Bounds and heuristic algorithms for the bin packing problem with minimum color fragmentation

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.