DOCSLIB.ORG

1. Introduction 2. Cutting and Packing Problems 3. Optimisation Techniques 4. Automated Packing Techniques 5. Computational Geom

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. Introduction 2. Cutting and Packing Problems 3. Optimisation Techniques 4. Automated Packing Techniques 5. Computational Geom 1. Introduction 2. Cutting and Packing Problems 3. Optimisation Techniques 4. Automated Packing Techniques 5. Computational Geometry, Benchmarks and Algorithms for Rectangular and Irregular Packing 6. Genetic Algorithms and Rectangular Packing 7. Genetic Algorithms and Irregular Packing 8. GENETIC ALGORITHMS AND 2D BIN PACKING..............................................208 8.1 Introduction................................................................................................................................................208 8.2 Comparison of the Packing Algorithms...................................................................................................208 8.3 Comparison of the Evolutionary Approaches .........................................................................................210 8.3.1 Genetic Algorithms versus Random Search........................................................................................210 8.3.2 Genetic Algorithms versus Naïve Evolution .......................................................................................211 8.3.3 Comparison of the Hybrid Combinations for the Genetic Algorithms................................................213 8.4 Comparison of the Meta-Heuristic and Heuristic Approaches..............................................................215 8.4.1 Genetic Algorithms versus Simulated Annealing................................................................................215 8.4.2 Genetic Algorithms versus Stochastic Optimisation ...........................................................................217 8.4.3 Meta-Heuristics and Local Search.......................................................................................................218 8.4.4 Comparison of the Meta-Heuristic and Heuristic Search Methods .....................................................219 8.5 Comparison with Commercial Nesting Software ....................................................................................221 8.6 Conclusions .................................................................................................................................................222 9. CONCLUSIONS AND FUTURE WORK...............................................................224 9.1 Conclusions .................................................................................................................................................224 9.2 Future Work...............................................................................................................................................226 10. REFERENCES AND BIBLIOGRAPHY.............................................................228 11. GLOSSARY ......................................................................................................239 8. Genetic Algorithms and 2D Bin Packing 8. Genetic Algorithms and 2D Bin Packing 8.1 Introduction The hybrid genetic algorithms developed for the 2D rectangular strip packing problems generate layouts of higher quality than the ones obtained with simple packing algorithms. In addition to that, it has been shown that the outcome is comparable or even better than some of the approaches used in the literature. The two commercial packages investigated in section 6.7 and 7.6 for regular and irregular packing tasks do not make an attempt to find the best set of objects. The hybrid algorithms developed for the 2D rectangular strip packing problems have therefore been extended for use with multiple objects. As mentioned in the previous chapters, the performance of the genetic algorithms is compared to random search. In order to establish the contribution of the cross-over operator to the search process a naïve evolution algorithm has been implemented. Since simulated annealing usually has achieved the best layouts in the case of rectangular strip packing, it also has been applied to the 2D bin packing problem. The comparison of the various hybrid algorithms also includes a local search heuristic, i.e. hill-climbing. Due to the lack of a comprehensive set of benchmark problems in the area of 2D bin packing, a direct performance evaluation of the hybrid algorithms has only been possible in respect to commercial software. For this purpose three problem categories have been created with each of them containing five test problems. The test problems of one category have the same number of items and have been created such that the total area of the items is similar (section 5.3.3.3). 8.2 Comparison of the Packing Algorithms Before the performance of the hybrid algorithms is compared, the simple packing algorithms have been tested on their own applying them to random and pre-ordered input sequences. Since the objective of the algorithm is to find the set of objects with the least total area, the outcome of the algorithms is evaluated on the basis of the average object utilisation. Solution Quality Figure 8.1 shows the object utilisation for the three test categories using the four packing routines with random input. As in the case of the strip packing problem the BLF-algorithm achieves the best layouts. This is due to its packing principle that allows nesting into enclosed areas of the partial layout. The utilisation achieved with this routine is up to 5% higher than the results of the next best algorithm. Among the sliding techniques the BL- and the BLLT-algorithm work equally well, with the BLLT- 208 8. Genetic Algorithms and 2D Bin Packing routine working better for the larger problems (i.e. M2 and M3). The BLD-routine shows the worst performance. 