In this section, we introduce some basic concepts and theorems for building uncertain random TCTP model. Let Γ be a nonempty set, \mathcal{L} a *σ*-algebra over Γ, and each element Ʌ in \mathcal{L} is called an event.

### Definition1.

(Liu [15]) The set function is called an uncertain measure if it satisfies:

Axiom 1 (Normality Axiom). \mathcal{M}\{\Gamma \}=1 for the universal set *Γ*.

Axiom 2 (Duality Axiom). \mathcal{M}\left\{\u0245\right\}+\mathcal{M}\left\{{\u0245}^{c}\right\}=1 for any event Ʌ.

Axiom 3 (Subadditivity Axiom). For every countable sequence of events Ʌ_{1}, Ʌ_{2}, ⋯, we have

\mathcal{M}\left\{\bigcup _{i=1}^{\infty}{\u0245}_{i}\right\}\le \underset{i=1}{\overset{\infty}{\Sigma}}\mathcal{M}\left\{{\u0245}_{i}\right\}.

Besides, the product uncertain measure on the product *σ*-algebra \mathcal{L} was defined by Liu [32] as follows:

*A* *x* *i* *o* *m* 4 (*P* *r* *o* *d* *u* *c* *t* *A* *x* *i* *o* *m*). Let ({\Gamma}_{k},{\mathcal{L}}_{k},{\mathcal{M}}_{k}) be uncertainty spaces for *k*=1,2,⋯. The product uncertain measure \mathcal{M} is an uncertain measure satisfying

\mathcal{M}\left\{\underset{k=1}{\overset{\infty}{\Pi}}{\u0245}_{k}\right\}=\underset{k=1}{\overset{\infty}{\u0245}}{\mathcal{M}}_{k}\left\{{\u0245}_{k}\right\}

where Ʌ_{
k
} are arbitrarily chosen events from {\mathcal{L}}_{k} for *k*=1,2,⋯, respectively.

### Definition2.

(Liu [15]) An uncertain variable is a measurable function *ξ* from an uncertainty space (\Gamma ,\mathcal{L},\mathcal{M}) to the set of real numbers, i.e., for any Borel set *B* of real numbers, the set

\{\xi \u03f5B\}=\{\Gamma \u03f5\Gamma \phantom{\rule{1em}{0ex}}\left|\phantom{\rule{1em}{0ex}}\xi (\Gamma )\u03f5B\right.\}

is an event.

### Definition3.

(Liu [32]) The uncertain variables *ξ*_{1}, *ξ*_{2},⋯, *ξ*_{
n
} are said to be independent if

\mathcal{M}\left\{\bigcap _{i=1}^{n}({\xi}_{i}\u03f5{B}_{i})\right\}=\underset{i=1}{\overset{n}{\u0245}}\mathcal{M}\{{\xi}_{i}\u03f5{B}_{i}\}

for any Borel sets *B*_{1}, *B*_{2},⋯, *B*_{
n
}.

Sometimes, to describe real-world optimization problems with uncertain parameters, it is sufficient to know the uncertainty distribution rather than the uncertain variable itself.

### Definition4.

(Liu [15]) The uncertainty distribution Φ of an uncertain variable *ξ* is defined by

\Phi \left(x\right)=\mathcal{M}\{\xi \le x\}

for any real number *x*.

### Definition5.

(Liu [16]) An uncertainty distribution Φ(*x*) is said to be regular if it is a continuous and strictly increasing function with respect to *x* at which 0<Φ(*x*)<1, and

\underset{x\to -\infty}{lim}\Phi \left(x\right)=0,\phantom{\rule{2em}{0ex}}\underset{x\to +\infty}{lim}\Phi \left(x\right)=1.

Liu [16] also defined the inverse function Φ^{-1} as the inverse uncertainty distribution of uncertain variable *ξ*.

### Theorem1.

(Liu [16]) Let *ξ*_{1}, *ξ*_{2},⋯, *ξ*_{
n
} be independent uncertain variables with regular uncertainty distributions Φ_{1},Φ_{2},⋯,Φ_{
n
}, respectively. If the function *f*(*x*_{1}, *x*_{2},⋯, *x*_{
n
}) is strictly increasing with respect to *x*_{1}, *x*_{2},⋯, *x*_{
m
} and strictly decreasing with respect to *x*_{m+1}, *x*_{m+2},⋯, *x*_{
n
}, then

\xi =f({\xi}_{1},{\xi}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{\xi}_{n})

is an uncertain variable with inverse uncertainty distribution

{\Phi}^{-1}\left(\alpha \right)=f\phantom{\rule{0.3em}{0ex}}\left({\Phi}_{1}^{-1}\left(\alpha \right),\cdots \phantom{\rule{0.3em}{0ex}},{\Phi}_{m}^{-1}\left(\alpha \right),{\Phi}_{m+1}^{-1}(1-\alpha ),\cdots \phantom{\rule{0.3em}{0ex}},{\Phi}_{n}^{-1}(1-\alpha )\right).

