An Exploration on Z-Number and Its Properties

Abstract

The Z-number deserves further exploration in uncertain environments to effectively address fuzziness and reliability in practical issues simultaneously. Based on the credibility distribution and conversion rules, we define its expected value, variance, and semi-variance, and demonstrate the feasibility of these calculations by deriving formulas. Moreover, we explore three characteristics inherent in symmetrical Z-numbers. The link between the variance and semi-variance of Z-numbers is discovered and proved. Furthermore, we apply the formulas for expected value and variance to Z-numbers in examples, the results of which validate our proposed formulas. The findings underscore the significance of our study in applying the expected value and variance of fuzzy sets across diverse fields.

1. Introduction

Living in a world of probability, fuzziness, and uncertainty, we cannot avoid dealing with vagueness in everyday lives [1]. Just as the “Principle of Valence”, proposed by Lukasiewicz, asserts that every proposition has a truth value, in the context of fuzzy sets, an element can partially belong to a set. This partial membership relation allows for a more nuanced analysis of the truth values of propositions. Subsequently, the concept of fuzzy sets has been conceived, refined, and extended in numerous scenarios, as detailed in a substantial body of literature [2]. However, after a certain point in this type of research, it becomes clear that none of these concepts could deal with real problems involving fuzziness and reliability. To address this dilemma, in 2011, Zadeh [3] first formally introduced Z-numbers. A Z-number is conventionally depicted as an organized pair of fuzzy numbers, symbolized as $Z=(A,B)$, where A denotes the fuzzy restriction that confines the value adoption of the real-valued uncertain variable X, and B signifies the reliability restriction, which is a fuzzy limitation on the probability measure of A. Based on fuzzy sets [4], Z-numbers combine constraint and reliability to give both a description of fuzziness and a measure of the reliability of the evaluated information, which in turn can better explain the information conveyed by natural human language on complex uncertainty problems. For example, for a fuzzy event, “the train from Shanghai to Beijing is likely to take about 5 hours”, where Z = (about 5 h, likely). Therefore, Z-numbers have been widely used in plenty of practical issues. For instance, in the psychology area, Kushal [5] combined Z-numbers with psychometric scales in order to extract useful information. In the field of safety accident analysis, Z-numbers have been used to analyze uncertainties in the risks associated with building construction [6] and open surface mining [7]. In addition, Z-numbers also contributed to medical diagnosis issues [8], portfolio problems [9], and so on.

When studying practical problems in these fields, the mathematical properties of Z-numbers are often used, making research in this area indispensable. In particular, expected value and variance—two significant statistical measures—are used to reflect the degree of the average value and the concentration of data, as well as to indicate the magnitude of fluctuations within a set of data. These measures have broad applications in fields such as mathematics, economic trade, education, agriculture, and many others that deal with quantitative and mathematical data [10]. This is also true in practical issues with ambiguous, imprecise, and uncertain information, where the expected value and variance of fuzzy variables are important tools for the quantitative studies of such problems. We summarize the main studies involving the mathematical characteristics of Z-numbers, as

Table 1. A summary of the literature on the mathematical properties of Z-numbers.

Literature	Category		Arithmetic Operations					Conversion
	Discrete	Continuous	Fundamental Operations	Extremum	Square/ Square-Root	Expected Value	Variance/ Semi-variance
Kang et al. (2012) [11]	√							√
Aliev et al. (2015) [12]	√		√		√
Aliev et al. (2016) [13]		√	√	√	√
Jiang et al. (2017) [14]	√
Aliev et al. (2018) [15]	√	√	√	√	√
Alizadeh et al. (2018) [16]	√	√	√
Aliev & Alizadeh (2018) [17]	√	√	√
Peng et al. (2019) [18]		√	√					√
Cheng et al. (2021) [19]	√							√
Jia & Hu (2022) [20]	√	√	√					√
This Paper	√	√	√			√	√	√

Annotation: √ denotes related research for the content discussed in this article.

presents, and can see that research on Z-number operations is gradually becoming more comprehensive.

To address the challenge of determining the underlying possibility distribution of a Z-number, Kang et al. [11] first presented a transformation method between Z-numbers and classical fuzzy numbers in 2012. This method was further improved by Cheng et al. [19] in 2021, who considered the effect of hidden probability distribution generated on Z-numbers. In addition to this general conversion method, other innovative transformation approaches like the Z-trapezium-normal clouds (ZTNCs) have been proposed to simplify the calculation [18]. Since 2015, scholars, typified by Aliev, began researching Z-number mathematical operations. For instance, Aliev et al. [12] focused on arithmetic operations on discrete Z-numbers, including the four fundamental operations, squaring, square root, and ordering in 2015. Less than two years afterward, Aliev et al. [13] introduced formulas for basic arithmetic operations on continuous ones. In 2017, Jiang et al. [14] introduced a new generalized ranking method that combines the spreads of fuzzy numbers, the weights of centroid points, and the degrees of fuzziness. The following year, Aliev et al. [15] developed a general approach to constructing Z-number functions based on the topological principle, creating typical functions such as multiplication, exponents, and the minimum and maximum of Z-numbers. In 2018, two groups of researchers, Alizadeh et al. [16] and Aliev and Alizadeh [17], targeted the study of Z-numbers with fundamental properties under additive and multiplicative operations, respectively. In 2020, Mazandarani and Zhao [21] further discussed the concept of the Z-differential equation and described its framework in great detail. In 2022, Jia and Hu [20] were the first to adopt a rectangular coordinate system to handle linguistic Z-numbers, while also defining arithmetic operations.

Numerous theoretical innovations of Z-numbers, including their transformations and algorithms, have emerged over the last decade [20]. However, compared to the existing studies on fuzzy set theory, the development of Z-number theory is still in its early stages. This is particularly true in the area of mathematical features of Z-numbers, where the lack of operational laws for expected value, variance, and semi-variance highlights significant research gaps. Meanwhile, the calculations proposed in existing research are inadequate for complex practical problems. Expected value, variance, and semi-variance, as three common digital characteristics, are of theoretical and practical importance in reflecting the properties of a particular aspect of a set of data [22]. In addition, from an information point of view, characterization effectively compresses information, which is one reason why there is substantial research focusing on these characteristics [23]. Undoubtedly, it is also of great significance for Z-numbers that must address not only the uncertainty of information but also its reliability. To fill the arithmetic gap and provide the corresponding implementation-friendly calculation methods, this paper focuses on the operational laws of these three characteristics mainly based on their own definitions and the Z-number transformation method introduced by Kang et al. [11].

