The Multi-Agent Planning Problem

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problems (MTSP above all) are solved (Section 3), the proposed GA-Inspired Descent method is described in. Section 4. .... tour, in order to 'equally' subdivide it, and having m starting subtours with similar costs. ..... [12] Carter, A. E. (2003).
The Multi-Agent Planning Problem Tam´as Kalm´ar-Nagy∗ Department of Fluid Mechnics, Faculty of Mechanical Engineering Budapest University of Technology and Economics Giovanni Giardini† SPS Italiana Pack Systems Bendeg´uz Dezs˝o Bak‡ Department of Fluid Mechnics, Faculty of Mechanical Engineering Budapest University of Technology and Economics

Abstract We describe the Multi-Agent 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 tours are sought such that each target is visited only once and by only one agent, minimizing the total cost of the visits. We propose a Genetic Algorithm Inspired Descent (GAID) method which evolves a set of subtours acted on by genetic operators. To validate the approach, the method has been benchmarked against MTSPs and routing problems. Numerical experiments demonstrate the good quality of solutions.

1

Introduction

Applications from space exploration [http://marsrovers.nasa.gov, 22, 3] and drone delivery to search and rescue problems [7, 40, 27, 11] have underlined the need to for a team of vehicles to plan a coordinated strategy to visit targets. It is important to realize a multi-agent planner for a team of autonomous vehicles to cooperatively explore their environment [17]. 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 well-known Multiple Traveling Salesmen Problem (MTSP) [35, 4, 23, 43], a generalization of the Traveling Salesman Problem (TSP) [28, http://www.tsp.gatech.edu/, 21] that can be stated as follow: Given n nodes (targets) and m salesmen (agents), the MTSP consists of finding m closed tours (that 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 [13], where MTSP is used for modeling the pre-printed insert scheduling problem. Planning problems have also been investigated through MTSP formulations, specifically in [46, 45], where a dynamic mission planning system for multiple mobile robots operating in unstructured environments is presented (analysis of planetary exploration), or in [50], 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 [http://neo.lcc.uma.es/radi aeb/WebVRP/, 38], 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). 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 Multi-Agent Planning Problem (MAPP): Given n nodes (targets) and m salesmen (agents) located at different depots, the MAPP seeks m tours such that each target is visited only once and by only one agent, while minimizing a given cost function.The paper presents a Genetic Algorithm Inspired Descent method (GAID) for obtaining good quality MAPP solutions. ∗ [email protected] † [email protected] ‡ [email protected]

1

The paper is organized as follows: after the MAPP formulation (Section 2), and an overview of how similar problems (MTSP above all) are solved (Section 3), the proposed GA-Inspired Descent method is described in Section 4. Results are then reported in Section 5, conclusions in Section 6.

2

The Multi-Agent Planning Problem: Formulation

Graph theory [8, 15] provides a natural framework to describe the Multi-Agent Planning Problem. Given V = {v1 , . . . , vm }, a set of m elements referred to as vertices (targets), and E = {(vi , v j )|vi , v j ∈ V }, a set of edges connecting vertices vi and v j , a graph G is defined as the pair (V, E). Here we consider only undirected graphs, i.e. graphs whose edges are unordered pairs with the symmetry relation (vi , v j ) = (v j , vi ). A complete graph is a graph where all vertices of V are connected to each other. The complete graph induced by the vertex set V is denoted by Km (V ), where m = |V | (|.| denotes the cardinality of a set) is the number of vertices. A graph G1 = (V1 , E1 ) is a subgraph of G (G1 ⊆ G) if V1 ⊆ V and E1 ⊆ E such that E1 = {(vi , v j ) | vi , v j ∈ V1 }.

(1)

A subgraph P = (V1 , E1 ) is called a path in G = (V, E) if V1 is a set of k vertices of the original graph and E1 = {(x1 , x2 ), (x2 , x3 ), . . . , (xk−1 , xk )} ⊆ E

(2)

is the set of k − 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 C = (V2 , E2 ) of G = (V, E) with V2 = {x1 , . . . , xk } ⊆ V,

(3)

E2 = {(x1 , x2 ), . . . , (xk−1 , xk ), (xk , x1 )} ⊆ E,

(4)

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 k in G will be denoted by Pk (G) and Ck (G), 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 G is called weighted if a weight (or cost) w(vi , v j ) is assigned to every edge (vi , v j ). A weighted graph G is called symmetric if w(vi , v j ) = w(v j , vi ). In this paper, the weight associated with each edge is the Euclidean distance between the corresponding vertices (locations), i.e. w(vi , v j ) = w(v j , vi ) =k r(vi ) − r(v j ) k . The total cost c(.) of a path P ∈ Pk (G) is the sum of its edge weights k

c(P) = ∑ w(xi , xi+1 ).

