On Guillotine Separable Packings for the Two-Dimensional Geometric Knapsack Problem

In two-dimensional geometric knapsack problem, we are given a set of n axis-aligned rectangular items and an axis-aligned square-shaped knapsack. Each item has integral width, integral height and an associated integral profit. The goal is to find a (non-overlapping axis-aligned) packing of a maximum profit subset of rectangles into the knapsack. A well-studied and frequently used constraint in practice is to allow only packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts that do not intersect any item of the solution. In this paper we study approximation algorithms for the geometric knapsack problem under guillotine cut constraints. We present polynomial time (1+?)-approximation algorithms for the cases with and without allowing rotations by 90 degrees, assuming that all input numeric data are polynomially bounded in n. In comparison, the best-known approximation factor for this setting is 3+? [Jansen-Zhang, SODA 2004], even in the cardinality case where all items have the same profit. Our main technical contribution is a structural lemma which shows that any guillotine packing can be converted into another structured guillotine packing with almost the same profit. In this packing, each item is completely contained in one of a constant number of boxes and ?-shaped regions, inside which the items are placed by a simple greedy routine. In particular, we provide a clean sufficient condition when such a packing obeys the guillotine cut constraints which might be useful for other settings where these constraints are imposed

An anytime tree search algorithm for two-dimensional two- and three-staged guillotine packing problems

[libralesso_anytime_2020] proposed an anytime tree search algorithm for the 2018 ROADEF/EURO challenge glass cutting problem (https://www.roadef.org/challenge/2018/en/index.php). The resulting program was ranked first among 64 participants. In this article, we generalize it and show that it is not only effective for the specific problem it was originally designed for, but is also very competitive and even returns state-of-the-art solutions on a large variety of Cutting and Packing problems from the literature. We adapted the algorithm for two-dimensional Bin Packing, Multiple Knapsack, and Strip Packing Problems, with two- or three-staged exact or non-exact guillotine cuts, the orientation of the first cut being imposed or not, and with or without item rotation. The combination of efficiency, ability to provide good solutions fast, simplicity and versatility makes it particularly suited for industrial applications, which require quickly developing algorithms implementing several business-specific constraints. The algorithm is implemented in a new software package called PackingSolver

Empaquetamiento de objetos regulares en un contenedor rectangular.

Objetivos y método de estudio: Determinar el efecto en el empaquetamiento rectangular teniendo como parámetros de control diferentes objetos y dos familias de desigualdades válidas. Usando métodos de solución exacta. Contribuciones y conclusiones: En el embalaje sin permitir la telescopia se notó que en los primeros 240 minutos de ejecución del solver (30,60,120 y 240), tanto las desigualdades validas usadas como los diferentes objetos no presentan significancia estadistica al 95%, para la variable respuesta GAP (valor P>0.05). Dicho fenomeno no es similar a los 480 y 600 minutos en donde el tipo de objeto si afecta significativamente al GAP (valor P<0.05), pudiendose dar este patron principalmente al crecimiento combinatorio del problema en esos instantes. Los resultados aqui presentados son los primeros en donde se empaqueta más de 10 objetos similares de diferentes tamaño en un contenedor rectangular, dado que la gran mayoria de los trabajos restringen la cantidad de tamaños diferentes a empaquetar a 8 aunado al analisis de fluctuación del GAP a lo largo del tiempousando analisis de varianza

Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

Un algoritmo FFD-Eficiente para resolver el problema de corte de guillotina con demanda no unitaria de requerimientos sobre stock de tamaño variado

Resuelve el problema Guillotine Cutting Stock Problem with Demand on Varied Stock (GCSP-DVS) a través de un algoritmo FFD-Eficiente variado (FFD-E 2DGV). Además, demuestra la capacidad del algoritmo propuesto para incidir en el ahorro significativo a través del reúso de materia prima reciclable para el proceso industrial de corte bidimensional. Asimismo, compendia los resultados del algoritmo propuesto aplicado al GCSP-DVS y los resultados comparativos entre el FFD y el FFD-E aplicado al GCSP-D; generando un banco inédito para instancias de cortes 2 dimensiones de tipo guillotina sobre stock de tamaño variado y otra de demostraciones numéricas comparativas del FFD-E respecto al FFD, respectivamente. Finalmente, implementa un sistema computacional parametrizable que ejecute el algoritmo propuesto y arroje reportes de solución del citado problema GCSP con demanda sobre stock variado (GCSP-DVS).Tesi