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

We propose several techniques to compute lower and upper bounds for this problem. For what concerns lower bounds, we present combinatorial techniques with guaranteed worst case and a more complex bound based on a column generation algorithm. We also present a technique to compute, in a fast heuristic way, dual information that is used to strengthen the convergence of the column generation. For what concerns upper bounds, we present a large set of constructive heuristics followed by a Variable Neighborhood Search algorithm. Our heuristic techniques are aimed at both computing upper bounds and strengthening the behavior of the lower bounds in a matheuristic fashion. Extensive computational tests show the effectiveness of the proposed algorithms.