Particle swarm optimization (PSO) algorithms have been widely used to solve a variety of optimization problems. Their success has motivated researchers to extend the use of these techniques to the multi-objective optimization field. However, most of these extensions have been used to solve multi-objective optimization problems (MOPs) with no more than three objective functions. Here, we propose a novel multi-objective PSO (MOPSO) algorithm characterized by the use of a recent approach that transforms a MOP into a linear assignment problem (LAP), with the aim of being able to solve many-objective optimization problems. Our proposed approach, called LAP based PSO (LAPSO), adopts the Munkres assignment algorithm to solve the generated LAPs and has no need of an external archive. LAPSO is compared with respect to three MOPSOs which are representative of the state-of-the-art in the area: the Optimized Multi-Objective Particle Swarm Optimizer (OMOPSO) the Speed-constrained Multiobjective Particle Swarm Optimizer (SMPSO) and a variant of the latter that uses the hypervolume indicator for its leader selection scheme (SMPSOhv). Our results indicate that LAPSO is able to outperform the MOPSOs with respect to which it was compared in most of the test problems adopted, specially when solving instances with more than three objectives.