Motivated by green manufacturing, this paper investigates a scheduling with rejection problem subject to an energy consumption constraint. Machines are associated with non-uniform energy consumption rates, defined as the energy consumed per unit time. Each job is either rejected with a rejection penalty or accepted and scheduled on some machine for processing, which incurs energy consumption. The problem aims to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs while the total energy consumption is bounded by a given threshold. In this paper, when the number of machines is part of the input, we develop the first $(2+ε)$-approximation algorithm for any fixed constant $ε$ and a simple QPTAS as well as a PTAS for uniform energy consumption rates. Moreover, we present an FPTAS when the number of machines is a fixed constant.

翻译：受绿色制造理念的驱动，本文研究了一类在能耗约束下带拒绝的调度问题。机器具有非均匀的能耗率，即单位时间内消耗的能量。每个工件要么被拒绝并产生拒绝惩罚，要么被接受并安排在某台机器上加工，此过程将消耗能量。该问题的目标是在总能耗不超过给定阈值的条件下，最小化被接受工件的完工时间（makespan）与被拒绝工件的总惩罚之和。本文中，当机器数量是输入的一部分时，我们针对任意固定常数 $ε$ 提出了首个 $(2+ε)$-近似算法，并针对均匀能耗率的情况给出了一个简单的拟多项式时间近似方案（QPTAS）以及一个多项式时间近似方案（PTAS）。此外，当机器数量为固定常数时，我们提出了一个完全多项式时间近似方案（FPTAS）。