Two-dimensional Disjunctively Constrained Knapsack Problem: Heuristic and exact approaches

- The two-dimensional knapsack problem with conflict graphs is solved.
- A greedy randomized heuristic with memory list and repacking is proposed.
- An integer formulation based on a location-allocation model is investigated.
- The formulation is strengthened with feasibility tests, bounds and valid cuts.
- The approaches are also extended for the version with complete shipment constraint.

This work deals with the 0-1 knapsack problem in its two-dimensional version considering a conflict graph, where each edge in this graph represents a pair of items that must not be packed together. This problem arises as a subproblem of the bin packing problem and in supply chain scenarios. We propose some integer programming formulations that are solved with a branch-and-cut algorithm. The formulation is based on location-allocation variables mixing the one- and two-dimensional versions of this problem. When a candidate solution is found, a feasibility test is performed with a constraint programming algorithm, which verifies if it satisfies the two-dimensional packing constraints. Moreover, bounds and valid cuts are also investigated. A heuristic that generates iteratively a solution and has components of Tabu search and Simulated Annealing approaches is proposed. The results are extended to consider complete shipment of items, where subsets of items all have to be loaded or left out completely. This constraint is applied in many real-life packing problems, such as packing parts of machinery, or when delivering cargo to different customers. Experiments on several instances derived from the literature indicate the competitiveness of our algorithms, which solved 99% of the instances to optimality requiring short computational time.