Some Similarity Measures for Triangular Fuzzy Number and Their Applications in Multiple Criteria Group Decision-Making

Abstract

We propose some similarity measures between two triangular fuzzy numbers (TFNs) based on the vector similarity measures in vector space, which can be used to aggregate the decision information with TFNs. A methodology for multiple criteria group decision-making (MCGDM) problems with triangular fuzzy information is proposed; the criteria values take the form of linguistic values, which can be converts to TFNs. According to the weighted similarity measures between each alternative and ideal alternative, it is easy to rank alternatives and select the most desirable alternative. Finally, we apply the proposed methods to an illustrative example of MCGDM; the numerical results show that our method is effective and practical. For comparison, we also apply our similarity measures method to solve the fuzzy decision-making problem in Wei (2011); our method has simpler computation and gets the same results more rapidly than the FLOWHM method.

1. Introduction

Fuzzy multiple criteria decision-making (FMCDM) is the process of ranking the feasible alternatives and selecting the best one by considering multiple criteria, in which the alternatives and criteria values are carried by fuzzy sets [1], intuitionistic fuzzy set [2, 3], triangular fuzzy number [4], triangular intuitionistic fuzzy number [5], interval-valued triangular fuzzy number [6], trapezoidal fuzzy number [7, 8], intuitionistic trapezoidal fuzzy number [9], internal-valued trapezoidal fuzzy number [10], and so on. As an important part of the modern decision science, some related methods have been successfully applied to fuzzy decision-making problems, for example, ordered weighted aggregation operators [11], weighted geometric aggregation operators [10], TOPSIS method [12–14], analytic hierarchy process method [15], grey relational analysis method [16, 17], and similarity measures [3, 18], and so forth. Over the last decades, many studies have been done on the concepts of similarity measures between two intuitionistic fuzzy sets. On the one hand, the similarity measures were defined based on distance models, such as the Hamming distance similarity method [19], the Hausdorff distance similarity measure [20], and the Euclidean distance similarity measure [3]. On the other hand, the intuitionistic fuzzy set was seen as a vector containing some elements, by using the vector similarity measures to define the similarity measures between two intuitionistic fuzzy sets, for example, the cosine similarity measures [21] and so forth. However, as an effective method and a wide range of application in various fields, very few researchers worked on similarity measures between two triangular fuzzy numbers and applied them to triangular fuzzy multiple criteria group decision-making.

Because the triangular fuzzy number (TFN) is intuitive, easy to use, computationally simple, and useful in promoting representation and information processing in a fuzzy environment, it was usefully applied to solve FMCDM problems, in which the criteria values are carried by the TFNs. Many methodologys have been proposed to solve FMCDM with TFNs in the literature; Fu [22] presented a fuzzy optimization method based on the concept of ideal and anti-ideal points to solve FMCDM problems with TFNs. A case study of reservoir flood control operation was given to demonstrate the proposed method’s effectiveness. Wei [23] proposed fuzzy induced ordered weighted harmonic mean operator to solve fuzzy group decision-making problems. In [24], Wei et al. investigated the multiple attribute decision-making problems with triangular fuzzy information and developed an generalized triangular fuzzy correlated averaging operator; numerical results showed the method was applicable. The above literatures mainly focus on the group decision-making with unknown information on criterion weight and on group member weight. However, in some actual decision-making process, the information about criterion weight and on group member weight are important. Therefore, with the defects of the methods in the literatures, we develop some similarity measures to solve FMCGDM problems with known information on criterion weight and on group member weight. In order to do so, the rest of this paper is set out as follows. In the next section, we introduce some basic concepts and operational laws of TFNs; three similarity measures between two TFNs are also defined. In Section 3, we present a triangular FMCGDM method based on similarity measures. Section 4 gives two illustrated examples to verify the developed approach; all numerical results show that our method is feasible and applicable. The paper ends with conclusion in Section 5.

