A hybrid approach based on genetic algorithms to solve the problem of cutting structural beams in a metalwork company

This work presents a hybrid approach based on the use of genetic algorithms to solve efficiently the problem of cutting structural beams arising in a local metalwork company. The problem belongs to the class of one-dimensional multiple stock sizes cutting stock problem, namely 1-dimensional multiple stock sizes cutting stock problem. The proposed approach handles overproduction and underproduction of beams and embodies the reusability of remnants in the optimization process. Along with genetic algorithms, the approach incorporates other novel refinement algorithms that are based on different search and clustering strategies.Moreover, a new encoding with a variable number of genes is developed for cutting patterns in order to make possible the application of genetic operators. The approach is experimentally tested on a set of instances similar to those of the local metalwork company. In particular, comparative results show that the proposed approach substantially improves the performance of previous heuristics.Gracia Calandin, CP.; Andrés Romano, C.; Gracia Calandin, LI. (2013). A hybrid approach based on genetic algorithms to solve the problem of cutting structural beams in a metalwork company. Journal of Heuristics. 19(2):253-273. doi:10.1007/s10732-011-9187-xS253273192Aktin, T., Özdemir, R.G.: An integrated approach to the one dimensional cutting stock problem in coronary stent manufacturing. Eur. J. Oper. Res. 196, 737–743 (2009)Alves, C., Valério de Carvalho, J.M.: A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem. Comput. Oper. Res. 35, 1315–1328 (2008)Anand, S., McCord, C., Sharma, R., et al.: An integrated machine vision based system for solving the nonconvex cutting stock problem using genetic algorithms. J. Manuf. 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O problema de corte de estoque unidimensional multiperíodo

The 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 only for the first period, since this is the solution to be, in fact, carried out. Finally, we conclude that the proposed model for multiperiod cutting stock problems allows flexibility on analyzing a solution to be put in practice. The multiperiod cutting problem can be a tool that provides the decision maker a wide view of the problem and it may help him/her on making decisions.O 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.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Universidade Federal de São Paulo (UNIFESP) Departamento de Ciência e TecnologiaUniversidade de São Paulo Inst. de Ciências Matemáticas e de ComputaçãoUNIFESP, Depto. de Ciência e TecnologiaSciEL

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm for a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints. These constraints are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on the fly within an efficient solution method. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm and then introduce the associated set of pricing subproblems in detail. The proposed methodology is demonstrated on numerical examples for the multi-stage cutting stock and the quadratic set covering problems

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems

Optimal surface cutting

Surface cutting problems in two dimensions are considered for nonrectangular items. An exact solution method is discussed. Outlines of several possible heuristic algorithms are also presented. For the heuristic methods a first approximation to the optimal solution is obtained by encompassing each item by a rectangle and then using some available strategy for this standard problem. Different approaches are then suggested for more accurate methods

Thecomposition of semi finished inventories at a solid board plant

A solid board factory produces rectangular sheets of cardboard in two different formats, namely large formats and small formats. The production process consists of two stages separated by an inventory point. In the first stage a cardboard machine produces the large formats. In the second stage a part of the large formats is cut into small formats by a separate rotary cut machine. Due to very large setup times, technical restrictions, and trim losses, the cardboard machine is not able to produce these small formats. The company follows two policies to satisfy customer demands for rotary cut format orders. When the company applies the first policy, then for each customer order an ‘optimal’ large format (with respect to trim loss) is determined and produced on the cardboard machine. In case of the second policy, a stock of a restricted number of large formats is determined in such a way that the expected trim loss is minimal. The rotary cut format order then uses the most suitable standard large format from the stock. Currently, the dimensions of the standard large formats in the semi finished inventory are based on intuitive motives, with an accent on minimizing trim losses. From the trim loss perspective it is most efficient to produce each rotary cut format from a specific large format. On the other hand, if there is only one large format in each caliper, the variety is minimal, but the trim loss might be inacceptably high. On average, the first policy results in a lower trim loss. In order to make efficiently use of the two machines and to meet customer’s due times the company applies both policies. In this paper we concentrate on the second policy, taking into account the various objectives and restrictions of the company. The purpose of the company is to have not too many different types of large formats and an acceptable amount of trim loss. The problem is formulated as a minimum clique covering problem with alternatives (MCCA), which is presumed to be NP-hard. We solve the problem by using an appropriate heuristic, which is built into a decision support system. Based on a set of real data, the actual composition of semi finished inventories is determined. The paper concludes with computational experiments.

The two-dimensional cutting stock problem within the roller blind production process

In this paper we consider a two-dimensional cutting stock problem encountered at a large manufacturer of window covering products. The problem occurs in the production process of made-to-measure roller blinds. We develop a solution method that takes into account the characteristics of the specific problem. In particular, we deal with the fact that fabrics may contain small defects that should end up with the waste. Comparison to previous practice shows significant waste reductions.cutting;trim loss;two-dimensional cutting stock problem