The Multiagent Planning Problem

. The classical Multiple Traveling Salesmen Problem is a well-studied optimization problem. Given a set of 𝑛 goals/targets and 𝑚 agents, the objective is to find 𝑚 round trips, such that each target is visited only once and by only one agent, and the total distance of these round trips is minimal. In this paper we describe the Multiagent Planning Problem, a variant of the classical Multiple Traveling Salesmen Problem: given a set of 𝑛 goals/targets and a team of 𝑚 agents, 𝑚 subtours (simple paths) are sought such that eachtargetisvisitedonlyonceandbyonlyoneagent.Weoptimizeforminimumtimeratherthanminimumtotaldistance;therefore theobjectiveistofindtheTeamPlaninwhichthelongestsubtourisasshortaspossible(amin–maxproblem).Weproposean easytoimplementGeneticAlgorithmInspiredDescent(GAID)methodwhichevolvesasetofsubtoursusinggeneticoperators. WebenchmarkedGAIDagainstotherevolutionaryalgorithmsandheuristics.GAIDoutperformedtheAntColonyOptimization andtheModifiedGeneticAlgorithm.EventhoughtheheuristicsspecificallydevelopedforMultipleTravelingSalesmenProblem (e.g., 𝑘 -split, bisection) outperformed GAID, these methods cannot solve the Multiagent Planning Problem. GAID proved to be much better than an open-source Matlab Multiple Traveling Salesmen Problem solver.


Introduction
Applications from space exploration [1][2][3] and drone delivery to search and rescue problems [4][5][6][7] have underlined the need to plan a coordinated strategy for a team of vehicles to visit targets.It is important to develop a multiagent planner for a team of autonomous vehicles to cooperatively explore their environment [8,9].We formulate the overall planning problem as finding a near-optimal set of paths that allows the team of agents to visit a given number of targets in the shortest amount of time.This problem is quite similar to the wellknown Multiple Traveling Salesmen Problem (MTSP) [10][11][12][13], a generalization of the Traveling Salesman Problem (TSP) [14][15][16], that can be stated as follows: given  nodes (targets) and  salesmen (agents), the MTSP consists of finding  closed tours (paths start and end at the starting point of the agents), such that each target is visited only once and by only one agent and the total cost of visiting all nodes is minimal.MTSP has been used for modeling many real situations, from scheduling activities of companies and industries to cooperative planning problems.See, for example, [17], where MTSP is used for modeling the preprinted insert scheduling problem.Planning problems have also been investigated through MTSP formulations, specifically in [18,19], where a dynamic mission planning system for multiple mobile robots operating in unstructured environments is presented (analysis of planetary exploration), or in [20], where the MTSP formulation is used to describe a path planning problem for a team of cooperative vehicles.An important and well-studied extension of the MTSP is the Vehicle Routing Problem [21,22], where a fleet of vehicles of different capacities, based at either one or several depots, must deliver different customer demands (the number of vehicles is often considered as a minimization criterion in addition to total traveled distance).
A long-term goal of this work is to endow a team of autonomous agents (drones) with the capability of cooperative motion planning.In this application, the time available for the solution is limited and real-time algorithms providing 2 Complexity good approximate solutions are required.In this work, the problem of planning a set of strategies for cooperatively exploring the environment with a fleet of vehicles is modeled as a variant of the classical MTSP, referred to as the Multiagent Planning Problem (MAPP): given  nodes (targets) and  salesmen (agents) located at different depots, the MAPP seeks  tours such that each target is visited only once and by only one agent that minimizes a given cost function (specified by (8) and ( 9)).The paper presents a Genetic Algorithm Inspired Descent (GAID) method for obtaining good quality MAPP solutions.
The paper is organized as follows: after describing the MAPP in detail (Section 2) and an overview of how similar problems (MTSP in particular) are solved (Section 3), the proposed GA-Inspired Descent method is described in Section 4. Results are reported in Section 5, and conclusions are drawn in Section 6.

