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The traveling salesman problem with drone resupply

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Abstract

This paper treats a variant of the famous Traveling Salesman Problem (TSP), which is extended to cover the peculiarities of a novel, drone-based distribution concept in last-mile logistics. In this context, the salesman represents the driver of a home delivery truck. Given a set of customers to be visited, the truck has a limited capacity, so that only a subset of shipments can be loaded on board when leaving the depot. The remainder of the shipments has to be resupplied to the truck on its tour by a single unmanned aerial vehicle (drone). In order to do so, the drone picks up shipments (one by one) at the depot and delivers them toward the truck. Our extension of the TSP aims at a tour of the truck through the set of given customers and a resupply schedule for the drone, so that the total delivery costs are minimized. We develop suited optimization approaches and apply them in static and dynamic problem settings. We, furthermore, benchmark the savings enabled by drone resupply with alternative home delivery options with and without drone support. Our results show that drone resupply promises substantial rewards when applied in the right delivery context.

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Correspondence to Michael Dienstknecht.

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Appendix: genetic algorithms to obtain truck tours

Appendix: genetic algorithms to obtain truck tours

With an efficient DP approach for DR on hand (see Sect. 4.1), we propose a GA that employs our DP in order to determine high-quality solutions for our TSP-DR. The basic idea is to have the GA determining suitable truck tours, which are evaluated in terms of total costs after complementing them with an optimal drone schedule found by our DP. To do so, we take up components of the GA developed by Snyder and Daskin (2006), which is still seen as the state-of-the-art when it comes to solving the TSP with a metaheuristic. We propose two variants with identical fitness functions, which only differ in the way a local search procedure is employed. As can be seen in our computational tests, these variants each bear their virtues and drawbacks. We focus on the common scheme first and highlight the differences afterward.

A truck tour is represented by a sequence of n numbers in [0, 1[. The represented tour visits customers in increasing order of their values in the sequence, where the i-th entry in the sequence is associated with customer i. In the terminology of our GAs, such a sequence p is an individual or a chromosome, and we will use all three terms interchangeably. Each sequence is evaluated by a fitness function, which will be detailed in a moment. The population size is 100 and is initialized with random sequences with a uniform distribution of numbers. When deriving the next generation from the current one, we take the 20 best sequences according to their fitness values and generate 10 new sequences randomly. The remaining 70 sequences are designed by a recombination of two randomly chosen parent sequences each. We adopt numbers one by one from one of the parent sequences choosing with 70% probability the one of the first parent sequence. The procedure terminates after at most 100 generations but is aborted prematurely, if there is no improvement achieved within ten consecutive generations. Each sequence is complemented by an optimal drone schedule found by employing our DP (Sect. 4.1). This allows us to derive an individual’s fitness value \(F \left( p \right)\) as its associated total costs according to equation (1).

Example (cont.). Consider the solution given in Fig. 2. The depicted truck tour could be encoded in our GA using the random key concept as, e. g., 0.12, 0.29, 0.34, 0.56, 0.71, 0.85. By complementing this sequence by an optimal drone schedule using our DP, we obtain a fitness value of \(31 \cdot \left( 2 + 1 \right) + 3 \cdot \left( 2 \cdot 4 + 2 \cdot 1 \right) = 123\).

Since it is well-known that local search routines applied to chromosomes can significantly contribute to the solution quality of GAs, e. g., see the computational study provided by Snyder and Daskin (2006), we implement a randomized version of 2opt. Randomization is incorporated by performing \((|V| \cdot \left( |V| - 1 \right) \cdot f)/2\) exchanges of two random arcs in a given truck tour and accepting the modified solution, if an improvement is achieved. Hence, the number of exchanges scales with the number of nodes and a factor f, which we set to \(f = 10\) based on preliminary tests. Although we have shown that our DP returns the optimal drone schedule for a given truck tour in polynomial time, it still consumes considerable time, especially if excessively executed in multiple runs of 2opt. Hence, we distinguish two variants of applying 2opt: either 2opt is applied on each altered chromosome of each generation or it is exclusively applied to the best sequence found at the very end of the evolutionary process. We will refer to the former version as GA-all and to the latter version as GA-best. Finally, we return the best solution found during the complete evolutionary process.

Algorithm 1 summarizes both versions of our GA, i. e., GA-best and GA-all. We control the use of 2opt in the evaluation function (Algorithm 2) by the Boolean indicator all. As can be seen in lines 9 to 19, we follow the lead of Snyder and Daskin (2006) in that we use only two operators for generating new individuals, namely the random generation of an individual from scratch and a standard crossover operator (Algorithm 3).

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Dienstknecht, M., Boysen, N. & Briskorn, D. The traveling salesman problem with drone resupply. OR Spectrum 44, 1045–1086 (2022). https://doi.org/10.1007/s00291-022-00680-1

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