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Universid Ade De Sã O Pa Mathematical models and heuristic methods for nesting problems Leandro Resende Mundim Tese de Doutorado do Programa de Pós-Graduação em Ciências de Computação e Matemática Computacional (PPG-CCMC) UNIVERSIDADE DE SÃO PAULO DE SÃO UNIVERSIDADE Instituto de Ciências Matemáticas e de Computação Instituto Matemáticas de Ciências SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________ Leandro Resende Mundim Mathematical models and heuristic methods for nesting problems Doctoral dissertation submitted to the Instituto de Ciências Matemáticas e de Computação – ICMC- USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Computer Science and Computational Mathematics. FINAL VERSION Concentration Area: Computer Science and Computational Mathematics Advisor: Profa. Dra. Marina Andretta Co-advisor: Prof. Dr. José Fernando da Costa Oliveira USP – São Carlos September 2017 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados fornecidos pelo(a) autor(a) Mundim, Leandro Resende M965m Mathematical models and heuristic methods for nesting problems / Leandro Resende Mundim; orientador Marina Andretta; coorientador José Fernando da Costa Oliveira. -- São Carlos, 2017. 166 p. Tese (Doutorado - Programa de Pós-Graduação em Ciências de Computação e Matemática Computacional) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2017. 1. Irregular Cutting and Packing Problems. 2. Mathematical Programming. 3. Heuristics. I. Andretta, Marina, orient. II. Oliveira, José Fernando da Costa, coorient. III. Título. Leandro Resende Mundim Modelos matemáticos e métodos heurísticos para os problemas de corte de itens irregulares Tese apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências – Ciências de Computação e Matemática Computacional. VERSÃO REVISADA Área de Concentração: Ciências de Computação e Matemática Computacional Orientador: Profa. Dra. Marina Andretta Coorientador: Prof. Dr. José Fernando da Costa Oliveira USP – São Carlos Setembro de 2017 To my family and my friends. ACKNOWLEDGEMENTS Agradeço a Deus por ter me dado saúde e tranquilidade para a conclusão deste trabalho. Quero agradecer a Bandeira da minha família: São João Batista, Santo Antônio e São Pedro, que sempre nos cuidam, protegem e nos guardam de todo mal. Os agradecimentos principais são direcionados aos meus pais, meus avós, minha irmã e demais familiares. Devo estender, os agradecimentos a todos os meus amigos, colegas e conhecidos. Sendo o principal autor desta tese, devo confessar que só atingi a qualidade aqui apre- sentada devido a meus queridos orientadores: Marina Andretta e José Fernando. Eles foram fundamentais para o desenvolvimento deste trabalho e meu crescimento profissional. Demais pesquisadores que contribuiram em partes pontuais deste trabalho e na minha formação foram: Thiago Queiroz, Maria Antónia, Franklina Toledo e Luiz Cherri. Um agradecimento especial ao ICMC e a CAPES que acreditaram no meu trabalho, durante o mestrado e doutorado. “He who thinks little, errs much.” (Leonardo da Vinci) ABSTRACT MUNDIM, L. R. Mathematical models and heuristic methods for nesting problems. 2017. 166 p. Tese (Doutorado em Ciências – Ciências de Computação e Matemática Computacional) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2017. Irregular cutting and packing problems, with convex and non-convex polygons, are found in many industries such as metal mechanics, textiles, of shoe making, the furniture making and others. In this thesis we study the two-dimensional version of these problems, where we want to allocate a set of items, without overlap, inside one or more containers, limited or unlimited, so as to optimize an objective function. In this document we study the knapsack problem, placement problem, strip packing problem, cutting stock problem and bin packing problem. For these problems, the heuristic methods and mathematical programming models are proposed and presented very promising results, surpassing in many cases the best results in the specialized literature. This thesis is organized as follows. In Chapter 1, we present a review of the studied problems, the value proposition for this thesis with the main contributions and ideas. In Chapter 2, we propose a metaheursitic for the strip packing problem with irregular items and circles. Then, in Chapter 3, we present a generic heuristic for the allocation of irregular items that may be weakly or strongly heterogeneous and will be allocated in a container (output maximization problems) or multiple containers (input minimization problems). In Chapter 4, we propose a solution method for the cutting stock problem with deterministic demand and stochastic demand. In Chapters 5 and 6, we present mathematical programming models for the strip packing problem. Finally, in Chapter 7, we present a conclusion and a concise direction for future works. Keywords: Irregular Cutting and Packing Problems, Mathematical Programming, Heuristics. RESUMO MUNDIM, L. R. Modelos matemáticos e métodos heurísticos para os problemas de corte de itens irregulares. 