Heuristic for Min-Max Heterogeneous Multi-Vehicle Multi-Depot Traveling Salesman Problem

TL;DR

This work addresses the min-max heterogeneous multi-vehicle multi-depot TSP with functional and structural heterogeneity by introducing a three-stage heuristic that extends the MD algorithm. The method combines speed-proportional load balancing (initialization), a savings-based local search to reduce the maximal tour, and a depot-perturbation step to escape local minima, using the LKH solver for routing. Empirical results on three vehicles and thirty targets show the heuristic achieves about $4\%$ average deviation from the MILP optimum while offering substantial runtimes advantages, and its performance remains robust across scenarios with distinct depots and shared-depot configurations. The approach holds practical value for surveillance and other logistics problems involving heterogeneous vehicle fleets and multiple starting points.

Abstract

In this article, a heuristic is proposed for a min-max heterogeneous multi-vehicle multi-depot traveling salesman problem (TSP), wherein heterogeneous vehicles start from given depot positions and need to cover a given set of targets. The vehicles should cover given targets such that the maximum tour time is minimized. In the considered problem, vehicles considered can be functionally heterogeneous, wherein specific targets must be covered by a particular vehicle, structurally heterogeneous, wherein the vehicles can travel at different speeds, or both. The proposed heuristic generalizes the MD heuristic for the min-max homogeneous multi-vehicle multi-depot TSP and has three stages: an initialization stage to generate a feasible solution, a local search stage in which the vehicle with the maximum tour time is improved, and a perturbation stage to break from a local minimum. The proposed heuristic is benchmarked with the optimal solution obtained by solving a mixed integer linear program using branch and cut for instances considering three vehicles covering thirty targets. Variations in the percentage of vehicle-target assignments and the number of vehicles starting at the same depot are studied to show the heuristic's effectiveness in producing high-quality solutions. It was observed that the heuristic generated feasible solutions within 4% of the optimum on average for the considered instances.

Figures (4)

• Figure 1: Depiction of perturbation for the $j\textsuperscript{th}$ vehicle
• Figure 2: Scenario 1: Vehicles $1, 2,$ and $3$ have speeds $1, 1.5, 2$, respectively, start at different depots covering thirty targets, twenty instances
• Figure 3: Tours of vehicles for instance 9 corresponding to $v_1 = 1, v_2 = 1.5, v_3 = 2,$$20\%$ target allocation, and vehicles start at different depots
• Figure 4: Scenario 2: Vehicles $1$ and $2$ have unit speeds and start at same depot, vehicle $3$ has $2$ units speed, thirty targets, twenty instances