Based on the definitions of uncertain variable and random variable, the concept of uncertain random variable is given as follows:

### Definition6.

(Liu [25]) An uncertain random variable is a function *ξ* from a chance space (\Gamma ,\mathcal{L},\mathcal{M})\times (\times ,\mathcal{A},Pr) to the set of real numbers such that {*ξ*ϵ*B*} is an event in \mathcal{L}\times \mathcal{A} for any Borel set *B*.

**Example 1** A random variable is a special uncertain random variable since any real value is a special uncertain variable.

**Example 2** The sum of a random variable and an uncertain variable is an uncertain random variable.

### Definition7.

(Liu [25]) Let (\Gamma ,\mathcal{L},\mathcal{M})\times (\times ,\mathcal{A},Pr)be a chance space, and \Theta \u03f5\mathcal{L}\times \mathcal{A} be an event. Then, the chance measure of Θ is defined as

\text{Ch}\{\Theta \}=\underset{0}{\overset{1}{\u2a1c}}Pr\{\Omega \u03f5\Omega \left|\mathcal{M}\right.\{\Gamma \u03f5\Gamma \left|\right.(\Gamma ,\Omega )\u03f5\Theta \}\ge x\}\mathrm{d}\mathrm{x.}

Liu [25] proved that the chance measure is self-dual.

### Definition8.

(Liu [25]) Let *ξ* be an uncertain random variable. Then, the chance distribution of *ξ* is defined by

\Phi \left(x\right)=\text{Ch}\{\xi \le x\}

for any *x*ϵ*R*.

For some special case, some operational laws are presented as follows:

### Theorem2.

(Liu [26]) Let *η*_{1}, *η*_{2},⋯, *η*_{
m
} be independent random variables with probability distributions Ψ_{1},Ψ_{2},⋯,Ψ_{
m
}, respectively, and *τ*_{1}, *τ*_{2},⋯, *τ*_{
n
} be uncertain variables (not necessarily independent). Then, the uncertain random variable *ξ*=*f*(*η*_{1}, *η*_{2},⋯, *η*_{
m
}, *τ*_{1}, *τ*_{2},⋯, *τ*_{
n
}) has a chance distribution

\Phi \left(x\right)=\underset{{R}^{m}}{\u2a1c}F(x;{y}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{y}_{m})\mathrm{d}{\Psi}_{1}\left({y}_{1}\right)\cdots \mathrm{d}{\Psi}_{m}\left({y}_{m}\right)

where *F*(*x*;*y*_{1},⋯, *y*_{
m
}) is the uncertainty distribution of the uncertain variable *f*(*y*_{1},⋯, *y*_{
m
}, *τ*_{1},⋯, *τ*_{
n
}) for any real numbers *y*_{1},⋯, *y*_{
m
}.

### Theorem 3.

(Liu [33]) Let *η*_{1}, *η*_{2},⋯, *η*_{
m
} be independent random variables with probability distributions Ψ_{1},Ψ_{2},⋯,Ψ_{
m
}, and *τ*_{1}, *τ*_{2},⋯, *τ*_{
n
} be independent uncertain variables with regular uncertainty distributions *Υ*_{1}, *Υ*_{2},⋯, *Υ*_{
n
}, respectively. If *f*(*η*_{1},⋯, *η*_{
m
}, *τ*_{1},⋯, *τ*_{
n
}) is strictly increasing with respect to *τ*_{1},⋯, *τ*_{
k
} and strictly decreasing with respect to *τ*_{k+1},⋯, *τ*_{
n
}, then

\text{Ch}\left\{\phantom{\rule{0.3em}{0ex}}f\right({\eta}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{\eta}_{m},{\tau}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{\tau}_{n})\le 0\}=\underset{{R}^{m}}{\u2a1c}G({y}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{y}_{m})\mathrm{d}{\Psi}_{1}\left({y}_{1}\right)\cdots \mathrm{d}{\Psi}_{m}\left({y}_{m}\right)

where *G*(*y*_{1},⋯, *y*_{
m
}) is the root *α* of the equation

\begin{array}{c}f\phantom{\rule{0.3em}{0ex}}\left({y}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{y}_{m},{{\rm Y}}_{1}^{-1}\left(\alpha \right),\cdots \phantom{\rule{0.3em}{0ex}},{{\rm Y}}_{k}^{-1}\left(\alpha \right),{{\rm Y}}_{k+1}^{-1}(1-\alpha ),\cdots \phantom{\rule{0.3em}{0ex}},{{\rm Y}}_{n}^{-1}(1-\alpha )\right)=0.\end{array}