Single-machine scheduling with simple linear deterioration 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 method. In this model, the processing time of a job is defined as a simple linear function of its starting 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.