An exact approach for the two-dimensional strip packing problem with defects

The paper studies the two-dimensional strip packing problem with defects (2DSPP_D), focusing on packing rectangular items orthogonally within a fixed-width, variable-height strip that includes defects. The objective is to minimize the height of the strips used. This problem is important because it appears in many real-world applications, such as cutting processes for materials like paper or steel coils, where the goal is to minimize waste, and in continuous berth allocation problems, where the objective is to minimize total unloading time. Although this problem has a wide range of practical applications, it is rarely discussed in the literature. In this paper, we present an integer programming formulation and an exact two-stage approach. In the first stage of the exact method, the 2DSPP_D is converted into a two-dimensional orthogonal placement problem (2DOPP) by fixing the strip height. In the second stage, we solve this placement problem using a Benders’ decomposition method. If the 2DOPP proves infeasible, we increase the strip height and repeat the algorithm. We employ customized preprocessing techniques, lower bounding methods, and valid inequalities to enhance the two-stage approach. Additionally, we propose a skyline-based adaptive iterative search heuristic algorithm that provides tight upper bounds for the 2DSPP_D, incorporating a randomized local search strategy and an adaptive search strategy to enhance algorithm effectiveness. Extensive computational results show that our approach proves optimal solutions for small and medium-sized benchmark instances within a reasonable time and achieves close gap values between upper and lower bounds for large benchmark instances.