A Comprehensive Survey on the Multiple Travelling Salesman Problem: Applications, Approaches and Taxonomy
AA Comprehensive Survey on the Multiple TravellingSalesman Problem: Applications, Approaches andTaxonomy
Omar Cheikhrouhou a, ∗ , Ines Khoufi b a College of CIT, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia. b SAMOVAR, T´el´ecom SudParis, Institut Polytechnique de Paris, France.CES Laboratory, University of Sfax, Tunisia.
Abstract
The Multiple Travelling Salesman Problem (MTSP) is among the most in-teresting combinatorial optimization problems because it is widely adoptedin real-life applications, including robotics, transportation, networking, etc.Although the importance of this optimization problem, there is no surveydedicated to reviewing recent MTSP contributions. In this paper, we aimto fill this gap by providing a comprehensive review of existing studies onMTSP. In this survey, we focus on MTSP’s recent contributions to bothclassical vehicles/robots and unmanned aerial vehicles. We highlight the ap-proaches applied to solve the MTSP as well as its application domains. Weanalyze the MTSP variants and propose a taxonomy and a classification ofrecent studies.
Keywords:
The Multiple Travelling Salesman Problem MTSP, MTSPApplications, MTSP variants, Taxonomy, Approaches, Robots, UAVs ∗ Corresponding author
Email addresses: [email protected] (Omar Cheikhrouhou), [email protected] (Ines Khoufi)
Preprint submitted to Journal Name February 25, 2021 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
Contents1 Introduction and Motivation 42 MTSP Application Fields 83 MTSP Definition and Variants 12 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
1. Introduction and Motivation
The Multiple Traveling Salesman Problem (MTSP) is a generalization ofthe well-known Traveling Salesman Problem (TSP), where multiple salesmenare involved to visit a given number of cities exactly once and return to theinitial position with the minimum traveling cost. MTSP is highly relatedto other optimization problems such as Vehicle Routing Problem (VRP) [1]and Task Assignment problem [2]. Indeed, MTSP is a relaxation of VRPwith neither considering the vehicle capacity or customer demands. MTSPalso shares some characteristics with the task assignment problem, however,MTSP does not allow multiple visits and sub tours. Thus, a solution toMTSP can be used to address VRP or Task Assignment optimization prob-lem.MTSP is one of the most important optimization problems, and it hasbeen applied in several real-life scenarios long ago. Depending on the appli-cation requirements, the salesmen in MTSP can be represented by groundvehicles such as robots or trucks, or by flying vehicles such as UnmannedAerial Vehicles (UAVs) known also as drones. Whereas the cities to be vis-ited by the salesmen can have different representations, such as customers intransportation and delivery services, sensor nodes for Wireless Sensor Net-works data collection, targets in military applications, victims in emergencymissions and critical sites in disaster management applications.Considering the importance of such optimization problem, we dedicatedthis survey to review, analyse and discuss recent contributions on MTSPwhile highlighting the application domains and the approaches applied tosolve the problem for both classical vehicles (i.e. ground robots, vehicles,4 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 trucks) and flying vehicles (i.e. UAVs and drones).In the context of classical vehicles, several surveys on vehicle routingproblem [3, 4, 5, 6] and traveling salesman problem [7, 8] have been proposedin the literature to review the different variants of those optimization prob-lems as well as the approaches applied to solve them. To name a few, theauthors in [7] presented a survey on routing problems for robotic systems.This survey discussed some routing solutions based on TSP and VRP andtheir extension to satisfy the robotic system’s constraints. These solutionsinclude the Dubins TSP (DTSP), a generalization of the TSP in which a pathis composed of Dubins curves; TSP with Neighborhoods (TSPN), where aneighborhood is associated with each city and the salesman needs to visitany neighborhood; the Multiple Traveling Salesman Problem (MTSP) andfinally the Generalized Asymmetric TSP (GATSP).The study in [8] introduced a comparative analysis of evolutionary al-gorithms for the Multi-Objective Travelling Salesman Problem (MOTSP).The authors focused on only five evolutionary algorithms, namely NSGA-II,NSGA-III, SPEA-2, MOEA/D and VEGA, in order to determine the mostsuitable algorithm for the MOTSP problem.Although it was cited in several studies, MTSP was only mentioned insome sections but was not the subject of any of the above surveys. Indeed,the only survey on MTSP proposed by Bektas [9] dates back to 2006, andwhich reviewed MTSP and its applications, highlighted some formulations,and described some exact and heuristic solutions.Therefore, there is a lack of analysis studies devoted to MTSP. To fillthis gap, the present survey is dedicated to review recent contributions that5 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 focus on either MTSP and its variants or on the use of MTSP to solve real-lifeproblems.In the context of flying vehicles, the study in [10] presented a surveyfocusing on UAV optimization problems and proposed different variants ofTSP and VRP for UAVs. Another interesting study is introduced in [11],where the authors reviewed contributions on UAV trajectory optimizationand UAV routing. They also provided a formal definition of the UAV Rout-ing and Trajectory Optimization Problem (UAVRTOP) that considers thekinematics and dynamics constraints of UAVs. Both surveys [10, 11], pro-posed taxonomy and classification of recent studies in terms of routing prob-lems, their variants, approaches used, applications, etc. So that, the readercan have a general idea about the existing variants of routing problem andtrajectory optimization for UAVs.The reader may also refer to the survey in [12] that can provide a fastpoint of entry into the topic of UAVs operations planning for civil applica-tions. In this survey, the authors summarized the important UAVs char-acteristics related to their operation planning, described planning problemsfor UAVs operations, and also planning problems for combined UAVs andother vehicles operations. They finally identified a number of new UAVsoptimization problems and discussed model extensions from the well-knownoptimization problem such as TSP. However, all these surveys [10, 11, 12]are limited to only UAVs context.In our paper, we propose a different survey that is complementary to theabove cited surveys, since we dedicated this study to MTSP.More precisely, our survey proposes the following contributions:6 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 • Presentation of the most important real-life applications of MTSP. • Analysis of the different MTSP existing variants and their formal de-scription. • A comprehensive review of recent contributions for ground and flyingvehicles while highlighting the application area and the approach usedto solve the MTSP for each contribution. • A Taxonomy and a classification of recent contributions on MTSP thathelps readers in their researches, and gives future research directions.
1. Introduction and Motivation - Motivation- Scope and Contribution- Roadmap
2. MTSP Application Fields3. MTSP Definition and Variants - Variants analysis- Variants formal description
4. MTSP Approaches5. Taxonomy, Classiffication and Analysis - An extended Taxonomy for MTSP- Classiffication of reviewed MTSP's solutions- Analysis of reviewed MTSP's solutions
6. Discussion and Future Directions - MTSP for Ground Vehicles and Robots - MTSP for UAVs
7. Conclusion
Figure 1: Paper Roadmap he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 The remainder of the survey is organized as follows. First, we mentionin Section 2, the different real-life applications of MTSP. This motivatesthe reader about the reviewed optimization problem as we show that it isused to model several real-life applications. Then, we present in Section 3 anoverview of MTSP, we analyze the main existing MTSP variants, and we givea formal description of them. After that, Section 4 is dedicated to reviewexisting solutions proposed to solve MTSP for ground vehicles and robots inSection 4.1, as well for flying vehicles in Section 4.2. In Section 5, we proposea taxonomy, classification and analysis of the reviewed MTSP studies. Then,in Section 6 we discuss these reviewed contributions and give some futuredirections. Finally, we conclude the survey in Section 7. Figure 1 presentsthe roadmap of the present survey.
2. MTSP Application Fields
For years, mobile robots, vehicles, and UAVs, which are aircraft oper-ating without a human pilot on board, have been considered as emergingtechnologies that have made many complex missions safer and easier. Inorder to achieve their missions, it is important to determine a path for eachvehicle that optimizes a given objective while considering some constraints.MTSP has been adopted in different real-life applications to obtain optimizedmultiple vehicle routes. The main applications of MTSP are summarized inFigure 2 and are as follows:
Transportation and Delivery . such as good distribution or parcel deliv-ery, or also bus transportation. Vehicles are in charge of transporting goods,parcels, or persons from a given location to another. In such an application,8 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 MTSPApplications
TransportationandDeliveryMulti-RobotTask Allo-cation andSchedulingWSN DataCollectionNetworkConnectivitySearch andRescuePrecisionagricultureDisastermanagement Cooperativemission MonitoringandSurveillance
Figure 2: MTSP Applications vehicle capacity and time constraints should be considered. In addition, afleet of trucks can efficiently transport valuable goods, whereas drones candeliver small parcels in a very short time [13, 14, 15, 16, 17]. With the emerg-ing of the UAVs technology, a new transportation service in which droneswork in tandem with trucks to deliver parcels to customers is proposed tosignificantly reduce the delivery time.9 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
WSN Data Collection and Network Connectivity . in wireless sensornetworks, additional sensor nodes may be deployed to forward the collecteddata until it reaches the sink. However, it is possible to use multiple mobilerobots to act as a mobile sink and, therefore, help to collect data providedby sensor nodes. This strategy of data collection allows minimizing energydissipation for sensor nodes located near the fixed sink and so increases thenetwork lifetime [18, 19, 20, 21, 22]. The authors in [22], proposed a multi-objective optimization model for the joint problem of data collection andenergy charging of sensor nodes. The proposed multi-objective model triesto optimize the total energy efficiency of the mobile robot and to reducethe average delay of data transmission of sensor nodes. In Delay TolerantNetwork, mobile robots can re-establish network connectivity by picking updata from a source node and then delivering it to the destination node. Insuch an application, a fleet of UAVs can be used instead of ground robots.The idea is that these unmanned vehicles fly over disconnected ground nodesin charge of collecting data and act as a relay [23, 24, 18]. Constraints onstorage capacity and data latency should be considered. Search and Rescue . when human lives are at risk, it is essential to opti-mize every second during the search and rescue operation [25, 26]. In such anapplication, the locations to visit are determined then the routes to followare optimized. With the emergence of the unmanned aerial vehicles tech-nologies, UAVs are becoming very practical and useful for search and rescueoperations [26]. Precision agriculture . mobile robots are widely used in precision agricul-ture, to ensure, for instance, crop monitoring or irrigation management. The10 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 farmer needs to provide an optimized path for each robot to reduce costsand improve production. Both ground vehicles and UAVs are of great helpto farmers [27, 28, 29]. The authors in [29], proposed a cloud-based archi-tecture composed of a WSN, mobile robots, and Cloud system to monitor agreenhouse region. The authors first generate candidate region monitoringpoints to be visited by mobile robots. Then, they generate the moving pathof mobile robots to reach these points. To compute an optimal robots’ path,the TSP and MTSP problems are used. Disaster management . after a forest fire, an earthquake, or an industrialaccident, mobile robots could help rescue teams in their mission [30, 31].For example, the UAV’s routes should be optimized in critical sites [32],such as during fire-fighting operations. The authors in [30] have proposeda cloud-based disaster management system. The system consists of a WSNdeployed across an area of interest that will be monitored, and in case ofa disaster, sensors nodes will report this information to a central stationlocated at the cloud side. An optimized rescue plan is modeled as an MTSPand then generated using the Analytical Hierarchy Process MTSP (AHP-MTSP) method [31]. Monitoring and Surveillance . large areas could be monitored by mobilerobots instead of sensor nodes. UAVs have been widely used in monitoringand surveillance applications. Most of the aerial vehicles used in such ap-plications are of small UAVs platforms. However, those UAVs have limitedenergy. In such a case, either the UAV tour must not exceed its maximumenergy or the UAV may make one or several refueling stops during its moni-toring mission [33]. 11 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Multi-Robot Task Allocation and Scheduling . In the robotic commu-nity, several work have modeled the Multi-Robot Task Allocation (MRTA)problem as an MTSP. Generally speaking, the MRTA consists of allocatinga set of tasks to each robot while optimizing some given metrics [9, 34, 35,36, 37, 38].