(5)

i=1

Analogously, for a cycle C ∈ Ck (G) k−1

c(C) =

∑ w(xi , xi+1 ) + w(xk , x1 ).

(6)

i=1

After having introduced the necessary notation, we are now in the position to formalize the combinatorial problems of interest. The Subtour Problem [18] is defined as finding a simple path P ∈ Pk (Kn+1 (V )) of length k, starting at vertex x1 = a and having the lowest cost c(P) = ∑ki=1 w(xi , xi+1 ). If k = n, the problem is equivalent to finding the ‘cheapest’ Hamiltonian path, where all the n targets in T are to be visited. The general Traveling Salesman Problem, or k-TSP, poses to find a simple cycle C ∈ Ck+1 (Kn+1 (V )) of minimal cost starting and ending at vertex a, visiting k targets. Let T = {t1 , . . . ,tn } be the set of n targets (goals) to be visited. The i-th target ti is an object located in Euclidean space whose position is specified by the vector r(ti ). Let A = {a1 , . . . , am } denote the set of m agents with position specified by r(ai ). The classical Multiple Traveling Salesman Problem can be formulated as follows. Let a denote the unique depot, i.e. ai = a. The augmented vertex set is V = T ∪ a and the configuration space of the problem is the complete graph Kn+1 (V ). Let Ci denote a cycle of length ki starting and ending at vertex a (the depot). The Multiple Traveling Salesmen Problem is to find m cycles (we also refer to these as tours) Ci of length ki m

C = {C1 , . . . ,Cm },

∑ ki = n + m,

i=1

2

(7)

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 m tours m

W (C) = ∑ W (Ci )

(8)

i=1

is minimal. The Multi-Agent Planning Problem can be formulated similarly. For the i-th agent, the augmented vertex set is given by Vi = T ∪ ai and the configuration space of the problem is the complete graph Kn+1 (Vi ). , with vi , v j ∈ Vi , rendering Kn+1 (Vi ) a weighted and symmetric graph. Let Pi denote a simple path (no repeated vertices) of length |Pi | = ki starting at vertex ai , the Multi-Agent Planning Problem (MAPP) is to find a set P of m pairwise disjoint (modulo the starting points) paths Pi m

P = {P1 , . . . , Pm },

∑ ki = n,

(9)

i=1

such that the length of the longest path

Wm (P ) = max W (Pi ) i

(10)

is minimal (a minmax problem). This means that m agents have to visit n targets in the least amount of time, with every target only visited once. Note that the agents can visit a different number of targets (no constraints are imposed on the path lengths ki ).

3

Overview of Solution Methods

The Multiple Traveling Salesman Problem (MTSP) and the Multi-Agent Planning Problem are notoriously difficult to solve due to their combinatorial nature. A common approach is to transform the studied 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 [48, 2, 41]) or approximate algorithms such as Genetic Algorithms, Simulated Annealing and Ant System [12, 32]. For example, in [20, 47] 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 [16, 29, 30], transforming the MTSP into an equivalent TSP can even be a harder problem to solve. Similar approaches are investigated in [43, 37, 5]. The first attempt to solve large-scale MTSPs is given in [16], where a branch-and-bound method (the most widely adopted technique for solving these combinatorial problems [33]) 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 [1] for solving an asymmetric MTSP up to 100 cities. Other solution methods have also been proposed, from simulated annealing [46] (here coupled with a generalpurpose interpreted grammar) to evolutionary algorithms. In [44], 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 [29], 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 [29] results are compared with a modified Genetic Algorithm that solves the equivalent TSP. Linear Programming is used in [30], and similarly to [29], the original MTSP is analyzed and solved. In both [30] and [29], the authors conclude that the original MTSP is easier to solve than the derived TSP. An important work is [36], where different local search heuristics are presented and compared. In [13, 14, 12, 9] 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 [47].

