A method for interval-valued intuitionistic fuzzy multiple attribute decision making based on fuzzy entropy

Abstract

With respect to multi-attribute decision making problem, in which attribute values are expressed in interval-valued intuitionistic fuzzy numbers, a decision making method based on fuzzy entropy and TOPSIS is presented. In this paper, the score value function and precision function of interval-valued intuitionistic fuzzy numbers are defined, and the attribute weight determination method based on improved fuzzy entropy is given. Then, by using the ranking method of interval-valued intuitionistic fuzzy numbers, the positive and negative ideal solutions of interval-valued intuitionistic fuzzy multiple attribute decision making problems are obtained. On this basis, an interval-valued intuitionistic fuzzy multi-attribute decision making method based on TOPSIS is proposed. Finally, the method is applied to the evaluation of local government public finance expenditure performance and its effectiveness is illustrated by a numerical example.

Keywords

Interval-valued intuitionistic fuzzy sets multi-attribute decision making TOPSIS method improved fuzzy entropy

1 Introduction

Fuzzy sets theory has been widely used in every fields of modern society since it was put forward by professor Zadeh [4] in 1965 [8]. Bulgarian scholar Atanassov [2] extended Zadeh’s fuzzy set and put forward the concept of intuitionistic fuzzy set in 1986, then in 1989, the intuitionistic fuzzy set is extended to interval intuitionistic fuzzy set by Atanassov and Gargov [3]. As both the degree of membership and that of non-membership of interval intuitionistic fuzzy sets are expressed in the form of subinterval of closed interval [0,1], Interval intuitionistic fuzzy multi-attribute decision-making method is more flexible and delicate in dealing with fuzzy information, thus it has been widely concerned by scholars at home and abroad.

The ranking of interval intuitionistic fuzzy numbers and the determination of attribute weights are two important links in interval intuitionistic fuzzy multi-attribute decision making. Zeshui Xu [24] defined the score value function and precision function of interval-valued intuitionistic fuzzy numbers in 2007 and put forward a ranking method of interval intuitionistic fuzzy number; Combined with membership uncertainty index and hesitation uncertainty index, the total ranking method of interval intuitionistic fuzzy numbers is given by Wang [22] in 2009; Lakshmana et al. [12] improved the score function of interval intuitionistic fuzzy sets on the basis of reference [24] in 2011; In 2013, Xiaohong Chen [15] defined interval intuitionistic fuzzy entropy and presented an interval intuitionistic fuzzy decision-making method with unknown attribute weight; Yanyan Wei [20] proposed an interval intuitionistic fuzzy number ranking method based on probability by defining the probability degree of the score value function and precision function in 2014. In the aspect of attribute weight determination, Hongan Zhou et al. [1] gave the attribute weight determination method based on the maximization of weighted attribute value deviation in 2008; Yanbing Gong [18] put forward an optimization model considering the deviation of the subjective and objective preference information, which is used to determine the attribute weight in 2008; In 2012, Yingjun Zhang et al. [21] presented a linear programming method based on interval intuitionistic fuzzy set accuracy function, which is used to solve the problem of attribute weight; Xiaobi Liu et al. [17] set up an attribute weight determination model based on the mean value, variance and the correlation degree between attributes in 2016; Gao Mingmei, Sheng Yin et al. [9] have given the attribute weight determination method based on fuzzy entropy in 2016 and 2018 respectively.

Interval-valued intuitionistic fuzzy sets can partly make up for the limitations of fuzzy sets in describing uncertainty information, and using fuzzy entropy to determine attribute weight is more objective and accurate. So, on the basis of related research, this paper put forward an interval intuitionistic fuzzy multi-attribute decision making method based on fuzzy entropy and TOPSIS, which adopts the improved interval intuitionistic fuzzy entropy to determine the attribute weights and uses the ranking rules of interval intuitionistic fuzzy numbers to determine the positive ideal solution and the negative rational solution of the multi-attribute decision making problem, and whose validity is verified by an example.

