Enhancing Cyber Defense Strategies with Discrete Multidimensional Z-Numbers: A Multi-Attribute Decision-Making Approach

With the rapid development of intelligent logistics and network environment, it has become an urgent problem to efficiently and accurately handle and analyse the huge amount of uncertain decision information. Traditional decision making methods often fail to make the best use of complex and incomplete information, especially in the field of cyber defence. To address these problems, this paper introduces a new mathematical tool, discrete multidimensional Z-numbers (MZs), for expressing and dealing with uncertainty and reliability in network defence decisions. In this paper, we first introduce the synthesis method of discrete multidimensional Z-numbers, which allows us to consider the uncertainty and reliability of multi-source information in network defence strategy. Then, the hidden probability of MZs is calculated by using the hidden probability model, and the multidimensional Z+−number is proposed, which improves the expressiveness of the model in dealing with information uncertainty. Based on MZs, we define a variety of utility functions and build a multi-attribute group decision framework for network defence around these functions. The MZs provides a novel perspective for analysing and designing response strategies, especially against highly adaptive and covert attacks. The method in this paper is verified by a case of network security assessment of an intelligent logistics company. The results show that this method can significantly improve the accuracy and efficiency of decision making, demonstrating its unique advantages and wide application potential in network defence. This integrated discrete multidimensional Z-number group decision method is not only suitable for intelligent logistics decision making, but also provides an efficient decision support tool for network defence, which helps to improve the intelligence and adaptability of network security defence.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

How to express uncertain information in the decision problem? Zadeh[1] presented Fuzzy Sets theory in 1965, which playing an important role in decision-making. Then, Atanassov[2] presented intuitionistic fuzzy sets (IFSs) that consider membership degree, non-membership degree, and hesitation. Torra[3] proposed hesitant fuzzy sets, which express hesitant information. Zadeh[4] defined type-2 fuzzy sets for representing uncertain flexibly. Liang et.al[5] showed the interval type-2 fuzzy sets. Chen et.al[6] introduced interval-valued hesitant fuzzy sets. However, nearly all of the proposed methods have used the membership function to describe the ambiguity of information.

In the field of intelligent logistics, these fuzzy logic and fuzzy set theories are widely used to solve complex decision problems such as path optimisation, inventory management and transport scheduling[7,8,9,10]. However, the challenge for intelligent logistics systems is how to effectively process and utilise large amounts of dynamic and uncertain information. Although existing fuzzy methods provide a framework for dealing with uncertainty, they usually rely on the subjective judgement of experts to define membership functions, which can lead to bias in decision outcomes. In addition, with the increasing amount of data and complexity of the decision environment, the traditional fuzzy set method may face the problem of insufficient computational efficiency and accuracy when dealing with large and high-dimensional data.

Therefore, the development of new methods that can more accurately describe and handle uncertain information in intelligent logistics has become the key to improving the quality and efficiency of system decision making. This requires not only taking into account the advantages of existing fuzzy set theory, but also overcoming its limitations in specific applications of intelligent logistics, such as introducing machine learning algorithms to automatically learn and adjust membership functions, or developing more advanced fuzzy processing techniques to cope with changing decision environments.

However, the effect of information reliability on decision making problems is important in the real world as well. Zadeh[4] proposed Z-number to embody reliability and uncertainty of decision information in 2011. Zadeh proposed Z-number which is a new fuzzy theoretic, and have capability to combine objective information with subjective understanding of cognitive information. A Z-number is expressed by a pair of ordered arrays A and B, where A represents the real value function of the uncertain variable X, and B is the measure of reliability about A.

In recent years,the study of Z-numbers have been increased by many scholars in some fields[11,12,13]. The current research on Z-numbers theoretical can be mainly divided into three aspect, as follows:

(a).: Basis theory[14,15,16]. The first aspect contain some concepts which are closely related to the concept of Z-numbers. Z-valuation[17] is expressed as an order triple $(X, A, B)$ , which is equivalent to an assignment statement X is $(A, B)$ (when A is not a singleton, X is an uncertain variable). $Z^{+}$ -number[18] is a distribution that combines the possibility and probability distributions of X. In other words, $Z^{+}$ -number is associated with what is referred to as a bimodal distribution. And $Z^{+}$ -number is usually represented as $Z^{+} = (A, R)$ or $(A, p_{X})$ or $(μ_{A}, p_{X})$ . Z-information[19] is a collection of Z-valuation. $Z^{+}$ -valuation[17] is indicated as $(X, A, p_{X})$ or $(X, μ_{A}, p_{X})$ , where $u_{A}$ is the membership function of A and $p_{X}$ is the probability distribution of X. Allahviranloo et al.[20] proposed the concept of Z-Advanced, which is a novel correlation concept about Z-numbers.
(b).: Language type Z-number calculation and related extension[21,22,23]. The second aspect is special type of the Z-number. The language type is expressed in the fuzzy restriction of the Z-number, such as language term sets and hesitation fuzzy set. Pal et al.[24] proposed an algorithm for Computing With Words (CWW) using Z-numbers and defined a operator for the evaluation which is the level of requirement satisfaction based on Z-number, and describe simulation experiments of CWW utilizing Z-numbers. Wang et al.[25] proposed the concept of a linguistic Z-number, they defined its distance measure and Choquet integral, and then put forward a TODIM method on linguistic Z-number. Wang et al.[26] introduced hesitant uncertain linguistic Z-numbers (HULZNs) based on Z-numbers and linguistic models. In addition, they defined operations and distance of HULZNs and a new MCGDM approach is developed by incorporating the power aggregation operators and the VIKOR model. Liu et al.[27] proposed a method of deriving knowledge of Z-numbers from the perspective of Dempster-Shafer theory. The method considers the Z-number generating from objective and subjective data using Dempster-Shafer theory.
(c).: Establishing a decision model based on Z-number[25,28,29]. The third aspect is establishing a algorithm based on Z-number. Kang et al. [30] proposed a method to convert Z-numbers into fuzzy number. Aliev et al. [18] proposed some algorithms about Z-numbers. They consider two approaches to decision making with Z-information. One is based on converting the Z-numbers to crisp number to determine the priority weight of each alternative. That would decrease some uncertain information during processing. The other one is based on Expected utility theory by using Z-numbers. The method of selecting Expected utility is a uncertain factor, it’s influence the effect of using Z-number. Kang et al. [31] proposed a utility function of Z-numbers. These decision methods, utility functions, or conversion methods may lose some information during the operation, and these shortcomings should be further considered. Kang et al.[32] proposed an environmental evaluation framework based on Dempster Shafer theory and Z-numbers, which is a new notion of the utility of fuzzy number to generate the basic probability assignment of Z-numbers.

Generally speaking, a random variable is associated with probability distributions and possibility, the possibility/probability consistency principle is expressed the weak connection of them[33]. The accurate distribution of a random variable is impossible to estimated, and similarly estimated the appear of possible situations in real problems cannot be easily achieved as well. However, it’s important that the different viewpoints and reliabilities are taken into consideration in decision making. Durbach and Calder [34] underline the significance of reliable information for stochastic multi-criteria acceptability analysis (SMAA)[35], input data that is preference information set as real-valued random variables. And now many researchers do not consider the hidden probability and reliability of Z-number in the Z-number multi-attribute decision problem. This paper will consider the important role of hidden probability and reliability in decision-making. Wang et al.[36] combined the concept of reliability to judge the decision-maker, and established the multi-criteria decision aiding model based on stochastic multi-criteria acceptability analysis. Thus, in this paper, we incorporate the reliability concept for decision-making judgment and establish a new hidden probability of

M Z s

on the basis of the above literature and the concept of

Z^{+} - n u m b e r

In order to study decision problems in quantitatively, in addition to considering the uncertainty of the potential probabilities in the

Z - n u m b e r

locale, you need to quantify the value of the consequences. Therefore, this paper defines the utility function from different perspectives to reflect the real value of consequences to decision-makers.

During decision analysis, there is the problem of how to describe or express the actual value of the consequences to the decision experts to reflect the preference order of the consequences in the decision experts′ mind, which is a reflection of the personality and values of decision experts, which are related to the social status, economic status, cultural quality, psychological and physiological (physical) states of decision-makers. Indecision theory, the actual value of consequences to decision experts, namely the order of preference of decision experts to consequences is described by utility. The utility is the quantization of preference is the number. Table 1 shows some kinds of research status of utility function in the decision model, the commonly used utility function formulas are given respectively from the four categories of mathematical analytic formula, construction criterion, content, and expression locale.

In addition to above, Mao et al.[49] discussed the interval-valued intuitionistic fuzzy entropy which reflects intuitionism and fuzziness of interval-valued intuitionistic fuzzy set (IvIFS) based on interval-valued intuitionistic fuzzy cross-entropy. They made the composite entropy is applied to multiple attributes decision-making by using the weighted correlation coefficient between IvIFSs and pattern recognition by a similarity measure transformed from the composite entropy. Yao et al.[50] discussed novel cross-entropy and entropy models on intuitionism and fuzziness of intuitionistic fuzzy sets (IFSs). They defined cross-entropy and symmetric cross-entropy based on intuitionistic factor and fuzzy factor for measure the discrimination of uncertain information. Sepehr et al.[51] used Z-number to express possibility and reliability in Earned value management(EVM), and Compared with traditional fuzzy EVM, ZEVM(Z-number EVM) was more advantageous. At the same time, the uncertainty in the project-related decision data which is sensitive to the cost duration fluctuation is considered. However, in practical problems, there are often some individual subjective factors of decision-makers, objective factors of the attributes, unpreventable error factors, and other uncertain factors, which are one of the characteristics of uncertain information itself. Therefore, the concept of information entropy and cross-entropy are introduced as utility functions. In this paper, three kinds of

M Z s

(

M Z s

in this paper all represent discrete multidimensional Z-numbers) utility functions are defined.

In summary, all the studies on Z-number are one-dimensional. Wang et al.[52] put forward the definition of multi-dimensional Z-number and the operation of multi-dimensional Z-number. Enlightened by this, the single-dimensional Z-number study is extended to the multi-dimensional, and the multi-dimensional Z has more practical significance than the single-dimensional Z-number. One-dimensional Z-number describes only one feature for an object, while

M Z s

can describe multiple features for an object at the same time. For example, for weather problems, a one-dimensional Z-number denotes

Z_{1} = (A_{1}, B_{1}) = (h o t, l i k e l y)

and

Z_{2} = (A_{2}, B_{2}) = (w i n d l e s s, a b s o l u t e)

, and

M Z s

denotes

M Z = ((h o t, w i n d l e s s), l i k e l y)

. To some extent,

M Z s

will make the evaluation information in decision-making more concise and clear.

This research explores an innovative multidimensional Z-number (

M Z s

) synthesis method and pioneers its application to solving complex decision problems in intelligent logistics. The following are the main innovation points and motivations of this work, with distinct characteristics and practical value:

Deep synthesis of multidimensional data: This paper introduces a new multidimensional Z-number synthesis technique for the first time. This technique not only combines the uncertainty processing ability of traditional Z-numbers, but also greatly improves the dimension and richness of data expression by integrating multidimensional information. This is particularly important in the intelligent logistics system, which deals with heterogeneous data from multiple sources.
Hidden probability model:We propose an innovative $M Z s$ hidden probability model and further define $M Z^{+}$ . This new concept, $M Z^{+}$ , adds more layers of information to the traditional $M Z s$ , making uncertainty management in the decision process more refined and efficient.
Multi-angle uncertainty quantification method: This study developed a variety of utility functions based on $M Z^{+}$ , which quantified uncertainty from multiple dimensions such as geometry, algebra, information entropy and cross entropy, providing diversified solutions for complex decision making. These methods not only enhance the explanatory power of the model, but also provide solid theoretical support for multi-attribute group decision making.

Through these innovations, this paper not only deepens the understanding and application of the concept of multidimensional Z-number, but also provides an efficient and reliable decision support tool for the field of intelligent logistics, showing a wide range of application potential and practical value.

