We investigate a two-machine job shop scheduling problem with optional job rejection. The target is to look for a feasible schedule for the set of accepted jobs so that the sum of the makespan of the accepted jobs and the total penalty of the rejected jobs is minimized. We propose an exact pseudo-polynomial dynamic programming algorithm, a greedy \(\frac{\sqrt{5}+1}{2}\)-approximation algorithm, an LP-based \(\frac{e}{e-1}\)-approximation algorithm, and a fully polynomial time approximation scheme to solve it. We demonstrate that the proportionate case is \(\mathcal{N}\mathcal{P}\)-hard and provide an \(O(n^2)\)-time algorithm for the agreeable case.

我们研究了具有可选作业拒绝的两台机器作业车间调度问题。目标是为已接受的作业集寻找可行的时间表，以使已接受作业的完成时间和已拒绝作业的总惩罚之和最小化。我们提出了一种精确的伪多项式动态规划算法、一种贪婪的\(\frac{\sqrt{5}+1}{2}\)逼近算法、一种基于 LP 的\(\frac{e}{e-1 }\) -近似算法，以及解决该问题的完全多项式时间近似方案。我们证明了比例情况是\(\mathcal{N}\mathcal{P}\)困难的，并为合适的情况提供了\(O(n^2)\)时间算法。