The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect

In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances.Two-dimensional cutting, defect, AND/OR-graph, Branch & Bound

The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect

In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances

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

Solving the two dimensional cutting problem using evolutionary algorithms with penalty functions

In this work a solution using evolutionary algorithms with penalty function for the non-guillotine cutting problem is presented. In this particular problem, the rectangular pieces have to be cut from an unique large object, being the goal to maximize the total value of cut pieces. Some chromosomes can hold pieces to be cut, but some pieces cannot be arranged into the object, generating infeasible solutions. A way to deal with this kind of solutions is to use a penalizing strategy. The used penalty functions have been originally developed for the knapsack problem and they are adapted for the cutting problem in this paper. Moreover, the effect on the algorithm performance to combine penalty functions with two different selection methods (binary tournament and roulette wheel) is studied. The algorithm uses a binary representation, one-point crossover, big-creep mutation and in order to evaluated the quality of solutions a placement routine is considered (Heuristic with Efficient Management of Holes). Experimental comparisons of the performance of the resulting algorithms are carried out using publicly available benchmarks to the non-guillotine cutting problem. We report on the high performance of the proposed models at similar (or better) accuracy with respect to existing algorithms.VI Workshop de Agentes y Sistemas Inteligentes (WASI)Red de Universidades con Carreras en Informática (RedUNCI

2차원 2단계 배낭문제에 대한 정수계획모형 및 최적해법

학위논문 (석사) -- 서울대학교 대학원 : 공과대학 산업공학과, 2021. 2. 이경식.In this thesis, we study integer programming models and exact algorithms for the two-dimensional two-staged knapsack problems, which maximizes the profit by cutting a single rectangular plate into smaller rectangular items by two-staged guillotine cuts. We first introduce various integer programming models, including the strip-packing model, the staged-pattern model, the level-packing model, and the arc-flow model for the problem. Then, a hierarchy of the strength of the upper bounds provided by the LP-relaxations of the models is established based on theoretical analysis. We also show that there exists a polynomial-size model that has not been proven yet as far as we know. Exact methods, including branch-and-price algorithms using the strip-packing model and the staged-pattern model, are also devised. Computational experiments on benchmark instances are conducted to examine the strength of upper bounds obtained by the LP-relaxations of the models and evaluate the performance of exact methods. The results show that the staged-pattern model gives a competitive theoretical and computational performance.본 논문은 2단계 길로틴 절단(two-staged guillotine cut)을 사용하여 이윤을 최대화하는 2차원 2단계 배낭 문제(two-dimensional two-staged knapsack problem: 이하 2TDK)에 대한 정수최적화 모형과 최적해법을 다룬다. 우선, 본 연구에서는 스트립패킹모형, 단계패턴모형, 레벨패킹모형, 그리고 호-흐름모형과 같은 정수최적화 모형들을 소개한다. 그 뒤, 각각의 모형의 선형계획완화문제에 대해 상한강도를 이론적으로 분석하여 상한강도 관점에서 모형들 간 위계를 정립한다. 또한, 본 연구에서는 2TDK의 다항크기(polynomial-size) 모형의 존재성을 처음으로 증명한다. 다음으로 본 연구는 2TDK의 최적해를 구하는 알고리즘으로써 패턴기반모형들에 대한 분지평가 알고리즘과 레벨패킹모형을 기반으로 한 분지절단 알고리즘을 제안한다. 단계패턴모형이 이론적으로도 가장 좋은 상한강도를 보장할 뿐만 아니라, 계산 분석을 통해 단계패턴모형을 기반으로 한 분지평가 알고리즘이 제한된 시간 내 좋은 품질의 가능해를 찾음을 확인하였다.Abstract i Contents iv List of Tables vi List of Figures vii Chapter 1 Introduction 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 Integer Programming Models for 2TDK 9 2.1 Pattern-based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Arc-flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Level Packing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter 3 Theoretical Analysis of Integer Programming Models 20 3.1 Upper Bounds of AF and SM(1;1) . . . . . . . . . . . . . . . . . . 20 3.2 Upper Bounds of ML, PM(d), and SM(d; d) . . . . . . . . . . . . . . 21 3.3 Polynomial-size Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 4 Exact Methods 33 4.1 Branch-and-price Algorithm for the Strip Packing Model . . . . . . . 34 4.2 Branch-and-price Algorithm for the Staged-pattern Model . . . . . . 39 4.2.1 The Standard Scheme . . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 The Height-aggregated Scheme . . . . . . . . . . . . . . . . . 40 4.3 Branch-and-cut Algorithm for the Modified Level Packing Model . . 44 Chapter 5 Computational Experiments 46 5.1 Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Upper Bounds Comparison . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 49 5.2.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 55 5.2.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Solving Instances to Optimality . . . . . . . . . . . . . . . . . . . . . 65 5.3.1 A Group of Small Instances . . . . . . . . . . . . . . . . . . . 65 5.3.2 A Group of Large Instances . . . . . . . . . . . . . . . . . . . 69 5.3.3 Class 5 Instances . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 6 Conclusion 77 Bibliography 79 국문초록 83Maste

2 stage guillotine cutting

Modello 1, modello di sezione 3.