94 Table 8.1: Relative difference in BL average utilisation between 92 BLLT BLF area-sorted and random 90 BLD input [%] 88 M1 M2 M3 86 BL 6.4 5.5 6.3 utilisation [%] BLLT 6.0 5.9 7.0 84 BLF 5.5 6.0 7.5 82 BLD 6.5 6.9 7.6 80 M1 M2 M3 problem category Figure 8.1: Average object utilisation for packing routines using random input The outcome of all packing routines can be improved by applying pre-ordered input sequences. Sorting the rectangles by decreasing area and perimeter works better than other sorting criteria (Appendix H, Table H.1 and Table H.2). The relative difference in the average object utilisation between the area-sorted and the random input has been between 5 and 7.6% for the problems tested (Table 8.1). Figure 8.2 shows the best layouts generated with the BLF- and the BLLT-algorithm input sequences that have been sorted by decreasing rectangle area. It should be noted that not all the unused objects are shown in Figure 8.2. Figure 8.2: Best layouts for problem M2a using BLF (left) and BL (right) with area-sorted input (not all the unused objects are shown) 209 8. Genetic Algorithms and 2D Bin Packing Time Performance The computation time of the four algorithms and consequently the relative performance are different from the one observed in the strip packing problem (Figure 6.3). Table 8.2 summarises the elapsed times of the routines for 1000 runs. Unlike in the case of the strip packing problems used in chapter 6 the BLF- algorithm does not have the longest run time. As expected, the two sliding techniques (BL and BLLT) have similar elapsed times, which are lower than the one of the BLF-algorithm. The BLF-routine operates faster than the third sliding technique, i.e. BLD-routine. Since overlap calculations are required in the BLD-routine, its time performance is poorer than that of the performance of the other two sliding techniques. Although the BLF-algorithm keeps track of the enclosed areas in the partial layout, this operation is still more efficient than the intersection checking carried out by the BLD- routine. This is mainly due to this type of 2D bin packing problem. Since the rectangle size is fairly large compared to the object size, the number of enclosed areas in the layout is usually small. In particular, since layout information is stored when a new item is positioned, intersection testing is still required but reduced to a minimum (section 5.5.4.5). A similar principle could be implemented in the BLD-routine in order to improve its time performance. Table 8.2: Average elapsed time of simple packing routines for 1000 runs [s] BL BLLT BLF BLD M1 3.7 3.5 3.9 6.0 M2 3.6 4.0 4.4 6.2 M3 14.6 14.2 15.3 16.0 8.3 Comparison of the Evolutionary Approaches In this section the performance of the evolutionary approaches is analysed. In order to investigate whether the implementation of the search process conducted by meta-heuristic methods works successfully, a comparison with random search is undertaken. Since the data structure for the bin packing problem consists of two chromosomes, which are manipulated separately by the operators of the meta-heuristic method, a series of experiments is conducted to establish the contribution of the cross-over and mutation operators to the search process. 8.3.1 Genetic Algorithms versus Random Search The representation of the 2D bin packing problem in the hybrid algorithms consists of two permutations describing the order of the items and the objects. Whereas cross-over and mutation operators manipulate both sequences, the rotation operator is only applied to the items. In order to analyse if this implementation of genetic algorithms can search the solution space successfully, a random search 210 8. Genetic Algorithms and 2D Bin Packing algorithm has been used to generate the permutations as well as the orientation of the items randomly. Both search processes have been simulated over the same number of iterations. Since the objective of this bin packing problem
Load more

Recommended publications
  • Process Optimization for Cutting Steel-Plates
    Process Optimization for Cutting Steel-Plates Markus Rothe, Michael Reyer and Rudolf Mathar Institute for Theoretical Information Technology, RWTH Aachen University, Kopernikusstraße 16, 52074 Aachen, Germany Keywords: Two-Stage Three-Dimensional Guillotine Cutting, Residual Bin-Packing Problem, Mixed Integer Program- ming Model, Reuseable Leftovers. Abstract: In this paper, we consider the two-stage three-dimensional guillotine cutting stock problem with usable left- over. There are two properties that distinguish our problem formulation from others. First, we allow the items to be rotated. Second, we consider the case in which leftover material is to be reused in subsequent production cycles. Third, we solve the problem in three dimensions. The optimization problem is formulated as a mixed integer linear program. To verify the approach, several examples show that our formulation performs well. 1 INTRODUCTION and (Gilmore and Gomory, 1961), utilize linear pro- gramming to solve cutting stock problems. P. Gilmore In many areas of industrial production large or long and Gomory investigate the one dimensional prob- pieces of material need to be cut into smaller ones. lem of cutting items from stock of several standard E.g., it needs to be decided from which reel of raw lengths. They devise the method of column gener- material a cable of a specific length will be produced. ation in their work. P. Gilmore and Gomory con- This is an example for a one-dimensional problem. tinued their work in (Gilmore and Gomory, 1965). An example for a two-dimensional problem is to cut Here, they extend their formulation to multistage cut- patterns from pieces of large leather.
