Solving the Firefighter Problem with two elements using a multi-modal Estimation of Distribution Algorithm

The Firefighter Problem (FFP) is an optimization problem of developing an optimal strategy for assigning firemen to nodes of a given graph in successive iterations of a simulation of spread of fires in the graph. This paper focusses on an extension of the original FFP, namely the Bi-Firefighter Problem (FFP2), where the second element (water) is introduced. FFP2 corresponds to the practical optimization problems, where more than one disease is spreading in the environment, and the objective is to minimize the total loss. Since the loss may come from two different sources, each of which causes different damages, the objective function is more complex than in the case of the original FFP. In this paper, an evolutionary approach to FFP2, the EA-FFP2 algorithm, based on a multi-modal Estimation of Distribution Algorithm (EDA), is proposed. EA-FFP2 was validated on a number of benchmark FFP2 instances that were also solved by the branch and bound algorithms or the heuristic local search algorithms run for a large number of iterations for a long time. Computational experiments confirmed that EA-FFP2 was capable of solving FFP2 and finding solutions close to the optima determined by the branch and bound algorithms or to the quasi-optima determined by exhaustive local search.