2 Preparatory knowledge

2.1 Interval intuitionistic fuzzy set

Definition 1. [2] Let X be a fixed set, then $\tilde{A} = < x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) > | x \in X$ is called an intuitionistic fuzzy set, where $μ_{\tilde{A}} (x)$ and $ν_{\tilde{A}} (x)$ mean the degree of membership and that of non-membership of element x belonging to $\tilde{A}$ respectively, namely, $μ_{\tilde{A}} : X \to [0, 1], x \in X \to μ_{\tilde{A}} (x) \in [0, 1]$ $ν_{\tilde{A}} : X \to [0, 1], x \in X \to ν_{\tilde{A}} (x) \in [0, 1]$ with the condition: $0 \leq μ_{\tilde{A}} (x) + ν_{\tilde{A}} (x) \leq 1, x \in X .$

Where $μ_{\tilde{A}} (x)$ represents the lower bound of the degree of positive membership derived from the evidence that the supporting element x belongs to the set $\tilde{A}$ and $ν_{\tilde{A}} (x)$ represents the lower bound of the degree of negative membership derived from the evidence that the opposing element x belongs to the set $\tilde{A}$ [11]. The set of all intuitionistic fuzzy sets in X is denoted as F (X).

Intuitionistic fuzzy sets can be simply denoted as $\tilde{A} = < x, μ_{\tilde{A}}, ν_{\tilde{A}} >$ or $\tilde{A} = < μ_{\tilde{A}}, ν_{\tilde{A}} > / x$ . For any x ∈ X, $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - ν_{\tilde{A}} (x)$ will be the intuitionistic index of element x in intuitionistic fuzzy set $\tilde{A}$ , which stands for the hesitancy degree of element x to $\tilde{A}$ . Obviously, if only x ∈ X, then 0 ≤ π_A (x) ≤1.

Definition 2. [3] Let X be a fixed set, and I_[0,1] represents the set of all closed subintervals on an interval [0, 1], then an interval-valued intuitionistic fuzzy set in X is given as the following: $\tilde{A} = < x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) > | x \in X$

Where $μ_{\tilde{A}}$ and $ν_{\tilde{A}}$ mean the interval-valued membership function and interval-valued non-membership function of $\tilde{A}$ respectively. $μ_{\tilde{A}} (x)$ and $ν_{\tilde{A}} (x)$ represent the interval-valued membership and interval-valued non-membership of the element x belongs to $\tilde{A}$ , namely, $μ_{\tilde{A}} : X \to I_{[0, 1]}, x \in X \to μ_{\tilde{A}} (x) \subseteq [0, 1]$ $ν_{\tilde{A}} : X \to I_{[0, 1]}, x \in X \to ν_{\tilde{A}} (x) \subseteq [0, 1]$

With the condition: $0 \leq sup {{μ}_{\tilde{A}} (x)} + sup {{ν}_{\tilde{A}} (x)} \leq 1, x \in X .$

The set of all interval-valued intuitionistic fuzzy sets on X is denoted as F_I (X).

For convenience, the upper and lower endpoints of interval-valued membership degree $μ_{\tilde{A}} (x)$ and interval-valued non-membership degree $ν_{\tilde{A}} (x)$ are recorded as $μ_{\tilde{AL}} (x)$ , $μ_{\tilde{AU}} (x)$ , $ν_{\tilde{AL}} (x)$ and $ν_{\tilde{AU}} (x)$ respectively, thus interval-valued intuitionistic fuzzy set $\tilde{A}$ can be expressed in interval-valued form as $\begin{matrix} \tilde{A} = {< x, [μ_{\tilde{AL}} (x), μ_{\tilde{AU}} (x)], [ν_{\tilde{AL}} (x), ν_{\tilde{AU}} (x)] \\ > | x \in X} . \end{matrix}$

Where $μ_{\tilde{AL}} (x) \in [0, 1]$ ; $μ_{\tilde{AU}} (x) \in [0, 1]$ ; $ν_{\tilde{AL}} (x)$ ∈[0, 1]; $ν_{\tilde{AU}} (x)$ ∈[0, 1]; $μ_{\tilde{AU}} (x)$ + $ν_{\tilde{AU}} (x) \leq 1$ .

