## 3 Modal Construction
### Basic Model Construction  
The modeling approach in this study proceeds in two distinct phases [36]. Initially, a deterministic multimodal transport route optimization model is developed in this section, serving as the foundational framework; this model is subsequently adapted to accommodate varying low-carbon policy requirements. Subsequently, uncertainties in both transportation costs and time are incorporated. Scenario-based robust regret minimization is employed to quantify freight cost fluctuations, while triangular fuzzy numbers address transport time variability. This integration culminates in a novel multimodal route optimization model capable of handling the hybrid uncertainty arising from these dual factors. (1) Model under the Carbon Tax Policy. Under carbon taxation policy frameworks, governmental authorities levy fiscal charges on enterprises' per-unit carbon emissions, thereby incorporating environmental externalities as quantifiable cost components into the optimization objective function. This study sets the carbon emission tax rate under the carbon tax policy as \(P_{t}\). \[C_{4}^{t}=P_{t}\cdot E \tag{11}\]\[min\ C^{t}=C_{1}+C_{2}+C_{3}+C_{4}^{t} \tag{12}\]
2. Model under the Carbon Trading Policy. Within carbon trading policy frameworks, regulatory authorities or designated organizations allocate enterprises specific carbon emission quotas. Enterprises may strategically manage these allowances through purchasing supplementary emission credits in carbon markets or generating fiscal offsets by selling surplus quotas, thereby achieving cost optimization through market-based emission right transactions. This study operationalizes the carbon trading mechanism by parameterizing the market equilibrium price as \(P_{s}\) within the emissions trading analytical model. This study sets the carbon emission quota as \(U_{s}\) under the carbon trading policy. \[C_{4}^{s}=P_{s}\cdot(E-U_{s})\] (13) \[minC^{s}=C_{1}+C_{2}+C_{3}+C_{4}^{s}\] (14)
3. Model under the Mandatory Carbon Emission Policy. Under mandatory carbon emission policy frameworks, regulatory authorities allocate enterprises specific carbon emission quotas through institutionalized allocation mechanisms. Enterprises are subject to rigorous emission constraints, with all production-related carbon outputs strictly prohibited from exceeding the predetermined maximum threshold. This regulatory imperative is mathematically operationalized as a carbon cap constraint within the analytical model, mandating that cumulative emissions across operational phases remain within the assigned quota boundary. The study parameterizes the emission allowance ceiling as \(U_{c}\) within this policy-specific modeling paradigm, thereby establishing an absolute compliance requirement for emission containment in multimodal transportation operations. \[C_{5}^{c}=0\] (15) \[minC^{c}=C_{1}+C_{2}+C_{3}\] (16) \[s.t.E\leq U_{c}\] (17)
4. Model under the Carbon Offset Policy. Under carbon offset policy frameworks, enterprises receive predetermined carbon emission allowances through institutional allocation mechanisms. Regulatory compliance mandates that entities exceeding allocated quotas must allocate financial resources to subsidize certified emission reduction initiatives to neutralize surplus emissions, while conversely prohibiting the monetization of unused allowances through market transactions when emissions fall below assigned thresholds, thereby eliminating potential cost offsets through carbon credit sales. This study operationalizes the carbon compensation mechanism by parameterizing the certified carbon price as \(P_{o}\) and initial allowance allocation as \(U_{o}\), establishing a unidirectional cost absorption model where enterprises bear full financial responsibility for emission exceedances without compensatory market-based revenue opportunities.