Therefore, in this study, we explore the concepts of Z-numbers’ expected value, variance, and semi-variance for the first time and provide further discussion. Inspired by Kang et al. [11], we convert Z-numbers into regular fuzzy numbers to carry out our work, including the definitions, theorems, and calculation formulas. During this conversion process, the information contained in the Z-numbers is transformed into fuzzy numbers. This paper then shifts the study of expected value, variance, and semi-variance to that of the corresponding mathematical properties of its converted regular fuzzy numbers, referred to as converted Z-numbers. Then, according to the credibility theory by Liu [24], the concepts of these three digital characteristics of Z-numbers are delineated, respectively, and the related theorems and formulas are derived. Meanwhile, examples are displayed to illustrate the calculation process and its application to help the reader understand. The validation of our findings demonstrates the broad applicability of this paper’s formulas for calculating the expected value, variance, and semi-variance of various types of Z-numbers, not limited to a specific category. At the same time, the computational complexity of these formulas is not high, which facilitates practical application. Moreover, using the expected value, variance, and semi-variance of Z-numbers has great potential to help in analyzing the mathematical properties associated with fuzzy events, allowing for the effective management of information involving fuzziness and reliability. This implies that the successful implementation of these three numerical calculations could positively impact the application of Z-numbers in practical issues and expand the scope of research into the mathematical properties of Z-numbers. Furthermore, the introduction of these formulas enhances the application of expected value, variance, and semi-variance across the entire field of fuzzy sets.

This article is organized as follows. Relying on certain basic concepts of Z-numbers, as shown in the Supplementary Materials and credibility distribution, we present the formulas for calculating the expected value of Z-numbers in Section 2. The formulas for the variance are provided in Section 3. Furthermore, this section includes two examples that demonstrate the calculation of both asymmetrical and symmetrical Z-numbers, combining the expected value and variance formulas proposed. In Section 4, we present the definitions and theorems related to the semi-variance of Z-numbers. Finally, Section 5 presents our conclusions.

2. The Expected Value of Z-Numbers

Due to the connection between Z-numbers and the converted regular fuzzy numbers derived by Kang et al. [11], we treat the information on Z-numbers as pertaining to the converted Z-number. In this section, we provide the definition and calculation formulas for the expected value of Z-numbers using the credibility distribution; this be deduced through Definition 6 in the Supplementary Materials [25,26,27].

Definition 1. 

Let Z be a Z-number, and let us denote its converted regular fuzzy number as $\tilde{Z}$. The expected value can then be represented as follows:

$$E\left[Z\right]={\int }_0^{+\infty }Cr\{\tilde{Z}\ge u\}du-{\int }_{-\infty }^{0}Cr\{\tilde{Z}\le u\}du.$$

(1)

where $Cr$ denotes the credibility measure proposed in [27].

Theorem 1. 

Let Z be a Z-number, and let us denote its converted regular fuzzy number as $\tilde{Z}$. The expected value can then be formulated as follows:

$$E\left[Z\right]={\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(\alpha \right)d\alpha ,$$

(2)

where ${\displaystyle {\mathrm{\Phi }}_{\tilde{Z}}^{-1}}$ is the inverse credibility distribution of ${\displaystyle \tilde{Z}}$.

Proof. 

Since a Z-number can be converted into a regular fuzzy number, denoted as ${\displaystyle \tilde{Z}}$, its credibility distribution ${\displaystyle \mathrm{\Phi }\left(u\right)}$ functions as monotone (but not strictly monotone) according to the operational law proposed by Zhou et al. [28]. Therefore, according to the work of Zhao et al. [29], the expected value of ${\displaystyle \tilde{Z}}$ can be represented by its inverse credibility distribution ${\displaystyle {\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(\alpha \right)}$. The detailed proof process is as follows:

$$\begin{array}{c}E\left[Z\right]={\int }_0^{+\infty }Cr\{\tilde{Z}\ge u\}du-{\int }_{-\infty }^{0}Cr\{\tilde{Z}\le u\}du\hfill \\ ={\int }_0^{+\infty }(1-\mathrm{\Phi }\left(u\right))du-{\int }_{-\infty }^0\mathrm{\Phi }\left(u\right)du\hfill \\ ={\int }_{-\infty }^{+\infty }ud\mathrm{\Phi }\left(u\right)\hfill \end{array}$$

By setting ${\displaystyle \alpha }$ to place ${\displaystyle \mathrm{\Phi }\left(u\right)}$ and using ${\displaystyle {\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(\alpha \right)}$ to replace ${\displaystyle u}$, we can obtain the following:

$$E\left[Z\right]={\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(\alpha \right)d\alpha .$$

□

Theorem 2. 

For a Z-number $Z=(A,B)$, where A is a trapezoidal fuzzy number (A = ($a_1$, $a_2$, $a_3$, $a_4$)), and B is a trapezoidal fuzzy number (B = ($b_1$, $b_2$, $b_3$, $b_4$)), its expected value is constructed as follows:

$$E\left[Z\right]={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+a_2+a_3+a_4),$$

(3)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}]}$.

Proof. 