The Multiagent Planning Problem: Notations
Graph theory [23,24] provides a natural framework to describe the Multiagent Planning Problem.Given  = {V 1 , . . ., V  }, a set of  elements referred to as vertices (targets), and A subgraph  = ( 1 ,  1 ) is called a path in  = (, ) if  1 is a set of  vertices of the original graph and is the set of  − 1 edges that connect those vertices.In other words, a path is a sequence of edges with each consecutive pair of edges having a vertex in common.Similarly, a subgraph  = ( 2 ,  2 ) of  = (, ), with is called a cycle.The length of a path or cycle is the number of its edges.The set of all paths and cycles of length  in  will be denoted by P  () and C  (), respectively.Paths and cycles with no repeated vertices are called simple.A simple path/cycle that includes every vertex of the graph is known as a Hamiltonian path/cycle.Graph  is called weighted if a weight (or cost) In this paper, the weight associated with each edge is the Euclidean distance between the corresponding vertices (locations); that is, (V  , V  ) = (V  , V  ) = ‖r(V  ) − r(V  )‖.The total cost (⋅) of a path  ∈ P  () is the sum of its edge weights Analogously, for a cycle  ∈ C  () After having introduced the necessary notation, we are now in the position to formalize the combinatorial problems of interest.The Subtour Problem [25] is defined as finding a simple path  ∈ P  ( +1 ()) of length , starting at vertex  1 =  and having the lowest cost () = ∑  =1 (  ,  +1 ).If  = , the problem is equivalent to finding the "cheapest" Hamiltonian path, where all the  targets in  are to be visited.The general Traveling Salesman Problem, or -TSP, poses to find a simple cycle  ∈ C +1 ( +1 ()) of minimal cost starting and ending at vertex , visiting  targets.
Let  = { 1 , . . .,   } be the set of  targets (goals) to be visited.The th target   is an object located in Euclidean space whose position is specified by the vector r(  ).Let  = { 1 , . . .,   } denote the set of  agents with position specified by r(  ).The classical Multiple Traveling Salesman Problem can be formulated as follows.Let  denote the unique depot; that is,   = .The augmented vertex set is  =  ∪  and the configuration space of the problem is the complete graph  +1 ().Let   denote a cycle of length   starting and ending at vertex  (the depot).The Multiple Traveling Salesmen Problem is to find  cycles (we also refer to these as tours) such that each target is visited only once (this also implies visitation by only one agent) and the sum of the lengths (costs) of all the  tours is minimal.The Multiagent Planning Problem can be formulated similarly.For the th agent, the augmented vertex set is given by   =  ∪   and the configuration space of the problem is the complete graph  +1 (  ).The weight associated with each edge is the Euclidean distance between the corresponding locations, that is, such that the length of the longest path is minimal (a min-max problem).In other words, the  agents have to visit  targets in the least amount of time, and every target is visited only once.The number of targets visited by each agent can be different, but each agent has to visit at least one target (i.e.,   ≥ 2).

Overview of Solution Methods
The Multiple Traveling Salesmen Problem (MTSP) and the Multiagent Planning Problem are notoriously difficult to solve due to their combinatorial nature (they are NP-hard).
A common approach is to transform the MTSP into an equivalent Traveling Salesman Problem, for which solutions can be found by exact methods (e.g., branch-and-bound algorithms and linear programming [26][27][28]) or approximate algorithms such as Genetic Algorithms, Simulated Annealing, and Ant System [29,30].For example, in [31,32] the authors proposed to transform the MTSP into an equivalent TSP by adding dummy cities and edges with ad hoc null or infinite costs.However, as stated in [33][34][35], transforming the MTSP into an equivalent TSP might yield an even harder problem to solve.Similar approaches are investigated in [13,36,37].
The first attempt to solve large-scale MTSPs is given in [33], where a branch-and-bound method (the most widely adopted technique for solving these combinatorial problems [38]) is applied to both Euclidean (up to 100 cities and 10 salesmen) and non-Euclidean problems (up to 500 cities and 10 salesmen).Branch-and-bound is also applied in [39] for solving an asymmetric MTSP up to 100 cities.
Other solution methods have also been proposed, for example, simulated annealing [18], Gravitational Emulation Local Search (GELS) algorithm [40], and evolutionary algorithms.In [41], different evolutionary algorithms, ranging from Genetic Algorithms to Particle Swarm and Monte-Carlo optimization, are compared.The MTSP with ability constraint is solved with an Ant Colony Optimization (ACO) algorithm in [34], where the MTSP is not translated into an equivalent TSP and the ACO algorithm is modified for dealing with the characteristics of the original problem.In [34] results are compared with a Modified Genetic Algorithm that solves the equivalent TSP.Linear Programming is used in [35], and similarly to [34], the original MTSP is analyzed and solved.In both [34,35], the authors conclude that the original MTSP is easier to solve than the derived TSP.An important work is [42], where different local search heuristics are presented and compared.In [17,29,43,44] Genetic Algorithms are used to minimize the sum of the salesmen path lengths together with the maximum distance traveled by each salesmen (to balance the agent workload).A Modified Genetic Algorithm on the equivalent TSP is used in [32].Shirafkan et al. [45] proposed a hybrid method incorporating simulated annealing and Genetic Algorithm to solve the multidepot MTSP.In this MTSP variant the starting location (depot) of each agent is fixed and multiple agents can have start from the same depot.The multiobjective MTSP variant is presented in [46] and Genetic Algorithm is used to find approximate solutions.In multiobjective MTSP the total distance traveled by the agents and the balance of agent working times are the objectives.Kaliaperumal et al. [47] also used Genetic Algorithm to solve the MTSP.They developed a brand new crossover operator called modified two-part chromosome crossover and obtained better results using this new operator.A new (branch-and-cut type) exact method is presented in [48] for the heterogeneous MTSP (some targets can be visited only by a specific agent).In [49] a heuristic search method is proposed to solve the Multiagent Path Finding problem, which is similar to MAPP, except that the endpoints of the tours are also fixed.These methods work well on the problem they were constructed for, but none of them is designed to solve the MAPP.In Section 5.1.2we benchmark our GAID against some of these methods in MTSP solving.In Section 5.2 we compare GAID against a Matlab solver which is capable of solving MAPP as well as MTSP.

