Maximum lateness minimization in one-dimensional bin packing

In the One-dimensional Bin Packing problem (1-BP) items of different lengths must be assigned to a minimum number of bins of unit length. Regarding each item as a job that requires unit time and some resource amount, and each bin as the total (discrete) resource available per time unit, the 1-BP objective is the minimization of the makespan C-max = max(j)(C-j). We here generalize the problem to the case in which each item j is due by some date d(j): our objective is to minimize a convex combination of C-max and L-max = max(j){C-j-d(j)}. For this problem we propose a time-indexed Mixed Integer Linear Programming formulation. The formulation can be decomposed and solved by column generation relegating single-bin packing to a pricing problem to be solved dynamically. We use bounds to (individual terms of) the objective function to address the oddity of activation constraints. In this way, we get very good gaps for instances that are considered difficult for the 1-BP.