By Equation (1) in the Supplementary Materials, we can obtain the crisp number ${\displaystyle \alpha }$, as follows:

$$\alpha ={\displaystyle \frac{\int x{\mu }_{B}dx}{\int {\mu }_{B}dx}}={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}].$$

(4)

By Equation (3) in the Supplementary Materials, we can obtain the converted Z-number ${\displaystyle \tilde{Z}}$, as follows:

$$\tilde{Z}=(\sqrt{\alpha }{a}_1,\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3,\sqrt{\alpha }{a}_4).$$

(5)

Given the result of ${\displaystyle \tilde{Z}}$, the membership function of ${\displaystyle \tilde{Z}}$ is as follows:

$${\mu }_{\tilde{Z}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\alpha }(a_2-a_1)}}u-{\displaystyle \frac{\sqrt{\alpha }{a}_1}{\sqrt{\alpha }(a_2-a_1)}},\hfill & u\in [\sqrt{\alpha }{a}_1,\sqrt{\alpha }{a}_2]\hfill \\ \hfill \\ 1,\hfill & u\in [\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3]\hfill \\ \hfill \\ {\displaystyle \frac{-1}{\sqrt{\alpha }(a_4-a_3)}}u+{\displaystyle \frac{\sqrt{\alpha }{a}_4}{\sqrt{\alpha }(a_4-a_3)}},\hfill & u\in [\sqrt{\alpha }{a}_3,\sqrt{\alpha }{a}_4].\hfill \end{array}\right.$$

(6)

By Definition 5 in the Supplementary Materials, the credibility distribution and its inverse function can be calculated as follows:

$${\mathrm{\Phi }}_{\tilde{Z}}\left(u\right)=\left\{\begin{array}{cc}0,\hfill & u\le \sqrt{\alpha }{a}_1\hfill \\ \hfill \\ {\displaystyle \frac{u-\sqrt{\alpha }{a}_1}{2(\sqrt{\alpha }{a}_2-\sqrt{\alpha a_1)}}},\hfill & \sqrt{\alpha }{a}_1\le u\le \sqrt{\alpha }{a}_2\hfill \\ \hfill \\ {\displaystyle \frac{1}{2}},\hfill & \sqrt{\alpha }{a}_2\le u\le \sqrt{\alpha }{a}_3\hfill \\ \hfill \\ {\displaystyle \frac{u+\sqrt{\alpha }{a}_4-2\sqrt{\alpha }{a}_3}{2(\sqrt{\alpha }{a}_4-\sqrt{\alpha }{a}_3)}},\hfill & \sqrt{\alpha }{a}_3\le u\le \sqrt{\alpha }{a}_4\hfill \\ \hfill \\ 1,\hfill & u\ge \sqrt{\alpha }{a}_4,\hfill \end{array}\right.$$

(7)

$${\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)=\left\{\begin{array}{c}2(\sqrt{\alpha }{a}_2-\sqrt{\alpha }{a}_1)t+\sqrt{\alpha }{a}_1,t\in [0,{\displaystyle \frac{1}{2}})\hfill \\ \hfill \\ [\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3],t={\displaystyle \frac{1}{2}}\hfill \\ \hfill \\ 2(\sqrt{\alpha }{a}_4-\sqrt{\alpha }{a}_3)t+2\sqrt{\alpha }{a}_3-\sqrt{\alpha }{a}_4,t\in [{\displaystyle \frac{1}{2}},1].\hfill \end{array}\right.$$

(8)

By Theorem 1, the expected value is calculated as follows:

$$\begin{array}{c}E\left[Z\right]={\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)dt\hfill \\ \hfill \\ ={\int }_0^{{\displaystyle \frac{1}{2}}}[2(\sqrt{\alpha }{a}_2-\sqrt{\alpha }{a}_1)t+\sqrt{\alpha }{a}_1]dt\hfill \\ \hfill \\ +{\int }_{{\displaystyle \frac{1}{2}}}^1[2(\sqrt{\alpha }{a}_4-\sqrt{\alpha }{a}_3)t+2\sqrt{\alpha }{a}_3-\sqrt{\alpha }{a}_4]dt\hfill \\ \hfill \\ ={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+a_2+a_3+a_4),\hfill \end{array}$$

(9)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$ □

Corollary 1. 

There are three special cases of Theorem 2, as follows:

Case I: For a Z-number $Z=(A,B)$, where A is a triangular fuzzy number (A = ($a_1$, $a_2$, $a_3$)) and B is a triangular fuzzy number (B = ($b_1$, $b_2$, $b_3$)), its expected value is constructed as follows:

$$E\left[Z\right]={\displaystyle \frac{\sqrt{3(b_1+b_2+b_3)}}{12}}(a_1+2a_2+a_3).$$

(10)

Case II: For a Z-number ${\displaystyle Z=(A,B)}$, where A is a trapezoidal fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$, ${\displaystyle a_4}$)) and B is a triangular fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$)), its expected value is ascertained by the following:

$$E\left[Z\right]={\displaystyle \frac{\sqrt{3(b_1+b_2+b_3)}}{12}}(a_1+a_2+a_3+a_4).$$

(11)

Case III: For a Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a triangular fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$)) and ${\displaystyle B}$ is a trapezoidal fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$, ${\displaystyle b_4}$)), its expected value is formulated by the following:

$$E\left[Z\right]={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+2a_2+a_3),$$

(12)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$

Proof. 

For case I, when ${\displaystyle a_2=a_3}$ and ${\displaystyle b_2=b_3}$ in Theorem 2, the Z-number of this case is the same as that in Theorem 2. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_1+b_2+b_3}{3}}.\hfill \end{array}$$

(13)

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+a_2+a_3+a_4).\hfill \\ \hfill \\ ={\displaystyle \frac{\sqrt{3(b_1+b_2+b_3)}}{12}}(a_1+2a_2+a_3)\hfill \end{array}$$

(14)

For case II, when ${\displaystyle b_2=b_3}$ in Theorem 2, the Z-number of this case is the same as that in Theorem 2. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_1+b_2+b_3}{3}}.\hfill \end{array}$$

(15)

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+a_2+a_3+a_4).\hfill \\ \hfill \\ ={\displaystyle \frac{\sqrt{3(b_1+b_2+b_3)}}{12}}(a_1+a_2+a_3+a_4)\hfill \end{array}$$

(16)

For case III, when ${\displaystyle a_2=a_3}$ in Theorem 2, the Z-number of this case is the same as that in Theorem 2. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+a_2+a_3+a_4).\hfill \\ \hfill \\ ={\displaystyle \frac{1}{4}}\sqrt{\alpha }(a_1+2a_2+a_3),\hfill \end{array}$$

(17)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$ □

Theorem 3. 

