A generalization for single-machine scheduling with deteriorating jobs to minimize earliness penalties

We consider a single-machine scheduling problem with deteriorating jobs in which the due dates are determined by the equal slack (SLK) method. By a deteriorating job, we mean that the job's processing time is an increasing function of its starting time. We model job deterioration as a function that is proportional to a linear function of time. The objective is to minimize the total weighted earliness penalty subject to no tardy jobs. We prove that two special cases of the problem remain polynomially solvable. The first case is the problem with equally weighted monotonous penalty objective function and the other case is the problem with weighted linear penalty objective function.