The Multiagent Planning Problem: Approximate Solution
Given  ≥ 1 agents and  known targets/cities to visit, the optimal team strategy (called Team Plan) is sought that allows the fleet to visit every target only once.We represent the Team Plan as a collection of  distinct subtours.Thus, given  agents and  targets, Team Plan P is defined as P = { 1 , . . .,   }, where   is the path of the th agent visiting   <  targets.Note that this representation allows the individual subtours to have different lengths.Moreover, the Multiagent Planning Problem can also be rewritten for each th agent to find the best possible subtour of length   <  that satisfies the imposed cost function.We propose an optimization technique we call the Genetic Algorithm Inspired Descent (GAID) method.Briefly, a Genetic Algorithm (GA) is an optimization technique used to find approximate solutions of optimization and search problems [50][51][52].Genetic Algorithms are a particular class of evolutionary methods that use techniques inspired by Darwin's theory of evolution and evolutionary biology, such as inheritance, mutation, selection, and crossover.In these systems, populations of solutions compete and only the fittest survive.
Similarly to the classical GA, GAID consists of two phases: initialization and evolution.In the initialization phase, the starting Team Plan is created (see Section 4.1), while the evolution phase (see Section 4.2) evolves Team Plan P toward a better quality final solution.Complexity 4.1.Initialization Phase.During the initialization phase, the starting Team Plan-a starting set of subtours-is created.Let  1 = { 1 , . . .,   } and  = { 1 , . . .,   } be the sets of  targets and the  agents, respectively.Without loss of generality, we assume that the order of planning is  1 ,  2 , . . .,   and that the starting subtours have similar lengths.A Team Plan is feasible if the subtours are pairwise disjoint (except possibly their starting points).
At first, a subtour  1 (with starting point  1 ) of length  1 is chosen.Then subtour  2 is chosen from the target set  2 that is simply obtained by discarding from  1 the targets visited by agent  1 .In general, the th agent plans a subtour   on the targets not yet allocated previously.Obviously, this process yields a feasible Team Plan P = { 1 , . . .,   }.
Here we utilize three simple initialization methods.
(i) Random initialization: for each agent the targets are selected at random from the unallocated ones.
(ii) Greedy initialization: the initial Team Plan is created using a greedy approach to form feasible starting subtours.Each agent selects its targets using a Nearest Neighbor heuristics: for a given a target, the next target will be the nearest one.
(iii) TSP-based initialization: for problems where the positions of the  agents are not imposed (the  agents can start from any target   ∈ ), a feasible starting Team Plan can be generated by clustering the TSP solution computed on the complete graph   ()."Clustering" is carried out by discarding  edges from the TSP tour, in order to subdivide it "fairly", and having  starting subtours with similar costs.This initialization method introduces a degree of complexity in the overall system, since a TSP solution must be computed.
The initialization phase is an important step in the optimization process, since it directly influences the efficacy of the algorithm.

