Hindawi Advances in Operations Research Volume 2020, Article ID 4053983, 10 pages https://doi.org/10.1155/2020/4053983 Research Article Inverting the Truck-Drone Network Problem to Find Best Case Configuration Robert Rich Industrial & Systems Engineering, Liberty University, Lynchburg, VA, USA Correspondence should be addressed to Robert Rich; rkrich@liberty.edu Received 16 October 2019; Revised 3 December 2019; Accepted 21 December 2019; Published 22 January 2020 Academic Editor: Demetrio Lagana` Copyright©2020RobertRich.(isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many industries are lookingfor ways toeconomically usetruck/rail/ship ﬁttedwith drone technologies to augmentthe “lastmile” delivery eﬀort. While drone technologies abound, few, if any studies look at the proper conﬁguration of the drone based on signiﬁcantfeaturesoftheproblem:deliverydensity,operatingarea,dronerange,andspeed.Here,weﬁrstpresentthetruck-drone problem and then invert the network routing problem such that the best case drone speed and range are ﬁtted to the truck for a given scenario based on the network delivery density. By inverting the problem, a business can quickly determine the drone conﬁguration (proper drone range and speed) necessary to optimize the delivery system. Additionally, we provide a more usable versionofthetruck-droneroutingproblemasamixedintegerprogramthatcanbeeasilyadoptedwithstandardizedsoftwareused to solvelinear programming.Furthermore, ourcomputational metaheuristicsandexperiments conducted insupport of thiswork are available for download. (e metaheuristics used herein surpass current best-in-class algorithms found in literature. drone delivers its only one package on a parallel path and is 1. Introduction practically useless—regardless of its range. Furthermore, (e use of drones or other parallel-constrained resources in that a very fast drone aﬀords no real improvement in de- conjunction with main delivery assets oﬀers potential per- livery time at all if it has only a short range. (e drone only formance improvements that may prove beneﬁcial. (e base becomes a helpful servant if the drone’s range and speed are problem for the truck-drone (DTSP) can be easily visualized correctlyproportionedtothetruck’sspeed.(us,itiseasyto as two shoppers working together to fetch goods from the imagine that there exists an optimal relationship between shelves of a supermarket as eﬃciently as possible. As one truck’s speed, drone range, drone speed, and the delivery shopper pushes the cart, the second shopper may stroll density of the network. alongsideormayseparateandoperateparallelfetchingitems For this problem, we assume the truck can launch the back to the cart. While the two shoppers may be tasked drone with only one parcel from any delivery location and independentlyinparalleloperation,therewillbetimeswhen then rendezvous w/drone downstream at an adjacent de- livery location while the drone delivers on a parallel path to it is more eﬃcient to walk together. Indeed, it is obvious that there is anoptimal set of routes for each shopper,butwhat is the truck. (is truck-drone routing is depicted (Figure 1). less intuitive is that the necessary speed and range of the (e remaining sections comprised herein as follows. parallel shopper will prevent any such optimization or may Section 2 discusses the inverted problem and other theo- signiﬁcantly delay the main shopper. Here, we examine this reticalinsightstosolvefor“best”casedronerangeanddrone relationship. speed. Section 3 discusses the literature surrounding the For a truck-drone (say UPS) parcel-delivery system, it is truck-drone problem. Section 4 deﬁnes a more usable easy to imagine that a very slow drone will aﬀord little or no version for themixedinteger programming(MIP).Section 5 beneﬁt to the truck. For at nearly every stop, the truck will formulates the truck-drone problem as an evolutionary waitidlywhilelookingforthedrone’sreturn.(elumbering algorithm (EA) type metaheuristic algorithm used to 2 Advances in Operations Research Truck-drone i t = max(2, 2α) dtsp Figure 2: Best case for truck and one-drone [1]. k/2 k/2 k/2 k/2 0 510 15 20 25 k/αk/α x coordinate (km) Figure 3: Practical best case parallelogram. Drone Truck of the triangle denote the delivery locations. We will call this Figure 1: Truck with single-drone parcel delivery. the practical best case. Using these triangles and simple parallelogram geom- conduct computational experiments. Section 6 performs etry, we can easily calculate the optimal drone speed and computation experiments for the EA against best-in-class drone range necessary to rendezvous with the truck at ex- found heuristics in literature, and Section 7 concludes the actly the same time at each parallel delivery operation. In research on ﬁndings as well as the direction of future such case, we would not need to spend any extra on un- research. needed resources while guaranteeing that the system will perform optimally. For the practical best case system, we calculate the de- 2. System Model: Theoretical Insights liverydensityas ρ � N/A. Usingdensity (ρ),wethenreplace For a three-node problem whereby node v denotes the N/A with N/((N/2) − 1)(κ /2α), where area of the paral- depot and parcel deliveries are to be made to nodes i and j; lelogram is denoted as A � ((N/2) − 1)(κ /2α). Since we are then, the best delivery time for the system is the max of the given the delivery density when given the problem scenario, drone’s time or the truck’s time. (e problem can easily be we can then invert the problem and solve for proper drone scaled by establishing that the truck’s speed is always one, capability (speed (α) and range (κ)) that solves the problem: and the drone’s speed is a multiple (or factor of truck’s ρ ∼ : α≥1, κ> α, N>4, ρ>0. (1) speed) as in (1 × α). Furthermore, since drone’sspeed α is as ((N/2) − 1) κ /2α ( ) factor of truck speed, then an optimal conﬁguration exists whendronespeedequalsdrone’srange κ.Inotherwords,we Furthermore, we conducted several computational ex- are not saturating our drone with unnecessary resources periments to better understand the relationship between (range or speed) to perform the delivery operation. As il- leandeliveriesandrandomlygeneratedstochasticsituations. lustrated (Figure 2),the truck launches the drone, movesout Foreachscenario,randomdeliverylocationswereuniformly a distance of 1 unit at rate of 1 unit/distance, and then distributed in the area of operation while drone speed (α), returnstorendezvouswiththedrone.(etotaldeliverytime range (κ), and operating area (A) were held constant. (e is t � max(2,2α), and an optimal conﬁguration exists when ′ number of deliveries N was perturbated ranging from 10 to range and speed for the drone are equal (2 � 2α and α � κ). 200 deliveries within the area of operation. Here, we wished However, most problems are not simple three-node to understand the percent improvement gained Π (in de- problems having one truck and one drone. For more ad- livery time) over a stand-alone truck (no drones) delivery vance network problems, the delivery density ρ becomes a system. We found that as the delivery density of the random critical component when solving for drone range and speed ′ scenario ρ � (N /A) approached or moved toward the opt (α, κ). Delivery density is deﬁned here as the number (N) of delivery density of the lean solution ρ � (N /((N/2) − required deliveries per area (A) of the delivery space 1)(κ /2α)) that performance time Π (over a stand-alone ρ � N/A. truck) improved. In such case, the delivery and routing Inordertobuildthecasetosolveforoptimalspeed αand portion of the problem for the randomly generated exper- range κ, we start with a “best case” or lean scenario. In such iments was solved using the metaheuristics described below, case, we would expect that approximately 50% of the de- while the delivery time for a stand-along truck was solved liveries will be made by the truck and 50% by the drone. using standard tsp metaheuristics. Furthermore, under these ideal conditions, a set of same size (e results (Figure 4) showed that as the problem delivery trianglescanbe constructedinsidetheareaofoperation A to density of the randomly generated scenario was close to the represent all the parcel deliveries (Figure 3), where vertices optimal delivery density (deﬁned by N, drone speed (α), and y coordinate (km) Advances in Operations Research 3 Performance of truck-drone when operating near optimal (3) 0.35 ≤1 − . 1 + α 0.3 (ii) (e minimum lower boundary for time improve- 0.25 ment (worst case) of truck-drone over the truck- 0.2 only solution defaults to the truck-only route time 0.15 or a tsp route time. 0.1 (iii) (e improvement (Π)over truck-only moves toward 0.05 optimalastheuniformlydistributed verticeswithina delivery area move toward the optimal density: Lean or optimal Diﬀerence between problem –0.05 density and optimal density opt ρ ∼ . (4) ((N/2) − 1) κ /2α –0.01 ( ) –0.03 –0.02 –0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Diﬀerence between problem density and optimal density (erefore,foradeliveryscenario,ifwearetobewithinthe vicinity of the “best set” of operating parameters to optimize Figure 4: Percent improvement Π over truck-only time as ρ opt Π , then system parameters are found by minimizing the moves toward optimal density ρ . diﬀerence between delivery density ρ and optimal density opt ρ .(eresultingdroneparametersandrelationshipbetween range (κ)) that the overall performance time Π improved the two done parameters α and κ become the lower without oversaturating the system with unneeded resources. boundaries for an optimal conﬁguration for the drone. Concretely, the abscissa (x-axis) denotes the diﬀerence between rho optimal based onalpha andkappa andthe problem density rho based on N and delivery area. (e ordinate (y-axis) denotes 3. Literature the improved performance over a truck-only delivery system Today, a vast body of literature exists on tsp and the closely (tsp-routed). (e graph shows that as the two densities are related vehicle routing problem (vrp). Several approaches to closertogether,theoverallperformanceofthesystemimproves. both problems can be found in