Combining dynamic programming with filtering to solve a four-stage two-dimensional guillotine-cut bounded knapsack problem

Abstract

The two-dimensional knapsack problem consists in packing a set of small rectangular items into a given large rectangle while maximizing the total reward associated with selected items. We restrict our attention to packings that emanate from a k-stage guillotine-cut process. We introduce a generic model where a knapsack solution is represented by a flow in a directed acyclic hypergraph. This hypergraph model derives from a forward labelling dynamic programming recursion that enumerates all non-dominated feasible cutting patterns. To reduce the hypergraph size, we make use of further dominance rules and a filtering procedure based on Lagrangian reduced costs fixing of hyperarcs. Our hypergraph model is (incrementally) extended to account for explicit bounds on the number of copies of each item. Our exact forward labelling algorithm is used to solve the max-cost flow model in the base hyper-graph with side constraints to model production bounds. Results of numerical comparison against existing approaches are reported on instances from the literature and on datasets derived from a real-world application.