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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References
Agatz N, Bouman P, Schmidt M (2018) Optimization approaches for the traveling salesman problem with drone. Transp Sci 52:965–981. https://doi.org/10.1287/trsc.2017.0791
Applegate DL, Bixby RE, Chvatal V, Cook WJ (2006) The traveling salesman problem: a computational study. Princeton University Press, United States
Arslan A, Agatz N, Kroon LG, Zuidwijk RA (2019) Crowdsourced delivery: a dynamic pickup and delivery problem with ad-hoc drivers. Transp Sci 53:222–235. https://doi.org/10.1287/trsc.2017.0803
Baskaran A, Kashef A (1996) Investigation of air flow around buildings using computational fluid dynamics techniques. Eng Struct 18:861–875. https://doi.org/10.1016/0141-0296(95)00154-9
Beckman BC, Bjone N (2017) Ground-based mobile maintenance facilities for unmanned aerial vehicles. U. S. Pat 9(718):564
Berg PW, Isaacs S, Blodgett KL (2016) Airborne fulfillment center utilizing unmanned aerial vehicles for item delivery. U. S. Pat 9(305):280
Bouman P, Agatz N, Schmidt M (2018) Dynamic programming approaches for the traveling salesman problem with drone. Networks 72:528–542. https://doi.org/10.1002/net.21864
Boysen N, Schwerdfeger S, Weidinger F (2018) Scheduling last-mile deliveries with truck-based autonomous robots. Eur J Oper Res 271:1085–1099. https://doi.org/10.1016/j.ejor.2018.05.058
Boysen N, Briskorn D, Fedtke S, Schwerdfeger S (2018) Drone delivery from trucks: drone scheduling for given truck routes. Networks 72:1–22. https://doi.org/10.1002/net.21847
Boysen N, Fedtke S, Schwerdfeger S (2021) Last-mile delivery concepts: a survey from an operational research perspective. OR Spectr 43:1–58. https://doi.org/10.1007/s00291-020-00607-8
Bretzke W-R (2013) Global urbanization: a major challenge for logistics. Logist Res 6:57–62. https://doi.org/10.1007/s12159-013-0101-9
Carlsson JG, Song S (2018) Coordinated logistics with a truck and a drone. Manage Sci 64:4052–4069. https://doi.org/10.1287/mnsc.2017.2824
Coutinho WP, Battarra M, Fliege J (2018) The unmanned aerial vehicle routing and trajectory optimisation problem, a taxonomic review. Comput Ind Eng 120:116–128. https://doi.org/10.1016/j.cie.2018.04.037
Daimler (2018) Vans & Drones: On-demand delivery of e-commerce goods. https://media.daimler.com/marsMediaSite/en/instance/print/Vans--Drones-in-Zurich-Mercedes-Benz-Vans-Matternet-and-siroop-start-pilot-project-for-on-demand-delivery-of-e-commerce-goods.xhtml?oid=29659302 (last access: May 2021)
Dayarian I, Savelsbergh M, Clarke JP (2020) Same-day delivery with drone resupply. Transp Sci 54:229–249. https://doi.org/10.1287/trsc.2019.0944
Dell’Amico M, Montemanni R, Novellani S (2020) Matheuristic algorithms for the parallel drone scheduling traveling salesman problem. Ann Oper Res 289:211–226. https://doi.org/10.1007/s10479-020-03562-3
Eun J, Song BD, Lee S, Lim DE (2019) Mathematical investigation on the sustainability of UAV logistics. Sustainability 11:5932. https://doi.org/10.3390/su11215932
Forbes (2008) In depth: Europe’s most congested cities. https://www.forbes.com/2008/04/21/europe-commute-congestion-forbeslife-cx_po_0421congestion_slide.html#17a562e86500 (last access: May 2021)
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York
Kim S, Moon I (2018) Traveling salesman problem with a drone station. IEEE Trans Syst Man Cybern 49:42–52. https://doi.org/10.1109/TSMC.2018.2867496
Khoufi I, Laouiti A, Adjih C (2019) A survey of recent extended variants of the traveling salesman and vehicle routing problems for unmanned aerial vehicles. Drones 3:66. https://doi.org/10.3390/drones3030066
Liu Y (2019) An optimization-driven dynamic vehicle routing algorithm for on-demand meal delivery using drones. Comput Op Res 111:1–20. https://doi.org/10.1016/j.cor.2019.05.024
Manjoo F (2016) Think amazon’s drone delivery idea is a gimmick? Think again. https://www.nytimes.com/2016/08/11/technology/think-amazons-drone-delivery-idea-is-a-gimmick-think-again.html?_r=0 (last access: May 2021)