surveys, reports, and papers Given good results of using the practical best case methodology [2–4]. As a rule, the vrp problem extents the tsp by adding as a basis to evaluate the performance of a system, we con- additional constraints. (ese constraints comprised such structed the following minimization problem to solve for the things as time windows, priorities, range, loiter times, per- proper drone speed and drone range based on number of missible-to-vehicle type route segments, load conﬁgurations, deliveries N and size delivery area of operations Akm . As and traﬃc patterns [5]. While many variants of the tsp/vrp such, minimization problem here is used to select system pa- problems exist in literature to include multivehicle mtsp, rameters α and κ thataresuitedforaparticularproblemdensity customer pickup and delivery problem, multiple synchro- ρ.Itisnotedthatduetothestochasticityoftheproblemthatthe nizationconstraintsvrp[6],multipledepotvehiclescheduling solution for range and speed are lower boundaries for the problem [7], and many-to-many milk run routing problem proper speed and range of the drone conﬁguration: [8], there exists only a handful of studies concerning the dtsp, opt ∗ min ρ − ρ and no studies exist that address the proper conﬁguration (capabilities) of a main tool (truck) and one or more assisting opt s.t. ρ � , tools when solving for the optimal network routing. ((N/2) − 1)(κ /2α) Work that addresses the main tool and a constrained assisting tool is the work of Agatz et al. [1] and Murray and ρ � , Chu [9]. Both of these authors propose the truck-drone problem as a type of main tool with assisting tools. (eir (2) works delineate the main diﬀerences between existing κ ≤ κ≤ κ , LB UB problems and their speciﬁc version of the tsp problem by α ≤ α≤ α , distinguishing the drone’s ability to travel with (on/in) the LB UB truck as well as to operate with the truck in several parallel N≥4, ρ , operations described as launch-deliver-rendezvous tasks. Before this demarcation, no problem dealt squarely with the ρ>0. uniqueparadigm.Concretely,thedtspdescribesthedroneas a parallel resource to the truck required to periodically In summary, the theorems for the optimal parameters rendezvous with the truck due to payload and distance for the truck-drone are as follows. restrictions. Murray and Chu [9] formally deﬁne the ﬂying (i) (ere exists a maximum theoretical upper boundary sidekick traveling salesman problem (FSTSP) as an NP-hard fortimeimprovementfactorwhichwillrarely,ifever, problem. (eir study suggests a mixed integer programming be reached [1]. (is is based on the comparison (MIP)approachaswell asmetaheuristic approach.(ey also between truck-drone and truck-only solution: consider a second similar hub type problem that addresses Performance improvement over a truck-only 4 Advances in Operations Research speed, and number of assisting tools are interdependent to the case where the customers are close enough to the depot to beserviceddirectlyfromthedepotbythedroneasthe parallel the node coordinates and thus determine the total possible ‘coverage’ area which may be diﬀerent for each operation. drone scheduling TSP (PDSTSP). Both Murray and Agatz discuss the mixed integer formulation for the optimal min- Furthermore, the speed and range of the assisting tool is time route. (ey allude to the fact that a drone is constrained signiﬁcant to the overall performance of the system. Herein, in range, capacity, and speed as it relates to the truck. wegiveaclosedformapproachtodeterminetheproperspeed However, they do adequately address this interrelationship. and range necessary for assisting tools to obtain an optimal Furthermore,Agatzonlyconsidersaslightlyalteredversionof performance ofthesystem withoutoversaturatingthesystem. the FSTSP and does not address the PDSTSP. Agatz, like (is is not currently found in any literature. Many other authors create variations of the main/assisting tool problem, Murray, considers the truck-drone in tandem as a team whereby the truck launches the drone, traverses to a separate but do not address these fundamental relationships. delivery location from the drone, and then rendezvous with the drone again. However, the main diﬀerence between the 4. Truck-Drone Mixed Integer Programming approaches is that Agatz et al. [1] require that the drone and trucktraversealongtheroadnetworksystem;aconstraintnot (e parallel resource truck-drone problem is recognized enforced by [9]. (ey do this to facilitate construction of when the associated second resource is constrained to re- heuristic approaches with approximations that guarantee a main at some proximity to the main vehicle, and when bound on the maximum achievable gain of the delivery separated, it will eventually rendezvous with the main ve- system over a “truck-only solution.” Murray and Chu [9] hicleatadownstreamlocation.Itispermittedtotemporarily formally deﬁne the ﬂying sidekick traveling salesman problem separate from the main resource, but must soon return due (FSTSP)asanNP-hardproblem.