Cooperative mission . when a swarm of vehicles/robots cooperates to ac-complish a given mission such as target attacks in military applications [32].In a cooperative mission, the computation of a vehicle’s route should takeinto account the other vehicle’ routes as well as collision avoidance. The tra-jectory of these vehicle must be optimized so that each vehicle can carry outits mission effectively, safely, and successfully. Notice that, in a cooperativemission, a cite may be visited several times by different robots. Some ofthe applications cited above may require cooperative robots such as disastermanagement [32] or parcel delivery using trucks and drones [13, 14, 15].
3. MTSP Definition and Variants
MTSP is widely studied and was originally defined as which, given a setof cities, one depot, m salesmen and a cost function (e.g. time or distance),MTSP aims to determine a set of routes for m salesmen minimizing the totalcost of the m routes, such that, each route starts and ends at the depot andeach city is visited exactly once by one salesman.MTSP has been applied to different application domains, which gave riseto new variants of this optimization problem. In this section, we analyze thedifferent variants of MTSP, and then we provide a formal description of themain ones. 12 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 In this paper, we extend MTSP variants presented in [9] and provide ananalysis of more metrics based on recent contributions on MTSP. The newvariants of MTSP result from considering the different characteristics of thesalesmen, the depot, the city, and the problem constraints and objectives.
Salesmen characteristics:. • The salesmen’s type: depending on the application, the travelling sales-men in MTSP can be salesmen, vehicles, robots, or Unmanned AerialVehicles (UAVs) known also as drones.Note: In what follows, we use the term salesmen, vehicles and robotsindifferently. Also, the term UAVs, drones, and flying vehicles are usedindifferently. • The number of salesmen m is strictly greater than 1; otherwise, theproblem is the same as TSP. m can be fixed or to be determined byMTSP. • Cooperative salesmen: several salesmen may cooperate to accomplishthe given mission. They could be of the same vehicle type, or they maybe combined, such as for delivery applications where trucks and droneswork in tandem to deliver parcels to customers.
Depot specifications:. • Single depot vs. Multiple depots: in the standard version of MTSP,only one depot should be considered, and its position is fixed. Since13 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 several salesmen are used, the existence of multiple depots could opti-mize the cost of the tours. In such a variant of MTSP, the salesman canstart its tour from a depot and join a different depot when it finishesits mission. • Fixed vs. Mobile depot: generally, in MTSP the depot is fixed. How-ever, in some applications, the depot can also be mobile. For example,a mobile depot can be a truck from which UAVs start and end theirtours. • Closed vs. Open path: in the classical MTSP, the salesman’s path isclosed since they have to start and end their tour from/at the samedepot position. However, in some applications, the salesman does notneed to return to depot and it can stay in the last visited city. Moreover,if multiple depots are considered, the salesman can join any of theexisting depots that may be different from its initial depot. • Refueling point: when the refueling is allowed, it can be performedeither at the depots or at some additional refueling positions.
Cities specifications:. • Standard MTSP: in the standard MTSP, all salesmen share the sameworkspace, i.e, they share all cities between them. • Colored MTSP: a new variant introduced by [39] called colored trav-eling salesman problem (CTSP), where there are two groups of cities.One group shared between all salesmen and one group of cities thatneed to be visited by a specific salesman.14 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
Objective function:. • Single objective vs. Multi-objective : MTSP can be used to optimize asingle objective or multiple objectives. Moreover, the main objectivesaddressed in the literature are: – Minimizing the total cost, in terms of (cumulative) distance ortime, of all tours. – Minimizing the maximum salesman’s tour cost. – Minimizing the mission time. – Minimizing the energy consumption. – Minimizing the number of salesmen. – Minimizing the additional cost such as the cost related to refuelingstops.It is worth noting that, the energy consumption and mission completiontime are highly dependent on the distance traveled, and therefore, the mainobjectives considered in the literature are: the total distance traveled, themaximum robot’s tour, and the balance between robots’ tour length.
Problem constraints:. • Energy constraint: the salesman consumes energy when moving. Ve-hicles like trucks do not have any constraint on their energy consump-tion, whereas other vehicles such as small UAVs have limited autonomy.Hence, if the vehicle cannot perform refueling during its mission, thecomputed tour will be constrained by the limited range of this vehicle.15 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 • Capacity constraint: during its mission, the salesman may carry parcelsor data. The same as for the energy constraint, the vehicle capacityconstraint is generally considered with small vehicles such as UAVs,which can only carry small parcel and may have limited data memorystorage. • Time window constraint: it corresponds to the time interval duringwhich the salesman needs to visit a given target. In some applications,it may correspond to data latency, where, the salesman has to pick updata from one site, carried it, and then, deliver it to another site.
As previously explained, MTSP has several variants. In this section, wegive a formal description of the main ones. For this purpose, we considera set of n Points of Interests (PoIs) or targets, saying { T , ..., T n } and a setof m robots { R , ..., R m } . The robots mission is to cooperatively visit thesetargets with an optimum cost.According to the objective function, two main formulations of MTSPgenerally exist, which are [31]:1. MinSum MTSP : in this MTSP variant, the objective function con-sists in minimizing the sum of all robots’ tour costs. Formally speaking,the MinSum variant is modeled as: minimize
T our Ri ∈ T OURS ( m (cid:88) i =1 C ( T our R i ))subject to : T our R i ∩ T our R j = ∅ , ∀ i (cid:54) = j, ≤ i, j ≤ m. i = m ∪ i =1 T our R i = { T j , ≤ j ≤ n } . (1)16 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 where C ( T our R i ) is the tour cost of robot R i and T OU RS is the set ofall possible Tours. Moreover, the two conditions in Equation 1 guaran-tee that all targets are visited, and that each target is visited by onlyone robot.2.
MinMax MTSP : in this variant, the objective function consists inminimizing the longest tour cost (such as in terms of distance or time)among all robots’ tours. This objective is highly adopted by stud-ies focusing, for example, on the mission completion time. Formallyspeaking, this variant is modeled as: minimize
T our Ri ∈ T OURS ( max j ∈ ...m { C ( T our R j ) } )subject to : T our R i ∩ T our R j = ∅ , ∀ i (cid:54) = j, ≤ i, j ≤ m i = m ∪ i =1 T our R i = { T j , ≤ j ≤ n } . (2)The M inSum is mostly used when the total travelled distance or therobots’ energy consumption should be minimized, whereas, the
M inM ax ismostly used when the mission completion time must be minimized. Otherobjective functions could be derived as linear combinations of the ones above[40].Moreover, the robots tour cost C ( T our R i ) formulation varies whether therobots start from a single depot or they are initially located at differentdepots, and whether the robots need to return to their initial depot (closedpath variant) or they can stop at the last visited target (open path variant).More precisely, let C ( T i , T j ) be the cost of traveling from target T i totarget T j , and C ( T our R i ) be the tour cost of robot R i , the robots tour costin each variant can be modeled as follows:17 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Single depot, closed path MTSP : In this variant, all robots startfrom the same depot, and they return to it once finishing their mission.Therefore, the robot R i starts its tour from the depot D , then visits the listof r assigned targets { T i , ..., T i r } in that order, and finally return to D . Thetour cost of robot R i is equal to: C ( T our R i ) = C ( D, T i ) + r − (cid:88) k =1 C ( T i k , T i k +1 ) + C ( T i r , D ) (3) Single depot, open path MTSP : In this variant, the robot stop atthe last visited target. The tour cost of robot R i is equal to: C ( T our R i ) = C ( D, T i ) + r − (cid:88) k =1 C ( T i k , T i k +1 ) (4) Multiple depots, closed path MTSP :In this variant, the m robots are initially located at m different depots,saying { D , ..., D m } . Therefore, the tour cost of robot R i is equal to: C ( T our R i ) = C ( D i , T i ) + r − (cid:88) k =1 C ( T i k , T i k +1 ) + C ( T i r , D i ) (5) Multiple depots, open path MTSP : In the open path variant ofthe multiple depots MTSP, the robots do not have to return to their initialdepots and therefore, the tour cost of robot R i is equal to: C ( T our R i ) = C ( D i , T i ) + r − (cid:88) k =1 C ( T i k , T i k +1 ) (6)For more details about MTSP formal description and MTSP possibleformulations we refer the reader to [31] and [9].18 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 In what follows, we identify two broad classes of MTSP, namely: (1)MTSP for salesman, vehicles, and robots, and (2) MTSP for UAVs. In-deed, UAVs based optimization problems have different characteristics andconstraints than classical vehicles and robots.