This paper is divided into eight sections. The second section is about the basic definitions of discrete

Z - n u m b e r

, discrete

Z^{+} - n u m b e r

, multidimensional Z-number and probability density distribution. In the third section, the method of synthesizing multidimensional Z-number is introduced, and the hidden probability model of

M Z s

is constructed, then the multidimensional

Z^{+} - n u m b e r

is proposed underlying

M Z s

and the hidden probability. The fourth section is divided into two subsections, which propose several formal utility functions of discrete

M Z s

from the perspective of mathematics and entropy and prove some properties. The fifth section is

M A G D M

method with hidden probability based on the utility function of

M Z s

. The sixth section is

M A G D M

case of a new energy car evaluation and explanation of the decision results. The seventh section is the analysis of

M A G D M

results, which contains sensitivity analysis with utility functions and synthesis operators and comparative analysis with the existing methods. The last is the conclusion of this paper.

2. Preliminaries

This section introduce some basic knowledge about discrete Z-number,

Z^{+} - n u m b e r

, multidimensional Z-numbers, and the probability density distribution in detail.

Definition 1. ([18]Discrete Z-number)Let X be a random variable, A and B are two discrete fuzzy numbers, where

μ_{A} : x_{1}, x_{2}, \dots, x_{n} \to [0, 1]

μ_{B} : b_{1}, b_{2}, \dots, b_{n} \to [0, 1]

For the membership function of A and B, respectively, where

x_{1}, x_{2}, \dots, x_{n} \in R

and

b_{1}, b_{2}, \dots, b_{n} \in [0, 1]

. A discrete Z-number is defined as an ordered pair of discrete fuzzy numbers

Z = (A, B)

on X, where A is the fuzzy restriction of X and B is the fuzzy restriction of the probability measure of A.

Definition 2. ([18]Discrete

Z^{+} - n u m b e r

) A discrete

Z^{+} - n u m b e r

, denoted as

Z^{+} = (A, R)

, where A is the fuzzy restriction and R is the probability distribution

p (x)

of X, expressed as:

μ = μ_{1} / x_{1} + μ_{2} / x_{2} + \dots + μ_{n} / x_{n}

p (x) = p_{1} ∖ x_{1} + p_{2} ∖ x_{2} + \dots + p_{n} ∖ x_{n}

where

μ_{i} / x_{i}

means that

μ_{i}, i = 1, \dots, n

, is the possibility that

X = x_{i}

. Similarly,

p_{i} ∖ x_{i}

is the probability that

X = x_{i}

.And the A plays the same role in

Z^{+} - n u m b e r

as it does in Z-numbers, the R plays the role of the probability distribution.

Definition 3. ([52]Multidimensional Z-numbers) Some random variables

X_{i} = (x_{i 1}, x_{i 2},

\dots, x_{i m_{k_{i}}})

defined on sample space

Ω

and

A_{i} \subseteq X_{i}

,then

(A_{1}, A_{2}, \dots, A_{n})

is called the n-dimension restriction vector, where

Ω = {(x_{1 k_{1}}, x_{2 k_{2}}, \dots, x_{n k_{n}}) | x_{i k_{i}} \in X_{i}, k_{i} = 1, 2, \dots, m_{k_{i}}, i = 1, 2, \dots, n}

. A multidimensional Z-numbers comprises multidimensional restriction vector,

(A_{1}, A_{2}, \dots, A_{n})

, and a fuzzy number, B, denoted as

M Z = ((A_{1}, A_{2}, \dots, A_{n}), B) .

Let

G = (A_{1}, A_{2}, \dots, A_{n})

, a multidimensional Z-numbers can be expressed as

M Z = (G, B)

For example, in the case of a comfortable weather with sun and temperature, the one-dimensional Z-numbers cannot express fully the condition of weather. Furthermore, we cannot judge whether good weather has from just one aspect of sun or breezy. It could be good weather or bad weather. Hence, we express the information in the form of multidimensional Z-numbers, (comfort weather, (sun and temperature around

26^{\circ} C

), likely), (hot weather, (sun and temperature above

30^{\circ} C

), likely).

Definition 4. ([52]Probability density distribution) The probability density distribution of a multidimensional Z-numbers is expressed as

p (x_{1 k_{1}}, x_{2 k_{2}}, \dots, x_{n k_{n}})

, where

\int \int \dots \int p (x_{1 k_{1}}, x_{2 k_{2}}, \dots, x_{n k_{n}}) d x_{1 k_{1}} \dots d x_{n k_{n}} = 1

and

p (x_{1 k_{1}}, x_{2 k_{2}}, \dots, x_{n k_{n}}) \geq 0

. This distribution is obtained by calculating

p_{X_{1}} (x_{1 k_{1}}),

p_{X_{2}} (x_{2 k_{2}}), \dots,

p_{X_{n}} (x_{n k_{n}})

, where

p_{X_{i}} (x_{i k_{i}}) \geq 0

and

\int p_{X_{i}} (x_{i k_{i}}) d x_{i k_{i}} = 1, i = 1, 2, \dots, n

. A probability distribution for X that belongs to the multidimensional restriction vector can be expressed as:

P {(x_{1}, \dots, x_{m}) \in G} = {\int \int \dots \int}_{G} p (x_{1 k_{1}}, x_{2 k_{2}}, \dots, x_{n k_{n}}) d x_{1 k_{1}} \dots d x_{n k_{n}}

where

G = (A_{1}, A_{2}, \dots, A_{m})

3. Synthesizing Discrete Multidimensional Z-numbers

In this section, the method of synthesizing discrete multidimensional Z-numbers is proposed. Multidimensional

Z^{+} - n u m b e r s

is defined as well.

Definition 5. (Synthesize Discrete Multidimensional Z-numbers) Let 2 one-dimensional discrete Z-numbers,

Z_{1} = (A_{1}, B_{1}) = (\int_{X} \frac{μ_{A_{1}} (a_{1})}{a_{1}}, \int_{X} \frac{μ_{B_{1}} (b_{1})}{b_{1}})

Z_{2} = (A_{2}, B_{2}) = (\int_{X} \frac{μ_{A_{2}} (a_{2})}{a_{2}}, \int_{X} \frac{μ_{B_{2}} (b_{2})}{b_{2}})

the method of synthesizing discrete two-dimensional Z-numbers as follow:

M Z (2) = ((A_{1}, A_{2}), B (2)) = (G, B (2))

where

G = A_{1} ⨂ A_{2} = {g | g = (a_{1}, a_{2}), a_{1} \in A_{1}, a_{2} \in A_{2}}

, and ⨂ is a synthesis operator, usually using the cartesian product as the operator. For example,

X = {a, b}

and

Y = {0, 1, 2}

are two sets, then:

X ⨂ Y = {(a, 0), (a, 1), (a, 2), (b, 0), (b, 1), (b, 2)}

Y ⨂ X = {(0, a), (1, a), (2, a), (0, b), (1, b), (2, b)}

And the membership function of G as follow:

μ_{G} (g) = μ_{A_{1}} (a_{1}) ⨁ μ_{A_{2}} (a_{2})

where ⨁ is a membership function synthesis operator, usually the following computing methods are used as membership function synthesis operators:

\begin{matrix} μ_{G} (g) & = μ_{A_{1}} (a_{1}) ⨁_{1} μ_{A_{2}} (a_{2}) = μ_{A_{1}} (a_{1}) ⋀ μ_{A_{2}} (a_{2}) \end{matrix}

(1)

\begin{matrix} μ_{G} (g) & = μ_{A_{1}} (a_{1}) ⨁_{2} μ_{A_{2}} (a_{2}) = \underset{a_{1} ⋀ a_{2} = ⊎ g_{i}}{⋁} {μ_{A_{1}} (a_{1}) ⋀ μ_{A_{2}} (a_{2})} \end{matrix}

(2)

where

⊎ g_{i} = \sum (a_{j}), j = 1, 2, \dots, n

Just like the calculation method of synthesizing two one-dimensional Z-numbers, the synthesis of

(n - m)

dimension Z-numbers and m dimension Z-numbers can obtain n dimensional Z-numbers.

Definition 6. (Hidden Probability of Discrete Multidimensional Z-numbers) Let

M Z (n) = (G, B (n))

is a discrete

M Z s

. A

Z^{+} - n u m b e r

is associated with a so-called bimodal distribution, namely a distribution which combines the possibility and probability distributions of X.[4]Informally, these distributions are compatible if the centroid of

μ_{A}

and

p_{X}

are coincident, that is,

\int_{R} u p_{X} (u) d u = \frac{\int_{R} u μ_{A} (u) d u}{\int_{R} μ_{A} (u) d u}

(3)

According to equation(3), the definition of probability and

Z^{+}

-number, the following linear programming model be established to solve hidden probability of discrete multidimensional Z-numbers.

m i n \sum_{i}^{N_{G}} {(μ_{G_{g_{i}}} p_{j} (g_{i}) - b_{j})}^{2}

\{\begin{matrix} \sum_{i}^{N_{G}} p_{j} (g_{i}) \bar{g_{i}} = \frac{\sum_{i}^{N_{G}} \bar{g_{i}} μ_{G} (g_{i})}{\sum_{i}^{N_{G}} μ_{G} (g_{i})} \\ \bar{g_{i}} = \frac{\sum_{k = 1}^{n} {(a_{1}, a_{2}, \dots, a_{n})}_{k}}{n}, (a_{1} \in A_{1}, \dots, a_{n} \in A_{n}) \\ \sum_{i}^{N_{G}} p_{j} (g_{i}) = 1 \\ p_{j} (g_{i}) \geq 0 \end{matrix}

Where

N_{G}

is the number of elements for G. According to the above hidden probability model, we can get a hidden probability matrix P,

P = {[p_{j} (g_{i})]}_{i \times j}

Then the hidden probability of

g_{i}

is calculated by the Equation (4)

p (g_{i}) = \frac{\sum_{j = 1}^{N_{B (n)}} p_{j} (g_{i})}{N_{B (n)}}

(4)

where

N_{B (n)}

is the number of elements for

B (n)

. The hidden probability of

M Z s

is expressed as follows:

P_{G} = \int_{Ω} p (g_{i})

Example 1. Let us consider the hidden probability of a two-dimensional Z-numbers

M Z (2) = (G, B (2))

. Given as:

G = \frac{0}{(1, 1)} + \frac{0.6}{(1, 2)} + \frac{1}{(1, 3)} + \frac{1}{(2, 1)} + \frac{0.5}{(2, 2)} + \frac{0}{(2, 3)}

B (2) = \frac{0}{0} + \frac{0.5}{0.1} + \frac{0.8}{0.2} + \frac{1}{0.3} + \frac{0.8}{0.4} + \frac{0.7}{0.5} + \frac{0.6}{0.6} + \frac{0.4}{0.7} + \frac{0.2}{0.8} + \frac{0.1}{0.9} + \frac{0}{1}

The hidden probability of matrix

M Z (2)

is calculated, that is written by the hidden probability matrix P as follows:

P = [\begin{matrix} 0.5048 & 0.0003 & 0.0002 & 0.0002 & 0.0004 & 0.4940 \\ 0.4359 & 0.0416 & 0.0250 & 0.0251 & 0.0498 & 0.4225 \\ 0.3678 & 0.0806 & 0.0518 & 0.0518 & 0.0961 & 0.3519 \\ 0.3022 & 0.1154 & 0.0828 & 0.0829 & 0.1301 & 0.2866 \\ 0.2384 & 0.1469 & 0.1167 & 0.1180 & 0.1543 & 0.2257 \\ 0.1798 & 0.1671 & 0.1571 & 0.1597 & 0.1659 & 0.1704 \\ 0.1292 & 0.1708 & 0.2080 & 0.2095 & 0.1600 & 0.1225 \\ 0.0884 & 0.1542 & 0.2724 & 0.2686 & 0.1330 & 0.0834 \\ 0.0570 & 0.1182 & 0.3493 & 0.3350 & 0.0895 & 0.0510 \\ 0.0267 & 0.0594 & 0.4114 & 0.4286 & 0.0486 & 0.0252 \\ 0.0002 & 0.0005 & 0.4833 & 0.5154 & 0.0004 & 0.0002 \end{matrix}]

Then the combined hidden probability of elements for

M Z (2)

is calculated,

p ((1, 1)) = \frac{0.5048 + 0.4359 + \dots + 0.0002}{11} = 0.2119

A similar algorithm can get the hidden probability of

M Z (2)

P_{G} = \frac{0.2119}{(1, 1)} + \frac{0.0959}{(1, 2)} + \frac{0.1962}{(1, 3)} + \frac{0.1995}{(2, 1)} + \frac{0.0935}{(2, 2)} + \frac{0.2030}{(2, 3)}

Definition 7. (Multidimensional

Z^{+} - n u m b e r s

) A

M Z^{+} - n u m b e r s

is associated with a so-called bimodal distribution, which is a distribution which combines the possibility and probability distributions of

X_{i} = {x_{i 1}, x_{i 2}, \dots, x_{i m_{k_{i}}}}

. Informally, these distributions are compatible if the centroid of

μ_{G}

and

p_{X_{i}}

are coincident, that is,

P_{G} = μ_{G} p_{X_{i}} = \int_{Ω} g p_{X_{i}} d g = \frac{\int_{Ω} g μ_{G} (g) d g}{\int_{Ω} μ_{G} (g) d g}

It is this relation that links the concept of

M Z s

to that of a

M Z^{+}

. More concretely,

M Z (n) = (G, B (n)) = M Z^{+} (G, μ_{G} \cdot p_{X_{i}} i s B (n))

Where the

p_{X_{i}}

is not know, but a restriction on

p_{X_{i}}

is expressed as:

μ_{G} \cdot p_{X_{i}} i s B (n) .