Masson R, Trentini A, Lehuédé F, Malhéné N, Péton O, Tlahig H (2017) Optimization of a city logistics transportation system with mixed passengers and goods. EURO J Transp Logist 6:81–109. https://doi.org/10.1007/s13676-015-0085-5
Murray CC, Chu AG (2015) The flying sidekick traveling salesman problem: optimization of drone-assisted parcel delivery. Transp Res Part C 54:86–109. https://doi.org/10.1016/j.trc.2015.03.005
Otto A, Agatz N, Campbell J, Golden B, Pesch E (2018) Optimization approaches for civil applications of unmanned aerial vehicles (UAVs) or aerial drones: A survey. Networks 72:411–458. https://doi.org/10.1002/net.21818
Pelletier S, Jabali O, Laporte G (2016) 50th anniversary invited article - goods distribution with electric vehicles: review and research perspectives. Transp Sci 50:3–22. https://doi.org/10.1287/trsc.2015.0646
Poikonen S, Wang X, Golden B (2017) The vehicle routing problem with drones: extended models and connections. Networks 70:34–43. https://doi.org/10.1002/net.21746
Poikonen S, Golden B, Wasil EA (2019) A branch-and-bound approach to the traveling salesman problem with a drone. INFORMS J Comput 31:335–346. https://doi.org/10.1287/ijoc.2018.0826
Poikonen S, Golden B (2020) The mothership and drone routing problem. INFORMS J Comput 32:249–262. https://doi.org/10.1287/ijoc.2018.0879
Renaud J, Boctor FF, Laporte G (1996) An improved petal heuristic for the vehicle routeing problem. J Op Res Soc 47:329–336. https://doi.org/10.1057/jors.1996.29
Reyes D, Savelsbergh MWP, Toriello A (2017) Vehicle routing with roaming delivery locations. Transp Res Part C 80:71–91. https://doi.org/10.1016/j.trc.2017.04.003
Rojas Viloria D, Solano-Charris EL, Muñoz-Villamizar A, Montoya-Torres JR (2021) Unmanned aerial vehicles/drones in vehicle routing problems: a literature review. Int Trans Oper Res 28:1626–1657. https://doi.org/10.1111/itor.12783
Ryan DM, Hjorring C, Glover F (1993) Extensions of the petal method for vehicle routeing. J Op Res Soc 44:289–296. https://doi.org/10.1057/jors.1993.54
Saleu RM, Deroussi L, Feillet D, Grangeon N, Quilliot A (2018) An iterative two-step heuristic for the parallel drone scheduling traveling salesman problem. Networks 72:459–474. https://doi.org/10.1002/net.21846
Savelsbergh MWP, Van Woensel T (2016) 50th anniversary invited article - city logistics: challenges and opportunities. Transp Sci 50:579–590. https://doi.org/10.1287/trsc.2016.0675
Sawadsitang S, Niyato D, Tan PS, Wang P (2019) Joint ground and aerial package delivery services: a stochastic optimization approach. IEEE Trans Intell Transp Syst 20:2241–2254. https://doi.org/10.1109/TITS.2018.2865893
Snyder LV, Daskin MS (2006) A random-key genetic algorithm for the generalized traveling salesman problem. Eur J Oper Res 174:38–53. https://doi.org/10.1016/j.ejor.2004.09.057
Song BD, Park K, Kim J (2018) Persistent UAV delivery logistics: MILP formulation and efficient heuristic. Comput Ind Eng 120:418–428. https://doi.org/10.1016/j.cie.2018.05.013
Statista (2021) Annual retail e-commerce sales growth worldwide from 2017 to 2024, https://www.statista.com/statistics/288487/forecast-of-global-b2c-e-commerce-growth/ (last access: May 2021)
Svensson O, Tarnawski J, Végh LA (2018) A constant-factor approximation algorithm for the asymmetric traveling salesman problem. Proc STOC 2018:204–213. https://doi.org/10.1145/3424306
Ulmer M, Thomas B (2018) Same-day delivery with heterogeneous fleets of drones and vehicles. Networks 72:475–505. https://doi.org/10.1002/net.21855
Vidal T (2022) Hybrid genetic search for the CVRP: open-source implementation and SWAP* neighborhood. Comput Op Res 140:105643. https://doi.org/10.1016/j.cor.2021.105643
Wang X, Poikonen S, Golden B (2017) The vehicle routing problem with drones: several worst-case results. Optim Lett 11:679–697. https://doi.org/10.1007/s11590-016-1035-3
Zarovy S, Costello M, Mehta A (2013) Experimental method for studying gust effects on micro rotorcraft. Proc Inst Mech Eng, Part G: J Aerosp Eng 227:703–713. https://doi.org/10.1177/0954410012440663
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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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DOI: https://doi.org/10.1007/s00291-022-00680-1