The novel entropy measurements of Z + -numbers and their application on multi-attribute decision making problem

Abstract

Z⁺-numbers, which carry more information than Z-numbers, are studied in this paper. Based on existed models, two more scientific and reasonable probability models of Z⁺-numbers are developed. In order to utilize Z⁺-numbers to solve practical problems, the α-cut set of Z⁺-numbers and corresponding utility function are proposed. Meanwhile, according to the structure of Z⁺-numbers, the entropy, cross-entropy and comprehensive cross-entropy are introduced to measure the uncertainty and fuzziness of Z⁺-numbers information. Furthermore, a linear programming model based on proposed three kinds of entropy is designed to obtain the weight vector of criteria in decision-making problems. Finally, we provide an example by selecting an optimal design of electricity vehicles charge station(DEVCS) combined the PROMETHEE method with Z⁺-numbers, and the feasibility of the proposed method are verified.

Keywords

entropy comprehensive cross-entropy PROMETHEE method multi-criteria decision-making

1 Introduction

To better handle uncertain information, Zadeh [1] proposed the concept of Z-numbers based on fuzzy set theory. Campared to fuzzy set, Z-numbers put up more outstanding capability to handle information which is uncertain, imprecise and incomplete. Hence, Z-numbers have attracted plenty of attention and studies from numerous scholars around the world. In the current study, studies about Z-numbers can be roughly classified three directions.

The first direction is about studies of the theory of Z-numbers. Aliev et al. [2, 3] respectively discussed the arithmetic of continuous Z-numbers and discrete Z-numbers. Aliev et al. [4] proposed a methodology to define functions on Z-numbers environment based on the extension principle. Pirmuhammadi et al. [5] discussed the parametric forms of Z-numbers and generalized differentiation based on generalized Hukuhara differentiation.

The second direction involves applications of Z-numbers. Marhamati et al. [6] combined integration of Z-numbers with bayesian decision theory and apply it to a breast cancer diagnosis problem. Kang et al. [7] introduced a total utility function defined on continuous Z-numbers context. Kang et al. [8] analysed the stable strategies in the evolutionary games environment based on the utility of Z-numbers which was defined in [7]. Aliev et al. [9] introduced a method to compute the sum of a large number of Z-numbers based on a predefined level of informativeness. Abiyev et al. [10] and Wu et al. [11] respectively utilized Z-numbers to solve the food security problem and the medical diagnosis problem. Khalif et al. [12] proposed the hybrid fuzzy multi-criteria decision-making model for Z-numbers based on the intuitive vectorial centroid.

The third direction concerns extended studies of Z-numbers. Banerjee et al. [13, 14] introduced Z^*-numbers and proposed a computational model for the endogenous arousal. Agakishiyev [15] suggested a method to solve supplier selection problem under Z-information context. Wang and Peng [16, 17] respectively introduced linguistic Z-numbers and hesitant uncertain linguistic Z-numbers. Peng et al. [18] and Yang et al. [19] used the hidden probability carried by Z-numbers which refers to Z⁺-numbers.

According to above review about Z-numbers, it can be easily find that Z-numbers has outstanding capability to solve decision-making problem which is under uncertain environment. However, there are plentiful excellent methods about decision-making problems. Brans et al. [20] proposed the preference ranking organization method for enrichment evaluations(PROMETHEE)method to select and rank projects with the help of preference functions. And Maity et al. [21] promoted the PROMETHEE method and proposed the PROMETHEE 2 method. One of merits of the PROMETHEE method is that the operation of the method is easy and convenient which just requires the criterion value of each project and weight vector of criteria. Roy B [22] developed the elimination and choice translating reality(ELECTURE) methods to rank the alternaltive. Based on [22], the extended ELECTURE methods, ELECTURE 1 method [23], the ELECTURE 2 method [24], and the ELECTURE 3 method [25], are respectively utilized to solve various decision-making problems in different environment. Moreover, the qualitative flexible multiple criteria method(QUALIFLEX) is introduced by Paelinck JHP [26], and Chen et al. [27] extended the QUALFIEX method under the interval type-2 fuzzy sets context.

Z-numbers can excellently deal with uncertain information, but how to measure the uncertainty of the cognitive information? In 1948, Shannon [28] proposed the concept of information entropy to measure the uncertainty. To measure the fuzziness of information, Zadeh [29] defined the fuzzy entropy by probability methods. Shang and Jiang [30] presented the fuzzy cross-entropy between two fuzzy sets. Burillo and Bustince [31] introduced intuitionistic fuzzy entropy defined on intuitionistic fuzzy sets(IFSs) environment. Mao et al. [32] designed the cross-entropy and symmetric cross-entropy of IFSs based on intuitionistic factor and fuzzy factor.

Based on the above introduction, the primary motivations of this paper are as follows:

(1) Z⁺-numbers contain both possibility distribution and probability distribution, hence, Z⁺-numbers have more outstanding capability to solve uncertain and fuzzy information compared to Z-numbers. As such, this study introduces three kinds of entropy which are defined on Z⁺-numbers context to measure the uncertainty and fuzziness of Z⁺-numbers.

(2) Since the probability distribution carried by Z⁺- numbers is hidden and unknown, we have no capability to straightly get data represented by Z⁺-numbers. For this reason, this study develops two probability models which are more scientific and appropriate based on two existed linear programming models.

(3) Real-life information always puts up incomplete and partially reliable, which makes decision-making problem difficult, while Z⁺-number can solve this problem. Therefore, this study establishes the PROMETHEE method with Z⁺-numbers to illustrate the practicability of Z⁺-numbers.

This paper is organized as follows: In section 2, we briefly review some definitions. In section 3, the Z⁺- numbers are briefly introduced and two improved probability model are proposed to obtain a Z⁺-number from a Z-number. In section 4, the α-cut set of Z⁺-n- umbers and three kinds of entropy defined on Z⁺-nu- mbers are proposed. In section 5, the PROMETHEE method combined with Z⁺-numbers is introduced. In section 6, the introduced method in section 5 is applied to a multi-criteria decision-making problem, and the feasibility of the method is discussed in section 7. Finally, section 8 presents some conclusions of this paper.

2 Preliminary

In this section, we briefly review some related concepts, including fuzzy sets, discrete fuzzy sets, Z-numbers, and discrete Z-numbers.

Definition 2.1 (Fuzzy set). [3] An arbitrary fuzzy set A, defined for universe X may be given as:

$A = {〈 x, μ_{A} (x) 〉 | x \in X}$ where μ_A : X → [0, 1]is the membership function of A, and the membership value μ_A (x) describes the degree of belongingness of x ∈ X to A.

Definition 2.2 (Discrete fuzzy set). [3] A fuzzy subset A of the real line R with membership function μ_A : R → [0, 1] is a discrete fuzzy set if its support set is finite, i.e. there exist x₁, . . . , x_n ∈ R with x₁ < x₂ < . . . < x_n,such that supp(A)={x₁, . . . , x_n} and there exist natural numbers s, t with 1 ≤ s ≤ t ≤ n satisfying the following conditions:

μ_A (x_i) =1 for any natural number i with s ≤ i ≤ t.

μ_A (x_i ≤ μ_A (x_j)for each natural numbers i, j with 1 ≤ i ≤ j ≤ s.

μ_A (x_i ≥ μ_A (x_j) for each natural numbers i, j with t ≤ i ≤ j ≤ n.

Definition 2.3(Z-numbers). [1] A Z-number is an ordered pair of fuzzy numbers represented by Z=(A, B) and associated with a real-valued uncertain variable, X, with the first component, A, playing the role of fuzzy restriction on the values which X may take, written as X is A. And the second component, B, measures the reliability of A.

Typically, Zadeh [1] pointed out that if X is a random variable, then X is A refers to a fuzzy event in real line. The probability of the event, p, can be expressed as:

$p = \int_{R} μ_{A} (u) p_{X} (u) du$ where p_X represents the probability density function of X, which is underlying. In effect, the second component, B, can be interpreted as:

$Prob (X is A) is B$

What is deserved to underscored is that the probability density function, p_X, is actually unknown. But we know that the restriction on p_X may be expressed as:

$\int_{R} μ_{A} (u) p_{X} (u) du is B$ And this is the basis that we can calculate the specific values of p_X in discrete environment.

Definition 2.4(Discrete Z-numbers). [3] A discrete Z-number is an ordered pair Z=(A, B) where A is a discrete fuzzy number playing a role of a fuzzy restriction on values of X, and B is a discrete fuzzy number with a membership function μ_B : b₁, . . . , b_n → [0, 1] , b₁, . . . , b_n ∈ [0, 1], playing a role of reliability of the event that X is A.

The introduction of part of notations which will appear to the following paper is shown in Table 1.

