzy similarity measure. Section 3 presents fuzzy TOPSIS and how it can be used with fuzzy similarity. There we also introduce three possible FPIS and FNIS criteria. In Section 4 numerical example is used to demonstrate how proposed method can be used and also three criteria for FPIS and FNIS are used to compare the results together with the previously presented fuzzy TOPSIS. Finally some discussion is given at the end of the paper.

2. Basic Concepts for Generalized Fuzzy Numbers, Linguistic Variables, and Fuzzy Similarity

Fuzzy logic is a logic that allows vagueness, imprecision, and uncertainty. Zadeh (1965) [21] introduced the concept of fuzzy sets and the respective theory that can be regarded as the extension of the classical set theory. One of the fundamental mathematical constructs is the similarity measure. In the same way as the notion of the fuzzy subset generalizes that of the classical subset, the concept of similarity can be considered as being a multivalued generalization of the classical notion of equivalence [22].

Wei and Chen introduced a fuzzy similarity measure [20] for generalized fuzzy numbers. They tested several similarity methods for generalized fuzzy numbers [23–26] and demonstrated successfully benefits of this measure. First we next briefly review basic concepts of generalized fuzzy numbers [27] and then introduce the similarity measure used later in fuzzy decision-making.

Chen [27] represented a generalized trapezoidal fuzzy number 𝐴 as 𝐴=(𝑎,𝑏,𝑐,𝑑;𝑤), where 𝑎,𝑏,𝑐, and 𝑑 are real values and 0<𝑤≤1 as shown in Figure 1.

Figure 1 

A generalized trapezoidal fuzzy number 𝐴=(𝑎,𝑏,𝑐,𝑑;𝑤).

The membership function 𝜇_𝐴 satisfies the following conditions [27]:(1)𝜇_𝐴 is a continuous mapping from the universe of discourse 𝑋 to the closed interval in [0,1];(2)𝜇_𝐴=0, where −∞<𝑥≤𝑎;(3)𝜇_𝐴 is monotonically increasing in [𝑎,𝑏];(4)𝜇_𝐴=𝑤, where 𝑏≤𝑥≤𝑐;(5)𝜇_𝐴 is monotonically decreasing in [𝑐,𝑑];(6)𝜇_𝐴=0, where 𝑑≤𝑥<∞.

Assume that we have two generalized trapezoidal fuzzy numbers 𝐴 and 𝐵, where 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑤_𝑎) and 𝐵=(𝑏₁,𝑏₂,𝑏₃,𝑏₄,𝑤_𝑏).

Now, for example, addition ⊕ for the generalized trapezoidal fuzzy numbers is𝐴⊕𝐵==𝑎𝑎₁₁,𝑎₂,𝑎₃,𝑎₄,𝑤_𝐴⊕𝑏₁,𝑏₂,𝑏₃,𝑏₄,𝑤_𝑏+𝑏₁,𝑎₂+𝑏₂,𝑎₃+𝑏₃,𝑎₄+𝑏₄,min𝑤_𝑎,𝑤_𝑏(1) and subtraction ⊖𝐴⊖𝐵==𝑎𝑎₁₁,𝑎₂,𝑎₃,𝑎₄,𝑤_𝐴⊖𝑏₁,𝑏₂,𝑏₃,𝑏₄,𝑤_𝑏−𝑏₄,𝑎₂−𝑏₃,𝑎₃−𝑐₂,𝑎₄−𝑏₁;min𝑤_𝑎,𝑤_𝑏,(2) for example, multiplication ⊗ and division ⊖ of fuzzy numbers see, for example, [27] or [20].

Notice that these operations for positive trapezoidal fuzzy numbers follows the same arithmetic rules as also earlier used in concept of using fuzzy TOPSIS, for example, in [18, 19] with the exception that height of the fuzzy number can be now between 0<𝑤≤1. In this paper we are using normalized fuzzy numbers, where 𝑤=1, so basically arithmetic rules are the same as in [28]. Basically 𝑤 can be omitted so that normalized fuzzy number with height equaling one can be represented as 𝐴=(𝑎,𝑏,𝑐,𝑑). Here this generalized form for fuzzy numbers is used to show generalized form of fuzzy similarity given by Wei and Chen [20].

