The Multiple Traveling Salesman Problem (MTSP) with a single depot is a generalization of the well-known Traveling Salesman Problem (TSP) that involves an additional parameter, namely, the number of salesmen. In the MTSP, several salesmen at the depot need to visit a set of interconnected targets, such that each target is visited precisely once by at most one salesman while minimizing the total length of their tours. An equally important variant of the MTSP, the min-max MTSP, aims to distribute the workload (length of the individual tours) among salesmen by requiring the longest tour of all the salesmen to be as short as possible, i.e., minimizing the maximum tour length among all salesmen. The min-max MTSP appears in real-life applications to ensure a good balance of workloads for the salesmen. It is known in the literature that the min-max MTSP is notoriously difficult to solve to optimality due to the poor lower bounds its linear relaxations provide. In this paper, we formulate two novel parametric variants of the MTSP called the "fair-MTSP". One variant is formulated as a Mixed-Integer Second Order Cone Program (MISOCP), and the other as a Mixed Integer Linear Program (MILP). Both focus on enforcing the workloads for the salesmen to be equitable, i.e., the distribution of tour lengths for the salesmen to be fair while minimizing the total cost of their tours. We present algorithms to solve the two variants of the fair-MTSP to global optimality and computational results on benchmark and real-world test instances that make a case for fair-MTSP as a viable alternative to the min-max MTSP.

翻译：多旅行商问题（MTSP）是经典旅行商问题（TSP）的推广，其引入了一个额外参数，即旅行商的数量。在MTSP中，位于仓库的多个旅行商需要访问一组相互连接的目标点，每个目标点最多被一名旅行商恰好访问一次，目标是最小化所有路径的总长度。MTSP的一个重要变体——最小化最大路径MTSP，旨在通过要求所有旅行商中最长路径尽可能短（即最小化所有旅行商中的最大路径长度）来均衡各旅行商的工作负载（即个体路径长度）。实际应用中，最小化最大路径MTSP用于确保旅行商工作负载的良好平衡。文献研究指出，由于其线性松弛提供的下界较弱，最小化最大路径MTSP在求解最优解时极为困难。本文提出了两种新型参数化MTSP变体，称为"公平-MTSP"。一种变体建模为混合整数二阶锥规划（MISOCP），另一种建模为混合整数线性规划（MILP）。两种模型均侧重于强制旅行商工作负载的公平性，即在最小化总路径成本的同时，实现旅行商路径长度的公平分配。我们提出了求解两种公平-MTSP变体至全局最优的算法，并在基准测试实例和真实世界测试实例上的计算结果证明了公平-MTSP作为最小化最大路径MTSP可行替代方案的有效性。