According to definition 3, the uncertain information is expressed by

M Z^{+}

are more than

M Z s

, therefore, which is defined multidimensional

Z^{+}

-numbers, denoted as:

M Z^{+} = (G, R (n))

Definition 8. (Reliability Measurement of Discrete Multidimensional Z-numbers) Assuming a discrete two-dimensional Z-numbers

M Z (2)

is composed of two one-dimensional Z-numbers

Z_{1} = (A_{1}, B_{1})

and

Z_{2} = (A_{2}, B_{2})

, the reliability

B (2)

M Z (2)

is defined as follows:

B (2) = \int_{X} \frac{μ_{B (2)} (\tilde{b_{k}})}{\tilde{b_{k}}}

(5)

where

\tilde{b_{k}} = \sum_{i}^{N_{A_{1}}} \sum_{j}^{N_{A_{2}}} μ_{G_{k}} (a_{i}, a_{j}) \cdot p_{k} (a_{i}, a_{j})

(6)

μ_{B (2)} (\tilde{b_{k}}) = p_{k} (A_{1}) \hat{\oplus} p_{k} (A_{2})

(7)

where

\hat{\oplus}

is the specific budget of membership composition operator of probability measurement is as follows,

p_{1} \hat{\oplus} p_{2} = min {p_{1}, p_{2}, | p_{1} - p_{2} |}

. And

N_{A_{1}}, N_{A_{2}}

are the number of elements for

A_{1}, A_{2}

\hat{p_{k}} (a_{i}, a_{j})

is joint probability by Joint Density, the computing method as follow:

\hat{p_{k}} (a_{i}, a_{j}) = p_{k} (a_{i}) + p_{k} (a_{j}) - p_{k} (a_{i}) \cdot p_{k} (a_{j})

p_{k} (A_{1})

and

p_{k} (A_{2})

are synthesize probability of

A_{1}

and

A_{2}

, respectively.

p_{k} (A_{1}) = \frac{1}{N_{A_{1}}} \sum_{i = 1}^{N_{A_{1}}} μ_{A_{1}} {(a_{i})}_{k}^{p_{A_{1}} {(a_{i})}_{k}}

(8)

p_{k} (A_{2}) = \frac{1}{N_{A_{2}}} \sum_{i = 1}^{N_{A_{2}}} μ_{A_{2}} {(a_{i})}_{k}^{p_{A_{2}} {(a_{i})}_{k}}

(9)

The reliability measure

B (n)

n-dimension Z-numbers can be obtained by repeating the combination of the reliability measure of n-dimension Z-numbers and m-dimension Z-numbers.

In this paper, the method of calculating the probability measure of discrete

M Z s

uses different synthesize operators for the membership function

\hat{\oplus}

. The influence of membership degree is considered in the synthesize hidden probability. In reference [52], the product of the hidden probability of G and the membership degree is taken as the synthesize probability (we call it "power function probability"). In this study, the exponential form (called "exponential probability") is used. The calculation method of "power function probability" has no effect on the composite probability of G, which is a membership degree, when the potential probability

p = 0

. In contrast, the method proposed in this paper fully considers the influence of the membership degree of G on synthesis.

Example 2. Let us consider the reliability measurement 2 discrete two-dimensional Z-numbers

M Z {(2)}_{1} = (G_{1}, B {(2)}_{1})

and

M Z {(2)}_{2} = (G_{2}, B {(2)}_{2})

. Given as:

G_{1} = \frac{0.1}{(1, 1)} + \frac{0.3}{(1, 2)} + \frac{0.4}{(1, 3)} + \frac{0.8}{(2, 1)} + \frac{1}{(2, 2)} + \frac{0.6}{(2, 3)} + \frac{0.2}{(3, 2)}

B {(2)}_{1} = \frac{0}{0} + \frac{0.3}{0.1} + \frac{0.8}{0.2} + \frac{1}{0.3} + \frac{0.9}{0.4} + \frac{0.8}{0.5} + \frac{0.5}{0.6} + \frac{0.4}{0.7} + \frac{0.3}{0.8} + \frac{0.1}{0.9} + \frac{0}{1}

G_{2} = \frac{0.2}{(1, 1)} + \frac{1}{(1, 2)} + \frac{0.7}{(2, 1)} + \frac{0.1}{(2, 2)}

B {(2)}_{2} = \frac{0}{0} + \frac{0.3}{0.1} + \frac{0.5}{0.2} + \frac{0.6}{0.3} + \frac{0.7}{0.4} + \frac{0.8}{0.5} + \frac{0.9}{0.6} + \frac{1}{0.7} + \frac{0.9}{0.8} + \frac{0.8}{0.9} + \frac{0}{1}

Four-dimensional Z-numbers

M Z (4)

can be synthesized by Definition 5,6,7:

M Z (4) = M Z {(2)}_{1} ⨂ M Z {(2)}_{2} = (G, B (4))

This example uses the synthesis operator

⨁_{2}

(Equation (2)) mentioned above to calculate the membership of G,as follows Table 2,

G = \frac{0.1}{(1, 1, 1, 1)} + \frac{0.2}{(1, 2, 1, 1)} + \dots + \frac{0.1}{(2, 3, 2, 2)}

According to the

M Z s

hidden probability model Definition 6, the hidden probabilities of

M Z_{1}

and

M Z_{2}

are calculated. Here, one of them is selected as an example to calculate the hidden probability, as shown in Table 3.

p_{1} = \frac{0.2341}{(1, 1)} + \frac{0.1284}{(1, 2)} + \frac{0.0540}{(1, 3)} + \frac{0.0427}{(2, 1)} + \frac{0.0369}{(2, 2)} + \frac{0.0712}{(2, 3)} + \frac{0.4328}{(3, 2)}

p_{2} = \frac{0.4146}{(1, 1)} + \frac{0.0869}{(1, 2)} + \frac{0.1339}{(2, 1)} + \frac{0.3646}{(2, 2)}

According to the Equation (8) and (9), calculate the synthesize probability of

G 1

and

G 2

P (G_{1}) = \frac{1}{7} \sum_{i = 1}^{7} p_{G_{1}} {(g_{1 i})}^{μ_{G_{1}} (g_{1 i})} = 0.6642

P (G_{2}) = \frac{1}{4} \sum_{i = 1}^{4} p_{G_{2}} {(g_{2 i})}^{μ_{G_{2}} (g_{2 i})} = 0.6448

According to the Equation (6), the reliability measure of synthesize

M Z s

is calculated as follows:

\tilde{b} = \sum_{i = 1}^{7} \sum_{j = 1}^{4} μ_{G} (g_{i}, g_{j}) \cdot \hat{p} (g_{i}, g_{j}) = 0.2153

According to the Equation (7), the reliability measure membership of synthesize Z-numbers is calculated as follows:

μ_{B (4)} (0.2153) = P (G_{1}) \hat{\oplus} P (G_{2}) = m i n {0.6642, 0.6448, | 0.6642 - 0.6448 |} = 0.0194

Than

B (4)

is calculated as follows:

B (4) = \frac{0.0716}{0.4623} + \frac{0.0716}{0.4623} + \frac{0.0605}{0.4364} + \frac{0.0194}{0.2153} + \frac{0.0241}{0.3107} + \frac{0.0622}{0.2621} + \frac{0.0879}{0.2272} + \frac{0.0838}{0.2091} + \frac{0.0857}{0.2527} +

\frac{0.0705}{0.1840} + \frac{0.0255}{0.1770}

4. The Different Forms Utility Function of Discrete $M Z s$

The

M Z s

contain hidden information is a question to worth studying. This paper proposes a variety of forms for utility function to measurement

M Z s

contains uncertain information. In this section, the math forms utility function and the entropy forms utility function of

M Z s

are proposed. The math forms are divided into algebraic and geometric, the entropy forms are split into information entropy, self-cross entropy, and cross-entropy.

4.1. The Math Forms Utility Function of Discrete $M Z s$

This subsection based on the angle of geometry and algebra, two forms of utility functions are defined to measure the uncertainty information contained in

M Z s

In this study, the utility function proposed is mainly aimed at the MAGDM problem of linguistic discrete

M Z s

. The utility function in algebraic form considers the influence of the hidden probability p and probability measure B for discrete

M Z s

on the utility function from an algebraic perspective. According to the irreducible equation in the algebraic function, the algebraic utility function is defined, which is suitable for the decision problem of discrete

M Z s

Definition 9. (The Algebraic Form of Discrete

M Z s

) Let a n-dimension Z-numbers

M Z (n)

, the algebraic form of

M Z (n)

utility function is defined as follows:

A U (M Z) = \frac{1}{n} [\sum_{i = 1}^{N_{G}} {(\bar{g_{i}})}^{p (g_{i})} + \sum_{j = 1}^{N_{B (n)}} {(\tilde{b_{j}})}^{μ_{B (n)} (\tilde{b_{j}})}]

(10)

where

\bar{g_{i}} = \frac{\sum_{k = 1}^{n} {(a_{1}, a_{2}, \dots, a_{n})}_{k}}{n}, (a_{1} \in A_{1}, \dots, a_{n} \in A_{n})

. And

N_{G}

N_{B (n)}

are the number of elements for G,

B (n)

The geometric utility function is based on the spatial structure, and the discrete

M Z s

are vectorized to calculate the angle with the unit vector. The geometric utility function is suitable for discrete Z-numbers type decision problems without calculating the hidden probability. In this study, the geometric utility function only considers the fuzzy limitation of discrete

M Z s

and the influence of membership degree.