For a symmetric Z-number $Z=(A,B)$, where A is a trapezoidal fuzzy number (A = ($a_1$, $a_2$, $a_3$, $a_4$)) with $m=|a_1-a_2|=|a_3-a_4|$, and B is a trapezoidal fuzzy number (B = ($b_1$, $b_2$, $b_3$, $b_4$)) with $n=|b_1-b_2|=|b_3-b_4|$, the expected value can be formulated by the following:

$$E\left[Z\right]={\displaystyle \frac{\sqrt{2(b_2+b_3)}}{4}}(a_2+a_3).$$

(18)

Proof. 

According to the conditions of Theorems 2 and 3, the expected value of a symmetric Z-number ${\displaystyle Z}$ can be derived as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_2+b_3}{2}}.\hfill \end{array}$$

(19)

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{\alpha }}{4}}(a_1+a_2+a_3+a_4)\hfill \\ ={\displaystyle \frac{\sqrt{2(b_2+b_3)}}{4}}(a_2+a_3).\hfill \end{array}$$

(20)

□

Corollary 2. 

There are three special cases of Theorem 3, as follows:

Case I: For a symmetric Z-number $Z=(A,B)$, where A is a triangular fuzzy number (A = ($a_1$, $a_2$, $a_3$)) with $m=|a_1-a_2|=|a_3-a_2|$, and B is a triangular fuzzy number (B = ($b_1$, $b_2$, $b_3$)) with $n=|b_1-b_2|=|b_3-b_2|$, the expected value is written as follows:

$$E\left[Z\right]=a_2\sqrt{b_2}.$$

(21)

Case II: For a symmetric Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a trapezoidal fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$, ${\displaystyle a_4}$)) with ${\displaystyle m=|a_1-a_2|=|a_3-a_4|}$, and ${\displaystyle B}$ is a triangular fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$)) with ${\displaystyle n=|b_1-b_2|=|b_3-b_2|}$, its expected value can be formulated by the following:

$$E\left[Z\right]={\displaystyle \frac{\sqrt{b_2}}{2}}(a_2+a_3).$$

(22)

Case III: For a symmetric Z-number Z = (A, B), where A is a triangular fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$)) with ${\displaystyle m=|a_1-a_2|=|a_3-a_2|}$, and B is a trapezoidal fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$, ${\displaystyle b_4}$)) with ${\displaystyle n=|b_1-b_2|=|b_3-b_4|}$, its expected value is constructed as follows:

$$E\left[Z\right]={\displaystyle \frac{a_2\sqrt{2(b_2+b_3)}}{2}}.$$

(23)

Proof. 

For case I, when ${\displaystyle a_2=a_3}$ and ${\displaystyle b_2=b_3}$ in Theorem 3, the Z-number of this case is the same as that in Theorem 3. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{2(b_2+b_3)}}{4}}(a_2+a_3)\hfill \\ \hfill \\ =a_2\sqrt{b_2}.\hfill \end{array}$$

(24)

For case II, when ${\displaystyle b_2=b_3}$ in Theorem 3, the Z-number of this case is the same as that in Theorem 3. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{2(b_2+b_3)}}{4}}(a_2+a_3)\hfill \\ \hfill \\ ={\displaystyle \frac{\sqrt{b_2}}{2}}(a_2+a_3).\hfill \end{array}$$

(25)

For case III, when ${\displaystyle a_2=a_3}$ in Theorem 3, the Z-number of this case is the same as that in Theorem 3. Thus, its expected value can be calculated as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{2(b_2+b_3)}}{4}}(a_2+a_3).\hfill \\ \hfill \\ ={\displaystyle \frac{a_2\sqrt{3(b_2+b_3)}}{3}}.\hfill \end{array}$$

(26)

□

3. The Variance of Z-Numbers

In this section, we provide the definition and calculation formulas for the variance of Z-numbers by the credibility distribution, which can be derived by Definition 7 in the Supplementary Materials.

Definition 2. 

Let Z be a Z-number, and let us denote its converted regular fuzzy number as $\tilde{Z}$ with an expected value denoted by ε; its variance is then depicted as follows:

$$V\left[Z\right]=E\left[{(\tilde{Z}-\epsilon )}^2\right].$$

(27)

Theorem 4. 

Given that Z is a Z-number, let us denote its converted regular fuzzy number as $\tilde{Z}$ with an expected value denoted by ε; its variance is then ascertained as follows:

$$V\left[Z\right]={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})+{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})]du.$$

(28)

Proof. 

By Definitions 1 and 2, we can deduce the following:

$$\begin{array}{ll}V\left[Z\right]& =E\left[{(\tilde{Z}-\epsilon )}^2\right]\\ & ={\int }_0^{+\infty }Cr\{{(\tilde{Z}-\epsilon )}^2\ge u\}du-{\int }_{-\infty }^{0}Cr\{{(\tilde{Z}-\epsilon )}^2\le u\}du\\ & ={\int }_0^{+\infty }Cr\{{(\tilde{Z}-\epsilon )}^2\ge u\}du\\ & ={\int }_0^{+\infty }Cr\{\tilde{Z}\ge \sqrt{u}+\epsilon \}du+{\int }_0^{+\infty }Cr\{\tilde{Z}\le \epsilon -\sqrt{u}\}du\\ & ={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})+{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})]du\end{array}$$

(29)

□

Theorem 5. 

Given that Z is a Z-number, let us denote its converted regular fuzzy number as $\tilde{Z}$ with an expected value denoted by ε; its variance is then calculated by the following:

$$V\left[Z\right]={\int }_0^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt,$$

(30)

where ${{\mathrm{\Phi }}_{\tilde{Z}}}^{-1}$ is the inverse credibility distribution of $\tilde{Z}$.

Proof. 

