On fuzzy TOPSIS method based on alpha level sets

1 Introduction

Multicriteria decision-making (MCDM) often involves decision-makers (DMs)’ subjective judgments and preferences such as qualitative criteria ratings and the weights of criteria. For example, three candidates are being considered for a professor position. The attributes used are creativity, maturity, communication skill and number of publications. The performance scores for the first three attributes are not quantifiable, rather they are represented by linguistic terms such as “good”, “average”, “poor” etc. The fourth attribute can be some integer numbers. This MADM problem contains a mixture of fuzzy and crisp data. Most of the real world MADM problems are of this type. To make a decision analysis under fuzzy environment, the classic TOPSIS method proposed by Hwang and Yoon [1] has been extensively extended by many researchers to deal with fuzzy multic-riteria decision-making problems [2–40].

In [11], a fuzzy TOPSIS method was proposed based on alpha level sets and a nonlinear programming (NLP) solution procedure was presented. The fuzzy decision matrix was represented by using α-cuts, thus making it possible to use all permissible forms of fuzzy values to represent the parameters of the decision matrix. The relationship between the fuzzy TOPSIS method and fuzzy weighted average (FWA) was also discussed. Three numerical examples including an application to bridge risk assessment were investigated using the proposed fuzzy TOPSIS method to illustrate its applications and the differences from the other procedures. Their method was claimed to be a good means of evaluation and more appropriate than other evaluation methods in the case that the fuzzy decision matrix was of the same dimension and needs no normalization. This paper provides a derivation process to prove the fact that their NLP models is not correct in such situation, and propose a new pair of NLP models for computing the relative closeness of fuzzy TOPSIS. The purpose of this paper is not to propose any new fuzzy TOPSIS method for fuzzy decision making, but to give a note pointing out the NLP problems with Wang and Elhag’s method to avoid any possible misapplications and to bring forward the corrections to their fuzzy TOPSIS method.

The rest of the paper is organized as follows. Section 2 gives a brief review of Wang and Elhag’s fuzzy TOPSIS method in the case that the fuzzy decision matrix is of the same dimension and needs no normalization. The reason is analysed in Section 3 and the corrections to their NLP models are also suggested for computing the relative closeness of fuzzy TOPSIS in the case that the fuzzy decision matrix is of the same dimension. Section 4 investigates an numerical examples including a manufacturer of transmission cables for motorcycles needs to select a supplier of metallic components to show the application of the fuzzy TOPSIS method and its differences from the aggregation operator-based MADM method. The paper is concluded in Section 5.

2 Preliminaries

TOPSIS (for the Technique for Order Preference by Similarity to Ideal Solution) was developed by Hwang and Yoon [1] as an alternative to the ELECTRE method and can be considered as one of its most widely accepted variants. The basic concept of this method is that the selected alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution in some geometrical sense. The TOPSIS method assumes that each criterion has a tendency of monotonically increasing or decreasing utility. Therefore, it is easy to define the ideal and negative-ideal solutions. The Euclidean distance approach was proposed to evaluate the relative closeness of the alternatives to the ideal solution. Thus, the preference order of the alternatives can be derived by a series of comparisons of these relative distances.

Suppose a MCDM problem has n alternatives, A₁, A₂, …, A _n , and m decision criteria/attributes, C₁, C₂, …, C _m . Each alternative is evaluated with respect to the m criteria/attributes. All the values/ratings assigned to the alternatives with respect to each criterion form a decision matrix denoted by X = (x _ij ) _n×m. Let W = (w₁, w₂, …, w _m )  ^T be the relative weight vector about the criteria, satisfying w _j  ⩾ 0, ∑ j = 1 m w j = 1 . Then the TOPSIS method can be summarized as follows:

Normalize the decision matrix X = (x _ij ) _n×m using the equation below: r ij = x ij ∑ k = 1 n x kj 2(1) where i = 1, 2, …, n ; j = 1, 2, …, m, r _ij is the normalized criteria/attribute value/rating.

