## 3 Model Processing
### Defuzzification Procedure  
Step 1: Defuzzification of the fuzzy objective function.  
We converted the initial objective, i.e., Equation (1), into Equations (19)-(21), where \(\xi\) is an auxiliary variable and represents the CE budget of the IMT prepared by the receiver [22]. Equation (20) means that the CE costs of the PIMR should not exceed the CE budget of the receiver.  
\[\min\sum_{(i,j)\in A}\sum_{r\in S_{ij}}c_{ijr}\cdot q\cdot x_{ijr}+\sum_{j\in N }\sum_{r\in S_{j}}\sum_{s\in S_{j}}c_{j}^{rs}\cdot q\cdot y_{j}^{rs}+c_{pen} \cdot q\cdot w+c_{pen}\,^{\prime}\cdot q\cdot w^{\prime}+\xi \tag{19}\]  
\[\widetilde{p}\cdot(\sum_{(i,j)\in A}\sum_{r\in S_{ij}}e_{ijr}\cdot d_{ijr} \cdot q\cdot x_{ijr}+\sum_{j\in N}\sum_{r\in S_{j}}\sum_{s\in S_{j}}e_{j}^{rs} \cdot q\cdot y_{j}^{rs})\leq\xi \tag{20}\]  
\[\xi\geq 0 \tag{21}\]  
Step 2: Defuzzification of the fuzzy constraints.  
We converted Equations (12), (13), and (20), which contain fuzzy parameters, into the following equations [7]:  
\[Poss\big{\{}x_{ijr}\cdot q\leq\widetilde{\eta}_{ijr}\big{\}}\geq\lambda\ \forall(i,j)\in A\ \forall r\in S_{ij} \tag{22}\]  
\[Poss\big{\{}y_{j}^{rs}\cdot q\leq\widetilde{\eta}_{jr}^{rs}\big{\}}\geq \lambda\ \forall j\in N\backslash\{n_{o},n_{d}\}\ \forall r\in S_{j}\ \forall s\in S_{j} \tag{23}\]  
\[Poss\big{\{}\widetilde{p}\cdot(\sum_{(i,j)\in A}\sum_{r\in S_{ij}}e_{ijr} \cdot d_{ijr}\cdot q\cdot x_{ijr}+\sum_{j\in N}\sum_{r\in S_{j}}\sum_{s\in S_{ j}}e_{j}^{rs}\cdot q\cdot y_{j}^{rs})\leq\xi\big{\}}\geq\lambda \tag{24}\]In Equations (23) and (24), _Poss_ represents the possibility that the fuzzy event in \(\{\ \}\) holds. \(\lambda\) denotes the optimization level that should be prescribed by the receiver, and \(\lambda\in[0,1]\). Equations (23) and (24) mean that the possibility that the fuzzy constraints in the actual transportation hold should not be lower than a minimum optimization level given by the receiver. According to [PERSON] et al. [7] and [PERSON] et al. [23], we have the following relationships for an interval fuzzy number \(\widetilde{\alpha}=(\alpha^{-},\alpha^{+})\), a deterministic number \(\beta\), and an optimization level \(\lambda\).  
\[Poss\{\widetilde{\alpha}\geq\beta\}\geq\lambda\Leftrightarrow(1-\lambda) \cdot\alpha^{+}+\lambda\cdot\alpha^{-}\geq\beta \tag{25}\]  
\[Poss\{\widetilde{\alpha}\leq\beta\}\geq\lambda\Leftrightarrow(1-\lambda) \cdot\alpha^{-}+\lambda\cdot\alpha^{+}\leq\beta \tag{26}\]  
Based on the demonstration of Equations (25) and (26), Equations (23) and (24) can be rewritten as follows:  
\[(1-\lambda)\cdot\eta_{ijr}^{+}+\lambda\cdot\eta_{ijr}^{-}\geq x_{ijr}\cdot q \ \forall(i,j)\in A\ \forall r\in S_{ij} \tag{27}\]  
\[(1-\lambda)\cdot\eta_{j}^{rs+}+\lambda\cdot\eta_{j}^{rs-}\geq y_{j}^{rs}\cdot q \ \forall j\in N\backslash\{n_{o},n_{d}\}\ \forall r\in S_{j}\ \forall s\in S_{j} \tag{28}\]  
\[\big{[}(1-\lambda)\cdot p^{-}+\lambda\cdot p^{+}\big{]}\cdot(\sum_{(i,j)\in A }\sum_{r\in S_{ij}}e_{ijr}\cdot d_{ijr}\cdot q\cdot x_{ijr}+\sum_{j\in N}\sum_ {r\in S_{j}}e_{j}^{rs}\cdot q\cdot y_{j}^{rs})\leq\xi \tag{29}\]  
Step 3: Construction of a solvable crisp model.  
After the above processing, we can obtain a crisp reformation of the initial model given in Section 2.3 and simultaneously conserve the linearity of the model. A parametric linear optimization model that takes Equation (19) as the objective and Equations (2)-(11), (14)-(18), (21), and (27)-(29) as the constraints is established to deal with the GIM-RPMTWIFCCTR. In this study, we utilize the mathematical programming software Lingo to implement the branch-and-bound algorithm to solve the crisp model and obtain its GOS.