Degree of similarity between two generalized fuzzy numbers can be computed as𝑆𝐴,𝐵=1−∑^4𝑖=1||𝑎_𝑖−𝑏_𝑖||×min𝑃4𝐴,𝑃𝐵+min𝑤_𝑎,𝑤_𝑏max𝑃𝐴,𝑃𝐵+max𝑤_𝑎,𝑤_𝑏,(3) where 𝑆(𝐴,𝐵)∈[0,1]; 𝑃(𝐴) and 𝑃(𝐵) are defined as follows:𝑃𝐴=+𝑎𝑎₁₄−𝑎₂²+𝑤^2𝑎+𝑎₃−𝑎₄²+𝑤^2𝑎+𝑎₃−𝑎₂𝑃𝐵=−𝑎₁,+𝑏𝑏₁₄−𝑏₂²+𝑤^2𝑏+𝑏₃−𝑏₄²+𝑤^2𝑏+𝑏₃−𝑏₂−𝑏₁,(4) where𝑃(𝐴) and 𝑃(𝐵) denote the perimeters of the generalized trapezoidal fuzzy numbers 𝐴 and 𝐵, respectively. The larger the value of 𝑆(𝐴,𝐵) is the more the similarity is between the generalized fuzzy numbers 𝐴 and 𝐵.

Notice that in normalized form where height equals one (𝑤=1) this becomes𝑆𝐴,𝐵=1−∑^4𝑖=1||𝑎_𝑖−𝑏_𝑖||4×mmainx𝑃𝑃𝐴𝐴,,𝑃𝑃𝐵𝐵,(5) where 𝑆(𝐴,𝐵)∈[0,1]; 𝑃(𝐴) and 𝑃(𝐵) are defined as follows:𝑃𝐴=+𝑎𝑎₁₄−𝑎₂²+1+𝑎₃−𝑎₄²+1+𝑎₃−𝑎₂𝑃𝐵=−𝑎₁,+𝑏𝑏₁₄−𝑏₂²+1+𝑏₃−𝑏₄²+1+𝑏₃−𝑏₂−𝑏₁.(6)

3. Extended TOPSIS to the Fuzzy Environment and Using Similarity instead of Closeness Evaluation in Supplier Selection

In this paper we follow basically the same process as given by Chen et al. [19] but when making the final evaluation and ranking we are using similarity instead of closeness criteria, and we do not use vertex method to calculate the distance between two fuzzy ratings but we use fuzzy similarity instead. About the vertex method for calculating the distance and closeness criteria see [18, 19]. Later both are tested with numerical example next into the supplier-selection problem with extended TOPSIS.

