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Cláudio Manuel Martins Alves Cutting and Packing: Problems, Models

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Cláudio Manuel Martins Alves Cutting and Packing: Problems, Models Universidade do Minho Escola de Engenharia Cláudio Manuel Martins Alves Cutting and Packing: Problems, Models and Exact Algorithms Tese submetida à Universidade do Minho para a obtenção do grau de Doutor no Ramo de Engenharia de Produção e Sistemas, área de Investigação Operacional Trabalho efectuado sob a orientação de Professor Doutor José Manuel Vasconcelos Valério de Carvalho Departamento de Produção e Sistemas da Escola de Engenharia da Universidade do Minho Fevereiro de 2005 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Universidade do Minho: RepositoriUM Summary Different integer programming models and exact algorithms for hard cutting and pack- ing problems are addressed. We consider in particular the combinatorial problems of this family that are defined over a single dimension. The optimization procedures rely on tools from the field of integer programming, namely column generation, cutting planes and branch-and-bound. Integrating the three into the same algorithm is not straightforward, and has been used with some success only recently. The combined method is known as branch-and-price-and-cut. It requires a great part of customiza- tion, which is directly related to the specific problem that is being tackled. We consider three variants of the standard one-dimensional Cutting Stock Problem, and its packing counterpart, the Bin-Packing Problem. We study the case for which more than a single large object is available, and an optimal cutting or packing has to be found so as to minimize the total length or capacity used. A related problem, which con- sists in selecting the best assortment of large objects subject to cardinality constraints, is also investigated. The problem of maximizing the homogeneity of the plans, and hence reducing the number of setups involved with its execution, is addressed. This combinatorial problem is commonly referred to as the Pattern Minimization Problem, and is particularly hard. We propose to improve an existing model, and describe how to derive new valid cutting planes for it using an original approach. Finally, we describe an exact solution algorithm for the Ordered Cutting Stock Problem. This problem has been formulated recently. It consists in finding the best assignment of small items to the stock rolls, avoiding breaks among pre-defined lots of items. Three integer pro- gramming models are presented, along with two families of cutting planes and their respective separation algorithms. Different strategies are explored to improve the convergence of the column genera- tion algorithm, when applied to one-dimensional cutting and packing problems. New dual cutting planes are presented, and we describe how existing ones can be extended to the whole branch-and-bound search tree, for a given branching scheme. Alternative methods based on model aggregation are also explored. Two strategies are proposed. The first is based on a simple row aggregation scheme, while the second relies on a more sophisticated double aggregation of rows and columns. All the procedures proposed were coded and tested. Various computational ex- i ii periments were conducted to evaluate their performance, using, with this purpose, problem instances from the literature, and randomly generated instances. The results are presented and discussed throughout the thesis. R´esum´e Au long de cette th`ese, nous ´etudions plusieurs probl`emes de d´ecoupe et d’empaquetage dont la difficult´eest reconnue. Nous consid´eronsen particulier les probl`emescombi- natoires `aune seule dimension. Les algorithmes d’optimisation que nous explorons s’appuient sur des m´ethodes du domaine de la programmation enti`ere,notamment la g´en´eration de colonnes, la m´ethode des plans coupants et la m´ethode de s´eparation et ´evaluation. L’int´egration de ces trois m´ethodes dans un mˆemealgorithme ne s’effectue pas directement. En fait, cette int´egration n’a ´et´er´eussie avec succ`esque tr`esr´ecemment. L’algorithme qui en r´esulte est connu sous le nom de m´ethode de s´eparation, g´en´eration de colonnes et plans coupants. Compte tenu du probl`emequi est abord´e, beaucoup d’adaptations sont n´ecessaires pour parvenir `aun algorithme performant. Nous consid´eronsici les variantes des probl`emesstandards `aune dimension de d´ecoupe et d’empaquetage. Nous ´etudions le cas dans lequel plusieurs objets de grande dimension sont disponibles, et une solution optimale doit ˆetretrouv´eede mani`ere`a minimiser la largeur totale, ou capacit´e,utilis´ee.Nous explorons ´egalement le probl`eme qui consiste `achoisir le meilleur lot d’objets de grande dimension quand il existe des restrictions de cardinalit´e.La maximisation de l’homog´en´eit´edes solutions de d´ecoupe, et correspondante minimisation du nombre de changements