# Optimization of Multimodal Transport Paths Considering a Low-Carbon Economy Under Uncertain Demand
## 3 Materials and Methods
### Model Transformation Under Different Low-Carbon Policies  
Based on various low-carbon regulations, a low-carbon multimodal transportation route optimization model is constructed, i.e., using carbon trading, carbon taxes, carbon footprint requirements and carbon compensation policies as conversion tools. The total amount of carbon emissions produced by multimodal transportation are transformed into constraints, carbon emission costs or carbon emission benefits and put into the model. Correspondingly, they are labeled as Model A, Model B, Model C and Model D, which are described as follows. 1. Model A--Choice model under mandatory carbon emission policy. In multimodal transportation, the mandatory carbon emission policy usually includes the regulation and limitation of the carbon emissions of each transportation mode. This requires that the choice of the optimal transportation path for multimodal transportation must satisfy the constraint that the actual carbon emissions are less than or equal to the cap on carbon emissions. Therefore, under the mandatory carbon emission policy, the total carbon emissions are transformed into a constraint and modeled as follows:  
\[\text{min}C=C_{1}+C_{2} \tag{13}\]  
\[s.t.\ E\leq U_{c} \tag{14}\]  
At the same time, Constraints (7) to (12) are established. Equation (13) is aimed at minimizing the total cost of transportation, where the total cost of transportation is equal to the sum of the transportation cost and waiting cost. Equation (14) suggests that the overall carbon emission must be equal to or less than the carbon emission limit, where E is the total amount of carbon emissions converted by individual road segments and nodes and \(U_{c}\) is the upper limit of carbon emission for the transportation task under the mandatory carbon emission. 2. Model B--Choice Model under a Carbon Tax Policy. A carbon tax is a tax level by the government on carbon emissions. In multimodal transportation, the government taxes each unit of actual carbon emissions at the tax rate \(R_{t}\). At this point, carbon emissions are converted into carbon costs and added to the total transportation costs. \[\text{min}C=C_{1}+C_{2}+R_{t}.\ E \tag{15}\]  
The constraints are the same as Equations (7)-(12). Equation (15) indicates that the total cost of transportation is minimized under the carbon tax policy. 3. Model C--Choice model under carbon trading policy. A carbon trading policy means that the government sets total carbon emission allowances \(U_{s}\) and then allocates these allowances to enterprises or industries. Firms can buy and sell these allowances on the market; if firms emit less than their allowances, they can sell their excess allowances, while if they emit more than their allowances, they need to buy additional allowances. \[\text{min}C=C_{1}+C_{2}+R_{s}.Q_{s} \tag{16}\]  
\[s.t.\ E+Q_{s}=U_{s} \tag{17}\]  
\[Q_{s}\in R \tag{18}\]  
At the same time, Equations (7)-(12) hold. Goal Function (16) shows how to minimize all costs, including those associated with transportation, waiting costs and carbon trading costs or benefits. \(R_{s}\) is the carbon trading price. Constraint (17) is the actual carbon emission and is equal to the carbon emission amount after trading. \(Q_{s}\) is the emission quota of the transportation task under the carbon trading policy. Constraint (18) is the actual purchased carbon emission amount. Model D--Selection Model under the Carbon Offset Policy. The principle of carbon offsetting is similar to carbon trading, except that when the actual carbon emissions are greater than the carbon offsetting allowances \(U_{0}\), the enterprise must purchase the difference in carbon emissions to meet the requirements and incur carbon offsetting costs. However, when the carbon emissions are less than the carbon offsetting allowances, the enterprise cannot sell the excess carbon offsetting allowances to generate revenue. \[\text{minC}=\text{C}_{1}+\text{C}_{2}+\text{R}_{0}\cdot Q_{0} \tag{19}\]  
\[s.t.\,\text{Q}_{0}=\left\{