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Scheduling deteriorating jobs to minimize makespan. (English) Zbl 0936.90026
Summary: We consider a single-machine problem of scheduling $n$ independent jobs to minimize makespan, in which the processing time of job ${J}_{j}$ grows by ${w}_{j}$ with each time unit its start is delayed beyond a given common critical date $d$. This processing time is ${p}_{j}$ if ${J}_{j}$ starts by $d$. We show that this problem is NP-hard, give a pseudopolynomial algorithm that runs in $O\left(nd{\sum }_{j=1}^{n}{p}_{i}\right)$ time and $O\left(nd\right)$ 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 $O\left({n}^{2}d\left(D-d\right){\sum }_{j=1}^{n}{p}_{j}\right)$ time and $O\left(nd\left(D-d\right)\right)$ space, the other runs in $O\left(nd{\sum }_{j=1}^{n}{w}_{j}{\left({\sum }_{j=1}^{n}{p}_{j}\right)}^{2}\right)$ time and $O\left(nd{\sum }_{j=1}^{n}{w}_{j}{\sum }_{j=1}^{n}{p}_{j}\right)$ space.
##### MSC:
 90B35 Scheduling theory, deterministic 90C39 Dynamic programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut