Modeling and solving cutting and packing problems

• Autores: Germán Pantoja Benavides
• Directores de la Tesis: Francisco Parreño Torres (dir. tes.), David Álvarez Martínez (codir. tes.)
• Lectura: En la Universidad de los Andes (Colombia) ( Colombia ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: Antonio Martínez Sykora (presid.), Luis Miguel Escobar Falcón (secret.), Camilo Hernando Gómez Castro (voc.)
• Enlaces
  • Tesis en acceso abierto en: hdl.handle.net
• Resumen

  • This thesis addresses three complex and practically relevant Cutting and Packing Problems (C&PPs): the Container Loading Problem (CLP), the Two-Dimensional Irregular Strip Packing Problem (2D-ISPP), and the Three-Dimensional Irregular Strip Packing Problem (3D-ISPP). These problems are central to logistics, manufacturing, and material optimization, where maximization of space utilization contributes to cost efficiency and environmental sustainability.

    Chapter 1 provides a general introduction to the three problems considered, highlighting their industrial importance and relevance as optimization challenges. It sets the stage for the methodological contributions developed in the remaining chapters.

    Chapter 2 presents a hybrid GRASP–Beam Search algorithm to solve the CLP under practical constraints, including valid box orientations, multidrop sequencing, load-bearing limitations, and vertical stability. The methodology incorporates static-mechanical equilibrium (SME) equations, enhanced with areaproportional constraints to resolve underdetermined systems. A tree-based structure is used to efficiently update the weight distribution and center of gravity of boxes as new items are packed. A novel mergingand-expanding mechanism for managing maximal spaces is also introduced. The results indicate that the proposed method achieves competitive performance and, in certain cases, exceeds existing methodologies.

    Chapter 3 presents the Normalized Direct Trigonometry Model (NDTM), a Mixed-Integer Linear Programming (MILP) formulation designed to solve the 2D-ISPP. The model enforces containment and non-overlapping constraints using trigonometric relationships and incorporates a separation constraint that ensures a minimum distance between each pair of items. Compared to existing direct trigonometry-based formulations, the NDTM demonstrates superior performance in terms of both optimality and feasibility.

    Chapter 4 proposes the Iterative Simplification Matheuristic (ISM) as a solution strategy for the 2D-ISPP, specifically designed to address the computational limitations of directly applying the NDTM to instances with high geometric complexity. The ISM iteratively solves the 2D-ISPP using the NDTM across a hierarchy of geometric representations, starting from coarse approximations (bounding boxes) and progressively refining toward the original shapes. These approximations are generated with a polygon simplification algorithm based on a greedy tree search, guided by an area similarity parameter that controls the fidelity of each simplification. In addition, each simplified polygon is decomposed into convex subpolygons using a Convex Decomposition Algorithm (CDA) that incorporates a convex polygon merging procedure to reduce the number of convex components. Computational experiments demonstrate that ISM consistently outperforms the direct use of NDTM in complex instances, providing a scalable and efficient approach for solving large-scale irregular packing problems.

    Chapter 5 extends the NDTM to three dimensions, resulting in the NDTM3D formulation for addressing the 3D ISPP. This model introduces a novel non-intertwining constraint designed to prevent entanglement between non-convex polyhedra with holes. Although NDTM3D effectively solves small-scale instances, its applicability is limited in larger cases due to the rapid growth of the search space. Then, a Constructive–Local Search Matheuristic (CLSM) is proposed, which incorporates NDTM3D in the constructive and local search phases. The CLSM achieves competitive performance in benchmark instances, demonstrating its potential to tackle complex three-dimensional irregular packing problems.

    This thesis addresses three industrially relevant and computationally challenging packing problems through the development of mathematical programming models, metaheuristic strategies, and integrated matheuristic frameworks. The proposed methodologies yield robust and efficient solutions, advancing the state of the art in C&PPs and establishing a solid foundation for future research in operations research.