New approaches to evaporation in ant colony optimization algorithms E. Foundas A. Vlachos Department of Informatics University of Piraeus Piraeus 85 34 Greece Abstract In Ant Colony Algorithms (ACA), artificial ants construct a solution by building a path on a construction graph. The ants behavior is specified by defining start states, construction rules, transition rules, pheromone update rules evaporation mechanisms. In this paper we introduce new approaches, based on the generation function geometrical, algebric explanation of the convergence, for deriving the evaporation mechanisms in Ant Colony Optimization algorithms. Keywords phrases : Ant colony optimization, ant colony algorithms, generation function, evaporation, velocity of evaporation, geometrical-algebric convergence.. Introduction Ant Colony Optimization (ACO) meta-heuristics have proved to be remarkably successful in solving a range of discrete optimization problems. ACO algorithms have been applied to an increasingly wide range of problems, including the Traveling Salesman Problem (TSP) [], Vehicle Routing Problem (VRP) [2], Quadric Assignment Problem (QAP) [3], bus driver scheduling [4], network routing [5], sequential ordering [6] graph colouring [7]. E-mail: efountas@unipi.gr E-mail: AVlachos@unipi.gr Journal of Interdisciplinary Mathematics Vol. 9 (2006), No., pp. 79 84 c Taru Publications
80 E. FOUNDAS AND A. VLACHO ACO algorithms are based on a probabilistic model called pheromone model that is used to model the pheromone trails. Artificial ants construct a solution by building a path on a construction graph G = (C, L) where the elements of L (called connections) fully connect C (set of components). To avoid a quick convergence [8], of all the ants towards a sub-optimal path, an exploration mechanism is added, similar to real pheromone trails, artificial pheromone trails evaporation. The pheromone evaporation is triggered by the environment it is used as a tool to avoid search stagnation to allow the ants to explore new space regions. In this paper we introduce new mathematical approaches, based on the generation function geometrical, algebric explanation of the convergence, for deriving the evaporation mechanism in Ant Colony Optimization algorithms. The paper is organized as follows. In section 2 we present the generation function for finding the pheromone evaporation in time t. Section 3 presents the geometrical explanation of convergence of evaporation. Finally, in section 4 we introduce the algebric explanation of convergence of evaporation. 2. Evaporation of pheromone For the pheromone evaporation, in time t, ρ t, we know that the following relation is satisfied 0 ρ t. with For the time t +, the ρ t+ relation can be written like the following: ρ t+ ρ t + ( ρ t ) βρ t (2.) 0 ρ t, 0, 0 β where, β constant. We will prove that: 0 ρ t+. (2.2) Proof. Indeed, it is ρ t+ = ρ t + ( ρ t ) βρ t ρ t + ( ρ t ) + 0 ρ t = ρ t+ = ρ t + ( ρ t ) βρ t ρ t + 0 ( ρ t ) ρ t = 0.
ANT COLONY OPTIMIZATION ALGORITHMS 8 Thus the relation (2.2) is true. Considering the relation (2.2) we write the alternative form of the equation (2.): ρ t = kρ t + (2.3) where k is the velocity of pheromone evaporation, for which we can write the following relations: k = β, 0 k. (2.4) The generation function for ρ t is: f (x) = f (x) = t= = k ρ t x t (k ρ t + )x t = k f (x) = k x f (x) = ρ t x t+ + a ρ t x t + a = k x f (x) + a a (x)(). The function f (x) can be written: thus : t= a (x)() A x + B A = f (x) ak k, B = a k k = k k x + (kx) t + ρ t x t + a x t. x t
82 E. FOUNDAS AND A. VLACHO So: thus: f (x) ρ t = ( k k kt + ) x t k k kt + a ( ρ 0 ) k t +, if k = ρ t = ρ 0, if k =. Easily results that the Probability Density Function (PDF) of the relation (2.3) is: ( f (t) t )( ) 0 ln k k t k t + 0, if k =, 0 t t 0 (2.5) its Cumulative Distribution Function (CDF) is: ( F(t) = t ) ( ) 0 ln k k t 0 ln k (kt ) + t. (2.6) 3. Geometrical explanation of convergence we have For the equation: ρ t = k ρ t + a, 0 k 0 a (3.) lim ρ t = t, if 0 k, = 0 0, if 0 k, a = 0 the convergence of the equation (3.) is the value = 0 (Figure )., when 0 < k < The point B (0, ρ ) corresponds in the first expression ρ = kρ 0 +, whereas the point B 2 (0, ρ 2 ) corresponds to the second expression ρ 2 ( = kρ + ). Continuing the operation, finally the cutting point T, a of the bisectional ρ t = ρ t the line ρ t = kρ t + k is the convergence of the sequence.
ANT COLONY OPTIMIZATION ALGORITHMS 83 t B 2 B M 0 B 0 0 A 0 A t Figure Diagram explanation of convergence 4. Algebric explanation of convergence For the equation (3.) we have, for t =, 2, 3,..., t: ρ ρ 2 = kρ 0 + = k 2 ρ 0 + ( + k) ρ 3 = k 3 ρ 0 + ( + k + k 2 )...... ρ t = k t ρ 0 + ( + k +... + k t ) thus ρ t = k t ρ 0 + t ρ t = lim ρ t = t + ( ρ 0 ) k t +, if 0 < k <, = 0 0, if 0 < k <, a = 0.
84 E. FOUNDAS AND A. VLACHO References [] M. Dorigo L. M. Gambardella, Ant colonies for the travelling salesman problem, Biosystems, Vol. 43 (997), pp. 73 8. [2] C. Fountas A. Vlachos, Ant colonies optimization (ACO) for the solution of the vehicle routing problem (VRP), Journal of Information Optimization Sciences, Vol. 26 () (2005), pp. 35 42. [3] V. Maniezzo A. Coloarni, The ant system applied to the quadratic assignment problem, IEEE Transaction on Knowledge Data Engineering, Vol. (999), pp. 769 778. [4] P. Forsyth A. Wren, An ant system for bus driver scheduling, presented at 7th International Workshop on Computer Aided Scheduling of Public Transport, Boston, USA, 997. [5] G. N. Varela M. C. Sinclair, Ant colony optimization for virtual wavelength path routing wavelength allocation, presented at Congress on Evolutionary Computation (CEC 99), Washington DC, USA, 999. [6] L. M. Gambardella M. Dorigo, HAS-SOP: an hybrid ant system for the sequential ordering problem, IDSIA, Lugano, Switzerl, Technical Report IDSIA--99, 997. [7] D. Costa A. Hertz, Ants can colour graphs, Journal of the Operations Research Society, Vol. 48 (997), pp. 295 305. [8] M. Dorigo G. D. Caro, The ant colony optimization mataheuristic, in New Ideas in Optimization, D. Corne, M. Dorigo F. Glover (eds.), London: McGraw-Hill, pp. 32, 999. Received August, 2005