The multiple traveling salesman problem in presence of drone- and robot-supported packet stations

Highlights

• We introduce a new drone- and robot-assisted routing problem.

• The problem is formulated as a mixed-integer linear program.

• We introduce three matheuristic algorithms for solving the problem.

• The results of our extensive computational experiments are presented.

• We investigate the effects of using drones and robots on energy consumption.

Abstract

In this paper, we introduce the multiple Traveling Salesman Problem with Drone Stations (mTSP-DS), which is an extension to the classical multiple Traveling Salesman Problem (mTSP). In the mTSP-DS, we have a depot, a set of trucks, and some packet stations that host a given number of autonomous vehicles (drones or robots). The trucks start their mission from the depot and can supply some packet stations, which can then launch and operate drones/robots to serve customers. The goal is to serve all customers either by truck or by drones/robots while minimizing the makespan. We formulate the mTSP-DS as a mixed integer linear programming (MILP) model to solve small instances. To address larger instances, we first introduce two variants of a decomposition-based matheuristic. Afterwards, we suggest a third approach that is based on populating a solution pool with several restarts of an iterated local search metaheuristic, which is followed by determining the best combination of tours using a set-partitioning model. To verify the performance of our algorithms, we conducted extensive computational experiments. According to the numerical results, we observe that the use of drone stations leads to considerable savings in delivery time compared to traditional mTSP solutions. Furthermore, we investigated the energy consumption of trucks and drones. Indeed, depending on the energy consumption coefficients of trucks and drones as well as on the distance covered by drones, the mTSP-DS can also achieve energy savings in comparison to mTSP solutions.

Introduction

Drones, also known as unmanned aerial vehicles, have the potential to transform many industries and thus may also have a social impact (Floreano & Wood, 2015). Indeed, infrastructure, agriculture, transport, and security sector offer the best financial opportunities for the use of drones (see Otto, Agatz, Campbell, Golden, & Pesch, 2018 and references therein). Within the transport area, drones could change the way that packages are delivered to the customers. In this context, a loaded drone, carrying a light and small- or medium-sized parcel, could fly to a customer, deliver the package, and then return to its starting point.

Due to technical limitations, drones, unlike trucks, can usually only transport one parcel at a time (Agatz, Bouman, & Schmidt, 2018). As a result, the total distance covered for deliveries by drones will be greater than for traditional deliveries by truck. Nevertheless, it is possible to reduce delivery costs by using drones, e.g., since the delivery drones operate autonomously, using drones can reduce costs by decreasing labor costs. Considering that acquisition costs for drones are also relatively low in comparison to trucks, drones allow for higher parallel operations, which in turn can lead to reduced delivery times. Another factor that has a positive effect on delivery times is that drones are not tied to a road network. Indeed, this enables drones to avoid congestion, especially in densely populated urban areas. Finally, drones have also benefits in terms of CO${}_2$ emissions. According to Goodchild & Toy (2018), drones emit less CO${}_2$ than trucks when customers are close to the depot or when only a few customers need to be supplied.

Although many technological advancements have been made in recent years, drone usage is still accompanied by shortcomings that need to be considered. For example, the size and the weight of parcels that can be transported by drones are limited. An alternative to drones, which circumvents some of these problems, are delivery robots such as those made known by, e.g., Starship Technologies (Figliozzi, 2020). In fact, robots move either on roads or sidewalks, reaching speeds of up to 30 km/h and 6 km/h, respectively, and are thus slower than trucks and drones (Moeini, Salewski, 2020, Schermer, Moeini, Wendt, 2020). However, robots offer some advantages, e.g., the use of sidewalks may reveal shortcuts that trucks cannot use. Moreover, compared to drones, robots can transport much heavier packages and, some types of robots can carry several packages at once. Furthermore, robots are more energy-efficient than drones and trucks. One problem of drones and robots alike is their relatively short range. However, this problem can be overcome, for example, by the use of drone stations.

A drone station is a facility that can be used to launch and operate drones (Kim & Moon, 2018). When a truck delivers parcels to a drone station, they are automatically processed and loaded into a drone. Furthermore, the batteries of returned drones could be replaced and charged in such stations. In practice, drone stations have already been tested by DHL (DHL, 2018). Even though the concept of drone stations can also be transferred to robots, for the sake of easing the readability of the paper, we will use the term drone station throughout this paper.

The aforementioned technical problems also have implications on the way that drones are approached from an operational perspective that we address in this paper. More precisely, the contributions of this paper are as follows:

• We introduce the multiple Traveling Salesman Problem with Drone Stations (mTSP-DS), a tour planning model that integrates drones and drone stations into the classical multiple Traveling Salesman Problem (mTSP). In addition, the mTSP-DS also incorporates elements from parallel machine scheduling (see, e.g., Cheng & Sin, 1990) and facility location problems (Cornuéjols, Nemhauser, & Wolsey, 1990). In the next sections, we define the mTSP-DS and formulate it as a mixed integer linear program (MILP).

• The MILP model of the mTSP-DS can be solved by any standard MILP solver, e.g., Gurobi Optimizer. But, this is possible only on small instances because the runtime grows too fast due to the complexity of the mTSP-DS. To overcome this issue, we first introduce two variants of a decomposition-based matheuristic. Afterwards, we present a two-phase matheuristic that is based on populating a solution pool and using a set-partitioning model. Recently, a similar approach has proven successful for truck and trailer routing problems(Accorsi & Vigo, 2020), which share some similarities with the mTSP-DS.

• We report the results of our extensive computational experiments, which prove the efficiency of the introduced algorithms as well as the usefulness of drones and drone stations in last-mile delivery. According to the numerical results, we observe not only that the algorithms are able to provide high-quality solutions, but also we show that the use of drone stations and drones leads to considerable savings in delivery time compared to traditional mTSP solutions. Furthermore, we show that depending on the energy consumption coefficients of trucks and drones, as well as on the distance covered by drones, the mTSP-DS can also achieve energy savings in comparison to mTSP solutions.

The remainder of this paper is structured as follows. Section 2 is devoted to an overview of the existing literature regarding the use of drones in last-mile delivery from a tour planning perspective. In Section 3, we provide a precise and formal description of the mTSP-DS as well as its MILP formulation. Afterwards, we describe our algorithms in Section 4. We report the computational experiments and their numerical results in Section 5. Finally, some concluding remarks are drawn in Section 6.