Moreover, for each interval-valued intuitionistic fuzzy set $\tilde{A}$ in X, if $\begin{matrix} π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - ν_{\tilde{A}} (x) = [1 - μ_{\tilde{AU}} (x) \\ - ν_{\tilde{AU}} (x), 1 - μ_{\tilde{AL}} (x) - ν_{\tilde{AL}} (x)] . \end{matrix}$

Then $π_{\tilde{A}} (x)$ is called the interval-valued hesitation or interval valued intuitionistic fuzzy index of the element x belongs to an interval-valued intuitionistic fuzzy set $\tilde{A}$ .

2.2 Distance of interval intuitionistic fuzzy sets

Definition 3. [23] If $\tilde{A} \in F (X)$ and $\tilde{B} \in F (X)$ are two interval-valued intuitionistic fuzzy sets, then the weighted Hamming distance $d (\tilde{A}, \tilde{B})$ of interval intuitionistic fuzzy sets $\tilde{A}$ and $\tilde{B}$ is given as the following:

$\begin{matrix} d (\tilde{A}, \tilde{B}) = | μ_{\tilde{AU}} (x_{j}) - μ_{\tilde{BU}} (x_{j}) |] \\ + | ν_{\tilde{AL}} (x_{j}) - ν_{\tilde{BL}} (x_{j}) | \\ + | ν_{\tilde{AU}} (x_{j}) - ν_{\tilde{BU}} (x_{j}) |] \end{matrix}$ (1)

2.3 Score function and precision function of interval-valued intuitionistic fuzzy numbers

Definition 4. [5] Let $\tilde{α} = ([μ_{\tilde{α} L}, μ_{\tilde{aU}}], [ν_{\tilde{α} L}, ν_{\tilde{aU}}])$ be an interval-valued intuitionistic fuzzy number, the score $s (\tilde{α})$ and accuracy $h (\tilde{α})$ of interval-valued intuitionistic fuzzy number $\tilde{α}$ are defined as:

$\begin{matrix} s (\tilde{α}) = \frac{μ_{\tilde{α} L} + μ_{\tilde{α} U} - ν_{\tilde{α} L} - ν_{\tilde{α} U}}{4} \\ (1 + \frac{1}{μ_{\tilde{α} L} + μ_{\tilde{α} U} - μ_{\tilde{α} L} ν_{\tilde{α} L} + μ_{\tilde{α} U} ν_{\tilde{α} U}}) \end{matrix}$ (2)

$h (\tilde{α}) = \frac{μ_{\tilde{α} L} + μ_{\tilde{α} U} + ν_{\tilde{α} L} + ν_{\tilde{α} U}}{2}$ (3)

Definition 5. [5,19, 5,19] If ${\tilde{α}}_{1} = ([μ_{{\tilde{α}}_{1} L}, μ_{{\tilde{a}}_{1} U}]$ , $[ν_{{\tilde{α}}_{1} L}$ , $ν_{{\tilde{a}}_{1} U}])$ and ${\tilde{α}}_{2} = ([μ_{{\tilde{α}}_{2} L}, μ_{{\tilde{a}}_{2} U}], [ν_{{\tilde{α}}_{2} L}, ν_{{\tilde{a}}_{2} U}])$ are two interval valued intuitionistic fuzzy numbers, then whose ranking rules are as follows:

If $s ({\tilde{α}}_{1}) < s ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1}$ is smaller than ${\tilde{α}}_{2}$ , denoted by ${\tilde{α}}_{1} <$ ${\tilde{α}}_{2}$ ;

If $s ({\tilde{α}}_{1}) = s ({\tilde{α}}_{2})$ ,

If $h ({\tilde{α}}_{1}) = h ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1}$ is equal to ${\tilde{α}}_{2}$ , denoted by ${\tilde{α}}_{1} =$ ${\tilde{α}}_{2}$ ;

If $h ({\tilde{α}}_{1}) < h ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1}$ is smaller than ${\tilde{α}}_{2}$ , denoted by ${\tilde{α}}_{1} <$ ${\tilde{α}}_{2}$ ;

If $h ({\tilde{α}}_{1}) > h ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1}$ is bigger than ${\tilde{α}}_{2}$ , denoted by ${\tilde{α}}_{1} >$ ${\tilde{α}}_{2}$ .

