# Linear Programming and its Applications

Springer Science & Business Media, Aug 15, 2007 - Business & Economics - 380 pages
Based on earlier work by a variety of authors in the 1930s and 1940s, the simplex method for solving linear programming problems was developed in 1947 by the American mathematician George B. Dantzig. Helped by the computer revolution, it has been described by some as the overwhelmingly most significant mathematical development of the last century. Owing to the simplex method, linear programming (or linear optimization, as some would have it) is pervasive in modern society for the planning and control of activities that are constrained by the availability of resources such as manpower, raw materials, budgets, and time. The purpose of this book is to describe the field of linear programming. While we aim to be reasonably complete in our treatment, we have given emphasis to the modeling aspects of the field. Accordingly, a number of applications are provided, where we guide the reader through the interactive process of mathematically modeling a particular practical situation, analyzing the consequences of the model formulated, and then revising the model in light of the results from the analysis.

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### Contents

 A1 Matrix Algebra 1 A2 Systems of Simultaneous Linear Equations 5 A3 Convexity 23 B1 Algorithms and Time Complexity Functions 31 B2 Examples of Time Complexity Functions 37 B3 Classes of Problems and Their Relations 41 11 A Short History of Linear Programming 45 12 Assumptions and the Main Components of Linear Programming Problems 48
 53 Column Generation 219 6 POSTOPTIMALITY ANALYSES 225 61 Graphical Sensitivity Analysis 227 62 Changes of the RightHand Side Values 232 63 Changes of the Objective Function Coefficients 240 64 Sensitivity Analyses in the Presence of Degeneracy 245 65 Addition of a Constraint 248 66 Economic Analysis of an Optimal Solution 252

 13 The Modeling Process 53 14 The Three Phases in Optimization 57 15 Solving the Model and Interpreting the Printout 60 21 The Diet Problem 67 22 Allocation Problems 71 23 Cutting Stock Problems 75 24 Employee Scheduling 80 25 Data Envelopment Analysis 82 26 Inventory Planning 85 27 Blending Problems 89 28 Transportation Problems 91 29 Assignment Problems 102 A Case Study 107 31 Graphical Concepts 311 The Graphical Solution Technique 129 312 Four Special Cases 138 321 The Algebraic Solution Technique 143 322 Four Special Cases Revisited 158 41 The Fundamental Theory of Duality 166 42 PrimalDual Relations 183 43 Interpretations of the Dual Problem 198 5 EXTENSIONS OF THE SIMPLEX METHOD 203 52 The Upper Bounding Technique 212
 7 NONSIMPLEX BASED SOLUTION METHODS 261 71 Alternatives to the Simplex Method 262 72 Interior Point Methods 273 81 Reformulations of Variables 811 Lower Bounding Constraints 295 812 Variables Unrestricted in Sign 296 82 Reformulations of Constraints 298 831 Minimize the Weighted Sum of Absolute Values 301 832 Bottleneck Problems 306 833 Minimax and Maximin Problems 313 834 Fractional Hyperbolic Programming 320 9 MULTIOBJECTIVE PROGRAMMING 325 91 Vector Optimization 327 921 The Weighting Method 337 922 The Constraint Method 339 93 Models with Exogenous Achievement Levels 341 931 Reference Point Programming 342 932 Fuzzy Programming 346 933 Goal Programming 351 94 Bilevel Programming 359 REFERENCES 363 SUBJECT INDEX 377 Copyright

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