## Methodology  
The robust two-stage stochastic optimization (RTSO) method brings risk aversion into stochastic programming methods, and finds robust schemes for system management [33; 34]. The RTSO method can deal with the stochastic uncertainties of real-world management problems, analyze the policy scenarios associated with economic penalties when the predefined policies of the first-stage are violated, capture the variability of the second-stage costs that are greater than the expected values, and evaluate trade-offs between system economy and risk [23; 24]. An RTSO method is formulated as follows:  
\[\text{Min}f=C_{T_{1}}X+\sum\limits_{h=1}^{s}p_{h}D_{T_{2}}Y_{h}+\rho_{1}\sum \limits_{h=1}^{s}p_{h}\;V \tag{1}\]  
subject to  
\[A_{r}X\leq B_{r},r\in M;\;M=1,2,\ldots,\;m_{1} \tag{2}\]  
\[A_{r}X+A^{\prime}{}_{rh}Y_{h}\geq w^{\prime}{}_{ih},\;i\in M;\;i=\;1,2,\; \ldots,\;m_{2};\;h=1,\;2,\;\ldots,\;s \tag{3}\]  
\[x_{j}\geq 0,\;x_{j}\in X;j=1,\;2,\;\ldots,\;n_{1} \tag{4}\]  
\[y_{ih}\geq 0,\;y_{jh}\in Y;j=1,\;2,\;\ldots,\;n_{2};\;h=1,\;2,\ldots,\;v \tag{5}\]  
\[V=D_{T_{2}}Y_{h}-\sum\limits_{h=1}^{s}p_{h}D_{T_{2}}Y_{h}+2\theta_{h},\;h=1,2, \;\ldots,s \tag{6}\]  
\[v_{jh}\geq 0,v_{jh}\in V;\;j=1,2,\;\ldots,\;n_{2};\;h=1,2,\;\ldots,s \tag{7}\]  
where, \(X\) and \(Y_{h}\) denote the decision variables of the first-stage and the second-stage, respectively; \(p_{h}\) are occurrence probability of scenario \(h\), \(\sum\limits_{h=1}^{H}p_{h}=1\); \(C\) represent coefficients of \(X\) and \(D_{h}\) are coefficients of \(Y_{h}\); \(w^{\prime}{}_{ih}\) are random variables with probability levels \(p_{h}\); \(A_{r}\) is the fixed coefficient of \(X_{i}\)\(A_{rh}^{\prime}\) are coefficients of \(Y_{h}\). \(B_{r}\) represent the boundary vectors of the right-hand side of constraints. \(\theta_{h}\) denote slack variables used for achieving looser constraints. \(\rho_{1}\) denotes a goal programming weight of stochastic uncertainties; the managers can regulate the variability of the stochastic recourse cost through adjusting the \(\rho_{1}\) level [24; 35]. When \(\rho_{1}\)= 0, the RTSO model becomes a conventional TSP, which indicates that the managers adopt risk-neutral attitudes and the variability of the stochastic uncertain recourse costs is not considered. However, when \(\rho_{1}\) = 1, the managers adopt risk- aversiveattitudes and the variability of the second-stage cost is considered. \(V\) denotes the deviation of an expected value from the given scenario's cost [3]. Besides, constraints (6) and (7) can define the positive variability of the recourse costs. However, RTSO is inefficient in addressing the uncertainties expressed by fuzzy sets. Thus, fuzzy possibilistic programming (FPP) is joined to RTSO as a hybrid robust stochastic-fuzzy optimization (RSFO) model as follows:  
\[\text{Min}\;\widetilde{f}=\widetilde{C}_{T_{1}}X+\sum_{h=1}^{s}p_{h} \widetilde{D}_{T_{2}}Y_{h}+\rho_{1}\sum_{h=1}^{s}p_{h}\;V \tag{8}\]  
subject to  
\[Cr\{\widetilde{B}|A_{r}X\leq\widetilde{B}_{r}\}\geq\alpha,r\in M;\;M=1,2, \ldots,\;m_{1} \tag{9}\]  
\[A_{r}X+A^{\prime}{}_{rh}Y_{h}\geq w^{\prime}{}_{ih},\;i\in M;\;i=\;1,2,\; \ldots,\;m_{2};\;h=1,\;2,\;\ldots,\;s \tag{10}\]  
\[A_{r}X\leq B_{r},r\in M;\;M=1,2,\ldots,\;m_{1} \tag{2}\]0  
\[A_{r}X\leq B_{r},r\in M;\;M=1,2,\ldots,\;m_{1} \tag{2}\]1  
\[A_{r}X\leq B_{r},r\in M;\;M=1,2,\ldots,\;m_{1} \tag{2}\]2  
\[A_{r}X\leq B_{r},r\in M;\;M=1,2,\ldots,\;m_{1} \tag{2}\]3  
where \(Cr\{\cdot\}\) represents the credibility measure of a fuzzy event in \(\{\cdot\}\); \(\widetilde{C}_{T_{1}}\) and \(\widetilde{D}_{T_{2}}\) are cost coefficients expressed as a triangular fuzzy number; \(\widetilde{B}_{r}(B_{r}{}^{1},B_{r}{}^{2},B_{r}{}^{3})\) is the boundary vectors of the right-hand side of constraints, which expresses as triangular fuzzy sets with its membership functions \(\mu(\widetilde{B}_{r})\). \(\alpha\) denotes the predetermined confidence-level.