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.

Column Generation is an insightful overview of the state of the art in integer programming column generation and its many applications. The volume begins with "A Primer in Column Generation" which outlines the theory and ideas necessary to solve large-scale practical problems, illustrated with a variety of examples. Other chapters follow this introduction on "Shortest Path Problems with Resource Constraints," "Vehicle Routing Problem with Time Window," "Branch-and-Price Heuristics," "Cutting Stock Problems," each dealing with methodological aspects of the field. Three chapters deal with transportation applications: "Large-scale Models in the Airline Industry," "Robust Inventory Ship Routing by Column Generation," and "Ship Scheduling with Recurring Visits and Visit Separation Requirements." Production is the focus of another three chapters: "Combining Column Generation and Lagrangian Relaxation," "Dantzig-Wolfe Decomposition for Job Shop Scheduling," and "Applying Column Generation to Machine Scheduling." The final chapter by François Vanderbeck, "Implementing Mixed Integer Column Generation," reviews how to set-up the Dantzig-Wolfe reformulation, adapt standard MIP techniques to the column generation context (branching, preprocessing, primal heuristics), and deal with specific column generation issues (initialization, stabilization, column management strategies).

1 Introduction.- 1.1. Purpose of the Investigation.- 1.2. Methodology Used.- 1.3. Structure of the Book.- 2 Cutting and Packing Problems as Geometric-Combinatoric Problems.- 2.1. Basic Logical Structure.- 2.2. Phenomena of Cutting and Packing.- 2.2.1. Cutting and Packing in Spatial Dimensions.- 2.2.2. Cutting and Packing in Abstract Dimensions.- 2.2.3. Related Problems.- 2.3. Delimitation in Investigation.- 3 The Treatment of Cutting and Packing Problems in the Literature.- 3.1. Models as Idealized Images of Actual Phenomena.- 3.2. Sources on Cutting and Packing Problems.- 3.2.1. Differentiation According to Thematic Criteria.- 3.2.2. Differentiation According to Bibliographical Criteria.- 3.3. Delimitation of Investigated Literature.- 4 Systematic Catalogue of Properties for the Characterization of Cutting and Packing Problems.- 4.1. Basis for Characteristic Properties.- 4.2. Design of the Catalogue.- 4.3. Characteristics Based on the Logical Structure.- 4.3.1. Dimensionality.- 4.3.2. Type of Assignment.- 4.3.3. Characteristics of Large Objects and Small Items.- 4.3.4. Pattern Restrictions.- 4.3.5. Objectives.- 4.3.6. Status of Information and Variability of Data.- 4.3.7. Solution Methods.- 4.4. Reality-Based Characteristics.- 4.4.1. Kind of Objects and Items, and Branch of Industry.- 4.4.2. Planning Context.- 4.4.3. Software.- 4.5. Overview.- 5 Types of Cutting and Packing Problems in the Literature.- 5.1. Principles of Type Definition.- 5.2. Hierarchical Catalogue of Types.- 5.2.1. General Types.- 5.2.2. Special Types.- 5.2.3. Summarized Description of the Hierarchy of Types.- 5.3. Properties of the Derived Problem Types.- 6 Bin Packing Types (BP).- 6.1. One-dimensional Bin Packing Type (BP1).- 6.2. Two-dimensional Bin Packing Types (BP2).- 6.2.1. BP2-Type with a Heterogeneous Assortment of Large Objects.- 6.2.2. BP2-Type with a Homogeneous Assortment of Large Objects.- 6.3. Actual Bin Packing Problems.- 7 Cutting Stock Types (CS).- 7.1. One-dimensional Cutting Stock Types (CS1).- 7.1.1. CS1-Type with Continuous Quantity Measurement of Large Objects.- 7.1.2. CS1-Types with Discrete Quantity Measurement of Large Objects.- 7.1.2.1. Discrete CSl-Type with a Homogeneous Assortment of Large Objects.- 7.1.2.2. Discrete CSl-Type with a Heterogeneous Assortment of Large Objects.- 7.2. Two-dimensional Cutting Stock Types (CS2).- 7.2.1. CS2-Type with Non-rectangular Small Items.- 7.2.2. CS2-Types with Rectangular Small Items.- 7.2.2.1. Rectangular CS2-Types with Only One Large Object per Figure.- 7.2.2.2. Rectangular CS2-Types with Guillotine Patterns.- 7.2.2.3. Rectangular CS2-Type with Nested Patterns.- 7.3. Three-dimensional Cutting Stock Type (CS3).- 7.4. Actual Cutting Stock Problems.- 8 Knapsack Types (KS).- 8.1. One-dimensional Knapsack Type (KS1).- 8.2. Two-dimensional Knapsack Type (KS2).- 8.3. Three-dimensional Knapsack Type (KS3).- 8.4. Actual Knapsack Problems.- 9 Pallet Loading Types (PL).- 9.1. Two-dimensional Pallet Loading Type (PL2).- 9.2. Three-dimensional Pallet Loading Type (PL3).- 9.3. Actual Pallet Loading Problems.- 10 Conclusions.- I. A Bibliography of Further C&P-Problems.- A. Published Surveys.- B. Literary References not Closely Analysed.- C. Most Recent Sources.- II. Brief Description of the Characteristics.- III. LARS Data Base System.- List of Abbreviations for the Journals.- I. General Literature.- II. C&P-Literature.

