Equitable Routing -- Rethinking the Multiple Traveling Salesman Problem

TL;DR

This paper addresses equitable routing in the MTSP by introducing two parametric fairness formulations, $oldsymbol{ ext{F}}^{oldsymbol{\varepsilon}}$ (SOC-based) and $oldsymbol{ ext{F}}^{oldsymbol{\Delta}}$ (Gini-based), enabling a controllable fairness-cost trade-off. It develops exact solution methods, including a branch-and-cut framework with specialized separation for sub-tour constraints and outer-approximation schemes for the $p$-norm and SOC facets, yielding global optimality. The authors also connect these fair MTSP variants to bi-objective formulations, enabling Pareto-front computation between total tour length and fairness, and demonstrate practical gains over the min-max MTSP in both benchmark and real-world EV-routing contexts. The results indicate that the proposed fair formulations provide a continuum of fairness levels with favorable computational properties and meaningful practical impact for workload balancing in routing and fleet management. Overall, fair-MTSP variants offer a promising, scalable alternative to min-max MTSP for equitable workload distribution with quantifiable cost of fairness and accessible Pareto-front analysis.

Abstract

The Multiple Traveling Salesman Problem (MTSP) generalizes the Traveling Salesman Problem (TSP) by introducing multiple salesmen tasked with visiting a set of targets from a single depot, ensuring each target is visited exactly once while minimizing total tour length. A key variant, the min-max MTSP, seeks to balance workloads by minimizing the longest tour among salesmen. However, this problem is challenging to solve optimally due to weak lower bounds from linear relaxations. This paper introduces two novel parametric variants of the MTSP, termed "fair-MTSP". One variant is modeled as a Mixed-Integer Second Order Cone Program (MISOCP), and the other as a Mixed Integer Linear Program (MILP). Both variants aim to distribute tour lengths equitably among salesmen while minimizing overall costs. We develop algorithms to achieve global optimality for these fair-MTSP variants. We present computational results based on benchmark and real-world scenarios, particularly in electric vehicle fleet management and routing. Furthermore, we also show that the algorithmic approaches presented for the fair-MTSP variants can be directly 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. The findings support fair-MTSP as a promising alternative to the min-max MTSP, emphasizing fairness in workload distribution.

Figures (4)

• Figure 1: An illustration of the outer approximation procedure. The region shaded in red is the convex constraint, and the violated point is the solution obtained when solving the relaxed problem. This point can be projected onto the surface of the convex curve in many ways; two trivial projections (a) and (b) are shown. For each point, a tangent outer approximates the convex region (shaded in red).
• Figure 2: The Pareto fronts for the two bi-objective formulations, the total tour length for the "min-sum" and "min-max" MTSP for the eil51 instance with $m=5$ are shown.
• Figure 3: Cost of fairness of the optimal solutions of $\varepsilon$-F-MTSP and $\Delta$-F-MTSP as we vary $\varepsilon$ and $\Delta$ from 0 to 1 for the eil51 instance with $m=5$.
• Figure 4: (a) shows the graph of the "Seattle" instance with the red dot representing the depot and the blue dots representing the different targets. $4$ electric vehicles are initially stationed at the depot. The path for each vehicle obtained by solving the different variants of the MTSP is shown in different colors in (b)--(i). (b) and (c) show the optimal min-sum and min-max MTSP tours, respectively. The second and third rows show optimal solutions for the $\Delta$-F-MTSP and $\varepsilon$-F-MTSP, respectively, with different $\Delta$ and $\varepsilon$ values. The time taken to compute the optimal solution in all the illustrations is provided in parentheses.

Theorems & Definitions (13)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• Proposition 3
• proof
• Proposition 4
• Proposition 5