Linear Programming and Network Flows

The authoritative guide to modeling and solving complex problems with linear programming--extensively revised, expanded, and updated

The only book to treat both linear programming techniques and network flows under one cover, Linear Programming and Network Flows, Fourth Edition has been completely updated with the latest developments on the topic. This new edition continues to successfully emphasize modeling concepts, the design and analysis of algorithms, and implementation strategies for problems in a variety of fields, including industrial engineering, management science, operations research, computer science, and mathematics.

The book begins with basic results on linear algebra and convex analysis, and a geometrically motivated study of the structure of polyhedral sets is provided. Subsequent chapters include coverage of cycling in the simplex method, interior point methods, and sensitivity and parametric analysis. Newly added topics in the Fourth Edition include:
* The cycling phenomenon in linear programming and the geometry of cycling
* Duality relationships with cycling
* Elaboration on stable factorizations and implementation strategies
* Stabilized column generation and acceleration of Benders and Dantzig-Wolfe decomposition methods
* Line search and dual ascent ideas for the out-of-kilter algorithm
* Heap implementation comments, negative cost circuit insights, and additional convergence analyses for shortest path problems

The authors present concepts and techniques that are illustrated by numerical examples along with insights complete with detailed mathematical analysis and justification. An emphasis is placed on providing geometric viewpoints and economic interpretations as well as strengthening the understanding of the fundamental ideas. Each chapter is accompanied by Notes and References sections that provide historical developments in addition to current and future trends. Updated exercises allow readers to test their comprehension of the presented material, and extensive references provide resources for further study.

Linear Programming and Network Flows, Fourth Edition is an excellent book for linear programming and network flow courses at the upper-undergraduate and graduate levels. It is also a valuable resource for applied scientists who would like to refresh their understanding of linear programming and network flow techniques.

