## 2 Preliminaries  
We lay the groundwork for our analysis by presenting the essential ideas and key findings of uncertainty theory. Envision a non-empty set, \(\Gamma\), \(\mathcal{L}\) is the \(\sigma\)-algebra over \(\Gamma\). \(\mathcal{M}\) is the uncertain measure which maps \(\mathcal{L}\) to \(\mathbb{R}\) (0,1), i.e., \(\mathcal{M}:\mathcal{L}\mapsto[0,1]\). In order to define the uncertain measure axiomatically, each event \(\Lambda\) of \(\mathcal{L}\) is assigned a belief degree \(\mathcal{M}\{\Lambda\}\) such that \(\mathcal{M}\{\Lambda\}\in[0,1]\). If \(\mathcal{M}\) satisfies the conditions of countable subadditivity, normality, and duality ([PERSON] [18]), we consider \(\mathcal{M}\) as an uncertain measure. In this scenario, the triplet \((\Gamma,\mathcal{L},\mathcal{M})\) represents the uncertainty space. Both the product uncertain measure and the product measure axiom were presented by [PERSON] [34]. If there is an arbitrary event, \(\Lambda_{p}\), to each \(\mathcal{L}_{p}\), where \(p=1,2,...\), for all uncertainty spaces \(\left(\Gamma_{p},\ \mathcal{L}_{p},\ \mathcal{M}_{p}\right)\), then \(\mathcal{M}\) is said to be a product uncertain measure if  
\[\mathcal{M}\left\{\prod_{p=1}^{\infty}\Lambda_{p}\right\}=\bigwedge_{p=1}^{ \infty}\mathcal{M}_{p}\{\Lambda_{p}\}. \tag{1}\]  
**Definition 1**. : ([PERSON] [18]). _In an uncertainty space \(\left(\Gamma,\mathcal{L},\mathcal{M}\right)\), an uncertain variable, \(\xi\), is defined by a measurable function from \(\left(\Gamma,\mathcal{L},\mathcal{M}\right)\) to \(\mathbb{R}\). In simple terms, for each set, \(\mathcal{B}\in\mathbb{R}\), where \(\mathcal{B}\) is the Borel set, the set \(\left\{\xi\in\mathcal{B}\right\}=\left\{\lambda\in\Gamma|\xi(\lambda)\in \mathcal{B}\right\}\) is an event._  
**Definition 2**. : (Liu [18]). _A zigzag uncertain variable, \(\xi\), denoted by \(\mathcal{Z}(p,q,r)\), where \(p<q<r\) and \(p,q,r\in\mathbb{R}\), is an uncertain variable with the uncertainty distribution as shown in Equation (2)._  
\[\Phi(t)=\left\{ \begin{array}{ll}0&:\!if\ t<p\\ \frac{t-p}{2(q-p)}&:\!if\ p\leq t<q\\ \frac{t+7-2q}{2(r-q)}&:\!if\ q\leq\ t<r\\ 1&:\!if\ t\geq r\end{array} \right. \tag{2}\]  
_When dealing with mathematical operations involving independent uncertain variables, the inverse uncertainty distribution becomes quite important. For any, \(\alpha\), in the interval \([0,1]\), a distribution, \(\Phi\), is said to be regular if and only if its inverse function, \(\Phi^{-1}(\alpha)\), exists. Consequently, the zigzag uncertain variable displays a regular uncertainty distribution._  
**Definition 3**. : (Liu [18]). _For any Borel sets \(\mathcal{B}_{p}\in\mathbb{R}\)_  
\[\mathcal{M}\left\{\bigcap_{p=1}^{n}\left(\xi_{p}\in\mathcal{B}_{p}\right) \right\}=\bigwedge_{p=1}^{n}\mathcal{M}\left\{\xi_{p}\in\mathcal{B}_{p}\right\}, \tag{3}\]  
_when each \(\xi_{p}\) is the independent uncertain variable. Furthermore, a strictly increasing real-valued function, \(f(x_{1},x_{2},\ldots,x_{n})\), adheres the following properties:_  
(i) \(f(x_{1},x_{2},\ldots,x_{n})\leq f(y_{1},y_{2},\ldots,y_{n})\)_, where \(x_{i}\leq y_{i},i=1,2,\ldots n\) (ii) \(f(x_{1},x_{2},\ldots,x_{n})<f(y_{1},y_{2},\ldots,y_{n})\), where \(x_{i}<y_{i},i=1,2,\ldots n\)_  
**Theorem 1**. : (Liu [35]). _Let \(\zeta_{1},\zeta_{2},\ldots,\zeta_{r}\) be the independent uncertain variables with regular uncertainty distributions \(\ \psi_{1},\psi_{2},\ldots,\psi_{r}\), respectively. If the function \(\zeta=f(\zeta_{1},\zeta_{2},\ldots,\zeta_{r})\) is continuous and strictly growing, then \(\zeta\) can be considered an uncertain variable that follows an inverse uncertainty distribution_  
\[\psi^{-1}(\alpha)=f\left(\psi_{1}^{-1}(\alpha),\psi_{2}^{-1}(\alpha),\ldots, \psi_{r}^{-1}(\alpha)\right).\]  
**Example 1**. : _Imagine \(\zeta\) as a zigzag uncertain variable with a regular uncertainty distribution \(\psi\), and identify it as \(\mathcal{Z}(p,q,r)\).