Table 1
Introdution table with part of notations

Notation Introduction

μ _A Membership function of fuzzy number A

p _X Probability density/mass function of random variable X

α threshold value

$Z_{α}^{+}$ α cut set of Z⁺-numbers

U(·) Utility of ·

f function of measurement of fuzziness

E(·) Entropy of ·

CE(·) Cross entropy of ·

CCE(·) Comprehensive cross entropy of ·

S Alternative set of decision-making

C Criteria set of decision-making

W Weight set of each criterion

Notation	Introduction
μ _A	Membership function of fuzzy number A
p _X	Probability density/mass function of random variable X
α	threshold value
$Z_{α}^{+}$	α cut set of Z⁺-numbers
U(·)	Utility of ·
f	function of measurement of fuzziness
E(·)	Entropy of ·
CE(·)	Cross entropy of ·
CCE(·)	Comprehensive cross entropy of ·
S	Alternative set of decision-making
C	Criteria set of decision-making
W	Weight set of each criterion

3 Probability models of Z⁺-numbers

In this section, we will briefly introduce Z⁺-numbers and two improved probability models to get a Z⁺- number from an arbitrary given discrete Z-numbers.

3.1 Introduction of Z⁺-numbers

Definition 3.1(Z⁺-numbers). A Z⁺-number, Z⁺, is a combination of a fuzzy number, A, and a random number R, written as an ordered pair Z⁺=(A, R), where the first component, A, plays a same role as it does in Z-numbers, and R represents the probability density function of a random number.

The concept of Z⁺-numbers is closely related to the concept of Z-numbers. In fact, R can be viewed as the hidden probability density function of X, i.e., a Z⁺-number can be equivalently expressed as (A, p_X) or (μ_A, p_X), where μ_A indicates the membership function of fuzzy number, A.

Here, Zadeh [1] indicated that μ_A and p_X are compatible if their centroids are coincident, that is

$\int_{R} {up}_{X} (u) du = \frac{\int_{R} u μ_{A} (u) du}{\int_{R} μ_{A} (u) du}$

Meanwhile, the scalar product of μ_A and p_X, μ_A · p_X, is the probability measure of A. More concretely,

$μ_{A} \cdot p_{X} = P_{A} = \int_{R} μ_{A} (u) p_{X} (u) du$ And this is the relation between a Z-number and a Z⁺-number.

Espacially, in discrete environment, μ_A and p_X respectively represent discrete fuzzy number and probability mass function. More concretely,

$\begin{matrix} μ = μ_{1} / u_{1} + . . . + μ_{n} / u_{n} \\ p = p_{1} / u_{1} + . . . + p_{n} / u_{n} \end{matrix}$ where μ_i and p_i, i = 1, . . . , n, mean the possibility and probability of X=u_i respectively. Equivalently, a Z⁺-number can be expressed as:

$Z^{+} = (μ_{1}, p_{1}) / u_{1} + . . . + (μ_{n}, p_{n}) / u_{n}$ And this paper will mainly aim at the Z⁺-numbers under discrete environment.

3.2 Two improved probability models

In reality, since the probability distribution of X is hidden and unknown, we are incapable to straightly describe an object by a Z⁺-number. But we have enough capability to describe the object by the form of Z-numbers. According to this truth, how to covert a Z-number to a Z⁺-number is an important research direction, for we can easily find that a Z⁺-number carries more information than a Z-number by comparing the structure of a Z⁺-number with a Z-number, i.e., we only know the possibility distribution of X by Z-numbers, but we can get both possibility and probability distribution of X from Z⁺-numbers.

In the current study, the primary method to obtain a Z⁺-number is linear programming models by various object functions and constraint conditions, but these methods have some lacks. For instance, a linear programming model based on a Z-number, Z=(A, B), which is introduced in [19] is defined as: $\min \sum_{i = 1}^{n} v_{i} * (x_{i} - EX)^{2} {\begin{matrix} x_{1} v_{1} + x_{2} v_{2} + . . . + x_{n} v_{n} = \frac{\sum_{i = 1}^{n} x_{i} c_{i}}{\sum_{i = 1}^{n} c_{i}} \\ c_{1} v_{1} + c_{2} v_{2} + . . . + c_{n} v_{n} = b_{0} \\ v_{1} + v_{2} + . . . v_{n} = 1 \\ v_{i} \geq 0, \forall i = 1, 2, . . ., n \end{matrix}$ Where v_i = p_X (x_i) and c_i = μ_A (x_i) respectively represent the possibility distribution and probability mass function, and $μ_{B} (b_{0}) = \max_{b} μ_{B} (b_{0})$ .

The author of [19] simply chosen b₀ in the second constraint condition by which has a maximum membership values. More clearly,

$μ_{B} (b_{0}) = max {μ_{B} (b_{1}), μ_{B} (b_{2}), . . ., μ_{b_{B}} (b_{m})},$ Where b₀ ∈ {b₁, b₂, . . . , b_m}.

But what if there exists a $b_{0}^{'}$ satisfied that $b_{0}^{'} \neq b_{0}$ and $μ_{B} (b_{0}^{'}) = μ_{B} (b_{0})$ ? The [19] appears to incapable to solve this case, for how to choose a proper one between b₀ and $b_{0}^{'}$ is a quite troublesome problem.

What’s more, little information carried by the second component, B, is utilized in the second constraint condition of [19], and the rest of information is ignored. Actually, the solution of the [19] may be unconvinced to solve uncertain phenomenon because of the incomplete utilization of information carried by Z-numbers.

Based on the above two views of the model proposed in [19], a modified linear programming model is given as: $\min \sum_{i = 1}^{n} v_{i} * (x_{i} - EX)^{2} {\begin{matrix} x_{1} v_{1} + x_{2} v_{2} + . . . + x_{n} v_{n} = \frac{\sum_{i = 1}^{n} x_{i} c_{i}}{\sum_{i = 1}^{n} c_{i}} \\ c_{1} v_{1} + c_{2} v_{2} + . . . + c_{n} v_{n} = b_{E} \\ v_{1} + v_{2} + . . . v_{n} = 1 \\ v_{i} \geq 0, \forall i = 1, 2, . . ., n \end{matrix}$ (M-1) Where $b_{E} = \sum_{i = 1}^{m} b_{i} μ_{B} (b_{i}) / \sum_{i = 1}^{m} μ_{B} (b_{i})$ . b_E can be regarded as the weighted sum of supp (B) or the expectation of b ∈ supp (B). Actually, b_E can solve the two above questions to a certain degree, for b_E is an unique value and the calculation of b_E is under the sufficient employ of information carried by fuzzy number, B.Example 3.1. Let us consider a discrete Z-number Z=(A, B) given as: $\begin{matrix} A = \frac{0}{1} + \frac{0.12}{2} + \frac{0.4}{3} + \frac{0.7}{4} + \frac{1}{5} + \frac{0.45}{6} + \frac{0.1}{7}, \end{matrix}$ $\begin{matrix} B = \frac{0}{0} + \frac{0.3}{0.2} + \frac{0.65}{0.4} + \frac{1}{0.6} + \frac{0.56}{0.8} + \frac{0.2}{1} . \end{matrix}$ We can solve the model(M-1) in MATLAB, and the result is as follows:

$P = \frac{0}{1} + \frac{0}{2} + \frac{0.12}{3} + \frac{0.5384}{4} + \frac{0}{5} + \frac{0.3416}{6} + \frac{0}{7} .$

As shown in Fig.1, the difference of probabilities calcualated by model(M-1) and original model respectively is small, for the gap between b_E and b₀ is small.

Fig. 1

Comparision of probability between the model(M-1) and original model.

An another linear programming model designed in [3] is defined as: $c_{1} v_{1} + c_{2} v_{2} + . . . + c_{n} v_{n} \to b_{l} {\begin{matrix} v_{1} + v_{2} + . . . + v_{n} = 1 \\ v_{1}, v_{2}, . . ., v_{n} \geq 0 \end{matrix}$ Where v_i = p_X (x_i) and c_i = μ_A (x_i) respectively represent the possibility distribution and probability mass function, and b_l ∈ supp (B) , l = 1, 2, . . . , m of a discrete fuzzy number B are values of probability measure of A.

However, there is still a question about the model in [3]. According to the object function of the above model, we can obviously get m different discrete probability mass function, denoted as P = {P₁, P₂, . . . , P_m} where P_i is a n-dimension vector, but how can we choose a reasonable distribution from P? And a problem similar to the model in [19] is that the model in [3] also utilizes little information of the second component, B. In order to solve these two question, a solution is given as follow: $\min (\sum_{i = 1}^{n} c_{i} v_{ij} - b_{j})^{2} {\begin{matrix} v_{1 j} + v_{2 j} + . . . + v_{nj} = 1 \\ v_{1 j}, v_{2 j}, . . ., v_{nj} \geq 0 \end{matrix} P_{w} = \frac{\sum_{j = 1}^{m} μ_{B} (b_{j}) * P_{j}}{\sum_{j = 1}^{m} μ_{B} (b_{j})}$ (M-2) Where P_j = (v_1j, v_2j, . . . , v_nj) is a n-dimension vector, μ_B (b_i) is membership value of B, P_w is named as the weighted probability mass function of X, and $μ_{B} (b_{i}) / \sum_{i = 1}^{m} μ_{B} (B_{i})$ can be regard as the weight of P_i.