4. MTSP Approaches
As its name suggests, MTSP was originally designed to optimize travel-ing salesman problems. It was then generalized to handle optimization tasksof ground vehicles or robots. However, over recent decades, UAVs, the newflying vehicle, have emerged. UAVs were first used in dangerous militarymissions to ensure the safety of pilots. Subsequently, these flying vehicleshave attracted a great deal of interest in various civilian applications, andtheir use continues to grow. In the literature, several recent studies haveproposed different approaches to solve MTSP. Most of the approaches pro-posed for flying vehicles are extended approaches of those applied for groundvehicles while considering UAVs constraints such as energy consumption ortheir limited carrying capacity.In this section, we propose to study and analyse the different approachesapplied to solve MTSP for ground vehicles (including salesmen and robots)and flying vehicles separately, in order to help the reader to understand theissues of each type of vehicles and to guide him/her in choosing the mostappropriate approach to his/her optimization problem. These approachesare shown in Figure 3. 19 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
MTSPApproachesDeterministic Meta-heuristic Market-based OthersBranch-and-cutConstraintProgramming..etc GAPSOACOABCTabu Search CentralAuctioneerDistributedAuctioneer Game TheoryFuzzy LogicAHPMemeticSAA..etc
Figure 3: MTSP Approaches Taxonomy
This section is dedicated to review existing contributions that focused onMTSP for ground vehicles, robots, or salesmen in the general context. Wehave classified these solutions according to the used optimization approaches,as shown in Figure 3, and they are as following:
Also known as exact methods which are able to reach to optimal solu-tion for a given optimization problem. In general, these methods are verytime-consuming due to the complexity of the calculations. Therefore, exactapproaches are adopted in only few papers. The authors in [41], transformedthe Multiple Depot Multiple Traveling Salesman Problem (MDMTSP) into20 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 a Single Depot Asymmetric TSP problem. In this paper, the cost of trav-eling between targets must satisfy the triangle inequality. To transform theproblem from a multiple depot to a single depot MTSP, a set of extra nodesis added such that each new node is considered as a depot. Once the trans-formation is done, standard TSP exact methods are applied to solve theproblem.MDMTSP with heterogeneous vehicles has also been solved in [42] usingan exact algorithm. The authors first introduced the integer linear program-ming (ILP) formulation. Then, they proposed a customized branch-and-cutalgorithm that reached the optimal solution within 300 seconds using aninstance of 100 targets and 5 vehicles.Another study in [43] used constraint programming (CP) to formulateand to optimally solve MTSP by applying global constraints, interval vari-ables, and domain filtering algorithms. However, the proposed approach istime-consuming since the execution time was more than 2 hours to solve aninstance of 51 cities and 3 salesmen.
The most used meta-heuristic algorithms in the literature are; GeneticAlgorithm (GA), Ant Colony Optimization (ACO), Practical Swarm Opti-mization (PSO), and Artificial Bee Colony (ABC) techniques.
GA based approaches . The principle of GA techniques is based on thenatural selection and genetics to generate the optimum solution from previousgeneration. The process starts from an initial random solutions (generations).Then, a fitness function is computed to evaluate the performance of the so-21 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 lutions at each iteration. After that, the process selects two parent solutionsand computes a crossover/mutation operation to produce two new solutionswhich will be inserted in the next generation. If the computed child solu-tions have better fitness values, they will replace the parent solutions. Thisprocess of selection, crossover, and mutation is repeated until the new gener-ation reaches the population size. This completes one iteration (generation).The algorithm continues until the number of certain generations (decided byuser) is reached or the solution quality is no longer improvedIn the GA approach a solution is represented as a chromosome. Thereare different chromosome representation techniques. The main ones used inthe literature are summarized in Figure 4.The authors in [45] proposed a two-part chromosome coding (Figure 4d)and developed a new crossover method. The proposed crossover method wascompared to existing ones, such as the ordered crossover operator (ORX), thecycle crossover operator (CX), and the Partial matched crossover operator(PMX). The considered objectives are optimizing both the total distancetraveled and the maximum tour length. The performance evaluation provedthe efficiency of the proposed crossover method in generating a better solutionquality.The authors in [46] aims to build a genetic algorithm solution to solveMTSP. They first compared six different crossover operators, namely the cy-cle crossover (CX), partial matched crossover, order crossover (OX), edgerecombination crossover (ERX), alternating position crossover (AEX), se-quential constructive crossover (SCX). Experimental analysis on TSPLIBbenchmark instances [47] of various sizes showed the performance of the22 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.2998539, IEEE Access
Author Name: Preparation of Papers for IEEE Access (February 2017)
2 VOLUME XX, 2017
II. Basic theory
As mentioned earlier, this paper mainly deals with MTSP in the case of multiple start depots and closed paths. Considering the shortcomings of Reference [25], we introduced a reproduction mechanism to improve the solution performance.
A. MTSP WITH MULTIPLE DEPOTS AND CLOSED PATHS
The MTSP we considered can be briefly described as follows: Give an undirected graph , G V A = ( ) , which is an ordered pair , G V A = ( ) comprising a set V of vertices, together with a set A of arcs. Let m represent the total number of salespersons. The objective function is to partition V into m nonempty subsets mi i S = { } , and find a minimum cost circuit passing through each vertex of each subset i S exactly once. The MTSP objective function is as follows: Minimize ( ) ii m ni ij jni j x x − += = + (1) The first sum represents cycling through the m salespersons. The second sum represents cycling through the total cities that the ith salesperson has visited (the index of the first city that the ith salesperson has visited is 1, and the last city index is i n ). , 1 ij j x + indicates the distance between city j and j + that the ith salesperson visited. ,1 i in x indicates the distance between the last city i n and the first city that the ith salesperson visited. The value of i n should not be less than the specified minimum number of cities for each salesperson. , 1 ij j x + is equal to +1, ij j x . B. PGA FOR THE MTSP
It is very important to design a better chromosome representation method based on the problem in genetic algorithm. A great genetic algorithm chromosome design should be able to minimize or eliminate redundant solutions from candidate solutions. The emergence of redundant solutions not only enlarges the understanding space, but also reduces the search efficiency. When using genetic algorithm to solve MTSP, there are two common forms of chromosome design, called "one-chromosome" [15] and "two-chromosome" [16,17]. We set n as the number of cities and m as the number of traveling salespersons and encode n cities to integers from 1 to n , then their forms are as follows: ( one-chromosome ) ( two-chromosome ) FIGURE 1. Two traditional encoding methods
The first part of
Fig. 1 . is a one-chromosome with 12 cities and 4 travelers. The design of a one-chromosome is to add m − virtual points, such as -1, -2, -3 above. By the calculation, we can see that the solution space of one-chromosome is ( 1)! n m + − , but there are many redundant solutions, such as the same solution expressed by the path of any two travelers. The second part of Fig. 1 is a two-chromosome with the same number of cities and travelers. Two-chromosome is designed to represent the traveling salesperson's path with one chromosome and the corresponding city-affiliated traveling salesperson with another chromosome. The calculation shows that the solution space is ! n n m . The redundant solution and solution space of two chromosome were significantly larger than one-chromosome design. In order to reduce the size of solution space and eliminate or reduce the number of redundant solutions, now we generally use a two-part chromosome coding based on breakpoint sets [18], as shown follow: (
Two-part chromosome ) (
Two-part chromosome based on breakpoint sets ) FIGURE 2. New coding method and its improvement
The first part of
Fig. 2 is a two-part chromosome. The first part is the path of n cities, and the second part is the number of m cities each traveler passes through. The calculation shows that the solution space is ! mn n C −− .The order of magnitude of the two-part chromosome solution space is significantly smaller than the traditional chromosome, and the number of redundant solutions is also less. Now we often use this way of coding base on salesman1 salesman2 salesman3 salesman4 Route
Belongs to1 1 1 3 42 2 3 4 2 3 4
Route salesman1 salesman2 salesman3 salesman4
Route
Break (a) Example of the one chromosome encoding
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.2998539, IEEE Access
Author Name: Preparation of Papers for IEEE Access (February 2017)
2 VOLUME XX, 2017
II. Basic theory
As mentioned earlier, this paper mainly deals with MTSP in the case of multiple start depots and closed paths. Considering the shortcomings of Reference [25], we introduced a reproduction mechanism to improve the solution performance.
A. MTSP WITH MULTIPLE DEPOTS AND CLOSED PATHS
The MTSP we considered can be briefly described as follows: Give an undirected graph , G V A = ( ) , which is an ordered pair , G V A = ( ) comprising a set V of vertices, together with a set A of arcs. Let m represent the total number of salespersons. The objective function is to partition V into m nonempty subsets mi i S = { } , and find a minimum cost circuit passing through each vertex of each subset i S exactly once. The MTSP objective function is as follows: Minimize ( ) ii m ni ij jni j x x − += = + (1) The first sum represents cycling through the m salespersons. The second sum represents cycling through the total cities that the ith salesperson has visited (the index of the first city that the ith salesperson has visited is 1, and the last city index is i n ). , 1 ij j x + indicates the distance between city j and j + that the ith salesperson visited. ,1 i in x indicates the distance between the last city i n and the first city that the ith salesperson visited. The value of i n should not be less than the specified minimum number of cities for each salesperson. , 1 ij j x + is equal to +1, ij j x . B. PGA FOR THE MTSP
It is very important to design a better chromosome representation method based on the problem in genetic algorithm. A great genetic algorithm chromosome design should be able to minimize or eliminate redundant solutions from candidate solutions. The emergence of redundant solutions not only enlarges the understanding space, but also reduces the search efficiency. When using genetic algorithm to solve MTSP, there are two common forms of chromosome design, called "one-chromosome" [15] and "two-chromosome" [16,17]. We set n as the number of cities and m as the number of traveling salespersons and encode n cities to integers from 1 to n , then their forms are as follows: ( one-chromosome ) ( two-chromosome ) FIGURE 1. Two traditional encoding methods
The first part of
Fig. 1 . is a one-chromosome with 12 cities and 4 travelers. The design of a one-chromosome is to add m − virtual points, such as -1, -2, -3 above. By the calculation, we can see that the solution space of one-chromosome is ( 1)! n m + − , but there are many redundant solutions, such as the same solution expressed by the path of any two travelers. The second part of Fig. 1 is a two-chromosome with the same number of cities and travelers. Two-chromosome is designed to represent the traveling salesperson's path with one chromosome and the corresponding city-affiliated traveling salesperson with another chromosome. The calculation shows that the solution space is ! n n m . The redundant solution and solution space of two chromosome were significantly larger than one-chromosome design. In order to reduce the size of solution space and eliminate or reduce the number of redundant solutions, now we generally use a two-part chromosome coding based on breakpoint sets [18], as shown follow: (
Two-part chromosome ) (
Two-part chromosome based on breakpoint sets ) FIGURE 2. New coding method and its improvement
The first part of
Fig. 2 is a two-part chromosome. The first part is the path of n cities, and the second part is the number of m cities each traveler passes through. The calculation shows that the solution space is ! mn n C −− .The order of magnitude of the two-part chromosome solution space is significantly smaller than the traditional chromosome, and the number of redundant solutions is also less. Now we often use this way of coding base on salesman1 salesman2 salesman3 salesman4 Route
Belongs to1 1 1 3 42 2 3 4 2 3 4
Route salesman1 salesman2 salesman3 salesman4
Route
Break (b) Example of the two chromosomes encoding
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.2998539, IEEE Access
Author Name: Preparation of Papers for IEEE Access (February 2017)
2 VOLUME XX, 2017
II. Basic theory
As mentioned earlier, this paper mainly deals with MTSP in the case of multiple start depots and closed paths. Considering the shortcomings of Reference [25], we introduced a reproduction mechanism to improve the solution performance.
