On the Min-cost Traveling Salesman Problem with Drone (1509.08764v3)

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.

• The paper introduces a cost-centric variant of the TSP that integrates drones to minimize overall operational expenses using MILP and heuristic models.
• It presents a robust GRASP algorithm with a novel split procedure that outperforms traditional heuristics in cost-saving across 100-customer instances.
• The study’s findings demonstrate practical benefits for last-mile delivery optimization and pave the way for further multi-modal logistics research.

Analysis of the Min-cost Traveling Salesman Problem with Drone

The paper "On the Min-cost Traveling Salesman Problem with Drone" introduces a novel variant of the Traveling Salesman Problem (TSP) which integrates the use of drones for logistics optimization in last-mile delivery. This research highlights the development and mathematical formulation of a problem known as the Traveling Salesman Problem with Drone (TSP-D), wherein the objective shifts from minimizing completion time, as seen in previous studies, to minimizing overall operational costs. These costs encompass both direct transportation expenses and the costs incurred from the waiting times of vehicles.

Methodology

The authors propose a detailed mathematical model for the new variant, the min-cost TSP-D, and provide two heuristic algorithms tailored for solution development: a Greedy Randomized Adaptive Search Procedure (GRASP) and a locally adaptive heuristic adapted from the work of Murray and Chu (2015), here referred to as TSP-LS. The formulation encompasses integer programming techniques and heuristic methods, with GRASP providing a robust framework with elements such as the split procedure and local search optimizations.

The core of this work lies in the formulation of the problem where the roles of both truck and drone are considered collaboratively, exploiting their strengths while mitigating their weaknesses. The authors propose a MILP model, offering a comprehensive mathematical representation of the min-cost TSP-D. The GRASP methodology developed utilizes a novel split algorithm to convert a feasible TSP tour into a TSP-D solution, enhancing both solution quality and computational efficiency.

Numerical Experiments

The paper features extensive empirical evaluations across various instances with up to 100 customers. These experiments compare TSP-D solutions with traditional TSP tours and assess the effectiveness of the proposed algorithms, particularly focusing on the cost-saving potential with drone integration. The results demonstrate that the GRASP algorithm outperforms the TSP-LS heuristic in terms of solution quality, albeit with higher computational demands. The GRASP strategy showed consistent performance, with the extent of cost savings being influenced by the operational parameters such as drone range and vehicle speed ratios.

Implications and Future Directions

The introduction of the min-cost TSP-D objective is a significant development in routing problems, reflecting practical considerations in logistics where operational costs are a priority. This paper opens up further exploration into variants incorporating more realistic constraints and multiple vehicles or drones. The results demonstrate potential applicability in commercial settings, such as courier services or small parcel deliveries, where the balance between time efficiency and cost-effectiveness is critical.

Future developments in this area could cite this research to springboard discussions into multi-modal logistics optimization and extended heuristic approaches that incorporate machine learning for predictive routing. The split methodology and local search operators proposed here serve as potential templates for enhancing other combinatorial optimization problems.

In conclusion, this paper provides a comprehensive examination of the TSP-D from a cost-centric perspective, deploying advanced heuristic methods that demonstrate viable improvements over conventional solutions. The research underscores the growing importance of drone logistics and sets a foundation for continuous evolution in the field of operations research and transportation logistics.