Here we consider the importance weights of various criteria and the ratings of qualitative criteria as linguistic variables. Because linguistic assessments merely approximate the subjective judgement of decision-maker, we are using linear trapezoidal membership functions to capture the vagueness of these linguistic assessments [19, 29–34]. These linguistic variables can be expressed in positive trapezoidal fuzzy numbers, as in Figures 2 and 3. The importance weight of each criterion can be either directly assigning or indirectly using pairwise comparison [35–37]. In this paper linguistic variables shown in Figures 2 and 3 are used to evaluate the importance of the criteria and the ratings of alternatives with respect to qualitative criteria. For example, the linguistic variable “Medium Good” can be represented as (5, 6, 7, 8) and “Good” as (7, 8, 8, 9). Membership functions would be in these cases𝜇_MediumGood(𝑥)=⎧⎪⎪⎪⎨⎪⎪⎪⎩0𝑥−5𝑥<5,1𝑥6−−5865≤≤𝑥𝑥≤≤76,,𝜇07−8𝑥7≤>𝑥8,≤8,(7)_Good(𝑥)=⎧⎪⎪⎪⎨⎪⎪⎪⎩0𝑥−7𝑥<7,1𝑥8−−79𝑥7≤=𝑥8,≤8,08−9𝑥8≤>𝑥9.≤9,(8) Supplier selection in supply chain system is considered to be a group multiple-criteria decision-making (GMCDM) problem, which may be described by the following sets:(i)a set of 𝐾 decision-makers called 𝐸={𝐷₁,𝐷₂,…,𝐷_𝐾};(ii)a set of 𝑚 possible suppliers called 𝐴={𝐴₁,𝐴₂,…,𝐴_𝑚};(iii)a set on 𝑛 criteria, 𝐶={𝐶₁,𝐶₂,…,𝐶_𝑛}, with which supplier performance is measured;(iv)a set of performance ratings of 𝐴_𝑖(𝑖=1,2,…,𝑚) w.r.t. criteria 𝐶_𝑗(𝑗=1,2,…,𝑛) called 𝑋={𝑥_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛}.

Figure 2 

Linguistic variables for importance weight of each criterion.

Figure 3 

Linguistic variables for ratings.

Assume that a decision group has 𝑘 decision-makers, and the fuzzy ratings of each decision-maker 𝐷_𝑘(𝑘=1,2,…,𝐾) are represented using positive trapezoidal fuzzy number 𝑋_𝑘(𝑘=1,2,…,𝐾) with membership function 𝜇𝑋_𝑘(𝑥). Let the fuzzy ratings of all decision-makers be trapezoidal fuzzy numbers 𝑅_𝑘=(𝑎_𝑘,𝑏_𝑘,𝑐_𝑘,𝑑_𝑘),𝑘=1,2,…,𝐾. Then the aggregated fuzzy rating can be defined as𝑅=(𝑎,𝑏,𝑐,𝑑),(9) where𝑎=min_𝑘𝑎_𝑘𝑏=1𝐾^𝐾_𝑘=1𝑐=1𝑏_𝑘,𝐾^𝐾_𝑘=1𝑐_𝑘𝑑=max_𝑘𝑑_𝑘.(10) Let the fuzzy rating and importance weight of the 𝑘th decision-maker be 𝑥̂_𝑖𝑗𝑘=(𝑎_𝑖𝑗𝑘,𝑏_𝑖𝑗𝑘,𝑐_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘) and 𝑤=(𝑤_𝑗𝑘1,𝑤_𝑗𝑘2,𝑤_𝑗𝑘3,𝑤_𝑗𝑘4);𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, respectively. Hence, the aggregated fuzzy ratings (𝑥̂_𝑖𝑗) of alternatives with respect to each criterion can be calculated as𝑥̂_𝑖𝑗=𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝑑_𝑖𝑗,(11) where𝑎_𝑖𝑗=min_𝑘𝑎_𝑖𝑗𝑘𝑏_𝑖𝑗=1𝐾^𝐾_𝑘=1𝑐_𝑖𝑗𝑏_𝑖𝑗𝑘,=1𝐾^𝐾_𝑘=1𝑐_𝑖𝑗𝑘𝑑_𝑖𝑗=max_𝑘𝑑_𝑖𝑗𝑘.(12) In similar manner the aggregated fuzzy weight (𝑤_𝑗) of each criterion can be calculated as𝑤_𝑗=𝑤_𝑗1,𝑤_𝑗2,𝑤_𝑗3,𝑤_𝑗4,(13) where𝑤_𝑗1=min_𝑘𝑤_𝑗𝑘1𝑤_𝑗2=1𝐾^𝐾_𝑘=1𝑤_𝑗3𝑤_𝑗𝑘2,=1𝐾^𝐾_𝑘=1𝑤_𝑗𝑘3𝑤_𝑗4=max_𝑘𝑤_𝑗𝑘4.(14) As stated, a supplier-selection problem can be concisely expressed in matrix form as𝑋=⎛⎜⎜⎜⎝𝑥̂¹¹𝑥̂₁₂…𝑥̂_1𝑛𝑥̂₂₁𝑥̂₂₂…𝑥̂_2𝑛𝑥⋮̂_𝑚1⋮⋮𝑥̂_𝑚2…𝑥̂_𝑚𝑛𝑊=𝑤₁⎞⎟⎟⎟⎠,,𝑤₂,…,𝑤_𝑛,(15) where 𝑥̂_𝑖𝑗=(𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝑑_𝑖𝑗) and 𝑤_𝑗=(𝑤_𝑗1,𝑤_𝑗2,𝑤_𝑗3,𝑤_𝑗4) can be approximated by positive trapezoidal fuzzy numbers.

