In this paper, we consider several scheduling problems with rejection on \(m\ge 1\) identical machines. Each job is either accepted and processed on the machines, or it is rejected by paying a certain rejection cost. The objective is to minimize the sum of the k-th power of the makespan of accepted jobs and the total rejection cost of rejected jobs, where \(k>0\) is a given constant. We also introduce the conception of “job splitting" in our problems. First, we consider the single machine scheduling problem, i.e., \(m=1\). When job splitting is allowed, we propose an \(O(n\log n)\)-time optimal algorithm for the problem. When job splitting is not allowed, we show that this problem is polynomially solvable when \(k\in (0,1]\) and it becomes binary NP-hard when \(k>1\). Furthermore, for the NP-hard problem, we propose a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS). Finally, we also extend our problems and some results to \(m\ge 2\) identical parallel machines.

在本文中，我们考虑了在\(m\ge 1\)台相同机器上拒绝的几个调度问题。每个作业要么被机器接受并处理，要么通过支付一定的拒绝成本来拒绝。目标是最小化已接受作业的完工时间的k次方和已拒绝作业的总拒绝成本之和，其中\(k>0\)是给定常数。我们还在问题中引入了“作业拆分”的概念。首先，我们考虑单机调度问题，即\(m=1\)。当允许作业拆分时，我们提出一个\(O(n\log n)\)-问题的时间最优算法。当不允许作业分割时，我们表明，当\(k\in (0,1]\)时，这个问题是多项式可解的，并且当\(k>1\)时，这个问题变成二元 NP 困难问题。此外，对于 NP-针对这一难题，我们提出了伪多项式动态规划算法和完全多项式时间近似方案（FPTAS）。最后，我们还将我们的问题和一些结果扩展到\(m\ge 2\)个相同的并行机。