Bi-objective Approximation Scheme for Makespan and Reliability Optimization on Uniform Parallel Machines

We study the problem of scheduling independent tasks on a set of related processors which have a probability of failure governed by an exponential law. We are interested in the bi-objective analysis, namely simultaneous optimization of the makespan and the reliability. We show that this problem can not be approximated by a single schedule. A similar problem has already been studied leading to a $\big $ -approximation algorithm (i.e.for any fixed value of the makespan, the obtained solution is optimal on the reliability and no more than twice the given makespan). We provide an algorithm which has a much lower complexity. This solution is finally used to derive a (2 + i¾?, 1)-approximation of the Pareto set of the problem, for any i¾?> 0.