One-dimensional bin packing with pattern-dependent processing time

In this paper the classical one-dimensional bin packing problem is integrated with scheduling elements: a due date is assigned to each item and the time required to process each bin depends on the pattern being used. The objective is to minimize a convex combination of the material waste and the delay costs, both significant in many real-world contexts. We present a novel pattern-based mixed integer linear formulation suitable for different classical scheduling objective functions, and focus on the specific case where the delay cost corresponds to the maximum tardiness. The formulation is tackled by a branch-and-price algorithm where the pricing of the column generation scheme is a quadratic problem solved by dynamic programming. A sequential value correction heuristic (SVC) is used to feed with warm starting solutions the column generation which, in turn, feeds the SVC with optimal prices so as to compute refined feasible solutions during the enumeration. Computational tests show that both column generation and branch-and-price substantially outperform standard methods in computing dual bounds and exact solutions. Additional tests are presented to analyze the sensitivity to parameters’ changes.