For the purpose of measuring a fuzzy event, Zadeh [1] introduced the concept of the possibility measure. After that, Liu and Liu [10] further pioneered the credibility measure, based on which the credibility theory was constructed by Liu [20] as a branch of mathematics for studying the behavior of fuzzy phenomena. In this section, some basic knowledge of the credibility theory is reviewed involving fuzzy variable, credibility measure, credibility distribution, inverse credibility distribution, and expected value of fuzzy variables.

### Fuzzy variable

Suppose that Θ is a nonempty set, \(\mathscr {P}(\Theta)\) is the power set of Θ, and Pos is a possibility measure. Then the triplet \((\Theta, \mathscr {P}(\Theta),\operatorname{Pos})\) is called a possibility space. A fuzzy variable is defined as a function from a possibility space \((\Theta, \mathscr {P} (\Theta),\operatorname{Pos})\) to the set of real numbers.

### Definition 1

(Liu [21])

Let *ξ* be a fuzzy variable defined on the possibility space \((\Theta, \mathscr {P}(\Theta), \operatorname{Pos})\). Then its membership function is derived from the possibility measure by

$$ \mu(x) = \operatorname{Pos}\bigl\{ \theta\in\Theta\bigm| \xi(\theta) = x\bigr\} ,\quad x \in\Re. $$

(1)

### Example 1

A fuzzy variable *ξ* is called a triangular fuzzy number and denoted by \(\xi\sim \mathcal{T}(a, b, c)\) if its membership function is

$$ \mu(x)= \textstyle\begin{cases} \frac{x-a}{b-a} &\mbox{if } a\leq x\le b, \\ \frac{c-x}{c-b} &\mbox{if } b< x \leq c,\\ 0 &\mbox{otherwise}, \end{cases} $$

(2)

as depicted in Figure 1, where *a*, *b*, and *c* are real numbers satisfying \(a< b< c\).

### Example 2

A fuzzy variable *ξ* is called a Gaussian fuzzy number and denoted by \(\xi\sim\mathcal{N}(a, b)\) if its membership function is

$$ \mu(x)= e^{-(\frac{x-a}{b})^{2}} $$

(3)

as depicted in Figure 2, where *a* and *b* are real numbers satisfying \(b>0\).

### Credibility measure

Suppose that *ξ* is a fuzzy variable, *μ* is the membership function of *ξ*, and *r* is a real number. The fuzzy event \(\{\xi\leq r \}\) has the possibility [1] and the necessity [22] as follows:

$$ \operatorname{Pos}\{\xi\leq r \}=\sup_{x \leq r}\mu(x),\qquad \operatorname{Nec}\{\xi \leq r \}=1-\sup_{x > r}\mu(x). $$

(4)

### Theorem 1

(Liu [21])

*Let*
\((\Theta, \mathscr {P} (\Theta),\operatorname{Pos})\)
*be a possibility space*, *and*
*A*
*a set in*
\(\mathscr {P} (\Theta)\). *Then the credibility of*
*A*
*is defined by*

$$ \operatorname{Cr}\{A\}=\frac{1}{2}\bigl(\operatorname{Pos}\{A\}+\operatorname{Nec}\{A\}\bigr). $$

(5)

Note that from Eqs. (4) and (5) it is easy to deduce

$$ \operatorname{Cr}\{\xi\le r\}= \frac{1}{2} \Bigl(\sup _{x\le r}\mu(x)+1-\sup_{x>r}\mu(x) \Bigr). $$

(6)

Moreover, Liu and Liu [10] proved that the credibility measure Cr is self-dual, that is,

$$ \operatorname{Cr}\{A\}+\operatorname{Cr}\bigl\{ A^{c}\bigr\} =1 $$

(7)

for any set \(A\in \mathscr {P}(\Theta)\). Since a self-dual measure is thoroughly indispensable both theoretically and practically, the credibility measure is taken into consideration in this paper.

### Theorem 2

(Liu [21])

*The credibility measure is subadditive*, *that is*,

$$ \operatorname{Cr}\{A\cup B\}\leq \operatorname{Cr}\{A\}+\operatorname{Cr}\{B\} $$

(8)

*for any events*
*A*
*and*
*B*.

