## 3 Mathematical Model
### Model Formulation  
In the mathematical model, this study comprehensively considers TC, the number of electric vehicles (NEVs) and fuel vehicles used, carbon emissions and the rest time of drivers under time window constraints. It is assumed that fuel vehicles can have the capacity of continuously working and that electric vehicles need to re  
Figure 1: A schematic diagram of the framework for the supplier and factory. working during a period of time. Every transportation of a vehicle is a full load. In addition, no carbon emissions are produced by electric vehicles. If the number of combinations of electric and fuel vehicles cannot meet the transportation demand of materials within the time windows in a production cycle, production scheduling needs to add the number of vehicles until the transportation task is completed within each time window. Each supplier needs to provide an inventory of material in a production cycle. The purchase plan and transportation plan are drawn up in terms of each supplier's material inventory. The notations used to develop the mathematical model are summarised in Table 1. \begin{table}
\begin{tabular}{c l} \hline \hline
**Sets** & \\ \hline \(N\) & Set of supplier storage \\ \(E\) & Set of electric vehicles \\ \(F\) & Set of fuel vehicles \\ \(W\) & Set of time windows \\ \(S_{o}\) & Set of vehicles \\
**Parameters** & \\ \(i\) & Index for supplier storage, \(i\in\{1,\ 2,\ \ldots,N\}\) \\ \(e\) & Index for electric vehicle, \(e\in\{1,\ 2,\ \ldots,E\}\) \\ \(f\) & Index for fuel vehicle, \(f\in\{1,\ 2,\ \ldots,F\}\) \\ \(w\) & Index for time window, \(w\in\{1,\ 2,\ \ldots,W\}\) \\ \(v\) & Index for the vehicle, \(v\in\{1,\ 2,\ \ldots,S_{o}\}\) \\ \(c\) & Ton-kilometer TC \\ \(d_{ij}\) & Distance from supplier \(i\) to factory storage \\ \(s_{i}\) & Material storage of supplier storage \(i\) \\ \(C_{e}\) & Utilizing cost of electric vehicle \(e\) \\ \(C_{f}\) & Utilizing cost of fuel vehicle \(f\) \\ \(E_{f}\) & Carbon emissions of ton-km transportation of fuel vehicle \\ \(Q_{e},Q_{f}\) & Carry capacity of electric vehicle \(e\) and fuel vehicle \(f\) \\ \(A_{e},A_{f}\) & Constants \\ \(best\) & Best rest time for driver of electric vehicle between continuous working \\ \(T_{e}\) & Time per charge of electric vehicle \(e\) \\ \(T_{c}\) & Time of a scheduling cycle \\ \(C_{wt}\) & Continuous working time of electric vehicle driver \\ \(L_{t}\) & Loading time of vehicle \\ \(V\) & Velocity of vehicle \\ \([S^{w}\ E^{w}]\) & Start time \(S^{w}\) and end time \(E^{w}\) of \(with\) time window \\ \(M\) & Minimum total volume of transportation \\ \(S_{max}\) & maximum number of electric and fuel vehicles \\ \(RestTime\) & Rest time electric vehicle driver is required to between continuous working \\ \(D_{w}\) & Quantity demand of material of \(with\) time window \\ \(E_{em}\) & Endurance mileage of electric vehicle \(e\) \\ \(C_{q}\) & Carbon quota allocated to the factory storage \\
**Variables** & \\ \(x_{i}\) & 1 if supplier storage \(i\) is selected, otherwise 0 \\ \(y_{e}\) & 1 if \(e\) is selected to transport, otherwise 0 \\ \(y_{fo}^{n}\) & 1 if \(v\) is selected to transport for \(i\) at \(nth\), otherwise 0 \\ \(y_{f}\) & 1 if \(f\) is selected, otherwise 0 \\ \(S_{em}\) & Sustained mileage of electric vehicle \(e\) transportation \\ \(q_{e}\) & 1 if \(E_{em}\leq S_{em}\), otherwise 0 \\ \(D_{e}\) & Transportation distance of electric vehicle \(e\) \\ \(N_{e}\) & Charging NEV \(e\) \\ \(T_{m\
u}\) & Time of the \(nth\) departure of vehicle \(v\) from the factory storage \\ \(A_{m\
u}\) & Time of the \(nth\) arrival at the factory storage of vehicle \(v\) from supplier storage \\ \(T_{f}^{\
u}\) & transportation volume of electric \(e\) during the \(wh\) time window \\ \(T_{f}^{\
u}\) & transportation volume of electric \(f\) during the \(wh\) time window \\ \(A_{rt}\) & Actual rest time of electric vehicle driver between continuous working \\ \(S_{e}\) & Number of used electric vehicles \\ \(S_{f}\) & Number of used fuel vehicles \\ \hline \hline \end{tabular}
\end{table}