Example 3.2. Consider the same Z-number Z=(A, B) in example 3.1, we can also solve the model(M-2) in MATLAB and obtain a result as follows:

$\begin{matrix} P & = \frac{0.1002}{1} + \frac{0.0944}{2} + \frac{0.0967}{3} + \frac{0.1287}{4} \\ + \frac{0.3840}{5} + \frac{0.0995}{6} + \frac{0.0965}{7} \end{matrix}$

The comparision of probabilities calculated by model(M-2) and original model when b_l takes different value is shown in Fig. 2. As shown in the figure, the probabilities behave huge difference when b_l takes different value from 0 to 1, and the scence indicates the rationality of the probability calculated by model(M-2).

Fig. 2

Comparision of probability between the model(M-2) and original model.

What is deserved to mention is there is no technique to measure the difference of the calculated probability between the modified models and the original models up to now so far. Hence, we can only rely on discriminating the rationality of objective function and constraint conditions in a model. Meanwhile, the calculated probabilities by the two modified models and original models are estimated values since the hidden probability of a Z-number is unknown, therefore, it is normal to obtain different results from different models, though the difference is huge.

4 α-cut set and entropy of Z⁺-numbers

In this section, some new definitions about Z⁺-numbers will be proposed.

4.1 α-cut set of Z⁺-numbers

Firstly, let us recall the purpose that the definition of α-cut set in fuzzy set theory, i.e., to mutually convert fuzzy sets and classical sets, by what we can understand fuzzy set more clearly, and utilize fuzzy set more conveniently and adequately. For a similar purpose, we propose the α-cut set of Z⁺-numbers which converts a Z⁺-number into a classical binary set as follows:

Definition 4.1 (α-cut set of Z⁺-numbers). Assume a Z⁺-number, Z⁺=(μ_A, p_X) and let μ_A (x)→[0,1], α ∈ [0, 1], and $Z_{α}^{+}$ called the α-cut set of Z⁺-number, is denoted by Eq. (4.1):

$Z_{α}^{+} = {(x, p_{X} (x)) | x \in X, μ_{A} (x) \geq α}$ (4.1) Where p (x) means the probability of x ∈ X, μ_A is the membership function of X, and α is named as threshold value.

Property 4.1: ∀α, β ∈ [0, 1], $Z_{1}^{+}$ and $Z_{2}^{+}$ are two Z⁺-numbers, then we have:

(1) $α < β \Rightarrow Z_{1, β}^{+} \subseteq Z_{1, α}^{+},$

(2) $Z_{1}^{+} \subseteq Z_{2}^{+} \Leftrightarrow \forall α \in [0, 1], Z_{1, α}^{+} \subseteq Z_{2, α}^{+} .$

Proof: (1) Assume that α < β, then

$\begin{matrix} \forall x \in X, μ_{A} (x) \geq β > α \\ \Rightarrow & \forall (x_{0}, p_{X} (x_{0})) \in Z_{1, β}^{+}, (x_{0}, p_{X} (x_{0})) \in Z_{1, α}^{+} \\ \Rightarrow & Z_{1, β}^{+} \subseteq Z_{1, α}^{+} \end{matrix}$ (2) (i) Assume that $Z_{1}^{+} \subseteq Z_{2}^{+}$ , then

$\begin{matrix} \forall α \in [0, 1], (x_{0}, p_{X} (x_{0}) \in Z_{1, α}^{+}, \\ \Rightarrow & μ_{A_{1}} (x_{0}) \geq α \\ \Rightarrow & μ_{A_{2}} (x_{0}) \geq α \\ \Rightarrow & (x_{0}, p_{X} (x_{0}) \in Z_{2, α}^{+} . \end{matrix}$ Hence,

$Z_{1, α}^{+} \subseteq Z_{2, α}^{+}$ (ii) Assume that ∀α ∈ [0, 1], $Z_{1, α}^{+} \subseteq Z_{2, α}^{+}$ , if $Z_{1}^{+} ≰ Z_{2}^{+}$ , then

$\begin{matrix} \exists x_{0} \in X, s . t . μ_{A_{1}} (x_{0}) > μ_{A_{2} (x_{0})}, \\ \Rightarrow & \forall α_{0}, s . t . μ_{A_{1}} (x_{0}) \geq α_{0} \geq μ_{A_{2}} (x_{0}), \\ \Rightarrow & (x_{0}, p_{X} (x_{0}) \in Z_{1, α_{0}}^{+} \end{matrix}$ But $(x_{0}, p_{X} (x_{0})) \notin Z_{2, α_{0}}^{+}$ which contraries to the assumption. Hence

$Z_{1}^{+} \subseteq Z_{2}^{+} .$

Based on the definition of α-cut set of Z⁺-numbers, the utility of $Z_{α}^{+}$ is introduced as follows:

Definition 4.2(Utility of $Z_{α}^{+}$ -set). Assume a Z⁺-number, Z⁺ = (μ_A, p_X), the utility of $Z_{α}^{+}$ -set is denoted as U_α (Z⁺),

$U_{α} (Z^{+}) = \frac{\int_{x \in X_{α}} {xp}_{X} (x) dx}{\int_{x \in X_{α}} p_{X} (x) dx}$ (4.2) Where X_α = {x ∈ X|μ_A (x) ≥ α}, α ∈ [0, 1].

In discrete environment, the formula of the utility of $Z_{α}^{+}$ -set can be expressed as follows:

$U_{α} (Z^{+}) = \frac{\sum_{i = 1}^{n} x_{i} p_{X} (x_{i})}{\sum_{i = 1}^{n} p_{X} (x_{i})}$ (4.3) Where $(x_{i}, p_{X} (x_{i})) \in Z_{α}^{+}$ , α ∈ [0, 1], and n is the number of element of the $Z_{α}^{+}$ -set.

According to the definition of utility of $Z_{α}^{+}$ -set, the U_α (Z⁺) is actually measures the weighted average about the cut set of Z⁺-numbers under different fuzzy threshold α.

Example 4.1. Give a Z⁺-number, Z⁺=(μ_A, p_X), which has been calculated in Example 3.2, where

$\begin{matrix} A & = \frac{0}{1} + \frac{0.12}{2} + \frac{0.4}{3} + \frac{0.7}{4} + \frac{1}{5} + \frac{0.45}{6} + \frac{0.1}{7}, \\ P & = \frac{0.1002}{1} + \frac{0.0944}{2} + \frac{0.0967}{3} + \frac{0.1287}{4} \\ + \frac{0.3840}{5} + \frac{0.0995}{6} + \frac{0.0965}{7} . \end{matrix}$ Then when α = 0.3, we can get:

$Z_{0.3}^{+} = {(3, 0.0967), (4, 0.1287), (5, 0.3840), (6, 0.0995)}$

And the variation tendency of the utility of $Z_{α}^{+}$ -set with the changing of α ∈ [0, 1] in example 4.1 is shown in Fig. 1.

4.2 Entropy measurements of Z⁺-numbers

In this subsection, we will develop three kinds of entropy defined on Z⁺-numbers environment.

First of all, let’s recall the structure of Z⁺-numbers in which the first component indicates a possibility distribution and the second component indicates a probability density function. In the rest of this subsection, we suppose that the two components are independent and regard the probability density function as a special possibility distribution.

Definition 4.3(Measurement of fuzziness).A map f : X → [0, 1] is said to be a measure of fuzziness, if it meets the following conditions:

(1) if μ_A (x)(or $p_{X} (x)) = \frac{1}{2}$ , then f (x) =1;

(2) if μ_A (x)(or p_X (x)) =0 or μ_A (x)(or p_X (x)) =1, f (x) =0 ;

(3) ∀x, y ∈ X,if $0 \leq μ_{A} (x) \leq μ_{A} (y) \leq \frac{1}{2}$ or $\frac{1}{2} \leq μ_{A} (y) \leq μ_{A} (x) \leq 1$ , then 0≤ f (x) ≤ f (y) ≤1 ;

(4) ∀x, y ∈ X,if $0 \leq p_{X} (x) \leq p_{X} (x) \leq \frac{1}{2}$ or $\frac{1}{2} \leq p_{X} (x) \leq 1 - p_{X} (y) \leq 1$ ,then 0 ≤ f (x) ≤ f (y) ≤1 .

Where X is the universe, μ_A (x) is the membership function of a fuzzy number A, and p_X (x) is a probability density function of a random variable, X. The four conditions indicate that the closer μ_A (x)(or p_X (x)) is to $\frac{1}{2}$ , the stronger the fuzziness of x ∈ X is.

