Route optimization for open-close multiple travelling salesman problem with load-balancing constraint: A multi-chromosome based genetic algorithm

The Multiple Travelling Salesman Problem (MTSP) is one of the prominent combinatorial optimization problems with both theoretical interest and practical applications. However, its less-explored variants, such as the Open-Close Multiple Travelling Salesman Problem (OCMTSP), have received comparatively limited attention. In the OCMTSP, all salesmen commence their routes from a central depot, but unlike the classical MTSP, not all are required to return to the starting point upon completing their deliveries. Additionally, allowing any salesman to visit the maximum number of cities can lead to an imbalanced workload distribution among the salesmen. To address this imbalance, the current study incorporates a load balancing constraint into the OCMTSP framework, ensuring a fair distribution of cities among all salesmen. This extended problem variant is termed as Open-Close Multiple Travelling Salesman Problem with Load Balancing (OCMTSPLB). The primary objective of the OCMTSPLB is to minimize the total travel distance or cost incurred by the combined open and closed tours while maintaining balanced workloads. To solve this variant, the study proposes two distinct crossover based multi-chromosome Genetic Algorithm (GA) frameworks. Given the novelty of this problem, the algorithms are assessed using standardized benchmark instances from the TSPLIB. Experimental findings indicate that one of the proposed GA variants consistently achieves superior solution quality, a result further validated through non-parametric statistical tests.