This paper deals with computational aspects of attack trees, more precisely, evaluating the expected adversarial utility in the failure-free game, where the adversary is allowed to re-run failed atomic attacks an unlimited number of times. It has been shown by Buldas and Lenin that exact evaluation of this utility is an NP-complete problem, so a computationally feasible approximation is needed. In this paper we consider a genetic approach for this challenge. Since genetic algorithms depend on a number of non-trivial parameters, we face a multi-objective optimization problem and we consider several heuristic criteria to solve it.