Definition 10. (The Geometric Form of Discrete

M Z s

) Let a n-dimension Z-numbers

M Z (n)

, the first step is to vectoring the n-dimensional Z-numbers to get the n-dimensional Z-numbers vector:

\vec{M Z (n)} = ({\bar{g_{1}}}^{μ_{G} (g_{1})}, {\bar{g_{2}}}^{μ_{G} (g_{2})}, \dots, {\bar{g_{n}}}^{μ_{G} (g_{n})})

The geometric utility function of

M Z s

borrows the n-dimensional unit vector as a reference to measure and the n-dimensional unit vector defined in this study is

\vec{N} = (\frac{\sqrt{n}}{n}, \frac{\sqrt{n}}{n}, \dots, \frac{\sqrt{n}}{n})

. Then the geometric form utility function for

M Z (n)

is defined as follows:

\begin{matrix} G U (M Z) & = arccos < \vec{M Z (n)}, \vec{N} > \\ = arccos \frac{\vec{M Z (n)} \cdot \vec{N}}{| \vec{N} | + | \vec{M Z (n)} |} \end{matrix}

(11)

where

\vec{M Z (n)} \cdot \vec{N} = \frac{\sqrt{n}}{n} \cdot {\bar{g_{1}}}^{μ_{G} (g_{1})} + \frac{\sqrt{n}}{n} \cdot {\bar{g_{2}}}^{μ_{G} (g_{2})} + \dots + \frac{\sqrt{n}}{n} \cdot {\bar{g_{n}}}^{μ_{G} (g_{n})}

and

| \vec{N} | + | \vec{M Z (n)} | = 1 + \sqrt{{({\bar{g_{1}}}^{μ_{G} (g_{1})})}^{2} + {({\bar{g_{2}}}^{μ_{G} (g_{2})})}^{2} + \dots + {({\bar{g_{n}}}^{μ_{G} (g_{n})})}^{2}}

The utility function of the geometric angle for n-dimensional Z-numbers is to consider the limitation and membership degree of X and then select an n-dimensional unit vector as a reference after its vectorization, to figure out the included Angle between n-dimensional

\vec{M Z (n)}

vector and unit vector. There are many ways to choose the n-dimensional unit vector, and the standard chosen in this study is the n-dimensional unit vector with the same coordinate in each dimension.

The geometric utility of 2 three-dimensional Z-numbers are calculated, and the schematic diagram of the included angle is drawn by MATLAB in the rectangular coordinate system of space, as shown in Fig.1. where the yellow area represents the geometric utility of

M Z_{1} = (G_{1}, B {(3)}_{1})

and the purple area represents the geometric utility of

M Z_{2} = (G_{2}, B {(3)}_{2})

, where

G_{1} = \frac{0.1}{(1, 1, 1)} + \frac{0.2}{(1, 1.5, 2)} + \frac{0.8}{(1, 2, 3)}

and

G_{2} = (\frac{0.4}{(1, 1, 4)} + \frac{0.5}{(2, 4, 6)} + \frac{0.1}{(7, 8, 9)})

\vec{O B}

and

\vec{O C}

vectors are all the representations of

M Z_{1}

and

M Z_{2}

vectorized in the rectangular coordinate system in space, and

\vec{O A}

vector is the unit vector of positive three-dimensional that conforms to the same

x, y, z

coordinates.

Figure 1. Geometric Utility of Discrete

M Z s

Figure 1. Geometric Utility of Discrete

M Z s

4.2. The Entropy Forms Utility Function of Discrete $M Z s$

This subsection defined three entropy forms utility function of discrete

M Z s

. The entropy form utility function is inspired by the parameter entropy that characterizes the state of matter in thermodynamics. In thermodynamics, entropy indicates the degree of chaos in the system,and information entropy indicates the uncertainty of the source in information theory, and cross-entropy indicates the difference between the two probability distributions.

The entropy forms utility function are both suitable for discrete

M Z s

decision problems. However, the information entropy utility function and self-cross entropy utility function are only suitable for computing the same

M Z s

, and the cross-entropy utility function is suitable for 2 different discrete

M Z s

First, information entropy form. Based on the definition of information entropy, which is a quite abstract concept in mathematics information entropy is used to measure the uncertain information contained in

M Z s

, which is the probability of the occurrence of some information in

M Z s

. The higher the utility value of the form of information entropy for

M Z s

, the more uncertain information contained in

M Z s

Definition 11. (The Information Entropy Form of Discrete

M Z s

) Generally speaking, when a kind of information has a higher probability of appearing, it indicates that is spread more widely, or cited more. This study from the perspective of information uncertainty, information entropy can represent the uncertainty value of information. In this way, a standard to measure the uncertainty value of information and make more inferences about the uncertainty and reliability of

M Z (n)

to express information.

\begin{matrix} I U (M Z) = & \frac{1}{N_{G}} \sum_{i = 1}^{N_{G}} [- p (g_{i}) log (p (g_{i}))] + \frac{1}{N_{G}} \sum_{j = 1}^{N_{G}} [- μ_{G} (g_{j}) log (μ_{G} (g_{j}))] + \\ \sum_{k = 1}^{N_{B (n)}} [- (μ_{B (n)} ({\tilde{b}}_{k})) log (μ_{B (n)} ({\tilde{b}}_{k}))] \end{matrix}

(12)

In general, the base in the logarithm takes 2. Where

N_{G}, N_{B (n)}

are the number of elements for G and

B (n)

in Equation (12).

Property 1 Let

I U (M Z)

be the information entropy form utility function of discrete

M Z s

, it has the following properties:

(1): (Non-negative) $I U (M Z) \geq 0$ .
(2): (Symmetry) $I U (M Z)$ is about $p (g_{i}) = μ_{G} (g_{j}) = μ_{B (n)} = \frac{1}{2}$ symmetry.
(3): (Certainty)When $p (g_{i}), μ_{G} (g_{j}), μ_{B (n)} = 0 o r 1$ , $I U (M Z)$ is certainty.
(4): (Extreme value)When $p (g_{i}) = μ_{G} (g_{j}) = μ_{B (n)} = \frac{1}{2}$ , $I U (M Z)$ gets the extreme value.

Proof. In order to simplify the proof process, the

M Z s

information entropy utility function is deformed as follows:

I U (x, y, z) = (- x log x) + (- y log y) + (- z log z)

where

x = p (g_{i})

y = μ_{G} (g_{j})

z = μ_{B (n)}

, and

x, y, z \in [0, 1]

(1): Because $x, y, z \in [0, 1]$ , then $(- x log x), (- y log y)$ ,

$(- z log z) \in [0, + \infty)$ so $I U (x, y, z) \in [0, + \infty)$ , which is $I U (M Z) \in [0, + \infty)$ .
(2): For any $α, β, γ \in [0, \frac{1}{2}]$ ,

$- (\frac{1}{2} + α) log (\frac{1}{2} + α) = - (\frac{1}{2} - α) log (\frac{1}{2} - α)$

$- (\frac{1}{2} + β) log (\frac{1}{2} + β) = - (\frac{1}{2} - β) log (\frac{1}{2} - β)$

$- (\frac{1}{2} + γ) log (\frac{1}{2} + γ) = - (\frac{1}{2} - γ) log (\frac{1}{2} - γ)$

then We can get that the following equation:

$I U (\frac{1}{2} + α, \frac{1}{2} + β, \frac{1}{2} + γ) = I U (\frac{1}{2} + α, \frac{1}{2} + β, \frac{1}{2} - γ) =$

$I U (\frac{1}{2} + α, \frac{1}{2} - β, \frac{1}{2} + γ) = I U (\frac{1}{2} - α, \frac{1}{2} + β, \frac{1}{2} + γ) =$

$I U (\frac{1}{2} + α, \frac{1}{2} - β, \frac{1}{2} - γ) = I U (\frac{1}{2} - α, \frac{1}{2} + β, \frac{1}{2} - γ) =$

$I U (\frac{1}{2} - α, \frac{1}{2} - β, \frac{1}{2} + γ) = I U (\frac{1}{2} - α, \frac{1}{2} - β, \frac{1}{2} - γ)$
(3): When $x, y, z = 0 o r 1$ , $- x log x = - y log y = - z log z = 0$ so $I U (x, y, z)$ is certainty, as follows:

$I U (0, 0, 0) = I U (0, 0, 1) = I U (0, 1, 0) = I U (1, 0, 0) = 0$

$I U (1, 0, 1) = I U (1, 1, 0) = I U (0, 1, 1) = I U (1, 1, 1) = 0$
(4): Take the partial of $x, y, z$ with respect to $I U (x, y, z)$ , so that

        $\frac{\partial I U}{\partial x} = - (log x + 1) = 0$

        $\frac{\partial I U}{\partial y} = - (log y + 1) = 0$

        $\frac{\partial I U}{\partial z} = - (log z + 1) = 0$

$I U (x, y, z)$ gets the extreme value, that is $x = y = z = \frac{1}{2}$ , the function of $I U$ will exist extreme value,

$I U {(x, y, z)}_{m a x} = I U (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}) = \frac{3}{2}$

then we can calculate the $I U {(M Z)}_{m a x}$ ,

$I U {(M Z)}_{m a x} = 1 + \frac{N_{B (n)}}{2}$

Hence, the proof of property 1 is now completed.

Second, self-cross entropy. Cross-entropy measures the difference between the probability distributions of two variables. Inspired by this, limiting of probability with the element and the hidden probability with

M Z s

were taken as two probability distributions for

M Z s

respectively, and the utility value in the form of self-crossing entropy was used to measure the difference between

M Z s

and

M Z^{+}

. Therefore, the larger the utility value of the form of self-crossing entropy, the greater the difference between

M Z s

and

M Z^{+}

, that is,

M Z s

contain more uncertain information.

Definition 12. (The Self-cross Entropy Form of Discrete

M Z s

) Self-cross entropy measures

M Z s

and

M Z^{+}

. Based on the perspective of cross entropy, we consider the reliability and hidden probability of the

M Z s

as the two probability distributions of the discrete multidimensional Z-numbers

M Z (n)

to calculate the utility function value in the form of cross entropy for

M Z s

. The multidimensional Z-numbers

M Z (n)

utility function has defined as follows.

\begin{matrix} S U (M Z) = & \sum_{i = 1}^{N_{G}} [{(p (g_{i}))}^{α} | ln {(\frac{2 {(p (g_{i}))}^{α}}{{(p (g_{i}))}^{α} + {(μ_{G} (g_{i}))}^{α}})}^{α} | \\ + {(μ_{G} (g_{i}))}^{β} | ln {(\frac{2 {(μ_{G} (g_{i}))}^{β}}{{(p (g_{i}))}^{β} + {(μ_{G} (g_{i}))}^{β}})}^{β} |] \end{matrix}

(13)

where

α

and

β

satisfy

α, β \geq 0

and

α + β = 1

Property 2 The self-cross entropy form utility of

M Z (n)

has three properties as follows:

(1)

(No negative) For any

α

and

β

satisfy

α, β \geq 0

and

α + β = 1

then

0 \leq S U \leq 2 n l n 2

;

(2)

(Symmetry) When

p (g_{i}) = μ (g_{i})

S U = 0

;

(3)

(Monotonic) When

α = β = 0.5

S U

exist monotonic,

a.: $S U$ is monotonically increasing on $p (g_{i})$ , monotonously decreasing on $μ_{G} (g_{i})$ when $p (g_{i}) > μ_{G} (g_{i})$ ,
b.: $S U$ is monotonically increasing on $μ_{G} (g_{i})$ , monotonously decreasing on $p (g_{i})$ when $p (g_{i}) < μ_{G} (g_{i})$ .

