A cutting plane method and a parallel algorithm for packing rectangles in a circular container

• Published in: European journal of operational research Vol. 303; no. 1; pp. 114 - 128
• Main Authors: Silva, Allyson, Coelho, Leandro C., Darvish, Maryam, Renaud, Jacques
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 16.11.2022
• Subjects: Circular container; Combinatorial optimization; Cutting plane method; Packing; Parallel computing; Parallel computing; Circular container; Packing; Combinatorial optimization; Cutting plane method
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2022.02.023

•We propose the first linear model to pack rectangles in a circle.•A cutting plane method approximates a circular container from a square.•The feasibility of packings is effectively done with an adapted version of the model.•Optimal or new best known solutions for almost all instances from the literature.•Optimal solutions for instances with 30 items vs. 8 items from the literature. We study a two-dimensional packing problem where rectangular items are placed into a circular container to maximize either the number or the total area of items packed. We adapt a mixed-integer linear programming model from the case with a rectangular container and design a cutting plane method to solve this problem by adding linear cuts to forbid items from being placed outside the circle. We show that this linear model allows us to prove optimality for instances larger than those solved using the state-of-the-art non-linear model for the same problem. We also propose a simple parallel algorithm that efficiently enumerates all non-dominated subsets of items and verifies whether pertinent subsets fit into the container using an adapted version of our linear model. Computational experiments using large benchmark instances attest that this enumerative algorithm generally provides better solutions than the best heuristics from the literature when maximizing the number of items packed. Instances with up to 30 items are now solved to optimality, against the eight-item instance previously solved.