Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{$k$-times bin-packing ($k$BP)}, where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by $k$-times bin-packing for some finite $k$. We also show that $k$-times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem.

翻译：给定不同尺寸的物品和固定的箱容量，装箱问题要求将这些物品装入最少数量的箱子中，使得每个箱子内物品尺寸之和不超过容量。我们定义了一种称为\emph{k次装箱（kBP）}的新变体，其目标是将物品装箱，使得每个物品恰好出现k次，且位于k个不同的箱子中。我们将一些现有的装箱近似算法推广以求解kBP，并分析其性能比。研究kBP的动机源于\emph{公平电力分配}问题。在许多发展中国家，总电力需求高于供电能力。我们证明，对于任意有限k，每个电力分配问题都可以通过k次装箱求解。我们还表明，k次装箱可用于以公平且高效的方式分配电力。具体而言，我们实现了First-Fit和First-Fit Decreasing装箱算法的推广版本以求解kBP，并将这些推广算法应用于实际电力需求数据。我们证明，我们的推广算法在相同问题上优于现有的启发式解决方案。