## 3 Model Formulation
### Model Development
#### 3.4.1 Objective Function Analysis of Model  
(1) Transportation Cost  
The transportation cost mainly includes vehicle use cost, fuel used in the process of transportation, driver's cost, and other factors. For the convenience of research, the transportation cost is considered in two parts, referring to the use and consumption cost of vehicles per kilometre and driver's cost. The transportation cost of a refrigerated vehicle in the first-stage and second-stage routes can be expressed as follows:  
\[C_{1}=c_{1}\sum_{k\in K}\sum_{i,j\in M}X^{k}_{ij}d_{ij}+c^{\prime}_{1}\sum_{k \in K}\sum_{i,j\in M}(X^{k}_{ij}\frac{d_{ij}}{v_{1}}+s^{k}_{j}) \tag{1}\]  
\[C^{\prime}_{1}=c_{2}\sum_{g\in M_{1}}\sum_{l\in L_{g}}\sum_{i,j\in N_{g}}Z_{g} d_{ij}x^{l}_{ijg}+c^{\prime}_{2}\sum_{g\in M_{1}}\sum_{l\in L_{g}}\sum_{i,j\in N _{g}}Z_{g}(\frac{d_{ij}}{v_{2}}x^{l}_{ijg}+s^{l}_{j}) \tag{2}\]  
Commodities in the cold-chain distribution are easily spoiled and therefore need to be kept in an appropriate low-temperature environment. The quality of perishable goods gradually declines or can lose value with time. When the quality of the product declines to a certain extent, spoilage cost will be incurred. The cost of damage is divided into two parts: the damage cost of goods accumulated over time in the process of transportation and the damage cost incurred when opening the door in the process of unloading. Because the first-stage delivery is usually carried out in an enclosed environment, spoilage cost will be minimal. We only considered the damage cost in the second-stage distribution. Hence, the total damage cost can be expressed as:  
\[C^{\prime}_{2}=P_{1}\sum_{g\in M_{1}}\sum_{l\in L_{g}}\sum_{j\in N_{g}}Z_{g} y^{l}_{ig}(1-e^{-\theta_{1}(t^{l}_{ij}-t^{l}_{bg})})u^{l}_{j}+P_{1}\sum_{g\in M _{1}}\sum_{l\in L_{g}}\sum_{j\in N_{g}}Z_{g}y^{l}_{ig}(1-e^{-\theta_{2}s^{l}_{ j}})b^{l}_{j} \tag{3}\]  
The first part represents the cost of cargo damage in the transportation process, and the second part represents the cost of cargo damage in the unloading process. (3) Refrigeration Cost  
The cost of storage associated with maintaining the temperature and humidity inside the carriage is called the refrigeration cost. The refrigeration methods employed in refrigerated vehicles on the market today mainly include liquid nitrogen refrigeration, mechanical refrigeration, dry ice refrigeration, and cold plate refrigeration. The refrigeration method considered in this study is mechanical refrigeration, and the cost generated in the refrigeration process is related to fuel consumption, time, and weight of goods. Literature [31; 32] proposed a formula for calculating the fuel consumption: \(W=\omega\frac{eP_{e}}{\xi}*10^{-3}\). The refrigeration cost includes the cost associated with the energy consumption of the vehicle in maintaining a low-temperature environment during delivery and the cost of additional energy supplied to the refrigeration system during the unloading process. The refrigeration cost in the first-stage and second-stage route can be expressed as follows:  
\[C_{3}=\theta_{3}W_{1}\sum_{k\in K}\sum_{i,j\in M}\frac{d_{ij}x_{ij}^{k}u_{j}^{ k}}{v_{1}}+\theta_{3}W_{2}\sum_{k\in K}\sum_{j\in M}x_{j}^{k}b_{j}^{k}s_{j} \tag{4}\]  
\[C_{3}^{\prime}=\theta_{3}W_{1}\sum_{g\in M_{1}}\sum_{l\in L_{g}}\sum_{i,j\in N _{g}}Z_{g}\frac{d_{ij}x_{ij}^{l}u_{j}^{l}}{v_{2}}+\theta_{3}W_{2}\sum_{g\in M _{1}}\sum_{l\in L_{g}}\sum_{j\in N_{g}}Z_{g}s_{j}^{l}b_{j}^{l} \tag{5}\]  
where, \(W\) is the fuel consumption (g/h); \(\omega\) is the power utilization coefficient of the refrigerator; \(\varepsilon\) is the fuel consumption rate (g/kW-h); \(P_{e}\) is the effective power of the refrigerator (kW); \(\xi\) is the specific gravity of the fuel; \(W_{1}\) is the fuel oil consumption during transportation; and \(W_{2}\) is the fuel consumption during unloading. The first part represents the refrigeration cost of refrigerated vehicles during transportation, and the second part represents the refrigeration cost of the refrigerated vehicles during unloading.