Lower and upper bounds for the Bin Packing Problem with Fragile Objects

We are given a set of items, each characterized by a weight and a fragility, and a large number of uncapacitated bins. Our aim is to find the minimum number of bins needed to pack all items, in such a way that in each bin the sum of the item weights is less than or equal to the smallest fragility of the items in the bin. The problem is known in the literature as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when one has to assign cellular calls to available channels by ensuring that the total noise in a channel does not exceed the noise acceptance limit of a call. We propose several techniques to compute lower and upper bounds for such problem. For what concerns lower bounds, we develop combinatorial techniques and a column generation algorithm. For upper bounds, we present a large set of heuristics followed by a Variable Neighborhood Search algorithm. We also show a few techniques to compute in a fast and heuristic way dual information that is then used to strengthen the valid lower bounds. One of this attempts improves the convergence of the column generation approach. Extensive computational tests show that the use of these heuristics techniques, both to compute upper bounds and to improve the lower bounds, is particularly effective.