Modified particle swarm optimization in a time-dependent vehicle routing problem: minimizing fuel consumption

In this paper, a new mathematical model for a time-dependent vehicle routing problem (TDVRP) is introduced as one of the sub-problems in the green VRP field. The model results in reducing carbon emissions by minimizing travel times, while congested routes are avoided and customers are served at the time of the day with an optimized speed. Hence, this paper presents a new bi-objective mathematical model in such a way that the total travel times and total fuel consumption are minimized in a time-dependent situation by considering the air drag, roads' physical condition, the physical features of each vehicle, and the weight and load of the vehicles. To solve such a hard problem, a new method based on the modified particle swarm optimization (PSO) algorithm is proposed and the results are compared with the standard PSO on well-known benchmark problems with constant travel speeds and time-dependent travel speeds. The results show the capability of the proposed optimization algorithm for handling this problem.