A study on ant colony systems with fuzzy pheromone dispersion

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1 A stuy on ant colony systems with fuzzy pheromone ispersion Louis Gacogne LIP6 104, Av. Kenney, Paris, France Sanra Sanri IIIA/CSIC Campus UAB, Bellaterra, Spain Abstract We escribe here a sensibility analysis to help tune the parameters of an ant colony moel where ants leave a fuzzy trace of pheromone to mark their track an its neighborhoo. We test it with ifferent parameters on classical numeric an symbolic problems. Keywors: Ant colonies, Optimization, Fuzzy Sets Theory. 1 Introuction Ant colonies is a heuristic inspire from nature to solve optimization problems. When an ant fins a source of foo, it leaves a trail pheromone on its way back to the nest. The accumulation of small iniviual reinforcements on the paths allow an ant colony to fin the best way between a source of foo an the nest. When the source is exhauste, no new pheromone is eposite an the pheromone trails slowly isappear as the pheromone slowly evaporates. In the same way, ant colonies systems [3] fin the optimal solution to a problem by the reinforcement of goo solutions. In ant colonies systems, each ant (or its path) represents a solution, an weights assigne to each solution represent pheromone. A mechanism that increases weights on the solutions correspon to pheromone reinforcement an a mechanism that ecreases weights correspon to pheromone evaporation. In some approaches, only one solution receives an increase of pheromone [4], as if only the leaing ant on a group woul be allowe to eposit pheromone. In both approaches, ranomly traces of iniviuals lea, after generations, to a best iniviual representing an optimal solution. In [1], an ant colony system is presente in which ants eposit pheromone not only on the best path foun on a turn but also on the paths close to it. In this formalism, the ants are sai to leave a fuzzy trace, as the closer a path is to the one that represents a goo solution, the larger the amount of pheromone it receives. This is a irect analogy to the what happens in nature, as the chemical particles of pheromone isperse themselves in the air on a region aroun the eposit. The ifficulty to use these formalisms in some optimization problems is to fin a suitable notion of closeness among paths, such as the case of the Traveling Salesman Problem (TSP) [2]. This formalism is inicate for problems with large search spaces an has been employe successfully in a real worl inverse problem, the estimation of chlorophyl profiles in offshore ocean water [6]. This problem is mae harer by the fact that the cost of the computation of the gooness of each caniate solution is very high. The use of the ant colony systems with fuzzy pheromone ispersion allows the search space to be well explore with only a small amount of solutions being actually examine. Here we are intereste on the task of tuning the parameters of ant colony systems with fuzzy pheromone ispersion on some optimization problems. We aress the solution of a symbolic problem, the Gauss Queens Problem, an of two numerical problems, using well-known functions in optimization literature, the Rastrigin an tripo functions. We focus in a ispersion func-

2 tion that can be moele by a linear fuzzy set an verify the relation between the number of ants an the raius of the ispersion. This work is organize as follows. In Section 2 we escribe the fuzzy pheromone ispersion algorithm use in this work. In Section 3, we present the results of the experiments mae on the Rastrigin, the tripo an Gauss queens problems, an Section 4 brings the conclusion. 