2. Preliminaries

We consider the following well-know description of a triangular fuzzy number 𝛼.

Definition 1 (see [25]). A triangular fuzzy number 𝛼 can be defined by a triplet (𝑎₁, 𝑎₂, and 𝑎₃). The membership function 𝜇_𝛼(𝑥) is 𝜇_𝛼(𝑥)=0,𝑥<𝑎₁,𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,𝑥−𝑎₃𝑎₂−𝑎₃,𝑎₂≤𝑥≤𝑎₃,0,𝑎₃<𝑥, (1) where 0≤𝑎₁≤𝑎₂≤𝑎₃≤1, 𝑎₁ and 𝑎₃ stand for the lower and upper values of the support of 𝛼, respectively, and 𝑎₂ stands for the modal values. The membership functions can be seen in Figure 1.

Figure 1 

Membership functions of TFNs.

Let 𝛼=(𝑎₁,𝑎₂,𝑎₃) and 𝛽=(𝑏₁,𝑏₂,𝑏₃) be two TFNs and 𝛾 a positive scalar number; the basic operational laws related to TFNs are shown as 𝛼⊕𝛽=𝑎₁+𝑏₁,𝑎₂+𝑏₂,𝑎₃+𝑏₃,𝛼⊗𝛽=𝑎₁𝑏₁,𝑎₂𝑏₂,𝑎₃𝑏₃,𝛾⊗𝛼=𝛾𝑎₁,𝛾𝑎₂,𝛾𝑎₃,1𝛼=1𝑎₁,1𝑎₂,1𝑎₃. (2)

In the vector space, there are some similarity measures between two vectors, which successfully apply to various fields, such as pattern recognition, description and classification of complex structured objects, faculty recruitment, and fuzzy assignment problems. In this section, we introduce three important vector similarity measures.

Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝑌=(𝑦₁,𝑦₂,…,𝑦_𝑛) be two vectors of length 𝑛, where all the coordinates are positive; three important similarity measures are defined as 𝐽(𝑌,𝑍)=𝑋𝑌‖𝑋‖²₂+‖𝑌‖²₂=∑−𝑋𝑌_𝑛𝑖=1𝑥_𝑖𝑦_𝑖∑^𝑛_𝑖=1𝑥²_𝑖+∑^𝑛_𝑖=1𝑦²_𝑖−∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖,𝐸(𝑌,𝑍)=2𝑋𝑌‖𝑋‖²₂+‖𝑌‖²₂=2∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖∑^𝑛_𝑖=1𝑥²_𝑖+∑^𝑛_𝑖=1𝑦²_𝑖,𝐶(𝑌,𝑍)=𝑋𝑌‖𝑋‖₂‖𝑌‖₂=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖∑^𝑛_𝑖=1𝑥²_𝑖∑^𝑛_𝑖=1𝑦²_𝑖. (3)

The three parameters 𝑎_𝑖 (𝑖=1,2,and3) in TFN 𝛼=(𝑎₁,𝑎₂,and𝑎₃) can be considered as one vector representation with three elements. Based on the extension of the similarity measures in vector space, the similarity measures between two TFNs are shown in Definition 2.

Definition 2. Let 𝛼=(𝑎₁,𝑎₂,𝑎₃) and 𝛽=(𝑏₁,𝑏₂,𝑏₃) be two TFNs, where 0≤𝑎₁≤𝑎₂≤𝑎₃≤1, 0≤𝑏₁≤𝑏₂≤𝑏₃≤1; the similarity measures between two TFNs can be defined as follows: 𝑆^𝐽=∑𝛼,𝛽^3𝑖=1𝑎_𝑖𝑏_𝑖∑³_𝑖=1𝑎²_𝑖+∑³_𝑖=1𝑏²_𝑖−∑³_𝑖=1𝑎_𝑖𝑏_𝑖,𝑆^𝐸𝛼,𝛽=2∑³_𝑖=1𝑎_𝑖𝑏_𝑖∑³_𝑖=1𝑎²_𝑖+∑³_𝑖=1𝑏²_𝑖,𝑆^𝐶=∑𝛼,𝛽^3𝑖=1𝑎_𝑖𝑏_𝑖∑³_𝑖=1𝑎²_𝑖∑³_𝑖=1𝑏²_𝑖. (4)(5)(6)
The three similarity measures between two TFNs in Definition 2 satisfies the following properties, as in Theorem 3.