### Credibility distribution

### Definition 2

(Liu [21])

The credibility distribution \(\Phi: \Re\rightarrow[0,1]\) of a fuzzy variable *ξ* is defined by

$$ \Phi(x)=\operatorname{Cr}\bigl\{ \theta\in\Theta\bigm| \xi(\theta)\leq x\bigr\} . $$

(9)

Here, \(\Phi(x)\) means the credibility when *ξ* takes a value less than or equal to *x*, which was proved to be a nondecreasing function on ℜ with \(\Phi(-\infty)=0\) and \(\Phi(+\infty)=1\) by Liu [20].

### Example 3

Suppose that \(\xi\sim \mathcal{T}(a, b, c)\) is a triangular fuzzy number with the membership function in Eq. (2). Then the credibility distribution of *ξ* is

$$ \Phi(x)= \textstyle\begin{cases} 0&\mbox{if }x\leq a,\\ (x-a)/2(b-a)&\mbox{if }a< x\leq b,\\ (x+c-2b)/2(c-b)&\mbox{if }b< x\leq c,\\ 1&\mbox{if }x>c; \end{cases} $$

(10)

see Figure 3.

### Example 4

Suppose that \(\xi\sim\mathcal{N}(a, b)\) is a Gaussian fuzzy number with the membership function in Eq. (3). Then the credibility distribution of *ξ* is

$$ \Phi(x)= \textstyle\begin{cases} \frac{1}{2}e^{-(\frac{x-a}{b})^{2}}&\mbox{if }x\leq a,\\ 1-\frac{1}{2}e^{-(\frac{x-a}{b})^{2}}&\mbox{if }x> a; \end{cases} $$

(11)

see Figure 4.

### Regular credibility distribution

### Definition 3

(Zhou *et al.* [19])

A credibility distribution Φ is said to be regular if it is a continuous and strictly increasing function with respect to *x* at which \(0<\Phi(x)<1\) and if

$$ \lim_{x\rightarrow-\infty}\Phi(x)=0, \qquad\lim_{x\rightarrow +\infty} \Phi(x)=1. $$

### Definition 4

(Zhou *et al.* [19])

Let *ξ* be a fuzzy variable with a regular credibility distribution Φ. Then the inverse function \(\Phi^{-1}\) is called the inverse credibility distribution of *ξ*.

### Example 5

The inverse credibility distribution of a triangular fuzzy number \(\xi \sim \mathcal{T}(a, b, c)\) is

$$ \Phi^{-1}(\alpha)= \textstyle\begin{cases} (2b-2a)\alpha+a&\mbox{if }\alpha< 0.5,\\ (2c-2b)\alpha+2b-c,&\mbox{if }\alpha\ge 0.5, \end{cases} $$

(12)

as shown in Figure 5.

### Example 6

The inverse credibility distribution of a Gaussian fuzzy number \(\xi \sim\mathcal{N}(a, b)\) is

$$ \Phi^{-1}(\alpha)= \textstyle\begin{cases} a-b\sqrt{-\ln(2\alpha)}&\mbox{if }\alpha\leq0.5,\\ a+b\sqrt{-\ln(2-2\alpha)}&\mbox{if }\alpha> 0.5, \end{cases} $$

(13)

as shown in Figure 6.

### Theorem 3

(Zhou *et al.* [19])

*Let*
\(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\)
*be independent fuzzy variables with regular credibility distributions*
\(\Phi_{1}, \Phi_{2}, \ldots, \Phi _{n}\), *respectively*. *If*
\(f(x_{1}, x_{2}, \ldots, x_{n})\)
*is strictly increasing with respect to*
\(x_{1}, x_{2}, \ldots, x_{m}\)
*and strictly decreasing with respect to*
\(x_{m+1}, x_{m+2}, \ldots, x_{n}\), *then*
\(\xi=f(\xi_{1}, \ldots, \xi_{m}, \xi_{m+1},\ldots, \xi_{n})\)
*is a fuzzy variable with an inverse credibility distribution*

$$\Phi^{-1}(\alpha) = f \bigl(\Phi^{-1}_{1}(\alpha), \ldots , \Phi^{-1}_{m}(\alpha),\Phi^{-1}_{m+1}(1- \alpha),\ldots, \Phi ^{-1}_{n}(1-\alpha) \bigr). $$

### Expected value

In 2002, the following general definition of expected value for fuzzy variables was provided by Liu and Liu [10] via the credibility distribution.