2 The algorithm for function optimization In the following, we escribe the fuzzy pheromone ispersion algorithm we use in the remaining of this text, base in [1]. For any positive function f from [a, b] n to R, it aims at fining the arguments for f that lea to the global minimum. An ant is characterize by its path, that represents its solution to the problem in han. Each solution is a vector v = (v 1, v 2,..., v n ), where the i-th position contains the value for the i-th argument of f accoring to that solution. The path of an ant is a set of eges that coify the solution, i.e. each ege represents in fact an argument of f. The value for each ege on a path is taken from a iscrete set, as escribe below. The quality of the solution escribe by vector v is simply calculate as f(v). Roughly speaking, in any generation, each ant chooses the eges of its path, accoring to probabilities that are proportional to the amount of pheromone eposite on them. At the en of this step, for each ant k, we take the solution v(k) encoe by its path an calculate its quality f(v(k)). Then, each ant allowe to leave pheromone eposits an amount of pheromone on its path that is proportional to the quality of the solution associate to it an, moreover, it leaves smaller quantities of that amount on paths consiere to be close to its own path. This algorithm has thus two important aspects: 1. the better is an ant that is allowe to eposit pheromone, the bigger the trace it leaves; 2. the closer is an ege to the path of an ant that is allowe to eposit pheromone, the more pheromone this ege receives. Let f be a positive function [a, b] n R, an let φ be the ilatation from omain [0, 1] to omain [a, b], efine as φ(x) = a + x (b a). Let be a positive integer; we create a iscrete set of + 1 values in [0, 1] as U = {0, 1, 2,..., 1}. Matrix τ contains the amount of pheromone on each ege at any moment of the execution of the algorithm: each position τ i,j represents assigning the j-th value from U (i.e. j U) to the i-th component of the path. Thus the value store in position τ i,j (an amount of pheromone) is a weighing for the choice of taking value φ( j ) as the i-th argument of f. Here, contrary to the approach use in [1], the initial values store in matrix τ are obtaine at ranom. As sai previously, an ant eposits a certain amount of pheromone on its path an smaller quantities of this amount on eges close to its path. Here, the notion of closeness is implemente using the iscretization in U; the eges consiere to be close to an ege (i, j) are those in set {(i, j ) j j m an j U}, where the neighborhoo raius m N is a small natural number (0 m << 2 ). Therefore, given that (i, j) is in the ant s path, an ege (i, j ) will receive extra pheromone if j is one of the m values either below or above j in set U. Note that usual scheme (no pheromone ispersion) is moele by m = 0. We use a fuzzy set G with membership function G : N [0, 1] to moel the notion of ispersion of pheromone on the eges in the neighborhoo of a path: the closer an ege to the path, the larger its membership to G an the larger the extra quantity of pheromone it receives. Various types of convex fuzzy membership functions can be use for G. In the algorithm below, we use a triangular fuzzy set < c m, c, c + m >, with core equal to point value c an support given by values of N in interval [c m, c + m], an efine as k N, G c,m (k) = max(0, 1 c k m ). In this work we aress two mechanisms for the eposition of pheromone. In the first mechanism, all ants eposit pheromone in their corresponing paths an neighborhoos. In the secon mechanism, only the ant with the best solution in each generation is allowe to eposit pheromone (this ant can be thought of as the leaer of its gener-

3 ation). In algorithm FAC escribe bellow, each generation consists of a set of m ants. We also have: τ is a matrix n ( + 1) of float numbers. ants is a m n matrix in N, that contains the paths of the ants consiere in a generation. val is a vector of length m, containing the values f(v k ), where v k correspons to the solution obtaine through the k-th ant, i.e. f(v k ) = f(φ( ants k,0 ),..., φ( ants k,n 1 )). vm is the best value of val an nv is the number of evaluations of function f. Algorithm F AC(m,, m, α, ρ, eval max, ɛ, tts) m, an m are efine as above; α an ρ are ant colony parameters inicating the accentuation of probabilities an the evaporation rate, respectively; eval max is the maximal number of evaluations of function f; ɛ > 0 is a lower boun for function f; tts is a boolean variable; if tts = true then all ants leave pheromone, if tts = f alse only the leaer oes it. Initialize matrix τ with a ranom value in ]0,1]. While vm > ɛ an nv < eval max o 1. Creation of ant paths: for i = 0 to n 1 o let normp be the normalize vector of probabilities obtaine from (τ i ) α. for k = 0 to m 1 o create ants k,i U accoring to probabilities normp. 