(eirstudysuggestsamixed to range or operational constraints. integer programming (MIP) as well as the metaheuristic In order to establish a generalized mathematical for- approach. (ey also consider a second similar hub-type mulation and improve the tractability and software mod- problemthataddressesthecasewherethecustomersareclose eling (Lingo/Lindo ) of this problem, we assume the enough to the depot to be serviced directly from the depot by following: (1) number, location, and distances between the drone as the parallel drone scheduling TSP (PDSTSP). customers (nodes) are known and deterministic, (2) each More recently Agatz et al. [10] presented an exact solution node must be visited no more than once by a vehicle or a approachforthetruck-droneproblemdenotedasthetraveling group of vehicles, (3) vehicles must traverse along arcs salesman problem with drone (TSP-D) based on dynamic (edges) between nodes, (4) vehicles separate and rendezvous programming. (ey modeled the problem ﬁrst as an integer atnodelocations,notalongarcspacebetweennodes,(5)any program and then developed multiple route-ﬁrst, cluster- vehicle may separate from the group to deliver to a node second heuristics based on local search and dynamic pro- before it must rendezvous with the main vehicle (or group) gramming. (eir insights suggested that larger problem in- at a downstream node; once rendezvous has occurred, each stances could be solved by dynamic programming better than vehicle is again permitted to separate to another delivery other mathematical programming approaches found in lit- node and then rendezvous again at a downstream node, and erature. (ey show worst case approximation ratios for their (6) each delivery node has demand of one unit; thus, one heuristics and compare the performance to optimal solutions sortie of drone or truck is capable of delivering a parcel. for smaller instances. (ey applied their heuristics to several Given wehave one drone(y) operating in adjacent space artiﬁcial instances with diﬀering characteristics and sizes to of the main delivery truck (x), then if drone (y) is used for show substantial improvements over the truck-only solution. delivery, and it shall be launched from the truck at node i, Perhaps, the problem herein most similar is the covering traverses to a delivery node (customer) at j, and rendezvous traveling salesmen problem described by Current and with the delivery truck at third node k, whereby the binary Schilling [11]. (e covering problem ﬁnds the shortest tour tuple variable (y � 1) ﬂags that the route segment is used ijk for the “tsp” network by reducing the problem to a subset of by drone. After launching drone, the truck (x) has two the total nodes. In such case, if the adjacent nodes can be options: (1) traverses directly from launch node to the reached or “covered” within range then they can be grouped rendezvous node (x � 1) as depicted (Figure 5) and (2) ijk as a single node or stop, thus reducing the ﬁnal optimal tour traverses to yet another delivery node j and then proceeds as well as the number of permutations of the problem onto the drone rendezvous node (x � 1). ijk signiﬁcantly. For the covering problem, they propose a (etruck-dronedtspproblem’smaterialelementscanbe metaheuristic based solution to solve larger problem sets. described mathematically as a network graph G � (V, E), Several generalizations and extensions to the covering where V denotes the customers-delivery-stops and E de- problem can be found in literature [12]. For the main tool notes the edges between stops. Each vertex V � {1, . . . , |V|} with assisting tools, it is not enough to bypass the unvisited and N denotes the cardinality of |V| or total number stops. or ‘covered’ nodes for three reasons: (1) ﬁrst, the main and An edge E is described by two vertices i, j, whereas two assistingtoolrequiresthatthetruckanddronerendezvousat joined edges are described as three vertices i, j, k , where the end of each operation, and thus their end of operation vertex j is the vertex between i and k. Furthermore, binary timing is a factor of the total time. (2) Secondly, the covered variables (x , y )denotewhetheranedgetriplicateisused ijk ijk nodes ‘within-range’ are subject to the number of assisting byatruck x oradrone y ;and x or y � 1ifused,else ijk ijk ijk ijk tools ﬁtted to the main tool. (3) And, thirdly, the range, x or y � 0. (e binary variable x � 1 if truck traverses ijk ijk ijk Advances in Operations Research 5 y y h h ihk ihk k k i i x x iik or ikk ijk (a) (b) Figure 5: Truck and one-drone problem depiction: (a) truck option 1 and (b) truck option 2. E(i, j)andthenfurthertraversesto E(j, k).Ifadronetravels x + x + x ijk kij jki on a truck in a dormant conﬁguration, the drone is assigned j∈V j∈V k∈V k∈V k∈V j∈V the truck’s i, j, k tuple index variables y � 1; else, a drone ijk i≠k k≠j j≠i (9) mayserveinanactivedeliveryroleandthereforebeassigned + y ≥1, ∀i ∈ V, binary variable y � 1 denoting the drone was launched kij ihk j∈V k∈V from vertex i, delivered to h, and subsequently recovered at vertex k. An important characteristic of the problem here is ⎧ ⎪ x ≤1 that the ﬁrst and last vertex index of the