A. MTSP WITH MULTIPLE DEPOTS AND CLOSED PATHS
The MTSP we considered can be briefly described as follows: Give an undirected graph , G V A = ( ) , which is an ordered pair , G V A = ( ) comprising a set V of vertices, together with a set A of arcs. Let m represent the total number of salespersons. The objective function is to partition V into m nonempty subsets mi i S = { } , and find a minimum cost circuit passing through each vertex of each subset i S exactly once. The MTSP objective function is as follows: Minimize ( ) ii m ni ij jni j x x − += = + (1) The first sum represents cycling through the m salespersons. The second sum represents cycling through the total cities that the ith salesperson has visited (the index of the first city that the ith salesperson has visited is 1, and the last city index is i n ). , 1 ij j x + indicates the distance between city j and j + that the ith salesperson visited. ,1 i in x indicates the distance between the last city i n and the first city that the ith salesperson visited. The value of i n should not be less than the specified minimum number of cities for each salesperson. , 1 ij j x + is equal to +1, ij j x . B. PGA FOR THE MTSP
It is very important to design a better chromosome representation method based on the problem in genetic algorithm. A great genetic algorithm chromosome design should be able to minimize or eliminate redundant solutions from candidate solutions. The emergence of redundant solutions not only enlarges the understanding space, but also reduces the search efficiency. When using genetic algorithm to solve MTSP, there are two common forms of chromosome design, called "one-chromosome" [15] and "two-chromosome" [16,17]. We set n as the number of cities and m as the number of traveling salespersons and encode n cities to integers from 1 to n , then their forms are as follows: ( one-chromosome ) ( two-chromosome ) FIGURE 1. Two traditional encoding methods
The first part of
Fig. 1 . is a one-chromosome with 12 cities and 4 travelers. The design of a one-chromosome is to add m − virtual points, such as -1, -2, -3 above. By the calculation, we can see that the solution space of one-chromosome is ( 1)! n m + − , but there are many redundant solutions, such as the same solution expressed by the path of any two travelers. The second part of Fig. 1 is a two-chromosome with the same number of cities and travelers. Two-chromosome is designed to represent the traveling salesperson's path with one chromosome and the corresponding city-affiliated traveling salesperson with another chromosome. The calculation shows that the solution space is ! n n m . The redundant solution and solution space of two chromosome were significantly larger than one-chromosome design. In order to reduce the size of solution space and eliminate or reduce the number of redundant solutions, now we generally use a two-part chromosome coding based on breakpoint sets [18], as shown follow: (
Two-part chromosome ) (
Two-part chromosome based on breakpoint sets ) FIGURE 2. New coding method and its improvement
The first part of
Fig. 2 is a two-part chromosome. The first part is the path of n cities, and the second part is the number of m cities each traveler passes through. The calculation shows that the solution space is ! mn n C −− .The order of magnitude of the two-part chromosome solution space is significantly smaller than the traditional chromosome, and the number of redundant solutions is also less. Now we often use this way of coding base on salesman1 salesman2 salesman3 salesman4 Route
Belongs to1 1 1 3 42 2 3 4 2 3 4
Route salesman1 salesman2 salesman3 salesman4
Route
Break (c) Example of the two-part chromosome encoding (with breaks)
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.2998539, IEEE Access
Author Name: Preparation of Papers for IEEE Access (February 2017)
2 VOLUME XX, 2017
II. Basic theory
As mentioned earlier, this paper mainly deals with MTSP in the case of multiple start depots and closed paths. Considering the shortcomings of Reference [25], we introduced a reproduction mechanism to improve the solution performance.
A. MTSP WITH MULTIPLE DEPOTS AND CLOSED PATHS
The MTSP we considered can be briefly described as follows: Give an undirected graph , G V A = ( ) , which is an ordered pair , G V A = ( ) comprising a set V of vertices, together with a set A of arcs. Let m represent the total number of salespersons. The objective function is to partition V into m nonempty subsets mi i S = { } , and find a minimum cost circuit passing through each vertex of each subset i S exactly once. The MTSP objective function is as follows: Minimize ( ) ii m ni ij jni j x x − += = + (1) The first sum represents cycling through the m salespersons. The second sum represents cycling through the total cities that the ith salesperson has visited (the index of the first city that the ith salesperson has visited is 1, and the last city index is i n ). , 1 ij j x + indicates the distance between city j and j + that the ith salesperson visited. ,1 i in x indicates the distance between the last city i n and the first city that the ith salesperson visited. The value of i n should not be less than the specified minimum number of cities for each salesperson. , 1 ij j x + is equal to +1, ij j x . B. PGA FOR THE MTSP
It is very important to design a better chromosome representation method based on the problem in genetic algorithm. A great genetic algorithm chromosome design should be able to minimize or eliminate redundant solutions from candidate solutions. The emergence of redundant solutions not only enlarges the understanding space, but also reduces the search efficiency. When using genetic algorithm to solve MTSP, there are two common forms of chromosome design, called "one-chromosome" [15] and "two-chromosome" [16,17]. We set n as the number of cities and m as the number of traveling salespersons and encode n cities to integers from 1 to n , then their forms are as follows: ( one-chromosome ) ( two-chromosome ) FIGURE 1. Two traditional encoding methods
The first part of
Fig. 1 . is a one-chromosome with 12 cities and 4 travelers. The design of a one-chromosome is to add m − virtual points, such as -1, -2, -3 above. By the calculation, we can see that the solution space of one-chromosome is ( 1)! n m + − , but there are many redundant solutions, such as the same solution expressed by the path of any two travelers. The second part of Fig. 1 is a two-chromosome with the same number of cities and travelers. Two-chromosome is designed to represent the traveling salesperson's path with one chromosome and the corresponding city-affiliated traveling salesperson with another chromosome. The calculation shows that the solution space is ! n n m . The redundant solution and solution space of two chromosome were significantly larger than one-chromosome design. In order to reduce the size of solution space and eliminate or reduce the number of redundant solutions, now we generally use a two-part chromosome coding based on breakpoint sets [18], as shown follow: (
Two-part chromosome ) (
Two-part chromosome based on breakpoint sets ) FIGURE 2. New coding method and its improvement
The first part of
Fig. 2 is a two-part chromosome. The first part is the path of n cities, and the second part is the number of m cities each traveler passes through. The calculation shows that the solution space is ! mn n C −− .The order of magnitude of the two-part chromosome solution space is significantly smaller than the traditional chromosome, and the number of redundant solutions is also less. Now we often use this way of coding base on salesman1 salesman2 salesman3 salesman4 Route
Belongs to1 1 1 3 42 2 3 4 2 3 4
Route salesman1 salesman2 salesman3 salesman4
Route
Break (d) Example of the two-part chromosome encoding (with cities per salesman)
Figure 4: Example of chromosome representation (12 Cities and 4 salesmen) in GA solu-tions [44] he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 crossover methods. Moreover, the experimental study showed that the se-quential constructive crossover outperforms other crossover methods.Recently, some studies focused on Partheno Genetic Algorithms (PGA) [48,44]. The study in [48] introduced two partheno genetic algorithms. The firstone is PGA with roulette and elitist selection, and it proposed four new typesof mutation operations. The second one is called IPGA, and it proposed tobind together selection and mutation, where a wider mutation operator areused. The authors used a sequence encoding method to describe the realpopulation by considering robot’s route and breakpoint in the chromosome(Figure 4c). This chromosome is divided into parts: the first part repre-sents the robot path and the second part corresponds to the breakpointportion. The solutions were compared with a PSO solution for based onspecific TSPLIB benchmarks [47]. Simulation results showed that IPGA hasthe best performance.The authors in [44], studied the advantages and drawbacks of PGA insolving MDMTSP and reported the defect resulting from the lack of localinformation of individuals in the population. To resolve this defect, the au-thors proposed to integrate the reproduction mechanism in the Invasive WeedAlgorithm (IWO). The resulting algorithm, called RIPGA for the ImprovedPartheno-Genetic algorithm with Reproduction mechanism, was comparedwith GA solutions to prove its efficiency in avoiding local convergence.Some real applications need to optimize more than one criteria such asminimizing the robot’s energy consumption, the mission completion time,etc. Therefore, the approach used to solve MTSP may need to optimizeseveral objectives simultaneously. This kind of problem is known as the24 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Multi-objective MTSP (MOMTSP). Indeed, the authors in [49] used theNon-dominated Sorting Genetic Algorithm (NSGA-II) to provide a solutionto the multi-objective MTSP. The solution aims to optimize two objectives,namely: minimizing the total distance traveled and balancing the travelingtimes between the salesmen. The solution computes a set of non-dominatedsolutions. However, the authors did not explain how to compute the travelingtime.One more study in [50] proposed a multi-objective optimization problemcalled Multi-Robot Deploying wireless Sensor nodes problem (MRDS), wheremultiple robots have to deploy sensor nodes in given positions. In MRDSthree objectives are considered; minimizing the mission time (i.e. the sensorsdeployment time), minimizing the number of used robots, and balancingrobots tours time. The authors solved MRDS problem based on the NSGA-II algorithm.
PSO based approaches . The particle swarm optimization approach is oneof the best-known meta-heuristics and has many similarities with geneticalgorithms. PSO first initiates a population of random solutions, then itupdates the generation until an optimum is reached. However, unlike thegenetic algorithm, PSO does not use crossover or mutation operations toevolve the population. In PSO, possible solutions, known as particles andcharacterized by their velocity, move through the problem space followingthe current optimal particles.The particle swarm optimization approach was applied in the study pro-posed in [51]. In this study, the authors addressed the problem of coopera-tive multi-robot task assignment and formulated it as MTSP. The solution25 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 aims to minimize both the total distance traveled and the maximum robot’stour cost. The authors extended the standard PSO to deal with severalobjectives. For that purpose, the author proposed two strategies, namelya Pareto front refinement strategy, which deletes inferior solutions, and aprobability-based leader selection strategy. The authors compared the pro-posed approach with well known existing multi-objective approaches suchas OMOPSO[52], SPEA2[53], NSGA-II[54], and SMPSO[55] and proved itssuperiority.In the same context, the authors in [56] also addressed the MRTA prob-lem and proposed DDPSO for dynamic and distributed PSO. The solutionconsists of two phases. In the first one, it groups tasks into clusters. Then,in the second phase, it assigns clusters to robots. The proposed solution wascompared to distributed PSO and GA.