3 Problem description and decision-making method

3.1 Problem description

Let Y ={ Y₁, Y₂, ⋯ , Y_m } be a finite set of feasible alternatives, and G ={ G₁, G₂, ⋯ , G_n } be a finite set of attributes to evaluate each feasible alternative. Suppose the weight vector of attribute G_j (j = 1, 2, ⋯ , n) is ω ={ ω₁, ω₂, ⋯ , ω_n }. For each feasible alternative Y_i ∈ Y, the evaluation value of attribute G_j ∈ G can be expressed as ${\tilde{F}}_{ij} = < [μ_{ijL}, μ_{ijU}], [ν_{ijL}, ν_{ijU}] >$ (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n), where ${\tilde{F}}_{ij}$ is interval-valued intuitionistic fuzzy set, [μ_ijL, μ_ijU] reflects the extent to which the alternative Y_i ∈ Y satisfies the attribute G_j ∈ G, and [ν_ijL, ν_ijU] indicates the extent to which the alternative Y_i ∈ Y does not satisfy the attribute G_j ∈ G [13]. Thus the matrix F_I = < [μ_ijL, μ_ijU] , [ν_ijL, ν_ijU] > _m×n is the interval-valued intuitionistic fuzzy decision matrix for the multi-attribute decision making problem (see Table 1). Now the problem is how to get an effective decision analysis method to sort all the alternatives by determining the attribute weight ω ={ ω₁, ω₂, ⋯ , ω_n } according to interval intuitionistic fuzzy decision matrix F_I.

Table 1
Interval-valued intuitionistic fuzzy decision matrix F_I

G₁ G₂ … G_n

Y₁ < [μ_11L, μ_11U] , [ν_11L, ν_11U]> < [μ_12L, μ_12U] , [ν_12L, ν_12U]> … < [μ_1nL, μ_1nU] , [ν_1nL, ν_1nU]>

Y₂ < [μ_11L, μ_11U] , [ν_11L, ν_11U]> < [μ_22L, μ_22U] , [ν_22L, ν_22U]> … < [μ_2nL, μ_2nU] , [ν_2nL, ν_2nU]>

⋮ ⋮ ⋮ ⋮ ⋮

Y_m < [μ_m1L, μ_m1U] , [ν_m1L, ν_m1U]> < [μ_m2L, μ_m2U] , [ν_m2L, ν_m2U]> … < [μ_mnL, μ_mnU] , [ν_mnL, ν_mnU]>

	G₁	G₂	…	G_n
Y₁	< [μ_11L, μ_11U] , [ν_11L, ν_11U]>	< [μ_12L, μ_12U] , [ν_12L, ν_12U]>	…	< [μ_1nL, μ_1nU] , [ν_1nL, ν_1nU]>
Y₂	< [μ_11L, μ_11U] , [ν_11L, ν_11U]>	< [μ_22L, μ_22U] , [ν_22L, ν_22U]>	…	< [μ_2nL, μ_2nU] , [ν_2nL, ν_2nU]>
⋮	⋮	⋮	⋮	⋮
Y_m	< [μ_m1L, μ_m1U] , [ν_m1L, ν_m1U]>	< [μ_m2L, μ_m2U] , [ν_m2L, ν_m2U]>	…	< [μ_mnL, μ_mnU] , [ν_mnL, ν_mnU]>

3.2 Attribute weight determination method based on Fuzzy Entropy

Fuzzy entropy is the extension of Shannon information entropy in the field of fuzzy mathematics, which is used to explain the amount of information contained in fuzzy sets. The more the information is, the lower the fuzziness is, and the more information it can provide to decision makers. Interval-valued intuitionistic fuzzy entropy is a generalization of fuzzy entropy in interval-valued intuitionistic fuzzy set. Since Xiaozhi Guo [14,16, 14,16] put forward the axiomatic definition of interval-valued intuitionistic fuzzy entropy in 2004, many scholars have studied the definition and calculating formula of interval-valued intuitionistic fuzzy entropy [6,7, 6,7].