truck and the drone ijk ⎪ j∈V must always be the same. (e distance matrix D and the ⎪ k∈V i≠k constructionoftripletdistancesastuple d denotes thecost ijk or distance of traversing the E(i, j) and E(j, k). (e subtour ⎪ x ≤1 jik elimination variable u ensures thatsequenceof j follows i in j∈V ∀ i ∈ V , (10) k∈V the event that E(i, j) is part of the solution; as such, it re- ⎪ j≠k stricts subtour formations. ⎪ x ≤1 (e minimax objective function minimizes the max ⎪ jki j∈V (Z � z ) time of the truck or drone as each visit ⎪ k∈V i∈C j∈C ij ⎪ j≠i three nodes in an operation. As such, z is used to evaluate ij each vehicle’s time, where i, j are the ﬁrst and last nodes of the three visited nods for truck (i, k, j) or drone’s (i, h, j). ⎧ ⎪ y ≤1, Since z is greater than or equal to the truck or drone time, jik ij ∀ i ∈ V, (11) ⎪j∈V we force z to evaluate each vehicle’s time using ⎪ k∈V ij j≠i z ≥ x d which denotes truck time while ij k∈C ikj ikj z ≥ y d (1/α) denotes drone time divided by speed ij h∈C ihj ihj factor α which is a factor of truck’s speed: x � x , ∀i ∈ V, ikj jki k∈V k∈V (12) i∈V j∈V Minimize Z � min z ij i≠j i≠j (5) i∈Vj∈V ⎧ ⎪ z ≥ x d ⎛ ⎝ ⎞ ⎠ ⎪ ij ikj ikj u ≥ u + x − (N − 2) 1 − x ⎪ j i ikj ikj ⎪ k∈V k∈V k∈C (13) subjectto ∀i, j ∈ V, i≠ j , (6) + (N − 3) x , ∀i, j ∈ V, jki ⎪ ihj k∈V ⎪ z ≥ y ij ihj h∈V y d < κ + Mx , ∀i, j, k ∈ V, (14) ijk ijk ijk x �� 0 ikj i � k, k � j, ∀i, j, k ∈ V , (7) ∀d ≥0, i, j, k∀ V, (15) y �� 0 ijk ikj x � y � 1, ∀i, j ∈ V, i≠ j ikj ihj , (8) ∀x , y ∈ ∀i, j, k ∈ V, u ∈ {1,2, . . . , N}. (16) ijk ijk k∈V h∈V 1 6 Advances in Operations Research Equation (5) minimizes the sum of the max time for any (1) Gene mutation (tour mutation) ﬁrst copies the route triplet segment starting at i and ending at j. Equation ﬁttest member of the group of ﬁve within the seed tournament to replace the four less ﬁt (6) mandates that the sum of the max time is the max of either the truck time or the drone time for any given op- members. eration described as a triplet where an operation could be a (2) Each of the four less ﬁt members (now identical truck-only or truck-drone. Equation (7) mandates that no to the ﬁttest) are then slightly mutated to im- triplet can contain i � j � k. Each operation must be a move prove ﬁtness. where node i is diﬀerent from node j or node k. Ensures that (3) Mutations comprised(a) randomly selecting and vehicle loitering. Equation (8) enforces the concept of an swapping two nodes within the tour, (b) reverse operation whereby a truck and a drone must operate with ordering of the tour between two nodes, (c) same launch and recovery node; however, the delivery node sliding a tour segment down between nodes to can be diﬀerent. Equation (9) forces every node to be visited leftorright,and(d)replacingthelastnodeinthe byatruckoradroneoracombinationthereof.Equation(10) tour with any other node. constrains a truck to visit a city once and only once. (e) Repeat step (b) until convergence. Stopping condi- Equation(11)constrainsadronetoonlyvisitacitytodeliver tion is based on a predetermined iteration budget, at most once. Equation (12) constrains each tuple segment tolerance, or saturation found in improvements. entered by the truck shall exit the same by truck. Equation (13) is a subtour elimination constraint to force the se- (f) Return ﬁttest member of the entire population. quencing of the route such that no subtour is possible. A seed tournament genetic algorithm’s strength lies in the Equation (14) forces all drone deliveries within the drone’s abilityto retainmultiplepathstowardoptimizationduring the delivery operation range κ (launch, delivery, and rendez- process which is critical for any network routing problem. vous) unless riding in a truck in which case Mx disables ijk Furthermore, because there are multiple members in a seed the constraint. Equation (15) constrains all Euclidean dis- tournament,thealgorithmallowsforvariousprovenmutation tances to greater than zero. Equation (16) assigns the var- methodologies to be performed on the members of the seed iables xand yasbinary;andtheutilitysequencingvariable u tournament. In this case, the swap, ﬂip, and slide mutations is an integer between one and the total number of nodes (or have proven to be robust, fast, and extremely accurate for customer deliveries). many routing problems including the tsp, multiple-truck tsp, as well as vehicle routing problems (vrp). (e performance of the algorithm is based on the un- 5. Truck-Drone Evolutionary derlying theoretical principles: Algorithm Approach (1) By initializing a relatively large population (i.e., 5n) (e tournament-based evolutionary algorithm (EA) here of randomly permuted tours, multiple paths (seed adopts a cluster-during-routing approach to solve the tournaments) increase the probability of an optimal truck-drone problem. More accurately, it assigns both convergence. truck and drone labels during the routing process. (is is (2) By saving the ﬁttest gene in a seed, and then slightly much diﬀerent from all other algorithms found in liter- perturbing (mutating) the best gene (tour) found in