    [Show full text]
  • Algorithms for 3D Guillotine Cutting Problems: Unbounded Knapsack, Cutting Stock and Strip Packing∗
    Algorithms for 3D Guillotine Cutting Problems: Unbounded Knapsack, Cutting Stock and Strip Packing∗ Thiago A. de Queiroz, Flávio K. Miyazawa, Instituto de Computação, IC, UNICAMP, 13084-971, Campinas, SP e-mail: [email protected], [email protected] Yoshiko Wakabayashi, Instituto de Matemática e Estatística, IME, USP 05508-090, São Paulo, SP e-mail: [email protected] Eduardo C. Xavier Instituto de Computação, IC, UNICAMP, 13084-971, Campinas, SP e-mail: [email protected] ABSTRACT We present algorithms for the following three-dimensional (3D) guillotine cutting problems: Un- bounded Knapsack, Cutting Stock and Strip Packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the Un- bounded 3D Knapsack problem, we extend the recurrence formula proposed by Beasley for the Rectangular Knapsack Problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the Unbounded Knapsack problem in which the cuts must be staged. For the 3D Cutting Stock problem and its variants in which the bins have differ- ent sizes (and the cuts must be staged), we present column generation based algorithms. Modified versions of the algorithms for the 3D Cutting Stock problems with stages are then used to build algorithms for the 3D Strip Packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of mod- erate size. KEYWORDS. Guillotine cutting; three-dimensional cutting stock; unbounded knapsack; strip packing; column generation.
    [Show full text]
  • Contributions to the Two-Dimensional Guillotine Cutting Stock Problem
    UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS DEPARTAMENTO DE ESTATÍSTICA E MATEMÁTICA APLICADA MESTRADO EM MODELAGEM E MÉTODOS QUANTITATIVOS RONALDO LAGE PESSOA CONTRIBUTIONS TO THE TWO-DIMENSIONAL GUILLOTINE CUTTING STOCK PROBLEM FORTALEZA 2017 RONALDO LAGE PESSOA CONTRIBUTIONS TO THE TWO-DIMENSIONAL GUILLOTINE CUTTING STOCK PROBLEM Dissertação apresentada ao Curso de Mestrado em Modelagem e Métodos Quantitativos do Departamento de Estatística e Matemática Aplicada do Centro de Ciências da Universidade Federal do Ceará, como requisito parcial à obtenção do título de mestre em Modelagem e Métodos Quantitativos. Área de Concentração: Otimização e Inteligência Computacional Orientador: Prof. Dr. Bruno de Athayde Prata Co-Orientador: Prof. Dr. Carlos Diego Rodrigues FORTALEZA 2017 RONALDO LAGE PESSOA CONTRIBUTIONS TO THE TWO-DIMENSIONAL GUILLOTINE CUTTING STOCK PROBLEM Dissertação apresentada ao Curso de Mestrado em Modelagem e Métodos Quantitativos do Departamento de Estatística e Matemática Aplicada do Centro de Ciências da Universidade Federal do Ceará, como requisito parcial à obtenção do título de mestre em Modelagem e Métodos Quantitativos. Área de Concentração: Otimização e Inteligência Computacional Aprovada em: BANCA EXAMINADORA Prof. Dr. Bruno de Athayde Prata (Orientador) Universidade Federal do Ceará (UFC) Prof. Dr. Carlos Diego Rodrigues (Co-Orientador) Universidade Federal do Ceará (UFC) Prof. Dr. Albert Einstein F. Muritiba Universidade Federal do Ceará (UFC) Prof. Dr. Reinaldo Morabito Universidade Federal de São Carlos (UFSCar) AGRADECIMENTOS Agradeço inicialmente a Deus que, por intermédio de Nossa Senhora, permitiu que o presente trabalho pudesse ser finalizado no prazo e qualidade desejada, mesmo com todas as barreiras pessoais que surgiram durante o seu desenvolvimento. A minha esposa Larissa, pela paciência, compreensão e o esforço extra nos cuidados do nosso filho, tendo ele nascido durante o desenvolvimento desta pesquisa.
