# Introduction to Stochastic Programming

Springer Science & Business Media, 2 lut 2000 - 421
The aim of stochastic programming is to find optimal decisions in problems which involve uncertain data. This field is currently developing rapidly with contributions from many disciplines including operations research, mathematics, and probability. Conversely, it is being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks. This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability. The authors aim to present a broad overview of the main themes and methods of the subject. Its prime goal is to help students develop an intuition on how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems. The first chapters introduce some worked examples of stochastic programming and demonstrate how a stochastic model is formally built. Subsequent chapters develop the properties of stochastic programs and the basic solution techniques used to solve them. Three chapters cover approximation and sampling techniques and the final chapter presents a case study in depth. A wide range of students from operations research, industrial engineering, and related disciplines will find this a well-paced and wide-ranging introduction to this subject.

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### Spis treści

 Introduction and Examples 3 11 A Farming Example and the News Vendor Problem 4 12 Financial Planning and Control 20 13 Capacity Expansion 28 14 Design for Manufacturing Quality 37 15 Other Applications 42 Uncertainty and Modeling Issues 49 22 Deterministic Linear Programs 51
 64 Nonlinear Programming in Simple Recourse Problems 225 65 Other Nonlinear ProgrammingBased Methods 231 Multistage Stochastic Programs 233 71 Nested Decomposition Procedures 234 72 Quadratic Nested Decomposition 244 73 Other Approaches to Multiple Stages 251 Stochastic Integer Programs 253 82 Simple Integer Recourse 262

 23 Decisions and Stages 52 24 TwoStage Program with Fixed Recourse 54 25 Random Variables and Risk Aversion 61 26 Implicit Representation of the Second Stage 63 27 Probabilistic Programming 64 28 Relationship to Other DecisionMaking Models 67 29 Short Reviews 73 Basic Properties 81 Basic Properties and Theory 83 31 TwoStage Stochastic Linear Programs with Fixed Recourse 84 32 Probabilistic or Chance Constraints 103 33 Stochastic Integer Programs 109 34 TwoStage Stochastic Nonlinear Programs with Recourse 122 35 Multistage Stochastic Programs with Recourse 128 The Value of Information and the Stochastic Solution 137 42 The Value of the Stochastic Solution 139 43 Basic Inequalities 140 44 The Relationship between EVPI and VSS 141 45 Examples 144 46 Bounds on EVPI and VSS 145 Solution Methods 153 TwoStage Linear Recourse Problems 155 51 The LShaped Method 156 52 Feasibility 163 53 The Multicut Version 166 54 Bunching and Other Efficiencies 169 55 Inner Linearization Methods 174 56 Basis Factorization Methods 179 57 Special CasesSimple Recourse and Network Problems 192 Nonlinear Programming Approaches to TwoStage Recourse Problems 199 62 The Piecewise Quadratic Form of the LShaped Method 206 63 Methods Based on the Stochastic Program Lagrangian 215
 83 Binary FirstStage Variables 268 84 Other Approaches 276 Approximation and Sampling Methods 283 Evaluating and Approximating Expectations 285 91 Direct Solutions with Multiple Integration 286 92 Discrete Bounding Approximations 288 93 Using Bounds in Algorithms 296 94 Bounds in ChanceConstrained Problems 301 95 Generalized Bounds 305 96 General Convergence Properties 323 Monte Carlo Methods 331 101 General Results for Sampled Problems 332 102 Using Sampling in the LShaped Method 335 103 Stochastic QuasiGradient Methods 343 Uses with Analytical and Empirical Observations 349 Multistage Approximations 353 111 Bounds Based on the Jensen and EdmundsonMadansky Inequalities 354 112 Bounds Based on Aggregation 359 113 Bounds Based on Separable Responses 362 114 Bounds for Specific Problem Structures 366 A Case Study 373 Capacity Expansion 375 121 Model Development 376 122 Demand Distribution Modeling 382 123 Computational Comparisons 383 Sample Distribution Functions 385 A2 Continuous Random Variables 386 References 387 Author Index 411 Subject Index 415 Prawa autorskie

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