ACO based approaches . The Ant colony optimization approach is a population-based meta-heuristic to solving combinatorial optimization problems. ACOis inspired by the capability of real ant colonies to efficiently organize thefood-seeking behavior of the colony by using chemical pheromone-tracks tocommunicate between ants. In the population, each individual is an artificialagent that progressively and stochastically constructs a solution to a givenoptimization problem. In the following, we review contributions based onthe ACO approach to solve MTSP.The authors in [57] have modeled the task assignment problem for multi-ple unmanned underwater vehicles as a constrained MTSP. The solution aimsto minimize the total traveled distance and the total turning angle, whichresult in minimizing the vehicles energy consumption. Moreover, the solution26 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 enforces that the number of targets per vehicle must not exceed one. Theproposed solution consists in two steps. First, the solution determines thenumber of targets to be assigned to each vehicle. Then, a proposed MultipleAnt Colonies System (MACS) method was used to solve the multi-objectiveMTSP. Experimental results proved that the proposed Multiple ACS outper-forms the classical ACS.In [58], the authors addressed the bi-criteria MTSP and adapted the ACOto solve the multi-objective single depot MTSP.In [59], the authors proposed the mission-oriented ant team ACO (MOAT-ACO) algorithm. The MOATACO algorithm aims to minimize the total dis-tance and to achieve load balance. In this solution, the authors endowed antswith directional awareness and loyal-to-duty characteristics, and proposed amission pheromone to mimic a biological instinct. As a consequence, thisrelieved the entanglement between ants tours, and thus minimizes one of theobjective, namely the total distance. To reach the second objective namelythe load balance between tour, an ant firing rule was introduced, which allowsthe slowest ant to join harder working ones.The authors in [60] addressed the MRTA problem and proposed an ACObased algorithm for multi-objective MTSP. The ACO method was inte-grated with a sequential variable neighborhood descent to further improvethe Pareto set solutions locally. The solution aims to minimize both thetotal distance traveled and the maximum tour length simultaneously, andwas compared to NSGA-II, MOEA/D-ACO [61] and FL-MTSP [62]. Al-though the solution outperforms these methods in terms of solution quality,it is slower than FL-MTSP. The authors in [63] addressed the MTSP with27 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 time window and proposed a hybrid solution by integrating ACO with theminimum spanning 1-tree to provide the optimal solution.
ABC based approaches . Like the ACO approach, the artificial bee colonyis an optimization method focused on the honey bee swarm ’s intelligentfood-seeking behaviour. ABC is also a population-based method in whichthe location of a food source (i.e. population) is a potential solution tothe optimization problem and the quantity of nectar from a food sourcerepresents the fitness of the associated solution.Venkatesh et al. [64] used the ABC algorithm to solve the single depotMTSP aiming to minimize the total traveled distance (MinSum), and themaximum traveled distance (MinMax). The authors proposed also to use alocal search to improve the obtained results. The same authors addressed thecolored MTSP in [65] and also proposed an ABC based solution. A furtherstudy on the colored MTSP was addressed in [66]. The authors modifiedthe ABC algorithm and introduced the generating neighbourhood solution(GNS) to solve MTSP.
Hybrid approaches . In the literature, several studies have proposed hybridalgorithms that combine different meta-heuristics and techniques to moreeffectively solve the multiple traveling salesman problem.For instance, the study in [61] proposed a hybrid approach combining theACO algorithm with the multi-objective evolutionary algorithm based ondecomposition (MOEA/D) to resolve the multi-objective MTSP. The ideaof the solution is to divide the multi-objective MTSP into several mono-objective sub-problems. Then, each mono-objective sub-problem is assigned28 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 to an ant. The ants are organized into groups, and each one has several neigh-boring ants. Each group is associated with a pheromone matrix, and eachant has a heuristic knowledge matrix. Moreover, each ant is charged withseeking the optimal solution to its assigned sub-problem. For that reason,the ant utilizes its heuristic knowledge matrix, its group pheromone matrix,and its current solution. The critical problems surrounding this method arethe ambiguity of time convergence and the difficulty of implementation.A new hybrid algorithm for large scale MTSP was proposed in [67]. Theproposed solution, called AC-PGA for Ant Colony-PGA, is obtained by in-tegrating PGA and ACO. More precisely, the algorithm first utilizes PGA todetermine the best value of the salesmen’s depots and the number of cities tobe visited by each salesman. Then, it exploits ACO to compute the shortestpath for each salesman.Further study in [68] addressed the scheduling and routing of caregiversin a home health-care problem and formulated it as an MTSP with timewindow. They proposed a hybrid method combining ACO with memeticalgorithm [69]. The solution aims to minimize the total traveling time andto balance the working time of caregivers (salesman).
Market-based approaches, also known as auction-based, can be eithercentralized with a central auctioneer that receives bids and assigns the citiesto the salesman with the lowest cost, or distributed where the bidding processis shared between the different salesmen.The authors in [70] proposed a market based distributed and dynamicalgorithm for MTSP, where the salesmen are robots. Thus the problem is29 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 called Multiple Traveling Robots Problem (MTRP). In this solution, eachrobot chooses its own targets in a progressive and distributed manner, as fol-lows. Firstly, each robot uses the shortest distance as a cost function to selectthe appropriate targets. Then, it declares a single-item auction of its targetvisiting schedule. Choosing the best robot for a given task is done thanks toan auction based protocol called CNP (for Contract Net Protocol). The re-sults of the Webots simulation showed that the proposed solution is efficientin terms of scalability, total path length, and communication overhead.In [71], the multi-robot task assignment problem (MRTA) was formulatedas MTSP, and solved using the K-means clustering technique with an auctionprocess. The solution aims to optimize the total distance traveled and tobalance the robots tours’ length. First, n groups of tasks are formed usingthe K-means algorithm, where n is the number of robots. Then, each robotcomputes the cost of visiting each cluster formed in the previous step. Finally,in the auction step, the robots bid on clusters, and each cluster is assignedto the robot with the lowest cost. However, the algorithm’s complexity isfairly high because it considers all possible combinations of cluster-robotsassignments. Therefore this approach might not be appropriate to solvelarge scale instances.Inspired by the Consensus Based Bundle Algorithm (CBBA) [72], andthe Market Based Approach with Look-ahead Agents (MALA) [73], the au-thors of [74] introduced a market-based solution for MTSP. The solution isan repetitive market procedure, where robots perform the following stepsin each iteration: market auction, agent-to-agent trading, agent switch, andagent relinquish step. Every robot selects the best tasks in the market auc-30 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 tion stage, based on the objective cost. Robots arbitrarily examine the tasksof other robots in the agent-to-agent trade step to verify if they can exe-cute any of these tasks at a lower price. During the agent switch phase, thealgorithm attempts to explore solutions that are not in the minimum of lo-cality. The algorithm ends after a number of iterations without performanceimprovement.Cheikhrouhou et al. proposed a market-based approach, called Move-and-Improve, in [75, 76]. The proposed solution involves the robots cooperatingin allocating targets and incrementally eliminating possible overlap. Theconcept is simple: a robot moves and tries to optimize its solution at everystep while communicating with its neighbours. Move-and-Improve approachconsists of four main phases: (1) initial target allocation, (2) tour construc-tion, (3) negotiation of conflicting targets, (4) solution improvement. Move-and-Improve was simulated using MATLAB and the Webots simulator anddeployed on real robots using the Robot Operating System (ROS).Another clustering market-based algorithm (CM-MTSP) [77] was pro-posed for the multi-objective MTSP. The solution consists of three steps,namely: clustering, auctioning, and improvement. In the clustering step, acentral server group targets into clusters using k-means. Then, the serverannounces the already formed cluster one by one and robots bids on eachcluster by computing the cost and sends it to the server. The server finallyassigns the cluster to the robot with the lowest cost. The improvement stepaims to optimize another objectives, namely the maximum distance traveledand the mission completion time. This improvement is achieved through thepermutation of clusters between robots. In this algorithm, the authors assign31 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 a cluster to each robot. However, varying the number of clusters might leadto better results. Moreover, the authors tackled the multi-objective problemin a simultaneous way that might not necessarily give the best result. In what follows, we review some MTSP contributions that adopted dif-ferent techniques such as probability, game theory, fuzzy logic, analyticalhierarchy process, etc.To solve the MDMTSP, the authors introduced a method using proba-bility collectives in [78], in which the vehicles are represented as autonomousagents and vehicles route as a strategy.Khoufi et al. [79] proposed a multi-objective optimization problem todetermine the robots tours responsible for collecting data from wireless sensornodes and delivering this data to the depot. The proposed optimizationproblem should meet some constraints such as the data delivery latency, therobots energy, and the limited number of robots. This optimization problemis solved based on game theory approach. The proposed theory game is acoalition formation game that optimizes the maximum tour time, the numberof robots, and balance the robots tours.To address the multi-objective MTSP, a fuzzy logic-based solution (FL-MTSP) was introduced by Trigui et al. [62]. The solution considers twoobjectives, MinSum and MinMax. The FL-MTSP solution was comparedwith a GA based solution [34] to prove its efficiency.Recently, Cheikhrouhou et al. [80, 31] proposed AHP-MTSP, an Analyt-ical Hierarchy Process (AHP) based approach. AHP-MTSP first computes aweight for each considered objectives. These weights are computed based on32 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 user preferences and using the AHP method [81]. Then, the different objec-tives are aggregated into a single function, as a sum of the different weightedobjectives. The comparison of AHP-MTSP to several methods includingFL-MTSP [62] and CM-MTSP [77] proves its superiority.Inspired by the work of [45], the authors in [82] suggested a ModifiedTwo-Part Wolf Pack Search (MTWPS) method revised by the two-part chro-mosome encoding method and the transposition and extension operation forsolving MTSP. The solution aims to minimize both the total distance traveledand the maximum tour.Venkatesh et al. [64] proposed a meta-heuristic approach based on aninvasive weed optimization algorithm. To further improve the solution, alocal search method was also used.Table 1 summarizes the different discussed MTSP solutions proposed forground vehicles and robots.
Table 1: Summary of MTSP Solutions for Ground Vehicles and Robots.
Class Ref. Objectives Techniques DescriptionExact [41] MinSum Transformationto TSP The authors transformed a MultipleDepot, MTSP into a Single DepotAsymmetric TSP if the cost of theedges satisfy the triangle inequality.[42] MinSum branch-and-cutalgorithm The authors proposed an ILP formu-lation and then used a customizedbranch-and-cut algorithm[43] MinSum Constraint Pro-gramming The authors formulated and solvedMTSP using CPMeta-heuristic [45] MinSum andMinMax GA The authors used a two-part chromo-some encoding and introduced a newcrossover operator he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 1: Summary of MTSP Solutions for Ground Vehicles and Robots.
Class Ref. Objectives Techniques Description[46] MinSum GA The study Considered six differentcrossover operators separately in orderto find optimal solutions[48] MinSum andMinMax Partheno GA Two partheno genetic algorithms(PGA). • PGA with roulette selection andelitist selection • IPGA, that binds together selectionand mutation.[44] MinSum Partheno GA The authors combine PGA with IWOand proposed RIPGA[49] • MinSum • the salesmen’sworking times NSGA-II A multi-objective NSGA-II.[50] • MinMax • the number ofrobots • the standarddeviation be-tween tours. NSGA-II A multi-objective MTSP called Multi-Robot Deploying wireless Sensor nodesproblem (MRDS), where multiplerobots have to deploy sensor nodes ina given positions.[51] MinSum andMinMax PSO The algorithm extends the standardPSO to support multiple objectives.[38] MinSum PSO The algorithm groups tasks into clus-ter and then assign cluster to robots[57] • MinSumthe total turningangle • While balanc-ing the number oftargets assignedto each vehicle ACO The algorithm first specifies the num-ber of task then solve MTSP usingACO.[58] • MinSum • BalancingRobots’ tours ACO The algorithm used ACO to generatepareto front solutions. he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 1: Summary of MTSP Solutions for Ground Vehicles and Robots.