Evolution Phase.
The evolution phase evolves the Team Plan, trying to design a strategy where the overall time is reduced (minimizing the cost  max (P); see (9)).
This phase has the same mechanism of a classical Genetic Algorithm [50] with one important difference: there is only one Team Plan P to be evolved (i.e., the population size is 1).
At every evolution/generation step, a set of operators (see Section 4.3) is applied to the subtours   ∈ P. If P improves, it is kept for the next generation step, otherwise the previous Team Plan is restored (we expect that a simulated annealingtype modification will be even more efficient).The flowchart of the GAID evolution phase is shown in Figure 1.

Team Plan Operators and
Boosting.The evolution of P toward a near-optimal multiagent strategy is accomplished by combining the genetic materials of its subtours through the application of genetic-like operators.Three different operators have been designed: crossover, mutation, and migration.The operators are applied in a predefined order, and their application depends on a given probability, as shown in Figure 1.
The crossover operator (Figure 2) is applied with probability  crossover and combines the genetic materials of two selected subtours (called parents), replacing them with the two newly created ones (the offsprings).Parents are chosen using the best-worst selection with probability  best-worst (Figure 2(a)) or randomly (Figure 2(b)) with probability 1 −  best-worst .The parents are mated the classical way [50]: they are randomly halved (not necessarily at the same position) and the halves are simply swapped.
The Mutation Operator changes the Team Plan (with probability  mutation ) by randomly swapping two genes between two different subtours.The Migration Operator (applied with probability  migration ) removes a randomly chosen target from subtour   (of length   ) and adds it to subtour   (of length   ).The location at which this target is placed into subtour   is also chosen at random.Note that the lengths of the subtours change:   decreases while   increases.

Results
An extensive set of simulations were run to test the performance of the proposed GAID method.Unless otherwise specified, simulations were run for 150000 generation steps, while the crossover, mutation, migration, and boosting (2opt) operators were applied with probabilities  crossover = 0.7,  mutation = 0.4,  migration = 0.6, and  boost = 0.3, respectively.For the application of the crossover operator, the probability of best-worst selection is  best-worst = 0.5 (thus the probability for random selection is 1 −  best-worst = 0.5).

Comparison with Structured and Known Solutions.
In this section, GAID is compared with known results.In order to make the comparison meaningful, we also imposed the same constraints (if any) used in the referenced works.

An Example Problem with Known Solution.
In this section we present a 9-81 MAPP (9 agents, 81 targets), in which the agents and targets form 9 tightly packed clusters as shown in Figure 3.Each cluster contains an agent and 9 targets.Since the distances among the clusters are considerably longer than those among the points of a cluster, the optimal solution can be found by treating each cluster as a 1-9 MAPP (1 agent, 9 targets).The tour lengths are between 356.1 and 548.6, meaning that the optimal cost (the length of the longest tour) is 548.6 (see ( 9)).In Figure 3 we show the random and greedy initialization for this MAPP.In both cases some of the subtours contain a higher number of targets than others.We intentionally set the initialization this way to further challenge the algorithm.The initial tour lengths vary between 1944.9-15302.3for the random, and 175.3-1627.9 for the greedy initialization.Consequently, the initial Team Plan costs are 15302.3and 1627.9, respectively.The convergence of the Team Plan cost for both simulations is shown in Figure 4 together with the cost of the optimal solution (548.6).At the end of the simulations the costs of the subtours varied between 378.3-612.3 and 389.9-594.1 for solutions starting from randomly initialized and greedy initialized Team Plans, respectively.
The final Team Plans of the two simulations are shown in Figure 5.We see that those long edges which initially connected targets belonging to different clusters were completely eliminated.The longest subtour connects the targets in the middle-top cluster for both simulations.

Comparison with Evolutionary Algorithms.
We also compare our method with the results reported in [34], where an Ant Colony Optimization algorithm is compared with the Modified Genetic Algorithm (MGA) of [32].We note here that these methods cannot solve the MAPP, therefore six MTSPs, all taken from well-known TSPLIB [56], are solved.The number of agents is set to  = 5, and their starting location is the first target in the corresponding problem data file.Since a single depot is considered, only the greedy initialization method is used.In both [32,34], rounded costs are considered.Therefore, for meaningful comparison, we need to minimize the following cost function: In addition, the maximum length of the cycles,  max , is limited to 20 (note that since MTSPs are considered here, we need to modify the Team Plan using cycles instead of paths).Results based on 100 simulations are presented in Table 1.In this case, our results outperform those reported in the above  The GAID algorithm was tested on a MAPP which contains 81 targets forming 9 clusters on a 2200 × 2200 square domain.An agent was placed into each cluster (white diamonds); thus the number of agents is  = 9.The initial cost based on (9) is 15302.3for random and 1627.9 for greedy initialization.The longest subtour is highlighted with thick line.
references. Figure 6 also shows the solution obtained for the pr76 TSPLIB problem.

