# Ant Colony Optimization with Warm-Up
###### Abstract  
The Ant Colony Optimization (ACO) is a probabilistic technique inspired by the behavior of ants for solving computational problems that may be reduced to finding the best path through a graph. Some species of ants deposit pheromone on the ground to mark some favorable paths that should be used by other members of the colony. Ant colony optimization implements a similar mechanism for solving optimization problems. In this paper a warm-up procedure for the ACO is proposed. During the warm-up, the pheromone matrix is initialized to provide an efficient new starting point for the algorithm, so that it can obtain the same (or better) results with fewer iterations. The warm-up is based exclusively on the graph, which, in most applications, is given and does not need to be recalculated every time before executing the algorithm. In this way, it can be made only once, and it speeds up the algorithm every time it is used from then on. The proposed solution is validated on a set of traveling salesman problem instances, and in the simulation of a real industrial application for the routing of pickers in a manual warehouse. During the validation, it is compared with other ACO adopting a pheromone initialization technique, and the results show that, in most cases, the adoption of the proposed warm-up allows the ACO to obtain the same or better results with fewer iterations.  
heuristics; traveling salesman problem; TSP; ant colony optimization; metaheuristic; warm-up +
Footnote †: journal: Applied Soft Computing  
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current quantity of pheromone on each edge of the graph, and it, therefore, determines the probability to include each specific edge in a newly generated solution. In the classic version of the ACO, every time the algorithm is executed, all the elements of the pheromone matrix are set equal to a starting (generally low) value. Then, as the computed iterations increase, the pheromone on the most promising paths is increased and that on the less convenient paths is reduced. Although, in most real implementations, the graph of nodes is given and is the same in every execution. Furthermore, it was verified that, given a graph, in many cases, each time the ACO was executed, after a certain number of iterations, the pheromone matrix was always very similar. The warm-up procedure proposed in this paper aims to carry out a fine-tuning of the pheromone matrix on a specific graph, so that, every time the ACO is executed, it starts from an already weighted graph, where the promising paths were highlighted with a high level of pheromone and the bad paths excluded a priori. This process is supposed to reduce the number of iterations that the ACO requires to converge every time it is executed. The remainder of this paper is organized as follows. Firstly, a brief overview of the scientific contributions to ACO is presented in Section 2. Then, the warm-up procedure proposed in this paper is described in Section 2. The computational experiments are shown in Section 4, where the ACO with warm-up is compared to the classic ACO, and two other ACO versions that carry out an initialization of the pheromone matrix. Finally, the conclusions are presented in Section 5.