Proof. In order to simplify the proof process, we use x stands for

p (g_{i})

and y stands for

μ_{G} (g_{i})

, then formula (13) can be expressed as

S U = x^{α} | ln {(\frac{2 x^{α}}{x^{α} + y^{α}})}^{α} | + y^{β} | ln {(\frac{2 y^{β}}{x^{β} + y^{β}})}^{β} |

(1)

Because the hidden probability and membership degree of

M Z (n)

belong to

[0, 1]

, and two parameters

α, β \in [0, 1]

0 \leq x^{α} | ln (\frac{2 x^{α}}{x^{α} + y^{α}}) | \leq ln 2,

M Z (n)

has n items added to get

0 \leq \sum x^{α} | ln (\frac{2 x^{α}}{x^{α} + y^{α}}) | \leq n ln 2,

to be similarly,

0 \leq \sum y^{β} | ln (\frac{2 y^{β}}{x^{β} + y^{β}}) | \leq n ln 2,

therefor

0 \leq S U \leq 2 n ln 2

(2)

When

x = y

ln {(\frac{2 x^{α}}{x^{α} + y^{α}})}^{α} = 0

for

\forall α, β \in [0, 1]

and

α + β = 1

, therefore

S U = 0

(3)

When

α = β = 1

the formula can be expressed

a.: If $x > y$ , then $ln (\frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}) > 0$

$S U = \frac{1}{2} x^{\frac{1}{2}} ln \frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}} - \frac{1}{2} y^{\frac{1}{2}} ln \frac{2 y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}$

Solving the first-order partial derivative of x,

$\frac{\partial S U}{\partial x} = \frac{1}{4} x^{- \frac{1}{2}} [ln (\frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}) + \frac{2 y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}]$

$\frac{\partial S U}{\partial x} > 0$ is obviously, therefor $S U$ is monotonically increasing on x. To be similarly $\frac{\partial S U}{\partial y} < 0$ therefor $S U$ is monotonically decreasing on y.
b.: If $x < y$ , then $ln (\frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}) < 0$

$S U = - \frac{1}{2} x^{\frac{1}{2}} l n \frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}} + \frac{1}{2} y^{\frac{1}{2}} ln \frac{2 y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}$

Solving the first-order partial derivative of x,

$\frac{\partial S U}{\partial x} = - \frac{1}{4} x^{- \frac{1}{2}} [ln (\frac{2 x^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}) + \frac{2 y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}]$

$\frac{\partial S U}{\partial x} < 0$ is obviously, therefor $S U$ is monotonically decreasing on x. To be similarly $\frac{\partial S U}{\partial y} > 0$ therefor $S U$ is monotonically increasing on y.

Hence, the proof of property 2 is now completed.

Third, cross-entropy. The Definition 12 is self-cross entropy, it’s measure a

M Z (n)

, then the cross entropy form utility function is measure the uncertain information of tow

M Z s

. That is to measure the degree to which two

M Z_{1}

are superior to

M Z_{2}

Definition 13. (The Cross-Entropy Form of Discrete

M Z s

) Cross-entropy measures the difference between probability distributions of 2

M Z s

. There 2 n-dimensional

M Z s

M Z {(n)}_{1} = (G_{1}, B {(n)}_{1})

and

M Z {(n)}_{2} = (G_{2}, B {(n)}_{2})

, where the number of elements for

G_{1}

and

G_{2}

are equal express as

N_{G}

B {(n)}_{1}

and

B {(n)}_{2}

are similar. Then the defined as follows:

C U (M Z {(n)}_{1}, M Z {(n)}_{2}) =

\sum_{i = 1}^{N_{G}} {μ_{G_{1}}^{p} (g_{1} (i)) ln {[\frac{2 μ_{G_{1}}^{p} (g_{1} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p}} + \sum_{j = 1}^{N_{B (n)}} {μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) ln {[\frac{2 μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i))}{μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) + μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}]}^{q}}

C U (M Z {(n)}_{2}, M Z {(n)}_{1}) =

\sum_{i = 1}^{N_{G}} {μ_{G_{2}}^{p} (g_{2} (i)) ln {[\frac{2 μ_{G_{2}}^{p} (g_{2} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p}} + \sum_{j = 1}^{N_{B (n)}} {μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i)) ln {[\frac{2 μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}{μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) + μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}]}^{q}}

where

N_{G}

is the number of elements for

G_{1}

and

G_{2}

N_{B (n)}

is the number of elements for

B {(n)}_{1}

and

B {(n)}_{2}

p, q \geq 0

. Specially, when

μ_{G_{1}} (g_{1} (i)), μ_{G_{2}} (g_{2} (i)), μ_{B {(n)}_{1}} ({\tilde{b}}_{1} (i)), μ_{B {(n)}_{2}} ({\tilde{b}}_{2} (i)) = 0

, then

μ_{G_{1}}^{p} (g_{1} (i)) ln {[\frac{2 μ_{G_{1}}^{p} (g_{1} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p}, μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) ln {[\frac{2 μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i))}{μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) + μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}]}^{q},

μ_{G_{2}}^{p} (g_{2} (i)) ln {[\frac{2 μ_{G_{2}}^{p} (g_{2} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p}, μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i)) ln {[\frac{2 μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}{μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) + μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}]}^{q} = 0

However,

C U (M Z {(n)}_{1}, M Z {(n)}_{2})

is also not symmetric, so in analogy with Shang and Jiang[53], the symmetric cross-entropy utility function

D U (M Z {(n)}_{1}, M Z {(n)}_{2})

is defined as follows:

D U (M Z {(n)}_{1}, M Z {(n)}_{2}) = C U (M Z {(n)}_{1}, M Z {(n)}_{2}) + C U (M Z {(n)}_{2}, M Z {(n)}_{1})

D U (M Z {(n)}_{1}, M Z {(n)}_{2})

is called symmetric discrimination uncertain information measure for

M Z s

Property 3 Let

M Z {(n)}_{1}

and

M Z {(n)}_{2}

are 2 discrete n-dimensional Z-numbers,

D U (M Z {(n)}_{1},

M Z {(n)}_{2})

satisfies following properties:

(1): (Symmetry) $D U (M Z {(n)}_{1}, M Z {(n)}_{2}) = D U (M Z {(n)}_{2}, M Z {(n)}_{1})$ .
(2): (Negative) $D U (M Z {(n)}_{1}, M Z {(n)}_{2}) \geq 0$ .
(3): (Normative) $0 \leq D U (M Z {(n)}_{1}, M Z {(n)}_{2}) \leq 2 (N_{G} + N_{B (n)}) ln 2$ .

Proof.

(1): The proof is obviously.
(2): Because $μ_{G_{1}} (g_{1} (i))$ , $μ_{G_{2}} (g_{2} (i))$ , $μ_{B {(n)}_{1}} ({\tilde{b}}_{1} (i))$ and $μ_{B {(n)}_{2}} ({\tilde{b}}_{2} (i))$ are all members of $[0, 1]$ ,then the $D U (M Z {(n)}_{1}, M Z {(n)}_{2}) \geq 0$ be obviously.
(3): $μ_{G_{1}}^{p} (g_{1} (i)), μ_{B {(n)}_{1}}^{p} ({\tilde{b}}_{1} (j)) \in [0, 1]$ when $μ_{G_{1}} (g_{1} (i)), μ_{B {(n)}_{1}} ({\tilde{b}}_{1} (j)) \in [0, 1]$ and $p, q \in [0, + \infty)$ the $μ_{G_{2}}^{q} (g_{2} (i)), μ_{B {(n)}_{2}}^{p} ({\tilde{b}}_{2} (j)) \in [0, 1]$ be similar.

$l n {[\frac{2 μ_{G_{1}}^{p} (g_{1} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p} \in [0, ln 2],$

so

$μ_{G_{1}}^{p} (g_{1} (i)) ln {[\frac{2 μ_{G_{1}}^{p} (g_{1} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p} \in [0, ln 2]$

Then

$\sum_{i = 1}^{N_{G}} {μ_{G_{1}}^{p} (g_{1} (i)) ln {[\frac{2 μ_{G_{1}}^{p} (g_{1} (i))}{μ_{G_{1}}^{p} (g_{1} (i)) + μ_{G_{2}}^{p} (g_{2} (i))}]}^{p}} \in [0, N_{G} ln 2]$

And

$\sum_{j = 1}^{N_{B (n)}} {μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) ln {[\frac{2 μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i))}{μ_{B {(n)}_{1}}^{q} ({\tilde{b}}_{1} (i)) + μ_{B {(n)}_{2}}^{q} ({\tilde{b}}_{2} (i))}]}^{q}} \in [0, N_{B (n)} ln 2]$

that be similar. So we can figure out

$C U (M Z {(n)}_{1}, M Z {(n)}_{2}), C U (M Z {(n)}_{2}, M Z {(n)}_{1}) \in [0, (N_{G} + N_{B (n)}) ln 2]$

therefore

$0 \leq D U (M Z {(n)}_{1}, M Z {(n)}_{2}) \leq 2 (N_{G} + N_{B (n)}) ln 2 .$

Hence, the proof of property 3 is now completed.□

Example 3. Let us consider the

D U

of 2 three-dimensional Z-numbers

M Z {(3)}_{1} = (G_{1}, B {(3)}_{1})

and

M Z {(3)}_{2} = (G_{2}, B {(n)}_{2})

as follows:

G_{1} = \frac{0.1}{(1, 1, 1)} + \frac{0.25}{(1, 1, 2)} + \frac{0.5}{(1, 1, 3)} + \frac{0.75}{(1, 2, 1)} + \frac{1}{(1, 2, 2)} + \frac{0.8}{(1, 2, 3)} + \frac{0.6}{(1, 3, 1)} + \frac{0.4}{(1, 3, 2)} + \frac{0.2}{(1, 3, 3)}

G_{2} = \frac{0.25}{(1, 1, 1)} + \frac{0.3}{(1, 1, 2)} + \frac{0.5}{(1, 1, 3)} + \frac{0.7}{(1, 2, 1)} + \frac{0.8}{(1, 2, 2)} + \frac{0.9}{(1, 2, 3)} + \frac{1}{(1, 3, 1)} + \frac{0.5}{(1, 3, 2)} + \frac{0.3}{(1, 3, 3)}

B {(3)}_{1} = \frac{0.1}{0} + \frac{0.2}{0.1} + \frac{0.5}{0.2} + \frac{0.8}{0.3} + \frac{1}{0.4} + \frac{0.8}{0.5} + \frac{0.7}{0.6} + \frac{0.6}{0.7} + \frac{0.4}{0.8} + \frac{0.2}{0.9} + \frac{0.1}{1}

B {(3)}_{2} = \frac{0.05}{0} + \frac{0.2}{0.1} + \frac{0.6}{0.2} + \frac{0.9}{0.3} + \frac{1}{0.4} + \frac{0.7}{0.5} + \frac{0.5}{0.6} + \frac{0.4}{0.7} + \frac{0.3}{0.8} + \frac{0.1}{0.9} + \frac{0.05}{1}

The

D U

M Z {(3)}_{1}

and

M Z {(3)}_{2}

is calculated showed Table 4 with different

0 \leq p, q \leq 5

, and when

p = q

(

0 \leq p, q \leq 5

) the expressed by Figure 2 and Figure 3.

5. Multi-attribute decision making method with hidden probability based on $M Z s$ utility function

In this section, a multi-attribute decision making method(MAGDM) with hidden probability based on

M Z s

utility function is constructed. The weight of experts is unknown, the weight vector of experts and the ranking of attributes are calculated by MAGDM method.

The procedure of the novel MAGDM method with hidden probability based on discrete

M Z s

can be represented as Figure 4.

Figure 2. DU of 2 Discrete

M Z s

(

p \neq q

)

Figure 2. DU of 2 Discrete

M Z s

(

p \neq q

)

Figure 3. DU of 2 Discrete

M Z s

(

p = q

)

Figure 3. DU of 2 Discrete

M Z s

(

p = q

)

A multi-attribute group decision making (MAGDM) method based on a multi-dimensional Z-number (

M Z s

) utility function. Its main purpose is to select the optimal system by calculating and optimising the weighted utility of decision options in combination with the evaluation of decision experts, as shown in Algorithm 1. The input parameters are given in line 1. Lines 4 to 6 describe the process of defining the decision matrix

E_{i}

for each expert. Lines 7 to 9 explain how to synthesise a multidimensional decision matrix D containing all expert evaluations. Lines 10 to 12 involve calculating the utility value

u_{j i}

for each alternative for each expert. Lines 13 and 14 initialise the weight vector

ω

. Lines 13 and 14 initialise the weight vector

ω

. Lines 20 to 24 describe the process of building the optimal weight model, including the objective function and constraints. Lines 20 to 24 describe the process of building the optimal weight model, including the objective function and constraints.

6. Case Study

This section provides a comprehensive performance of the new energy car evaluation problem to highlight the applicability of the decision model based on

M Z s

utility and demonstrate its strengths.