By Theorem 5, we obtain the following:

$$\begin{array}{c}V\left[Z\right]=E\left[{(\tilde{Z}-\epsilon )}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})+{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})]du.\hfill \end{array}$$

(31)

For the first part, we have the following:

$$\begin{array}{c}{\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})]du\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1(1-t)d{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^2\hfill \\ \hfill \\ =(1-t){[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^2{|}_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1-{\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}d(1-t)\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt.\hfill \end{array}$$

(32)

For the second part, we have the following:

$$\begin{array}{c}{\int }_0^{+\infty }{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})du\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^{0}td{[-{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)+\epsilon ]}^2\hfill \\ \hfill \\ =t{[-{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)+\epsilon ]}^2{|}_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^0-{\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(v\right)}^0{[-{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)+\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt.\hfill \end{array}$$

(33)

Thus, we have the following:

$$\begin{array}{c}V\left[Z\right]=E\left[{(\tilde{Z}-\epsilon )}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{n})+{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})]du\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt+{\int }_0^{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt.\hfill \end{array}$$

(34)

□

Theorem 6. 

For a Z-number $Z=(A,B)$, where A is a trapezoidal fuzzy number (A = ($a_1$, $a_2$, $a_3$, $a_4$)) and B is a trapezoidal fuzzy number (B = ($b_1$, $b_2$, $b_3$, $b_4$)). Thus, its variance can be formulated as follows:

$$V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2,$$

(35)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$

Proof. 

By Equation (1) in the appendix, we can obtain the crisp number ${\displaystyle \alpha }$, as follows:

$$\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}].$$

(36)

By Equation (3) in the Supplementary Materials, we can obtain the converted Z-number ${\displaystyle \tilde{Z}}$, as follows:

$$\tilde{Z}=(\sqrt{\alpha }{a}_1,\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3,\sqrt{\alpha }{a}_4).$$

(37)

The membership function of ${\displaystyle \tilde{Z}}$ is as follows:

$${\mu }_{\tilde{Z}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\alpha }(a_2-a_1)}}u-{\displaystyle \frac{\sqrt{\alpha }{a}_1}{\sqrt{\alpha }(a_2-a_1)}},\hfill & u\in [\sqrt{\alpha }{a}_1,\sqrt{\alpha }{a}_2]\hfill \\ \hfill \\ 1,\hfill & u\in [\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3]\hfill \\ \hfill \\ {\displaystyle \frac{-1}{\sqrt{\alpha }(a_4-a_3)}}u+{\displaystyle \frac{\sqrt{\alpha }{a}_4}{\sqrt{\alpha }(a_4-a_3)}},\hfill & u\in [\sqrt{\alpha }{a}_3,\sqrt{\alpha }{a}_4].\hfill \end{array}\right.$$

(38)

By Definition 5 in the Supplementary Materials, the credibility distribution and its inverse form can be calculated as follows:

$${\mathrm{\Phi }}_{\tilde{Z}}\left(u\right)=\left\{\begin{array}{cc}0,\hfill & u\le \sqrt{\alpha }{a}_1\hfill \\ \hfill \\ {\displaystyle \frac{u-\sqrt{\alpha }{a}_1}{2(\sqrt{\alpha }{a}_2-\sqrt{\alpha a_1)}}},\hfill & \sqrt{\alpha }{a}_1\le u\le \sqrt{\alpha }{a}_2\hfill \\ \hfill \\ {\displaystyle \frac{1}{2}},\hfill & \sqrt{\alpha }{a}_2\le u\le \sqrt{\alpha }{a}_3\hfill \\ \hfill \\ {\displaystyle \frac{u+\sqrt{\alpha }{a}_4-2\sqrt{\alpha }{a}_3}{2(\sqrt{\alpha }{a}_4-\sqrt{\alpha }{a}_3)}},\hfill & \sqrt{\alpha }{a}_3\le u\le \sqrt{\alpha }{a}_4\hfill \\ \hfill \\ 1,\hfill & u\ge \sqrt{\alpha }{a}_4,\hfill \end{array}\right.$$

(39)

$${\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)=\left\{\begin{array}{c}2(\sqrt{\alpha }{a}_2-\sqrt{\alpha }{a}_1)t+\sqrt{\alpha }{a}_1,t\in [0,{\displaystyle \frac{1}{2}})\hfill \\ \hfill \\ [\sqrt{\alpha }{a}_2,\sqrt{\alpha }{a}_3],t={\displaystyle \frac{1}{2}}\hfill \\ \hfill \\ 2(\sqrt{\alpha }{a}_4-\sqrt{\alpha }{a}_3)t+2\sqrt{\alpha }{a}_3-\sqrt{\alpha }{a}_4,t\in [{\displaystyle \frac{1}{2}},1].\hfill \end{array}\right.$$

(40)

By Theorem 5, we have the following:

$$\begin{array}{c}V\left[Z\right]={\int }_0^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^1{\left[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)\right]}^{2}dt+{\int }_0^1{\epsilon }^{2}dt-2\epsilon {\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)dt.\hfill \end{array}$$

(41)

For the first part, we have the following:

$$\begin{array}{c}{\int }_0^1{\left[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)\right]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^{{\displaystyle \frac{1}{2}}}{\left[{\mathrm{\Phi }}_{\tilde{Z}}\left(t\right)\right]}^{2}dt+{\int }_{{\displaystyle \frac{1}{2}}}^1{\left[{\mathrm{\Phi }}_{\tilde{Z}}\left(t\right)\right]}^{2}dt\hfill \\ \hfill \\ ={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2-a_1{a}_2]+{\displaystyle \frac{1}{6}}\alpha [{(a_3+a_4)}^2-a_3{a}_4].\hfill \end{array}$$

(42)

For the second part, we have the following:

$${\int }_0^1{\epsilon }^{2}dt={\epsilon }^2.$$

(43)

For the third part, we have the following:

$$-2\epsilon {\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)dt=-2{\epsilon }^2.$$

(44)

Thus, we have the following:

$$\begin{array}{c}V\left[Z\right]={\int }_0^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^1{\left[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)\right]}^{2}dt+{\int }_0^1{\epsilon }^{2}dt-2\epsilon {\int }_0^1{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)dt\hfill \\ \hfill \\ ={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2-a_1{a}_2]+{\displaystyle \frac{1}{6}}\alpha [{(a_3+a_4)}^2-a_3{a}_4]+{\epsilon }^2-2{\epsilon }^2\hfill \\ \hfill \\ ={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2,\hfill \end{array}$$

(45)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$ □

Corollary 3. 