    [Show full text]
  • On Guillotine Separability of Squares and Rectangles
    On guillotine separability of squares and rectangles Arindam Khan Indian Institute of Science, Bengaluru, India [email protected] Madhusudhan Reddy Pittu 1 Indian Institute of Technology, Kharagpur, India [email protected] Abstract Guillotine separability of rectangles has recently gained prominence in combinatorial optimization, computational geometry, and combinatorics. Consider a given large stock unit (say glass or wood) and we need to cut out a set of required rectangles from it. Many cutting technologies allow only end-to-end cuts called guillotine cuts. Guillotine cuts occur in stages. Each stage consists of either only vertical cuts or only horizontal cuts. In k-stage packing, the number of cuts to obtain each rectangle from the initial packing is at most k (plus an additional trimming step to separate the rectangle itself from a waste area). Pach and Tardos [20] studied the following question: Given a set of n axis-parallel rectangles (in the weighted case, each rectangle has an associated weight), cut out as many rectangles (resp. weight) as possible using a sequence of guillotine cuts. They provide a guillotine cutting sequence that recovers 1/(2 log n)-fraction of rectangles (resp. weights). Abed et al. [1] claimed that a guillotine cutting sequence can recover a constant fraction for axis-parallel squares. They also conjectured that for any set of rectangles, there exists a sequence of axis-parallel guillotine cuts that recovers a constant fraction of rectangles. This conjecture, if true, would yield a combinatorial O(1)-approximation for Maximum Independent Set of Rectangles (MISR), a long-standing open problem.
    [Show full text]
  • Pattern Based Diving Heuristics for a Two-Dimensional Guillotine Cutting
    Pattern based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers François Clautiaux, Ruslan Sadykov, François Vanderbeck, Quentin Viaud To cite this version: François Clautiaux, Ruslan Sadykov, François Vanderbeck, Quentin Viaud. Pattern based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers. EURO Journal on Computational Optimization, Springer, 2019, 7 (3), pp.265-297. 10.1007/s13675-019-00113-9. hal- 01656179 HAL Id: hal-01656179 https://hal.archives-ouvertes.fr/hal-01656179 Submitted on 5 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Pattern based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers François Clautiauxa,b,˚, Ruslan Sadykovb,a, François Vanderbecka,b, Quentin Viauda,b aIMB, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France bINRIA Bordeaux - Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence, France Abstract We consider a variant of two-dimensional guillotine cutting-stock problem that arises when different bills of order (or batches) are considered consecutively. The raw material leftover of the last cutting pattern is not counted as waste as it can be reused for cutting the next batch.
    [Show full text]
  • A Heuristic for the 3-Staged 2D Cutting Stock Problem with Usable Leftover
    International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) A Heuristic for the 3-staged 2D Cutting Stock Problem with Usable Leftover Q.L. Chen L.P. Li School of Business Administration College of Computer and Electronic Information South China University of Technology Guangxi University Guangzhou, China Nanning, China College of Computer and Electronic Information Guangxi University Nanning, China Y. Chen School of Business Administration, South China University of Technology, Y.D. Cui Guangzhou, China College of Computer and Electronic Information College of Computer and Electronic Information Guangxi University Guangxi University Nanning, China Nanning, China X.Y. Lu College of Computer and Electronic Information Guangxi University Nanning, China Abstract—A heuristic approach is proposed for the 3-staged only homogenous strips and is referred to as the homogenous two-dimensional cutting stock problem with usable leftover, 3-staged cutting pattern with usable leftover (3HL pattern). where the unused rectangular segments can be returned to stock for future use. The approach is based on the recursive technique II. PROBLEM DESCRIPTION combined with beam search. A forward recursion is used to The paper proposes a heuristic algorithm to determine the generate normal segments that consist of homogenous strips, and cutting plan that consists of 3HL patterns, where a set of m a beam search heuristic is used to obtain the 3-staged cutting types of rectangular items are cut from stock plates of patterns considering usable leftover. The computational results show that the algorithm is effective in reducing material cost. dimensions L W , each item type mIi },,1{ , has a length li , a width wi , a demand di and a value vi .
