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This paper introduces a novel multi-objective algorithm (HMPSO) based on discrete particle swarm optimization (PSO) to solve vehicle routing problems with time windows (VRPTW). The presented HMPSO algorithm was combined with an advanced discrete PSO based on set and variable neighbourhood searches to find Pareto optimal routing solutions. These consisted of a complete routing schedule for serving the customers to minimize the two aims of travelling distance and number of vehicles. To increase the discrete PSO efficiency, a novel decoding scheme based on set was designed, and the variable neighbourhood local search was employed to explore new solutions. The experiment results were showed for a set of the Solomon's 56 VRPTW. The HMPSO algorithm was compared with some algorithms published in papers with the computational evaluations clearly supporting the high performance of the proposed HMPSO algorithm against other algorithms, and confirming that the HMPSO is an efficient algorithm because of a reasonable computational time and cost in solve VRPTW.

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