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 $k$-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 fair electricity distribution. In many developing countries, the total electricity demand is higher than the supply capacity. We 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箱子容量，装箱问题要求将这些物品装入尽可能少的箱子，使得每个箱子中物品尺寸之和不超过容量。我们定义了一种新变体，称为$k$次装箱问题（$k$BP），其目标是将物品装箱，使得每个物品恰好出现在$k$个不同的箱子中，每次出现一次。我们将一些现有的装箱近似算法推广到求解$k$BP，并分析其性能比。$k$BP的研究源于公平电力分配问题。在许多发展中国家，总电力需求高于供电容量。我们表明，$k$次装箱可用于以公平且高效的方式分配电力。特别是，我们实现了首次适应算法和首次适应递减装箱算法的推广版本来求解$k$BP，并将这些推广应用于实际电力需求数据。我们证明，我们的推广算法在解决同一问题时优于现有的启发式解决方案。