## 3 Establish an EFLP Model and Solution Method
### Robust Peer-to-Peer Model  
This section uses robust optimization theory to transform objective functions and constraints containing uncertainty parameters into uncertainty sets. Then, the objective functions and constraints in the previous deterministic model are equivalent to the robust optimization model with uncertainty sets. To represent the robust optimization model with the shortest total emergency rescue distance \(D(X)\) of the RDGT emergency rescue system, we consider the robust optimization model with the demand of demand point \(i\) under the elliptical uncertainty set.  
The elliptic uncertainty set is defined in the following form:  
\[U^{E}=\left\{\widetilde{\beta}_{i}\in R^{N}:\left(\widetilde{\beta}_{i}- \beta_{i}\right)^{T}\sum^{-1}\left(\widetilde{\beta}_{i}-\beta_{i}\right) \leq\Omega^{2}\right\}, \tag{9}\]  
where \(\left(\widetilde{\beta}_{i}-\beta_{i}\right)^{T}\sum^{-1}\left(\widetilde{ \beta}_{i}-\beta_{i}\right)\leq\Omega^{2}\) is an ellipse, and \(\widetilde{\beta}_{i}\) is the nominal value of \(\beta_{i}\) and the center of the ellipse. The positive definite matrix \(\sum\) is a matrix that describes the uncertainty relation of the objective function and constraint conditions and affects the direction of the ellipse. Value \(\Omega\) is the conservative degree, that is, the parameter of the conservative limit, which affects the size of the ellipse. Therefore, under elliptic uncertain sets, the constraints are equivalent to the following:  
\[\max_{\tilde{\beta}_{i}}\biggl{\{}\widetilde{\beta_{i}}^{T}x:\left(\widetilde{ \beta}_{i}-\beta_{i}\right)^{T}\sum^{-1}\biggl{(}\widetilde{\beta}_{i}-\beta_{ i}\biggr{)}\leq\Omega^{2}\biggr{\}}\leq b_{j},\forall i, \tag{10}\]  
where \(\sum\) is a positive definite matrix, and thus, \(\max_{\tilde{\beta}_{i}}\biggl{\{}\widetilde{\beta}_{i}^{T}x:\left(\widetilde {\beta}_{i}-\beta_{i}\right)^{T}\sum^{-1}\biggl{(}\widetilde{\beta}_{i}-\beta_ {i}\biggr{)}\leq\Omega^{2}\biggr{\}}\) is a convex problem solved by the Lagrange multipliers and Kuhn-Tucker (K-T) conditions. Let \(\beta_{i}^{*}\) be the optimal solution, \(Z^{*}\) be the optimal objective value, and \(\lambda^{*}\) be the Lagrange multiplier.  
Then, we have the following:  
\[L(\beta_{i}^{*},\lambda^{*}) =\beta_{i}^{*T}x+\lambda^{*}[\left(\widetilde{\beta}_{i}-\beta_{ i}\right)^{T}\sum^{-1}\biggl{(}\widetilde{\beta}_{i}-\beta_{i}\biggr{)}- \Omega^{2}]\] (11) s.t. \[-x+2\lambda^{*}\sum^{-1}[(\beta_{i}^{*}-\beta_{i})=0\] \[\lambda^{*}[\left(\widetilde{\beta}_{i}-\beta_{i}\right)^{T}\sum ^{-1}\biggl{(}\widetilde{\beta}_{i}-\beta_{i}\biggr{)}-\Omega^{2}]=0\] \[\lambda^{*}\geq 0\]  
This leads to the following:  
\[\beta_{i}^{*T}x=\beta_{i}x+\frac{\Omega}{\sqrt{x^{T}\sum x}}x^{T}\sum x=\beta_ {i}x+\Omega\sqrt{x^{T}\sum x} \tag{12}\]  
Therefore, the linear programming model corresponds to the equivalent elliptic uncertainty set model:  
\[\min cx+d\,\] (13) s.t \[\beta_{i}x+\Omega\sqrt{x^{T}\sum x}\leq b_{i},\forall i\] \[x\in X\]  
The elliptical robust optimization model is more complex, and its robust model can be adjusted by the safety parameter \(\Omega\) to improve the immunity of the original model.  
The deterministic siting model is transformed into a robust peer-to-peer model according to the elliptic robust optimization conclusion for the uncertain set:  
\[\min(Y),\] (14) s.t. \[\sum_{i=1}^{n}\beta_{i}\left(\sum_{j=1}^{m}X_{i,j}d(i,j)\right)+ \Omega\Bigl{(}X_{d}\sum(X_{d})^{T}\Bigr{)}^{1/2}\leq Y \tag{15}\] \[X_{d}=\left(\sum_{j=1}^{m}X_{0,j}d(0,j),\ldots,\sum_{i=1}^{m}X_ {n,j}d(n,j)\right)^{T}\] (16) \[X_{i,j}d(i,j)\leq\min\bigl{\{}d_{i},d_{j}\bigr{\}}\] (17) \[\sum_{i=1}^{n}\beta_{i}X_{i,j}+\Omega\Bigl{(}X_{,j}\sum(X_{,j})^{ T}\Bigr{)}\leq C_{j}\] (18) \[1\leq\sum_{j=1}^{m}X_{i,j}\leq k\] (19) \[\sum_{i=1}^{n}X_{i,j}\leq h\] (20) \[X_{i,j}\in\{0,1\} \tag{21}\]