Measurement of fuzziness has capability to measure the fuzziness(or uncertainty) of an arbitrary variable when its possibility distribution is known. Based of this reason, a new entropy formula to measure the degree of uncertain information for Z⁺-numbers is proposed.

Definition 4.4(Entropy of Z⁺-numbers). Assume a Z⁺-number, Z⁺=(μ_A, p_X), the entropy of Z⁺-numbers is defined as E (Z⁺),

$E (Z^{+}) = \frac{1}{2 ∥ X ∥ ln 2} \int_{x \in X} [ln (1 + f_{1} (x)) + ln (1 + f_{2} (x))] dx$ (4.4)

Where f₁ (x) and f₂ (x) are functions of measure of fuzziness, and f₁ (x) satisfies the conditions (1), (2), (3) in definition 4.3 while f₂ (x) satisfies the conditions (1), (2), (4), ∥X∥ represents the length of the universe X.

In discrete environment, the formula of entropy of Z⁺-numbers can be expressed as follows:

$E (Z^{+}) = \frac{1}{2 n ln 2} \sum_{i = 1}^{n} [ln (1 + f_{1} (x_{i})) + ln (1 + f_{2} (x_{i}))]$ (4.5) Where n represents the number of element in X.

Combining the definition 4.3 and definition 4.4, it is obvious that the purpose of entropy of Z⁺-numbers is measuring the fuzziness of Z⁺-numbers which contain both possibility and probability information. However, the bigger the E (Z⁺) is, the fuzzier the information contained by Z⁺-numbers are.

Suppose that n=1, then the variation tendency of E (Z⁺) with the changing of f₁ (x) and f₂ (x) is shown in Fig. 2.

Specially, in the rest of this paper, the measure function of fuzziness contained in definition 4.4 can be given as follows: $f_{1} (x_{i}) = 2 | μ_{A} (x_{i}) - χ_{\frac{1}{2}} (μ_{A} (x_{i})) |,$ (4.6) $f_{2} (x_{i}) = 2 | p_{X} (x_{i}) - χ_{\frac{1}{2}} (p_{X} (x_{i})) |,$ (4.7) Where $χ_{\frac{1}{2} (x)}$ is an eigenfunction defined as follows:

$χ_{\frac{1}{2}} (x) = {\begin{matrix} 0, & x \in [0, \frac{1}{2}]; \\ 1, & x \in (\frac{1}{2}, 1] . \end{matrix}$

Property 4.2:

(1) 0 ≤ E (Z⁺) ≤1,

(2) E (Z⁺) is monotonously increasing with f₁ (x_i) and f₂ (x_i), respectively.

Proof:

(1) According to definition 4.3, it can be easily found that f₁ (x_i), f₂ (x_i) ∈ [0, 1], i = 1, . . . , n, then, $ln (1 + f_{j} (x_{i})) \in [0, ln 2],$ Where j = 1, 2 and i = 1, 2, . . . , n. Hence,E (Z⁺) ∈ [0, 1] .

(2) We can easily find that ln(1 + f₁ (x_i)) and ln(1 + f₂ (x_i)) are monotonously increasing with f₁ (x_i) and f₂ (x_i) respectively. However, E (Z⁺) is monotonously increasing with f₁ (x_i) and f₂ (x_i), respectively.

For the same objective object expressed by Z⁺-numbers, if the objects are different, their interaction information also has different knowledge acquire, matching, reasoning expression and so on. Different interaction information will directly influence the uncertainty measure of the object itself. Therefore, a cross-entropy formula and a comprehensive cross-entropy formula are put forward to describe their relative uncertain degree of identification information and relative uncertainty measure based on Z⁺-numbers.

Definition 4.5 (Cross-entropy of Z⁺-numbers). Assume two Z⁺-numbers, $Z_{1}^{+}$ =(μ_{A
₁}, p_{X
₁}) with universe X₁ and $Z_{2}^{+}$ =(μ_{A
₂}, p_{X
₂}) with universe X₂, the cross-entropy among Z⁺-numbers is defined as Eq. (4.8). Where f₁₁ (x₁) and f₁₂ (x₂ satisfy the conditions (1), (2), (3) in Definition 4.3 while f₂₁ (x₁) and f₂₂ (x₂) satisfy the conditions (1),(2),(4), ∥X₁∥ and ∥X₂∥ respectively represent the length of universe X₁ and X₂.

In discrete environment, the formula of cross-entropy of Z⁺-numbers can be expressed as Eq.(4.9). Where x₁ ∈ X₁ and x₂ ∈ X₂, and ɛ is a positively infinitesimally small quantity to make sure that the denominator of the formula isn’t zero.

The definition of cross entropy of two Z⁺-numbers is for measuring the difference of fuzziness respect to possibility distribution and probability distribution of two different Z⁺-numbers. Similar to entropy of Z⁺-numbers, the bigger the $CE (Z_{1}^{+}, Z_{2}^{+})$ is, the bigger of the difference of the fuzziness between $Z_{1}^{+}$ and $Z_{2}^{+}$ . Similarly, in the rest of this paper, the measure function of fuzziness contained in definition 4.5 can be given as follows:

$\begin{matrix} f_{11} (x_{i}) = 2 | μ_{A_{1}} (x_{i}) - χ_{\frac{1}{2}} (μ_{A_{1}} (x_{i})) |, \\ f_{12} (x_{j}) = 2 | μ_{A_{2}} (x_{j}) - χ_{\frac{1}{2}} (μ_{A_{2}} (x_{j})) |, \\ f_{21} (x_{i}) = 2 | p_{X_{1}} (x_{i}) - χ_{\frac{1}{2}} (p_{X_{1}} (x_{i})) |, \\ f_{22} (x_{j}) = 2 | p_{X_{2}} (x_{j}) - χ_{\frac{1}{2}} (p_{X_{2}} (x_{j})) |, \end{matrix}$

Suppose that m=1, n=1 and f₂₁ (x)=0(or f₁₁ (x)=0), then the variation tendency of $CE (Z_{1}^{+}, Z_{2}^{+})$ with the changing of f₁₁ (x) and f₁₂ (x)(or with the changing of f₂₁ (x) and f₂₂ (x)) is shown in Fig. 3.

Fig. 3

Variation tendency of Utility with the changing of α.

Property 4.3:

(1) $0 \leq CE (Z_{1}^{+}, Z_{2}^{+}) \leq 1,$

(2) $CE (Z_{1}^{+}, Z_{2}^{+})$ is monotonously increasing with f₁₁ (x_i), f₂₁ (x_i), and monotonously decreasing with f₁₂ (x_i), f₂₂ (x_i), respectively.

Proof:

(2) We firstly proof the second property. Since the two parts of $CE (Z_{1}^{+}, Z_{2}^{+})$ have same form, they have same monotonicity. Hence, we just consider one $C E (Z_{1}^{+}, Z_{2}^{+}) = \frac{1}{2 | | X_{1} | | | | X_{2} | | I n 2} \int_{x_{1} \in X_{1}} \int_{x_{1} \in X_{1}} {\begin{matrix} f_{11} (x_{1}) I n (1 + \frac{f_{11} (x_{1})}{f_{11} (x_{1}) + f_{12} (x_{2})}) \\ + f_{21} (x_{1}) I n (1 + \frac{f_{21} (x_{1})}{f_{21} (x_{1}) + f_{22} (x_{2})}) \end{matrix}} d x_{2} d_{1}$ (4.8) $C E (Z_{1}^{+}, Z_{2}^{+}) = \frac{1}{2 m n I n 2} \sum_{j = 1}^{m} \sum_{i = 1}^{n} (\begin{matrix} f_{11} (x_{i}) I n (1 + \frac{f_{11} (x_{i})}{f_{11} (x_{i}) + f_{12} (x_{j}) + ε}) \\ + f_{21} (x_{i}) I n (1 + \frac{f_{21} (x_{i})}{f_{21} (x_{i}) + f_{22} (x_{j}) + ε}) \end{matrix})$ (4.9)

of them. Obviously, its monotonicity can be proved to be equivalent to the monotonicity of the follow functional equation: $F (x_{1}, x_{2}) = x_{1} ln (1 + \frac{x_{1}}{x_{1} + x_{2} + ɛ}),$ Where x₁, x₂ ∈ [0, 1]. Therefore, the partial derivatives of F (x₁, x₂) are solved as follows:

${\begin{matrix} \frac{\partial F}{\partial x_{1}} = ln \frac{2 x_{1} + x_{2} + ɛ}{x_{1} + x_{2} + ɛ} + \frac{x_{1} (x_{2} + ɛ)}{(2 x_{1} + x_{2} + ɛ) (x_{1} + x_{2} + ɛ)}, \\ \frac{\partial F}{\partial x_{2}} = \frac{- x_{1}}{(2 x_{1} + x_{2} + ɛ) (x_{1} + x_{2} + ɛ)} . \end{matrix}$ Since x₁, x₂ ∈ [0, 1] and ɛ > 0, we can easily find that $\frac{\partial F}{\partial x_{2}} \leq 0$ , i.e., F (x₁, x₂) is monotonicity decreasing with x₂. Next, we continue consider the second partial derivative about x₁ on $\frac{\partial F}{\partial x_{1}}$ :

