George B. Dantzig Mukund N. Thapa. Linear Programming. 1: Introduction. With 87 Illustrations. Springer

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1 George B. Dantzig Mukund N. Thapa Linear Programming 1: Introduction With 87 Illustrations Springer

2 Contents FOREWORD PREFACE DEFINITION OF SYMBOLS xxi xxxiii xxxvii 1 THE LINEAR PROGRAMMING PROBLEM SOME SIMPLE EXAMPLES MATHEMATICAL STATEMENT FORMULATING LINEAR PROGRAMS The Column (Recipe/Activity) Approach The Row (Material Balance) Approach EXAMPLES OF MODEL FORMULATION Product Mix Problem (Column Approach) Product Mix Problem (Row Approach) A Simple Warehouse Problem On-the-Job Training BOUNDS AXIOMS NOTES к SELECTED BIBLIOGRAPHY PROBLEMS 25 2 SOLVING SIMPLE LINEAR PROGRAMS TWO-VARIABLE PROBLEM TWO-EQUATION PROBLEM Graphical Solution The Dual Linear Program FOURIER-MOTZKIN ELIMINATION Illustration of the FME Process The Fourier-Motzkin Elimination Algorithm Fourier-Motzkin Elimination Theory INFEASIBILITY THEOREM NOTES & SELECTED BIBLIOGRAPHY 53 ix

3 CONTENTS 2.6 PROBLEMS 54 3 THE SIMPLEX METHOD GRAPHICAL ILLUSTRATION THE SIMPLEX ALGORITHM Canonical Form and Basic Variables Improving a Nonoptimal Basic Feasible Solution The Simplex Algorithm Theory Behind the Simplex Algorithm SIMPLEX METHOD The Method Phase I/Phase II Algorithm Theory Behind Phase I BOUNDED VARIABLES REVISED SIMPLEX METHOD Motivation Revised Simplex Method Illustrated Revised Simplex Algorithm Computational Remarks NOTES к SELECTED BIBLIOGRAPHY PROBLEMS 98 4 INTERIOR-POINT METHODS BASIC CONCEPTS PRIMAL AFFINE / DIKIN'S METHOD INITIAL SOLUTION NOTES к SELECTED BIBLIOGRAPHY PROBLEMS DUALITY DUAL AND PRIMAL PROBLEMS Von Neumann Symmetric Form Tucker Diagram Duals of Mixed Systems The Dual of the Standard Form Primal-Dual Feasible-Infeasible Cases DUALITY THEOREMS COMPLEMENTARY SLACKNESS OBTAINING A DUAL SOLUTION NOTES к SELECTED BIBLIOGRAPHY PROBLEMS 139

4 CONTENTS xi 6 EQUIVALENT FORMULATIONS RESTRICTED VARIABLES UNRESTRICTED (FREE) VARIABLES ABSOLUTE VALUES GOAL PROGRAMMING MINIMIZING THE MAXIMUM OF LINEAR FUNCTIONS CURVE FITTING PIECEWISE LINEAR APPROXIMATIONS Convex/Concave Functions Piecewise Continuous Linear Functions Separable Piecewise Continuous Linear Functions NOTES & SELECTED BIBLIOGRAPHY PROBLEMS PRICE MECHANISM AND SENSITIVITY ANALYSIS THE PRICE MECHANISM OF THE SIMPLEX METHOD Marginal Values or Shadow Prices Economic Interpretation of the Simplex Method The Manager of a Machine Tool Plant The Ambitious Industrialist Sign Convention on Prices INTRODUCING A NEW VARIABLE INTRODUCING A NEW CONSTRAINT COST RANGING CHANGES IN THE RIGHT-HAND SIDE CHANGES IN THE COEFFICIENT MATRIX THE SUBSTITUTION EFFECT OF NONBASIC ACTIVITIES ON BASIC ACTIVITIES NOTES AND SELECTED BIBLIOGRAPHY PROBLEMS TRANSPORTATION AND ASSIGNMENT PROBLEM THE CLASSICAL TRANSPORTATION PROBLEM Mathematical Statement Properties of the System STANDARD TRANSPORTATION ARRAY FINDING AN INITIAL SOLUTION Triangularity Rule The Least Remaining Cost Rule Vogel's Approximation Method Russel's Approximation Method Cost Preprocessing FAST SIMPLEX ALGORITHM FOR THE TRANSPORTATION PROBLEM Simplex Multipliers, Optimality, and the Dual 222

5 Xll CONTENTS Finding a Better Basic Solution Illustration of the Solution Process THE ASSIGNMENT PROBLEM EXCESS AND SHORTAGE Mathematical Statement Properties of the System Conversion to the Classical Form Simplex Multipliers and Reduced Costs PRE-FIXED VALUES AND INADMISSIBLE SQUARES THE CAPACITATED TRANSPORTATION PROBLEM NOTES & SELECTED BIBLIOGRAPHY PROBLEMS NETWORK FLOW THEORY TERMINOLOGY FLOWS AND ARC-CAPACITIES AUGMENTING PATH ALGORITHM FOR MAXIMAL FLOW CUTS IN A NETWORK SHORTEST ROUTE MINIMAL SPANNING TREE MINIMUM COST-FLOW PROBLEM THE NETWORK SIMPLEX METHOD THE BOUNDED VARIABLE PROBLEM NOTES к SELECTED BIBLIOGRAPHY PROBLEMS 304 A LINEAR ALGEBRA 315 A.l SCALARS, VECTORS, AND MATRICES 315 A.2 ARITHMETIC OPERATIONS WITH VECTORS AND MATRICES 317 A.3 LINEAR INDEPENDENCE 320 A.4 ORTHOGONALITY 321 A.5 NORMS 321 A.6 VECTOR SPACES 324 A.7 RANK OF A MATRIX 326 A.8 MATRICES WITH SPECIAL STRUCTURE 326 A.9 INVERSE OF A MATRIX 329 A. 10 INVERSES OF SPECIAL MATRICES 330 A.ll DETERMINANTS 331 A.12 EIGENVALUES 333 A.13 POSITIVE-DEFINITENESS 336 A. 14 NOTES & SELECTED BIBLIOGRAPHY 337 A.15 PROBLEMS 337

6 CONTENTS хш В LINEAR EQUATIONS 341 B.l SOLUTION SETS 341 B.2 SYSTEMS OF EQUATIONS WITH THE SAME SOLUTION SETS 343 B.3 HOW SYSTEMS ARE SOLVED 345 B.4 ELEMENTARY OPERATIONS 346 B.5 CANONICAL FORMS, PIVOTING, AND SOLUTIONS 349 B.6 PIVOT THEORY 354 B.7 NOTES к SELECTED BIBLIOGRAPHY 357 B.8 PROBLEMS 357 REFERENCES 361

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