Obtaining high convergence and uniform distributions remains a major challenge in most metaheuristic multi-objective optimization problems. In this article, a novel multi-objective particle swarm optimization (PSO) algorithm is proposed based on Gaussian mutation and an improved learning strategy. The approach adopts a Gaussian mutation strategy to improve the uniformity of external archives and current populations. To improve the global optimal solution, different learning strategies are proposed for non-dominated and dominated solutions. An indicator is presented to measure the distribution width of the non-dominated solution set, which is produced by various algorithms. Experiments were performed using eight benchmark test functions. The results illustrate that the multi-objective improved PSO algorithm (MOIPSO) yields better convergence and distributions than the other two algorithms, and the distance width indicator is reasonable and effective.

In this paper, we propose a vehicle routing problem with time windows (TWVRP). In this problem, we consider a hard time constraint that the fleet can only serve customers within a specific time window. To solve this problem, a hybrid ant colony (HACO) algorithm is proposed based on ant colony algorithm and mutation operation. The HACO algorithm proposed has three innovations: the first is to update pheromones with a new method; the second is the introduction of adaptive parameters; and the third is to add the mutation operation. A famous Solomon instance is used to evaluate the performance of the proposed algorithm. Experimental results show that HACO algorithm is effective against solving the problem of vehicle routing with time windows. Besides, the proposed algorithm also has practical implications for vehicle routing problem and the results show that it is applicable and effective in practical problems.