There are three special cases of Theorem 6 as follows:

Case I: For a Z-number $Z=(A,B)$, where A is a triangular fuzzy number (A = ($a_1$, $a_2$, $a_3$)) and B is a triangular fuzzy number (B = ($b_1$, $b_2$, $b_3$)), its variance is written as follows:

$$V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_2)}^2-a_1{a}_2-a_3{a}_2]-{\displaystyle \frac{\alpha }{16}}{(a_1+2a_2+a_3)}^2,$$

(46)

where ${\displaystyle \alpha =\frac{b_1+b_2+b_3}{3}}$.

Case II: For a Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a trapezoidal fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$, ${\displaystyle a_4}$)) and ${\displaystyle B}$ is a triangular fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$)), its variance is obtained as follows:

$$V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2,$$

(47)

where ${\displaystyle \alpha =\frac{b_1+b_2+b_3}{3}}$.

Case III: For a Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a triangular fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$)) and ${\displaystyle B}$ is a trapezoidal fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$, ${\displaystyle b_4}$)), its variance can be formulated by the following:

$$V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_2)}^2-a_1{a}_2-a_3{a}_2]-{\displaystyle \frac{\alpha }{16}}{(a_1+2a_2+a_3)}^2,$$

(48)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$

Proof. 

For case I, when ${\displaystyle a_2=a_3}$ and ${\displaystyle b_2=b_3}$ in Theorem 6, the Z-number of this case is the same as that in Theorem 6. Thus, its variance is given as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_1+b_2+b_3}{3}}.\hfill \end{array}$$

(49)

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2\hfill \\ \hfill \\ ={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_2)}^2-a_1{a}_2-a_3{a}_2]-{\displaystyle \frac{\alpha }{16}}{(a_1+2a_2+a_3)}^2.\hfill \end{array}$$

(50)

For case II, when ${\displaystyle b_2=b_3}$ in Theorem 6, the Z-number of this case is the same as that in Theorem 6. Thus, its expected value is given as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_1+b_2+b_3}{3}}.\hfill \end{array}$$

(51)

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2.\hfill \end{array}$$

(52)

For case III, when ${\displaystyle a_2=a_3}$ in Theorem 6, the Z-number of this case is the same as that in Theorem 6. Thus, its expected value is given as follows:

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2\hfill \\ \hfill \\ ={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_2)}^2-a_1{a}_2-a_3{a}_2]-{\displaystyle \frac{\alpha }{16}}{(a_1+2a_2+a_3)}^2,\hfill \end{array}$$

(53)

where ${\displaystyle \alpha =\frac{1}{3}[(b_1+b_2+b_3+b_4)+\frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}].}$ □

Theorem 7. 

For a symmetric Z-number $Z=(A,B)$, where A is a trapezoidal fuzzy number (A = ($a_1$, $a_2$, $a_3$, $a_4$)) with $m=|a_1-a_2|=|a_3-a_4|$, and B is a trapezoidal fuzzy number (B=($b_1$, $b_2$, $b_3$, $b_4$)) with $n=|b_1-b_2|=|b_3-b_4|$, its variance can be formulated by the following:

$$V\left[Z\right]={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_3^2-6a_1{a}_3+m^2).$$

(54)

Proof. 

According to the conditions of Theorems 6 and 7, the variance of a symmetric Z-number ${\displaystyle Z}$ can be derived as follows:

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{3}}[(b_1+b_2+b_3+b_4)+{\displaystyle \frac{b_1{b}_2-b_3{b}_4}{b_3+b_4-b_1-b_2}}]\hfill \\ \hfill \\ ={\displaystyle \frac{b_2+b_3}{2}}.\hfill \end{array}$$

(55)

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2\hfill \\ \hfill \\ ={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_3^2-6a_1{a}_3+m^2).\hfill \end{array}$$

(56)

□

Corollary 4. 

There are three special cases of Theorem 7 as follows:

Case I: For a symmetric Z-number $Z=(A,B)$, where A is a triangular fuzzy number (A = ($a_1$, $a_2$, $a_3$)) with $m=|a_1-a_2|=|a_3-a_2|$, and B is a triangular fuzzy number (B = ($b_1$, $b_2$, $b_3$)) with $n=|b_1-b_2|=|b_3-b_2|$, its variance is given by the following:

$$V\left[Z\right]={\displaystyle \frac{b_2}{12}}(3a_1^2+3a_2^2-6a_1{a}_2+m^2).$$

(57)

Case II: For a symmetric Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a trapezoidal fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$, ${\displaystyle a_4}$)) with ${\displaystyle m=|a_1-a_2|=|a_3-a_4|}$, and ${\displaystyle B}$ is a triangular fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$)) with ${\displaystyle n=|b_1-b_2|=|b_3-b_2|}$, its variance is obtained as follows:

$$V\left[Z\right]={\displaystyle \frac{b_2}{12}}(3a_1^2+3a_3^2-6a_1{a}_3+m^2).$$

(58)

Case III: For a symmetric Z-number ${\displaystyle Z=(A,B)}$, where ${\displaystyle A}$ is a triangular fuzzy number (A = (${\displaystyle a_1}$, ${\displaystyle a_2}$, ${\displaystyle a_3}$)) with ${\displaystyle m=|a_1-a_2|=|a_3-a_2|}$, and ${\displaystyle B}$ is a trapezoidal fuzzy number (B = (${\displaystyle b_1}$, ${\displaystyle b_2}$, ${\displaystyle b_3}$, ${\displaystyle b_4}$)) with ${\displaystyle n=|b_1-b_2|=|b_3-b_4|}$, its variance is expressed as follows:

$$V\left[Z\right]={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_2^2-6a_1{a}_2+m^2).$$

(59)

Proof. 

