Intuitionistic fuzzy TOPSIS multi-attribute decision making method based on revised scoring function and entropy weight method

Abstract

In multi-attribute decision-making problems, vague decision information is well-represented by intuitionistic fuzzy sets. However, many of the scoring functions of existing methods cannot always obtain a ranking for the alternatives. In this paper, a TOPSIS-based decision-making method is proposed for multi-attribute decision-making problems in which the attribute weights are unknown and the decision information is in the form of intuitionistic fuzzy numbers. First, a revised definition of the scoring function is introduced and used to solve the intuitionistic fuzzy entropy, which is then used to objectively determine the attribute weights. Second, intuitionistic fuzzy-weighted geometric operators are used to integrate the information. The positive and negative ideal solutions of the comprehensive attribute values are determined, and the similarities between each alternative and the positive and negative ideal solutions are calculated. Finally, the alternatives set is ranked by comparing the relative closeness of the alternatives. This proposed method increases the range of applications of the traditional entropy-weighted method. Moreover, it does not require the decision-maker to specify the attribute weights in advance. The results hence tend to be more objective. Examples comparing this method with existing TOPSIS-based methods illustrate its practicality.

Keywords

Score function fuzzy entropy TOPSIS

1. Introduction

In most decision-making problems, it has been traditionally assumed that all information is expressed in the form of crisp numbers. However, most of a decision-maker’s information is imprecise or uncertain. Hence, he or she cannot estimate his or her preferences using an exact numerical value. Since Zadeh [1] presented the theory of fuzzy sets in 1965, fuzzy-set theory has been successfully used for handling fuzzy decision-making problems. In fuzzy-set theory, the membership of an element to a fuzzy set is a single value between 0 and 1, and the degree of non-membership is just equal to 1 minus the degree of membership. In 1986, Atanassov [2] extended the concept of Zadeh’s fuzzy sets and introduced intuitionistic fuzzy sets (IFSs), whose characteristics consist of three important indexes: membership, non-membership, and hesitancy. Hence, these three dimensions can be used to describe the characteristics of complex systems so that the IFSs becomes a suitable tool for describing imprecise or uncertain decision information and dealing with hesitation and vagueness.

The weight of attributes is very important in intuitionistic fuzzy decision-making problems. In [3, 4], Liu et al. and Lakshmana et al. presented a new method for handling multiple attribute decision-making problems in which the weights of the attributes are known. In [5 –7], Qi et al., Wu et al., and Wang et al. investigated linear program models for determining the weights of attributes provided by each decision-maker expressed with fuzzy data. In [8], Ye presented a multi-criteria fuzzy decision-making method based on entropy weight-based correlation coefficients of interval-valued intuitionistic fuzzy sets.

Subsequently, [9, 10] Saini et al. and Tian et al. introduced a multiple criteria decision-making (MCDM) method based on intuitionistic fuzzy entropy. In [11], Chen and Tan developed a technique for handling fuzzy MCDM problems based on fuzzy set theory. This technique considers only the degrees of membership and non-membership for each attribute. Moreover, Hong et al. presented a method for fuzzy MCDM based on a score function under the fuzzy environment [12]. Although the score function provides a better ranking of the alternatives, it ignores hesitancy. The importance of hesitancy has, however, been addressed by others [13 –16]. In [17], Wang et al. proposed four improved score functions to effectively rank IFSs for multi-attribute decision making; however, this method has a high number of indicators that makes it difficult to calculate the results. The studies [3 , 19] employ score and accuracy functions for solving multi-attribute decision-making problems under an intuitionistic fuzzy environment, but these methods cannot distinguish the preferred order of alternatives in some situations. In [20], Wang et al. developed a score function for ranking interval-valued IFSs to solve multi-attribute decision-making problems, but this function cannot always distinguish the preferred order of alternatives either.

A classic multi-attribute decision-making approach, TOPSIS (technique for order performance by similarity to ideal solution) was first developed by [21]. The essential concept of the TOPSIS approach is that the most preferred alternative should have not only the shortest distance from the positive ideal solution, but also the farthest distance from the negative ideal solution.

Several studies have investigated the integration of fuzzy theory and TOPSIS. In [22], Jin et al. extended TOPSIS to deal with interval-valued intuitionistic fuzzy decision-making problems. In [23], John et al. extended TOPSIS to solve multi-attribute group decision-making problems under triangular intuitionistic fuzzy sets using correlation coefficients. In [24], Fayek et al. extended the application of fuzzy numbers, fuzzy relative importance scores, fuzzy relative weights, and the fuzzy TOPSIS method in prioritized aggregation. In [25], an intuitionistic fuzzy distance measure between two triangular intuitionistic fuzzy numbers was developed for TOPSIS. In [26], Ridvan et al. extended the TOPSIS method to the MCDM problem with single valued neutrosophic information. In [27], Onar et al. determined factor weights using interval type-2 fuzzy AHPs. They then select the best strategy using hesitant fuzzy TOPSIS.

However, despite these studies, there exists little investigation into the combination of intuitionistic fuzzy number-based entropy, TOPSIS, and extended score functions to solve multi-attribute decision-making problems.

In this paper, we tackle the problem that some score functions for intuitionistic fuzzy numbers cannot always distinguish the order of preferred alternatives when the attribute weights are unknown. A new intuitionistic fuzzy TOPSIS multi-attribute decision-making method is hence proposed based on a revised scoring function and entropy weights. Moreover, it is compared with other methods to demonstrate its effectiveness and advantages. The proposed method is a generalization of the traditional entropy-weighted method and does not require the decision-maker to specify the attribute weights in advance.

