Batching work and rework processes with limited deterioration of reworkables
Computers and Operations Research
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Single machine parallel-batch scheduling with deteriorating jobs
Theoretical Computer Science
Integrated production and delivery scheduling with disjoint windows
Discrete Applied Mathematics
Single-machine makespan minimization scheduling with nonlinear shortening processing times
Computers and Operations Research
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In this paper we consider the problem of scheduling n jobs on a single machine, where the jobs are processed in batches and the processing time of each job is a step function depending on its waiting time, which is the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. For job i, if its waiting time is less than a given threshold value D, then it requires a basic processing time a i ; otherwise, it requires an extended processing time a i +b i . The objective is to minimize the completion time of the last job. We first show that the problem is NP-hard in the strong sense even if all b i are equal, it is NP-hard even if b i =a i for all i, and it is non-approximable in polynomial time with a constant performance guarantee Δ $\mathcal {P}=\mathcal{NP}$ . We then present O(nlog驴n) and O(n 3F驴1log驴n/F F ) algorithms for the case where all a i are equal and for the case where there are F, F驴2, distinct values of a i , respectively. We further propose an O(n 2log驴n) approximation algorithm with a performance guarantee $\Delta\le1+\lfloor\frac{m^{*}}{2}\rfloor/m^{*}\le3/2$ for the general problem, where m * is the number of batches in an optimal schedule. All the above results apply or can be easily modified for the corresponding open-end bin packing problem.