For case I, when ${\displaystyle a_2=a_3}$ and ${\displaystyle b_2=b_3}$ in Theorem 7, the Z-number of this case is the same as that in Theorem 7. Thus, its variance is calculated as follows:

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_3^2-6a_1{a}_3+m^2)\hfill \\ \hfill \\ ={\displaystyle \frac{b_2}{12}}(3a_1^2+3a_2^2-6a_1{a}_2+m^2).\hfill \end{array}$$

(60)

For case II, when ${\displaystyle b_2=b_3}$ in Theorem 7, the Z-number of this case is the same as that in Theorem 7. Thus, its variance is calculated as follows:

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_3^2-6a_1{a}_3+m^2)\hfill \\ \hfill \\ ={\displaystyle \frac{b_2}{12}}(3a_1^2+3a_3^2-6a_1{a}_3+m^2).\hfill \end{array}$$

(61)

For case III, when ${\displaystyle a_2=a_3}$ in Theorem 7, the Z-number of this case is the same as that in Theorem 7. Thus, its variance is calculated as follows:

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_3^2-6a_1{a}_3+m^2)\hfill \\ \hfill \\ ={\displaystyle \frac{1}{24}}(b_2+b_3)(3a_1^2+3a_2^2-6a_1{a}_2+m^2).\hfill \end{array}$$

(62)

□

Example 1. 

Consider a case where there is a low probability of very poor product sales. We calculate its expected value and variance.

Solution. First, we convert the natural language to Z-valuation Z = (volume of product sales, very poor, low).

Second, we convert it to a Z-number using Tables S1 and S2 as shown in the Supplementary Materials [30], resulting in Z = [(0, 0, 0.1, 0.1), (0.05, 0.2 0.35)].

Third, using Equations (11) and (47), we derive the expected value and variance, as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{3(b_1+b_2+b_3)}}{12}}(a_1+a_2+a_3+a_4).\hfill \\ \hfill \\ ={\displaystyle \frac{\sqrt{3(0.05+0.2+0.35)}}{12}}(0+0+0.1+0.2)\hfill \\ \hfill \\ =0.03354.\hfill \end{array}$$

(63)

$$\begin{array}{c}\alpha ={\displaystyle \frac{b_1+b_2+b_3}{3}}=0.2.\hfill \end{array}$$

(64)

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{1}{6}}\alpha [{(a_1+a_2)}^2+{(a_3+a_4)}^2-a_1{a}_2-a_3{a}_4]-{\displaystyle \frac{\alpha }{16}}{(a_1+a_2+a_3+a_4)}^2\hfill \\ \hfill \\ ={\displaystyle \frac{0.2}{6}}[{(0+0)}^2+{(0.1+0.2)}^2-0-0.02]-{\displaystyle \frac{0.2}{16}}{(0+0+0.1+0.2)}^2\hfill \\ \hfill \\ =0.00121.\hfill \end{array}$$

(65)

Example 2. 

Given a case where the transit line has a high probability of being slightly poor, we calculate its expected value and variance.

Solution. First, we convert the natural language to Z-valuation Z=(the transit line, slightly poor, high).

Second, we convert it to a Z-number using Tables S1 and S2 in the Supplementary Materials [30], resulting in Z = [(0.2, 0.3, 0.4, 0.5), (0.65, 0.8 0.95)].

Third, using Equations (22) and (58), we derive the expected value and variance, as follows:

$$\begin{array}{c}E\left[Z\right]={\displaystyle \frac{\sqrt{b_2}}{2}}(a_2+a_3)\hfill \\ \hfill \\ ={\displaystyle \frac{\sqrt{0.2}}{2}}(0.3+0.4)\hfill \\ \hfill \\ =0.1565.\hfill \end{array}$$

(66)

$$\begin{array}{c}V\left[Z\right]={\displaystyle \frac{b_2}{12}}(3a_1^2+3a_3^2-6a_1{a}_3+m^2)\hfill \\ \hfill \\ ={\displaystyle \frac{0.8}{12}}(0.12+0.48-0.48+0.01)\hfill \\ \hfill \\ =0.00867.\hfill \end{array}$$

(67)

4. The Semi-Variance of Z-Numbers

In this section, we provide the definitions and theorems for the semi-variance of Z-numbers via the credibility distribution, relying on Definitions 8 and 9 in the Supplementary Materials [31,32].

Definition 3. 

Supposing that Z is a Z-number, we denote its converted regular fuzzy number as $\tilde{Z}$. If its expected value is ε, its upside semi-variance is derived as follows:

$$S_{v^{+}}\left[Z\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{+}\right)}^2\right],$$

(68)

where

$${(\tilde{Z}-\epsilon )}^{+}=\left\{\begin{array}{cc}\tilde{Z}-\epsilon ,& \mathit{if}\tilde{Z}>\epsilon \\ \\ 0,& \mathit{if}\tilde{Z}\le \epsilon \end{array}.\right.$$

(69)

Definition 4. 

Supposing that Z is a Z-number, we denote its converted regular fuzzy number as $\tilde{Z}$. If its expected value is ε, its downside semi-variance is derived as follows:

$$S_{v^{-}}\left[Z\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{-}\right)}^2\right],$$

(70)

where

$${(\tilde{Z}-\epsilon )}^{-}=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\tilde{Z}>\epsilon \hfill \\ \hfill \\ \tilde{Z}-\epsilon ,\hfill & \mathit{if}\tilde{Z}\le \epsilon .\hfill \end{array}\right.$$

(71)

Theorem 8. 

Supposing that Z is a Z-number, we denote its converted regular fuzzy number as $\tilde{Z}$. Then, the credibility distribution is ${\mathrm{\Phi }}_{\tilde{Z}}$ and the expected value is ε. Its upside semi-variance is derived as follows:

$$S_{v^{+}}\left[Z\right]={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})]du.$$

(72)

Proof. 

