A modified particle swarm optimization algorithm for a vehicle scheduling problem with soft time windows

This article constructed a vehicle scheduling problem (VSP) with soft time windows for a certain ore company. VSP is a typical NP-hard problem whose optimal solution can not be obtained in polynomial time, and the basic particle swarm optimization(PSO) algorithm has the obvious shortcoming of premature convergence and stagnation by falling into local optima. Thus, a modified particle swarm optimization (MPSO) was proposed in this paper for the numerical calculation to overcome the characteristics of the optimization problem such as: multiple constraints and NP-hard. The algorithm introduced the “elite reverse” strategy into population initialization, proposed an improved adaptive strategy by combining the subtraction function and “ladder strategy” to adjust inertia weight, and added a “jump out” mechanism to escape local optimal. Thus, the proposed algorithm can realize an accurate and rapid solution of the algorithm’s global optimization. Finally, this article made typical benchmark functions experiment and vehicle scheduling simulation to verify the algorithm performance. The experimental results of typical benchmark functions proved that the search accuracy and performance of the MPSO algorithm are superior to other algorithms: the basic PSO, the improved particle swarm optimization (IPSO), and the chaotic PSO (CPSO). Besides, the MPSO algorithm can improve an ore company’s profit by 48.5–71.8% compared with the basic PSO in the vehicle scheduling simulation.