Lower bounds for three-dimensional multiple-bin-size bin packing problems

The three-dimensional multiple-bin-size bin packing problem (MBSBPP) is the problem of packing a set of boxes into a set of bins when several types of bins of different sizes and costs are available and the objective is to minimize the total cost of the bins used for packing the boxes. We present a study of lower bounds for this packing problem. We have developed new bounds based on integer programming formulations of some relaxations of the original problem. These formulations are enhanced with logical considerations. The proposed bounds are compared with other existing bounds in an extensive computational study, including two- and three-dimensional instances with up to 100 boxes, some of them taken from the literature and others adapted from the classical Bin Packing Problem. The proposed bounds improve the results of previous bounds by more than 10%, though at a higher computational cost.