# SVD Enhanced Metaheuristic for Plug-In Hybrid Electric Vehicle Routing Problem
## 4 Methodology
### Memetic Algorithm
This work was partly supported by Fundamental Research Funds for the Xi'an Science and Technology Project (201805045 YD23 CG29), the Natural Science Basic Research Plan in Shaanxi Province of China(2018 JM5156) and the Shaanxi Key Project of Research and Development Plan(2019 ZDLGY03-01)

Authors declare no conflict of interest. The notations is introduced as follows: \begin{tabular}{l l} Sets & \\ \(Z\) & Set of power source, \(Z=\{e,f\}\) \\ \(M\) & Set of vehicles \\ \(I\) & Set of customers \\ \(I_{0}\) & Set of customers vertices and depot, \(I_{0}=I\cup\{v_{0}\}\) \\ \(S_{e}\) & Set of electric stations \\ \end{tabular} 

Figure 5: The graphics of the comparison result. \begin{tabular}{l l} \(S_{f}\) & Set of fueling stations \\ \(\Phi_{e}\) & Set of dummy electric stations \\ \(\Phi_{f}\) & Set of dummy fuel stations \\ \(S_{0}\) & Set of stations vertices including depot, fuel stations, electric stations and dummy stations, \\ \(S_{0}=\{v_{0}\}\cup S_{e}\cup S_{f}\cup\Phi_{e}\cup\Phi_{f}\) \\ \(V\) & Sets of vertices(without depot) and dummy vertices, \(V=I\cup S_{0}\) \\ \(V_{0}\) & Sets of vertices and dummy vertices, \(V_{0}=I_{0}\cup S_{0}\) \\ Coefficients \\ \(n\) & Number of customers \\ \(m\) & Number of vehicles \\ \(p_{j}\) & Service time at vertex \(j\); if \(i\in I\), then \(p_{j}\) is the service time at the customer vertex; if \(i\in S_{0}\), then \(p_{j}\) is the refueling time or recharging time at the station vertex \\ \(d_{ij}\) & Distance from vertex \(i\) to vertex \(j\) \\ \(t_{ij}\) & Time of arrival at vertex \(j\) from vertex \(i\) \\ \(Q_{kz}\) & Vehicle k's tank capacity of power type \(z\) \\ \(r_{kz}\) & Vehicle k's consumption rate (gallons per mile) on power type \(z\) \\ \(c_{z}\) & Cost rate for power source type \(z\), which includes \(c_{e}\) and \(c_{f}\) \\ \(T_{max}\) & Pre-determined maximum time for travel from depot until return to depot \\ \(C\) & A large positive constant value \\ Decision variables \\ \(x_{ijk}\) & Binary variable equal to 1 if vehicle \(k\) travels from vertex \(i\) to vertex \(j\); otherwise, 0 \\ \(y_{jkz}\) & The remaining tank Level of vehicle k's power type \(z\) upon arrival to vertex \(j\) \\  & Time variable specifying the time of arrival of a vehicle at vertex \(j\), initialized to zero upon departure \\ \(\tau_{j}\) & from the depot \\ \(f_{ijkz}\) & Vehicle k's consumption from vertex \(i\) to vertex \(j\) using power type \(z\) (electric or fuel) \\ \end{tabular} 

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