The rest of this paper is organized as follows. Section 2 provides a brief introduction to the basic concepts of intuitionistic fuzzy sets. In Section 3, an extension of similarity is proposed and new functions for score and accuracy based on intuitionistic fuzzy numbers are presented. Section 4 presents an intuitionistic fuzzy TOPSIS multi-attribute decision-making method based on revised the scoring function and entropy weights. In Section 5, an example is used to illustrate the effectiveness of the proposed method. Moreover, a sensitivity analysis indicates that this ranking is easily affected by changes in weights. The conclusions are drawn in Section 6.

2. Preliminaries

This section briefly introduces some basic concepts on intuitionistic fuzzy sets. The intuitionistic fuzzy weighted geometric and averaging operators are also given.

Definition 1. [2, 28] An intuitionistic fuzzy set (IFS) A in the universe of discourse X is defined with the form $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X},$ (1)

where μ_A : x → [0, 1] , ν_A : x → [0, 1].

The numbers μ_A (x) and ν_A (x) denote the membership degree and non-membership degree of x to A, respectively.

With the condition $0 \leq μ_{A} (x) + ν_{A} (x) \leq 1, \forall x \in X .$

Furthermore, the function π_A (x) =1 - μ_A (x) - ν_A (x) is called the hesitation degree of x to A. It is obvious that 0 ≤ π_A ≤ 1 or each x ∈ X.

Obviously, each ordinary fuzzy set may be written as ${〈 x, μ_{A} (x), 1 - μ_{A} (x) 〉 | x \in X} .$ (2)

That is, fuzzy sets may be reviewed as particular cases of IFSs.

Note that A is a crisp set if and only if for ∀x ∈ X, either μ_A (x) =0, ν_A (x) =1 or μ_A (x) =1, ν_A (x) =0 .

For convenience of notation, IFSs (X) is used to denote the set of IFSs in X.

IFSs A and B in a universe of discourse X can be represented by A = (μ_A (x) , ν_A (x)) and B = (μ_B (x) , ν_B (x)). Moreover, we introduce the following relations and operations over fuzzy sets [29]: $\begin{matrix} (1) \bar{A} = (ν_{A} (x), μ_{A} (x)); \\ (2) A \cap B = (\min {μ_{A} (x), μ_{B} (x)}, min {ν_{A} (x), ν_{B} (x)}); \\ (3) A \cup B = (\max {μ_{A} (x), μ_{B} (x)}, max {ν_{A} (x), ν_{B} (x)}); \\ (4) A \oplus B = (μ_{A} (x) + μ_{B} (x) - μ_{A} (x) μ_{B} (x), ν_{A} (x) ν_{B} (x)); \\ (5) A \otimes B = (μ_{A} (x) μ_{B} (x), ν_{A} (x) + ν_{B} (x) - ν_{A} (x) ν_{B} (x)); \\ (6) λ A = (1 - (1 - μ_{A} (x))^{λ}, ν_{A}^{λ} (x)), λ > 0; \\ (7) A^{λ} = (μ_{A}^{λ} (x), 1 - (1 - ν_{A} (x))^{λ}), λ > 0 . \end{matrix}$

Definition 2. [29] Let A_i = (μ_{A
_i} (x) , ν_{A
_i} (x)) , (i = 1, 2, …, n) be IFSs, where 0 ≤ μ_{A
_i} (x) , ν_{A
_i} (x) ≤1. Let ω_i be the weight of A_i, where 0 ≤ ω_i ≤ 1, (1 ≤ i ≤ n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . The intuitionistic fuzzy weighted averaging operator IFWA_ω of the IFSs is defined as follows: $\begin{matrix} I F W A_{ω} (A_{1}, A_{2}, \dots, A_{n}) = ω_{1} A_{1} \oplus ω_{2} A_{2} \oplus \dots \oplus ω_{n} A_{n} \\ = {1 - \prod_{i = 1}^{n} {(1 - μ_{A_{i}} (x))}^{ω_{i}}, \prod_{i = 1}^{n} ν_{A_{i}}^{ω_{i}} (x)} \end{matrix}$ (3)

Definition 3. [30] Let A_i = (μ_{A
_i} (x) , ν_{A
_i} (x)) , (i = 1, 2, …, n) be IFSs, where 0 ≤ μ_{A
_i} (x) , ν_{A
_i} (x) ≤1. Let ω_i be the weight of attributes, where 0 ≤ ω_i ≤ 1, (1 ≤ i ≤ n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . The intuitionistic fuzzy weighted geometric operator IFWG_ω of the IFSs is defined as follows: $\begin{matrix} I F W G_{ω} = (A_{1}, A_{2}, \dots, A_{n}) = A_{1}^{ω_{1}} \otimes A_{2}^{ω_{2}} \otimes \dots \otimes A_{n}^{ω_{n}} \\ = {\prod_{j = 1}^{n} μ_{A_{j}}^{ω_{j}} (x), 1 - \prod_{j = 1}^{n} {(1 - ν_{A_{j}} (x))}^{ω_{j}}} . \end{matrix}$ (4)