In fuzzy MCDM problems, criteria/attribute values and the relative weights are usually characterized by fuzzy numbers. A fuzzy number is a convex fuzzy set, characterized by a given interval of real numbers, each with a grade of membership between 0 and 1. The most commonly used fuzzy numbers are triangular fuzzy numbers, whose membership functions are respectively defined as μ A ˜ ( x ) = { x - a b - a , a ⩽ x ⩽ b , d - x d - b , b ⩽ x ⩽ d , 0 , others(8)

For brevity, triangular fuzzy number A ˜ is often denoted as (a, b, d), if a = b = d, then the triangular fuzzy number becomes real number b. That’s to say that any real number r is expressed as the triangular fuzzy number (r, r, r). According to Zadeh’s extension principle [41], a fuzzy number A can also be expressed by its intervals, namely A ˜ = ⋃ α α A α , 0 < α ⩽ 1(9) where A _α is referred to as α-level sets or α-cuts of the fuzzy number A ˜ , and A α = { x ∈ R | μ A ˜ ( x ) ⩾ α } = [ A α L , A α U ](10) where A α L = min { x ∈ R | μ A ˜ ( x ) ⩾ α } , A α U = max { x ∈ R | μ A ˜ ( x ) ⩾ α } .

Suppose A₁, A₂, …, A _n are n possible alternatives among which decision makers have to choose, C₁, C₂, …, C _m are criteria with which alternatives performance are measured, x ¯ ij is a fuzzy number representing the rating of alternative A _i with respect to criterion C _j . Let X ¯ = ( x ¯ ij ) n × m be a interval decision matrix, and W ˜ = ( w ˜ 1 , w ˜ 2 , … , w ˜ m ) T be fuzzy weights characterized by μ w ˜ ij ( i = 1 , 2 , … , n ; j = 1 , 2 , … , m ) . A MCDM problem with interval data can be concisely expressed in matrix format as below:

X ¯ = ( x ¯ 11 x ¯ 12 ⋯ x ¯ 1 m x ¯ 21 x ¯ 22 ⋯ x ¯ 2 m ⋮ ⋮ ⋱ ⋮ x ¯ n 1 x ¯ n 1 ⋯ x ¯ nm )

If all the criteria/attributes, C₁, C₂, …, C _m , are assessed using the same set of fuzzy linguistic variables, then the fuzzy decision matrix X ¯ is of the same dimension and therefore needs no normalization. Wang and Elhag [11] tried to combine the TOPSIS method and the extension principle to solve fuzzy MCDM problems. Accordingly, they presented a fuzzy TOPSIS method based on α-level sets, which can be summarized as follows:

The fuzzy decision matrix X ¯ is of the same dimension and therefore needs no normalization.

In this section, we deduce a pair of NLP models for computing the fuzzy relative closeness of each alternative, and prove that Wang and Elhag’s NLP models (13) and (14) are therefore not correct.

By setting different α levels, let ( x ij ) α = [ ( x ij ) α L , ( x ij ) α U ] be the α-level sets of the triangular fuzzy numbers x ˜ ij = ( a ij , b ij , d ij ) (i = 1, 2, ⋯, ., n ; j = 1, 2, ⋯, m), and ( w j ) α = [ ( w j ) α L , ( w j ) α U ] (j = 1, 2, ⋯, m) be α-level sets of the weight w ˜ ij (j = 1, 2, ⋯, m), respectively, where

( x ij ) α L = a ij + ( b ij - a ij ) α(16) ( x ij ) α U = d ij - ( d ij - b ij ) α(17)

Then the fuzzy relative closeness of the ith alternative can be equivalently rewritten as

RC i = ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 + ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 , i = 1 , 2 , … , n(18) where ( x ij ) α L ⩽ x ij ⩽ ( x ij ) α U , ( w j ) α L ⩽ w j ⩽ ( w j ) α U , j = 1, 2, …, m.

Obviously, RC _i is an interval in this situation, whose lower and upper bounds can be captured by the following pair of fractional programming models: { RC i L = min { ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 + ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 } s . t . ( x ij ) α L ⩽ x ij ⩽ ( x ij ) α U , ( w j ) α L ⩽ w j ⩽ ( w j ) α U , j = 1 , 2 , … , m .(19){ RC i U = max { ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 + ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 } s . t . ( x ij ) α L ⩽ x ij ⩽ ( x ij ) α U , ( w j ) α L ⩽ w j ⩽ ( w j ) α U , j = 1 , 2 , … , m .(20)

Due to the fact that

∂ RC i ∂ x ij = ( x ij - x j - ) ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 + ( x j * - x ij ) ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 ( ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 + ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 ) 2 ∑ j = 1 m [ w j ( x ij - x j - ) ] 2 ∑ j = 1 m [ w j ( x ij - x j * ) ] 2 w j 2(21)

According to Equations (11) and (12), in the case of j ∈ Q _b , x j - = min i a ij , x j * = max i d ij , then x ij - x j - ⩾ ( x ij ) α L - x j - = a ij + ( b ij - a ij ) α - min i a ij ⩾ ( b ij - a ij ) α ⩾ 0 , and x j * - x ij ⩾ x j * - ( x ij ) α U = max i d ij - ( d ij - ( d ij - b ij ) α ) = ( max i d ij ) - d ij + ( d ij - b ij ) α ⩾ ( d ij - b ij ) α ⩾ 0 .

