We consider a single-machine problem of scheduling n independent jobs to minimize makespan, in which the processing time of job Jj grows by wj with each time unit its start is delayed beyond a given common critical date d. This processing time is pj if Jj starts by d. We show that this problem is NP-hard, give a pseudopolynomial algorithm that runs in O(nd Σnj=1 pj) time and O(nd) space, and develop a branch-and-bound algorithm that solves instances with up to 100 jobs in a reasonable amount of time. We also introduce the case of bounded deterioration, where the processing time of a job grows no further if the job starts after a common maximum deterioration date D > d. For this case, we give two pseudopolynomial time algorithms: one runs in O(n2d(D – d) Σnj=1 pj) time and O(nd(D – d)) space, the other runs in O(nd Σnj=1 wj(Σnj=1 pj)2) time and O(nd Σnj=1 wj Σnj=1 pj) space.