To avoid complexity of mathematical operations in a decision process, the linear scale transformation is used to transform the various criteria scales into comparable scales. The set of criteria can be divided into benefit criteria (the larger the rating, the greater the preference) and cost criteria (the smaller the rating, the greater the preference). Therefore, the normalized fuzzy decision matrix can be represented as𝑅=𝑟̂_𝑖𝑗_𝑚×𝑛,(16) where 𝐵 and 𝐶 are the sets fo benefit criteria and cost criteria, respectively, and𝑟̂_𝑖𝑗=𝑎^𝑖𝑗𝑑^⊕𝑗,𝑏^𝑖𝑗𝑑^⊕𝑗,𝑐^𝑖𝑗𝑑^⊕𝑗,𝑑^𝑖𝑗𝑟̂_𝑖𝑗𝑑^⊕𝑗,𝑗∈𝐵,=𝑎^⊖𝑗𝑎_𝑖𝑗,𝑎^⊖𝑗𝑏_𝑖𝑗,𝑎^⊖𝑗𝑐_𝑖𝑗,𝑎^⊖𝑗𝑑_𝑖𝑗,𝑗∈𝐶,(17) where 𝑑^⊕𝑗=max_𝑖(𝑑_𝑖𝑗),𝑗∈𝐵, and 𝑎^⊖𝑗=min_𝑖(𝑎_𝑖𝑗),𝑗∈𝐶. This normalization method is designed to preserve the property in which the elements are normalized trapezoidal fuzzy numbers.

Considering the importance of each criterion, the weighted normalized fuzzy decision matrix can now be expressed as𝑉=𝑣̂_𝑖𝑗_𝑚×𝑛,(18) where 𝑣̂_𝑖𝑗=𝑟̂_𝑖𝑗(⋅)𝑤_𝑗. According to the weighted normalized fuzzy decision matrix, normalized positive trapezoidal fuzzy numbers can also approximate the elements 𝑣̂_𝑖𝑗,for all𝑖,𝑗. Then, the fuzzy positive-ideal solution (FPIS, 𝐴^⊕) and fuzzy negative-ideal solution (FNIS, 𝐴^⊖) needs to be defined. Here we consider three alternatives which later will be compared.