Class Ref. Objectives Techniques Description[59] MinSum ACO Four techniques were introduced inthe search process of the ant teams:mission pheromone, path pheromone,greedy factor, and Max–Min ant firingscheme[60] • MinSum • MinMax ACO The ACO method was integrated witha sequential variable neighborhood de-scent to further improve the Pareto setsolutions locally.[63] MinSum ACO The ACO was improved by the min-imum spanning 1-tree to solve MT-SPTW.[64] • MinSum • MinMax ABC The authors solved SDMTSP by ABCand local search.[65] MinSum ABC The authors addressed the CTSP.[66] MinSum ABC The authors solved the large scaleCTSP with ABC and GNS.[61] • MinSum • MinMax • MOEA/D • ACO The algorithm is an integration ofACO with MOEA/D[67] • MinSum • With the con-straint of mini-mum and maxi-mum number ofcities to be visited • PGA • ACO First, PGA is used to determine theoptimal salesmen’s depots and numberof cities per salesman. Then, ACO isused to compute the shortest route foreach salesman.[68] • MinSum • Balancing thetravelling Time • ACO • Memetic The authors addressed the multi-objective MTSPTW using hybrid ap-proach.Market-based [70] MinSum Contract NetProtocol (CNP) An auction-based distributed and dy-namic algorithm[71] • MinSum • Balancing theworkload betweenthe robots • K-means forclustering • Auction-basedfor allocation ofclusters To solve the balanced multi-robot taskallocation problem it was modeled asan MDMTSP he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 1: Summary of MTSP Solutions for Ground Vehicles and Robots.
Class Ref. Objectives Techniques Description[74] • MinSum • MinMax Consensus-BasedBundle Algo-rithm A market-based algorithm with itera-tive process[75,76] • MinSum • MinMax Move and im-prove The algorithm involves the coopera-tion of the robots to allocate targetsand remove possible overlap incremen-tally.[77] • MinSum • MinMax • Minimizingthe missioncompletion time Clustering The algorithm divides the locationsinto clusters and then allocates eachcluster to the best robot.Others [78] MinSum Probability col-lectives The authors used the Rosenbrock func-tion in which the coupled variablesare considered as autonomous agentsworking collectively to achieve thefunction optimum[79] • MinMaxMinimizing thenumber of robots • Maximizingthe fairness interm of tourduration Game theory A multi objective optimization prob-lem to determine the robots tours incharge of collecting data from wirelesssensor nodes and delivering this datato the depot.[62] • MinSum • MinMax Fuzzy logic The algorithm combines the objectivesinto a single fuzzy metric, transform-ing the problem to a single objectiveoptimization problem.[80,31] • MinSum • MinMax • Balancing thetour length AHP The algorithm used the AHP tech-nique to combine the objectives into aglobal function, and then optimize thisglobal function. he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 1: Summary of MTSP Solutions for Ground Vehicles and Robots.
Class Ref. Objectives Techniques Description[82] • MinSum • MinMax Two-part wolfpack search The two-part wolf pack search algo-rithm was modified by the two-partchromosome encoding approach andthe transposition and extension oper-ation
In this section, we review recent studies that used MTSP in a UAV contextand provide a classification of these MTSP solutions according to the adoptedapproach while highlighting their application areas.Table 2 summarizes the different reviewed MTSP solutions proposed forUAVs.
In transport and delivery application, the authors in [83] are the first toformally define the problem of combining an aerial vehicle with a truck forparcel delivery. Then, they extended their study in [13] to introduce themultiple flying sidekicks traveling salesman problem (mFSTSP), in which atruck and a fleet of UAVs are deployed to deliver small parcels to customers.This problem is formulated as a mixed-integer linear program (MILP) andsolved via Gurobi for small size problems.In the same context of parcel delivery, the authors in [14] were based onthe study in [83] and proposed the Multiple Traveling Salesman Problem withDrones (MTSPD), which is based on both UAVs and trucks in the last-miledelivery. The proposed model is a variant of MTSP. In this optimizationproblem, multiple drones and multiple trucks perform deliveries together.37 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
The objective of MTSPD is to minimize the arrival time of both vehicles,trucks and drones, at the depot after delivering all parcels to customers. Tosolve MTSPD, the authors proposed a mixed integer programming (MIP)formulation and obtained solution using IBM-CPLEX [84].Due to computation complexity, the optimization problems introducedin [13] and [14] were solved optimally for only small instances. However,both studies applied heuristics to solve their problem for medium and largeinstances. We reviewed these heuristics in the next section.Another study in [15] extended the problem of multiple traveling salesmanproblem proposed in [83]. In this study, a drone is in charge of dropping theparcel when it arrives at the customer and then it can either flies to anothercustomer to pickup a new parcel or return directly to the depot to start newdelivery. This problem is modeled as an unrelated parallel machine schedul-ing (PMS), and it integrates multiple depots with multiple trucks and drones,constrained by time-window, drop-pickup synchronization, and multi-visit.To solve this problem, a novel application of constraint programming ap-proach is proposed in order to minimize the maximum time needed to satisfyall delivery tasks.In the context of attacking multiple targets, the problem of coopera-tive trajectory planning integrated target assignment for multiple UnmannedCombat Aerial Vehicles (UCAV) is studied in [32]. The problem is formu-lated as dynamics-constrained, multiple depots, multiple traveling salesmanproblem with neighborhoods (DC MDMTSPN), which is a variant of MTSP.The proposed problem take into account the battlefield environment con-straints (e.g. threat avoidance) and the UCAV dynamics model (e.g. avoid38 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 collision with each other) and was solved based on a two-phases approach. Inthe first phase, the authors construct the directed graph and then transformthe original problem into asymmetric TSP (ATSP). In the second phase,they use the Lin − Kernighan Heuristic (LKH) searching algorithm to solvethe ATSP.The study in [85] introduced a problem formulation of the heterogeneousmulti-UAV task assignment problem as multiple traveling salesman problem.The proposed problem considers situations that include the traditional offlinecentralized situation and the offline with parameter uncertainty situation.The authors proposed a novel digraph-based deadlock-free algorithm as wellas a Modified Two-part Wolf Pack Search algorithm to solve the deterministicoffline problem efficiently.
Meta-heuristics have been applied to solve many NP-hard optimizationproblems. In this section, we provide a review of contributions that solvedMTSP based on a meta-heuristic such as Genetic Algorithm and Tabu Search.Notice that many studies proposed combined heuristics where the problemis solved following several phases and based on different techniques (e.g.clustering, meta-heuristic) to reduce the problem complexity.
GA based approaches . In the context of trucks and drones in the last-mile delivery, the authors in [14], solved optimally MTSPD for small sizeinstances and also proposed a new heuristic called Adaptive Insertion algo-rithm (ADI) to solve large size instances. The principle of ADI is to firstbuild an initial solution and second to improve it by applying removal and39 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 insertion operators to construct the MTSPD solution. To perform the sec-ond phase of ADI, the authors were based on the genetic algorithm, thecombined K − means/Nearest Neighbor and the random Cluster/Tour. Sim-ulation results showed that, in small size instances, both solvers and thegenetic meta-heuristic, GA − ADI, reached the optimal solution. Notice thatGA-ADI solved the MTSPD problem significantly faster than the solver IBM-CPLEX [84]. However, in large size instances, the authors used only theGA − ADI heuristic and they demonstrated the efficiency of using multipleUAVs to reduce the delivery time.The authors in [23] proposed a study addressing UAVs as DTN relays,and introduced a proactive scheme called Deadline Triggered Pigeon withTravelling Salesman Problem with Deadlines (DTP-TSP-D). In this prob-lem, UAVs can communicate with one or a cluster of nodes, pick up datafrom a ground node while considering it capacity of carrying messages fromone location to another and then hovering until being triggered to delivermessages directed to other ground node. A genetic algorithm is used todetermine the optimized UAVs tours in terms of delivery time.A similar study in [24] proposed to use a UAV as a message ferry nodethat is in charge of traveling among disconnected nodes in a DTN networkto deliver their messages. The authors introduced a multiple message ferryUAVs optimization problem which is a variant of MTSP so that the optimalpath planning minimizes the message delivery delay. To obtain the optimalpath planning solution, the genetic algorithm is adopted to solve the problemin a feasible time. In the genetic algorithm, clusters of nodes are built,each cluster will be assigned to a UAV, then the UAV’s path visiting all40 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 nodes inside a cluster is determined based on the traffic flows between nodesand a load of messages nodes in those nodes in order to optimize messagedelivery delay in the network. In a small network, this genetic algorithmprovides good solution as those obtained with an exhaustive search approachbut in shorter run time. In addition, the proposed optimization problemoutperforms the traditional MTSP solution in terms of message delivery delayin DTN.The study in [18], addressed the problem of data collection using multiplemobile sinks in large scale WSN, in order to save energy throughout thenetwork and to increase the packet delivery rate. To solve this problem,the authors proposed a two-phase heuristic. In the first phase, k clusters ofsensor nodes are formed, where k is the number of UAVs acting as mobilesinks. In the second phase, the path of each UAV is determined based on asmooth path construction (SPC) algorithm. In SPC, the genetic algorithmis used to determine each UAV’s tour visiting each sensor node inside thecluster exactly once. Then, the smooth tours are computed based on UAV’sturning constraints.When cooperative UAVs operate in hazardous areas to perform multipletasks, optimizing the number of UAVs deployed, as well as the trajectory ofeach of them, would help to accomplish the mission in a minimum amountof time. In this context, the study in [86] proposed the Multi − UAVs tomulti-tasks, task assignment and path planning problem which is a variantof MTSP that optimizes the number of UAVs for the given time constraintand task set. To solve this problem, a coordinated optimization algorithmcombining the genetic algorithm and cluster algorithm, is introduced in this41 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 paper. The authors proposed first the K-means clustering algorithm to buildclusters of multiple tasks, where each cluster will be assigned to one UAV.Second, adjacent tasks of the same cluster are grouped in order to reduce thenumber of locations to be visited by the UAV. After that, a TSP optimizationproblem will be solved based on the genetic algorithm while considering timeconstraints. The comparison evaluation between the coordinated optimiza-tion algorithm and GA showed that the proposed coordinated optimizationalgorithm is more effective than GA.In the context of monitoring applications where UAVs are in charge ofcovering multiple regions, the study in [87] proposed the Energy ConstrainedMultiple Traveling Salesman Problem for Coverage Path Planning (EMTSP-CPP). To solve this problem, the authors introduced a modified version of thegenetic algorithm while considering the new representation of the individual(chromosome), as well as a new version of crossover and mutation operationand, finally, a constraint-aware fitness function. The authors demonstratedthat the modified genetic algorithm has a better performance compared toother approaches.In [26], a multi-objective UAV path planning for search and rescue wasproposed. The proposed solution is based on genetic algorithm and aims tominimize the completion time. The solution divides the environment intocells and, then, uses MTSP to guarantee that each cell is searched by exactlyone UAV.In precision agriculture, UAVs can be deployed to spray fields with pes-ticides. In this context, the study in [88] introduced the mission assignmentand the path-planning problem with Multi-Quadcopters which are a specific42 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 type of UAVs. The proposed problem was formulated as the MTSP opti-mization with the objective is to reduce the mission completion time whileconsidering a constraint on the Quadcopter’s battery capacity limitation. Tosolve this problem, the authors proposed a hierarchal approach in which aninner-and-outer loop structure is employed. Indeed, the inner loop is basedon the genetic algorithm while the outer loop uses a nonlinear programmingmethod is based on the optimal results obtained by the inner loop. Theefficiency of the proposed approach was illustrated based on performancecomparisons to a conventional approach.