Theorem 3. The similarity satisfies the following properties: (1)0≤𝑆(𝛼,𝛽)≤1, (2)𝑆(𝛼,𝛽)=𝑆(𝛽,𝛼), (3)if 𝛼=𝛽, that is, 𝑎_𝑖=𝑏_𝑖, 𝑖=1,2, and 3. 𝑆(𝛼,𝛽)=1.

Proof. Firstly, we prove 𝑆^𝐽(𝛼,𝛽) satisfies the above properties, as follows.(P1) It is obvious that 𝑆^𝐽(𝛼,𝛽)≥0. Thus, we only need to prove 𝑆^𝐽(𝛼,𝛽)≤1. By using the basic mathematical equations, 2𝑎_𝑖𝑏_𝑖≤𝑎²_𝑖+𝑏²_𝑖, (7) we get ³_𝑖=1𝑎²_𝑖+³_𝑖=1𝑏²_𝑖−³_𝑖=1𝑎_𝑖𝑏_𝑖≥2𝑎_𝑖𝑏_𝑖−𝑎_𝑖𝑏_𝑖≥𝑎_𝑖𝑏_𝑖. (8) Taking (8) in (4), we get 𝑆^𝐽(𝛼,𝛽)≤1.(P2) It is obvious that the equation is true.(P3) When 𝛼=𝛽, that is, 𝑎_𝑖=𝑏_𝑖, for 𝑖=1,2, and 3, 𝑆^𝐽∑𝛼,𝛽^3𝑖=1𝑎_𝑖𝑎_𝑖∑³_𝑖=1𝑎²_𝑖+∑³_𝑖=1𝑎²_𝑖−∑³_𝑖=1𝑎_𝑖𝑎_𝑖=1. (9)
In the same way, we can prove 𝑆^𝐸(𝛼,𝛽) and 𝑆^𝐶(𝛼,𝛽) satisfy the properties in Theorem 3.
Therefore, we have finished the proofs.

3. Triangular Fuzzy Multiple Criteria Group Decision-Making Method Based on Similarity Measures

In this section, we present a handling method for TFNs multiple criteria group decision-making problems.

Let 𝐶={𝐶₁,𝐶₂,…,𝐶_𝑛} be a set of criteria and 𝐴={𝐴₁,𝐴₂,…,𝐴_𝑚} a set of alternatives. Suppose we invite 𝑝 experts to make the judgement, and let 𝐺={𝐺₁,𝐺₂,…,𝐺_𝑝} be a set of experts. They are expected to give the linguistic value of TFNs (see Table 1). The vectors of alternative 𝐴_𝑖 given by the expert 𝐺_𝑘 are represented by following TFNs: 𝑣^𝑘_𝑖=𝐶₁,𝑎^𝑘_𝑖11,𝑎^𝑘_𝑖12,𝑎^𝑘_𝑖13,𝐶₂,𝑎^𝑘_𝑖21,𝑎^𝑘_𝑖22,𝑎^𝑘_𝑖23𝐶,…,_𝑛,𝑎^𝑘_𝑖𝑛1,𝑎^𝑘_𝑖𝑛2,𝑎^𝑘_𝑖𝑛3, (10) for 𝑖=1,2,…,𝑚, 𝑘=1,2,…,𝑝.

Table 1 

Linguistic values of TFNs.

