Single-Machine Scheduling Problems with Bounded Convex Resource Consumption Functions

We consider three single-machine scheduling problems in which the processing time of each job is inversely proportional to the power of of the resource consumption amount. Given a set of jobs and the resource lower and upper bounds for each job, the problem is to determine the job sequence and the number of resources allocated to each job. We consider three different objectives. The first (and the third) objective is to minimize the sum of the makespan (and total completion time, respectively) and the total resource consumption cost. The second is to minimize the makespan subject to the total resource consumption amount. We show that the three problems are polynomially solvable.