Scheduling deteriorating jobs to minimize makespan

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 $O(nd \sum^n_{j=1}p_j)$ 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) $ \sum^n_{j=1}p_j)$ time and O(nd(D − d)) space, the other runs in $O(nd \sum^n_{j=1}) w_j(\sum^n_{j=1}$ pj)2) time and $O(nd \sum^n_{j=1} w_j \sum^n_{j=1}p_j)$ pj) space. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 511–523, 1998