The prize-collecting single machine scheduling with bounds and penalties

This study investigates the prize-collecting single machine scheduling with bounds and penalties (PC-SMS-BP). In this problem, a set of n jobs and a single machine are considered, where each job \(J_j\) has a processing time \(p_{j}\), a profit \(\pi _{j}\) and a rejection penalty \(w_{j}\). The upper bound on the processing number is U. The objective of this study is to find a feasible schedule that minimizes the makespan of the accepted jobs and the total rejection penalty of the rejected jobs under the condition that the number of the accepted jobs does not exceed a given threshold U while the total profit of the accepted jobs does not fall below a specified profit bound \(\varPi \). We first demonstrate that this problem is NP-hard. Then, a pseudo-polynomial time dynamic programming algorithm and a fully polynomial time approximation scheme (FPTAS) are proposed. Finally, numerical experiments are conducted to compare the effectiveness of the two proposed algorithms.