In this paper, we study the {\em green bin packing} (GBP) problem where $β\ge 0$ and $G \in [0, 1]$ are two given values as part of the input. The energy consumed by a bin is $\max \{0, β(x-G) \}$ where $x$ is the total size of the items packed into the bin. The GBP aims to pack all items into a set of unit-capacity bins so that the number of bins used plus the total energy consumption is minimized. When $β= 0$ or $G = 1$, GBP is reduced to the classic bin packing (BP) problem. In the {\em constrained green bin packing} (CGBP) problem, the objective is to minimize the number of bins used to pack all items while the total energy consumption does not exceed a given upper bound $U$. We present an APTAS and a $\frac 32$-approximation algorithm for both GBP and CGBP, where the ratio $\frac 32$ matches the lower bound of BP. Keywords: Green bin packing; constrained green bin packing; approximation scheme; offline algorithms

翻译：本文研究{\em 绿色装箱}（GBP）问题，其中$β\ge 0$和$G \in [0, 1]$是作为输入给定的两个参数。每个箱子的能耗为$\max \{0, β(x-G) \}$，其中$x$表示装入该箱子的物品总尺寸。GBP的目标是将所有物品装入一组单位容量的箱子，使得所用箱子数量与总能耗之和最小化。当$β= 0$或$G = 1$时，GBP退化为经典装箱（BP）问题。在{\em 约束绿色装箱}（CGBP）问题中，目标是在总能耗不超过给定上界$U$的条件下，最小化装载所有物品所需的箱子数量。我们针对GBP和CGBP分别提出了一个近似多项式时间方案（APTAS）和一个$\frac 32$近似算法，其中$\frac 32$的近似比与BP问题的下界相匹配。关键词：绿色装箱；约束绿色装箱；近似方案；离线算法