$\frac{\partial^{2} F}{\partial x_{1}^{2}} = \frac{(x_{2} + ɛ) (2 x_{2}^{2} + 3 x_{1} x_{2} + ɛ)}{(x_{1} + x_{2} + ɛ)^{2} (2 x_{1} + x_{2} + ɛ)^{2}} \geq 0$ Hence, $\frac{\partial F}{\partial x_{1}}$ is monotonicity decreasing with x₁, meanwhile,

$min (\frac{\partial F}{\partial x_{1}}) = \frac{\partial F}{\partial x_{1}} |_{x_{1} = 0} = 0$ Hence

$\frac{\partial F}{\partial x_{1}} \geq min (\frac{\partial F}{\partial x_{1}}) = 0$ However, F (x₁, x₂) is monotonicity increasing with x₁. Equivalently, $CE (Z_{1}^{+}, Z_{2}^{+})$ is monotonously increasing with f₁₁ (x_i), f₂₁ (x_i), and monotonously decreasing with f₁₂ (x_i), f₂₂ (x_i), respectively.

(1) Next, we proof the first property. According to the second property, we can easily find that

$0 \leq f_{t 1} (x_{i}) ln \frac{f_{t 1} (x_{i})}{f_{t 1} (x_{i}) + f_{t 2} (x_{j})} \leq ln 2$ Where t=1,2,i=1,2,...,n and j=1,2,...,m. Hence, $0 \leq CE (Z_{1}^{+}, Z_{2}^{+}) \leq 1$ is obvious.

In realistic multi-attribute decision-making problem, there usually have more than two alternatives, and we should consider not only the merits and demerits of the alternative itself, but also the relative uncertainty of measurements between alternatives. Based on this view, a kind of comprehensive cross-entropy among Z⁺-numbers, to measure the uncertainty between one alternative and other alternatives on a same attribute, is introduced.

Definition 4.6 (Comprehensive cross-entropy of Z⁺-numbers). Assume n alternatives which are represented by $Z_{i}^{+}$ , i=1,...,n on a common attribute, the comprehensive cross-entropy of $Z_{i}^{+}$ is defined as: $C C E (Z_{i}^{+}) = \frac{1}{2 (n - 1)} \sum_{\begin{matrix} j = 1 \\ i \neq j \end{matrix}}^{n} [C E (Z_{i}^{+}, Z_{j}^{+}) + C E (Z_{j}^{+}, Z_{i}^{+})]$ (4.10)

Property 4.4: $0 \leq CCE (Z_{i}^{+}) \leq 1$ .

Proof: According to Property 5.3, the conclusion can be easily proved.

The effect of comprehensive cross entropy of Z⁺-numbers is measuring the total differece between a Z⁺-number, $Z_{i}^{+}$ , and a series of Z⁺-numbers, $Z_{j}^{+}$ , j=1,...,n, i ≠ j in a set represented by Z⁺-numbers. However, the bigger the $CCE (Z_{i}^{+})$ is, the bigger the total difference of a Z⁺-number respect to a Z⁺-number set.

Example 4.2. Give two Z⁺-numbers, $Z_{1}^{+} = (A_{1}, R_{1})$ and $Z_{2}^{+} = (A_{1}, R_{2})$ which have been calculated in example 3.1 and example 3.2 respectively, where

$\begin{matrix} A_{1} & = \frac{0}{1} + \frac{0.12}{2} + \frac{0.4}{3} + \frac{0.7}{4} + \frac{1}{5} + \frac{0.45}{6} + \frac{0.1}{7}, \\ R_{1} & = \frac{0}{1} + \frac{0}{2} + \frac{0.12}{3} + \frac{0.5384}{4} + \frac{0}{5} + \frac{0.3416}{6} + \frac{0}{7}, \\ R_{2} & = \frac{0.1002}{1} + \frac{0.0944}{2} + \frac{0.0967}{3} + \frac{0.1287}{4} \\ + \frac{0.3840}{5} + \frac{0.0995}{6} + \frac{0.0965}{7} \end{matrix}$ What’s more, Eq.(4.6) and Eq.(4.7) are employed to be the fuzziness measurement functions. Then, the fuzziness measurements of A₁, R₁ and R₂ when X takes different value are as follows,

$\begin{matrix} A_{1}^{F} & = \frac{0}{1} + \frac{0.24}{2} + \frac{0.8}{3} + \frac{0.6}{4} + \frac{0}{5} + \frac{0.9}{6} + \frac{0.2}{7}, \\ R_{1}^{F} & = \frac{0}{1} + \frac{0}{2} + \frac{0.24}{3} + \frac{0.9232}{4} + \frac{0}{5} + \frac{0.6832}{6} + \frac{0}{7}, \\ R_{2}^{F} & = \frac{0.2004}{1} + \frac{0.1888}{2} + \frac{0.1934}{3} + \frac{0.2574}{4} \\ + \frac{0.7680}{5} + \frac{0.1990}{6} + \frac{0.1930}{7} \end{matrix}$

And the entropy of $Z_{1}^{+}$ and $Z_{2}^{+}$ can be respectively calculated by Eq.(4.5) as follows,

$E (Z_{1}^{+}) = 0.3593, E (Z_{2}^{+}) = 0.3902$

And the cross entropy of $Z_{1}^{+}$ and $Z_{2}^{+}$ can be calculated by Eq.(4.9) as follows,

$CE (Z_{1}^{+}, Z_{2}^{+}) = 0.2516, CE (Z_{2}^{+}, Z_{1}^{+}) = 0.3467$

And the comprehensive cross entropy of $Z_{1}^{+}$ and $Z_{2}^{+}$ can be calculated by Eq.(4.10) as follows,

$CCE (Z_{1}^{+}) = CCE (Z_{2}^{+}) = 0.2992$

In a multi-attribute decision-making problem, we always hope that the uncertainty and fuzziness of an alternative are as small as possible while the criterion value is as big as possible. According to this view, a linear programming method based on the principle of maximum deviation to determine weights of criteria combined with entropy of Z⁺-numbers and comprehensive cross-entropy of Z⁺-numbers.

Assume an alternative set S = {s₁, s₂, . . . , s_n}, and a criteria set C = {c₁, c₂, . . . , c_m}, then the importance weight of each criteria can be solved by a lineal programming model defined as follows: $\max \sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | \begin{matrix} \frac{U (Z_{ij}^{+})}{E (Z_{ij}^{+}) + CCE (Z_{ij}^{+})} - \\ \frac{U (Z_{kj}^{+})}{E (Z_{kj}^{+}) + CCE (Z_{kj}^{+})} \end{matrix} | w_{j} s . t . \sum_{j = 1}^{m} w_{j}^{2} = 1, w_{j} \geq 0$ (M-3) Where w_j represents importance weight of the j-th criterion, $Z_{ij}^{+}$ represents the value of the i-th alternative under the j-th criterion,and $U (Z_{ij}^{+})$ , $E (Z_{ij}^{+})$ , $CCE (Z_{ij}^{+})$ respectively represent corresponding criterion value, entropy and comprehensive cross-entropy.

5 PROMETHEE Method under Z⁺-numbers environment

In this section, we will present the steps of the PROMETHEE method combined with Z⁺-numbers.

PRPMETHEE method(denoted as P-method for convenience) is an outranking method based on a multi-criteria decision-making problem. By introducing the preference function which can map the difference of criterion values of two alternatives to 0-1 interval, P-method can compare the preference degree of each pair of alternatives. What’s more, the operation of P-method is extremely sample that P-method just requires two additional types of information, i.e., criterion value of each alternative and weight value of each attribute.

Let S={s₁, s₂, . . . , s_n} be the alternative set, C={c₁, c₂, . . . , c_m} be the criteria set, W={w₁, w₂, . . . , w_m} be the weight set of each criterion, and Z_ij = (A_ij, B_ij) be the value of the i-th alternative under the j-th criterion where Z_ij is a Z-number and i=1,2,...,n, j=1,2,...,m, then the basic steps of P-method combined with Z⁺-numbers are as follows:

Step 1: According to model(M-1)or model(M-2), convert Z-numbers, Z_ij, to Z⁺-numbers $Z_{ij}^{+}$ .

Step 2: Give a threshold value α ∈ [0, 1], calculate the utility value of $Z_{ij}^{+}$ based on Eq.(4.2), and denote it as $U_{α} (Z_{ij}^{+})$ .