Comparison with Heuristics.
We compare our method against the results reported in [42], where different heuristics for solving MTSP instances are proposed and compared.
In [42], a no-depot MTSP variant is considered (the agents do not have a predefined starting location), and the min-max optimization problem is solved ((9) is minimized).In addition, the number of salesmen is fixed and no constraints on the plan lengths are imposed.Note that since we are comparing our method with MTSP results, we need to modify our Team Plan accordingly, using cycles instead of paths.In addition, since in [42] rounded distances are considered, for meaningful comparison the cost function (see (9)) is modified accordingly: For each test case, our results are compared with only the best ones of [42].We also report the name of the Heuristic with which each referenced solution has been obtained (please refer to [42] for a more detailed description of the adopted heuristic methods).Tables 2 and 3 show the results obtained by initializing the Team Plan with either the greedy or the TSP-based initialization methods, respectively.Results are averaged over 100 simulations.
In each test case, GAID returns good solutions, regardless of the applied initialization method.In general, our best solutions are close to the ones from the literature and in a few cases (e.g., in the berlin52 or the pr264 problems) are even better.The greedy initialization method seems to provide a better initial Team Plan in those problems where the distribution of targets has a well-defined structure (the pr264 problem), while in "small-size" problems (i.e., berlin52) the TSP-based method is preferable (even if the obtained improvement does not justify its required complexity, time,  and computational costs).Even though these heuristics are superior in solving MTSP compared to GAID, they cannot solve MAPP, while our method can solve both.Figure 7 shows the solutions obtained by applying the greedy (Figure 7(a)) and the TSP-based (Figure 7(b)) initialization methods for the kroA100 problem (with  = 5 agents).Compared with [42], the TSP-based method and the greedy method result in final solutions that are only 3.6% and 7.2% worse than the cited one, respectively.

Comparison with Other Software.
In this section we test GAID against a freely available Matlab MTSP solver based on a Genetic Algorithm [57].
The number of agents is fixed, and the minimum path size,  min = 2, is imposed (this way, solutions with paths composed of only one target are avoided).The cost function, which is specified by (9), is minimized, and targets are randomly generated over the unit square map.In all simulations,  11) is minimized, and the comparison between the starting and the final Team Plans is shown.In (a) the starting Team Plan, with cost equal to 194618, is shown, while in (b) the final Team Plan, with cost equal to 152722, is reported.These solutions are 15.4% and 2.6% better than the referenced ones, obtained by implementing an Ant Colony Optimization method and a Modified Genetic Algorithm, respectively.the greedy initialization method is adopted.Since the Matlab MTSP solver is based only on a Genetic Algorithm and no heuristics is used, we also run a set of tests with  boost = 0%.
Table 4 shows the obtained results, averaged over 100 simulations.Our method clearly outperforms the Matlab solver (clearly, with the 2-opt method the results are much better).Figure 8 shows two examples where the Matlab MTSP solver solution is compared with the GAID ones.

Solution for a Large Example
Problem.We tested our algorithm on a 20-400 MAPP.The initial Team Plan was constructed by means of the greedy initialization method, such that each subtour visits 10 targets.The simulation was ran for 1000000 generations.
Figures 9 and 10 summarize the results.The initial cost of 1.94 was reduced by 43.5% (to 1.08) by the end of the 150000th generation and by a further 5.7% (to 0.94) till the end of the simulation.

