The Coloured Travelling Salesman Problem (CTSP), a generalised version of the Multiple Travelling Salesman Problem (MTSP), has been proposed to model some real-world applications. This work proposes a discrete ITÔ (DITÔ) algorithm to solve CTSP. It combines the continuous ITÔ stochastic process with Ant Colony Optimisation (ACO) algorithm. First, the stochastic drift and volatility terms of ITÔ process are designed to be suitable for solving combinatorial optimisation problems, such as TSP and CTSP. And then, inspired by ACO, the generative model for a feasible solution of CTSP is constructed via the distance between cities and the experience gathered in the searching process. The experiments results show that the performance of DITÔ is superior to some state-of-the-art meta-heuristic algorithms in terms of the quality of the solution and computational time.

The convergence speed and the diversity of the population plays a critical role in the performance of particle swarm optimization (PSO). In order to balance the trade-off between exploration and exploitation, a novel particle swarm optimization based on the hybrid learning model (PSO-HLM) is proposed. In the early iteration stage, PSO-HLM updates the velocity of the particle based on the hybrid learning model, which can improve the convergence speed. At the end of the iteration, PSO-HLM employs a multi-pools fusion strategy to mutate the newly generated particles, which can expand the population diversity, thus avoid PSO-HLM falling into a local optima. In order to understand the strengths and weaknesses of PSO-HLM, several experiments are carried out on 30 benchmark functions. Experimental results show that the performance of PSO-HLM is better than other the-state-of-the-art algorithms.

The vehicle routing problem with time windows (VRPTW) is a well-known NP-Hard combinatorial optimization problem, which is frequently encountered in transportation and logistics scenarios. When traditional algorithms solve this problem, there are some problems, such as slow convergence speed. In this paper, a Self-competitive Particle Swarm Optimization (ScPSO) algorithm is proposed, which regulates the learning direction of the next generation of particles based on their degree of self-competition. When the particle’s self-competition degree is low, it is compelled to learn from the individual optimal solution using the self-competition selection probability, thereby achieving rapid convergence. ScPSO uses the nonlinear inertia weight adaptive strategy to adjust the search preference in the search process, and it can balance the relationship between exploration and exploitation. Meanwhile, Random greedy heuristic selection and variable neighborhood search strategies are used in initial solution construction and constraint restriction. Finally, ScPSO is tested on 56 Solomon 100-customers benchmark problems and compared with four state-of-the-art VRPTW algorithms and best-known solutions reported on Solomon’s webpage.The running results of the ScPSO algorithm are better than four state-of-the-art VRPTW algorithms and best-known solutions reported on Solomon’s webpage. The experimental results show that ScPSO can solve VRPTW problems efficiently.