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：单基地多旅行商问题（Multiple Traveling Salesman Problem, MTSP）是经典旅行商问题（Traveling Salesman Problem, TSP）的推广，引入了额外参数即旅行商数量。在MTSP中，基地处的若干旅行商需访问一组相互关联的节点，每个节点仅由一名旅行商访问一次，同时最小化所有路径的总长度。MTSP的一个同等重要的变体——最小化最大距离MTSP（min-max MTSP）——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），旨在均衡分配各旅行商的工作负载（个体路径长度）。min-max MTSP出现在实际应用中，以确保旅行商的工作负载良好平衡。文献表明，min-max MTSP因其线性松弛提供的下界较差，在求解最优性方面极为困难。本文提出了两种称为“公平MTSP”（fair-MTSP）的新型参数化变体：一种构造为混合整数二阶锥规划（Mixed-Integer Second Order Cone Program, MISOCP），另一种构造为混合整数线性规划（Mixed Integer Linear Program, MILP）。两者均着力于强制旅行商的工作负载实现公平，即在最小化总路径成本的同时，使各旅行商的路径长度分布公平。我们提出了两种公平MTSP变体的全局最优求解算法，并在基准测试和实际测试实例上给出了计算结果，论证了公平MTSP作为min-max MTSP可行替代方案的有效性。