3. Proposed similarity measure, score function, and accuracy function

3.1. Similarity measure

Let A_i = (μ_A (x_i) , ν_A (x_i)) , (i = 1, 2, …, n) be IFSs, where 0 ≤ μ_A (x_i) , ν_A (x_i) ≤1. Let ω_i be the weight of A_i, where 0 ≤ ω_i ≤ 1, (1 ≤ i ≤ n), and $\sum_{i = 1}^{n} ω_{i} = 1$ . From [18], the degree of similarity between intuitionistic fuzzy numbers A and B can be evaluated by the function S (A, B): $\begin{matrix} S (A, B) = \sum_{j = 1}^{n} ω_{j} \frac{2 - \min {μ_{j}^{-}, ν_{j}^{-}} - \min {μ_{j}^{+}, ν_{j}^{+}}}{2 + \max {μ_{j}^{-}, ν_{j}^{-}} + \max {μ_{j}^{+}, ν_{j}^{+}}} \\ μ_{j}^{-} = | μ_{A}^{-} (x_{j}) - μ_{B}^{-} (x_{j}) |; μ_{j}^{+} = | μ_{A}^{+} (x_{j}) - μ_{B}^{+} (x_{j}) |; \\ ν_{j}^{-} = | ν_{A}^{-} (x_{j}) - ν_{B}^{-} (x_{j}) |; ν_{j}^{+} = | ν_{A}^{+} (x_{j}) - ν_{B}^{+} (x_{j}) | . \end{matrix}$ (5)

To overcome the drawbacks of the similarity measure proposed above, this paper proposed a new measure. Assumes that there are two IFSs A and B in X ={ x₁, x₂, …, x_n }. Then, the degree of similarity between them can be calculated as follows: $\begin{matrix} S (A, B) = \\ \sum_{j = 1}^{n} ω_{j} \frac{2 - \min {| μ_{A} (x_{j}) - μ_{B} (x_{j}) |, | ν_{A} (x_{j}) - ν_{B} (x_{j}) |}}{2 + \max {| μ_{A} (x_{j}) - μ_{B} (x_{j}) |, | ν_{A} (x_{j}) - ν_{B} (x_{j}) |}} \end{matrix}$ (6)

Theorem 1. A real-valued function S : IFS (X) × IFS (X) → [0, 1] is called a similarity measure on IFS (X) because it satisfies the following axiomatic requirements [31]:

0 ≤ S (A, B) ≤1;

S (A, B) =1 if and only if A = B;

S (A, B) = S (B, A);

If A ⊆ B ⊆ C, then S (A, C) ≤ S (A, B) and S (A, C) ≤ S (B, C) .

Proof.

Let $\begin{matrix} μ_{A} = μ_{A} (x_{j}); μ_{B} = μ_{B} (x_{j}); μ_{C} = μ_{C} (x_{j}); \\ ν_{A} = ν_{A} (x_{j}); ν_{B} = ν_{B} (x_{j}); ν_{C} = ν_{C} (x_{j}) . \end{matrix}$

Axiom 1) 0 ≤ S (A, B) ≤1:

Here, $\begin{matrix} 0 \leq min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} \\ \leq \max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} \end{matrix},$

and then $\begin{matrix} 0 \leq 2 - min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} \\ \leq 2 + max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} . \end{matrix}$

As a result, 0 ≤ S (A, B) ≤1.

Axiom 2) S (A, B) =1 if and if only A = B:

If μ_A = μ_B, ν_A = ν_B,

then

S (A, B) =1 and A = B.

If S (A, B) =1, then $\begin{matrix} 2 - \min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} \\ = 2 + \max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |}, \forall x_{j} \in X^{'} \end{matrix}$

hence, $| μ_{A} - μ_{B} | + | ν_{A} - ν_{B} | = 0,$

and $min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} = {\begin{matrix} | μ_{A} - μ_{B} | \\ | ν_{A} - ν_{B} | \end{matrix},$ $max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} = {\begin{matrix} | μ_{A} - μ_{B} | \\ | ν_{A} - ν_{B} | \end{matrix} .$

Thus, $| μ_{A} - μ_{B} | = | ν_{A} - ν_{B} | and | μ_{A} - μ_{B} | + | ν_{A} - ν_{B} | = 0,$

hence

μ_A = μ_B, ν_A = ν_B and S (A, B) =1, if and only if A = B.

Axiom 3) S (A, B) = S (B, A): This axiom is obviously true.

Axiom 4) If A ⊆ B ⊆ C, then S (A, C) ≤ S (A, B) and S (A, C) ≤ S (B, C):

If A ⊆ B ⊆ C, then μ_A ≤ μ_B ≤ μ_C and ν_A ≥ ν_B ≥ ν_C, ∀x_i ∈ X. Thus, the following result can be obtained: ${\begin{matrix} | μ_{A} - μ_{C} | \geq | μ_{A} - μ_{B} | \\ | ν_{A} - ν_{C} | \geq | ν_{A} - ν_{B} |, \end{matrix}$

and $\begin{matrix} min {| μ_{A} - μ_{C} |, | ν_{A} - ν_{C} |} \\ \geq min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} \\ max {| μ_{A} - μ_{C} |, | ν_{A} - ν_{C} |} \\ \geq max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |} . \end{matrix}$

Thus, $\begin{matrix} \frac{2 - min {| μ_{A} - μ_{C} |, | ν_{A} - ν_{C} |}}{2 + max {| μ_{A} - μ_{C} |, | ν_{A} - ν_{C} |}} \\ \leq \frac{2 - min {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |}}{2 + max {| μ_{A} - μ_{B} |, | ν_{A} - ν_{B} |}} . \end{matrix}$

Hence, S (A, C) ≤ S (A, B) and S (A, C) ≤ S (B, C) are proved.