This book investigates in detail the two-dimensional packing and cutting problems in the field of operations research and management science. It introduces the mathematical models and intelligent solving algorithms for these problems, as well as their engineering applications. Most intelligent methods reported in this book have already been applied in reality, which can provide reference for the engineers. The presented novel methods for the two-dimensional packing problem provide a new way to solve the problem for researchers interested in operations research or computer science. This book also introduces three new variants of packing problems and their solving methods, which offer a different research direction. The book is intended for undergraduate and graduate students who are interested in the solving methods for packing and cutting problems, researchers investigating the application of intelligent algorithms, scientists studying the theory of the operations research and CAM software developers working on integration of packing and cutting problem.

This paper focuses on the study of the two-dimensional cutting or packing problem of irregular polygons with free rotation that must be organized within a set of identical plates with fixed dimensions. The main objective is to minimize the number of plates that can fit a specific demand of objects. Given the constraint of non-overlapping, the possibility of free rotations and the irregular shape of the objects, this is a combinatorial problem with NP-Hard complexity. A column generation procedure was proposed, in which the master problem will be in charge of selecting the subset of plates that will compose the solution, while the auxiliary problem will create each of the proposed plates to be given to the master problem as columns. For this, the auxiliary problem will use a two phased procedure with a constructive algorithm and a local search enhanced with swapping and insertion operators, made using Unity engine. To test the proposed algorithms, known instances for the cutting stock problem were selected. The solutions and computational times achieved were compared by benchmarking with the available work in the literature, obtaining satisfactory results. Further work will include the sequential use of the shown algorithms that produce good performance in order to produce a more specialized algorithm.

This book constitutes the refereed proceedings of the Third International Workshop on Experimental and Efficient Algorithms, WEA 2004, held in Angra dos Reis, Brazil in May 2004. The 40 revised full papers presented together with abstracts of two invited talks were carefully reviewed and selected from numerous submissions. The book is devoted to the areas of design, analysis, and experimental evaluation of algorithms. Among the topics covered are scheduling, heuristics, combinatorial optimization, evolutionary optimization, graph computations, labeling, robot navigation, shortest path algorithms, flow problems, searching, randomization and derandomization, string matching, graph coloring, networking, error detecting codes, timetabling, sorting, energy minimization, etc.

An accessible treatment of the modeling and solution of integer programming problems, featuring modern applications and software In order to fully comprehend the algorithms associated with integer programming, it is important to understand not only how algorithms work, but also why they work. Applied Integer Programming features a unique emphasis on this point, focusing on problem modeling and solution using commercial software. Taking an application-oriented approach, this book addresses the art and science of mathematical modeling related to the mixed integer programming (MIP) framework and discusses the algorithms and associated practices that enable those models to be solved most efficiently. The book begins with coverage of successful applications, systematic modeling procedures, typical model types, transformation of non-MIP models, combinatorial optimization problem models, and automatic preprocessing to obtain a better formulation. Subsequent chapters present algebraic and geometric basic concepts of linear programming theory and network flows needed for understanding integer programming. Finally, the book concludes with classical and modern solution approaches as well as the key components for building an integrated software system capable of solving large-scale integer programming and combinatorial optimization problems. Throughout the book, the authors demonstrate essential concepts through numerous examples and figures. Each new concept or algorithm is accompanied by a numerical example, and, where applicable, graphics are used to draw together diverse problems or approaches into a unified whole. In addition, features of solution approaches found in today's commercial software are identified throughout the book. Thoroughly classroom-tested, Applied Integer Programming is an excellent book for integer programming courses at the upper-undergraduate and graduate levels. It also serves as a well-organized reference for professionals, software developers, and analysts who work in the fields of applied mathematics, computer science, operations research, management science, and engineering and use integer-programming techniques to model and solve real-world optimization problems.

The conference aims at forming a unique platform to bring together academicians and practitioners from industrial engineering and management engineering as well as from other disciplines working on production function applying the tools of operational research and production/operational management. Topics treated include: computer aided manufacturing, industry 4.0, big data and analytics, flexible manufacturing systems, fuzzy logic, industrial applications, information technologies in production management, optimization, production economy, production planning and control, productivity and performance management, project management, quality management, risk analysis and management, supply chain management.