On the convergence of Ant Colony Optimization with stench pheromone

Ant Colony Optimization (ACO) has proved to be a powerful metaheuristic for combinatorial optimization problems. From a theoretical point of view, the convergence of the ACO algorithm is an important issue. In this paper, we analyze the convergence properties of a recently introduced ACO algorithm, called ACO with stench pheromone (ACO-SP), which can be used to solve dynamic traffic routing problems through finding the minimum cost routes in a traffic network. This new algorithm has two different types of pheromone: the regular pheromone that is used to attract artificial ants to the arc in the network with the lowest cost, and the stench pheromone that is used to push ants away when too many ants converge to that arc. As a first step of a convergence proof for ACO-SP, we consider a network with two arcs. We show that the process of pheromone update will transit among different modes, and finally stay in a stable mode, thus proving convergence for this given case.