4

The Multi-Agent Planning Problem: Approximate Solution

Given m ≥ 1 agents and n 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 m distinct subtours. Thus, given m agents and n targets, Team Plan P is defined as P = {P1 , . . . , Pm }, where Pi is the path of the i-th agent visiting ki < n targets. Note that this representation allows the individual subtours to have different lengths. Moreover, the Multi-Agent Planning Problem can also be rewritten for each i-th agent to find the best possible Subtour of length ki < n 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 [19, 10, 31]. Genetic Algorithms are a particular class of evolutionary

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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. Similar 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 (hopefully) good quality final solution.

4.1

Initialization Phase

During the initialization phase, the starting Team Plan -a starting set of subtours- is created. Let T1 = {t1 , . . . ,tn } and A = {a1 , . . . , am } be the sets of n targets and the m agents, respectively. Without loss of generality, we assume that the order of planning is a1 , a2 , . . . , am 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 P1 (with starting point a1 ) of length k1 is chosen. Then subtour P2 is chosen from the target set T2 that is simply obtained by discarding from T1 the targets visited by agent a1 . In general, the i-th agent plans a subtour Pi on the targets not yet allocated previously. Obviously, this process yields a feasible Team Plan P = {P1 , . . . , Pm }. Here we utilize two simple initialization methods: • 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. • TSP-based Initialization: for problems where the positions of the m agents are not imposed (the m agents can start from any target ti ∈ T ), a feasible starting Team Plan can be generated by clustering the TSP solution computed on the complete graph Kn (T ). ‘Clustering’ is carried out by discarding m edges from the TSP tour, in order to ‘equally’ subdivide it, and having m 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.

4.2

Evolution Phase

The evolution phase evolves the Team Plan, trying to design a strategy where the overall time is reduced (minimizing the cost Wm (P), see cost function (10)). This phase has the same mechanism of a classical Genetic Algorithm [19] 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 Pi ∈ P . If P improves, it is kept for the next generation step, otherwise the previous Team Plan is restored. The logical flowchart of the GAID evolution phase is shown in Figure 1.

4.3

Team Plan Operators

The evolution of P toward a near-optimal multi-agent strategy is accomplished by combining the genetic materials of its subtours through the application of genetic-like operators. Three different operators have been designed: the crossover, the mutation and the 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) combines the genetic materials of two selected subtours (called parents), replacing them with the two newly created ones (called offspring). At every generation step the crossover operator is applied only once, thus only two subtours are chosen and mated. Depending on a given probability (pselection ), parents can be chosen either randomly (see Figure 2(a)) or deterministically (see Figure 2(b)). In the deterministic case, the mating process always occurs between the subtours with maximum and minimum costs. Once selected, parents are mated in the classical way [19]: they are randomly halved (not necessarily at the same target) and their halves are simply swapped. The Mutation Operator changes the Team Plan by randomly swapping two genes between two different subtours. The Migration Operator consists of moving a randomly chosen target from subtour Pi (of length ki ) to subtour Pj (of length k j ). Note that the lengths of the subtours change: ki decreases while k j increases.

4

Figure 1: Flowchart of the evolution phase, together with the sequence of operators.

(a) Random selection.

(b) Deterministic selection.

Figure 2: Crossover operator: two subtours are randomly (a) or deterministically (b) selected and mated through the crossover operator. In the deterministic case, the best and the worst subtours are selected.

With probability pboost , each subtour is processed with a heuristic boosting technique. The 2-opt method [34, 6, 42] is adopted here to directly improve the quality of subtours (it replaces subtours with better ones from their ‘neighborhood’). The 2-opt method determines whether the inequality w(xi , xi+1 ) + w(x j , x j+1 ) > w(xi , x j ) + w(xi+1 , x j+1 )

(11)

between the four vertices xi , xi+1 , x j and x j+1 of a path/subtour holds, in which case edges (xi , xi+1 ) and (x j , x j+1 ) are replaced with the edges (xi , x j ) and (xi+1 , x j+1 ), respectively. This method provides a shorter path without intersecting edges.

5

Results

First, in Figure 3 we show the results of a 4 agent, 24 target simulation. The targets constitute 4 tightly packed groups, thus the optimal (or a good approximating solution) must not contain an edge which connects two targets belonging to different groups. The initial Team Plan was generated by the greedy method. The solution at the 82th generation is also shown. 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 (2-opt) operators were applied with probabilities pcrossover = 0.7, pmutation = 0.4, pmigration = 0.6 and pboost = 0.3, respectively. When the crossover operator is applied, the probability of deterministically selecting two parents is pselection = 0.5. For each simulation, the initialization method is either greedy or TSP-based, as indicated.