The weight vector of the experts is 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑝); for each alternative 𝐴_𝑖, the group preference vector can be calculated by 𝑉_𝑖=𝐶₁,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖11,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖12,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖13,𝐶₂,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖21,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖22,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖23,…,𝐶_𝑛,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖𝑛1,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖𝑛2,^𝑝_𝑘=1𝜆_𝑘𝑎^𝑘_𝑖𝑛3. (11)

For the different critera 𝐶_𝑗(𝑗=1,2,…,𝑛), the weight of a criterion is a triangular fuzzy weight, which is obtained by the decision maker, as 𝑤_𝑗=(𝑎_𝑗1,𝑎_𝑗2,𝑎_𝑗3)(𝑗=1,2,…,𝑛). The expected weight value 𝑤_𝑗(𝑗=1,2,…,𝑛) for a triangular fuzzy weight is obtained by the following equation: 𝐸𝑉𝑤_𝑗=𝑎_𝑗1+𝑎_𝑗2+𝑎_𝑗33. (12) Then we normalize the expected weight value 𝑤_𝑗(𝑗=1,2,…,𝑛) by the following formula: 𝑤_𝑗=𝐸𝑉𝑤_𝑗∑^𝑚_𝑖=1𝐸𝑉𝑤_𝑗. (13)

In FMCGDM environments, the concept of ideal point has been used to determine the best alternative in the decision set. Although the ideal alternative does not exist in real world, it does provide a useful theoretical construct to evaluate alternatives (Ye [26]).

Thus three weighted similarity measures between an alternative 𝐴_𝑖 and the ideal alternative 𝐴_𝑃 represented by the TFNs are defined as follows: 𝑆^𝐽𝐴_𝑖,𝐴_𝑃=^𝑛_𝑗=1𝑤_𝑗∑³_𝑙=1𝑎_𝑖𝑗𝑙𝑏_𝑝𝑗𝑙∑³_𝑙=1𝑎²_𝑖𝑗𝑙+∑³_𝑙=1𝑏²_𝑝𝑗𝑙−∑³_𝑙=1𝑎_𝑖𝑗𝑙𝑏_𝑝𝑗𝑙,𝑆^𝐸𝐴_𝑖,𝐴_𝑃=^𝑛_𝑗=1𝑤_𝑗2∑³_𝑙=1𝑎_𝑖𝑗𝑙𝑏_𝑝𝑗𝑙∑³_𝑙=1𝑎²_𝑖𝑗𝑙+∑³_𝑙=1𝑏²_𝑝𝑗𝑙,𝑆^𝐶𝐴_𝑖,𝐴_𝑃=^𝑛_𝑗=1𝑤_𝑗∑³_𝑙=1𝑎_𝑖𝑗𝑙𝑏_𝑝𝑗𝑙∑³_𝑙=1𝑎²_𝑖𝑗𝑙∑³_𝑙=1𝑏²_𝑝𝑗𝑙. (14)

The decision procedure for the proposed method can be summarized as follows.

Step 1. Give the characteristic of each alternative and criteria by the linguistic values of TFNs.

Step 2. Using (11), we can obtain the group preference vector for each alternative 𝐴_𝑖(𝑖=1,2,…,𝑚).

Step 3. Calculate the expected weight value 𝑤_𝑗 for a criterion 𝐶_𝑗(𝑗=1,2,…,𝑛) by using (12) and (13).

Step 4. From formula (14), we can get three weighted similarity measures for an alternative 𝐴_𝑖(𝑖=1,2,…,𝑚).

Step 5. Rank the alternatives, and select the best one(s) in accordance with the weighted similarity measure.