In general, ∂ RC i ∂ x ij > 0 for the equal signs in the inequality are taken under different conditions. According to Equations (11) and (12), in the case of j ∈ Q _c , x j - = max i d ij , x j * = min i a ij , then x ij - x j - ⩽ ( x ij ) α U - x j - = d ij - ( d ij - b ij ) α - max i d ij ⩽ - ( d ij - b ij ) α ⩽ 0 and x j * - x ij ⩽ x j * - ( x ij ) α L = min i a ij - ( a ij + ( b ij - a ij ) α ) = ( min i a ij ) - a ij - ( b ij - a ij ) α ⩽ - ( b ij - a ij ) α ⩽ 0 .

Generally, ∂ RC i ∂ x ij < 0 for the equal signs in the inequality are taken under different conditions.

Therefore, RC _i is a monotonically increasing function of x _ij (j ∈ Q _b ), and a monotonically decreasing function of x _ij (j ∈ Q _c ), which means RC _i reaches its maximum at x ij = ( x ij ) α U (j ∈ Q _b ), x ij = ( x ij ) α L (j ∈ Q _c ) and arrives at its minimum when x ij = ( x ij ) α L (j ∈ Q _b ), x ij = ( x ij ) α U (j ∈ Q _c ). So, the above pair of fractional programming models (19) and (20) can be simplified as

{ min { ∑ j ∈ Q b [ w j ( ( x ij ) α L - x j - ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α U - x j - ) ] 2 ∑ j ∈ Q b [ w j ( ( x ij ) α L - x j - ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α U - x j - ) ] 2 + ∑ j ∈ Q b [ w j ( ( x ij ) α L - x j * ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α U - x j * ) ] 2 } s . t . ( w j ) α L ⩽ w j ⩽ ( w j ) α U , j = 1 , 2 , … , m .(22) { max { ∑ j ∈ Q b [ w j ( ( x ij ) α U - x j - ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α L - x j - ) ] 2 ∑ j ∈ Q b [ w j ( ( x ij ) α U - x j - ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α L - x j - ) ] 2 + ∑ j ∈ Q b [ w j ( ( x ij ) α U - x j * ) ] 2 + ∑ j ∈ Q c [ w j ( ( x ij ) α L - x j * ) ] 2 } s . t . ( w j ) α L ⩽ w j ⩽ ( w j ) α U , j = 1 , 2 , … , m .(23)

Obviously, the new pair of NLP models (22) and (23) are corroborative ones and different from Wang and Elhag’s (13) and (14).

In this section, we examine a numerical example using the fuzzy TOPSIS method based on the new pair of NLP models (22) and (23).

Reconsider the example investigated by Lima Junior [42], in which a manufacturer of transmission cables for motorcycles needs to select a supplier of metallic components used in a variety of transmission cables. To select the best alternative, five potential suppliers are evaluated against five decision criteria. The evaluation of the potential suppliers in each criterion is made based on linguistic judgments given by the decision makers (DMs), a group of employees from the quality and purchase are as of the company. The criteria are defined by the DMs, as follows:

Quality (C₁): related to quality of conformance, quality management and after sale service quality.

The relative importance weights of the five criteria are described using linguistic variables such as Moderately important, Important, Very important etc., which are defined in Table 1. The ratings (i.e. criteria values) are also characterized by linguistic variables such as Low, Medium, High etc., which are defined in Table 2. Table 3 presents the linguistic judgments of the weights of the criteria and the ratings of the alternatives for the three decision makers involved in the selection process. The linguistic variables shown in Table 3 are converted into triangular fuzzy number (TFN). Table 4 presents the parameters of the TFN resulting from aggregation of the judgments presented in Table 3, and it represents the fuzzy decision matrix, where the aggregated fuzzy numbers are obtained by averaging the fuzzy opinions of the three DMs. That is x ˜ ij = 1 3 ∑ k = 1 3 x ˜ ij ( k ) and w ˜ j = 1 3 ∑ k = 1 3 w ˜ j ( k ) , where x ˜ ij ( k ) and w ˜ j ( k ) are the ratings and the relative importance weight given by the kth DM, k = 1, 2, 3.