The two-dimensional cutting stock problem within the roller blind production process

In this paper we consider a two-dimensional cutting stock problem encountered at a large manufacturer of window covering products. The problem occurs in the production process of made-to-measure roller blinds. We develop a solution method that takes into account the characteristics of the specific problem. In particular, we deal with the fact that fabrics may contain small defects that should end up with the waste. Comparison to previous practice shows significant waste reductions.cutting;trim loss;two-dimensional cutting stock problem

Algorithms for cutting and packing problems

Orientador: Flávio Keidi MiyazawaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Problemas de Corte e Empacotamento são, em sua maioria, NP-difíceis e não existem algoritmos exatos de tempo polinomial para tais se for considerado P ¿ NP. Aplicações práticas envolvendo estes problemas incluem a alocação de recursos para computadores; o corte de chapas de ferro, de madeira, de vidro, de alumínio, peças em couro, etc.; a estocagem de objetos; e, o carregamento de objetos dentro de contêineres ou caminhões-baú. Nesta tese investigamos problemas de Corte e Empacotamento NP-difíceis, nas suas versões bi- e tridimensionais, considerando diversas restrições práticas impostas a tais, a saber: que permitem a rotação ortogonal dos itens; cujos cortes sejam feitos por uma guilhotina; cujos cortes sejam feitos por uma guilhotina respeitando um número máximo de estágios de corte; cujos cortes sejam não-guilhotinados; cujos itens tenham demanda (não) unitária; cujos recipientes tenham tamanhos diferentes; cujos itens sejam representados por polígonos convexos e não-convexos (formas irregulares); cujo empacotamento respeite critérios de estabilidade para corpos rígidos; cujo empacotamento satisfaça uma dada ordem de descarregamento; e, cujos empacotamentos intermediários e final tenham seu centro de gravidade dentro de uma região considerada "segura". Para estes problemas foram propostos algoritmos baseados em programação dinâmica; modelos de programação inteira; técnicas do tipo branch-and-cut; heurísticas, incluindo as baseadas na técnica de geração de colunas; e, meta-heurísticas como o GRASP. Resultados teóricos também foram obtidos. Provamos uma questão em aberto levantada na literatura sobre cortes não-guilhotinados restritos a um conjunto de pontos. Uma extensiva série de testes computacionais considerando instâncias reais e várias outras geradas de forma aleatória foram realizados com os algoritmos desenvolvidos. Os resultados computacionais, sendo alguns deles comparados com a literatura, comprovam a validade dos algoritmos propostos e a sua aplicabilidade prática para resolver os problemas investigadosAbstract: Several versions of Cutting and Packing problems are considered NP-hard and, if we consider that P ¿ NP, we do not have any exact polynomial algorithm for solve them. Practical applications arises for such problems and include: resources allocation for computers; cut of steel, wood, glass, aluminum, etc.; packing of objects; and, loading objects into containers and trucks. In this thesis we investigate Cutting and Packing problems that are NP-hard considering theirs two- and three-dimensional versions, and subject to several practical constraints, that are: that allows the items to be orthogonally rotated; whose cuts are guillotine type; whose cuts are guillotine type and performed in at most k stages; whose cuts are non-guillotine type; whose items have varying and unit demand; whose bins are of variable sizes; whose items are represented by convex and non-convex polygons (irregular shapes); whose packing must satisfy the conditions for static equilibrium of rigid bodies; whose packing must satisfy an order to unloading; and, whose intermediaries and resultant packing have theirs center of gravity inside a safety region; Such cutting and packing problems were solved by dynamic programming algorithms; integer linear programming models; branch-and-cut algorithms; several heuristics, including those ones based on column generation approaches, and metaheuristics like GRASP. Theoretical results were also provided, so a recent open question arised by literature about non-guillotine patterns restricted to a set of points was demonstrated. We performed an extensive series of computational experiments for algorithms developed considering several instances presented in literature and others generated at random. These results, some of them compared with the literature, validate the approaches proposed and suggest their applicability to deal with practical situations involving the problems here investigatedDoutoradoDoutor em Ciência da Computaçã