TS based approaches . The tabu search approach is a meta-heuristic basedon the local search methods and used for mathematical optimization. Localsearch methods tend to be stuck in sub-optimal solutions. Therefore, Tabusearch improves the performance of these methods by exploring the solutionspace beyond local optimality.In the field of monitoring and surveillance application, the study in [33]proposed the two-stage Fuel-Constrained Multiple-UAV Routing Problem(FCMURP). The FCMURP is a generalization of the multiple traveling sales-man problem and a variant of the vehicle routing problem for UAVs with fuelconstraints. In FCMURP, multiple depots are considered, and they also rep-resent refueling points. However, there is only one depot from where eachUAV tour starts and ends. The two-stage FCMURP works as follows: Inthe first stage, MTSP is adopted to build a tour for each UAV. However, ifany UAV does not have sufficient energy to complete its tour, it can make arefueling stop at any depot. In the second stage, additional refueling stopsare added to the first-stage tours to ensure feasibility for the achievement of43 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 the energy consumption. In the first stage of FCMURP, the objective is tominimize the sum of the distance traveled by all UAVs whereas the objectiveof the second stage is to minimize the expected traveling distance for addi-tional refueling stops. To solve FCMURP, the authors proposed the SampleAverage Approximation (SAA) approach. In small instances, the SAA ap-proach converges to optimal solutions for the two-stage model, however, inmedium and large instances, the SAA takes so much time to converge. Tocope with this problem, the authors proposed to solve FCMURP based ona tabu search-based heuristic and they showed that this heuristic provideshigh quality solutions.
In trucks and drones delivery application, the authors in [13] proposed athree-phase heuristic to solve the multiple flying sidekicks traveling salesmanproblem (mFSTSP), with 100 customers and 4 drones. In the first phase,some customers are selected to be served by the truck and others will beserved by UAVs. Then, the truck tour is determined based on the travelingsalesman optimization problem. In the second phase, each customer is as-signed to a specific UAV and both truck and UAV tours are identified. Inthe third phase, the authors solved a MILP to determine the exact timing ofthe scheduled operations for the truck and the UAVs. The flight endurancefor the heterogeneous UAVs is modeled as a function of their battery size,payload, travel distance, and flight phases. Finally, a local search proce-dure is executed to improve the solution quality. The authors showed thatfor problems of realistic size the three-phase heuristic provides high-qualitysolutions with reasonable execution time.44 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
In the context of task allocation and scheduling, the authors in [85] for-mulated the multi-UAV task assignment problem as MTSP. The presentedproblem considers various situations, including multiple consecutive tasks,heterogeneous UAVs, time-sensitive, and uncertainty. To solve this problemwith parameter uncertainty, several methods were introduced, such as robustoptimization method, the duality theory and a novel combined algorithm, in-cluding the classical interior point method and the modified two-part wolfpack search algorithm. The authors also proposed an online hierarchicalplanning algorithm to solve the online problem with the time-sensitive un-certainty. Finally, they conducted several numerical simulations and physicalexperiments to check the efficiency of their presented algorithms.Table 2 summarizes existing solutions to MTSP for UAVs.
Table 2: Summary of MTSP Solutions for UAVs
Class Ref. Objectives Techniques DescriptionExact [13] MinMax (time) Formulatedas MILP andsolved via Gurobiversion 7.0.1 A Truck and drones in the last-mile de-livery. Due to the computation com-plexity the problem was solved opti-mally for small instances.[14] MinMax (time) Formulated asMixed IntegerProgramming(MIP) and solverby Cplex solver Trucks and drones parcel delivery forlast mile. The problem was solved op-timally for small instances.[15] MinMax (time) Modeled as PMS.Constraintsprogrammingapproach Trucks and drones in the last-mile de-livery. The problem integrates: • multiple depots, • multiple trucks and drones, • time-window’s constraint, • drop-pickup synchronization, • multi-visit. he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 2: Summary of MTSP Solutions for UAVs
Class Ref. Objectives Techniques Description[32] MinMax (dis-tance) Transformationto ATSP Cooperative mission for multiple tar-get attack. The (DC MDMTSPN) issolved based on two phases: • the problem is transformed to ATSP, • the Lin − Kernighan Heuristic (LKH)searching algorithm is used to solve theATSP.UAV’s dynamic constraints are consid-ered.[85] Min (distance) • Digraph-baseddeadlock-freealgorithm • MTWPS algo-rithm Multi-UAV task allocation andscheduling. A novel digraph-baseddeadlock-free algorithm as well asModified Two-part Wolf Pack SearchMTWPS algorithm are proposed tosolve the deterministic offline problemefficiently.Meta-heuristic [14] MinMax (time) ADI heuristiccombined with: • GA, • K − means/Nearest Neigh-bor, • Random Clus-ter/Tour. Trucks and drones in the last-mile de-livery. The ADI’s heuristic builds firstan initial solution and second improvesit by applying removal and insertionoperators to construct the MTSPD so-lution.[24] Minimizing themessage deliverydelay GA Message Delivery in DTN.In the genetic algorithm: • clusters of nodes are built, • each cluster is assigned to a UAV, • each UAV’s tour is determined suchthat the message delivery delay is op-timized. he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 2: Summary of MTSP Solutions for UAVs
Class Ref. Objectives Techniques Description[23] • Maximizingthe delivery ratio • Minimizing theaverage deliverydelay. • GA • Proactiveschemes: Mes-sage Ferrying andHoming Pigeon Message Delivery in DTN. In the pro-posed scheme each UAV switches be-tween two modes: • Ferry mode: each UAV follows aferry route in its cluster of groundnodes. This route is computed basedon GA. • Pigeon mode: each UAV delivers theData collected following a route com-puted based on GA[18] • Maximizingthe delivery rate • Minimizing theenergy consump-tion. • Clustering • GA Data collection using UAVs. Two-phase Heuristic • Clusters of nodes are formed • Construction of UAVs’ smooth pathinside each cluster based on Genetic al-gorithm.[86] MinSum (dis-tance) • GA • Clustering Cooperative UAVs operating in haz-ardous areas. The problems is solvedas follows : • the K − means algorithm to buildclusters of multiple tasks, each clusteris assigned to one UAV. • adjacent tasks of the same clusterare grouped to reduce the UAV path. • GA is used to determine the UAVs’tours.[26] Minimizing thecompletion time GA A multi-objective UAV path planningfor search and rescue missions is pro-posed and solved based on the geneticalgorithm. he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 2: Summary of MTSP Solutions for UAVs
Class Ref. Objectives Techniques Description[33] • MinSum (dis-tance) • SAA • Tabu Search Monitoring and surveillance applica-tion. The two-stage FCMURP worksas follows: • first stage MTSP is adopted to builda tour for each UAV. • second stage, additional refuelingstops are added to satisfy energy con-sumption.FCMURP is solved using: • SAA for small instances • Tabu Search for meduim and largeinstances[87] • MinSum • MinMax GA Covering multiple regions using UAVs.A problem called Energy ConstrainedMTSP for Coverage Path Planning(EMTSP-CPP) is proposed and solvedusing a modified Genetic algorithm.[88] MinMax (time) Hierarchal ap-proach based onGA In precision agriculture, Multi-Quadcopters are deployed to sprayfields with pesticides. The limitedbattery capacity is considered. • A hierarchal approach using inner-and-outer loop structure • The innerloop is based on the genetic algorithm • The outer loop uses a nonlinearprogramming method he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 Table 2: Summary of MTSP Solutions for UAVs
Class Ref. Objectives Techniques DescriptionOthers [13] MinMax (time) Three-phaseheuristic Truck and drones in the last-mile deliv-ery. The problem was solved for largeinstances based on 3 phases heuristic: • phase 1: dividing customer’s deliv-ery tasks between the truck and thedrones. • phase 2: assigning each customer toa specific UAV and identifying truckand UAV tours. • phase 3: determining the schedule ofeach truck and drone’s activities.[85] MinSum (dis-tance) • Hierarchicalplanning algo-rithm • a novel com-bined algorithmincluding theclassical interiorpoint methodand the MTWPSalgorithm Multi-UAV task allocation andscheduling. The problem to solveconsiders various situations includingmultiple consecutive tasks, hetero-geneous UAVs, time-sensitive anduncertainty.
5. Taxonomy, Classification and Analysis
In this section, we first provide an extended taxonomy for MTSP, whichis based on the MTSP variants, the applied optimization approaches, andthe application domains for which the solution was proposed. After that,the previously reviewed solutions are classified according to this proposedtaxonomy. This classification presents an overview of the existing MTSPstudies which can help the readers to select the suitable MTSP variant for agiven application, as well as the approach used to solve the problem. Finally,49 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 we conclude this section by an analysis study of the reviewed papers, basedon the metrics provided in our classification.
