Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths

This paper deals with the classical one-dimensional integer cutting stock problem, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. This process is carried out in order to optimize a given objective function (e.g., minimizing waste). Our study deals with a case in which there are several stock lengths available in limited quantities. Moreover, we have focused on problems of low demand. Some heuristic methods are proposed in order to obtain an integer solution and compared with others. The heuristic methods are empirically analyzed by solving a set of randomly generated instances and a set of instances from the literature. Concerning the latter. most of the optimal solutions of these instances are known, therefore it was possible to compare the solutions. The proposed methods presented very small objective function value gaps. (C) 2008 Elsevier Ltd. All rights reserved.

An evolutionary algorithm for the one-dimensional cutting stock problem

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The usable leftover one-dimensional cutting stock problem-a priority-in-use heuristic

We consider a one-dimensional cutting stock problem in which the material not used in the cutting patterns, if large enough, is kept for use in the future. Moreover, it is assumed that leftovers should not remain in stock for a long time, hence, such leftovers have priority-in-use compared to standard objects (objects bought by the industry) in stock. A heuristic procedure is proposed for this problem, and its performance is analyzed by solving randomly generated dynamic instances where successive problems are solved in a time horizon. For each period, new demands arise and a new problem is solved on the basis of the information about the stock of the previous periods (remaining standard objects in the stock) and usable leftovers generated during those previous periods. The computational experiments show that the solutions presented by the proposed heuristic are better than the solutions obtained by other heuristics from the literature. © 2012 The Authors. International Transactions in Operational Research © 2012 International Federation of Operational Research Societies.

The one-dimensional cutting stock problem with usable leftovers - A survey

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A genetic algorithm for the one-dimensional cutting stock problem with setups

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem`s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution-VSS-and the expected value of perfect information-EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.

Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm - for staged patterns - also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms. (c) 2007 Elsevier B.V. All rights reserved.

The cutting stock problem in the wood industry

This thesis proposes a solution for board cutting in the wood industry with the aim of usage minimization and machine productivity. The problem is dealt with as a Two-Dimensional Cutting Stock Problem and specific Combinatorial Optimization methods are used to solve it considering the features of the real problem.

The integrated lot sizing and cutting stock problem in a furniture factory

This work studies the integrated lot sizing and cutting stock problem, where the goal is to capture the dependency that exists between two important decisions in the production process, in order to economize raw materials and also reduce production and inventory costs. The integrated lot sizing and cutting stock problem is studied in a small furniture factory that produces wardrobes, dressing tables and cupboards and the lot sizing and cutting stock decisions are taken by the production manager. A column-generation technique is used to solve a linear relaxation of the proposed model. The computational results, using real data from the factory, show that it is possible to reduce total inventory and raw material costs when integrated planning is used. © 2013 IFAC.

Precise solutions of the one-dimensional Monge-Kantorovich problem

Plakhov, A.Y., (2004) 'Precise solutions of the one-dimensional Monge-Kantorovich problem', Sbornik: Mathematics 195(9) pp.1291-1307 RAE2008

A Lagrangian relaxation approach to a coupled lot-sizing and cutting stock problem

Industrial production processes involving both lot-sizing and cutting stock problems are common in many industrial settings. However, they are usually treated in a separate way, which could lead to costly production plans. In this paper, a coupled mathematical model is formulated and a heuristic method based on Lagrangian relaxation is proposed. Computational results prove its effectiveness. (C) 2009 Elsevier B.V. All rights reserved.

A Lagrangian relaxation approach to a coupled lot-sizing and cutting stock problem

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A method of fundamental solutions for the one-dimensional inverse Stefan problem

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional inverse Stefan problem for the heat equation by extending the MFS proposed in [5] for the one-dimensional direct Stefan problem. The sources are placed outside the space domain of interest and in the time interval (-T, T). Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate and stable results can be obtained efficiently with small computational cost.

Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed. © 2014 IMACS.