2. Evaluation: for k = 0 to m 1 o val(k) = f(v k ); let vm be the lowest value in val an b the ant evaluate with vm (the best ant). 3. Evaporation: matrix τ becomes the prouct of itself with (1 ρ). 4. Pheromone Upating: if tts then (* upating with all ants *) for k = 0 to m 1 o for i = 0 to n 1 o x ants k,i for j = max(0, x m) to min(, x + m) o τ i,j τ i,j + G x,m(j) val(k) else (* upating only with the best ant b *) for i = 0 to n 1 o x ants b,i for j = max(0, x m) to min(, x + m)) o τ i,j τ i,j + G x,m(j) vm 5. Incrementation of nv with nv + m. At the en of an execution, the algorithm elivers the best value (v, f(v)) an the number of evaluations of f taken to reach it. 3 Tests an results In the following we present sensibility analyses on the parameters of the algorithm above regaring three problems, two numerical (Rastrigin an tripo functions) an one symbolic (Gauss queens problem). For all the problems, the results presente are the average of 100 trials. Unless state otherwise, we have use evaporation rate ρ = 0.1 an accentuation coefficient α =.5. In all our experiments, the scheme in which only the best ant leaves pheromone fare much worse than the case where all ants leave pheromone an, for this reason, in the figures we only illustrate this secon scheme (tts = true). 3.1 Experiments with the Rastrigin function The Rastrigin function is efine as f R (x) = [x 2 i cos(2πx i )]/400, i=1,n where n is the number of imensions of vector x an x [ 30, 30]. This function is epicte for the one-imensional case in Figure 1.

4 3.1.2 Evaporation rate Figure 1: Rastrigin function (single imension). In each generation, the pheromone is ecrease losing a part ρ of it. Figure 3 shows the average number of evaluations the optimum with m = 10, varying ρ in [0, 0.9]. The best results are always roughly aroun ρ = 0.1 to 0.4. We performe experiments for this function in imension 1, using a iscretization factor of = 100. With this factor, the optimal value for f R is 10 4, reache with x = Neighborhoo raius We stuie the behavior of the neighborhoo raius m for function f R : Figure 2.a (respec. Figure 2.b) epicts the average number of evaluations taken to reach the optimum using between 50 to 250 (respec. 2 an 30 ) ants. We can see that the best results are obtaine with a very small number of ants, an that there is no nee to use raius values larger than 8. Figure 3: Number of evaluations per evaporation rate for the 1-imensional Rastrigin function Accentuation of probabilities coefficient ants 50 ants 150 We varie the accentuation of probabilities parameter α for the Rastrigin function in imension 1, using m = 10. The experiments confirm that the best values are aroun α = 0.5 (see Figure 4). a) m b) Figure 2: Number of evaluations per neighborhoo raius for the 1-imensional Rastrigin function: a) 50 to 250 ants an b) 2 to 30 ants. In these experiments, the average number of evaluations taken to reach the minimum in the case of no fuzzy trace use (m = 0) is about twice as high as the ones using fuzzy trace, an for this reason this case is not shown in the graphics. Figure 4: Number of evaluations per accentuation of probabilities for the 1-imensional Rastrigin function. 3.2 Experiments with the tripo function The tripo function is very simple, but has three minima (see Figure 5), an is quite ifficult to minimize with stochastic heuristics [5]. It is efine as:

5 if y < 0 then f T (x, y) = x + y + 50 else if x < 0 then f T (x, y) = 1 + x y 50 else f T (x, y) = 2 + x 50 + y 50. Figure 6 shows the number of evaluations taken by the algorithm to reach the optimum for the tripo function with x an y in omain [ 100, 100], varying the number of ants an the neighborhoo raius m, using iscretization factor = Experiments with a symbolic problem The Gauss queens problem consists in placing n queens on a n n chessboar, in such a way that the queens woul not be able attack one another, consiering the movements the queen is allowe to take in chess. The possible configurations can be represente by a sequence of n