Definition 6. Let $\begin{matrix} \tilde{A} = {< x_{i}, [μ_{AL} (x_{i}), μ_{AU} (x_{i})], [ν_{AL} (x_{i}), ν_{AU} (x_{i})] \\ | x_{i} \in X}, \end{matrix}$ $\tilde{B} = {< x_{i}, [μ_{BL} (x_{i}), μ_{BU} (x_{i})],$ [ν_BL (x_i) , ν_BU (x_i)]|x_i ∈ X } are two interval-valued intuitionistic fuzzy sets in X, then the function E: IIFS(X)⟶[0,1] is called interval Intuitionistic Fuzzy Entropy, if it meets the following conditions:

$E (\tilde{A}) = 0$ , if and only if $\tilde{A}$ is a clear set, that is [μ_AL (x_i) , μ_AU (x_i)] = [1, 1] , [ν_AL (x_i) , μ_AU (x_i)] = [0, 0] or [μ_AL (x_i) , μ_AU (x_i)] = [0, 0] , [ν_AL (x_i) , μ_AU (x_i)] = [1, 1].

$E (\tilde{A}) = 1$ , if and only if x_i ∈ X, μ_AL (x_i) = ν_AL (x_i) and μ_AU (x_i) = ν_AU (x_i).

$E (\tilde{A}) = E ({\tilde{A}}^{c})$ , for any x_i ∈ X.

$E (\tilde{A}) \leq E (\tilde{B})$ , If interval-valued intuitionistic fuzzy sets $\tilde{B}$ are more fuzzy than $\tilde{A}$ .

Definition 7. If $\tilde{A} = {< x_{i}, [μ_{AL} (x_{i}), μ_{AU} (x_{i})],$ [ν_AL (x_i) , ν_AU (x_i)] | x_i∈ X } is any interval-valued intuitionistic fuzzy set, then,

$\begin{matrix} E (\tilde{A}) = \frac{1}{n} \sum_{i = 1}^{n} cos \frac{1}{4} [| μ_{AL}^{2} (x_{i}) - ν_{AL}^{2} (x_{i}) | \\ + | μ_{AU}^{2} (x_{i}) - ν_{AU}^{2} (x_{i}) |] π \end{matrix}$ (4) is called an interval-valued intuitionistic fuzzy entropy.

Equation (4) can also be written as:

$\begin{matrix} E (\tilde{A}) = \frac{1}{n} \sum_{i = 1}^{n} cos \frac{1}{4} [| (μ_{AL} (x_{i}) - ν_{AL} (x_{i})) \\ (1 - π_{AL} (x_{i}) | + | (μ_{AU} (x_{i}) - ν_{AU} (x_{i})) \\ (1 - π_{AU} (x_{i})) |] π \end{matrix}$ (5)

It can be seen from Equation (5) that $E (\tilde{A})$ not only includes the deviation between membership μ_AL (x_i) - ν_AL (x_i) and that of non-membership μ_AU (x_i) - ν_AU (x_i), but also contains the information on hesitation degree π_AL (x_i) and π_AU (x_i).

From Equation (5), fuzzy entropy E_j of attribute G_j (j = 1, 2, ⋯ , n) can be obtained, through which the weight ω_j of attribute G_j (j = 1, 2, ⋯ , n) can be calculated:

$ω_{j} = \frac{1 - E_{j}}{n - \sum_{j = 1}^{n} E_{j}}, j = 1, 2, \dots, n$ (6)

3.3 Steps of interval-valued intuitionistic fuzzy multi-attribute decision making

Based on the above analysis, an interval-valued intuitionistic fuzzy decision-making method based on improved fuzzy entropy is proposed in the case of unknown attribute weights. The decision-making steps are as follows:

Step 1. Determines the alternative set Y = {Y₁, Y₂, ⋯, Y_m} and attribute set G ={ G₁, G₂, ⋯ , G_n } of the multi-attribute decision-making problem, obtains the interval intuitionistic fuzzy characteristic information about the attribute G_j ∈ G in the multi-attribute decision-making alternative Y_i ∈ Y, and constructs the interval-valued intuitionistic fuzzy decision-making matrix F_I = < [μ_ijL, μ_ijU] , [ν_ijL, ν_ijU] > _m×n;