Step 3: Calculate the difference between two alternatives with the given threshold value α:

$d_{ik}^{j} = U_{α} (Z_{ij}^{+}) - U_{α} (Z_{kj}^{+})$ (5.1) Where $U_{α} (Z_{ij}^{+})$ represents the criterion value of the i-th alternative on the j-th criteria, and i, k=1,2,...,n, j=1,2,...,m, i ≠ k. If $U_{α} (Z_{ij}^{+})$ is better than $U_{α} (Z_{kj}^{+})$ , then $d_{ik}^{j}$ > 0, i.e., $U_{α} (Z_{ij}^{+})$ > $U_{α} (Z_{kj}^{+})$ ; and if $U_{α} (Z_{ij}^{+})$ is indifference with $U_{α} (Z_{kj}^{+})$ , then $d_{ik}^{j}$ =0, i.e., $U_{α} (Z_{ij}^{+})$ = $U_{α} (Z_{kj}^{+})$ .

Step 4: Give a preference function $P (d_{ik}^{j})$ to map $d_{ik}^{j}$ into 0-1 interval. More concretely, $P (d_{ik}^{j})$ actually represents the preference degree of two alternatives under a same criterion, and the preference function is as follows:

(1)if c_j is a benefit attribute, then $P (x) = {\begin{matrix} 0, & if x \leq 0; \\ 1 - {exp}^{- \frac{x^{2}}{2 {exp}^{2}}}, & if x > 0 . \end{matrix}$ (5.2)

(2)if c_j is a cost attribute, then $P (x) = {\begin{matrix} 0, & if x \geq 0; \\ 1 - {exp}^{- \frac{x^{2}}{2 {exp}^{2}}}, & if x < 0 . \end{matrix}$ (5.3)

Step 5: According to Eq.(4.4), Eq.(4.8), Eq.(4.9) and model(M-3), calculate the weight matrix W={w₁, w₂, . . . , w_m} of each criterion under the threshold value α.

Step 6: Calculate the aggregated preference value taking into account the criteria weights: $π (s_{i}, s_{k}) = [\sum_{j = 1}^{m} w_{j} P (d_{ik}^{j})] / \sum_{j = 1}^{m} w_{j}$ (5.4)

Step 7: Calculate the leaving(positive) flow, entering(neg- ative) flow, and net flow for i-th alternative, respectively, $ϕ^{+} (s_{i}) = \frac{1}{n - 1} \sum_{j = 1}^{n} π (s_{i}, s_{j}) (i \neq j)$ (5.5) $ϕ^{-} (s_{i}) = \frac{1}{n - 1} \sum_{j = 1}^{n} π (s_{j}, s_{i}) (i \neq j)$ (5.6) $ϕ (s_{i}) = ϕ^{+} (s_{i}) - ϕ^{-} (s_{i})$ (5.7)

Step 8: Rank the alternatives:

(1) s_iP⁺s_j, if ϕ⁺ (s_i) > ϕ⁺ (s_j),

(2) s_iI⁺s_j, if ϕ⁺ (s_i) = ϕ⁺ (s_j),

(3) s_iP^-s_j, if ϕ^- (s_i) < ϕ^- (s_j),

(4) s_iI^-s_j, if ϕ^- (s_i) = ϕ^- (s_j).

Where s_iP⁺s_j represents that s_i is dominates s_j based on leaving flow, s_iI⁺s_j represents that s_i is indifferent to s_j based on leaving flow, s_iP^-s_j represents that s_i is dominates s_j based on entering flow, and s_iI^-s_j represents that s_i is indifferent to s_j based on entering flow.

The PROMETHEE 1 proposed a partial ranking of alternatives:

(1) s_iP¹s_j, if s_iP⁺s_j and s_iP^-s_j, or if s_iP⁺s_j and s_iI⁺s_j, or if s_iP^-s_j and s_iI^-s_j,

(2) s_iI¹s_j, if s_iI⁺s_j and s_iI^-s_j,

(3) s_iR¹s_j, others

Where s_iP¹s_j represents that s_i is dominates s_j, s_iI¹s_j represents that s_i is indifferent to s_j and s_iR¹s_j represents that the relation between s_i and s_j is incomparable.

The PROMETHEE 2 proposed a complete ranking of alternatives:

(1) s_iP²s_j, if ϕ_{s
_i} > ϕ_{s
_j},

(2) s_iI²s_j, if ϕ_{s
_i} = ϕ_{s
_j}.

Where s_iP²s_j represents that s_i is dominates s_j, and s_iI²s_j represents that s_i is indifferent to s_j.

The procedure of PROMETHEE Method combined with Z⁺-numbers can be represented as Fig. 6.

Fig. 4

Variation tendency of E (Z⁺) with the changing of f₁ (x) and f₂ (x) when n = 1.

Fig. 5

Variation tendency of $CE (Z_{1}^{+}, Z_{2}^{+})$ with the changing of f₁₁ (x)(or f₂₁ (x)) and f₁₂ (x)(or f₂₂ (x)) when m, n=1.

Fig. 6

Procedure of PROMETHEE Method based on Z⁺-numbers environment.

6 An application

With the fast-growing economy, Chinese vehicle population(mainly diesel and petrol cars) has dramatically increased in the past decades. As a consequence, air pollution, vehicle emission and oil consumption get increasingly serious. Hence, the development of new energy vehicles gets more and more significant. However, the electricity vehicle(EV) is an important component among all of new energy vehicles, for electricity is one of commonest energy in reality and can improve energy security and minimize greenhouse gas emission. In order to promoting the adoption of EV in China, the design of EV charge station(DEVCS) is extremely important. However, DEVCS problem is influenced by quantitative and qualitative factors.

To determine an optimal and acceptable EV charge station, there are eight alternatives of DEVCS problem denoted by S={s₁, s₂, . . . , s₈}. And all of alternatives are influenced by five criteria denoted by C={c₁, c₂, c₃, c₄, c₅}, where c₁ refers to charging time, c₂ refers to conversion efficiency of battery electric energy, c₃ refers to total compatibility volume of available motorcycle type, c₄ refers to installation and operating costs, including maintenance and electricity costs and c₅ refers to risk of environment hazards to surrounding areas. What’s more, c₁, c₄, c₅ are cost criteria, and c₂, c₃ are benefit criteria. The weight vector of five criteria, denoted by W=(w₁, w₂, w₃, w₄, w₅), are completely unknown. The criterion values for alternative s_i under criteria c₁, c₂, c₃, c₄ are displayed as Z-numbers, while under criterion c₅ are displayed as real values, which are shown in Table 1.

Step 1: According to Model(M-2), convert Z_ij=(A_ij, B_ij) to $Z_{ij}^{+}$ =(A_ij, R_ij), i = 1, 2, . . . , 8,j = 1, 2, 3, 4, and the results are shown in Table 2 and the detail values of probability mass function are shown in Table A1 of appendix.

Table 2
Preliminary assessment of eight candicate DEVCSs

c ₁ c ₂ c ₃ c ₄ c ₅

s ₁ (A₁₁, B₁₁) (A₁₂, B₁₂) (A₁₃, B₁₃) (A₁₄, B₁₄) 1

s ₂ (A₂₁, B₂₁) (A₂₂, B₂₂) (A₂₃, B₂₃) (A₂₄, B₂₄) 1

s ₃ (A₃₁, B₃₁) (A₃₂, B₃₂) (A₃₃, B₃₃) (A₃₄, B₃₄) 2

s ₄ (A₄₁, B₄₁) (A₄₂, B₄₂) (A₄₃, B₄₃) (A₄₄, B₄₄) 3

s ₅ (A₅₁, B₅₁) (A₅₂, B₅₂) (A₅₃, B₅₃) (A₅₄, B₅₄) 1

s ₆ (A₆₁, B₆₁) (A₆₂, B₆₂) (A₆₃, B₆₃) (A₆₄, B₆₄) 3

s ₇ (A₇₁, B₇₁) (A₇₂, B₇₂) (A₇₃, B₇₃) (A₇₄, B₇₄) 2

s ₈ (A₈₁, B₈₁) (A₈₂, B₈₂) (A₈₃, B₈₃) (A₈₄, B₈₄) 2

	c ₁	c ₂	c ₃	c ₄	c ₅
s ₁	(A₁₁, B₁₁)	(A₁₂, B₁₂)	(A₁₃, B₁₃)	(A₁₄, B₁₄)	1
s ₂	(A₂₁, B₂₁)	(A₂₂, B₂₂)	(A₂₃, B₂₃)	(A₂₄, B₂₄)	1
s ₃	(A₃₁, B₃₁)	(A₃₂, B₃₂)	(A₃₃, B₃₃)	(A₃₄, B₃₄)	2
s ₄	(A₄₁, B₄₁)	(A₄₂, B₄₂)	(A₄₃, B₄₃)	(A₄₄, B₄₄)	3
s ₅	(A₅₁, B₅₁)	(A₅₂, B₅₂)	(A₅₃, B₅₃)	(A₅₄, B₅₄)	1
s ₆	(A₆₁, B₆₁)	(A₆₂, B₆₂)	(A₆₃, B₆₃)	(A₆₄, B₆₄)	3
s ₇	(A₇₁, B₇₁)	(A₇₂, B₇₂)	(A₇₃, B₇₃)	(A₇₄, B₇₄)	2
s ₈	(A₈₁, B₈₁)	(A₈₂, B₈₂)	(A₈₃, B₈₃)	(A₈₄, B₈₄)	2

Note: The detail value of (A_ij, B_ij) is shown in Table A1 in [18].

