Equitable Routing--Rethinking the Multiple Traveling Salesman Problem

TL;DR

This paper introduces fairness-driven variants of the Multiple Traveling Salesman Problem, promoting equitable workload distribution while maintaining cost efficiency, with algorithms guaranteeing global optimality demonstrated through computational experiments.

Contribution

The paper proposes two new parametric fairness variants of the MTSP and develops algorithms that ensure global optimality for these formulations.

Findings

Algorithms effectively solve fairness-constrained MTSP variants.

Computational experiments validate the effectiveness on benchmark and real-world data.

Fairness variants can generate Pareto fronts balancing total cost and workload equity.

Abstract

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 $\Delta$-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 $\Delta$-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 in which one objective minimizes the total tour length and the other balances the lengths of the individual tours. Overall, these fairness-constrained MTSP variants provide a practical and flexible alternative to the min-max MTSP.