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(a) 4 agents, 24 targets with greedy initialization

(b) Solution found at the 82th generation

14000

14000

12000

12000

10000

10000 Y COORDINATE

Y COORDINATE

Figure 3: The GAID algorithm was tested on a 4 agent, 24 target case. Greedy initial Team Plan and final Team Plan are shown.

8000

6000

8000 6000

4000

4000

2000

2000

0 0

5000

10000 X COORDINATE

15000

0 0

20000

(a) pr76 TSPLIB problem: starting Team Plan.

5000

10000 X COORDINATE

15000

20000

(b) pr76 TSPLIB problem: final Team Plan.

Figure 4: The MTSP is solved for the pr76 TSPLIB target configuration. The maximum plan length is constrained to kmax = 20 targets and m = 5 agents are considered. Cost function (12) 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. This solution is 15.4% and 2.6% better than the referenced ones, obtained by implementing an Ant Colony Optimization method and a Modified Genetic Algorithm, respectively.

5.1

Comparison with Structured and Well-Known Solutions

In this Section, GAID has been compared with known results. In order to make the comparison meaningful, we imposed the same constraints adopted in the referenced works. 5.1.1

Comparison with Evolutionary Algorithms

First we compare our method with the results reported in [29], where an Ant Colony Optimization algorithm is compared with the Modified Genetic Algorithm (MGA) of [47]. Six MTSPs, all derived from well-known TSPLIB instances [39], are solved. The number of agents is set to m = 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 [29, 47], rounded costs are considered. Therefore, for meaningful comparison, we need to minimize the following cost function: m

W (C) = ∑ round (W (Ci )).

(12)

i=1

In addition, the maximum length of the cycles, kmax , is limited (note that since MTSPs are considered here, we need to ‘modify’ the Team Plan using cycles instead of paths). Results are shown in Table 1, and they are based on an average of 100 simulations. In this case, our results outperform those reported in the above references. Figure 4 also shows the solution obtained for the pr76 TSPLIB problem.

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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. kmax is the maximum number of targets an agent can visit. For each case, a fixed number of m = 5 agents is considered, starting from the same location (one-depot problem). Only the greedy initialization method is used.

TSPLIB Problem pr76 pr152 pr226 pr299 pr439 pr1002

ACO Min Mean 178597 180690 130953 136341 167646 170877 82106 83845 161955 165035 382198 387205

kmax 20 40 50 70 100 220

MGA Min Mean 157444 160574 127839 133337 166827 178501 82176 85796 173839 183698 427269 459179

Proposed Method Min Mean 152722 156503.9 114698 126128.8 152198 158073.9 70059 71705.1 136169 138655.5 311492 319240.4

Difference of Means ACO MGA −15.4% −2.6% −8% −5.7% −8% −12.9% −16.9% −19.6% −15.6% −24.5% −17.5% −30.5%

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

TSPLIB Problem berlin52

Number of Agents 4 5 6

kroA100

4 5 6

bier127 pr264

5.1.2

4 6 4 6

Method bisection k-split SGH bisection SGH k-split k-center k-means bisection k-split bisection k-means bisection

Heuristics Minimum 2182 1713 1531 1476 5955 4629 4200 4230 32423 22815 12705 8526 8739

Mean 2204.3 1739.7 1585 1643.1 6096.7 5025.9 4429.4 4234.6 32757.5 23071.7 12705 9131.6 9051.6

Greedy Method Minimum Mean 2183 2422.4 1825 2160.4 1611

1905.8

6000 4964

6690.3 5620.1

4370

5038.4

32434 24608 12196

35740.9 26993.3 13830.3

9000

10256.3

Errors Minimum Mean 0.04% 9.8% 6.5% 24.2% 20.2% 9.1% 0.7% 9.7% 7.2% 11.8% 4% 18.9% 0.03% 9.1% 7.8% 16.9% −4.1% 8.8% 5.5% 13.3%

Comparison with Heuristics

We compare our method against the results reported in [36], where different heuristics for solving MTSP instances are proposed and compared. In [36], a no-depot MTSP variant is considered (the agents do not have a predefined starting location), and the minmax optimization problem is solved (cost function (10) 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 [36] rounded distances are considered, for meaningful comparison cost function (10) is modified accordingly:

