In this paper, we consider a recently introduced packing problem in which a given set of weighted items with colors has to be packed into a set of identical bins, while respecting capacity constraints and the number of available bins, and minimizing the total number of times that colors appear in the bins. We review exact methods from the literature and present a fast lower bounding procedure that, in some cases, can also provide an optimal solution. We theoretically study the worst-case performance of the lower bound and the effect of the number of available bins on the solution cost. Then, we computationally test our solution method on a large benchmark of instances from the literature: quite surprisingly, all of them are optimally solved by our procedure in a few seconds, including those for which the optimal solution value was still unknown. Thus, we introduce additional harder instances, which are used to evaluate the performance of a constructive heuristic method and of a tabu search algorithm. Results on the new instances show that the tabu search produces considerable improvements over the heuristic solution, with a limited computational effort.

中文翻译：

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具有最小颜色碎片的装箱问题的界限和启发式算法


在本文中，我们考虑最近引入的打包问题，其中一组给定的带有颜色的加权项目必须打包到一组相同的垃圾箱中，同时尊重容量限制和可用垃圾箱的数量，并最小化总次数颜色出现在垃圾箱中。我们回顾了文献中的精确方法，并提出了一种快速下界程序，在某些情况下，它也可以提供最佳解决方案。我们从理论上研究了下界的最坏情况性能以及可用箱数对解决方案成本的影响。然后，我们在文献中的大量实例基准上计算测试我们的解决方法：令人惊讶的是，所有这些实例都在几秒钟内通过我们的程序得到最佳解决，包括那些最佳解决方案值仍然未知的实例。因此，我们引入了额外的更难的实例，用于评估构造性启发式方法和禁忌搜索算法的性能。新实例的结果表明，禁忌搜索在计算量有限的情况下比启发式解决方案产生了相当大的改进。