ature. (e best-in-class found in literature perform an the seed group of ﬁve tours ensures the solution entireroutingoperationﬁrst,andthenthealgorithmlabels never gets worse while promoting improvements at truck or drone. Conversely, the algorithm herein denoted each iteration. as EA creates a population matrix of randomly permuted routes whereby each node in a tour is evaluated as a (3) By autoassigning the drone to any “within range” potential drone-delivery node unless that node is out of node, the use of the drone is maximized throughout range. Since a population of many randomly generated the routing process while simultaneously reducing tours is evaluated simultaneously, any node within drone the truck’s overall tour length. (e risk of assigning range is autoassigned and labeled drone. (e EA performs the wrong node to a drone is mitigated by ﬁrst the following process steps: initializing thepopulationwith random permutation and then maintaining multiple paths toward (a) Randomlypermutesapopulation Pof mtourswhere optimization. eachtourdenotedasagenomesequence (1,2, . . . , n) for n delivery nodes in the tour. (4) Multiple path random search is much faster than (b) Determines the ﬁtness for each population member having to calculate the greediness or the exactness of (tours) based on total tour delivery time. All ﬁtness each neighborhood within reach as in other algo- times are saved for seed tournament. rithms. (erefore, the algorithm relies on compu- tational speed and iterations without the burden of (c) (e total population is divided into groups of ﬁve unnecessary calculations. tours each to conduct a set of seed tournaments. (d) Foreachofthegroups,thebestmemberwithintheseed For thealgorithm,werandomly permutea populationof group(oftheﬁve)ischosenasthesinglegenetomutate tours denoted as the initial population matrix P comprising for the remaining four members of the seed group. mtourseachhaving nnodesinthetour(nlengthoftour).In Advances in Operations Research 7 such case, P[1, :] denotes the ﬁrst tour (1) in the population 6.1. Comparison Study 1 (Smaller Problem Sizes: Against Best- matrix and all the nodes (:) for that ﬁrst tour. Whereas in-Class). For the sake of simplicity, we generate node coordinates within an (x, y) Cartesian coordinate system variable R denotes the ﬁrst tour in population P[1, :], then 1 th R also denotes the i node within the ﬁrst tour or route. and assume truck or drone distance Euclidean distance (e function D denotes a distance function that properly based on the uniform distribution. In the original com- calculates the distance between segments of the route. (e parison problem [13], the metaheuristic comparison ex- p p distance function nomenclature D(R , R ) denotes the perimentsdrewfromthreediﬀerentdistributions:uniformly i i+2 distance between nodes (i) and node (i + 2) found in route (random) distributed nodes, 1-center Gaussian-distributed R ∈ P[p, :]. Furthermore, the neighborhood function nodes, and 2-center Gaussian-nodes. In our study, we an- N(R ) denotes the next three nodes or route segments for alyzed the results obtained from the diﬀerent distributions R as in [R , R , R ] which is used to evaluate whether or and found that the underlying distribution had no statistical i i i+1 i+2 not a segment is reachable by the drone given the constraint relevance for our EA metaheuristic; therefore, for simplicity, dronerange. In such case, the algorithm’s distance function weadoptedtheuniformdistributionfromwhichtodrawour experiments. Concretely, all nodes for problem comparisons can handle the distance between two nodes, or a neigh- borhood of three nodes as in D(N(R )). (e returned herein were drawn from the uniform distribution with from {0, 1, 2, . . ., 100}. distance from D function is then divided by the speed of the truck or the drone to arrive at time. (e table (Table 1) below provides results for 10 ran- domly generated instances with 10 nodes of each instance type. For each instance the optimality delta gap is deﬁned as 6. Metaheuristic Comparisons objectivevalueheuristic − optimalobjectivevalue Δ � . As stated, our evolutionary algorithm (EA) (Algorithm 1) optimalobjectivevalue metaheuristic uses a cluster-during-routing approach, whereby (17) each drone is autoassigned to any next delivery node in the tour if it is within range. (is forces the drone to be highly Results proved the EA outperformed other methods utilized. Next, we initialize the population with multiple found in literature. (e EA herein proved optimal for 10/10 randomly permuted routes to improve potential optimization instances for similar problem sizes found in current liter- paths thus helping to preclude a local optima. At each iter- ature. As such, we increased the problem size, constructed a ation, each tour in the population is slightly improved using constraint based model (CP) model in order to compare to mutationswhichcomprisedrandomnodeswaps,randomtour optimal, and conducted additional experiments. segment ﬂip, and random segment slide right or left. Concretely, our cluster-during-routing EA is robust, fast, and