Let fuzzy positive-ideal solution be𝐴^⊕=𝑣^⊕1̂,𝑣^⊕2̂,…,𝑣^⊕𝑛̂(19) and fuzzy negative-ideal solution be𝐴^⊖=𝑣^⊖1̂,𝑣^⊖2̂,…,𝑣^⊖𝑛̂.(20) Criteria for choosing these was given by Chen et al. [19] as𝑣^⊕𝑗̂=max_𝑖𝑣_𝑖𝑗4,𝑣^⊖𝑗̂=min_𝑖𝑣_𝑖𝑗1.(21) Here, for example, fuzzy positive ideal is chosen so that one finds the maximum values from fuzzy decision matrix with respect to suppliers and used the fourth value from fuzzy number (𝑑 from 𝐴=(𝑎,𝑏,𝑐,𝑑)), and also replaces 𝑎,𝑏, and 𝑐 with 𝑑 value (See e.g., numerical example in [19]). Fuzzy negative ideal is chosen in similar manner now just using minimum and replacing 𝑏,𝑐, and 𝑑 with 𝑎. Notice that this seems a bit arbitrary since membership values at points 𝑑 or 𝑎 are zero and there we are actually using crisp numbers presented in fuzzy way instead of fuzzy numbers. Following this type of thinking one clear and simple intuitive proposal for the choice of fuzzy positive ideal would be simply the set of ones and for negative ideals a set of zeros. Third proposal which intuitively makes most sense would be that instead of replacing values for 𝑎,𝑏, and 𝑐 with 𝑑 (max) or 𝑏, 𝑐, and 𝑑 with 𝑎 (min) as done in [19] one simply finds the maximum and minimum fuzzy numbers from fuzzy decision matrix with respect to suppliers𝑣^⊕𝑗̂=max_𝑖𝑣_𝑖𝑗∶,𝑣^⊖𝑗̂=min_𝑖𝑣_𝑖𝑗∶.(22) These three suggestions will be compared later.

Next in method described in [19] distance matrix would be calculated from 𝐴^⊕ and 𝐴^⊖ to each alternative supplier and after this a closeness coefficient is defined to determine the ranking order of all possible suppliers. Distances are calculated as𝑑^⊕𝑖=^𝑛_𝑗=1𝑑𝑑_𝑣𝑣̂^𝑖𝑗,𝑣^⊕𝑗̂,𝑖=1,2,…,𝑚,_⊖𝑖=^𝑛_𝑗=1𝑑_𝑣𝑣̂^𝑖𝑗,𝑣^⊖𝑗̂,𝑖=1,2,…,𝑚,(23) where 𝑑_𝑣(,)̇̇ is the distance measurement between two fuzzy numbers. The closeness coefficient is then calculated by𝐶𝐶_𝑖=𝑑^⊖𝑖𝑑^⊕𝑖+𝑑^⊖𝑖,𝑖=1,2,…,𝑚.(24) Here we propose alternative way. Instead of calculating a distance matrix for both ideal solutions we can calculate similarity matrix using fuzzy similarity (see (5)) and after this simply compute average of the similarities and use this as a similarity measure to make the ranking𝑆^⊕𝑖=1𝑛^𝑛_𝑗=1𝑆_𝑣𝑣̂^𝑖𝑗,𝑣^⊕𝑗̂.(25) Assessment status in Table 1 has also been used to evaluate the status of the suppliers. For this five linguistic variables with respect to the sub-intervals are defined to divide the assessment status of supplier into five classes. The decision rules for five classes are shown in Table 1. This can be clearly also modified to suite also similarity assessment.

Table 1 

Approval status table.

According to Table 1, it means the following.(i)If 𝑆_𝑖∈[0,0.2), then supplier 𝐴_𝑖 belongs to Class I, and the assessment status of supplier 𝐴_𝑖 is “not recommend”.(ii)If 𝑆_𝑖∈[0.2,0.4), then supplier 𝐴_𝑖 belongs to Class II, and the assessment status of supplier 𝐴_𝑖 is “recommend with high risk”.(iii)If 𝑆_𝑖∈[0.4,0.6), then supplier 𝐴_𝑖 belongs to Class III, and the assessment status of supplier 𝐴_𝑖 is “recommend with low risk”.(iv)If 𝑆_𝑖∈[0.6,0.8), then supplier 𝐴_𝑖 belongs to Class IV, and the assessment status of supplier 𝐴_𝑖 is “approved”.(v)If 𝑆_𝑖∈[0.8,1.0], then supplier 𝐴_𝑖 belongs to Class V, and the assessment status of supplier 𝐴_𝑖 is “approved and preferred to recommend”.

So basically assessment status from Table 1 can be used, and in addition if any two suppliers belong to the same class of assessment status, the supplier with the highest similarity value is used to determine the rank. Notice that to these values and to approvement status given in [19] the choice of FPIS and FNIS is effecting the status considerably as later will be shown by the given example. But before going to this study proposed algorithm is summarized as follows.