Figure 4. The novel

M C G D M

method with hidden probability based on

M Z s

Figure 4. The novel

M C G D M

method with hidden probability based on

M Z s

New energy car is a hot topic nowadays, which use new energy instead of ancient energy. In daily life, people use the car which consumes a lot of oil, however, the oil belongs to fossil energy that is limited in the earth. Therefore the new energy car can help humans save on the planet’s limited non-renewable energy. Meanwhile, the car will emission harmful gases such as carbon dioxide and nitrogen, the new energy car not.

There are many choices for new energy cars in the car market, but their comprehensive performance evaluation is an important problem. The comprehensive performance evaluation of new energy cars can be conducted inconsistently with their comprehensive effects on the environment, society, and economy. However, those effects have many uncertain and are difficult to quantify[54]. Therefore, the

M Z s

be used to express that in this case study.

A new energy car company evaluates the comprehensive performance of four new energy cars (denote by

A = {a_{1}, a_{2}, a_{3}, a_{4}})

. The performance of a new energy car contains energy per mile index, mileage utilization index, daily average economic index, regional applicability index, seasonal applicability index, operational reliability index, regulatory system reliability index, failure pre-alarm safety index, accident frequency evaluation index. That is divided into four comprehensive performances are economic index

c_{1}

, environmental applicability index

c_{2}

, reliability index

c_{3}

, safety index

c_{4}

. A professional team that includes three experts (denoted by

E = {e_{1}, e_{2}, e_{3}}

) is invited to assist in the evaluation.

6.1. Multi-attribute Group Decision Model Based on Discrete $M Z s$ Utility

1) Decision question description.

There are 4 alternatives under 4 criteria and 3 decision experts, denote

A = {a_{1}, a_{2},

a_{3}, a_{4}}

and

C = {c_{1}, c_{2}, c_{3}, c_{4}}

and

E = {e_{1}, e_{2}, e_{3}}

. Table 5 and Table 6 respectively show the discrete fuzzy numbers corresponding to the linguistic term set. And the decision matrix of three decision experts as Table 7, Table 8, Table 9.

Algorithm 1 MAGDM with Hidden Probability Based on

M Z^{'} s

Utility Function.

Input: decision experts

e_{i}

i = 1, . . ., l

, alternatives

s_{j}

j = 1, . . ., m

, criteria

c_{k}

k = 1, . . ., n

Define decision matrix Z for each expert: fori from 1 to l do

E_{i} = {[Z_{j k}^{i}]}_{m \times n} = {[(A_{j k}^{i}, B_{j k}^{i})]}_{m \times n}

end for Synthesize multidimensional decision matrix: fori from 1 to l do

D = {[M Z {(n)}_{j i}^{l}]}_{m \times l} = {[(A_{j i}, B_{j i})]}_{m \times l}

end for Calculate the utility of MZs: fori from 1 to l do

U = {[u_{j i}]}_{m \times l}

end for Initialize weight vector:

ω = [ω_{1}, ω_{2}, \dots, ω_{l}]

Calculate the deviation and total deviation: fori from 1 to l do

V_{j i} = \sum_{s = 1}^{n} | u_{j i} \cdot ω_{i} - u_{s i} \cdot ω_{i} |

V_{i} = \sum_{j = 1}^{m} V_{j i} = \sum_{j = 1}^{m} \sum_{s = 1}^{n} | u_{j i} \cdot ω_{i} - u_{s i} \cdot ω_{i} |

end for Build the optimization weight model: maxV = max(

\sum_{i = 1}^{l} \sum_{j = 1}^{n} \sum_{s = 1}^{n} | u_{j i} - u_{s i} | \cdot ω_{i}

) subject to the constraints:

m a x V = m a x {\sum_{i = 1}^{l} \sum_{j = 1}^{n} \sum_{s = 1}^{n} | u_{j i} - u_{s i} | \cdot ω_{i}}

\{\begin{matrix} \sum_{i = 1}^{l} ω_{i} = 1 0 \leq ω_{i}^{2} \leq 1 \end{matrix}

Compare the weighted utility of each alternative:

\bar{U} = {[{\bar{U}}_{A_{j}}]}_{1 \times n} = {[\sum_{i = 1}^{l} u_{j i} \cdot ω_{i}]}_{1 \times n}

Return: weighted utility argmax(

\bar{U}

)

2) Synthesize multidimensional decision matrix.

Using the discrete

M Z s

synthesize method above, the three expert decision matrices are combined into a four-dimensional Z-numbers matrix (ie, a matrix of 4 rows and 1 column) and the four-dimensional Z-numbers of the three experts are combined into one expert decision matrix D.

The

E_{j k}^{i} = [Z_{j k_{1}}^{i}, Z_{j k_{2}}^{i}, Z_{j k_{3}}^{i}, Z_{j k_{4}}^{i}] = [(s_{j k_{1}}, s_{j k_{1}}^{'}), (s_{j k_{2}}, s_{j k_{2}}^{'}), (s_{j k_{3}}, s_{j k_{3}}^{'}), (s_{j k_{4}},

s_{j k_{4}}^{'})]

synthesize

M Z s

obtain

M Z {(4)}_{j k}^{i}

(because the paper is limits the value show as Table 10 express

G_{j k}

M Z {(4)}_{j k}^{i}

). Where

⨁_{1}

is selected as synthesis operator, as follows:

The

E_{11}^{1}

as an example showed as follows:

G_{11}^{1} = \frac{0}{(4, 4, 4, 5)} + \frac{0}{(4, 4, 4, 5.25)} + \dots \frac{0}{(4, 4, 4, 5.5)} + \dots + \frac{0}{(6, 6, 6, 7)}

where the number of elements for

G_{11}^{1}

9^{4} = 6561

, which is permutation and combination about the discrete fuzzy number. Therefore, the uncertain information of the synthesize

M Z s

is more than Z-numbers.

B {(4)}_{11}^{1} = \frac{0}{0.72} + \frac{0.2}{0.75} + \frac{0.3}{0.78} + \frac{0.5}{0.81} + \frac{0.5}{0.84} + \frac{0.5}{0.87} + \frac{0.3}{0.9} + \frac{0.2}{0.93} + \frac{0}{0.9}

3) Calculate the utility of discrete $M Z s$ .

We calculate the utility values of the geometric form, algebraic form, and information entropy form of the discrete

M Z s

, such as the matrix

A U, G U, I U

. Since the utility value in the form of cross-entropy is not applicable in this case, the decision result in this form is not considered.

A U = [\begin{matrix} 9.563 & 9.0041 & 8.5427 \\ 8.6875 & 8.8214 & 9.6257 \\ 8.9792 & 8.7425 & 8.2541 \\ 9.2534 & 8.4025 & 9.7348 \end{matrix}]

G U = [\begin{matrix} 1.3471 & 1.6034 & 1.5146 \\ 1.5342 & 1.5201 & 1.6094 \\ 1.4862 & 1.6427 & 1.3452 \\ 1.4971 & 1.3958 & 1.4635 \end{matrix}]

I U = [\begin{matrix} 1.8509 & 2.5581 & 3.3491 \\ 2.5421 & 3.5490 & 3.1442 \\ 3.1342 & 1.7509 & 2.5461 \\ 1.4509 & 3.1682 & 3.5451 \end{matrix}]

Step 4: Calculate the weight of decision experts.

According to the optimal expert weight model, the expert weight vector is calculated.

ω^{A U}

ω^{G U}

and

ω^{I U}

represent the expert weight in algebraic form, geometric form, and information entropy form utility function respectively.

ω^{A U} = [0.3251, 0.4275, 0.8575]

ω^{G U} = [0.3002, 0.4563, 0.9516]

ω^{I U} = [0.5587, 0.6752, 0.4527]

5) Compare the weighted utility of each alternative.

{\bar{U}}^{A U} = [{\bar{U}}_{A_{1}}^{A U}, {\bar{U}}_{A_{2}}^{A U}, {\bar{U}}_{A_{3}}^{A U}, {\bar{U}}_{A_{4}}^{A U}] = [13.8985, 14.4372, 13.8566, 14.0458]

{\bar{U}}^{G U} = [{\bar{U}}_{A_{1}}^{G U}, {\bar{U}}_{A_{2}}^{G U}, {\bar{U}}_{A_{3}}^{G U}, {\bar{U}}_{A_{4}}^{G U}] = [2.3587, 2.26452, 2.2227, 2.27524]

{\bar{U}}^{I U} = [{\bar{U}}_{A_{1}}^{I U}, {\bar{U}}_{A_{2}}^{I U}, {\bar{U}}_{A_{3}}^{I U}, {\bar{U}}_{A_{4}}^{I U}] = [4.3487, 5.3903, 4.5411, 4.4852]

The decision results of

M Z s

synthesized by operators

⨁_{1}

and

⨁_{2}

are shown in Table 11.

6.2. The Explanation of Decision Results

The decision results are expressed by Table 11. when the synthesis operator

⨁_{1}

is chosen, the decision results of the geometric form and information entropy form both is

a_{2} ≻ a_{4} ≻ a_{1} ≻ a_{3}

, the decision result of the algebraic form is

a_{4} ≻ a_{2} ≻ a_{1} ≻ a_{3}

. And when the synthesis operator

⨁_{2}

is chosen, the decision results of the geometric form and algebraic form both is

a_{2} ≻ a_{4} ≻ a_{3} ≻ a_{1}

, however the decision result of the information entropy form is

a_{2} ≻ a_{4} ≻ a_{1} ≻ a_{3}

As shown in Table 11, the ranking results change when different forms of utility for

M Z s

under the same synthesis operator. And the ranking results vary slightly when the same form utility of

M Z s

with different synthesis operator. This shows that the model based on the

M Z s

utility function proposed in this study is sensitive to the changes in different forms of

M Z s

utility function and the synthesis operator of synthesis discrete

M Z s

. From this, the conclusion can be drawn both factors play a critical role in determining the final ranking of alternatives.

Generally speaking, the more complicated the synthesis operators of multidimensional Z-number is chosen, the more precise the decision model results. Inversely, the easier the synthesis operators for discrete

M Z s

are chosen, the weaker the accuracy of the decision model results. This shows the decision model according to different synthesis operators and a suitable utility form is chosen by actual decision problems. Therefore, real-world group decision-making problems involve complicated information and experts from distinct backgrounds with different degrees of expertise. And it is important and reasonable to take these factors into consideration.

7. The Analysis of MCGDM Results

7.1. Sensitivity Analysis with Utility Functions and Synthesis Operators

In this subsection, according two factors (synthesis operator and different forms utility function) of MCGDM model to analysis the influences with Table 11.

When the synthesis operator

⨁_{1}

is selected, the optimal ranking results of the utility function in the form of geometric information entropy are consistent. Therefore, it can be inferred that the uncertain information ratio of these two forms of utility function measures

M Z s

is consistent, which leads to the consistent ranking result of the scheme. We can’t simply assume that the uncertain information of

M Z s

measured by different forms of the utility function is equal to the ranking result of the scheme. We can only infer that the ratio of the uncertain information measured by them is equal when the ranking result is the same.

When the synthesis operator

⨁_{2}

is selected, the utility functions of algebraic and geometric forms get the same scheme ranking result, and the information entropy form is slightly different. Operator

⨁_{2}

is more complex than operator

⨁_{1}

. Therefore, during the synthesis of

M Z s

, uncertain information will be increased or decreased, thus leading to different sorting results.

Next, let’s analyze the effect of choosing the same form of utility function on the decision result. The difference between the comprehensive utility values of the optimal scheme and the suboptimal scheme obtained by the utility function in the form of algebra, geometry and information entropy are shown as follow:

Δ {\bar{U}}_{12}^{A U} = {\bar{U}}_{A_{4}}^{A U} - {\bar{U}}_{A_{2}}^{A U} = 0.0383

Δ {\bar{U}}_{12}^{G U} = {\bar{U}}_{A_{2}}^{G U} - {\bar{U}}_{A_{4}}^{G U} = 0.0025

Δ {\bar{U}}_{12}^{I U} = {\bar{U}}_{A_{2}}^{I U} - {\bar{U}}_{A_{4}}^{I U} = 0.05677

So it can be known that the difference between the comprehensive utility values obtained by the geometric utility function is the smallest. That is to say, the selection differentiation between the optimal scheme and the sub-optimal scheme is also the smallest.