is able to solve large problems optimally better than any 6.2. Computational Study 2 (Larger Problem Sizes: MIP, CP, other found in literature. To illustrate, we compared its and EA). (e study was run on a 64-bit version of performance to mathematical procedures (MIP), artiﬁcial Xumbutu 15.04 on virtualBox 4.3.12 hypervisor with intelligence constraint-based programming procedures windows 7 as the host OS. (e EA was coded in the (CP), and the ‘best-in-class’ heuristics conducted by Agatz MATLAB . (e ﬁles we created for experimental purpose et al. [13]. To prove optimality, we compared the EA to were made available at Mathworks ﬁle exchange closed form mixed integer programming approaches for (dtsp_ga_basic) for evaluation/comparison. (e hardware smaller size problems less than 15 nodes and then to IBM conﬁguration consisted of an Intel Core i7-4770 CPU with constraint-basedprogramming(CP)forlargerproblemsizes 16GB of RAM. And, the MIP and CP were all coded in IBM OPL 12.8.0 on a personal computer with an Intel Core i5- up to 80 delivery nodes. (e EA was compared to the following methods: 3537 @ 2.5 Ghz processor and 8GB RAM. A total of 10 test problem instances having nodes (be- (1) Closed form mixed integer programming (MIP) for tween 10 and 100) were randomly generated and then av- smaller problems (less than 12 nodes) to prove eraged. As in other studies, the drone is assumed to be twice optimality. the speed of the truck (α � 2). In this case, all customers are (2) Constraint-based programming (CP) for larger distributed across an 8 mile square region. (e maximum problems (less than 80 nodes) to prove optimality. ﬂight endurance (drone range) is set as 14 miles as drone range. Table 2 summarizes the computational results of test (3) MST-gp-all (route mst ﬁrst, cluster with greedy al- problem instances. Column 1 shows diﬀerent job sizes gorithm, all iterative improvements) (numberofnodes n).Columns2–4,5–7,and8–10recordthe (4) MST-ep-all (route mst ﬁrst, cluster with exact par- elapsed computational processing time (elapse ti.), objective titioning dynamic programming with all iterative function (f), denoting total time, and gap of MIP, CP, and improvements) EA, respectively, from optimal. (e gap is calculated against (5) TSP-gp-all (route tsp ﬁrst, cluster using greedy al- the best solution if optimality guaranteed. However, if op- gorithm with all iterative improvements) timalitynotguaranteedforlargerproblems,gapiscalculated (6) TSP-ep-all (route tsp ﬁrst, cluster using exact par- against best of MIP, CP, and EA delivery time. In such case, titioning with all iterative improvements) IBM CP and LINDO/LINGO MIP optimization software 8 Advances in Operations Research DATA: a population P matrix of initially randomly generated tours where tour R � P[1, :] Size of the population is m and the length of any tour is n. (e distance function Distance (launch, deliver, rendezvous) determines the total distance between the nodes in the RESULT: an∼optimal tour for the nodes [1: n]. FOR iter 1 to Budget LOOP FOR p in m LOOP (for each population of tours) R ⟵ P[p, :]; (a tour within the population of tours) R ⟵ P[p, 1: n, 1, 2, 3]; (add wrap back around to depot to tour) time ⟵0; (initialize total tour time for this tour) WHILE i≤ n DO (go through each of the nodes in tour) case⟵1; (initialize default case: truck carries drone and delivers) launch⟵ R (i); (truck launch node) deliver⟵ R (i+1); (drone delivery node) rendezvous⟵ R (i+1); (truck and drone rendezvous node) launch fathom⟵ R (i+1); (check next operation truck launch node) deliver fathom⟵ R (i+2); (check next operation drone delivery node) rendezvous fathom⟵ R (i+3); (check next operation truck/drone rendezvous node) drone dist⟵Distance (launch, deliver, rendezvous) truck dist⟵Distance (launch, , rendezvous) IF drone dist<range AND i+1≤ n THEN case⟵2; fathom 1⟵max[(drone dist)/(drone speed), (truck dist)/(truck speed)] drone dist 2⟵Distance (fathom launch, fathom deliver, fathom rendezvous) truck dist 2⟵Distance (fathom launch, , fathom rendezvous) fathom 2⟵max[(drone dist 2)/(drone speed), (truck dist 2)/(truck speed)] IF drone dist 2<range AND fathom 2<fathom 1 AND i+2≤ n THEN case⟵1; (save drone for next operation, set to truck deliver for this iteration) drone dist⟵0; (no drone distance) END IF ELSE drone dist⟵0; (out of drone range. . .) END IF SWITCH CASE CASE �� 1 (truck delivers) truck dist:�Distance (launch, , deliver); (ﬁnd truck distance to next node) k⟵ k+1; CASE �� 2 (truck and drone deliver) truck dist:�Distance (launch, , rendezvous); (ﬁnd truck distance to rendezvous) drone dist:�Distance (launch, deliver, rendezvous); (ﬁnd drone distance for operation) k⟵ k+2; (two nodes satisﬁed) END CASE p p time �time +max[(drone dist)/(drone speed), (truck dist)/(truck speed)] (capture and record the total time for population member p) END WHILE LOOP END FOR LOOP P⟵randomly shuﬄe rows in population matrix P for a tournament (do not change tours) FOR p �5:5: m LOOP (select groups comprised of 5 tours each from the population P of size m) Best time⟵Get Best Time (P[p, :]) for the group of the 5 tours Best Id⟵Find route id of the group of 5 having the best time (Best Time) Pʹ[p , :]⟵Replace all tours in population group P(p, :) with ﬁttest tour of the 5 tours. 