Just-in-time two-dimensional bin packing

Abstract

This paper considers the on-time guillotine cutting of small rectangular items from large rectangular bins. Items assigned to a bin define the bins’ processing time. Consequently, an item inherits the completion time of its assigned bin. Any deviation of an item’s completion time from its due date causes either earliness or tardiness penalties. This just-in-time two-dimensional bin packing problem (JITBP) combines two difficult discrete optimization problems: Bin packing and total weighted earliness tardiness single machine scheduling. It is herein modeled as an integrated constraint program, augmented with two sets of logically redundant constraints that speed the search. The first set uses the concept of dual feasible functions. It focuses on bin packing feasibility. The second is the result of a linear program that schedules filled bins on a single machine. As an alternative to this integrated model, this paper proposes two decomposition cut-and-check approaches that define the master problem (MP) as a relaxation of JITBP where the items are reduced to dimensionless entities. They then reestablish the geometric feasibility of the MPs’ solutions by iteratively augmenting MP with Benders cuts generated from the subproblems. The two approaches are similar in concept except that one implements MP as a constraint program (CP) while the second implements it as a mixed-integer program (MIP). Because JITBP is computationally challenging, we test all approaches under a number of heuristic assumptions, which include a maximum runtime for the MIP and CP solvers. The results provide computational evidence that the integrated constraint programming approach performs relatively well, and outperforms the decomposition approach whose MP is a CP. However, both approaches are outperformed by the decomposition approach whose MP is a warm-started MIP.