Wm (C) = max round (W (Ci )). i

(13)

For each test case, our results are compared with only the best ones of [36]. We also report the name of the Heuristic with which each referenced solution has been obtained (please refer to [36] for a more detailed description of the adopted heuristic methods). Tables 2 and 3 show the results obtained by initializing the Team Plan either with the greedy or the TSP-based initialization methods, respectively. Results are averaged over 100 simulations. In each test case, GAID returns good solutions, independent of the applied initialization method. In general, our best solutions are closer to the ones from the literature, and, in a few cases (for example in the berlin52 or the pr264 problems), 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). Figure 5 shows the solutions obtained by applying the greedy (Figure 5 (a)) and the TSP-based (Figure 5 (b)) initialization methods for the kroA100 problem (with m = 5 agents). In this case, if compared with [36], the TSPbased method and the greedy method result in final solutions that are only 3.6% and 7.2% worse than the cited one, respectively.

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

TSPLIB Problem berlin52

Number of Agents 4 5 6

kroA100

4 5 6 4 6 4

bier127 pr264

6

Method bisection k-split SGH bisection SGH k-split k-center k-means bisection k-split bisection k-means bisection

Heuristics Minimum 2182 1713 1531 1476 5955 4629 4200 4230 32423 22815 12705 8526 8739

Mean 2204.3 1739.7 1585 1643.1 6096.7 5025.9 4429.4 4234.6 32757.5 23071.7 12705 9131.6 9051.6

1576

1801.2

6000 4796

6443.6 5314.6

4310

4693.9

32948 24290 13400

34606 26497.1 19135.3

8629

14395.8

Errors Minimum Mean −4.5% 7% 5.3% 15.8% 13.6% 6.7% 0.7% 5.7% 3.6% 5.7% 2.6% 10.8% 1.6% 5.6% 6.4% 14.8% 5.4% 50.6% 1.2% 59%

2000

Y COORDINATE

Y COORDINATE

2000

TSP-based Method Minimum Mean 2088 2359.6 1804 2014.8

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(a) kroA100 problem: greedy initialization.

2000 X COORDINATE

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(b) kroA100 problem: TSP-based initialization.

Figure 5: Solving the MTSP by the greedy and the cluster initialization methods. In both cases, m = 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 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.

5.2

Comparison with Other Software

In this Section we test GAID against a freely available Matlab MTSP solver based on a Genetic Algorithm [49]. The number of agents is fixed, and the minimum path size, kmin = 2, is imposed (this way, solutions with paths composed of only one target are avoided). Cost function (10) is minimized, and targets are randomly generated over the unit square map. In all simulations, 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 pboost = 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). Figures 6 shows two examples where the Matlab MTSP solver solution is compared with the GAID ones. Table 4: Comparison between the proposed GAID method and a Matlab MTSP code. For each test, kmin = 2 and the greedy initialization method is used. Targets are randomly generated over the unit square.

Test case 100 targets 10 agents 200 targets 20 agents

Min

Matlab Code Mean Max

7.14

8.19

9.21

15.64

17.5

19.65

pboost 30% 0% 30% 0%

Proposed Method Min Mean 6.37 6.72 6.67 7.14 9.55 9.89 10.05 10.6

8

Max 7.11 7.58 10.25 11.23

Min −10.7% −6.5% −39% −35.7%

Cost Variation Mean Max −17.9% −22.8% −12.8% −17.7% −43.5% −47.8% −39.4% −42.8%

1

Y COORDINATE

Y COORDINATE

1

0.5

0 0

0.5 X COORDINATE

0.5

0 0

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(a) Matlab solution.

0.5 X COORDINATE

1

(b) Proposed method.

Figure 6: GAID has been compared with the Matlab MTSP solver. In this example, MAPP is solved. 100 targets are randomly distributed over the unit square map. m = 10 agents are considered. The minimum path length is kmin = 2. Cost function (10) 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.

6

Conclusions and Future Work

This paper describes the Multi-Agent Planning Problem, a variant of the classical Multiple Traveling Salesman Problem. Our solution method is based on a simplified Genetic Algorithm (only one gene is evolved), that is initialized in two different ways: a greedy (Nearest Neighbor) and a TSP-based approach. The importance of this work is to provide a team of autonomous agents the capability of cooperative motion planning. The results presented here show the success of the approach, demonstrating how a simple method can solve otherwise hard combinatorial problems.

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