The two-dimensional knapsack problem with splittable items in stacks

The two-dimensional knapsack problem consists in packing rectangular items into a single box such that the total value of packed is maximized. In this article, we restrict to 2-stage non-exact guillotine cut packings and consider variant with splittable items: each item can be horizontally as many times needed, may contain only portion an item. This arises semifluid items, like tubes small radius, which has property behave fluid one direction, solid other directions. addition, are stable stacks, is, at most laid on top another item, necessarily wider than itself. We establish NP-hard, propose integer linear formulation. exhibit very strong dominance properties structure extreme solutions, call canonical packings. enables us design polynomial time algorithms for some special cases pseudo-polynomial algorithm general case. also develop Fully Polynomial Time Approximation Scheme (FPTAS) case where height does not exceed box. Finally, numerical results reported assess efficiency our algorithms.