2017. 166 p. Tese (Doutorado em Ciências – Ciências de Computação e Matemática Computacional) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2017. Os problemas de corte e empacotamento de itens irregulares, polígonos convexos e não convexos, são encontrado em diversas indústrias, tais como a metal-mecânica, a têxtil, a de calçados, a moveleira e outras. Nesta tese estudamos a versão bidimensional destes problemas, na qual desejamos alocar um conjunto de itens, sem sobreposição, no interior de um ou mais recipientes, limitados ou ilimitados, de modo a otimizar uma função objetivo. Neste trabalho estudamos o problema da mochila, o problema do assentamento, o problema empacotamento em faixa, o problema de corte de estoque e o problema de empacotamento de contêineres. Para estes problemas, os métodos heurísticos e modelos de programação matemática propostos e apresentam resultados muito promissores, ultrapassando em muitos casos os melhores resultados da literatura especializada. Esta tese esta organizada da seguinte maneira. No Capítulo 1, apresentamos uma revisão dos problemas estudados, a proposta de valor deste doutorado com as principais contribuições e ideias. No Capítulo 2, propomos uma meta-heursítica para o problema de empacotamento em faixa para itens irregulares e círculos. Em seguida, no Capítulo 3 apresentamos uma heurística genérica para a alocação de itens irregulares que podem ser fracamente ou fortemente heterogêneos e serão alocados em um recipiente (problema de maximização de saída) ou de múltiplos recipientes (problemas de minimização de entrada). O Capítulo 4 propõem um método de solução para o problema de corte de estoque com demanda conhecida e demanda estocástica. Nos Capítulos 5 e 6 apresentamos modelos de programação matemática para o problema de corte de itens irregulares em faixa. Finalmente, no Capítulo 7, apresentamos a conclusão e uma sucinta direção para os trabalhos futuros. Palavras-chave: Problemas de Corte e Empacotamento de Itens Irregulares, Programação Matemática, Métodos Heurísticos. LIST OF FIGURES Figure 1 – Types of basic problems, extracted from Wäscher, Haubner and Schumann (2007)...................................... 27 Figure 2 – Representation extracted from Oliveira and Ferreira(1993).......... 29 Figure 3 – Example of phi-function of circles i and j................... 29 Figure 4 – Values of D-function and its implications with the classical coordinate system, i.e. x-coordinates grow to the right and y-coordinates grow upperwards. 30 Figure 5 – Example of the relative position of two items via direct trigonometry, ex- tracted from Oliveira and Ferreira(1993). .................. 30 Figure 6 – The no-fit polygon for convex polygons, extracted from Burke et al. (2007). 31 Figure 7 – Representation of polygons by circles, extracted from Jones(2014). 31 Figure 8 – Bottom-left heuristics: (a) without allocating items in holes, (b) allocating items in holes. Extracted from Dowsland, Vaid and Dowsland(2002) . 32 Figure 9 – Evaluation criteria for layouts. Extracted from Oliveira, Gomes and Ferreira (2000)...................................... 33 Figure 10 – Criteria for ordering items. Extracted from Oliveira, Gomes and Ferreira (2000)...................................... 33 Figure 11 – Neighborhood of the sequence proposed by Gomes and Oliveira(2002). 34 Figure 12 – Example of horizontal (a) and vertical (b) search, to obtain a new solution (c). Extracted from Egeblad, Nielsen and Odgaard(2007)............. 35 Figure 13 – Vision of the cuckoo search algorithm proposed by Elkeran(2013). 36 Figure 14 – Example of irregular bin from the leather cutting industry presented by Alves et al. (2012)................................... 36 Figure 15 – The blue regions represent the positions where the item can be allocated in (a) discrete methods, (b) semi-continuous methods and (c) continuous methods. 37 Figure 16 – Four irregular shaped items according to Segenreich and Braga(1986). 48 Figure 17 – The inner-fit raster of some irregular shaped items are represented by the blue dots in the bin’s grid. Note that the inner-fit raster of each item contains only points of the grid where such items can be packed generating
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