Preface. ONE: INTRODUCTION.
1. 1 The Linear Programming Problem.
1. 2 Linear Programming Modeling and Examples.
1. 3 Geometric Solution.
1. 4 The Requirement Space.
1. 5 Notation.
Exercises.
Notes and References.
TWO: LINEAR ALGEBRA, CONVEX ANALYSIS, AND POLYHEDRAL SETS.
2. 1 Vectors.
2. 2 Matrices.
2. 3 Simultaneous Linear Equations.
2. 4 Convex Sets and Convex Functions.
2. 5 Polyhedral Sets and Polyhedral Cones.
2. 6 Extreme Points, Faces, Directions, and Extreme Directions of Polyhedral Sets: Geometric Insights.
2. 7 Representation of Polyhedral Sets.
Exercises.
Notes and References.
THREE: THE SIMPLEX METHOD.
3. 1 Extreme Points and Optimality.
3. 2 Basic Feasible Solutions.
3. 3 Key to the Simplex Method.
3. 4 Geometric Motivation of the Simplex Method.
3. 5 Algebra of the Simplex Method.
3. 6 Termination: Optimality and Unboundedness.
3. 7 The Simplex Method.
3. 8 The Simplex Method in Tableau Format.
3. 9 Block Pivoting.
Exercises.
Notes and References.
FOUR: STARTING SOLUTION AND CONVERGENCE.
4. 1 The Initial Basic Feasible Solution.
4. 2 The Two-Phase Method.
4. 3 The Big-M Method.
4. 4 How Big Should Big-M Be?
4. 5 The Single Artificial Variable Technique.
4. 6 Degeneracy, Cycling, and Stalling.
4. 7 Validation of Cycling Prevention Rules.
Exercises.
Notes and References.
FIVE: SPECIAL SIMPLEX IMPLEMENTATIONS AND OPTIMALITY CONDITIONS.
5. 1 The Revised Simplex Method.
5. 2 The Simplex Method for Bounded Variables.
5. 3 Farkas' Lemma via the Simplex Method.
5. 4 The Karush-Kuhn-Tucker Optimality Conditions.
Exercises.
Notes and References.
SIX: DUALITY AND SENSITIVITY ANALYSIS.
6. 1 Formulation of the Dual Problem.
6. 2 Primal-Dual Relationships.
6. 3 Economic Interpretation of the Dual.
6. 4 The Dual Simplex Method.
6. 5 The Primal-Dual Method.
6. 6 Finding an Initial Dual Feasible Solution: The Artificial Constraint Technique.
6. 7 Sensitivity Analysis.
6. 8 Parametric Analysis.
Exercises.
Notes and References.
SEVEN: THE DECOMPOSITION PRINCIPLE.
7. 1 The Decomposition Algorithm.
7. 2 Numerical Example.
7. 3 Getting Started.
7. 4 The Case of Unbounded Region X.
7. 5 Block Diagonal or Angular Structure.
7. 6 Duality and Relationships with other Decomposition Procedures.
Exercises.
Notes and References.
EIGHT: COMPLEXITY OF THE SIMPLEX ALGORITHM AND POLYNOMIAL-TIME ALGORITHMS.
8. 1 Polynomial Complexity Issues.
8. 2 Computational Complexity of the Simplex Algorithm.
8. 3 Khachian's Ellipsoid Algorithm.
8. 4 Karmarkar's Projective Algorithm.
8. 5 Analysis of Karmarkar's Algorithm: Convergence, Complexity, Sliding Objective Method, and Basic Optimal Solutions.
8. 6 Affine Scaling, Primal-Dual Path-Following, and Predictor-Corrector Variants of Interior Point Methods.
Exercises.
Notes and References.
NINE: MINIMAL-COST NETWORK FLOWS.
9. 1 The Minimal-Cost Network Flow Problem.
9. 2 Some Basic Definitions and Terminology from Graph Theory.
9. 3 Properties of the A Matrix.
9. 4 Representation of a Nonbasic Vector in Terms of the Basic Vectors.
9. 5 The Simplex Method for Network Flow Problems.
9. 6 An Example of the Network Simplex Method.
9. 7 Finding an Initial Basic Feasible Solution.
9. 8 Network Flows with Lower and Upper Bounds.
9. 9 The Simplex Tableau Associated with a Network Flow Problem.
9. 10 List Structures for Implementing the Network Simplex Algorithm.
9. 11 Degeneracy, Cycling, and Stalling.
9. 12 Generalized Network Problems.
Exercises.
Notes and References.
TEN: THE TRANSPORTATION AND ASSIGNMENT PROBLEMS.
10. 1 Definition of the Transportation Problem.
10. 2 Properties of the A Matrix.
10. 3 Representation of a Nonbasic Vector in Terms of the Basic Vectors.
10. 4 The Simplex Method for Transportation Problems.
10. 5 Illustrative Examples and a Note on Degeneracy.
10. 6 The Simplex Tableau Associated with a Transportation Tableau.
10. 7 The Assignment Problem: (Kuhn's) Hungarian Algorithm.
10. 8 Alternating Path Basis Algorithm for Assignment Problems.
10. 9 A Polynomial-Time Successive Shortest Path Approach for Assignment Problems.
10. 10 The Transshipment Problem.
Exercises.
Notes and References.
ELEVEN: THE OUT-OF-KILTER ALGORITHM.
11. 1 The Out-of-Kilter Formulation of a Minimal Cost Network Flow Problem.
11. 2 Strategy of the Out-of-Kilter Algorithm.
11. 3 Summary of the Out-of-Kilter Algorithm.
11. 4 An Example of the Out-of-Kilter Algorithm.
11. 5 A Labeling Procedure for the Out-of-Kilter Algorithm.
11. 6 Insight into Changes in Primal and Dual Function Values.
11. 7 Relaxation Algorithms.
Exercises.
Notes and References.
TWELVE: MAXIMAL FLOW, SHORTEST PATH, MULTICOMMODITY FLOW, AND NETWORK SYNTHESIS PROBLEMS.
12. 1 The Maximal Flow Problem.
12. 2 The Shortest Path Problem.
12. 3 Polynomial-Time Shortest Path Algorithms for Networks Having Arbitrary Costs.
12. 4 Multicommodity Flows.
12. 5 Characterization of a Basis for the Multicommodity Minimal-Cost Flow Problem.
12. 6 Synthesis of Multiterminal Flow Networks.
Exercises.
Notes and References.
BIBLIOGRAPHY.
INDEX.