1,2,3,4,5 Pʹ [p , :]⟵keep (do not mutate) the ﬁttest tour of the group of 5 (keep for next iteration) Pʹ[p , :]⟵Mutate the other 4 less ﬁt tours as follows: 2,3,4,5 (1) tour 1: randomly select 2 points in the route Pʹ[p , :] to swap (2) tour 2: randomly select 2 points Pʹ[p , :]to reverse all nodes in between (3) tour 3: randomly select 2 points in route Pʹ[p , :]to slide to left and replace last with ﬁrst (4) tour 4: replace the ﬁrst and last nodes Pʹ[p , :]with two randomly selected nodes END FOR LOOP P⟵update P with all mutations P′ END FOR LOOP RETURN best time and best route found in population ALGORITHM 1: Evolutionary algorithm. Advances in Operations Research 9 Table 1: Comparison against best-in-class heuristics. Comparison against “best-in-class” heuristic solutions to the optimal solution (uniform n �10 nodes, α � 2, averaged over 10 instances). Nodes: Cartesian coordinates—uniform distribution (10 nodes) Δfromoptimal Avg. Max #opt MTSP-gp-all [13] 2.0 6.4 1/10 MTSP-ep-all [13] 0.7 2.7 4/10 TSP-gp-all [13] 1.6 3.1 1/10 TSP-ep-all [13] 0.4 2.3 6/10 EA 0.00 0.00 10/10 Table 2: Elapsed time and objective function value according to diﬀerent job sizes. 1 2 3 4 5 6 7 8 9 10 MIP CP EA Nodes Elapse time f Gap Elapsed time f Gap Elapsed time f Gap ∗ ∗ ∗ 10 1s 228.0 0.0% 1s 228.0 0.0% 1s 228.0 0.0% 20 54s 285.5 0.0% 20s 285.5 0.0% 5s 285.5 0.0% 30 1834s 400.0 8.4% 10s 369.0 0.0% 15s 374.0 1.4% 40 1692s 621.0 44.6% 55s 429.5 0.0% 30s 434.0 1.0% 50 — 118s 466.0 0.0% 45s 478.5 2.7% 60 — 422s 15.5 4 0.0% 60s 419.5 1.0% 70 — 1398s 511.5 0.0% 90s 504.0 1.7% 80 — 1745s 523.0 0.0% 120s 546.5 4.5% 90 — — 120s 638.0 0.0% 100 — — 120s 655.0 0.0% Best solution in bold. Optimal values, no valid bound found within the time limit of 1800s. cannot guarantee optimality for jobs (number of nodes) (e study also gives a simpliﬁed version of the MIP not greater than 20+ or when computer computational time is found in any other study as well as a useful and easily above 1800 seconds on current computer systems. In such implemented metaheuristic necessary to solve for the case, CP reports the best solution it ﬁnds and gives a optimal route and optimal time for the truck-drone probability of being optimal (i.e., up to 80 job instances). problem. We used a simple single chromosome evolu- Conversely, EA consistently delivered an eﬃcient solution tionary algorithm (EA) metaheuristic to test each case. (e with the shortest computational time. Larger sized problems EA was modeled as function within the MATLAB de- require more computer processing capability for compari- velopment environment language, and the ﬁles were made son studies to optimal. available at Mathworks ﬁle exchange (dtsp_ga_basic) for All the test instances and MIP and CP logs are located at evaluation. (e EA was tested against current best-in-class the following link: https://drive.google.com/open? heuristics found in literature and signiﬁcantly surpassed id�19ZZ9ukEwSSfjCqNID1P1kPsGMOyMefXC. them in terms of accuracy as well as computational time performance. 7. Conclusions In conclusion, this research answers the questions of expected eﬃciencies in time would be expected given a (e current literature lacks any usable information in terms truck-drone conﬁguration as well as ﬁnding on “what is the of thehighest yield regions of the design space (speed, range, proper conﬁguration in terms of drone range and drone and number of drones) for the truck-drone conﬁgurations, speed for a truck-drone situation” given a typical delivery and any company considering the use of a delivery-drone scenario. It gives business a foundation to evaluate a variety has no way to guide the selection of drone parameters of conﬁgurations against their typical daily last-mile parcel- (speedfactor αandrange κ) without some visibility into the delivery scenarios. (e work also opens several additional high yield regions of the operating space. (e lean geometry questions for future research. (e most obvious questions approach shown herein inverts the problem space to solve involve the design space time and/or eﬃciencies of 1-truck for practical best case drone speed and drone range given a which comprised many drones. randomly generated scenario. As such, it is quite useful for practical design decisions regarding the proper/most eﬃ- Data Availability cient drone speed and range to achieve the maximum de- siredyieldoverthetruck-onlysolution.Assuch,itservesasa (e metaheuristic codes and algorithms are available at way to screen out the inadvertent selection of low per- MATLAB.com and also from the corresponding author forming designs. upon request. 10 Advances in Operations Research Conflicts of Interest (e author declares that there are no conﬂicts of interest. References [1] N. Agatz, P. Bouman, and M. 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Published: Jan 22, 2020
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