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# A Linear Programming Approach to the Cutting Stock Problem—Part II

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## Abstract

In this paper, the methods for stock cutting outlined in an earlier paper in this Journal [Opns Res 9, 849--859 1961] are extended and adapted to the specific full-scale paper trim problem. The paper describes a new and faster knapsack method, experiments, and formulation changes. The experiments include ones used to evaluate speed-up devices and to explore a connection with integer programming. Other experiments give waste as a function of stock length, examine the effect of multiple stock lengths on waste, and the effect of a cutting knife limitation. The formulation changes discussed are i limitation on the number of cutting knives available, n balancing of multiple machine usage when orders are being filled from more than one machine, and m introduction of a rational objective function when customers' orders are not for fixed amounts, but rather for a range of amounts. The methods developed are also applicable to a variety of cutting problems outside of the paper industry.
... O PCE tem uma ampla gama de aplicações industriais e tem sido estudado extensivamente nas últimas décadas. Os trabalhos de [6] e [5] são considerados os pioneiros e apresentaram uma técnica de geração de colunas para a solução, o que viabilizou a aplicação em problemas reais. ...
... Dessa forma, como os atrasos e adiantamentos estão inteiramente ligados a essas variáveis, consegue-se determinar, para essas variáveis, uma solução inteira. 5 ...
... Gilmore and Gomory [16] used linear programming methods to build mathematical models. The postponed column generation method, improved in 1963 [17] and 1965 [18], is also applicable to the problem of twodimensional plate cutting problems and even multidimensional problems. ...
Article
Dalam pembangunan proyek konstruksi, penggunaan material yang tidak optimal dapat menimbulkan waste material konstruksi yang berdampak terhadap tingginya biaya proyek dan menjadi limbah bagi lingkungan. Salah satu material konstruksi yang sangat banyak digunakan dan menimbulkan waste ialah besi beton bertulang, terutama apabila variasi potongan tulangan cukup banyak. Waste tersebut diakibatkan pemotongan tulangan yang tidak optimal. Integer linear programming merupakan metode yang terbukti efektif dalam menyelesaikan banyak permasalahan optimasi yang bersifat linear. Namun, masih sedikit yang mengaplikasikan integer linear programming pada problem optimasi pemotongan besi tulangan dengan banyak variasi potongan tulangan. Oleh karena itu, penelitian ini akan mengevaluasi sejauh mana tingkat keberhasilan optimasi metode integer linear programming dalam upaya untuk memperoleh waste besi beton paling minimum dengan bantuan program MATLAB. Data yang digunakan ialah besi beton berdiameter 13mm pada pembangunan sebuah proyek gudang. Setelah dilakukan proses optimasi, penggunaan besi beton berdiameter 13mm dapat dihemat sebanyak 59 lonjor (setara dengan penghematan 5,97%).
Chapter
The one-dimensional cutting stock problem considers only one dimension in the cutting process and consists of cutting a set of objects available in stock to produce the desired items in specified quantities and sizes. Therefore, the cutting process can generate leftovers - which can be reused in a new demand - or losses, which are discarded. In this context, the objective of this work is to present a methodology for generating a numerical data set, considering items demand data and cut objects, for the problem of classifying leftovers or losses from the cutting stock process. The paper presents an algorithm to generate such a set and its evaluation using Machine Learning methods, as Logistic Regression, Naive Bayes, Decision Trees and Random Forests. These methods are trained and validated using statistical measures. The provided dataset is available to be used in supervised training algorithms for classification tasks. Results show the performance of the Machine Learning methods, which are evaluated using stratified k-fold cross validation and specific statistical measures. Since the analysis indicate good performance, we can also conclude that the generated data set is robust and it can be used in other classification tasks.
Chapter
Wireless Sensor Networks (WSNs) are systems with great potential for applications in the most diverse areas such as industry, security, public health, and agriculture. In general, for a WSN to achieve high performance, multiple criteria must be considered, such as coverage area, connectivity, and energy consumption. In this work an Integer Programming (IP) model to solve a Sensor Allocation Problem (SAP) is presented. The IP model considers a heterogeneous WSN and deterministic locations to positioning of sensors. The proposed model was validated using the IBM ILOG CPLEX solver. A several computational experiments were performed, and an analysis through small and medium-sized instances of the problem under study are presented and discussed. The proposed model presents good results given the problem premises, constraints and considered objectives, achieving 0.0099% optimality gap for the best scenarios where networks are fully connected and are feasible to implement. Other suboptimal evaluated scenarios with denser distribution of sensor nodes depict about 0.04% of isolated node positioning, spite maintaining overall balance between energy consumption and coverage. Therefore, the proposed model shows promise for achieving practical solutions, i.e., those with implementation feasibility in most considered heterogeneous network scenarios.
Chapter
This paper presents a model aimed at solving a Multi-period Cutting Stock Problem (MPCSP) for a sawmill at a strategic-operational level. The study seeks to minimize raw material and storage costs to manufacture square wood planks for packing batches of wood boards and panels. For these purposes, the study used real data from an actual Chilean sawmill. The resolution of the Gurobi model allows a $$20.4\%$$ cost reduction when comparing the empirical method with the model implemented in the sawmill.KeywordsMulti-periodOne-dimensional cutting stock problemSawmillIndustrial application
Article
Interest in integrating lot-sizing and cutting stock problems has been increasing over the years. This integrated problem has been applied in many industries, such as paper, textile and furniture. Yet, there are only a few studies that acknowledge the importance of uncertainty to optimise these integrated decisions. This work aims to address this gap by incorporating demand uncertainty through stochastic programming and robust optimisation approaches. Both robust and stochastic models were specifically conceived to be solved by a column generation method. In addition, both models are embedded in a rolling-horizon procedure in order to incorporate dynamic reaction to demand realisation and adapt the models to a multistage stochastic setting. Computational experiments are proposed to test the efficiency of the column generation method and include a Monte Carlo simulation to assess both stochastic programming and robust optimisation for the integrated problem. Results suggest that acknowledging uncertainty can cut costs by up to 39.7%, while maintaining or reducing variability at the same time.
Article
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In this paper, an automotive spring factory is studied to optimize its hardening process. The assignment of items to the hardening furnace is treated as a one-dimensional cutting stock problem, an approach not found in the literature. To make a feasible decision in this assignment, the activity that follows the furnace, i.e. the bending of the items, is also analyzed. In order to consider practical constraints of the company, as the position of items on the furnace, the proposed mathematical model is based on an arc flow formulation and it is validated through instances with real and random data. A heuristic approach was developed to simulate the company's decision, and to compare the random instances results. Results with real data demonstrate that the model found, in viable computational time, a solution significantly better than that of current company practice, increasing the production by 51.2%. This increase was mainly made possible by a 71.5% reduction in wasted space in the furnace and by a 26.2% reduction of time spent on setups. In random instances, the mathematical model also far outperformed the company's practice, finding the optimal solution in 98.9% of the cases. It was identified that computational time is the most sensitive criterion to the variation in the parameters and the length of the items is the parameter that most influences the results.
Article
Short cut computational methods are developed for solving systems whose matrices may be generally described as block triangular.
Article
Zusammenfassung Es wird ein spezielles nichtlineares Programm, die Maximierung eines Quotienten zweier linearer Funktionen, dargestellt und mit Hilfe einer verÄnderten Simplex-Methode gelöst.