### Definition 5

(Liu and Liu [10])

Let *ξ* be a fuzzy variable. Then the expected value of *ξ* is defined by

$$ E[\xi]= \int_{0}^{+\infty} \operatorname{Cr}\{\xi\geq r\} \,\mathrm{d}r- \int_{-\infty }^{0}\operatorname{Cr}\{\xi\leq r\} \,\mathrm{d}r, $$

(14)

provided that at least one of the two integrals is finite.

### Theorem 4

(Zhou *et al.* [19])

*Let*
*ξ*
*be a fuzzy variable with the inverse credibility distribution*
\(\Phi^{-1}\). *If its expected value exists*, *then*

$$ E[\xi]= \int_{0}^{1}\Phi^{-1}(\alpha) \,\mathrm{d}\alpha. $$

(15)

### Example 7

According to Eqs. (12) and (15), the expected value of a triangular fuzzy number \(\xi\sim \mathcal{T}(a, b, c)\) can be calculated as

$$\begin{aligned} E[\xi] &= \int_{0}^{0.5} \bigl((1-2\alpha)a+2\alpha b \bigr) \,\mathrm{d}\alpha + \int_{0.5}^{1} \bigl((2-2\alpha)b+(2\alpha-1)c \bigr) \,\mathrm{d}\alpha \\ & =\frac{a+2b+c}{4}. \end{aligned}$$

Provided that \(b-a=c-b\), which means that *ξ* is a symmetric triangular fuzzy number, it is clear that the expected value is \(E[\xi ] = b\).

### Example 8

According to Eqs. (13) and (15), the expected value of a Gaussian fuzzy number \(\xi \sim\mathcal{N}(a, b)\) can be calculated as

$$\begin{aligned} E[\xi] &= \int_{0}^{0.5} \bigl(a-b\sqrt{-\ln(2\alpha )} \bigr) \,\mathrm{d}\alpha + \int_{0.5}^{1} \bigl(a+b\sqrt{-\ln(2-2\alpha)} \bigr) \,\mathrm{d}\alpha \\ &= 0.5a-b \int_{0}^{0.5}\sqrt{-\ln(2\alpha)} \,\mathrm{d}\alpha + 0.5a+b \int_{0.5}^{1}\sqrt{-\ln(2-2\alpha)} \,\mathrm{d}\alpha \\ &= a-0.5b \int_{0}^{1}\sqrt{-\ln t} \,\mathrm{d}t- 0.5b \int_{1}^{0}\sqrt {-\ln t} \,\mathrm{d}t \\ &=a. \end{aligned}$$

### Theorem 5

(Zhou *et al.* [19])

*Let*
\(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\)
*be independent fuzzy variables with regular credibility distributions*
\(\Phi_{1}, \Phi_{2}, \ldots, \Phi _{n}\), *respectively*. *If*
\(f(x_{1}, x_{2}, \ldots, x_{n})\)
*is strictly increasing with respect to*
\(x_{1}, x_{2}, \ldots, x_{m}\)
*and strictly decreasing with respect to*
\(x_{m+1}, x_{m+2}, \ldots, x_{n}\), *then the expected value of the fuzzy variable*
\(\xi=f(\xi_{1}, \ldots, \xi_{m}, \xi_{m+1},\ldots, \xi_{n})\)
*is*

$$E[\xi] = \int_{0}^{1} f \bigl(\Phi^{-1}_{1}( \alpha), \ldots , \Phi^{-1}_{m}(\alpha),\Phi^{-1}_{m+1}(1- \alpha),\ldots, \Phi ^{-1}_{n}(1-\alpha) \bigr) \,\mathrm{d}\alpha. $$

### Theorem 6

(Liu and Liu [10])

*Let*
*ξ*
*and*
*η*
*be independent fuzzy variables with finite expected values*. *Then for any real variables*
*a*
*and*
*b*, *we have*

$$ E[a\xi+b\eta]=aE[\xi]+bE[\eta]. $$

(16)