In the following, we propose an extended taxonomy for MTSP. First, westart by enumerating the attributes considered in this taxonomy based onthree criteria, namely: MTSP’s variants, approaches, and application fields.We select the most common attributes for the MTSP variants which arepreviously described in Section 3. We also select the different approachesapplied to solve MTSP (cited in Section 4). Moreover, the different MTSPapplication fields are previously detailed in Section 2. • MTSP Variants – Salesman: ∗
1- Salesmen ∗
2- Robots ∗
3- Vehicles ∗
4- UAVs – Depot: ∗
5- Single ∗
6- Multiple ∗
7- Refueling point – Cities: ∗
8- Standard MTSP 50 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 ∗
9- Colored TSP – Problem Constraints: ∗
10- Energy ∗
11- Capacity ∗
12- Time window – Problem Objectives: ∗
13- Single-objective optimization problem ∗
14- Multi-objective optimization problem • Approaches –
15- Exact algorithm –
16- Genetic Algorithm (GA) –
17- Particle Swarm Optimization (PSO) –
18- Ant Colony Optimization (ACO) –
19- Artificial Bee Colony (ABC) –
20- Tabu Search (TS) –
21- Clustering algorithm (e.g. K − means, Nearest neighbor) –
22- Market-based –
23- Others ∗ a- Fuzzy logic ∗ b- Game theory ∗ c- ADI heuristic 51 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 ∗ d- Three-phases heuristic ∗ e- Sample Average Approximation (SAA) ∗ f- Lin − Kernighan Heuristic (LKH) ∗ g- Proactive scheme ∗ h- Probability ∗ i- Analytical Hierarchy Process ∗ j- Two-Part Wolf Pack Search ∗ k- Memetic • Applications –
24- Transportation and Delivery –
25- Data Collection –
26- Search and Rescue –
27- Precision agriculture –
28- Disaster management –
29- Monitoring and Surveillance –
30- Multi-Robot Task Allocation and scheduling –
31- Cooperative mission –
32- General context
We classify the existing studies reviewed in our survey according to thepreviously proposed taxonomy as illustrated in Table 3.52 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 R e f Y e a r V a r i a n t s A pp r oa c h e s A pp li c a t i o n s S a l e s m a n D e p o t C i t i e s C o n s t . O b j . G r o und V e h i c l e s a nd R o b o t ss o l u t i o n s [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88) h (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88) b (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88) a (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88) j (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88) i (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) k (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 R e f Y e a r V a r i a n t s A pp r oa c h e s A pp li c a t i o n s S a l e s m a n D e p o t C i t i e s C o n s t . O b j . U AV s o l u t i o n s [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) f (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) g (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) e (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) j (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) j (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) c (cid:88) [ ] (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) d (cid:88) T a b l e : A c l a ss i fi c a t i o n o f t h e r e v i e w e dp a p e r s he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 In what follows, we present a detailed analysis of these classified solutions.
Ground Vehicles71%UAVs29% (a) Vehicles types (b) Publication years
GA36%ACO18%Exact18%
MarketBased11%
ABC7%PSO4%TS2% Other27% (c) Approaches
Transportation and Delivery9% Data Collection9%Search and Rescue5%Precision agriculture2%Disaster management4%Monitoring and Surveillance4%MRTA13%Cooperative mission7%General 47% (d) Applications
Figure 5: Statistical study of the reviewed papers
In order to better understand and analyse the classification proposed inTable 3, we propose in Figure 5 a statistical study for the papers cited in oursurvey in terms of the vehicle types, the applied approaches, the applicationfields, and the publication years.We can see that 71% of the studies focusing on MTSP are in the context of55 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 ground vehicles compared with 29% for flying vehicles as shown in Figure 5a.This can be explained by the fact that MTSP studies for ground vehicles havebeen published since 2007, however, the oldest paper focusing on UAVs andcited in our survey was published in 2015 as illustrated in Figure 5b. MTSPfor ground vehicles and robots is more studied than MTSP for UAVs, sincethis flying vehicles are considered as an emerging technology and their usein civilian areas is relatively recent.Moreover, as shown in Figure 5c, the GA approaches is the most usedapproach in both types of vehicles (36% of papers have used GA). The sec-ond most used approaches are exact and ACO approaches (used by 18% ofpapers). Then, it comes the Market-based approach (used by 11% of papers).Moreover, as shown in Table 3, for the ground vehicles context, the ma-jority of papers consider the multiple depots variant, however, for the UAVscontext the majority of papers consider the single depot variant of MTSP.Indeed, the number of depots used depends on the application domain andalso on the type of vehicles. For example, UAVs have a limited autonomycompared with ground vehicles, and these UAVs are generally deployed in anon-extended area. This justifies the need of a single depot to charge andlaunch these UAVs. Furthermore, this single depot can be mobile in order tooptimize both the scheduled time and the energy of the UAVs, for exampleby using trucks and UAVs for the last mile delivery as proposed in [13, 14].Moreover, what characterize UAVs context is that some papers such as[32, 33] consider refueling point in the path of UAVs.Another characteristic of the UAVs solutions is that they consider moreconstraints, such as the energy constraint in [33, 88, 87, 13] and the time win-56 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 dow constraint in [15, 23, 85, 86]. This is justified by the fact that UAVs havemore constrained resources (i.e. especially in terms of energy) than groundvehicles. For ground vehicles, the energy is considered as an optimizationobjective (i.e. to be minimized) rather than a constraint. Moreover, con-trary to ground vehicles context, where the single objective as well as themulti-objective variant of MTSP are fairly considered, in the UAVs contextthe multi-objective variant is rarely considered (only Ref. [26]). Indeed, themost considered objective in UAVs solutions is minimizing the mission time.
Figure 6: Distribution of solutions per approaches for each vehicle type
Regarding the used approaches, as shown in Figure 6, the genetic algo-rithm approach is the most used one in both types of vehicles. The secondmost used approach for ground vehicles is the ACO meta-heuristic. Then, itcomes the Market-based approach. It is worth noting that the GA approachwas combined with other techniques in several papers, such as GA+ACO in[61, 67], or GA+memetic algorithm in [68]. Moreover, for the UAVs context,57 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 the GA approach is widely used with the clustering technique such as in[18, 23, 24, 86, 14]. In addition, we note from Table 3 that MTSP solutionsproposed for the UAVs context are more complex and are based on the inte-gration of several approaches. Indeed, UAVs are limited by their energy andload/carrying capacity, and can operate in tandem with trucks, as some pa-pers suggest. The optimization problem therefore becomes complex. In orderto optimize the expected time while taking into account the constraints ofUAVs and the requirements of the application, many studies have proposedto solve the optimization problem in several steps to reduce the complexityof the problem. To do this, these studies have applied or combined severalapproaches for these different stages.In terms of application fields (Figure 5d), the most papers proposed forground vehicles considered the general context, without being limited to aparticular application area, except some papers such as [57, 71, 60, 38, 51],which are proposed for the MRTA problem. However, UAVs solutions presenta diversity of applications areas (Table 3).
6. Discussion and Future Directions
The previous two sections have been devoted to reviewing contributionsproposed for MTSP. Indeed, we have provided an overview of the differentapproaches proposed in the literature to address MTSP while highlightingthe application’s areas.Even though MTSP is very relevant for real-life applications, we pointedout that several studies have solved the general context MTSP without con-sidering a specific application. However, when the study is within a given58 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 context such as parcel delivery, data collection, monitoring, and surveillance,etc., new variants of MTSP are proposed. These variants consider differenttypes of vehicles (e.g. robots, vehicles, trucks, and UAVs) along with newcharacteristics of the depot and the cities to be visited.For instance, UAVs are characterized by their low cost, high mobility, andcan easily be deployed to carry out a search and rescue mission or monitora given area. However, in the field of transport, ground vehicles and trucksare used because they have great autonomy and load capacity. It shouldbe noted that several studies have been based on heterogeneous vehiclesand have proposed solutions where trucks and UAVs are used together toaccomplish a given task.Based on the application requirements, additional constraints are alsoconsidered in MTSP, such as vehicle capacity, energy consumption, and timewindow, which are commonly used in vehicle routing problems. Even ifMTSP becomes similar to VRP when considering these constraints, in oursurvey, we only reviewed papers in which the problem is formulated as MTSP.In the reviewed studies, several exact and heuristic approaches have beenproposed, either to solve MTSP directly or after transformation to TSP.Although exact approaches give the optimal solution, they are usefulonly for very small instances due to the NP-hardness of the problem. Meta-heuristic approaches, including the genetic algorithm (GA), ant colony opti-mization (ACO), artificial bee colony (ABC), and particle swarm optimiza-tion (PSO), have been extensively explored to efficiently solve MTSP. Nev-ertheless, the execution time is very long when the problem scales, makingit unsuitable for real-time applications. We pointed out that the genetic al-59 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 gorithm has been the most widely used meta-heuristic, however, in recentyears, there has been a tendency to apply the ACO algorithm.Besides, several studies proposed to solve MTSP based on hybrid algo-rithms that combine the use of a meta-heuristic with, for example, a localsearch or clustering algorithm. The problem is then solved in different phases,so that, the computation complexity is reduced and the convergence time be-comes reasonable.The market-based approach has also been adopted to solve MTSP, as itcan deal effectively with dynamic system changes and does not require priorknowledge of all system states to provide a solution.The path followed by the vehicles to accomplish a given mission mustmeet two important conditions. On the one hand, the vehicle trajectorymust be feasible, hence the need to take into account vehicle constraintssuch as endurance, kinematics (e.g. speed and acceleration), and systemdynamics. On the other hand, the safety of the vehicle must be ensured byavoiding collisions with obstacles as well as collisions between vehicles.For the sake of simplicity, many studies did not consider these vehiclecharacteristics and proposed optimization problems that relax these con-straints. Only a few contributions on UAV’s applications have included thevehicle constraints in the problem model [11] or have considered collisionavoidance in the solution [32].Future studies on the problem of MTSP optimization need to pay moreattention to vehicle characteristics and constraints to provide effective solu-tions for real-life applications.Unlike vehicles and trucks, ground robots and UAVs might have limited60 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369 energy. Some studies have focused on UAV’s endurance, however, an energyconsumption model for robots and UAVs needs to be proposed in futurecontributions.As for the classical MTSP, studies on MTSP for UAVs can provide so-lutions for the data Benchmark [89] to perform comparisons between thedifferent approaches proposed to solve MTSP and to push research studiesin that direction.
7. Conclusion
The multiple travelling salesman problem is one of the most interest-ing combinatorial optimization problems due to its ability to describe andformulate real-life applications. Indeed, this survey showed that MTSP isused to formulate optimization problems in several fields, including trans-portation and delivery, data collection, search and rescue, multi-robot taskallocation and scheduling, etc. Although the MTSP importance, there is alack of a survey that describes existing solutions. This paper aims to fill thisgap by providing a comprehensive review of existing and recent solutions forMTSP. We have divided existing solutions into two broad classes: (1) MTSPfor vehicles and robots and (2) MTSP for UAVs or drones. Moreover, eachclass’s solutions are classified according to the optimization approaches used,such as exact, meta-heuristic, market-based, etc. This paper also proposed ataxonomy of studied solutions according to several criteria, including MTSPvariants, approaches, applications, etc. Finally, it is worth noting that MTSPremains a promising research field, especially for UAVs based applications,where new optimizing problems are emerging.61 he final version of this paper is published in Computer ScienceReview, https://doi.org/10.1016/j.cosrev.2021.100369
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