## Methodology
\(Cr\{\widetilde{B}|A_{r}X\leq\widetilde{B}_{r}\}\geq\alpha\) denotes the credibility of satisfying \(A_{r}X\leq\widetilde{B}_{r}\) is higher than or equal to confidence-level \(\alpha\). According to reference [29], the detailed solution procedures of model (2) can be summarized as: firstly, transforming objective function (8) into its expected value form; secondly, converting constraints (9) into their crisp equivalents. A series of solutions can be obtained under difference confidence-levels [25]. However, an FPP algorithm cannot ensure the minimization of objective function under all conditions because the expected objective value is used as the objective function. This may result in significant deviations of the optimized decision solutions, even in the event of system optimization failure. Robust fuzzy possibilistic programming (RFPP) was developed by [PERSON] et al. [32] to overcome the above-mentioned drawbacks of FPP methods. Additionally, the RFPP method is scarcely applied to EES systems, which are commonly used to plan water resource allocation, and solid management [25; 27]. Therefore, this study developed a dual robust stochastic fuzzy optimization (DRSFO) method through integrating TSP, RTSP, FPP, and RFPP methods in a single framework, which can be used to effectively tackle fuzzy and stochastic uncertainties as well as their combinations, capture the associated risks from fuzzy and stochastic uncertainties, as well as thoroughly analyze the trade-offs between system costs and reliability. In detail, it is formulated as follows:\[ \begin{array}{ll}\text{Min}\;\widetilde{f}&=\overline{f}+\lambda_{1}\cdot(f_{ \text{max}}-\overline{f})+\lambda_{2}\cdot(\overline{f}-f_{\text{min}})+\rho_{ 2}\cdot[B^{R}(\alpha)-B_{r}{}^{1}]\\ &=0.25\cdot(C_{T_{1}}^{1}+2\cdot C_{T_{1}}^{2}+C_{T_{1}}^{3})\cdot X+\lambda_{1 }\cdot[C_{T_{1}}^{3}-0.25\cdot(C_{T_{1}}^{1}+2\cdot C_{T_{1}}^{2}+C_{T_{1}}^{3 })]\cdot X\\ &+\lambda_{2}\cdot[0.25\cdot(C_{T_{1}}^{1}+2\cdot C_{T_{1}}^{2}+C_{T_{1}}^{3})- C_{T_{1}}^{1}]\cdot X\\ &+0.25\cdot(D_{T_{2}}^{1}+2\cdot D_{T_{2}}^{2}+D_{T_{2}}^{3})\cdot\sum\limits_{h =1}^{\infty}p_{h}\cdot Y_{h}\\ &+\lambda_{1}\cdot[D_{T_{2}}^{3}-0.25\cdot(D_{T_{2}}^{1}+2\cdot D_{T_{2}}^{2}+ D_{T_{2}}^{3})]\cdot\sum\limits_{h=1}^{\infty}p_{h}\cdot Y_{h}\\ &+\lambda_{2}\cdot[0.25\cdot(D_{T_{2}}^{1}+2\cdot D_{T_{2}}^{2}+D_{T_{2}}^{3}) -D_{T_{2}}^{1}]\cdot\sum\limits_{h=1}^{\infty}p_{h}\cdot Y_{h}\\ &+\rho_{1}\cdot\sum\limits_{h=1}^{s}p_{h}\;V+\rho_{2}\cdot[B^{R}(\alpha)-B_{r }{}^{1}]\end{array}  \tag{15}\]  
subject to:  
\[Cr\{\widetilde{B}|A_{r}X\leq\widetilde{B}_{r}\}\geq\alpha,r\in M;\;M=1,2,\ldots,\;m_{1} \tag{16}\]  
\[A_{r}X+A^{\prime}{}_{rh}Y_{h}\geq w^{\prime}{}_{ih},\;i\in M;\;i=1,2,\;\ldots,\;m_{2};\;h=1,\;2,\;\ldots,\;s \tag{17}\]  
\[x_{j}\geq 0,\;x_{j}\in X;j=1,\;2,\;\ldots,\;n_{1} \tag{18}\]  
\[y_{jh}\geq 0,\;y_{jh}\in Y;j=1,\;2,\;\ldots,\;n_{2};\;h=1,\;2,\ldots,\;v \tag{19}\]  
\[V=D_{T_{2}}Y_{h}-\sum\limits_{h=1}^{s}p_{h}D_{T_{2}}Y+2\theta_{h},\;h=1,2, \;\ldots,s \tag{20}\]  
\[v_{jh}\geq 0,v_{jh}\in V;\;j=1,2,\;\ldots,\;n_{2};\;h=1,2,\;\ldots,s \tag{21}\]  
where \(\overline{f}\) denotes the expected value of objective function (8); \(f_{\text{max}}\) and \(f_{\text{min}}\) are the maximum value and minimum value of objective function (8); \(B^{R}(\alpha)\) is the maximum values of all potential values, i.e., \(B^{R}(\alpha)=\sup\{B|B=\mu^{-1}(\alpha)\}\), \(\mu^{-1}\) is the inverse of \(\mu\).