    [Show full text]
  • Two-Dimensional Bin Packing Problem with Guillotine Restrictions
    Alma Mater Studiorum Universit`adi Bologna DEI - Dipartimento di Ingegneria dell'Energia Elettrica e dell'Informazione \Guglielmo Marconi" Dottorato di Ricerca in Automatica e Ricerca Operativa Ciclo XXVI Settore concorsuale di afferenza: 01/A6 - RICERCA OPERATIVA Settore scientifico disciplinare: MAT/09 - RICERCA OPERATIVA Two-Dimensional Bin Packing Problem with Guillotine Restrictions Enrico Pietrobuoni Coordinatore Relatore Prof. Daniele Vigo Prof. Andrea Lodi Ing. Michele Monaci Esame Finale 2015 Contents List of Figuresv 1 Outline1 2 The Two-Dimensional Bin Packing Problem3 2.1 Introduction..................................3 2.2 Cutting and Packing.............................4 2.3 Rectangle Packing Problem.........................4 2.4 Applications..................................6 2.5 Models.....................................7 2.5.1 One-dimensional bin packing problem...............7 2.5.2 Two-dimensional bin packing problem...............7 2.5.3 ILP models for level packing....................9 2.6 The asymptotic and the absolute worst-case performance ratios..... 11 2.7 Upper Bounds................................ 11 2.7.1 Strip packing............................. 12 2.7.2 Bin packing: Two-phase heuristics................. 15 2.7.3 Bin packing: One-phase level heuristics.............. 17 2.7.4 Bin packing: One-phase non-level heuristics............ 18 2.7.5 Metaheuristics............................ 19 2.7.6 Approximation algorithms...................... 23 2.8 Lower Bounds................................. 24 2.9 Exact Algorithms............................... 28 3 Two-Dimensional Bin Packing: the 2BPjOjG case 31 3.1 Introduction.................................. 31 3.1.1 Our goals............................... 32 3.1.2 Definitions.............................. 33 3.1.3 Convexification Algorithm...................... 34 3.1.4 Algorithm and assumptions..................... 37 3.2 Smallest non-separable pattern....................... 38 3.2.1 Rows and Intersections....................... 38 3.3 Blocked Ring................................
    [Show full text]
  • Two-Dimensional Cutting Problem
    Two-Dimensional Cutting Problem 3. Btaiewicz, M. Drozdowski, B. Soniewicki, R. Walkowiak CP-91-009 June 1991 Collaborative Papers report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. International Institute for Applied Systems Analysis A-2361 Laxenburg Austria EIIIASA Telephone: +43 2236 715210 o Telex: 079 137 iiasa a Telefax: +43 2236 71313 Foreword This Collaborative Paper is one of a series which presents the different software packages designed and implemented for interactive decision support. These packages constitute the outcome of the contracted study agreement between the System and Decision Sciences Program at IIASA and several Polish scientific institutions. The theoretical part of these results is presented in the IIASA Collaborative Paper CP-90-008 entitled Contributions to Methodology and Techniques of Decision Analysis (First Stage), edited by Andrzej Ruszczyriski, Tadeusz Rogowski and Andrzej P. Wierzbicki. The distributable versions of the software are usually tailored for the illustration of methodology and possible applications. However, for most of these software packages there exists a version made for a specific application and it is possible to modify each software package for a specific real-life application (if the corresponding mathematical programming model is of the type for which a particular package has been designed). All software developed within the scientific cooperation mentioned above is available either at distribution cost or free of charge for scientific non-commercial usage by insti- tutions and individuals from the countries which are members of IIASA.
    [Show full text]
  • Heuristics for Two-Dimensional Rectangular Guillotine Cutting Stock
    Thailand Statistician July 2016; 14(2): 147-164 http://statassoc.or.th Contributed paper Heuristics for Two-Dimensional Rectangular Guillotine Cutting Stock Kimseng Tieng [a], Supphakorn Sumetthapiwat [b], Aussadavut Dumrongsiri [a] and Chawalit Jeenanunta* [a] [a] Sirindhorn International Institute of Technology, Thammasat University, Pathumthani, Thailand. [b] Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand. *Corresponding author; e-mail: [email protected] Received: 14 August 2015 Accepted: 28 December 2015 Abstract Two-Dimensional Rectangular Guillotine Cutting Stock Problem (2DRGCSP) is one of the most significant problems in the manufacturing industries. A set of small rectangular size of paper, aluminum rolls, glasses, fiber glasses, or plastic are required to cut from a set of big size rectangular sheet of its raw materials. A guillotine cut is used, where the sheet is cut from one side to another side without changing the direction of the blade to produce the strips. Then each strip is cut again to produce the small rectangular size of panels that match with the required size, and the number of the panels must satisfy with the demand. This paper presents the heuristic techniques to solve the cutting problems. The heuristic techniques create the good cutting patterns such that the waste of the sheets is minimized and demands for each panel are satisfied. There are five proposed heuristics, developed to solve this problem. They are 2D simple heuristic cutting (2DSHC), 2D horizontal construction (2DHC), 2D vertical construction (2DVC), 2D horizontal improvement (2DHI), and 2D vertical improvement (2DVI). The testing instances are created from the real problems in the Printed Circuit Board (PCB) Company.