3.2. Proposed score and accuracy functions

In this subsection, modified score and accuracy functions are defined and some of their theoretical properties established. Note that the new core function satisfies all the basic properties of the previously proposed score functions.

Let A_i = (μ_A (x_i) , ν_A (x_i)) and B_i = (μ_B (x_i) , ν_B (x_i)) be IFSs of alternatives A_i and B_i. The score function from [12] as follows:

Definition 4. [12] Hong and Choi’s score function S (A_i) and accuracy function H (A_i) are defined as follows: ${\begin{matrix} S (A_{i}) = μ_{A} (x_{i}) - ν_{A} (x_{i}) \\ H (A_{i}) = μ_{A} (x_{i}) + ν_{A} (x_{i}) \end{matrix},$ (7)

where S (A_i) ∈ [-1, 1] and H (A_i) ∈ [0, 1]. Larger values of S (A_i) indicate that alternative A_i better satisfies the decision makers’ requirements. If the scores S (A_i) of all A_i are equal, then a decision maker can draw his or her conclusions using the value of H (A_i). Larger values of H (A_i) indicate a greater degree of accuracy of A_i [3].

If S (A_i) < S (B_i), then A_i is smaller than B_i, denoted by A_i < B_i;

If S (A_i) > S (B_i), then A_i is greater than B_i, denoted by A_i > B_i;

If S (A_i) = S (B_i), then the following is true.

If H (A_i) < H (B_i), then A_i is smaller than B_i, denoted by A_i < B_i.

If H (A_i) > H (B_i), then A_i is greater than B_i, denoted by A_i > B_i.

If H (A_i) = H (B_i), then A_i = B_i.

In [3], although the score and accuracy function provide a better ranking of the alternatives, they ignore the hesitancy of IFS, which means that the information provided by an IFS is not fully utilized in the ranking process.

Example 1. If the IFS values for the two alternatives are S (A) = (0.6, 0) and S (B) = (0.7, 0.1), 60% of the decision makers support alternative A, no decision makers are against it, but 40% of the decision makers abstain from expressing their support or non-support. Moreover, 70% of the decision makers support alternative B, 10% disagree with alternative B, and 20% abstain from expressing an opinion. From Equation (6), S (A) = S (B) =0.6, H (A) =0.6, and H (B) =0.8, where H (B) > H (A), which indicates that alternative B is better than alternative A. However, in reality, this result is obviously wrong. It is important to consider that the degree of hesitancy of decision makers will be influenced by the supporters and opponents of an alternative, and this hesitancy could then turn into opposition or support. Many other score functions were subsequently proposed by Wu et al. [14], Lin et al. [33], and Wang et al. [19], but these score function have some limitations. It has hence been widely concluded that the existing score and accuracy functions are not always suitable for ranking alternatives.

Thus, there is a need to design a more accurate score function for determining the best alternative among feasible alternatives. We hence propose the following score function.

Let A_i = (μ_A (x_i) , ν_A (x_i)) be an intuitionistic fuzzy number. The proposed score function of an intuitionistic fuzzy value is defined as ${\begin{matrix} S (A_{i}) = [μ_{A} (x_{i}) + μ_{A} (x_{i}) π_{A} (x_{i})] \\ - [ν_{A} (x_{i}) + ν_{A} (x_{i}) π_{A} (x_{i})] \\ H (A_{i}) = [μ_{A} (x_{i}) + μ_{A} (x_{i}) π_{A} (x_{i})] \\ + [ν_{A} (x_{i}) + ν_{A} (x_{i}) π_{A} (x_{i})] \end{matrix},$

where the values μ_A (x_i), ν_A (x_i), and π_A (x_i) denote the degree of membership, non-membership, and hesitation.

Theorem 2. Let A_i = (μ_A (x_i) , ν_A (x_i)) be an intuitionistic fuzzy set and S (A_i) be a score function of A_i. For an IFS, the score function satisfies the following:

-1 ≤ S (A) ≤1;

S (A) =1 if and only if A = (1, 0) and S (A) = -1 if and only if A = (0, 1);

S (A) =0 if and only if μ_A = ν_A.

Proof. Let μ_A = μ_A (x_j); ν_A = ν_A (x_j).

Axiom 1) -1 ≤ S (A) ≤1:

Because $\begin{matrix} S (A) = [μ_{A} + μ_{A} π_{A}] - [ν_{A} + ν_{A} π_{A}] \\ = (μ_{A} - ν_{A}) [2 - (μ_{A} + ν_{A})] \\ = (1 - ν_{A})^{2} - (1 - μ_{A})^{2} \end{matrix}$

and $0 \leq (1 - μ_{A})^{2} \leq 1, 0 \leq (1 - ν_{A})^{2} \leq 1,$

then $- 1 \leq S (A) \leq 1 .$

The score function hence satisfies Axiom 1).

Axiom 2) S (A) =1 if and only if A = (1, 0) and S (A) = -1 if and only if A = (0, 1):

When S (A) =1, $\begin{matrix} S = [μ_{A} + μ_{A} π_{A}] - [ν_{A} + ν_{A} π_{A}] \\ = (1 - ν_{A})^{2} - (1 - μ_{A})^{2} . \end{matrix}$

Thus, 1 - ν_A = 1, 1 - μ_A = 0, ν_A = 0, and μ_A = 1. Obviously, if A = (1, 0), then S (A) =1, so the proof of Axiom 3 is not repeated here.