Step 2: Let α be 0.3, and calculate the utility value of $Z_{ij}^{+}$ under the threshold value α based on Eq.(4.1) and Eq.(4.3). And let the criterion values for a_i under c₅ be the corresponding utility values,which are shown in Table 3.

Table 3

Result of Z-numbers converting to Z⁺-numbers

	c ₁	c ₂	c ₃	c ₄
s ₁	(A₁₁, R₁₁)	(A₁₂, R₁₂)	(A₁₃, R₁₃)	(A₁₄, R₁₄)
s ₂	(A₂₁, R₂₁)	(A₂₂, R₂₂)	(A₂₃, R₂₃)	(A₂₄, R₂₄)
s ₃	(A₃₁, R₃₁)	(A₃₂, R₃₂)	(A₃₃, R₃₃)	(A₃₄, R₃₄)
s ₄	(A₄₁, R₄₁)	(A₄₂, R₄₂)	(A₄₃, R₄₃)	(A₄₄, R₄₄)
s ₅	(A₅₁, R₅₁)	(A₅₂, R₅₂)	(A₅₃, R₅₃)	(A₅₄, R₅₄)
s ₆	(A₆₁, R₆₁)	(A₆₂, R₆₂)	(A₆₃, R₆₃)	(A₆₄, R₆₄)
s ₇	(A₇₁, R₇₁)	(A₇₂, R₇₂)	(A₇₃, R₇₃)	(A₇₄, R₇₄)
s ₈	(A₈₁, R₈₁)	(A₈₂, R₈₂)	(A₈₃, R₈₃)	(A₈₄, R₈₄)

Note: The detail value of R_ij is shown in Table A1 in section appendix.

Step 3: Calculate the difference between two alternatives using Eq.(5.1) and map the difference into 0-1 interval using preference function displayed as Eq.(5.2) and Eq.(5.3). The preference degrees that the alternative s_i relatives to s_j are shown in Table A2 of appendix.

Step 4: According to Eq.(4.4), Eq.(4.8), Eq.(4.9) and Table 2, respectively calculate the entropy and comprehensive cross-entropy of Z⁺-numbers, and the results are shown in Table 4.

Table 4

Utility value of Z⁺-numbers when α = 0.3

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	s ₈
c ₁	43.3019	36.7808	39.1054	39.0192	39.0950	38.6153	40.2448	43.2751
c ₂	87.6051	89.3797	83.7944	84.4774	84.7828	86.5925	82.6228	85.4133
c ₃	34.9574	34.9964	29.7199	28.9005	34.3738	28.6815	31.2109	34.1342
c ₄	1.6458	1.7414	1.7860	1.6888	1.8695	1.7260	2.0395	1.5976
c ₅	1	1	2	3	1	3	2	2

Step 5: For the criterion values of eight alternatives under the criterion c₅ are real values denoted as U_i5, i=1,2,...,8, which are certain information, the model(M-3) can’t straightly calculate the weight vector of five criteria. Therefore, the model(M-3) can be adjusted as follows: $\max \begin{matrix} \sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | \begin{matrix} \frac{U (Z_{ij}^{+})}{E (Z_{ij}^{+}) + CCE (Z_{ij}^{+})} - \\ \frac{U (Z_{kj}^{+})}{E (Z_{kj}^{+}) + CCE (Z_{kj}^{+})} \end{matrix} | w_{j} \\ + \sum_{i = 1}^{8} \sum_{k = 1}^{8} | U_{i 5} - U_{k 5} | w_{5} \end{matrix} s . t . \sum_{j = 1}^{m} w_{j}^{2} = 1, w_{j} \geq 0$ (M-4) And the result of the weight vector under the threshold value α = 0.3 is

$W_{0.3} = [0.3801, 0.7361, 0.3372, 0.0041, 0.0109] .$

Step 6: Accord to Eq.(5.4), Eq.(5.5), Eq.(5.6) and Eq.(5.7), respectively calculate the aggregated preference value, the leaving flow,entering flow end net flow of eight alternatives under threshold value α=0.3, the result are shown in Table 5.

Table 5

Entropy and comprehensive cross-entropy of eight candidate DEVCSs for first four criteria

	Entropy				Comprehensive cross-entropy
	c ₁	c ₂	c ₃	c ₄	c ₁	c ₂	c ₃	c ₄
s ₁	0.4503	0.4279	0.4257	0.2775	0.2314	0.2323	0.2275	0.1992
s ₂	0.2507	0.3496	0.3580	0.4203	0.2017	0.2214	0.2149	0.2171
s ₃	0.4800	0.3843	0.4147	0.4871	0.2369	0.2237	0.2225	0.2377
s ₄	0.4146	0.4569	0.4355	0.3200	0.2211	0.2405	0.2266	0.2070
s ₅	0.3508	0.5014	0.3628	0.3766	0.2071	0.2485	0.2139	0.2091
s ₆	0.4422	0.4158	0.3439	0.3586	0.2252	0.2320	0.2123	0.2033
s ₇	0.3560	0.4400	0.3882	0.3385	0.2104	0.2363	0.2157	0.2037
s ₈	0.4559	0.4252	0.4901	0.4358	0.2290	0.2293	0.2412	0.2209

Step 7: Rank the eight alternatives by PROMETHEE 2t1 method: $s_{2} ≻ s_{1} ≻ s_{5} ≻ s_{6} ≻ s_{8} ≻ s_{4} ≻ s_{3} ≻ s_{7}$

Similarly, we can obtain the ranking relation of the eight candidate DEVCSs when α get other values which are shown as Table 8.

The net flow, leaving flow and entering flow of eight candidate DEVCSs and their relative ranking tendency under different threshold values α are shown in Fig. 7.

Fig. 7

Net flow,leaving flow and entering flow of each alternative under different α.

7 Feasibility analysis and discussion

In this section, we will demonstrate the feasibility of the P-method combined with Z⁺-numbers from three aspect.

Aspect I: In the P-method combined with Z⁺-num- bers, the optimal alternative won’t be changed when replace a non-optimal alternative by a worse alternative.

Let a non-optimal alternative s₁ be replaced by a worse alternative $s_{1}^{'}$ with the threshold value α = 0.3, i.e., suppose that the utility of $s_{1}^{'}$ under the five criteria are 43.3019, 82.6228, 28.6815, 2.0395 and 3 respectively. Using the same steps with unchanged weight vector of criteria in section 6, we can calculate the net flow of each alternatives as [-0.6561, 0.9718, -0.1926, -0.1499, 0.2120, 0.1290, -0.2945, -0.0198], and the ranking relation of eight alternatives is $s_{2} ≻ s_{5} ≻ s_{6} ≻ s_{8} ≻ s_{4} ≻ s_{3} ≻ s_{7} ≻ s_{1}^{'}$ by PROMETHEE 2 method. Thus, the relative position of eight alternatives in this ranking order is as same as the original problem in section 4 with α=0.3 except alternative $s_{1}^{'}$ ,and $s_{1}^{'}$ is exactly the last one in this ranking order. Similary, the same result can be obtained for other six non-optimal alternatives. Hence, the optimal alternative won’t be changed when replacing a non-optimal alternative by a correspondingly worse alternative in the P-method combined with Z⁺-numbers.

Aspect II: The P-method combined with Z⁺-num- bers has a transitive property, i.e., if s_i ≻ s_k and s_k ≻ s_l, then s_i ≻ s_l. The transitive property can be easily find by Step 8 in section 5.

Aspect III: Comparative analysis.

To demonstrate the validity of the P-method combine with Z⁺-numbers, we conducted a comparative analysis against the two methods in reference [19] in which we used same data. The comparative result is shown in Table 7.

Accord to the comparison in Table 7 and combined with the ranking results in Table 6, we can easily find that the three methods have same optimal alternative S₂. When α=0.2 and α=0.3, the first three alternative in our ranking result are also same as SMAA method and SMAA-2 method proposed in reference [19], i.e., s₂,s₁ and s₅. What is different to the two methods proposed in [19] is that the P-method combined with Z⁺-numbers provides more ranking schemes to decision-maker, so that the decision-maker can choose one from these schemes based on their own concrete requirement. What’s more,the introduced method makes the decision more flexible, i.e., we have a series of alternative schemes to ensure that they can adjust the original scheme immediately and reduce decision-maker’s loss, when decision-maker find that the original scheme is unbefitting because of some unexpected factors which will influence the practical decision, for instance, the change of government policies.

