A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths

Belov, Gleb; Scheithauer, Guntram

doi:10.1016/s0377-2217(02)00125-x

Cited by 106 publications

(90 citation statements)

References 9 publications

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“…The slave is solved by a specifically tailored branch-and-bound method that takes into account the dual variables associated with the additional constraints. The method was improved in Belov and Scheithauer [29], and then embedded into a branch-and-price algorithm in Belov and Scheithauer [30]. The resulting algorithm directly branches on the variables associated with the patterns, selecting the variable whose fractional value is closer to 0.5.…”

Section: Chapter 2 Bpp and Csp: Mathematical Models And Exact Algorimentioning

confidence: 99%

“…, K. The objective is now to pack all the items into a minimum cost set of bins that respects the availability and the capacity of each bin type. The problem was extensively studied in the literature and various exact and heuristic approaches were proposed, see, e.g., Belov and Scheithauer [29], Alves and Valério de Carvalho [10], and Hemmelmayr et al [152]). …”

Section: Variable Sized Bppmentioning

confidence: 99%

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Mathematical models and decomposition algorithms for cutting and packing problems

Delorme¹

2017

4OR-Q J Oper Res

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Section: Chapter 2 Bpp and Csp: Mathematical Models And Exact Algorimentioning

confidence: 99%

Section: Variable Sized Bppmentioning

confidence: 99%

Mathematical models and decomposition algorithms for cutting and packing problems

Delorme¹

2017

4OR-Q J Oper Res

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“…Em caso de múltiplos períodos, a combinação de itens também pode ser melhorada por permitir que itens possam ser cortados antes do período no qual são demandados. Embora o problema com vários tipos de objetos em estoque seja bem conhecido, relativamente poucos trabalhos são encontrados na literatura (Belov & Scheithauer, 2002;e Holthaus, 2002). O problema de corte de estoque com demanda em múltiplos períodos também foi pouco explorado na literatura, apenas por Gramani & França (2006).…”

Section: Introductionunclassified

O problema de corte de estoque unidimensional multiperíodo

Poldi

Arenales

2010

Pesqui. Oper.

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ResumoO problema de corte de estoque multiperíodo surge imerso no planejamento e programação da produção em empresas que têm um estágio de produção caracterizado pelo corte de peças. As demandas dos itens ocorrem em períodos diversos de um horizonte de planejamento finito, sendo possível antecipar ou não a produção de itens. Os objetos não utilizados em um período ficam disponíveis no próximo, juntamente com possíveis novos objetos adquiridos ou produzidos pela própria empresa. Um modelo de otimização linear inteira de grande porte é proposto, cujo objetivo pondera as perdas nos cortes, os custos de estocagem de objetos e itens. O método simplex com geração de colunas foi especializado para resolver a relaxação linear. Experiências computacionais preliminares mostram que ganhos efetivos podem ser obtidos, quando comparado com a solução lote-por-lote, tipicamente utilizada na prática. No entanto, em problemas práticos, uma solução fracionária não é aplicável. Então, foram desenvolvidas duas abordagens para o arredondamento da solução para o problema de corte de estoque multiperíodo. Tais procedimentos são baseados em horizonte rolante, que basicamente, consiste em tentar encontrar uma solução inteira apenas para o primeiro período, já que esta será uma solução implementada na prática; para os demais períodos pode haver mudança na demanda, por exemplo, a chegada de novos pedidos ou o cancelamento de pedidos. Finalmente, concluímos que o modelo proposto para o problema de corte de estoque multiperíodo permite flexibilidade na análise da solução a ser posta em prática. O modelo multiperíodo pode ser uma ferramenta que fornece ao tomador de decisões uma ampla visão do problema e pode auxiliá-lo na tomada de decisão.Palavras-chave: problema de corte de estoque; otimização linear e inteira; geração de colunas. AbstractThe Multiperiod Cutting Stock Problem arises embedded in the production planning and programming in many industries which have a cutting process as an important stage. Ordered items have different due date over a finite planning horizon. A large scale integer linear optimization model is proposed. The model makes possible to anticipate or not the production of items. Unused objects in inventory in a period become available to the next period, added to new inventory, which are acquired or produced by the own company. The mathematical model's objective considers the waste in the cutting process, and costs for holding objects and final items. The simplex method with column generation was specialized to solve the linear relaxation. Some preliminary computational experiments showed that the multiperiod model could obtain effective gains when compared with the lot-for-lot solution, which is typically used in practice. However, in real world problems, the fractional solution is useless. So, additionally, two rounding procedures are developed to determine integer solutions for multiperiod cutting stock problems. Such procedures are based on a rolling horizon scheme, which roughly means, find an integer solution o...

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“…Such simple cuts were preferred because they do not destroy the structure of the pricing problem. Scheithauer et al [11] and Belov and Scheithauer [12,7] used Gomory fractional and mixed-integer cuts to strengthen the relaxation. The non-linear dependence of cut coefficients on the elements of each column make the pricing problem rather difficult: the objective function of the knapsack becomes non-linear.…”

Section: Branching Schemes and Cutting Planesmentioning

confidence: 99%

“…The first-rank cut (3) is an inequality and it must have a positive multiplier in a linear combination for (3) to be valid. However, this can lead to very large coefficients in cuts of higher ranks [12]. Thus, we follow the way of [7] and turn (3) into an equation by introducing a slack variable s 1 :…”

Section: Representation Of Cutsmentioning

confidence: 99%

Gomory Cuts from a Position-Indexed Formulation of 1D Stock Cutting

Belov

Scheithauer

Alves

et al.

Intelligent Decision Support

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Abstract. Most integer programming problems can be formulated in several ways. Some formulations are better suited for solution by exact methods, because they have either (i) a strong LP relaxation, (ii) few symmetries in the solution space, or both. However, solving one formulation, we can often branch and/or add cutting planes which are implicitly based on variables of other formulations, working in fact on intersection of several polytopes. Traditional examples of this approach can be found in, e.g., (capacitated) routing and network planning where decomposed models operate with paths or trees, and thus need to be solved by column generation, but original models operate on separate edges. We consider such a 'capacity-extended formulation', the so-called arc-flow model, of the 1D cutting stock problem. Its variables are known to induce effective branching constraints leading to small and stable branch&bound trees. In this work we explore Chvátal-Gomory cuts on its variables. The results are positive only for small instances. Moreover, we compare the results to the cuts constructed on the variables of the direct model. The latter are more involved but also more effective.

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A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths

Cited by 106 publications

References 9 publications

Mathematical models and decomposition algorithms for cutting and packing problems

Mathematical models and decomposition algorithms for cutting and packing problems

O problema de corte de estoque unidimensional multiperíodo

Gomory Cuts from a Position-Indexed Formulation of 1D Stock Cutting

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