In the same form of the utility function, the utility function in the form of information entropy is the most stable among different synthesis operators. Since information entropy is used to measure the uncertainty of information in probability, we think that the utility function in the form of information entropy is most appropriate to measure the uncertainty of

M Z s

In general, different forms of utility functions measure the hidden uncertainty contained in

M Z s

from different perspectives. Different synthesis operators use different ideas to synthesize

M Z s

, providing information that one-dimensional Z-number cannot directly present. Decision makers can choose the form of synthesis operator and utility function according to their own subjective needs.

7.2. Comparative Analysis with the Existing

Several other methods were compared with our novel method and analyzed to further demonstrate the validity. This analysis is based on the case study described in subsection 6.1. The MCGDM method in references[26,29].

The method of reference[56], the A and B of Z-numbers were transformed to triangular fuzzy numbers that use a multiplication operation convert to crisp values. The final priority weight was obtained and rank all alternatives, the answer of all alternatives as

ω = (ω_{1}, ω_{2}, ω_{3}, ω_{4}) = (0.7153, 0.2457, 0.2012, 0.5124)

, which is the ranking was

a_{1} ≻ a_{4} ≻ a_{2} ≻ a_{3}

The method of reference[25], the linguistic Z-numbers were addressed directly by the extended

T O D I M

method. Then, the result of comprehensive evaluation value of each alternative can be calculated as follows:

ξ = (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) = (0, 0.8156, 0.4235, 1)

and the ranking result was

a_{4} ≻ a_{2} ≻ a_{3} ≻ a_{1}

The method of reference[29], the linguistic values with Z-numbers are transformed to

T r F N s

. And the closeness coefficient vector for all alternatives by the modified

T O P S I S

method of the reference which is

C C = (C C_{1}, C C_{2}, C C_{3}, C C_{4}) = (0.1246, 0.1153, 0.4216, 0.5235)

. Therefore, the ranking result was

a_{4} ≻ a_{3} ≻ a_{1} ≻ a_{2}

The method of reference[55], the dominance degree of two Z-numbers was defined based on the outranking relation. According to the translating reality III (ELECTRE III) and qualitative flexible multiple criteria method (QUALIFLEX) with the dominance degree obtain the ranking result as follows:

a_{4} ≻ a_{1} ≻ a_{2} ≻ a_{3}

. The ranking obtained by different methods are displayed in Table 12.

The decision results in Table 11 and Table 12, in which the rankings obtained by the four different methods are inconsistent with the method of this paper. The major reason is the uncertain environment and the handling of uncertainly. The existing method is a decision matrix based on a one-dimensional Z-number. That difference is decision matrix is synthesized discrete

M Z s

by synthesis operator combine with the hidden probability of discrete

M Z s

in this paper. The reference[29,56] damaged and distorted the uncertain and essence of Z-number by translated A, B to fuzzy number. They ignore the probability information of Z-number according to the reference[57,58]. In this paper, the one-dimensional Z-number is expressed

M Z s

by a new synthesis method, then the hidden probability of

M Z s

is calculated by the hidden probability model and different form utility function are defined consider the restriction and hidden probability of

M Z s

In addition, the linguistic scale functions were used to deal with Z-number with linguistic variables in reference[25,26]. That shortcoming was the randomness were ignored by treating processes. The outranking method used in reference[55] is a relation model, which considers the non-compensation principle among criteria to some extent, then they modeled pseudo-criteria by introducing three thresholds and implemented interaction with experts in the decision-making process. The outranking contain subjective component and the choice with three thresholds will affect the result of ranking.

The comparative analysis results indicate that the proposed method can be applied successfully to the evaluation of new energy cars, which is to identify more convincing and reasonable outcomes than existing methods. According to synthesis several criterions of each alternative become

M Z s

, then contain with the different forms of utility function based hidden probability of

M Z s

to make decision obtain the ranking. Because this method considered the hidden probability and it is more applicable than previous methods based on function models.

8. Conclusions

To address the problem of multiple experts assessing multiple attributes and multiple criteria for decision-making, this study introduced and used

M Z s

. First, the synthesize method of discrete

M Z s

is defined. Second, the hidden probability model of discrete

M Z s

is calculated and the multidimensional

Z^{+} - n u m b e r

(

M Z^{+}

) is proposed underlying

M Z s

and the hidden probability. In this paper, we consider not only the membership of discrete

M Z s

, but also the hidden probability. In the calculation of the probability measure of discrete

M Z s

, the "exponential probability" synthesis operator is defined, considering the influence of the hidden probability of

M Z s

is 0. With the increase of hidden probability, the value of "exponential probability" is smaller, so as to reduce the influence of membership degree in the calculation process. Next defined some different forms utility of discrete

M Z s

. Then MAGDM method based on

M Z s

utility function is constructed. Finally, the decision model was tested using a new energy car evaluation problem, and it was further validated through analyses and conclusion.

The main contributions of this study can be summarized as follows. First, the proposal of the synthesizing method for discrete

M Z s

can help synthesize uncertain information contained in multiple single or

M Z s

. In this way, extend from one-dimensional definitions to multidimensional spaces. And the hidden probability of

M Z s

was calculated by the probability model, the membership function, and probability density function within

M Z s

were reflected explicitly, the corresponding numerical characters were extracted.Second, four forms of utility function expressing the information expressed by a discrete

M Z s

in a single number is very intuitive and concise. The first and second are the innovation in this study. Finally, the proposed approach is fairly flexible and suitable with multi-attribute group decision problem.

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Table 1. Research Status of Utility Function in Decision Model

I: Mathematical Structure	Common Utility Function Formula	Applications
continued table
A.Linear[37]	$\{\begin{matrix} 0 x < x_{m i n} \\ \frac{x - x_{m i n}}{x_{m a x} - x_{m i n}} x_{m i n} < x < x_{m a x} \\ 1 o t h e r w i s e \end{matrix}$	Users choose the radio access network which meets their data transfer terms best.
B.Nonlinear
Power Function[38]	$- {(1 - a D)}^{2}, D \leq \frac{1}{a}$	Describe customers’ behavior.
Exponential[39]	$e^{x - k}, 0 \leq x \leq k$	Decide the "best" network interface and "best" time moment to handoff.
Sigmoid[40]	$\frac{{(x / x_{m})}^{c}}{1 + {(x / x_{m})}^{c}}$	Identify the characteristics of the wireless systems to quantify the quality of service and to analytically investigate the users’ satisfaction.
Logarithm[41]	$ln (x)$ or $ln (1 + c x)$	Estimating current network conditions.
$I I$ : Construction Criterions	Common Utility Function Formula	Applications
A.Probabilistic Equivalent
Expected[42]:	$\int μ {p_{Ψ}^{+} (.), Ψ} p_{Ψ} (Ψ \| x) d Ψ$	Design an experiment to maximize the expected information to be gained from it.
Preference Relations[43]	$U (\sum_{j = 1}^{n} ω_{i j}) U (\frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} ω_{i j}})$	Generalization of numerous representative utility functions.
Entropy[44]	$\sum_{m = 1}^{K} p (m \| y_{t}, d_{t}) \int \int p (z, θ_{m} \| m, z, y_{t}, d_{t}, d)$
	$l o g [\frac{p (z, θ_{m} \| m, z, y_{t}, d_{t}, d)}{p (z, θ_{m} \| m, z, y_{t}, d_{t}, d)} p (m \| y_{t}, d_{t})] d z d θ_{m}$	The utility will prefer designs for which the outcome Z is most uncertain after removing the uncertainty due to experimental error.
B.Certainty Equivalents
Multiple Aspirations[45]	$\{\begin{matrix} 1 c_{i} f_{i} (X) \leq \underset{̲}{g_{i}} \\ \frac{\bar{g_{i}} - c_{i} f_{i} (X)}{\bar{g_{i}} - \underset{̲}{g_{i}}} \underset{̲}{g_{i}} \leq c_{i} f_{i} (X) \leq \bar{g_{i}} \\ 0 \bar{g_{i}} \leq c_{i} f_{i} (X) \end{matrix}$	This utility function is characterized as the left linear utility function
	$\{\begin{matrix} 1 c_{i} f_{i} (X) \leq \underset{̲}{g_{i}} \\ \frac{c_{i} f_{i} (X) - \bar{g_{i}}}{\bar{g_{i}} - \underset{̲}{g_{i}}} \underset{̲}{g_{i}} \leq c_{i} f_{i} (X) \leq \bar{g_{i}} \\ 0 \bar{g_{i}} \leq c_{i} f_{i} (X) \end{matrix}$	(LLUF) and the right linear utility function (RLUF), multiple aspirations with utility functions.
C.Gain Equivalents[46]
Marginal Utility	$k_{1} (r q + q_{m})$ , $r < 0$	Online checkout and web hosting workloads.
Isoelastic Utility	$K_{2} \{\begin{matrix} \frac{q^{1 - α}}{1 - α}, α \neq 1 \\ ln (q), α = 1 \end{matrix}$	Backend and content delivery workloads.
$I I I$ : Analysis Contents	Common Utility Function Formula	Applications
A.Risk Attitude[38]	$1 - e^{- a D}$ , $a > 0$	Describe the risk preferences of the customers.
B.Consequences of Preference[47]	$\sum_{i = 1}^{n} w_{i} ϕ_{i} (x_{i})$	A multi-pattern utility function.
C.Measurable Value[46]	$U_{0} - α p - β t$	Subjectively measure profit, customer satisfaction, and cloud resource utilization.
$I V$ : Language Environment	Common Utility Function Formula	Applications
A.Fuzzy Number[47]	$\frac{1}{α - 1} [1 - {(\frac{D (x) - D_{m i n}}{D_{m a x} D_{m i n}})}^{α}]$	A single-pattern utility function.
B.TFN [43]	$\frac{a_{1} + 2 a_{2} + a_{3}}{4}$	A triangular fuzzy number utility function.
C. IVIFN [48]
	General: $\frac{S^{'} {(A)}^{H^{'} (A) E^{'} (A)} - S^{'} (A)}{1 - S^{'} (A)} S^{'} (A)$	Describe the relationship between $S^{'} (A)$ , $H^{'} (A)$ , $E^{'} (A)$ .
	Weight: $[{(S^{' (1 - w_{1})})}^{H^{' 1 - w_{2}} \cdot E^{' w_{1} + w_{2}}}$	The importance of $S^{'} (A)$ , $H^{'} (A)$ ,
	$- S^{'}] \frac{S^{'}}{1 - S^{'}}$	$E^{'} (A)$ is taken into account.
	Probability: $\int_{0}^{1} \int_{0}^{1 - w_{2}} U_{(W e i g h t)} d w_{1} d w_{2}$	A probability-based utility function.
D. Z-number[31]	$\int_{0}^{1} \int_{0}^{1} \int_{- \frac{1}{2}}^{\frac{1}{2}} \int_{- \frac{1}{2}}^{\frac{1}{2}} (\frac{A_{1}}{2} + A_{2} x) e^{- A_{2}^{2}} (\frac{R_{1}}{2} + R_{2} x) e^{- R_{2}^{2}} d x d y d α d β$	Used to determine the ordering of Z-numbers, and applied in the application of multi-criteria decision making under uncertain environments.