To demonstrate the ability of the proposed score function to find the best alternative among the given alternatives, we discuss the following example (detailed in Table 1).

Table 1

Results obtained by different score functions

Original reference	Score function	Example	Original score function results	Proposed method results
[13]	$s = μ_{α} (1 + π_{α}) - π_{α}^{2}$	$\begin{matrix} α_{1} = (0.3, 0.6) \\ α_{2} = (0.3, 0.5) \end{matrix}$	Cannot get the order of these two alternatives	α₂ > α₁
[14]	s = μ_α - ν_α+	$\begin{matrix} α_{3} = (0.3, 0.6) \\ α_{4} = (0.3, 0.5) \end{matrix}$	α₄ > α₃	α₄ > α₃
	((e^μ_α-ν_α/e^μ_α-ν_α + 1) -1/2) π_α
[19]	s = μ_α - ν_α - π_α/2	$\begin{matrix} α_{5} = (0.3, 0.6) \\ α_{6} = (0.2, 0.3) \end{matrix}$	Cannot get the order of these two alternatives	α₅ > α₆
[32]	s = μ_α + μ_απ_α	$\begin{matrix} α_{7} = (0, 0.1) \\ α_{8} = (0, 0.9) \end{matrix}$	Cannot get the order of these two alternatives	α₈ > α₇
[33]	s = μ_α - π_α	$\begin{matrix} α_{9} = (0.3, 0.5) \\ α_{10} = (0.4, 0.2) \end{matrix}$	α₉ > α₁₀	α₁₀ > α₉

If IFS values for two alternatives are α₁ = (0.3, 0.6) and α₂ = (0.3, 0.5), then the order of alternatives cannot be distinguished by the score function of [13], but Equation (7) yields α₂ > α₁. The larger the value of S (α₂) indicates that alternative α₂ better fulfills the decision-makers’ requirements. Moreover, if the values of two other alternatives are α₃ = (0.3, 0.6) and α₄ = (0.3, 0.5), the order of alternatives obtained by the score function in [14] is the same as that of Equation (7), i.e., α₄ > α₃ which agrees with the actual situation. For two alternatives α₅ = (0, 0.1) and α₆ = (0, 0.9) the score function from [32] does not specify an order. In contrast, Equation (7) yields S (α₅) > S (α₆), so decision makers can obtain α₅ > α₆. For two alternatives α₇ = (0.3, 0.5) and α₈ = (0.4, 0.2) the order found by [33] is α₇ > α₈, However, this result differs from the actual situation and does not meet the decision-makers’ expectations. Equation (7) yields a larger value for S (α₈) and alternative α₈ appeases the decision-makers’ requirements. For two alternatives α₉ = (0.3, 0.6) and α₁₀ = (0.2, 0.3), decision makers cannot determine which alternative is better if the score function proposed in [19] is used. In contrast, Equation (7) gives α₁₀ > α₉, which meets the decision-makers’ requirements and agrees with the actual situation.

4. Proposed approach for MCDM

4.1. Entropy weight for intuitionistic fuzzy sets

Attributes weights play an important function in multi-attribute decision-making problems. Many re-searchers focus on multi-attribute decision-making problems with incomplete or unknown criterion weight information under an intuitionistic fuzzy environment. In [18], Shannon proposed the concept of entropy, which is a measure of the uncertainty in information, formulated in terms of probability theory. Thus, we proposed a multi-attribute fuzzy decision-making method using the entropy weights of IFSs to identify the best alternative. In this method, all the information provided by the decision makers is represented as an intuitionistic fuzzy decision matrix, where each of the elements is an IFSs, and the information about the attribute weights is completely unknown.

Definition 5. Because the source of the information is uncertain, we express the average uncertainty of the information sources as follows: $H_{S} (P) = - k \sum_{i = 1}^{n} p_{i} \ln p_{i},$ (9)

where k is a positive number, where usually k = 1.

For each alternative, the score function can be normalized as follows: $\bar{S} (A_{i}) = \frac{| S (A_{i}) |}{\sum_{i = 1}^{n} | S (A_{i}) |} .$ (10)

Subsequently, $H_{S} (X_{i}) = - \frac{1}{\ln n} \sum_{i = 1}^{n} \bar{S} (A_{i}) \ln \bar{S} (A_{i}) .$ (11)

When $\bar{S} (A_{i}) = 0$ and $| \bar{S} (A_{i}) | ln | \bar{S} (A_{i}) | = 0$ , then the weights of the attributes are $ω_{j} = \frac{1 - H_{S} (X_{j})}{\sum_{k = 1}^{m} (1 - H_{S} (X_{j}))}, j = 1, 2, \dots, m .$ (12)

4.2. Algorithm

The proposed method consists of the following steps.

Step 1. Construct an intuitionistic fuzzy decision matrix.

Step 2. Use the following to derive the score function values of the alternatives from the experts.

$\begin{matrix} S (A_{i}) = [μ_{A} (x_{i}) + μ_{A} (x_{i}) π_{A} (x_{i})] \\ - [ν_{A} (x_{i}) + ν_{A} (x_{i}) π_{A} (x_{i})] \end{matrix}$ (13)

Step 3. Use Equation (10) to obtain $\bar{S} (A_{i})$ .

Step 4. Calculate the attribute weights according to the following equations.

