A mathematical model for the electric vehicle routing with time windows considering queuing system at charging stations and alternative paths

Due to many damages that human activities have imposed on the environment, authorities, manufacturers, and industry owners have taken into account the development of supply chain more than ever. One of the most influential and essential human activities in the supply chain are transportation which green vehicles such as electric vehicles (EVs) are expected to generate higher economic and environmental impact. To this end, designing efficient routing scheme for the fleet of EVs is significant. A remarkable issue about EVs is their need to stations for charging their battery. Due to the existence of time limitations, more attention should be paid to time spent at the charging station, so considering the queuing system at charging stations makes more precise time calculations.  Furthermore, multi-graphs are more consistent with the characteristics of the transportation network. Hence, we consider alternative paths including two criterion cost and energy consumption in the network. First, we develop a mixed integer linear programming for the electric vehicle routing problem on a multi-graph with the queue in charging stations to minimize the traveling and charging costs. Since the proposed problem is NP-hard in a strong sense, we provide a simulated annealing algorithm to search the solution space efficiently and explore a large neighborhood within short computational time.  The efficiency of the model is investigated with numerical and illustrative examples. Then the sensitivity analysis is performed on the proposed model to indicate the importance of the queuing system and the impact of battery capacity on the penetration of EVs.