According to Wang and Elhag [11], the ideal solution ( A ˜ * ) and the negative ideal solution ( A ˜ - ) are defined as A ˜ * = { x 1 * , x 2 * , ⋯ , x 5 * } = { ( max j d ij | j ∈ Q b ) , ( min j a ij | j ∈ Q c ) } = { 10.0 , 3.33 , 10.0 , 10.0 , 10.0 }(24)A ˜ - = { x 1 - , x 2 - , ⋯ , x 5 - } = { ( min j a ij | j ∈ Q b ) , ( max j d ij | j ∈ Q c ) } = { 2.50 , 10.0 , 2.50 , 3.33 , 2.50 }(25)

As far as this example is concerned, due to the fact that the five criteria are all assessed using the same set of linguistic variables defined in Table 2. To make the fuzzy relative closeness of each alternative accurate enough, 11 α levels are set for computation, i.e. α=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0, respectively. All the NLP models determined by (22) and (23) are solved using the LINGO software package [43]. The results are presented in Table 5 from columns 2 to 6. The defuzzified values computed by Equation (15) are ( RC ˜ 1 ) ALC = 0.442 , ( RC ˜ 2 ) ALC = 0.6284 , ( RC ˜ 3 ) ALC = 6445 , ( RC ˜ 4 ) ALC = 0.4518 and ( RC ˜ 5 ) ALC = 0.4528 , which give the ranking of A₃ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₁, where the symbol means is superior or preferred to.

The relative closenesses of the five candidates obtained by Lima Junior [42] are 0.47 for A₁, 0.68 for A₂, 0.62 for A₃, 0.56 for A₄, and 0.50 for A₅. The reason Francisco’s approach leads to different ranking than ours is because his approach produces only a crisp point estimate for the relative closeness of each candidate, while our proposed fuzzy TOPSIS method generates a nearly accurate fuzzy estimate for the relative closeness of each candidate. It is obvious that our proposed fuzzy TOPSIS method provides much more information on the relative closeness of each candidate than Lima Junior et al’s approach.

In fuzzy decision theory, the weighted sum model (WSM) [44, 45] is the best known and simplest multi-criteria decision analysis (MCDA)/multi-criteria decision making method for evaluating a number of alternatives in terms of a number of decision criteria [46–53]. Due to the fact that all the criteria/attributes are assessed using the same set of fuzzy linguistic variables, then the fuzzy decision matrix X ˜ is of the same dimension. Therefore the fuzzy weighted average operator [54–60] also is used for supplier selection. There are benefit attributes C₁, C₃, C₄ and C₅ (i.e., the bigger the attribute values the better) and cost attribute C₂ (i.e., the smaller the attribute values the better) in multi-attribute decision making. Let A ˜ ij = ( a ij , b ij , d ij ) (i = 1, 2, …, n ; j = 1, 2, …, m) be triangular fuzzy numbers, first of all, we may transform the attribute values of cost type into the attribute values of benefit type by A ˜ ij = { ( a ij , b ij , d ij ) , j ∈ Q b ( d * - d ij , d * - b ij , d * - a ij ) , j ∈ Q c(26) where i = 1, 2, …, n, d * = max i , j { d ij } . The transformed fuzzy decision matrix is also denoted by A ˜ ij in Table 7.

To compare the proposed fuzzy TOPSIS method and the FWA method, presented in the columns 2 through 6 of Table 6 are the results obtained by the FWA method. It is very clear from Tables 5 and 6 that the proposed fuzzy TOPSIS method based on alpha-level sets and the FWA produce almost the same alpha-level sets and defuzzified values for the five candidates. There is significant difference between the two approaches. The main reason is the FWA method can be seen as a special case of the fuzzy TOPSIS method, where the fuzzy decision matrix must be normalized, the ideal solution and negative ideal solution are defined as {1, 1, …, 1} and {0, 0, …, 0}, respectively, and absolute distance is substituted to measure the differences between each alternative and the ideal solution as well as the negative ideal solution [11].

In this paper we have examined the fuzzy TOPSIS method developed by Wang and Elhag. In the case that the fuzzy decision matrix is of the same dimension and needs no normalization, it has been proved that the pair of NLP models are lack of theoretical evidence and may lead to erroneous decision results. We have provided a correct pair of NLP models for computing the relative closeness in this situation. The illustrated example for selecting the best supplier of metallic components has been examined using the proposed NLP models to fuzzy TOPSIS method and has been demonstrated its superiorities, rationalities, and the detailed implementation process.