$H_{S} (A_{i}) = - \frac{1}{\ln n} \sum_{i = 1}^{n} \bar{S} (A_{i}) \ln \bar{S} (A_{i})$ (14) $ω_{j} = \frac{1 - H_{S} (A_{i})}{\sum (1 - H_{S} (A_{i}))}$ (15)

Step 5. Use the IFWG_ω operator to derive the overall assessment values of the alternatives. $\begin{matrix} I F W G_{ω} = (A_{1}, A_{2}, \dots, A_{n}) = A_{1}^{ω_{1}} \otimes A_{2}^{ω_{2}} \otimes \dots \otimes A_{n}^{ω_{n}} \\ = {\prod_{j = 1}^{n} μ_{A_{j}}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{A_{j}})}^{ω_{j}}} \end{matrix}$ (16)

Step 6. Determine the positive-ideal solution $A_{i}^{+}$ and negative-ideal solution $A_{i}^{-}$ using $\begin{matrix} A_{i}^{+} = (max μ_{A} (x_{i}), min ν_{A} (x_{i})) \\ A_{i}^{-} = (min μ_{A} (x_{i}), max ν_{A} (x_{i})) . \end{matrix}$

Step 7. Calculate S (A_i, A⁺), which is the similarity between option A_i (i = 1, 2, …) and the positive-ideal solution A⁺. Similarly, calculate S (A_i, A^-) for the negative-ideal solution A^-. $\begin{matrix} S (A_{j}, A^{+}) = \sum_{j = 1}^{n} ω_{j} \\ \frac{2 - \min {| μ_{A} (x_{j}) - μ_{A^{+}} (x) |, | ν_{A} (x) - ν_{A^{+}} (x) |}}{2 + \max {| μ_{A} (x_{j}) - μ_{A^{+}} (x) |, | ν_{A} (x) - ν_{A^{+}} (x) |}} \end{matrix}$ (17) $\begin{matrix} S (A_{j}, A^{-}) = \sum_{j = 1}^{n} ω_{j} \\ \frac{2 - \min {| μ_{A} (x_{j}) - μ_{A^{-}} (x) |, | ν_{A} (x) - ν_{A^{-}} (x) |}}{2 + \max {| μ_{A} (x_{j}) - μ_{A^{-} (x) |, | ν_{A} (x) - ν_{A^{-}} (x) |}}} \end{matrix}$ (18)

Step 8. Use Equation (19) to calculate the relative similarity measure S (A_i) of A_i with respect to A⁺ and A^-, where $S (A_{i}) = \frac{S (A_{i}, A^{+})}{S (A_{i}, A^{+}) + S (A_{i}, A^{-})}, (i = 1, 2, \dots),$ (19)

and rank the alternatives according to the relative similarity in decreasing order. The best alternative is the one closest to the positive-ideal solution and farthest from the negative-ideal solution.

5. Numerical examples and sensitivity analysis

5.1. Numerical examples

To illustrate the application of the proposed multi-attribute decision-making method, an example from [15], which evaluates investment opportunities, is discussed. Suppose there is an investment company that wants to invest a sum of money in the best option for the construction of a railway track. Suppose that there are five alternatives: replacement of the soil, improve the soil quality, repair of the drainage facilities, heat preservation of the track bed, and laying EPS on the track bed. In addition, there are six attributes: quality of the soil, water content, frost damage level, height of the track bed, rated train speed, and rated train weight. Here, the attribute weights are unknown and the decision information is in the form of intuitionistic fuzzy numbers. The five alternatives were evaluated using intuitionistic fuzzy information by decision makers for the above attributes.

First, the results evaluated by the decision makers are used to create an intuitionistic fuzzy decision matrix (Table 2). Then, Equation (12) is used to derive the scores of the alternatives (Table 3). Equation (10) is used to obtain the normalized score matrix (Table 4).

Table 2
Intuitionistic fuzzy decision matrix for the example given in [15]

G1 G2 G3 G4 G5 G6

y1 (0.2,0.4) (0.6,0.2) (0.4,0.2) (0.7,0.1) (0.3,0.5) (0.5,0.2)

y2 (0.6,0.2) (0.5,0.1) (0.6,0.2) (0.6,0.1) (0.3,0.5) (0.4,0.1)

y3 (0.4,0.3) (0.7,0.1) (0.5,0.3) (0.6,0.1) (0.4,0.3) (0.3,0.1)

y4 (0.6,0.2) (0.5,0.1) (0.7,0.1) (0.3,0.1) (0.5,0.1) (0.7,0.1)

y5 (0.5,0.3) (0.3,0.3) (0.6,0.1) (0.6,0.1) (0.6,0.2) (0.5,0.2)

	G1	G2	G3	G4	G5	G6
y1	(0.2,0.4)	(0.6,0.2)	(0.4,0.2)	(0.7,0.1)	(0.3,0.5)	(0.5,0.2)
y2	(0.6,0.2)	(0.5,0.1)	(0.6,0.2)	(0.6,0.1)	(0.3,0.5)	(0.4,0.1)
y3	(0.4,0.3)	(0.7,0.1)	(0.5,0.3)	(0.6,0.1)	(0.4,0.3)	(0.3,0.1)
y4	(0.6,0.2)	(0.5,0.1)	(0.7,0.1)	(0.3,0.1)	(0.5,0.1)	(0.7,0.1)
y5	(0.5,0.3)	(0.3,0.3)	(0.6,0.1)	(0.6,0.1)	(0.6,0.2)	(0.5,0.2)

Here and in Tables 3–5, y1–y5 are alternatives and G1–G6 are attributes.