values taken from the set {1, 2,..., n 1}, where the value in the i-th position in the sequence inicates the line of the queen in the i-th column. A set of solutions for n from 4 to 9 are for instance (see Figure 7): chessboar 4 4 ( ) chessboar 5 5 ( ) chessboar 6 6 ( ) chessboar 7 7 ( ) chessboar 8 8 ( ) chessboar 9 9 ( ) Figure 5: Tripo function for x, y [ 100, 100]. The best results were obtaine with a large number of ants (from 100 to 500) an relatively small values for m (between 2 an 6). With m greater than 7, results from 50 to 1000 are quite similar. The use of a small number of ants (2 to 10) i not yiel goo results, particularly for small values of m (between 0 an 7). Figure 6: Number of evaluations per neighborhoo raius for the tripo function: 5 to 500 ants. Figure 7: A solution for the 7 7 queens problem. A solution to the problem can be foun by minimizing function f gauss, that computes the number of couples of queens in catching positions on the chessboar. For example, f gauss (1, 3, 2, 5, 5) = 4 an f gauss (2, 7, 5, 3, 1, 6, 4) = 0. The ants algorithm generates n-imensional vectors in [1, n] n an so we have use iscretization factor = n 1. In Figures 8.a an 8.b, we see that for a 7 7 chessboar, the best results are obtaine with a small number of ants an with raius aroun m = 4 (values for m = 0 an m = 1 are too large an are not plotte). We see that the use of a larger number of ants oes not yiel better results, but that even then the best value for m is still aroun 4. Note also that a very small number of ants (2 to 10) give results worse than for a large number of ants (100 to 350) in the case of no pheromone ispersion (m = 0).

6 a) of 7 7 points (> ), the best results are obtaine with a small number of ants (2 to 10) an m aroun 2 to 6. As future work, we inten to exten the sensibility analyses. In particular, we woul like to verify the sensibility of the algorithm in relation to other types of fuzzy set shapes for pheromone ispersion, such as the Gaussian curve. Acknowlegements This work has been partially supporte by the Generalitat e Catalunya uner grant 2005-SGR References b) Figure 8: Number of evaluations per neighborhoo raius for the 7 7 f gauss function: a) 2 to 10 ants an b) 50 to 500 ants. 4 Conclusion We presente a stuy on the tuning of parameters of ant colony systems with fuzzy pheromone ispersion on some optimization problems. In this algorithm, the closer is an ege to the path in which an ant eposits pheromone, the more pheromone this ege shall receive. Moreover, the amount of pheromone eposite by an ant is proportional to the quality of the solution it represents. Using a a linear fuzzy set we verify that usually only a relatively small number of ants an small values for the neighborhoo raius that characterizes the ispersion are neee to obtain goo results. In all experiments, no ispersion of pheromone always yiele inferior results. In the case of Rastrigin function, using here a very small search space, the best results were obtaine with the number of ants varying between 2 an 20 an with neighborhoo raius aroun 8. For the tripo function, with a search space of aroun points, one obtains goo results with 50 to 100 ants an m between 2 an 6. For the Gauss problem with 7 queens, with a search space [1] J. C. Becceneri, S. Sanri, Function optimization using ant colony systems with pheromone ispersion, Proc. IPMUÕ06 (11th Intl. Conf. on Info. Processing an Management of Uncertainty in Knowlegebase Systems), Paris (Fr), jul [2] J. C. Becceneri, S. Sanri, E.F.P. Luz, Ant colony systems with pheromone ispersion for TSP, Proc. 2n LNCC Meeting on Computational Moelling, Petrópolis (Br), aug [3] E. Bonabeau, M. Dorigo, an G. Theraulaz. From natural to artificial swarm intelligence. Oxfor university Press, [4] M. Dorigo, G. Di Caro, an L.M. Gambarella. Ant algorithms for iscrete optimization. Artificial Life, Vol. 5, No. 3, pp , [5] L. Gacogne, Methos to apply operators in a steay state evolution algorithm. In Proc. IPMU 2006, Paris (Fr), [6] R.P. Souto, H.F. e Campos Velho, S. Stephany, S. Sanri, Estimating Vertical Chlorophyll Concentration in Offshore Ocean Water Using a Moifie Ant Colony System, submitte.