Step 2. Calculate the fuzzy entropy of the attribute G_j (j = 1, 2, ⋯ , n) based on intuitionistic fuzzy multi-attribute decision matrix F_I, using Equation (4)

$\begin{matrix} E_{j} = \frac{1}{m} \sum_{i = 1}^{m} cos \frac{1}{4} [| (μ_{ijL} - ν_{ijL}) (1 - π_{ijL}) | \\ + | (μ_{ijU} - ν_{ijU}) (1 - π_{ijU}) |] π \end{matrix}$ (7)

Step 3. Using fuzzy entropy E_j of attribute G_j (j = 1, 2, ⋯ , n) to calculate its weight ω_j.

$ω_{j} = \frac{1 - E_{j}}{n - \sum_{j = 1}^{n} E_{j}}, j = 1, 2, \dots, n$ (8)

Step 4. According to the interval-valued intuitionistic fuzzy multi-attribute decision-making matrix F_I, the positive ideal point Y⁺ and the negative ideal point Y^- of multi-attribute solving problem are determined.

For attribute G_j ∈ G, the evaluation value of each alternative Y_i (i = 1, 2, ⋯ , m) is ${\tilde{F}}_{ij} = < [μ_{ijL}, μ_{ijU}], [ν_{ijL}, ν_{ijU}] >$ (i = 1, 2, ⋯ , m) (The j-th row elements of interval-valued intuitionistic fuzzy decision-making matrices), the values of the score and accuracy function are calculated, and the best evaluation value $Y_{j}^{+}$ and the worst evaluation value $Y_{j}^{-}$ are determined by the ranking method of interval-valued intuitionistic fuzzy numbers: $Y_{j}^{+} = [< μ_{jL}^{+}, μ_{jU}^{+}, ν_{jL}^{+}, ν_{jU}^{+}]$ > (j = 1,2, ⋯,n), $Y_{j}^{-} = [< μ_{jL}^{-}, μ_{jU}^{-}, ν_{jL}^{-}, ν_{jU}^{-}]$ > (j = 1,2, ⋯,n).

Then, the positive ideal point Y⁺ and the negative ideal point Y^- of multi-attribute solution problem can be obtained as follows:

$\begin{matrix} Y^{+} = (< [μ_{1 L}^{+}, μ_{1 U}^{+}], [ν_{1 L}^{+}, ν_{1 U}^{+}] >, \\ < [μ_{2 L}^{+}, μ_{2 U}^{+}], [ν_{2 L}^{+}, ν_{2 U}^{+}], \dots, \\ < [μ_{nL}^{+}, μ_{nU}^{+}], [ν_{nL}^{+}, ν_{nU}^{+}] >) \end{matrix}$ (9)

$\begin{matrix} Y^{-} = (< [μ_{1 L}^{-}, μ_{1 U}^{-}], [ν_{1 L}^{-}, ν_{1 U}^{-}] >, \\ < [μ_{2 L}^{-}, μ_{2 U}^{-}], [ν_{2 L}^{-}, ν_{2 U}^{-}], \dots, \\ < [μ_{nL}^{-}, μ_{nU}^{-}], [ν_{nL}^{-}, ν_{nU}^{-}] >) \end{matrix}$ (10)

Step 5. Calculate the weighted hamming distances $d_{i}^{+}$ and $d_{i}^{-}$ from each alternative Y_i (i = 1, 2, ⋯ , m) to the positive ideal point Y⁺ and the negative ideal point Y^-.