Table 6
Leaving flow,entering flow and net flow of eight candidate DEVCSs under α = 0.3

s ₁ s ₂ s ₃ s ₄ s ₅ s ₆ s ₇ s ₈

Leaving flow 0.2984 0.5896 0.0720 0.0813 0.1838 0.1889 0.0652 0.1398

Entering flow 0.1711 0.0000 0.2790 0.2553 0.1241 0.1658 0.3469 0.2768

Net flow 0.1273 0.5896 -0.2069 -0.1740 0.0596 0.0231 -0.2817 -0.1369

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	s ₈
Leaving flow	0.2984	0.5896	0.0720	0.0813	0.1838	0.1889	0.0652	0.1398
Entering flow	0.1711	0.0000	0.2790	0.2553	0.1241	0.1658	0.3469	0.2768
Net flow	0.1273	0.5896	-0.2069	-0.1740	0.0596	0.0231	-0.2817	-0.1369

Table 7

Ranking relation of eight candidate DEVCSsunder different α

α	Ranking relation
0.1	s₂ ≻ s₅ ≻ s₁ ≻ s₈ ≻ s₇ ≻ s₃ ≻ s₄ ≻ s₆
0.2	s₂ ≻ s₁ ≻ s₅ ≻ s₆ ≻ s₇ ≻ s₈ ≻ s₄ ≻ s₃
0.3	s₂ ≻ s₁ ≻ s₅ ≻ s₆ ≻ s₈ ≻ s₄ ≻ s₃ ≻ s₇
0.4	s₂ ≻ s₆ ≻ s₁ ≻ s₅ ≻ s₈ ≻ s₃ ≻ s₄ ≻ s₇
0.5	s₂ ≻ s₆ ≻ s₁ ≻ s₄ ≻ s₅ ≻ s₈ ≻ s₃ ≻ s₇
0.6	s₂ ≻ s₆ ≻ s₄ ≻ s₁ ≻ s₅ ≻ s₃ ≻ s₇ ≻ s₈
0.7	s₂ ≻ s₆ ≻ s₁ ≻ s₄ ≻ s₅ ≻ s₃ ≻ s₇ ≻ s₈
0.8	s₂ ≻ s₆ ≻ s₄ ≻ s₁ ≻ s₅ ≻ s₃ ≻ s₈ ≻ s₇
0.9	s₂ ≻ s₆ ≻ s₄ ≻ s₅ ≻ s₁ ≻ s₃ ≻ s₇ ≻ s₈

Table 8

Ranking results acquired by different methods

Methods	Ranking results
SMAA method in [19]	s₂ ≻ s₁ ≻ s₅ ≻ s₇ ≻ s₈ ≻ s₆ ≻ s₄ ≻ s₃
SMAA-2 method in [19]	s₂ ≻ s₁ ≻ s₅ ≻ s₇ ≻ s₈ ≻ s₆ ≻ s₄ ≻ s₃
Our method(α = 0.2)	s₂ ≻ s₁ ≻ s₅ ≻ s₆ ≻ s₇ ≻ s₈ ≻ s₄ ≻ s₃
Our method(α = 0.3)	s₂ ≻ s₁ ≻ s₅ ≻ s₆ ≻ s₈ ≻ s₄ ≻ s₃ ≻ s₇

Since the optimal alternatives of the proposed application are s₂ under all α from 0.1 to 0.9 with step 0.1, there is no selection about α. But how to choose a reasonable value of threshold value of α is still a key problem, for in other case, the optimal alternative may be different under different α, hence choosing a value of α which only depends on experience or preference is unscientific.

8 Conclusion

This paper introduced and used Z⁺-numbers to apply to multi-criteria decision-making problems in which the data is represented by Z-numbers. After reviewing the drawbacks of two existing linear programming model to obtain the probability mass function of Z⁺-numbers, two improved probability model are proposed. Then the α-cut set of Z⁺-numbers and the utility function of $Z_{α}^{+}$ are introduced to make sure that we can apply Z⁺-numbers to decision-making problems. Meanwhile,three kinds of entropy of Z⁺-numbers are developed to measure the uncertainty and fuzziness of Z⁺-numebers information.Furthermore, a linear programming model is designed to obtain a weight vector of each criterion in decision-making problem. Finally, we briefly introduced the PROMETHEE method and combined the PROMETHEE method with Z⁺-numbers and applied the new PROMETHEE method to a practical decision-making problem to indicate the validity of the method.

The main contributions of the study can be summarized as three points.

Two existing linear programming model of Z-numbers are improved to make the probability mass function of Z⁺-numbers more convincing.

Three kinds of entropy are proposed to adequately measure the uncertainty of a Z⁺-numebr.

We combine PROMETHEE method with Z⁺-numbers to provide with a series of decision-making schemes and make the decision process more flexible.

Footnotes

Appendix

Table A2

Preference value of s_i relatives to s_j when α = 0.3

		s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	s ₈
c ₁	s ₁	0	0	0	0	0	0	0	0
	s ₂	0.9347	0	0.3063	0.2875	0.3040	0.2037	0.5560	0.9424
	s ₃	0.6963	0	0	0	0	0	0.0841	0.6916
	s ₄	0.7110	0	0.0005	0	0.0004	0	0.0966	0.7064
	s ₅	0.6981	0	0	0	0	0	0.0856	0.6934
	s ₆	0.7738	0	0.0161	0.0110	0.0154	0	0.1645	0.7699
	s ₇	0.4687	0	0	0	0	0	0	0.4628
	s ₈	0.4891	0	0	0	0	0	0	0

c ₂	s ₁	0	0	0.6257	0.4842	0.4167	0.0670	0.8136	0.2775
	s ₂	0.1919	0	0.8789	0.8033	0.7607	0.4088	0.9545	0.6551
	s ₃	0	0	0	0	0	0	0.0887	0
	s ₄	0	0	0.0311	0	0	0	0.2077	0
	s ₅	0	0	0.0640	0.0063	0	0	0.2707	0
	s ₆	0	0	0.4113	0.2612	0.1988	0	0.6557	0.0898
	s ₇	0	0	0	0	0	0	0	0
	s ₈	0	0	0.1625	0.0576	0.0265	0	0.4096	0

c ₃	s ₁	0	0	0.8437	0.9165	0.0228	0.9304	0.6132	0.0448
	s ₂	0.0001	0	0.8480	0.9191	0.0259	0.9327	0.6208	0.0491
	s ₃	0	0	0	0.0444	0	0.0704	0	0
	s ₄	0	0	0	0	0	0.0032	0	0
	s ₅	0	0	0.7691	0.8683	0	0.8884	0.4918	0.0039
	s ₆	0	0	0	0	0	0	0	0
	s ₇	0	0	0.1397	0.3032	0	0.3514	0	0
	s ₈	0	0	0.7325	0.8433	0	0.8633	0.4391	0

c ₄	s ₁	0	0.0006	0.0013	0.0001	0.0034	0.0004	0.0104	0
	s ₂	0	0	0.0001	0	0.0011	0	0.0060	0
	s ₃	0	0	0	0	0.0005	0	0.0043	0
	s ₄	0	0.0002	0.0006	0	0.0022	0.0001	0.0083	0
	s ₅	0	0	0	0	0	0	0.0020	0
	s ₆	0	0	0.0002	0	0.0014	0	0.0066	0
	s ₇	0	0	0	0	0	0	0	0
	s ₈	0.0002	0.0014	0.0024	0.0006	0.0050	0.0011	0.0131	0

c ₅	s ₁	0	0	0.0654	0.2371	0	0.2371	0.0654	0.0654
	s ₂	0	0	0.0654	0.2371	0	0.2371	0.0654	0.0654
	s ₃	0	0	0	0.0654	0	0.0654	0	0
	s ₄	0	0	0	0	0	0	0	0
	s ₅	0	0	0.0654	0.2371	0	0.2371	0.0654	0.0654
	s ₆	0	0	0	0	0	0	0	0
	s ₇	0	0	0	0.0654	0	0.0654	0	0
	s ₈	0	0	0	0.0654	0	0.0654	0	0

Acknowledgment

The authors are highly thankful to any anonymous referee for their careful reading and insightful comments, and their constructive opinions and suggestions will generate an improved version of this article. The work is supported by the National Nature Science Foundation of China (No.61972437, No.71871001), the National Nature Science Foundation for Young of China (No.61806001), Anhui Provincial Natural Science Foundation of China, and the Project of Graduate Academic Innovation of Anhui University.

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The novel entropy measurements of Z + -numbers and their application on multi-attribute decision making problem

Abstract

Keywords

1 Introduction

2 Preliminary

3.1 Introduction of Z+-numbers

3.2 Two improved probability models

4.1 α-cut set of Z+-numbers

Footnotes

Appendix

Acknowledgment

References

3.1 Introduction of Z⁺-numbers

4.1 α-cut set of Z⁺-numbers