Deterministic Operations Research: Models and Methods in Linear Optimization

652 Pages·2010·15.78 MB·English

by  David J. Rader

CONTENTS
PREFACE
1 INTRODUCTION TO OPERATIONS RESEARCH
1.1 WHAT IS DETERMINISTIC OPERATIONS RESEARCH?
1.2 INTRODUCTION TO OPTIMIZATION MODELING
1.3 COMMON CLASSES OF MATHEMATICAL PROGRAMS
1.4 ABOUT THIS BOOK
EXERCISES
2 LINEAR PROGRAMMING MODELING
2.1 RESOURCE ALLOCATION MODELS
2.2 WORK SCHEDULING MODELS
2.3 MODELS AND DATA
2.4 BLENDING MODELS
2.5 PRODUCTION PROCESS MODELS
2.6 MULTIPERIOD MODELS: WORK SCHEDULING AND
INVENTORY
2.7 LINEARIZATION OF SPECIAL NONLINEAR MODELS
2.8 VARIOUS FORMS OF LINEAR PROGRAMS
2.9 NETWORK MODELS
EXERCISES
3 INTEGER AND COMBINATORIAL MODELS
3.1 FIXED-CHARGE MODELS
3.2 SET COVERING MODELS
3.3 MODELS USING LOGICAL CONSTRAINTS
3.4 COMBINATORIAL MODELS
3.5 SPORTS SCHEDULING AND AN INTRODUCTION TO
IP SOLUTION TECHNIQUES
EXERCISES
4 REAL-WORLD OPERATIONS RESEARCH
APPLICATIONS: AN INTRODUCTION 126
4.1 VEHICLE ROUTING PROBLEMS
4.2 FACILITY LOCATION AND NETWORK DESIGN
MODELS
4.3 APPLICATIONS IN THE AIRLINE INDUSTRY
EXERCISES
5 INTRODUCTION TO ALGORITHM DESIGN
5.1 EXACT AND HEURISTIC ALGORITHMS
5.2 WHAT TO ASK WHEN DESIGNING ALGORITHMS?
5.3 CONSTRUCTIVE VERSUS LOCAL SEARCH
ALGORITHMS
5.4 HOW GOOD IS OUR HEURISTIC SOLUTION?
5.5 EXAMPLES OF CONSTRUCTIVE METHODS
5.6 EXAMPLE OF A LOCAL SEARCH METHOD
5.7 OTHER HEURISTIC METHODS
5.8 DESIGNING EXACT METHODS: OPTIMALITY
CONDITIONS
EXERCISES
6 IMPROVING SEARCH ALGORITHMS AND
CONVEXITY
6.1 IMPROVING SEARCH AND OPTIMAL SOLUTIONS
6.2 FINDING BETTER SOLUTIONS
6.3 CONVEXITY: WHEN DOES IMPROVING SEARCH
IMPLY GLOBAL OPTIMALITY?
6.4 FARKAS’ LEMMA: WHEN CAN NO IMPROVING
FEASIBLE DIRECTION BE FOUND?
EXERCISES
7 GEOMETRY AND ALGEBRA OF LINEAR
PROGRAMS
7.1 GEOMETRY AND ALGEBRA OF “CORNER POINTS”
7.2 FUNDAMENTAL THEOREM OF LINEAR
PROGRAMMING
7.3 LINEAR PROGRAMS IN CANONICAL FORM
EXERCISES
8 SOLVING LINEAR PROGRAMS: SIMPLEX
METHOD
8.1 SIMPLEX METHOD
8.2 MAKING THE SIMPLEX METHOD MORE EFFICIENT
8.3 CONVERGENCE, DEGENERACY, AND THE SIMPLEX
METHOD
8.4 FINDING AN INITIAL SOLUTION: TWO-PHASE
METHOD
8.5 BOUNDED SIMPLEX METHOD
8.6 COMPUTATIONAL ISSUES
EXERCISES
9 LINEAR PROGRAMMING DUALITY
9.1 MOTIVATION: GENERATING BOUNDS
9.2 DUAL LINEAR PROGRAM
9.3 DUALITY THEOREMS
9.4 ANOTHER INTERPRETATION OF THE SIMPLEX
METHOD
9.5 FARKAS’ LEMMA REVISITED
9.6 ECONOMIC INTERPRETATION OF THE DUAL
9.7 ANOTHER DUALITY APPROACH: LAGRANGIAN
DUALITY
EXERCISES
10 SENSITIVITY ANALYSIS OF LINEAR
PROGRAMS
10.1 GRAPHICAL SENSITIVITY ANALYSIS
10.2 SENSITIVITY ANALYSIS CALCULATIONS
10.3 USE OF SENSITIVITY ANALYSIS
10.4 PARAMETRIC PROGRAMMING
EXERCISES
11 ALGORITHMIC APPLICATIONS OF DUALITY
11.1 DUAL SIMPLEX METHOD
11.2 TRANSPORTATION PROBLEM
11.3 COLUMN GENERATION
11.4 DANTZIG–WOLFE DECOMPOSITION
11.5 PRIMAL–DUAL INTERIOR POINT METHOD
EXERCISES
12 NETWORK OPTIMIZATION ALGORITHMS
12.1 INTRODUCTION TO NETWORK OPTIMIZATION
12.2 SHORTEST PATH PROBLEMS
12.3 MAXIMUM FLOW PROBLEMS
12.4 MINIMUM COST NETWORK FLOW PROBLEMS
EXERCISES
13 INTRODUCTION TO INTEGER
PROGRAMMING
13.1 BASIC DEFINITIONS AND FORMULATIONS
13.2 RELAXATIONS AND BOUNDS
13.3 PREPROCESSING AND PROBING
13.4 WHEN ARE INTEGER PROGRAMS “EASY?”
EXERCISES
14 SOLVING INTEGER PROGRAMS: EXACT
METHODS
14.1 COMPLETE ENUMERATION
14.2 BRANCH-AND-BOUND METHODS
14.3 VALID INEQUALITIES AND CUTTING PLANES
14.4 GOMORY’S CUTTING PLANE ALGORITHM
14.5 VALID INEQUALITIES FOR 0–1 KNAPSACK
CONSTRAINTS
14.6 BRANCH-AND-CUT ALGORITHMS
14.7 COMPUTATIONAL ISSUES
EXERCISES
15 SOLVING INTEGER PROGRAMS: MODERN
HEURISTIC TECHNIQUES
15.1 REVIEW OF LOCAL SEARCH METHODS: PROS AND
CONS
15.2 SIMULATED ANNEALING
15.3 TABU SEARCH
15.4 GENETIC ALGORITHMS
15.5 GRASP ALGORITHMS
EXERCISES
APPENDIX A BACKGROUND REVIEW
A.1 BASIC NOTATION
A.2 GRAPH THEORY
A.3 LINEAR ALGEBRA
REFERENCE
INDEX

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Library of Congress Cataloging-in-Publication Data:
Rader, David J., 1970—
Deterministic operations research : models and methods in linear optimization /
David J. Rader, Jr.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-48451-7 (cloth) 1. Operations research–Mathematics. 2.
Mathematical optimization. 3.
Linear programming. I. Title.
T57.6.R245 2010
519.7’2–dc22
2009054232
10 9 8 7 6 5 4 3 2 1