A column-and-row generation approach for the flying sidekick travelling salesman problem

Highlights

• A new graph representation for the Flying Sidekick TSP is proposed.

• An original Integer Programming formulation for this problem is provided.

• A branch-and-cut algorithm coupled with a column generation procedure is developed.

• Optimal solution of literature instances with 20 customers.

Abstract

The flying sidekick traveling salesman problem (FS-TSP) is a parcel delivery problem arising in the last-mile logistics, where the distribution plan of a driver-operated truck assisted by a drone (unmanned aerial vehicle, UAV) has to be defined. The FS-TSP is a variant of the TSP where routing decisions are integrated with customer-to-drone and customer-to-truck assignment decisions and truck-and-drone synchronization constraints. The objective is the minimization of the time required to serve all the customers, taking into account drone payload capacity and battery power constraints. In this work we provide a new representation of the FS-TSP based on the definition of an extended graph. This representation allows to model the problem by a new and compact integer linear programming formulation, where the synchronization issue is tackled in a column generation fashion, thus avoiding the usage of big-M constraints, representing one of the main drawbacks of the models present in literature. The proposed formulation has been solved by an exact approach which combines a Branch-and-Cut algorithm and a column generation procedure, strengthened by variable fixing strategies and new valid inequalities specifically defined for the problem. The proposed method has been experienced on a large set of benchmark instances. Computational results show that the proposed approach either is competitive or outperforms the best exact approach present in literature for the FS-TSP. Indeed, it is able to provide the optimal solution for all small size instances with 10 customers and for several medium size instances with 20 customers, some of them never solved before.

Introduction

In the last years, the attention of the scientific research community towards the usage of drones has continuously increased (Otto et al., 2018). The motivations behind this growing research interest can be mainly found in the continuous developments of drone technologies and in the subsequent exploration of new drone applications, driven by large corporations such as Amazon, Google, UPS, and DHL (Bezos, 2013, Cary and Bose, 2016, Reuters staff, 2018, Reuters staff, 2015, UPS staff, 2017). Among the others, great attention has been paid to the usage of drones in surveillance and security, disaster management and transportation fields (Shakhatreh et al., 2019). This has led to the definition of new and complex decision and optimization problems for which operations research methodologies represent a valuable support tool.

In this context, several studies have been made on the usage of drones in the last-mile logistic, showing the benefits in terms of emissions and completion time reduction that can be achieved in parcel delivery operations (Kille et al., 2019, Rojas Viloria et al., 2020).

The most studied drone assisted logistic system consists of a single truck and a single drone (or unmanned aerial vehicle, UAV) operating in tandem for the parcel delivery to the customers. In a nutshell, the system performs as follows: the truck and the drone must depart and return to a single depot, either in tandem or independently; each customer must be served once either by the truck or by the drone; the truck has an infinite capacity and acts as a mobile depot for the drone, replenishing its batteries and providing the parcels to be delivered. A more formal and detailed description of the system features, assumptions and operations will be provided in Section 3.

To the best of authors’ knowledge, the first work dealing with this hybrid working unit (truck-and-drone) is the one presented in Murray and Chu (2015), where it is referred to as Flying Sidekick Traveling Salesman Problem (FS-TSP). The FS-TSP is a variant of the TSP where routing decisions are integrated with customer-to-drone and customer-to-truck assignment decisions and truck-and-drone synchronization constraints. The objective is the minimization of the time required to serve all the customers (completion time), taking into account drone payload capacity and battery power constraints.

Several contributions have appeared in the last years providing integer and mixed-integer linear programming (ILP and MILP) formulations for the FS-TSP and modifications of it, where different objective functions and/or operating conditions are considered. However, most of these formulations suffer from dimensional drawbacks which make their solution impracticable even on small instances. This is mainly due to the synchronization issue which is generally modeled by the usage of big-M constraints. Indeed, these constraints make the usage of off-the-shelf optimization software ineffective and they would require the development of ad-hoc exact row and/or column generation solution approaches.

In this context, as will be discussed in the review provided in Section 2, even if the majority of the literature contributions present formulations for the specific truck-and-drone problem under investigation, then they mainly focus on heuristic solving approaches able to find good sub-optimal solutions with an acceptable computational burden.

The aims of this work is filling this gap by proposing an exact Branch-and-Cut algorithm coupled with a column generation procedure for the FS-TSP presented in Murray and Chu (2015).

Our method exploits a new representation of the FS-TSP based on the definition of an extended graph. This representation allows to model the problem by a new and compact ILP formulation, where the synchronization issue is tackled in a column generation fashion, thus avoiding the usage of big-M constraints. Moreover, the proposed method is strengthened by the usage of variable fixing strategies and new valid inequalities specifically defined for the problem.

To the best of our knowledge, the state of the art of exact approaches for the FS-TSP is represented by the work of Dell’Amico et al. (2019b). Our Branch-and-Cut algorithm has been experienced on the same set of benchmark instances used in Dell’Amico et al. (2019b) and the obtained computational results show that our approach either is competitive or outperforms it. Indeed, it is able to provide the optimal solution for all small size instances with 10 customers and several medium size instances with 20 customers, some of them never solved before. We underline that the maximum size of the tackled instances is coherent with the potential usage of the described truck-and-drone system. Indeed, such system can easily find real applications in low density inhabited (rural) areas due to the present flight regulation. In these areas the time required to travel distances represents the main part of the courier shift, so reducing the number of customers that can be served per day. In Karcz and Slusarczyk (2016) a real example of parcel delivery in rural areas is described and the same order of magnitude of customers is considered.

The paper is organized as follows: in Section 2, we discuss upon the most relevant studies focusing on exact solution methods for delivery problems with drones; in Section 3, we provide a detailed description of the FS-TSP and we present its extended graph representation; Section 4 is devoted to the presentation of the proposed ILP formulation; Section 5 introduces several variable fixing strategies and families of valid inequalities to strengthen the proposed model; Section 6 describes our exact approach to solve the problem; Section 7 proves the effectiveness of the proposed method showing the obtained computational results; finally, conclusions are given and future perspectives are discussed in Section 8.