Table 3

Score function matrix

	G1	G2	G3	G4	G5	G6
y1	-0.28	0.48	0.28	0.72	-0.24	0.39
y2	0.48	0.56	0.48	0.65	-0.24	0.45
y3	0.13	0.72	0.24	0.65	0.13	0.32
y4	0.48	0.56	0.72	0.32	0.56	0.72
y5	0.24	0	0.65	0.65	0.48	0.39

Table 4

Normalized score function matrix

	G1	G2	G3	G4	G5	G6
y1	0.174	0.207	0.118	0.241	0.145	0.172
y2	0.298	0.241	0.203	0.217	0.145	0.198
y3	0.081	0.310	0.101	0.217	0.079	0.141
y4	0.298	0.241	0.304	0.107	0.339	0.317
y5	0.149	0.000	0.274	0.217	0.291	0.172

Then, using Equations (14 and 15), the following values for the average information entropy H_S (G_i) and the attribute weights ω_i are obtained. $\begin{matrix} H_{S} (G_{1}) = 0.8445, H_{S} (G_{2}) = 0.7674 \\ H_{S} (G_{3}) = 0.8506, H_{S} (G_{4}) = 0.8800 \\ H_{S} (G_{5}) = 0.8296, H_{S} (G_{6}) = 0.8743 \end{matrix}$ $\begin{matrix} ω_{1} = 0.1631, ω_{2} = 0.2439, ω_{3} = 0.1566 \\ ω_{4} = 0.1259, ω_{5} = 0.1787, ω_{6} = 0.1318 \end{matrix}$

Equation (15) is used to determine the overall assessment values G_i (i = 1, 2, 3, 4, 5, 6) for alternatives y_i (i = 1, 2, 3, 4, 5) as follows: $\begin{matrix} α_{1} = {IFWG}_{ω} (y_{1}^{G_{1}}, y_{1}^{G_{2}}, y_{1}^{G_{3}}, y_{1}^{G_{4}}, y_{1}^{G_{5}}, y_{1}^{G_{6}}) \\ = (0.41, 0.29) \\ α_{2} = {IFWG}_{ω} (y_{2}^{G_{1}}, y_{2}^{G_{2}}, y_{2}^{G_{3}}, y_{2}^{G_{4}}, y_{2}^{G_{5}}, y_{2}^{G_{6}}) \\ = (0.48, 0.22) \\ α_{3} = {IFWG}_{ω} (y_{3}^{G_{1}}, y_{3}^{G_{2}}, y_{3}^{G_{3}}, y_{3}^{G_{4}}, y_{3}^{G_{5}}, y_{3}^{G_{6}}) \\ = (0.48, 0.21) \\ α_{4} = {IFWG}_{ω} (y_{4}^{G_{1}}, y_{4}^{G_{2}}, y_{4}^{G_{3}}, y_{4}^{G_{4}}, y_{4}^{G_{5}}, y_{4}^{G_{6}}) \\ = (0.53, 0.12) \\ α_{5} = {IFWG}_{ω} (y_{5}^{G_{1}}, y_{5}^{G_{2}}, y_{5}^{G_{3}}, y_{5}^{G_{4}}, y_{5}^{G_{5}}, y_{5}^{G_{6}}) \\ = (0.48, 0.22) \end{matrix}$

These results are used to obtain the positive- and negative-ideal solutions as $A^{+} = (0.53, 0.12), A^{-} = (0.41, 0.29) .$

Equations (16–18) are then used to calculate the similarity measures S (y_i, A⁺) and S (y_i, A^-) as well as the relative similarity measure S (y_i) as follows: $\begin{matrix} S (y_{1}, A^{+}) = 0.8605 S (y_{1}, A^{-}) = 0.8741 \\ S (y_{2}, A^{+}) = 0.9104 S (y_{2}, A^{-}) = 0.8682 \\ S (y_{3}, A^{+}) = 0.8947 S (y_{3}, A^{-}) = 0.8981 \\ S (y_{4}, A^{+}) = 0.9385 S (y_{4}, A^{-}) = 0.8474 \\ S (y_{5}, A^{+}) = 0.9073 S (y_{5}, A^{-}) = 0.8951 \end{matrix}$ $\begin{matrix} S (y_{1}) = 0.4961, S (y_{2}) = 0.5119 \\ S (y_{3}) = 0.4991, S (y_{4}) = 0.5255 \\ S (y_{5}) = 0.5034 \end{matrix}$

Finally, the relative similarity measure S (y_i) is used to rank the alternatives, yielding S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₃) ≻ S (y₁).

Therefore, y₄ is the best alternative. This is the same as that obtained in [15]. Note that in [15], the resulting order of alternatives is opposite that found using the method proposed in this paper, but this does not affect the final result.

We also present the ranking obtained by the methods given in [14 , 34] in Table 5. The results indicate that although different ranking methods were used, the same results were obtained for the best and worst choices.

Table 5

Ranking order of alternatives for different methods

Method	Ranking
Proposed	S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₃) ≻ S (y₁)
[14]	S (y₄) ≻ S (y₃) ≻ S (y₂) ≻ S (y₅) ≻ S (y₁)
[15]	S (y₄) ≻ S (y₅) ≻ S (y₂) ≻ S (y₃) ≻ S (y₁)
[16]	S (y₄) ≻ S (y₃) ≻ S (y₂) ≻ S (y₅) ≻ S (y₁)
[34]	S (y₄) ≻ S (y₃) ≻ S (y₂) ≻ S (y₅) ≻ S (y₁)

5.2. Sensitivity analysis

To determine the effects of possible changes in the weights of the criteria on intuitionistic fuzzy decision-making problems, referring to the method in [35], a sensitivity analysis was conducted. We use different values of the weights and assess the obtained ranking of the alternatives. The resulting rankings for altered weights were calculated, and the results are shown in Table 6.

