We consider a bicriteria permutation flow shop scheduling problem, where each operation’s processing time varies inversely with the power of the resource allocation assigned to the machine. The two criteria are the makespan and the total resource consumption cost, resulting in the formulation of four distinct models. The first model focuses on minimizing the sum of both criteria. In the second and third models, one criterion is minimized under a constraint imposed on the other. The fourth model aims to identify Pareto front. We show that these four models can be solved in polynomial time for the two-machine case; however, they become strongly NP-hard when extended to the three-machine case.