    [Show full text]
  • Two-Stage Two-Dimensional Guillotine Cutting Stock Problems with Usable Leftover∗
    Two-stage two-dimensional guillotine cutting stock problems with usable leftover∗ R. Andrade y E. G. Birginy R. Morabito z July 11, 2013x Abstract In this study we are concerned with the non-exact two-stage two-dimensional guillotine cutting problem considering usable leftovers, in which stock plates remainders of the cutting patterns (non-used material or trim loss) can be used in the future, if they are large enough to fulfill future demands of items (ordered smaller plates). This cutting problem can be charac- terized as a residual bin-packing problem because of the possibility of putting back into stock residual pieces, since the trim loss of each cutting/packing pattern does not necessarily repre- sent waste of material depending on its size. Two bilevel mathematical programming models to represent this non-exact two-stage two-dimensional residual bin-packing problem are presented. The models basically consist on cutting/packing the ordered items using a set of plates of min- imum cost and, among all possible solutions of minimum cost, choosing one that maximizes the value of the generated usable leftovers. Because of special characteristics of these bilevel models, they can be reformulated as one-level mixed integer programming models. Results of some numerical experiments are presented to show that the models represent appropriately the problem and to illustrate their performances. Key words: Two-stage two-dimensional guillotine cutting, residual bin-packing problem, resid- ual cutting-stock problem, bilevel programming, MIP models, leftovers. 1 Introduction Cutting problems are often found in different industrial processes where paper and aluminium rolls, glass and fibreglass plates, metal bars and sheets, hardboards, pieces of leather and cloth, etc., are cut in order to produce smaller pieces of ordered sizes and quantities.
    [Show full text]
  • Arxiv:2004.12619V1 [Math.OC] 27 Apr 2020 Higher Dimensions, Like, E.G., the (Three-Dimensional) Container Loading Problem (See Bortfeldt and W¨Ascher [28])
    Exact Solution Techniques for Two-dimensional Cutting and Packing Manuel Iori(1), Vin´ıciusL. de Lima(2), Silvano Martello(3), Fl´avioK. Miyazawa(2), Michele Monaci(3) (1) DISMI, University of Modena and Reggio Emilia (Italy) (2) Institute of Computing, University of Campinas (Brazil) (3) DEI "Guglielmo Marconi", University of Bologna (Italy) Abstract We survey the main formulations and solution methods for two-dimensional orthogonal cut- ting and packing problems, where both items and bins are rectangles. We focus on exact methods and relaxations for the four main problems from the literature: finding a packing with minimum height, packing the items into the minimum number of bins, finding a packing of maximum value, and determining the existence of a feasible packing. Keywords: Two-dimensional rectangle cutting and packing; Exact methods; Relaxations. 1 Introduction The number of publications on cutting and packing problems has been increasing considerably in recent years. In cutting problems, we are given a set of standardized stock units to be cut into smaller items so as to fulfill a given demand, while in packing problems a set of items has to be packed into one or more containers. These two classes of problems are strongly correlated, as packing an item into a container may be equivalent to cutting an item from a stock unit, and hence the same solution methods are often adopted. In the following, we will denote as bins both the stock units and the containers. In certain applications the unique container is a strip of (theoretically) infinite height. Cutting and packing problems have been widely studied in the literature both for their theoret- ical interest and their many practical applications, in which they appear in a number of different variants.
    [Show full text]
  • On Guillotine Separable Packings for the Two-Dimensional Geometric
    On Guillotine Separable Packings for the Two-dimensional Geometric Knapsack Problem Arindam Khan Indian Institute of Science, Bengaluru, India [email protected] Arnab Maiti Indian Institute of Technology, Kharagpur, India [email protected] Amatya Sharma Indian Institute of Technology, Kharagpur, India [email protected] Andreas Wiese Universidad de Chile, Santiago, Chile [email protected] Abstract 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 + ε)-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 + ε [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.
    [Show full text]