Table 6
Ranking order of alternatives for different weights.

ω ₁ ω ₂ ω ₃ ω ₄ ω ₅ ω ₆ Ranking

1 0.150 0.250 0.250 0.100 0.100 0.150 S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₃) ≻ S (y₁)

2 0.110 0.100 0.290 0.200 0.100 0.200 S (y₄) ≻ S (y₅) ≻ S (y₂) ≻ S (y₃) ≻ S (y₁)

3 0.070 0.330 0.150 0.150 0.150 0.150 S (y₄) ≻ S (y₂) ≻ S (y₃) ≻ S (y₁) ≻ S (y₅)

4 0.310 0.010 0.180 0.240 0.100 0.160 S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₁) ≻ S (y₃)

5 0.200 0.085 0.037 0.041 0.598 0.039 S (y₄) ≻ S (y₅) ≻ S (y₂) ≻ S (y₁) ≻ S (y₃)

	ω ₁	ω ₂	ω ₃	ω ₄	ω ₅	ω ₆	Ranking
1	0.150	0.250	0.250	0.100	0.100	0.150	S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₃) ≻ S (y₁)
2	0.110	0.100	0.290	0.200	0.100	0.200	S (y₄) ≻ S (y₅) ≻ S (y₂) ≻ S (y₃) ≻ S (y₁)
3	0.070	0.330	0.150	0.150	0.150	0.150	S (y₄) ≻ S (y₂) ≻ S (y₃) ≻ S (y₁) ≻ S (y₅)
4	0.310	0.010	0.180	0.240	0.100	0.160	S (y₄) ≻ S (y₂) ≻ S (y₅) ≻ S (y₁) ≻ S (y₃)
5	0.200	0.085	0.037	0.041	0.598	0.039	S (y₄) ≻ S (y₅) ≻ S (y₂) ≻ S (y₁) ≻ S (y₃)

From Table 6, it is apparent that the ranking orders obtained by different values of weights are different, in this example. The results in Table 6 show that the ranking of the alternatives is sensitive to the changes in weights. In other words, depending on different value of weights, the ranking orders of alternatives may be slightly different, and the results may lead to different decisions. Therefore, when attribute weights are unknown, it is important to choose an appropriate method for determining them. This proposed method does not require the decision-maker to specify the attribute weights in advance.

5.3. Related TOPSIS methods

In this subsection, the TOPSIS method proposed in this paper is compared with the extended TOPSIS methods in [26, 27]. In [26], Şahin et al. presented a method based on TOPSIS for multi-criteria neutrosophic group decision-making problems. In [27], Onar et al. also proposed an approach integrated with hesitant fuzzy TOPSIS to select the best strategy. Compared with the methods in [26, 27], the method proposed in this paper has the following three advantages:

In [26, 27], authors computed the relative closeness coefficient using a distance measure, but they cannot distinguish the preference order of alternatives when the two distance measures of alternatives A_i and A_j from the positive-ideal solution and negative-ideal solution are equal. In the proposed method, the relative closeness is calculated using similarity rather than distance. Hence, if there exist two different alternatives with equal distance measures, the proposed method can further distinguish these two alternatives. Consequently, it is more suitable for dealing with MCDM within a fuzzy environment.

In [26], the proposed method easily solves MCDM with single-valued neutrosophic data. However, converting linguistic terms to single-valued neutrosophic numbers will cause information loss. The TOPSIS method presented in this paper is hence more objective.

In [27], the authors used interval type-2 fuzzy AHP to determine the weights, but the entropy weight method in this paper is more convenient to calculate. The method proposed in this paper might enable more flexible decision making within a fuzzy environment.

6. Conclusion

In real-life situations, judgments provided by a decision maker are difficult to express as crisp numbers because they are usually fuzzy and uncertain. Moreover, some previously proposed score functions cannot always distinguish the preference order of alternatives, as illustrated by the examples in this paper.

Hence, in this work, a decision-making method was proposed that employs a new score function and an intuitionistic fuzzy TOPSIS method. This method solves multi-attribute decision-making problems where the attribute weights are unknown and the decision information is in the form of intuitionistic fuzzy numbers.

In this method, a revised definition of the scoring function is introduced and used to solve intuitionistic fuzzy entropy, which is then used to objectively solve for the attribute weights. Moreover, the alternatives set is ranked by comparing the relative closeness of each alternative. Comparisons between the new measure and some existing methods show that the proposed method is practical and effective.

Furthermore, this proposed method widens the scope of the traditional entropy-weighted methods and the decision-maker does not need to specify the attribute weights in advance. The results tend to be more objective, and the sensitivity analysis showed that slight changes in the attribute weights will change the ranking of the alternatives. Therefore, compared with existing subjective weight methods, the objective weight method proposed in this paper is more practical for multi-attribute decision making.

The method presented in this paper can be applied to problems in other fields, such as credit evaluation or green supplier selection. In future, we will use the entropy measure to determine the weights of experts in group decision problems within an interval-valued intuitionistic fuzzy environment and we will continue to apply the proposed method in other domains.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China under the grant No. 61773123.

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