The Multiple Traveling Salesman Problem (MTSP) extends the traveling salesman problem by assigning multiple salesmen to visit a set of targets from a common depot, with each target visited exactly once while minimizing total tour length. A common variant, the min-max MTSP, focuses on workload balance by minimizing the longest tour, but it is difficult to solve optimally due to weak linear relaxation bounds. This paper introduces two new parametric fairness-driven variants of the MTSP: the $\varepsilon$-Fair-MTSP and the $Δ$-Fair-MTSP, which promote equitable distribution of tour lengths while controlling overall cost. The $\varepsilon$-Fair-MTSP is formulated as a mixed-integer second-order cone program, while the $Δ$-Fair-MTSP is modeled as a mixed-integer linear program. We develop algorithms that guarantee global optimality for both formulations. Computational experiments on benchmark instances and real-world applications, including electric vehicle fleet routing, demonstrate their effectiveness. Furthermore, we show that the algorithms presented for the fairness-constrained MTSP variants can be used to obtain the pareto-front of a bi-objective optimization problem where one objective focuses on minimizing the total tour length and the other focuses on balancing the tour lengths of the individual tours. Overall, these fairness-constrained MTSP variants provide a practical and flexible alternative to the min-max MTSP.

翻译：多旅行商问题（MTSP）是旅行商问题的扩展，其将多个销售人员分配至从公共仓库出发访问一组目标点，每个目标点仅被访问一次，同时最小化总路径长度。一个常见变体——最小-最大MTSP通过最小化最长路径来关注工作量平衡，但由于线性松弛界较弱而难以精确求解。本文提出了两种新的参数化公平性驱动的MTSP变体：$\varepsilon$-公平-MTSP与$Δ$-公平-MTSP，它们在控制总体成本的同时促进路径长度的公平分配。$\varepsilon$-公平-MTSP被表述为混合整数二阶锥规划，而$Δ$-公平-MTSP被建模为混合整数线性规划。我们开发了保证两种模型全局最优性的算法。在基准算例和实际应用（包括电动汽车车队路径规划）上的计算实验验证了其有效性。此外，我们证明了所提出的公平约束MTSP变体算法可用于获得双目标优化问题的帕累托前沿，其中一个目标聚焦于最小化总路径长度，另一个目标则关注平衡各独立路径的长度。总体而言，这些公平约束的MTSP变体为最小-最大MTSP提供了一种实用且灵活的替代方案。