4. Illustrative Example

4.1. Example 1: Personnel Selection

In this section, a numerical example is presented to illustrate the application of the proposed method. Suppose that the human resources department of a company desires to hire a competent. After initial screening, four candidates (i.e., alternatives) 𝐴₁, 𝐴₂, 𝐴₃, and 𝐴₄ remain for further evaluation. In order to select the most suitable candidate, the decision maker invite five experts, who take into account the following five criteria: (1) emotional steadiness (𝐶₁); (2) oral communication skill (𝐶₂); (3) education experience (𝐶₃); (4) work experience (𝐶₄); (5) personality and self-confidence (𝐶₅). Assume that the five experts provide his/her preference information on candidates with regard to criteria by using a linguistic variable, as listed in Table 1.

The decision procedure for the above decision-making problem is as follows.

Step 1. Five experts provide his/her preference information on candidates with linguistic terms; we convert the linguistic variables to TFNs, as depicted in Table 2.

Table 2 

Preference values of alternative given by 5 experts for TFNs.

Step 2. The weight vector of five experts is 𝜆=(0.1,0.2,0.3,0.2,and0.2), and using (11), we can obtain the group preference vector for each alternative 𝐴_𝑖(𝑖=1,2,3,4), as follows: 𝑉₁=𝐶₁,(0.13,0.17,0.22),𝐶₂,(,𝐶0.36,0.46,0.56)₃,(0.39,0.54,0.69),𝐶₄,(,𝐶0.42,0.50,0.60)₅,(;𝑉0.52,0.60,0.60)₂=𝐶₁,(0.22,0.27,0.32),𝐶₂,(,𝐶0.54,0.64,0.74)₃,(0.60,0.75,0.90),𝐶₄,(,𝐶0.44,0.54,0.62)₅,(;𝑉0.18,0.26,0.26)₃=𝐶₁,(0.18,0.23,0.27),𝐶₂,(,𝐶0.28,0.36,0.46)₃,(0.69,0.84,0.99),𝐶₄,(,𝐶0.16,0.24,0.34)₅,(;𝑉0.48,0.58,0.58)₄=𝐶₁,(0.21,0.26,0.31),𝐶₂,(,𝐶0.34,0.42,0.52)₃,(0.57,0.72,0.87),𝐶₄,(,𝐶0.28,0.38,0.48)₅,(0.46,0.56,0.56). (15)

Step 3. The weight value of criteria 𝐶_𝑖 is represented by the following TFNs: 𝑤₁=𝐶₁,(,𝑤0.3,0.4,0.5)₂=𝐶₂,(,𝑤0.5,0.6,0.7)₃=𝐶₃,(,𝑤0.1,0.2,0.3)₄=𝐶₄,(,𝑤0.9,1.0,1.0)₅=𝐶₅,(0.4,0.5,0.7). (16)
The expected weight values can be obtained by (12). Then we normalize the expected weight value by (13) and get the weight vector 𝑊=(0.1481,0.2222,0.0741,0.3580,and0.1975).

Step 4. The ideal alternative is given by the decision maker as 𝐴_𝑝=𝐶₁,(0.40,0.50,0.60),𝐶₂,(,𝐶0.70,0.80,0.90)₃,(0.80,0.90,1.00),𝐶₄,(,𝐶0.70,0.80,0.90)₅,(0.50,0.60,0.70). (17) From formula (14), we can get three weighted similarity measure for an alternative 𝐴_𝑖(𝑖=1,2,…,4), as listed in Table 3.

Table 3 

Decision results of three weighted similarity measures for TFNs.

Step 5. From Table 3, we can see that all the proposed method have the same decision results. The decision results of different similarity measures demonstrate that the proposed method for FMCGDM problem is effective.
Let us discuss the different weight 𝑊^∗=(0.2,0.1,0.3,0.20,and0.2), and using the above 4 steps, we can get the decision results as follows.
In Table 4, it is easy to see that the three similarity measures get the same decision results; the best alternative is 𝐴₄. From Tables 3 and 4, for the different weights of the criteria, the decision results are different.

Table 4 

Decision results for different weights in the same numerical example.