$\begin{matrix} d_{i}^{+} = \frac{1}{4} \sum_{j = 1}^{n} ω_{j} [| μ_{ijL} - μ_{jL}^{+} | + | μ_{ijU} - μ_{jU}^{+} | \\ + | ν_{ijL} - ν_{jL}^{+} | + | ν_{ijU} - ν_{jU}^{+} |] \end{matrix}$ (11)

$\begin{matrix} d_{i}^{-} = \frac{1}{4} \sum_{j = 1}^{n} ω_{j} [| μ_{ijL} - μ_{jL}^{-} | + | μ_{ijU} - μ_{jU}^{-} | \\ + | ν_{ijL} - ν_{jL}^{-} | + | ν_{ijU} - ν_{jU}^{-} |] \end{matrix}$ (12)

Step 6. Calculate the degree of closeness c_i of the alternative Y_i (i = 1, 2, ⋯ , m).

$c_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{+}}, i = 1, 2, \dots, m$ (13)

Rank all the alternatives Y_i (i = 1, 2, ⋯ , m) according to the degree of closeness c_i, the larger the c_i is, the closer the alternative Y_i is to the positive ideal point, the farther away from the negative ideal point, and the better the alternative is.

4 Numerical example

In this selection, the proposed approach is applied to evaluate the performance of public fiscal expenditure of local government in China. Performance evaluation of public fiscal expenditure is to evaluate the economy, efficiency and effectiveness of fiscal expenditure activities, which can be evaluated from four dimensions: education expenditure performance G₁, pension expenditure performance G₂, employment expenditure performance G₃ and infrastructure construction expenditure performance G₄ [10, 19].

It is assumed that the interval-valued intuitionistic fuzzy evaluation results of attributes G_j (j = 1, 2, 3, 4) in 5 regions Y_i (i = 1, 2, 3, 4, 5) can be obtained through investigation and expert consultation, shown in Table 2. Try to rank the performance of government public fiscal expenditure in 5 regions Y_i (i = 1, 2, 3, 4, 5).

Table 2
Interval-valued intuitionistic fuzzy decision matrix F_I of performance evaluation of public fiscal expenditure

G₁ G₂ G₃ G₄

Y₁ < [0.4,0.5,0.3,0.4]> < [0.4,0.6],[0.2,0.4]> < [0.3,0.4],[0.4,0.5]> < [0.5,0.6],[0.1,0.3]>

Y₂ < [0.5,0.6],[0.2,0.3]> < [0.6,0.7],[0.2,0.3]> < [0.5,0.6],[0.3,0.4]> < [0.4,0.7],[0.1,0.2]>

Y₃ < [0.3,0.5],[0.3,0.4]> < [0.1,0.3],[0.5,0.6]> < [0.2,0.5],[0.4,0.5]> < [0.2,0.3],[0.4,0.6]>

Y₄ < [0.2,0.5],[0.3,0.4]> < [0.4,0.7],[0.1,0.2]> < [0.4,0.5],[0.3,0.5]> < [0.5,0.8],[0.1,0.2]>

Y₅ < [0.3,0.4],[0.1,0.3]> < [0.7,0.8],[0.1,0.2]> < [0.5,0.6],[0.2,0.4]> < [0.6,0.7],[0.1,0.2]>

	G₁	G₂	G₃	G₄
Y₁	< [0.4,0.5,0.3,0.4]>	< [0.4,0.6],[0.2,0.4]>	< [0.3,0.4],[0.4,0.5]>	< [0.5,0.6],[0.1,0.3]>
Y₂	< [0.5,0.6],[0.2,0.3]>	< [0.6,0.7],[0.2,0.3]>	< [0.5,0.6],[0.3,0.4]>	< [0.4,0.7],[0.1,0.2]>
Y₃	< [0.3,0.5],[0.3,0.4]>	< [0.1,0.3],[0.5,0.6]>	< [0.2,0.5],[0.4,0.5]>	< [0.2,0.3],[0.4,0.6]>
Y₄	< [0.2,0.5],[0.3,0.4]>	< [0.4,0.7],[0.1,0.2]>	< [0.4,0.5],[0.3,0.5]>	< [0.5,0.8],[0.1,0.2]>
Y₅	< [0.3,0.4],[0.1,0.3]>	< [0.7,0.8],[0.1,0.2]>	< [0.5,0.6],[0.2,0.4]>	< [0.6,0.7],[0.1,0.2]>