de patrons de d´ecoupe, sont ´egalement ´etudi´ees.Ce probl`emeest connu sous le nom de Probl`emede Minimisation des Patrons de D´ecoupes, et il est particuli`erement difficil. Nous montrons comment un mod`elequi a ´et´er´ecemment propos´epeut ˆetream´elior´e,et nous expliquons comment d´eriver des in´egalit´es valides en utilisant une approche originale. Finalement, nous d´ecrivons un algorithme de r´esolution exacte pour le probl`emede d´ecoupe ordonn´ee. Ce probl`emea ´et´eformul´er´ecemment. Il consiste `atrouver la meilleure association des petits et grands objets en ´evitant qu’il y ait des cassures entre les lots qui ont ´et´e pr´ed´efinis. Nous proposons trois mod`eles de programmation enti`ere,ainsi que deux familles de plans coupants et leurs algorithmes de s´eparation. Plusieurs strat´egies ayant pour but d’am´eliorer la convergence de la m´ethode de g´en´eration de colonnes sont explor´ees. Nous consid´erons le cas particulier o`ucette m´ethode est appliqu´ee `ades probl`emesde d´ecoupe et d’empaquetage `aune dimension. Nous pr´esentons de nouveaux plans coupants dans l’espace dual, et nous d´ecrivons comment ´etendre d’autres abordages pr´esent´esdans la litt´erature scientifique `al’arbre iii iv g´en´er´epar la m´ethode de s´eparation et d’´evaluation, en assumant un sch´emade s´eparation sp´ecifique. Des m´ethodes alternatives bas´eessur l’aggr´egation de mod`eles sont aussi explor´ees. Deux m´ethodes sont propos´ees. La premi`ereest bas´eesur un sch´emasimple d’aggr´egation de restrictions, tandis que la seconde s’appuie sur une aggr´egation plus sophistiqu´eede variables et de restrictions. Toutes les proc´edurespropos´eesont ´et´eprogramm´eeset test´ees.Plusieus exp´eriences computationelles ont ´et´emen´eespour ´evaluer leur performance en utilisant, `acet effet, des instances d´ecritesdans la litt´erature, et des instances g´en´er´eesal´eatoirement. Nous pr´esentons les r´esultats obtenus, et nous les discutons tout au long de la th`ese. Resumo Nesta tese, s˜aoapresentados diferentes modelos de programa¸c˜aointeira e algoritmos de resolu¸c˜ao exacta para problemas dif´ıceis de corte e empacotamento. Consideramos em particular os problemas combinat´orios dessa fam´ılia numa ´unicadimens˜ao. Os algoritmos de optimiza¸c˜ao que exploramos assentam em m´etodos do dom´ınioda pro- grama¸c˜ao inteira, nomeadamente a gera¸c˜ao de colunas, planos de corte e o m´etodo de parti¸c˜aoe avalia¸c˜ao sucessivas. Integrar esses trˆesm´etodos no mesmo algoritmo n˜ao ´eum exerc´ıciodirecto. Na realidade, isso foi conseguido s´omuito recentemente. O algoritmo resultante ´edesignado de parti¸c˜ao, gera¸c˜aode colunas e corte. Atendendo ao problema que ´eabordado, v´arias adapta¸c˜oes tˆemde ser efectuadas. Consideramos trˆesvariantes dos problemas standards de corte e empacotamento com uma dimens˜ao.Estudamos o caso para o qual mais de um tipo de rolos ou con- tentores est˜aodispon´ıveis, e uma solu¸c˜ao´optima tem de ser encontrada de forma a minimizar a largura ou capacidade total utilizada. Investigamos tamb´emum problema relacionado no qual deve ser escolhido o melhor lote de rolos ou contentores atendendo a restri¸c˜oesde cardinalidade. O problema no qual se pretende maximizar a homo- geneidade das solu¸c˜oesde corte ou empacotamento, e dessa forma minimizar o n´umero de mudan¸casque ocorrem quando se executa o plano, ´eanalizado. Esse problema combinat´orio ´etamb´emconhecido por Problema da Minimiza¸c˜ao de Padr˜oes,e ´ede dif´ıcilresolu¸c˜ao. Propomos melhorar um modelo existente, e descrevemos como derivar planos de corte v´alidos usando uma abordagem original. Finalmente, descrevemos um algoritmo de resolu¸c˜aoexacta para o problema de corte ordenado. Esse problema foi formulado muito recentemente. Consiste em determinar a melhor afecta¸c˜ao de itens aos rolos, evitando interrup¸c˜oesentre lotes pre-definidos de itens. Trˆesmodelos de pro- grama¸c˜ao inteira s˜aoapresentados, juntamente com duas fam´ılias de planos de corte e os seus respectivos algoritmos de separa¸c˜ao. Diferentes estrat´egias para melhorar a convergˆenciado m´etodo de gera¸c˜aode colu- nas s˜ao exploradas. Consideramos o caso no qual esse m´etodo ´eaplicado a problemas de corte e empacotamento a uma dimens˜ao. Novos cortes duais s˜ao apresentados, e descrevemos como cortes que j´aforam descritos na literatura podem ser extendi- dos `atotalidade da `arvore de parti¸c˜ao, assumindo um esquema de parti¸c˜aoespec´ıfico. M´etodos alternativos baseados em agrega¸c˜aode modelos s˜ao tamb´em explorados. Duas v vi estrat´egias s˜aopropostas. A primeira ´ebaseada num esquema simples de agrega¸c˜ao de restri¸c˜oes,enquanto a segunda assenta num esquema mais sofisticado de restri¸c˜oes e vari´aveis. Todos os procedimentos propostos foram codificados e testados. V´arias experiˆencias computacionais foram levadas a cabo, usando, para isso, instˆanciasdescritas na liter- atura, e instˆancias geradas aleatoriamente.
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