"There are different variants for the Cutting and Packing Problems (CPP), i.e. nesting problems consist of placing irregular small objects inside a bigger plate with fixed dimensions. This paper aims to solve a Two-dimensional Cutting Stock Problem (CSP) with irregular polygons that must be packed inside a set of identical plates with fixed dimensions. The main objective is to minimize the number of plates needed to pack all the polygons. This is a combinatorial problem with NP-hard complexity, as the main problem is based on a non-overlapping arrangement of the polygons. We used a column generation (CG) approach to find a good solution, and a simulation engine, Unity, to generate the columns. This simulation tool allowed us to verify and satisfy the non-overlapping constraints to get the final cutting patterns. The final solution is tested with the shapes1 instance found in the ES-ICUP website and then compared to one similar paper from literature. We did obtain similar results in terms of plates needed to pack all polygons but achieving a better computational time." -- Tomado del Formato de Documento de Grado.

Seminar paper from the year 2020 in the subject Business economics - Operations Research, grade: 1,3, University of Kaiserslautern, language: English, abstract: The Cutting Stock Problem (CSP) appears when a material has to be cut into smaller pieces and occurs in many branches of industry. On the one hand, the CSP belongs to the earliest studied problems through methods of Operational Research and on the other to the most intensively studied problems in combinatorial optimization. In the one-dimensional Cutting Stock Problem (1DCSP), there are typically identical pieces of a single standard length, called rolls, that need to be cut into smaller pieces lengthwise. Examples, where the cutting process is performed in one single dimension, can be found in the steel industry and the paper industry . The two-dimensional CSP (2DCSP) is classified into cutting of regular and irregular shapes and is often found in clothing and shoe-leather industries. A real-world application of a three-dimensional CSP (3DCSP) lies in the production of mattresses, where rubber blocks are cut into different types of orthogonal rectangular prisms. Methods of finding an optimal solution exist for the 1DCSP. Often in large problem instances, the required time for finding an optimal solution proliferates, and heuristics may turn out to be the more sensible option in this case. Nowadays, there are countless different ways to find acceptable solutions in a fast manner of time, among others, the column generation approach, which is the central component of the present work. This work is organized as follows. In Chapter 2, a brief overview of different formulations for the CSP is given. Furthermore, some known extensions of the classic CSP are presented, e.g., raw material, that consists of various sizes at the same time. CSP has many relatives, the closest is the Bin Packing Problem (BPP), where items are packed into bins as efficiently as possible. The third chapter shows the column generation technique for solving the CSP and provides the connection between a solution for the relaxed problem and an integer solution. In Chapter 4, different test instances of the CSP are compared using a column generation implementation solved in three different MIP solvers. The conclusion is provided in Chapter 5.

The essential introduction to the principles and applications of feedback systems—now fully revised and expanded This textbook covers the mathematics needed to model, analyze, and design feedback systems. Now more user-friendly than ever, this revised and expanded edition of Feedback Systems is a one-volume resource for students and researchers in mathematics and engineering. It has applications across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl Åström and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. Åström and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. Features a new chapter on design principles and tools, illustrating the types of problems that can be solved using feedback Includes a new chapter on fundamental limits and new material on the Routh-Hurwitz criterion and root locus plots Provides exercises at the end of every chapter Comes with an electronic solutions manual An ideal textbook for undergraduate and graduate students Indispensable for researchers seeking a self-contained resource on control theory

Now in widespread use, generalized additive models (GAMs) have evolved into a standard statistical methodology of considerable flexibility. While Hastie and Tibshirani's outstanding 1990 research monograph on GAMs is largely responsible for this, there has been a long-standing need for an accessible introductory treatment of the subject that also emphasizes recent penalized regression spline approaches to GAMs and the mixed model extensions of these models. Generalized Additive Models: An Introduction with R imparts a thorough understanding of the theory and practical applications of GAMs and related advanced models, enabling informed use of these very flexible tools. The author bases his approach on a framework of penalized regression splines, and builds a well-grounded foundation through motivating chapters on linear and generalized linear models. While firmly focused on the practical aspects of GAMs, discussions include fairly full explanations of the theory underlying the methods. Use of the freely available R software helps explain the theory and illustrates the practicalities of linear, generalized linear, and generalized additive models, as well as their mixed effect extensions. The treatment is rich with practical examples, and it includes an entire chapter on the analysis of real data sets using R and the author's add-on package mgcv. Each chapter includes exercises, for which complete solutions are provided in an appendix. Concise, comprehensive, and essentially self-contained, Generalized Additive Models: An Introduction with R prepares readers with the practical skills and the theoretical background needed to use and understand GAMs and to move on to other GAM-related methods and models, such as SS-ANOVA, P-splines, backfitting and Bayesian approaches to smoothing and additive modelling.

Global optimization aims at solving the most general problems of deterministic mathematical programming: to find the global optimum of a nonlinear, nonconvex, multivariate function of continuous and/or integer variables subject to constraints which may be themselves nonlinear and nonconvex. In addition, once the solutions are found, proof of its optimality is also expected from this methodology. Therefore, with these difficulties in mind, global optimization is becoming an increasingly powerful and important methodology. Essays and Surveys in Global Optimization is the most recent examination of its mathematical capability, power, and wide ranging solutions to many fields in the applied sciences.

This book aims to demonstrate and detail the pervasive nature of Discrete Optimization. The handbook couples the difficult, critical-thinking aspects of mathematical modeling with the hot area of discrete optimization. It is done with an academic treatment outlining the state-of-the-art for researchers across the domains of the Computer Science, Math Programming, Applied Mathematics, Engineering, and Operations Research. The book utilizes the tools of mathematical modeling, optimization, and integer programming to solve a broad range of modern problems.