[Paper Review] Heuristic methods for the Traveling Salesman Problem with Drone.

Quang Minh Ha, Yves Deville+2 more|arXiv (Cornell University)|Sep 29, 2015

Vehicle Routing Optimization Methods26 citations

TL;DR

This paper proposes a new variant of the TSP with Drone (TSP-D) that minimizes total operational costs, including transportation and waiting time penalties. It introduces a GRASP-based algorithm with a novel split procedure and local search, outperforming the TSP-LS method in solution quality within acceptable runtime on diverse problem instances.

ABSTRACT

Over the past few years, unmanned aerial vehicles (UAV), also known as drones, have been adopted as part of a new logistic method in the commercial sector called last-mile delivery. In this novel approach, they are deployed alongside trucks to deliver goods to customers to improve the quality of service and reduce the transportation cost. This approach gives rise to a new variant of the traveling salesman problem (TSP), called TSP with drone (TSP-D). A variant of this problem that aims to minimize the time at which truck and drone finish the service (or, in other words, to maximize the quality of service) was studied in the work of Murray and Chu (2015). In contrast, this paper considers a new variant of TSP-D in which the objective is to minimize operational costs including total transportation cost and one created by waste time a vehicle has to wait for the other. The problem is first formulated mathematically. Then, two algorithms are proposed for the solution. The first algorithm (TSP-LS) was adapted from the approach proposed by Murray and Chu (2015), in which an optimal TSP solution is converted to a feasible TSP-D solution by local searches. The second algorithm, a Greedy Randomized Adaptive Search Procedure (GRASP), is based on a new split procedure that optimally splits any TSP tour into a TSP-D solution. After a TSP-D solution has been generated, it is then improved through local search operators. Numerical results obtained on various instances of both objective functions with different sizes and characteristics are presented. The results show that GRASP outperforms TSP-LS in terms of solution quality under an acceptable running time.

Motivation & Objective

• To address the growing need for cost-efficient last-mile delivery using drones and trucks in logistics.
• To formulate a new TSP-D variant that minimizes total operational costs, including transportation and waiting time penalties.
• To develop efficient heuristic algorithms for solving large-scale TSP-D instances under the new objective function.
• To compare the performance of the proposed GRASP algorithm against the TSP-LS method in terms of solution quality and computational time.

Proposed method

• A mathematical formulation is developed to model the TSP-D variant with the objective of minimizing total operational costs, including waiting time penalties.
• The TSP-LS algorithm is adapted from Murray and Chu (2015), transforming an optimal TSP solution into a feasible TSP-D solution via local search heuristics.
• A novel split procedure is introduced to optimally decompose any TSP tour into a TSP-D solution, enabling efficient construction of initial solutions in the GRASP framework.
• The GRASP algorithm uses a greedy randomized construction phase followed by local search improvement to refine solutions.
• The split procedure ensures that each TSP tour is partitioned into truck and drone routes such that all customers are served and the objective function is minimized.
• Local search operators are applied to both algorithms to further improve solution quality after initial construction.

Experimental results

Research questions

• RQ1 How does the proposed GRASP-based algorithm compare to TSP-LS in terms of solution quality for the new TSP-D cost-minimization objective?
• RQ2 Can the proposed split procedure effectively and optimally convert a TSP solution into a feasible TSP-D solution under the new cost function?
• RQ3 What is the trade-off between solution quality and computational time when using GRASP versus TSP-LS on various TSP-D instances?
• RQ4 How do different instance sizes and characteristics affect the performance of the proposed algorithms?
• RQ5 To what extent does minimizing waiting time improve overall operational cost in drone-assisted delivery systems?

Key findings

• The GRASP-based algorithm consistently outperforms the TSP-LS method in terms of solution quality across all tested instances.
• The GRASP algorithm achieves better objective values while maintaining acceptable computational runtime, demonstrating its efficiency.
• The novel split procedure enables effective and optimal decomposition of TSP tours into TSP-D solutions, forming a strong foundation for the GRASP construction phase.
• Numerical results show that the GRASP approach is robust across diverse problem instances with varying sizes and characteristics.
• The inclusion of waiting time penalties in the objective function leads to more balanced and cost-effective drone-truck coordination strategies.
• The local search refinement step significantly improves solution quality in both algorithms, but the gains are more substantial in GRASP due to better initial solution quality.