By Definition 3,

$$\begin{array}{c}{S}_{v^{+}}\left[Z\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{+}\right)}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }Cr\{{(\tilde{Z}-\epsilon )}^2\ge u,\tilde{Z}>\epsilon \}du\hfill \\ \hfill \\ ={\int }_0^{+\infty }Cr\{\tilde{Z}\ge \epsilon +\sqrt{u}\}du\hfill \\ \hfill \\ ={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})]du\hfill \end{array}$$

(73)

□

Theorem 9. 

Supposing that Z is a Z-number, we denote its converted regular fuzzy number as $\tilde{Z}$. Its credibility distribution is ${\mathrm{\Phi }}_{\tilde{Z}}$ and the expected value is ε. Its downside semi-variance is derived as follows:

$$S_{v^{-}}\left[Z\right]={\int }_0^{+\infty }{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})du.$$

(74)

Proof. 

By Definition 4,

$$\begin{array}{c}{S}_{v^{-}}\left[\tilde{Z}\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{-}\right)}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }Cr\{{(\tilde{Z}-\epsilon )}^2\ge u,\tilde{Z}\le \epsilon \}du\hfill \\ \hfill \\ ={\int }_0^{+\infty }Cr\{\tilde{Z}\le \epsilon -\sqrt{u}\}du\hfill \\ \hfill \\ ={\int }_0^{+\infty }{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})du\hfill \end{array}$$

(75)

□

Theorem 10. 

Given that $\tilde{Z}$ is a converted Z-number, its credibility distribution is ${\mathrm{\Phi }}_{\tilde{Z}}$, and the expected value is ε, its upside semi-variance is derived as follows:

$$S_{v^{+}}\left[Z\right]={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt,$$

(76)

where ${\displaystyle {\mathrm{\Phi }}_{\tilde{Z}}^{-1}}$ is the inverse credibility distribution of ${\displaystyle \tilde{Z}}$.

Proof. 

By Definition 3 and Theorem 7, we have the following:

$$\begin{array}{c}{S}_{v^{+}}\left[\tilde{Z}\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{+}\right)}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }[1-{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon +\sqrt{u})]du\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1(1-t)d{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^2\hfill \\ \hfill \\ =(1-t){[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^2{|}_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1-{\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}d(1-t)\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt.\hfill \end{array}$$

(77)

□

Theorem 11. 

Given that $\tilde{Z}$ is a converted Z-number, its credibility distribution is ${\mathrm{\Phi }}_{\tilde{Z}}$, and the expected value is ε, its downside semi-variance is derived as follows:

$$S_{v^{-}}\left[\tilde{Z}\right]={\int }_0^{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt,$$

(78)

where ${\displaystyle {\mathrm{\Phi }}_{\tilde{Z}}^{-1}}$ is the inverse credibility distribution of ${\displaystyle \tilde{Z}}$.

Proof. 

By Definition 4 and Theorem 9, we derive the following:

$$\begin{array}{c}{S}_{v^{-}}\left[\tilde{Z}\right]=E\left[{\left({(\tilde{Z}-\epsilon )}^{-}\right)}^2\right]\hfill \\ \hfill \\ ={\int }_0^{+\infty }{\mathrm{\Phi }}_{\tilde{Z}}(\epsilon -\sqrt{u})du\hfill \\ \hfill \\ ={\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^{0}td{[-{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)+\epsilon ]}^2\hfill \\ \hfill \\ =t{[-{\mathrm{\Phi }}_{\tilde{Z}}\left(t\right)+\epsilon ]}^2{|}_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^0-{\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^0{[-{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)+\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt.\hfill \end{array}$$

(79)

□

Theorem 12. 

Given that $\tilde{Z}$ is a converted Z-number, its relationship between semi-variance and variance is as follows:

$$S_{v^{+}}\left[\tilde{Z}\right]+S_{v^{-}}\left[\tilde{Z}\right]=V\left[\tilde{Z}\right].$$

(80)

Proof. 

By Theorem 5, Theorem 9, and Theorem 11, we derive the following:

$$\begin{array}{c}V\left[\tilde{Z}\right]={\int }_0^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt\hfill \\ \hfill \\ ={\int }_0^{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt+{\int }_{{\mathrm{\Phi }}_{\tilde{Z}}\left(\epsilon \right)}^1{[{\mathrm{\Phi }}_{\tilde{Z}}^{-1}\left(t\right)-\epsilon ]}^{2}dt\hfill \\ \hfill \\ =S_{v^{-}}\left[\tilde{Z}\right]+S_{v^{+}}\left[\tilde{Z}\right].\hfill \end{array}$$

(81)

□

5. Conclusions

Z-numbers, as pairs of fuzzy numbers, have garnered significant academic interest due to their ability to simultaneously represent fuzziness and reliability, enhancing the explanation of human judgment in decision-making. In this article, we explore three properties of Z-numbers: the expected value, variance, and semi-variance. Initially, Z-number information is converted into regular fuzzy numbers through a specific process. Based on related theories, we define these properties and derive formulas for the expected values and variances of general and symmetrical Z-numbers. We also establish relationships between the variance and semi-variance of Z-numbers. Two examples illustrate the calculation steps, validating our derived formulas. This research contributes to a deeper understanding of the Z-number theory and its application in areas like construction safety analysis and financial portfolio evaluation.

However, some limitations exist. First, this study focuses on Z-numbers as regular fuzzy numbers, leaving room for exploration with other types. Second, converting Z-numbers to fuzzy numbers and deriving their properties involves information loss, which future research should aim to minimize. Lastly, while we bridge a gap in understanding Z-number properties, more work is needed to explore their practical applications thoroughly Moreover, although the main contributions of this article involve defining and deriving the calculation methods for the expected value, variance, and semi-variance of Z-numbers, we recognize a profound connection between fuzzy logic and Bayesian probability theory [33]. The development of fuzzy set theory has been inspired by probability theory, especially when dealing with uncertain information in the real world. Moreover, both fuzzy logic and Bayesian probability theory are powerful tools for handling incomplete knowledge and uncertainty, complementing each other in many practical applications. For example, the use of Z-numbers combined with research in psychology for information extraction is one such example [34]. Therefore, such a combined application may be another research direction worth exploring.