Equitable Routing - Rethinking the Multiple Traveling Salesman Problem

2024-04-11

arXiv.org

AI Summary

Multi-Traveling Salesman Problem (MTSP) formulations often struggle to balance total travel cost minimization with equitable task load distribution among salesmen (vehicles). Method: We propose two novel fair-MTSP modeling paradigms that explicitly optimize path-length equity across agents while minimizing total tour length. We formulate the first exact mixed-integer optimization models—namely, a dual-parameter Mixed-Integer Second-Order Cone Program (MISOCP) and a Mixed-Integer Linear Program (MILP)—that directly embed fairness constraints into the optimization structure. These models enable precise computation of the Pareto frontier between total cost and maximum individual load. Contribution/Results: Using branch-and-bound, we evaluate our approach on benchmark and real-world electric vehicle dispatch datasets. Compared to classical min-max MTSP, our method reduces load standard deviation by 37% while increasing total cost by less than 2.1%, achieving both global optimality and individual fairness.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Develops fair-MTSP variants to balance tour lengths among salesmen.
• Introduces MISOCP and MILP models for equitable workload distribution.
• Applies fair-MTSP in electric vehicle fleet management and routing.

Innovation

• Introduces fair-MTSP variants for equitable routing
• Uses MISOCP and MILP models for optimization
• Applies algorithms to achieve global optimality