Step 1. Form a committee of decision-makers, and then identify the evaluation criteria.

Step 2. Choose the appropriate linguistic variables for the importance weight of the criteria and the linguistic ratings for suppliers.

Step 3. Aggregate the weight of criteria to get the aggregated fuzzy weight 𝑤_𝑗 of criterion 𝐶_𝑗, and pool the decision-makers’ ratings to get the aggregated fuzzy rating 𝑥̂_𝑖𝑗 of supplier 𝐴_𝑖 under criterion 𝐶_𝑗.

Step 4. Construct the fuzzy decision matrix and the normalized fuzzy decision matrix.

Step 5. Construct weighted normalized fuzzy decision matrix.

Step 6. Determine FPIS.

Step 7. Calculate the similarity of each supplier from FPIS by calculating the similarity matrix, and then average similarity value for each supplier.

Step 8. according to the average similarity value we can get the assessment status of each supplier and determine the ranking order of all suppliers.

4. Numerical Example and Comparison

Next we use numerical example to compare fuzzy TOPSIS and proposed method using fuzzy similarity and also to test how three different FPIS and FNIS criteria are effecting the results. We use the same numerical example which was given in [19] to make this comparison clear. A high-technology manufacturing company desires to select a suitable material supplier to purchase the key components of new products. After preliminary screening, five candidates (𝐴₁,𝐴₂,𝐴₃,𝐴₄ and 𝐴₅) remain for further evaluation. A committee of three decision-makers, 𝐷₁,𝐷₂, and 𝐷₃, has been formed to select the most suitable supplier. Five benefit criteria are considered:(1)profitability of supplier (𝐶₁),(2)relationship closeness (𝐶₂),(3)technological capability (𝐶₃),(4)conformance quality (𝐶₄),(5)conflict resolution (𝐶₅).

The proposed method is applied to solve the problem, the computational procedure is summarized as follows:

Step 1. Three decision-makers use the linguistic weighting variables shown in Figure 2 to assess the importance of the criteria. The importance weights of the criteria can be seen in Table 2.

Table 2 

Importance weight of criteria from three decision-makers.

Step 2. Three decision-makers use the linguistic rating variables shown in Figure 3 to evaluate the ratings of suppliers with respect to each criterion. The resulting ratings are shown in Table 3.

Table 3 

Ratings of the five suppliers by three decision-makers under five criteria.

Step 3. Convert the linguistic evaluations in Tables 2 and 3 into trapezoidal fuzzy numbers and determine the fuzzy weight of each criterion.

Step 4. Normalize the fuzzy-decision matrix.

Step 5. Build the weighted normalized fuzzy-decision matrix as in Table 4.

Table 4 

Weighted normalized fuzzy decision matrix.

Step 6. Determine FPIS and FNIS. Now according to three different criteria we get three different FPIS and FNIS which are given in Table 5.

Table 5 

Fuzzy positive-ideal solutions and fuzzy negative-ideal solutions according to different criteria.

Step 7. Calculate the similarity of each supplier by using weighted normalized fuzzy decision matrix and FPIS by calculating the similarity matrix, and then average similarity value for each supplier, as in Tables 6 and 7.

Table 6 

Similarities between FPIS and weighted normalized fuzzy decision matrix.

Table 7 

Comparison of closeness coefficient (𝐶𝐶_𝑖) and similarity values (𝑆_𝑖) in approval status table with different ideal solution criteria.

Step 8. According to the average similarity we get the ranking 𝐴₂>𝐴₃>𝐴₄>𝐴₁>𝐴₅. By using the approval status Table 1 we also get status approved and prefered to supplier 𝐴₂ in case we use FPIS_id3. From other cases we get approval status approved. Further study about the status table can be seen in Table 7.