Conclusions and Future Work
This paper describes the Multiagent Planning Problem, a variant of the classical Multiple Traveling Salesman Problem.Our solution method uses a simplified Genetic Algorithm,   7: Solving the MTSP by the greedy and the cluster initialization methods.In both cases,  = 5 agents are considered, and the target set is kroA100.In (a), the greedy initialization method is used and the final obtained Team Plan is shown.The cost is 4964, and it is 7.2% worse than the referenced one.In (b), the TSP-based initialization method is used and the obtained final Team Plan is shown.The cost is 4796, which is only 3.6% worse than the referenced one.called Genetic Algorithm Inspired Descent (GAID), capable of solving both the MTSP and the MAPP.GAID takes an initial Team Plan and applies various genetic operators to decrease the length of the longest path.We benchmarked GAID against other evolutionary algorithms and heuristics.GAID outperformed the Ant Colony Optimization and the Modified Genetic Algorithm.Even though the heuristics specifically developed for MTSP (e.g., -split, bisection) outperformed GAID, these methods cannot solve MAPP.GAID proved to be much better than the Matlab MAPP solver.GAID is also easy to implement.
A long-term goal of this work is to endow a team of autonomous agents (drones) with the capability of cooperative motion planning.In this application, the time available for the solution is limited and fast algorithms providing good approximate solutions prevail.The results presented here show the success of the approach, demonstrating how a simple method can solve an otherwise hard combinatorial problem.The GAID algorithm was performed on a MAPP with 20 agents and 400 targets located in the unit square.The Team Plan cost was 1.91 after the initialization and was reduced to 1.08 by the end of the 150000th generation and to 0.94 by the end of the simulation.The longest subtour is highlighted by a thick line.

Figure 1 :Figure 2 :
Figure 1: Flowchart of the evolution phase, together with the sequence of operators.
Figure3: The GAID algorithm was tested on a MAPP which contains 81 targets forming 9 clusters on a 2200 × 2200 square domain.An agent was placed into each cluster (white diamonds); thus the number of agents is  = 9.The initial cost based on (9) is 15302.3for random and 1627.9 for greedy initialization.The longest subtour is highlighted with thick line.

Figure 4 :
Figure 4: Convergence of the Team Plan cost for the 9-81 MAPP shown in Figure 3.

Figure 5 :Figure 6 :
Figure 5: The GAID algorithm found 9 subtours which connect targets belonging to the same cluster for the 9-81 MAPP with both initialization strategies.The final costs are 612.3 and 594.1, respectively.

Figure
Figure7: Solving the MTSP by the greedy and the cluster initialization methods.In both cases,  = 5 agents are considered, and the target set is kroA100.In (a), the greedy initialization method is used and the final obtained Team Plan is shown.The cost is 4964, and it is 7.2% worse than the referenced one.In (b), the TSP-based initialization method is used and the obtained final Team Plan is shown.The cost is 4796, which is only 3.6% worse than the referenced one.

Figure 8 :
Figure8: GAID has been compared with the Matlab MTSP solver.In this example, MAPP is solved for 100 targets randomly distributed in the unit square. = 10 agents are considered.The minimum path length is  min = 2.The cost function (see(9)) is minimized.In (a) the solution obtained by running the MTSP Matlab solver is shown, with cost 7.92.In (b) our solution is shown with cost 6.37, which is 19.5% better than the Matlab MTSP solution.
Figure9: The GAID algorithm was performed on a MAPP with 20 agents and 400 targets located in the unit square.The Team Plan cost was 1.91 after the initialization and was reduced to 1.08 by the end of the 150000th generation and to 0.94 by the end of the simulation.The longest subtour is highlighted by a thick line.

Figure 10 :
Figure 10: Convergence of the Team Plan cost for the 20-400 MAPP shown in Figure 9.
rendering  +1 (  ) a weighted and symmetric graph.Let   denote a simple path (no repeated vertices) of length |  | =   starting at vertex   .The optimal solution of the Multiagent Planning Problem (MAPP) is a set P of  pairwise disjoint (modulo the starting points) paths   P = { 1 , . . .,   } ,

Table 1 :
Comparison between the proposed GAID, the Ant Colony Optimization, and the Modified Genetic Algorithm methods.Results are averaged over 100 simulations.The number of targets is included in the TSPLIB problem name. max is the maximum number of targets an agent can visit.For each case, the fixed number of  = 5 agents is considered, starting from the same location (one-depot problem).Only the greedy initialization method is used.

Table 2 :
Heuristics comparison for 100 simulations.The greedy initialization method was used.

Table 3 :
Heuristics comparison for 100 simulations using the TSP-based initialization.

Table 4 :
Comparison between the proposed GAID method and a Matlab MTSP code.For each test,  min = 2 and the greedy initialization method is used.Targets are randomly generated over the unit square.