First, from Equation (7), the fuzzy entropy E_j (j = 1, 2, 3, 4) of attributes G_j (j = 1, 2, 3, 4) are calculated through the interval-valued intuitionistic fuzzy decision matrix F_I as follows respectively: $E_{1} = 0.9851, E_{2} = 0.8571, E_{3} = 0.9790, E_{4} = 0.8729 .$

Then we can utilize Equation (8) to calculate the weight vector of the attribute G_j (j = 1, 2, 3, 4): $ω = (0.049, 0.467, 0.069, 0.415)^{T}$

According to the interval-valued intuitionistic fuzzy decision matrix F_I, from Equations (2) and (3), the most favorable value and the worst value of all attributes G_j (j = 1, 2, 3, 4) are determined as: $\begin{matrix} \begin{matrix} Y_{1}^{+} & = < [0.5, 0.6], [0.2, 0.3] >, \\ Y_{2}^{+} & = < [0.7, 0.8], [0.1, 0.2] >, \\ Y_{3}^{+} & = < [0.5, 0.6], [0.2, 0.4] >, \\ Y_{4}^{+} & = < [0.6, 0.7], [0.1, 0.2] >; \\ Y_{1}^{-} & = < [0.2, 0.5], [0.3, 0.4] >, \\ Y_{2}^{-} & = < [0.1, 0.3], [0.5, 0.6] > \\ Y_{3}^{-} & = < [0.2, 0.5], [0.4, 0.5] >, \\ Y_{4}^{-} & = < [0.2, 0.3], [0.4, 0.6] > . \end{matrix} \end{matrix}$

Then, the positive ideal point Y⁺ and the negative ideal point Y^- of multi-attribute solution problem can be obtained as follows: $\begin{matrix} \begin{matrix} Y^{+} = (< [0.5, 0.6], [0.2, 0.3] >, \\ < [0.7, 0.8], [0.1, 0.2] >, < [0.5, 0.6], [0.2, 0.4] >, \\ < [0.6, 0.7], [0.1, 0.2] >), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} Y^{-} = (< [0.2, 0.5], [0.3, 0.4] >, \\ < [0.1, 0.3], [0.5, 0.6] >, < [0.2, 0.5], [0.4, 0.5] >, \\ < [0.2, 0.3], [0.4, 0.6] >) . \end{matrix} \end{matrix}$

Using the Equations (11)–(13), the weighted hamming distances to the positive ideal point Y⁺ and the negative ideal point Y^- and the degree of closeness of each alternative can be calculated, shown in Table 3.

Table 3

Distances to the ideal point and the degree of closeness of each alternative

	$d_{i}^{+}$	$d_{i}^{-}$	c _i
Y₁	0.153	0.258	0.628
Y₂	0.081	0.331	0.803
Y₃	0.407	0.004	0.010
Y₄	0.073	0.329	0.818
Y₅	0.015	0.396	0.964

It can be seen from Table 3, c₅ > c₄ > c₂ > c₁ > c₃.

Therefore, the rank of the performance evaluation of public fiscal expenditure of these 5 districts is Y₅ ≻ Y₄ ≻ Y₂ ≻ Y₁ ≻ Y₃, thus the performance evaluation of public fiscal expenditure of the district Y₅ is the best.

5 Conclusion

In this paper, a multi-attribute decision making method based on fuzzy entropy and TOPSIS is proposed to solve the multi-attribute decision-making problem with attribute value as interval-valued intuitionistic fuzzy number. Interval-valued intuitionistic fuzzy sets are of strong flexibility and expression ability in describing fuzzy information. We use improved fuzzy entropy to determine attribute weights, determines the positive and negative rational solutions based on the score function and accuracy function of interval-valued intuitionistic fuzzy numbers, and then establishes the interval-valued intuitionistic fuzzy multi-attribute decision making model based on the TOPSIS method. Finally, an example is given to verify the practicability and effectiveness of the proposed method.

Footnotes

Acknowledgments

This paper is supported by the National Social Science Foundation of China (NO: 11BGL089), the Social Science Foundation of Hebei province of China (No. HB18GL008), and Beijing Intelligent Logistics System Collaborative Innovation Center (BILSCIC-2019KF-15).

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