4.1. Comparison of Different Ideal Solutions, Usage of Fuzzy Similarity, and Closeness Coefficient

In Table 7 we have compared the approval status classes and closeness and similarity values with respect to different ideal solution criteria. As can be seen from Table 7 the ranking order is the same in each case but the assessment status of the supplier is changing depending on which ideal solution criteria we choose. If we use ideal solution criteria 1 we get as results that with closeness criteria we get that suppliers 𝐴₂ and 𝐴₃ belong to class IV and others to class III. If we use similarity we get that suppliers from 𝐴₁ to 𝐴₄ belong to class IV and 𝐴₅ to class III. If we use the second ideal solution criteria we get the results other way around. With closeness coefficient suppliers from 𝐴₁ to 𝐴₄ belong to class IV and with similarity 𝐴₂ and 𝐴₃ belong to class IV and others to class III. Then if we choose the third criteria which seems most realistic and is truly using fuzzy trapezoidal numbers instead of presenting crisp numbers in a fuzzy way we get quite high assessments with similarity, now suppliers 𝐴₁ to 𝐴₄ belonging to class V and 𝐴₅ belonging to class IV. If we look at the closeness coefficient we get surprisingly different results as suppliers 𝐴₂ and 𝐴₃ get assessment status belonging to class V and 𝐴₄ belonging to class III, 𝐴₁ to class II, and 𝐴₅ to class I. When we consider how realistic these assessment status ratings are we first need to look to the decision-makers assessments. As can be seen from Table 3 decision-makers gave quite good ratings for the suppliers. There were several ratings as “very good” and even worse ratings were only “medium good”. Notice that even “Fair” which was in the middle was never used not to mention “Poor”. So decision-makers evaluated these suppliers with quite high ratings. This speaks to advantage of criteria three from fuzzy positive-ideal solution and the usage of similarity instead of closeness coefficient since with closeness coefficient rating, for example, for supplier 𝐴₅ would have been “do not recommend” even though all the decision-makers gave at least “medium good” rating for this supplier.

Having pointed this one also should notice that this approval status Table 1 was build more for closeness coefficient and using FPIS and FNIS with criteria 1; so to suggest appropriate approval status table from these experiments one could suggest to use similarity, criteria 3 for FPIS, and approval status given in Table 8.

Table 8 

Approval status table.

From this approval status Table 8 assessments status would be approved and prefered for 𝐴₂ and 𝐴₃ and approved for 𝐴₄ and 𝐴₁ and recommended with low risk for 𝐴₅. If one prefers to use FPIS criteria 2 and similarity then approval status Table 1 would be preferred. This experiment clearly shows that one should not just blindly use approval status table given in [19] (see also Table 1) but also consider the situation at hand and preferably even test different FPIS and FNIS criteria to get a better understanding about the supplier evaluation.

5. Conclusions and Discussion

In this paper a new approach is proposed to deal with the supplier selection problem in an uncertain environment. This problem to be solved is MCDM problem. In conventional MCDM methods, the ratings and the weights of attributes must be known precisely [17, 38]. This requirement however is not met in many situations since decision-makers judgements are often uncertain and cannot be estimated by an exact numerical value. This leads to the fact that the problem of selecting suppliers has many uncertainties and becomes more difficult. Fuzzy methods provides a solution for supplier selection problem [19]. However in evaluation of the supplier the evaluation depends heavily on the selection of appropriate fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS). In this paper we have proposed three different criteria for selecting FPIS and FNIS and studied how they affect supplier evaluation. Also a new proposal to fuzzy supplier evaluation and selection is given. This method uses fuzzy similarity to make supplier selection and evaluation. By using this method less computations are needed to supplier selection problem, and we do not need to calculate FNIS anymore when using similarity, but we can simply use given fuzzy similarity measure to calculate similarities between FPIS and suppliers. This simplifies the model and makes computations a bit easier. Also new assessment status Table 8 is proposed when similarity and third FPIS criteria are used. The experimental results show that the proposed approach is reliable and reasonable.