Single machine scheduling to minimise resource consumption cost with a bound on scheduling plus due date assignment penalties

We study a single-machine scheduling problem in a flexible framework, where both job processing times and due dates are decision variables to be determined by the scheduler. We consider the case where each of the job processing times is a convex decreasing function of the amount of non-renewable resource that is allocated to the corresponding processing operation. Moreover, we consider two of the more common due-date assignment methods. For each of the methods, our objective is to find a solution minimising the total resource consumption cost, given an upper bound on the value of the weighted number of tardy jobs plus due date assignment costs. Since the problem is known to be NP$ \mathcal NP $-hard, we focus on designing approximation algorithms.