4.2. Example 2: Investment Case Selection

Using the illustrative example in [23], an investment company wants to invest a sum of money in the best option. There is a panel with five possible alternatives to invest the money: 𝐴₁ is a car company; 𝐴₂ is a food company; 𝐴₃ is a computer company; 𝐴₄ is an arms company; 𝐴₅ is a TV company. The three experts (𝐺₁, 𝐺₂, and 𝐺₃) must take a decision according to the following four attributes: 𝐶₁ is the risk analysis; 𝐶₂ is the growth analysis; 𝐶₃ is the social-political impact analysis; 𝐶₄ is the environmental impact analysis.

Using our methods in Section 3, the decision procedure is as follows.

Step 1. The TFNs preference vector of the three experts is shown in Table 5.

Table 5 

The TFNs preference vector of the three experts.

Step 2. The weight vector of three experts is 𝜆=(0.2,0.5,and0.3), and using (11), we can obtain the group preference vector for each alternative 𝐴_𝑖(𝑖=1,2,3,4,5) as follows: 𝑉₁=𝐶₁,(0.611,0.661,0.711),𝐶₂,(,𝐶0.531,0.571,0.611)₃,(0.721,0.741,0.771),𝐶₄,(;𝑉0.431,0.461,0.491)₂=𝐶₁,(0.691,0.711,0.741),𝐶₂,(,𝐶0.481,0.581,0.641)₃,(0.396,0.511,0.486),𝐶₄,(;𝑉0.501,0.531,0.561)₃=𝐶₁,(0.761,0.781,0.791),𝐶₂,(,𝐶0.711,0.741,0.761)₃,(0.581,0.601,0.631),𝐶₄,(;𝑉0.741,0.761,0.771)₄=𝐶₁,(0.631,0.661,0.691),𝐶₂,(,𝐶0.781,0.791,0.811)₃,(0.791,0.801,0.811),𝐶₄,(;𝑉0.781,0.801,0.811)₅=𝐶₁,(0.591,0.601,0.621),𝐶₂,(,𝐶0.591,0.601,0.591)₃,(0.641,0.661,0.706),𝐶₄,(0.751,0.781,0.781). (18)

Step 3. From [23], the weight value of criteria vector 𝐶_𝑖 is 𝑊=(0.2,0.1,0.1,0.4).

Step 4. The ideal alternative is given by the decision maker as 𝐴_𝑝=𝐶₁,(0.44,0.47,0.50),𝐶₂,(,𝐶0.79,0.82,0.80)₃,(0.97,0.98,1.00),𝐶₄,(0.83,0.85,0.88). (19) From formula (14), we can get three weighted similarity measures for an alternative 𝐴_𝑖(𝑖=1,2,…,5), as listed in Table 6. For comparison, we also list the decision results of FIOWHM operator method in Table 6.

Table 6 

Decision results of different methods in the same numerical example.

Step 5. From Table 6, we can see that all the proposed methods have the same decision results, but our method has simpler computation and gets the same results more rapidly than the FLOWHM method.
The decision results of different methods demonstrate that the proposed method for FMCGDM problem is effective.

5. Conclusion

The similarity measure is an effective method to solve the FMCDM problem, but it rarely applies to triangular FMCDM problem. In this paper, we propose three similarity measures and three weighted similarity measures between two TFNs based on the vector similarity measures. Using the similarity measures model, we establish a method for FMCGDM with known information on criterion weight and group weight, in which the alternatives and criteria are given by triangular fuzzy information. Finally, a practical example is given to select the most suitable candidate in human resource evaluation system, by the weighted similarity measures between each alternative and ideal alternative; the ranking order of all alternatives can be determined, and the best one(s) can be easily identified as well. For comparison, we also apply our method to solve the triangular FMCGDM problem in [23]. The different methods have the same decision results, which show that our proposed method in this paper is applicable and effective.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (no. 71171202), the National Social Foundation of China (no. 12AZD109), and the Science Foundation for National Innovation Research Group of China (no. 70921001).