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

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

Problems, Models and Algorithms in One- and Two-Dimensional Cutting

Within such disciplines as Management Science, Information and Computer Science, Engineering, Mathematics and Operations Research, problems of cutting and packing (C&P) of concrete and abstract objects appear under various specifications (cutting problems, knapsack problems, container and vehicle loading problems, pallet loading, bin packing, assembly line balancing, capital budgeting, changing coins, etc.), although they all have essentially the same logical structure. In cutting problems, a large object must be divided into smaller pieces; in packing problems, small items must be combined to large objects. Most of these problems are NP-hard. Since the pioneer work of L.V. Kantorovich in 1939, which first appeared in the West in 1960, there has been a steadily growing number of contributions in this research area. In 1961, P. Gilmore and R. Gomory presented a linear programming relaxation of the one-dimensional cutting stock problem. The best-performing algorithms today are based on their relaxation. It was, however, more than three decades before the first `optimum? algorithms appeared in the literature and they even proved to perform better than heuristics. They were of two main kinds: enumerative algorithms working by separation of the feasible set and cutting plane algorithms which cut off infeasible solutions. For many other combinatorial problems, these two approaches have been successfully combined. In this thesis we do it for one-dimensional stock cutting and two-dimensional two-stage constrained cutting. For the two-dimensional problem, the combined scheme provides mostly better solutions than other methods, especially on large-scale instances, in little time. For the one-dimensional problem, the integration of cuts into the enumerative scheme improves the results of the latter only in exceptional cases. While the main optimization goal is to minimize material input or trim loss (waste), in a real-life cutting process there are some further criteria, e.g., the number of different cutting patterns (setups) and open stacks. Some new methods and models are proposed. Then, an approach combining both objectives will be presented, to our knowledge, for the first time. We believe this approach will be highly relevant for industry