DETERMINISTIC OPERATIONS RESEARCH

Transcription

1 DETERMINISTIC OPERATIONS RESEARCH Models and Methods in Optimization Linear DAVID J. RADER, JR. Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN WILEY A JOHN WILEY & SONS, INC., PUBLICATION

2 CONTENTS PREFACE xi 1 INTRODUCTION TO OPERATIONS RESEARCH 1 I. I What is Deterministic Operations Research? I 1.2 Introduction to Optimization Modeling Common Classes of Mathematical Programs About this Book 18 Exercises 19 2 LINEAR PROGRAMMING MODELING Resource Allocation Models Work Scheduling Models Models and Data Blending Models Production Process Models Multiperiod Models: Work Scheduling and Inventory Linearization of Special Nonlinear Models Various Forms of Linear Programs Network Models 56 Exercises 68 vii

3 Viii CONTENTS 3 INTEGER AND COMBINATORIAL MODELS Fixed-Charge Models Set Covering Models Models Using Logical Constraints Combinatorial Models Sports Scheduling and an Introduction to IP Solution Techniques 109 Exercises REAL-WORLD OPERATIONS RESEARCH APPLICATIONS: AN INTRODUCTION Vehicle Routing Problems Facility Location and Network Design Models Applications in the Airline Industry 139 Exercises INTRODUCTION TO ALGORITHM DESIGN Exact and Heuristic Algorithms What to Ask When Designing Algorithms? Constructive versus Local Search Algorithms How Good is our Heuristic Solution? Examples of Constructive Methods Example of a Local Search Method Other Heuristic Methods Designing Exact Methods: Optimality Conditions 183 Exercises IMPROVING SEARCH ALGORITHMS AND CONVEXITY Improving Search and Optimal Solutions Finding Better Solutions Convexity: When Does Improving Search Imply Global Optimality? Farkas' Lemma: When Can No Improving Feasible Direction be Found? 227 Exercises GEOMETRY AND ALGEBRA OF LINEAR PROGRAMS Geometry and Algebra of "Corner Points" Fundamental Theorem of Linear Programming Linear Programs in Canonical Form 250 Exercises 262

4 CONTENTS ix 8 SOLVING LINEAR PROGRAMS: SIMPLEX METHOD Simplex Method Making the Simplex Method More Efficient Convergence, Degeneracy, and the Simplex Method Finding an Initial Solution: Two-Phase Method Bounded Simplex Method Computational Issues 305 Exercises LINEAR PROGRAMMING DUALITY Motivation: Generating Bounds Dual Linear Program Duality Theorems Another Interpretation of the Simplex Method Farkas' Lemma Revisited Economic Interpretation of the Dual Another Duality Approach: Lagrangian Duality 338 Exercises SENSITIVITY ANALYSIS OF LINEAR PROGRAMS Graphical Sensitivity Analysis Sensitivity Analysis Calculations Use of Sensitivity Analysis Parametric Programming 375 Exercises ALGORITHMIC APPLICATIONS OF DUALITY Dual Simplex Method Transportation Problem Column Generation Dantzig-Wolfe Decomposition Primal-Dual Interior Point Method 432 Exercises NETWORK OPTIMIZATION ALGORITHMS Introduction to Network Optimization Shortest Path Problems Maximum Flow Problems Minimum Cost Network Flow Problems 470 Exercises 484

5 X CONTENTS 13 INTRODUCTION TO INTEGER PROGRAMMING Basic Definitions and Formulations Relaxations and Bounds Preprocessing and Probing When are Integer Programs "Easy?" 506 Exercises SOLVING INTEGER PROGRAMS: EXACT METHODS Complete Enumeration Branch-and-Bound Methods Valid Inequalities and Cutting Planes Gomory's Cutting Plane Algorithm Valid Inequalities for 0-1 Knapsack Constraints Branch-and-Cut Algorithms Computational Issues 547 Exercises SOLVING INTEGER PROGRAMS: MODERN HEURISTIC TECHNIQUES Review of Local Search Methods: Pros and Cons Simulated Annealing Tabu Search Genetic Algorithms GRASP Algorithms 570 Exercises 574 APPENDIX A BACKGROUND REVIEW 579 A.l Basic Notation 579 A.2 Graph Theory 581 A.3 Linear Algebra 583 REFERENCE 597 INDEX 603