[Paper Review] An Ant Colony System for the Team Orienteering Problem with Time Windows

Roberto Montemanni, Luca Maria Gambardella|arXiv (Cornell University)|May 12, 2023

Product Development and Customization11 citations

TL;DR

The paper proposes an Ant Colony System (ACS) algorithm for the Team Orienteering Problem with Time Windows (TOPTW), using a hierarchic HTOPTW model and a giant-tour solution representation, and shows competitive results on OPTW and TOPTW benchmarks.

ABSTRACT

This paper discusses a heuristic approach for Team Orienteering Problems with Time Windows. The method we propose takes advantage of a solution model based on a hierarchic generalization of the original problem, which is combined with an Ant Colony System algorithm. Computational results on benchmark instances previously adopted in the literature suggest that the algorithm we propose is effective in practice.

Motivation & Objective

• Motivate and formalize the TOPTW and its time window variant (TOPTW) as practical, NP-hard routing problems.
• Introduce a Hierarchic Team Orienteering Problem with Time Windows (HTOPTW) to aid solution construction and local search.
• Develop an Ant Colony System-based heuristic to construct and improve solutions under HTOPTW, aiming to enhance TOPTW performance.

Proposed method

• Represent solutions as a giant tour with depot copies to model multiple paths.
• Propose a constructive ACS phase where ants build feasible HTOPTW tours using a transition rule combining pheromone trails and a time-window aware heuristic η_ij (Equation 10).
• Update pheromones using the best-so-far solution (Equation 11) and apply pheromone evaporation to promote diversity (Equation 12).
• Embed a hierarchical HTOPTW structure to precompute and optimize additional tours beyond the top-m to furnish high-quality fragments for local search.
• Apply a specialized CROSS-exchange local search on the giant tour to refine solutions, with adaptive subchain lengths and non-improvement handling.
• Parameter settings include ρ=0.1, ψ=0.1, s=10, τ0=(ProfitFirst·n)^{-1}, 10 ants, and a local search window strategy (NI=5, LS_init=3, LS_wnd=3, LS_step=2).

Experimental results

Research questions

• RQ1 Can an Ant Colony System effectively solve the TOPTW and its time-window variant using HTOPTW as an auxiliary hierarchical model?
• RQ2 Does the giant-tour representation facilitate efficient construction and effective local search for TOPTW?
• RQ3 What is the impact of hierarchic optimization of non-top-m tours on the quality of the TOPTW solution?
• RQ4 How does the ACS-based method compare with existing metaheuristics (e.g., GVNS) on standard OPTW/TOPTW benchmarks?
• RQ5 Are the proposed parameter settings robust across instance classes and time-window widths?

Key findings

• The ACS approach achieves best-known solutions on many Solomon-based OPTW instances and improves lower bounds on several instances.
• ACS generally outperforms the Granular Variable Neighborhood Search (GVNS) in solution quality and is often faster on easy instances, with some variance on harder instances.
• On Cordeau-based instances with wide time windows, the method is less effective, highlighting sensitivity to time-window width and the need for local-search tuning for such cases.
• The HTOPTW hierarchical modeling and the giant-tour representation facilitate effective construction and a broad local search, contributing to improved TOPTW solutions.
• The authors provide extensive computational results across OPTW benchmarks (Solomon and Cordeau-derived) and report new best solutions and lower bounds where applicable.