Table 2. The membership of G

$G_{2}$ $G_{1}$		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)	(3,2)
		0.1	0.3	0.7	0.8	1	0.6	0.2
(1,1)	0.2	0.1	0.2	0.8	0.2	0.8	1	1
(1,2)	1	0.2	0.8	1	0.8	1	0.6	0.6
(2,1)	0.7	0.2	0.8	1	0.8	1	0.6	0.6
(2,2)	0.1	0.8	1	0.6	1	0.6	0.1	0.1

Table 3. Probability values of G

$P_{G_{2}}$ $P_{G_{1}}$		(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)	(3,2)
		0.2341	0.1284	0.0540	0.0427	0.0369	0.0712	0.4328
(1,1)	0.4146	0.5516	0.4898	0.4462	0.4396	0.4362	0.4563	0.6680
(1,2)	0.0869	0.3006	0.2041	0.1362	0.1258	0.1206	0.1519	0.4821
(2,1)	0.1339	0.3366	0.2451	0.1806	0.1709	0.1659	0.1956	0.5088
(2,2)	0.3646	0.5133	0.4462	0.3989	0.3917	0.3881	0.4098	0.6396

Table 4. The DU Values of

M Z {(3)}_{1}

and

M Z {(3)}_{2}

Table 4. The DU Values of

M Z {(3)}_{1}

and

M Z {(3)}_{2}

p	0.1	0.2	0.5	1	0.1	0.5	1	2
q	2	1	0.8	2.5	1	0.8	0.5	0.2
$D U$	0.2672	0.0910	0.0833	0.4843	0.0892	0.0833	0.1397	0.5076
p=q	0.1	0.2	0.5	1	2	3	4	5
$D U$	0.0007	0.0047	0.0469	0.2054	0.7719	1.6748	2.8718	4.2795

Table 5. Linguistic Terms in S and Corresponding Discrete Fuzzy Numbers[55]

Linguistic Terms in S	Discrete fuzzy numbers
Very Bad	0/0+0.25/0.25+0.5/0.5+0.75/0.75+1/1+0.75/1.25+0.5/1.5+0.25/1.75+0/2
Bad	0/1+0.25/1.25+0.5/1.5+0.75/1.75+1/2+0.75/2.25+0.5/2.5+0.25/2.75+0/3
Slightly Bad	0/2+0.25/2.25+0.5/2.5+0.75/2.75+1/3+0.75/3.25+0.5/3.5+0.25/3.75+0/4
Middle	0/3+0.25/3.25+0.5/3.5+0.75/3.75+1/4+0.75/4.25+0.5/4.5+0.25/4.75+0/5
Slightly Good	0/4+0.25/4.25+0.5/4.5+0.75/4.75+1/5+0.75/5.25+0.5/5.5+0.25/5.75+0/6
Good	0/5+0.25/5.25+0.5/5.5+0.75/5.75+1/6+0.75/6.25+0.5/6.5+0.25/6.75+0/7
Very Good	0/6+0.25/6.25+0.5/6.5+0.75/6.75+1/7+0.75/7.25+0.5/7.5+0.25/7.75+0/8

Table 6. Linguistic Terms in

S^{'}

and Corresponding Discrete Fuzzy Numbers[55]

Table 6. Linguistic Terms in

S^{'}

and Corresponding Discrete Fuzzy Numbers[55]

Linguistic Terms in $S^{'}$	Discrete Fuzzy Numbers
Strongly Uncertain	0/0+0.3/0.03+0.5/0.06+0.8/0.09+1/0.12+0.8/0.15+0.5/0.18+0.3/0.21+0/0.24
Uncertain	0/0.12+0.3/0.15+0.5/0.18+0.8/0.21+1/0.24+0.8/0.27+0.5/0.3+0.3/0.33+0/0.36
Somewhat Uncertain	0/0.24+0.3/0.27+0.5/0.3+0.8/0.33+1/0.36+0.8/0.39+0.5/0.42+0.3/0.45+0/0.48
Neutral	0/0.36+0.3/0.39+0.5/0.42+0.8/0.45+1/0.48+0.8/0.51+0.5/0.54+0.3/0.57+0/0.6
Somewhat Certain	0/0.48+0.3/0.51+0.5/0.54+0.8/0.57+1/0.6+0.8/0.63+0.5/0.66+0.3/0.69+0/0.72
Certain	0/0.6+0.3/0.63+0.5/0.66+0.8/0.69+1/0.72+0.8/0.75+0.5/0.78+0.3/0.81+0/0.84
Strongly Certain	0/0.72+0.3/0.75+0.5/0.78+0.8/0.81+1/0.84+0.8/0.87+0.5/0.9+0.3/0.93+0/0.96

Table 7. The Decision Expert

e_{1}

’s Rating

Table 7. The Decision Expert

e_{1}

’s Rating

	$c_{1}$	$c_{2}$	$c_{3}$	$c_{4}$
$a_{1}$	$(s_{5}, s_{4}^{'})$	$(s_{5}, s_{5}^{'})$	$(s_{5}, s_{3}^{'})$	$(s_{6}, s_{6}^{'})$
$a_{2}$	$(s_{3}, s_{5}^{'})$	$(s_{6}, s_{5}^{'})$	$(s_{3}, s_{5}^{'})$	$(s_{4}, s_{5}^{'})$
$a_{3}$	$(s_{5}, s_{6}^{'})$	$(s_{4}, s_{3}^{'})$	$(s_{4}, s_{2}^{'})$	$(s_{3}, s_{4}^{'})$
$a_{4}$	$(s_{6}, s_{5}^{'})$	$(s_{4}, s_{5}^{'})$	$(s_{6}, s_{3}^{'})$	$(s_{5}, s_{6}^{'})$

Table 8. The Decision Expert

e_{2}

’s Rating

Table 8. The Decision Expert

e_{2}

’s Rating

	$c_{1}$	$c_{2}$	$c_{3}$	$c_{4}$
$a_{1}$	$(s_{5}, s_{5}^{'})$	$(s_{4}, s_{4}^{'})$	$(s_{6}, s_{4}^{'})$	$(s_{4}, s_{5}^{'})$
$a_{2}$	$(s_{4}, s_{6}^{'})$	$(s_{3}, s_{3}^{'})$	$(s_{4}, s_{6}^{'})$	$(s_{6}, s_{3}^{'})$
$a_{3}$	$(s_{3}, s_{4}^{'})$	$(s_{3}, s_{5}^{'})$	$(s_{5}, s_{3}^{'})$	$(s_{4}, s_{6}^{'})$
$a_{4}$	$(s_{3}, s_{3}^{'})$	$(s_{3}, s_{2}^{'})$	$(s_{3}, s_{5}^{'})$	$(s_{4}, s_{4}^{'})$

Table 9. The Decision Expert

e_{3}

’s Rating

Table 9. The Decision Expert

e_{3}

’s Rating

	$c_{1}$	$c_{2}$	$c_{3}$	$c_{4}$
$a_{1}$	$(s_{4}, s_{5}^{'})$	$(s_{3}, s_{4}^{'})$	$(s_{4}, s_{3}^{'})$	$(s_{3}, s_{2}^{'})$
$a_{2}$	$(s_{6}, s_{3}^{'})$	$(s_{6}, s_{5}^{'})$	$(s_{5}, s_{4}^{'})$	$(s_{6}, s_{4}^{'})$
$a_{3}$	$(s_{3}, s_{5}^{'})$	$(s_{3}, s_{6}^{'})$	$(s_{3}, s_{3}^{'})$	$(s_{3}, s_{5}^{'})$
$a_{4}$	$(s_{6}, s_{2}^{'})$	$(s_{6}, s_{4}^{'})$	$(s_{6}, s_{5}^{'})$	$(s_{6}, s_{5}^{'})$

Table 10. The Membership of Synthesize

M Z {(4)}_{j k}^{i}

Table 10. The Membership of Synthesize

M Z {(4)}_{j k}^{i}

	$c_{2} (s_{k_{2}}) \to$	$k_{2} - 1$	$k_{2} - 0.75$	$k_{2} - 0.5$	$k_{2} - 0.25$	$k_{2}$	$k_{2} + 0.25$	$k_{2} + 0.5$	$k_{2} + 0.75$	$k_{2} + 1$
$c_{1} (s_{k_{1}}) ↓$		0	0.25	0.5	0.75	1	0.75	0.5	0.25	0
$k_{1} - 1$	0	0	0	0	0	0	0	0	0	0	0	$k_{3} - 1$
$k_{1} - 0.75$	0.25	0	0.25	0.25	0.25	0.25	0.25	0.25	0.25	0	0.25	$k_{3} - 0.75$
$k_{1} - 0.5$	0.5	0	0.25	0.5	0.5	0.5	0.5	0.5	0.25	0	0.5	$k_{3} - 0.5$
$k_{1} - 0.25$	0.75	0	0.25	0.5	0.75	0.75	0.75	0.5	0.25	0	0.75	$k_{3} - 0.25$
$k_{1}$	1	0	0.25	0.5	0.75	1	0.75	0.5	0.25	0	1	$k_{3}$
$k_{1} + 0.25$	0.75	0	0.25	0.5	0.75	0.75	0.75	0.5	0.25	0	0.75	$k_{3} + 0.25$
$k_{1} + 0.5$	0.5	0	0.25	0.5	0.5	0.5	0.5	0.5	0.25	0	0.5	$k_{3} + 0.5$
$k_{1} + 0.75$	0.25	0	0.25	0.25	0.25	0.25	0.25	0.25	0.25	0	0.25	$k_{3} + 0.75$
$k_{1} + 1$	0	0	0	0	0	0	0	0	0	0	0	$k_{3} + 1$
		$k_{4} - 1$	$k_{4} - 0.75$	$k_{4} - 0.5$	$k_{4} - 0.25$	$k_{4}$	$k_{4} + 0.25$	$k_{4} + 0.5$	$k_{4} + 0.75$	$k_{4} + 1$		$c_{3} (s_{k_{3}}) ↑$
		0	0.25	0.5	0.75	1	0.75	0.5	0.25	0	$\leftarrow c_{4} (s_{k_{4}})$

Table 11. Ranking of Schemes with Different Utility Function Forms and Synthesize Operator

Utility Function Form	Synthesis Operator $⨁_{1}$	Synthesis Operator $⨁_{2}$
AU	$a_{4} ≻ a_{2} ≻ a_{1} ≻ a_{3}$	$a_{2} ≻ a_{4} ≻ a_{3} ≻ a_{1}$
GU	$a_{2} ≻ a_{4} ≻ a_{1} ≻ a_{3}$	$a_{2} ≻ a_{4} ≻ a_{3} ≻ a_{1}$
IU	$a_{2} ≻ a_{4} ≻ a_{1} ≻ a_{3}$	$a_{2} ≻ a_{4} ≻ a_{1} ≻ a_{3}$

Table 12. Ranking results obtained by different methods

Methods	Ranking Results
Final priority weight[56]	$a_{1} ≻ a_{4} ≻ a_{2} ≻ a_{3}$
Comprehensive evaluation value[25]	$a_{4} ≻ a_{2} ≻ a_{3} ≻ a_{1}$
Closeness coefficient[29]	$a_{4} ≻ a_{3} ≻ a_{1} ≻ a_{2}$
ELECTRE III and QUALIFLEX[55]	$a_{4} ≻ a_{1} ≻ a_{2} ≻ a_{3}$

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Enhancing Cyber Defense Strategies with Discrete Multidimensional Z-Numbers: A Multi-Attribute Decision-Making Approach

Abstract

1. Introduction

2. Preliminaries

3. Synthesizing Discrete Multidimensional Z-numbers

4. The Different Forms Utility Function of Discrete M Z s

4.1. The Math Forms Utility Function of Discrete M Z s

4.2. The Entropy Forms Utility Function of Discrete M Z s

5. Multi-attribute decision making method with hidden probability based on M Z s utility function

6. Case Study

6.1. Multi-attribute Group Decision Model Based on Discrete M Z s Utility

6.2. The Explanation of Decision Results

7. The Analysis of MCGDM Results

7.1. Sensitivity Analysis with Utility Functions and Synthesis Operators

7.2. Comparative Analysis with the Existing

8. Conclusions

References

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4. The Different Forms Utility Function of Discrete $M Z s$

4.1. The Math Forms Utility Function of Discrete $M Z s$

4.2. The Entropy Forms Utility Function of Discrete $M Z s$

5. Multi-attribute decision making method with hidden probability based on $M Z s$ utility function

6.1. Multi-attribute Group Decision Model Based on Discrete $M Z s$ Utility