## 2 Mathematical Model
### Interval Fuzzy Linear Optimization Model  
Referring to the modeling structure presented by [PERSON] et al. (2016), this study proposes an IFLOM that integrates both a MTW and capacity and carbon tax rate uncertainty to deal with the GIMRP. Objective:  
\[ \begin{array}{l}\min\sum\limits_{(i,j)\in A}\sum\limits_{r\in S_{ij}}c_{ijr} \cdot q\cdot x_{ijr}+\sum\limits_{j\in N}\sum\limits_{r\in S_{j}}\sum\limits_{ s\in S_{j}}c_{ij}^{rs}\cdot q\cdot y_{j}^{rs}+c_{pen}\cdot q\cdot w+c^{ \prime}_{pen}\cdot q\cdot w^{\prime}\\ +\widetilde{p}\cdot(\sum\limits_{(i,j)\in A}\sum\limits_{r\in S_{ij}}e_{ijr} \cdot d_{ijr}\cdot q\cdot x_{ijr}+\sum\limits_{j\in N}\sum\limits_{r\in S_{j}} \sum\limits_{s\in S_{j}}e_{j}^{rs}\cdot q\cdot y_{j}^{rs})\\ \end{array}  \tag{1}\]  
Constraints:  
\[\sum\limits_{i\in N_{j}^{-}}\sum\limits_{r\in S_{ij}}x_{ijr}-\sum\limits_{k\in N _{j}^{-}}\sum\limits_{s\in S_{jk}}x_{jks}=\left\{ \begin{array}{ll}-1\;\;j=n_ {o}\\ 0\;\;\forall j\in N\backslash\{n_{o},n_{d}\}\\ 1\;\;j=n_{d}\end{array} \right. \tag{2}\]  
\[\sum\limits_{r\in S_{ij}}x_{ijr}\leq 1\;\forall(i,j)\in A \tag{3}\]  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]  
\[\sum\limits_{k\in N_{j}^{+}}x_{jks}=\sum\limits_{r\in S_{j}}y_{j}^{rs}\;\forall j \in N\backslash\{n_{o},n_{d}\}\;\forall s\in S_{j} \tag{5}\]  
\[\sum\limits_{i\in N_{j}^{-}}x_{ijr}=\sum\limits_{s\in S_{j}}y_{j}^{rs}\;\forall j \in N\backslash\{n_{o},n_{d}\}\;\forall r\in S_{j} \tag{6}\]  
\[z=t_{0}+\sum\limits_{(i,j)\in A}\sum\limits_{r\in S_{ij}}\frac{d_{ijr}}{v_{ ijr}}\cdot x_{ijr}+\sum\limits_{j\in N}\sum\limits_{r\in S_{j}}\sum\limits_{s \in S_{j}}t_{j}^{rs}\cdot q\cdot y_{j}^{rs} \tag{7}\]  
\[z\geq\theta \tag{8}\]  
\[z\leq\theta^{\prime} \tag{9}\]  
\[w\geq\pi-z \tag{10}\]  
\[w^{\prime}\geq z-\pi^{\prime} \tag{11}\]  
\[x_{ijr}\cdot q\leq\widetilde{\eta}_{ijr}\;\forall(i,j)\in A\;\forall r\in S_{ ij} \tag{12}\]  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]0  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]1  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]2  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]3  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]4  
\[\sum\limits_{r\in S_{j}}\sum\limits_{s\in S_{j}}y_{j}^{rs}\leq 1\;\forall j\in N \backslash\{n_{o},n_{d}\} \tag{4}\]5  
In the proposed model, Equation (1) is the optimization objective that aims to minimize the total costs (TCs) of accomplishing the transportation order. The TCs consist of transportation costs (i.e., the sum of the in-transit costs and the transfer costs), time costs (i.e., penalty costs that results from the violation of the receiver's expected time window), and CE costs (i.e., carbon tax when CTR is implemented to reduce CEs). Equation (2) is the flow equilibrium constraint, and it ensures that there are only containers departingfrom the origin or arriving at the destination, while the containers arriving at each transfer nodes must all depart from it. Equation (3) means that only one transportation mode can be selected to move the containers between two nodes in the route. Equation (4) indicates that only one transfer type can happen at a transfer node in the route. Equations (5) and (6) regulate two types of binary variables to ensure that the in-transit process and the transfer process are connected reasonably by the route. Equations (2)-(6) are the basic constraints to ensure that the PIMR smoothly connecting the in-transit process on the arcs and transfer process at the nodes can be generated to move the containers of the transportation order in an unsplittable manner [1]. Equation (7) calculates the delivery time of the containers to the destination.