Linear Programming and Network Flows

Linear Programming and Network Flows presents the problem of minimizing and maximizing a linear function in the presence of linear equality or inequality constraints. This text provides methods for modeling complex problems via effective algorithms on modern computers, optimization problems, and effective solution algorithms. The book also explores linear programming and network flows, polynomial-time algorithms, and geometric concepts. This modified edition includes new exercises, comments, and references on recent developments, like the geometry of cycling. This is the only text that covers both linear programming techniques and network flows for students.

Mokhtar S. Bazaraa, PHD, is Emeritus Professor at the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology. He is the coauthor of Nonlinear Programming: Theory and Algorithms, Third Edition and Linear Programming and Network Flows, Third Edition, both published by Wiley. John J. Jarvis, PHD, is Emeritus Professor at the H. Milton Stewart School of Industrial and Systems Engineering at Georiga Institute of Technology. A Fellow of the Institute of Industrial Engineers (IIE) and Institute for Operations Research and the Management Sciences (INFORMS), Dr. Jarvis is the coauthor of Linear Programming and Network Flows, Third Edition (Wiley). Hanif D. Sherali, PhD, is University Distinguished Professor and the W. Thomas Rice Chaired Professor of Engineering at the Virginia Polytechnic and State University. A Fellow of INFORMS and IIE, he is the coauthor of Nonlinear Programming: Theory and Algorithms, Third Edition and Linear Programming and Network Flows, Third Edition, both published by Wiley.

Preface	p. xi
Introduction	p. 1
The Linear Programming Problem	p. 1
Linear Programming Modeling and Examples	p. 7
Geometric Solution	p. 18
The Requirement Space	p. 2
Notation	p. 27
Exercises	p. 29
Notes and References	p. 42
Linear Algebra, Convex Analysis, and Polyhedral Sets	p. 45
Vectors	p. 45
Matrices	p. 51
Simultaneous Linear Equations	p. 61
Convex Sets and Convex Functions	p. 64
Polyhedral Sets and Polyhedral Cones	p. 70
Extreme Points, Faces, Directions, and Extreme Directions of Polyhedral Sets: Geometric Insights	p. 71
Representation of Polyhedral Sets	p. 75
Exercises	p. 82
Notes and References	p. 90
The Simplex Method	p. 91
Extreme Points and Optimality	p. 91
Basic Feasible Solutions	p. 94
Key to the Simplex Method	p. 103
Geometric Motivation of the Simplex Method	p. 104
Algebra of the Simplex Method	p. 108
Termination: Optimality and Unboundedness	p. 114
The Simplex Method	p. 120
The Simplex Method in Tableau Format	p. 125
Block Pivoting	p. 134
Exercises	p. 135
Notes and References	p. 148
Starting Solution and Convergence	p. 151
The Initial Basic Feasible Solution	p. 151
The Two-Phase Method	p. 154
The Big-M Method	p. 165
How Big Should Big-M Be?	p. 172
The Single Artificial Variable Technique	p. 173
Degeneracy, Cycling, and Stalling	p. 175
Validation of Cycling Prevention Rules	p. 182
Exercises	p. 187
Notes and References	p. 198
Special Simplex Implementations and Optimality Conditions	p. 201
The Revised Simplex Method	p. 201
The Simplex Method for Bounded Variables	p. 220
Farkas' Lemma via the Simplex Method	p. 234
The Karush-Kuhn-Tucker Optimality Conditions	p. 237
Exercises	p. 243
Notes and References	p. 256
Duality and Sensitivity Analysis	p. 259
Formulation of the Dual Problem	p. 259
Primal-Dual Relationships	p. 264
Economic Interpretation of the Dual	p. 270
The Dual Simplex Method	p. 277
The Primal-Dual Method	p. 285
Finding an Initial Dual Feasible Solution: The Artificial Constraint Technique	p. 293
Sensitivity Analysis	p. 295
Parametric Analysis	p. 312
Exercises	p. 319
Notes and References	p. 336
The Decomposition Principle	p. 339
The Decomposition Algorithm	p. 340
Numerical Example	p. 345
Getting Started	p. 353
The Case of an Unbounded Region X	p. 354
Block Diagonal or Angular Structure	p. 361
Duality and Relationships with other Decomposition Procedures	p. 371
Exercises	p. 376
Notes and References	p. 391
Complexity of the Simplex Algorithm and Polynomial-Time Algorithms	p. 393
Polynomial Complexity Issues	p. 393
Computational Complexity of the Simplex Algorithm	p. 397
Khachian's Ellipsoid Algorithm	p. 401
Karmarkar's Projective Algorithm	p. 402
Analysis of Karmarkar's Algorithm: Convergence, Complexity, Sliding Objective Method, and Basic Optimal Solutions	p. 417
Affine Scaling, Primal-Dual Path Following, and Predictor-Corrector Variants of Interior Point Methods	p. 428
Exercises	p. 435
Notes and References	p. 448
Minimal-Cost Network Flows	p. 453
The Minimal Cost Network Flow Problem	p. 453
Some Basic Definitions and Terminology from Graph Theory	p. 455
Properties of the A Matrix	p. 459
Representation of a Nonbasic Vector in Terms of the Basic Vectors	p. 465
The Simplex Method for Network Flow Problems	p. 466
An Example of the Network Simplex Method	p. 475
Finding an Initial Basic Feasible Solution	p. 475
Network Flows with Lower and Upper Bounds	p. 478
The Simplex Tableau Associated with a Network Flow Problem	p. 481
List Structures for Implementing the Network Simplex Algorithm	p. 482
Degeneracy, Cycling, and Stalling	p. 488
Generalized Network Problems	p. 494
Exercises	p. 497
Notes and References	p. 511
The Transportation and Assignment Problems	p. 513
Definition of the Transportation Problem	p. 513
Properties of the A Matrix	p. 516
Representation of a Nonbasic Vector in Terms of the Basic Vectors	p. 520
The Simplex Method for Transportation Problems	p. 522
Illustrative Examples and a Note on Degeneracy	p. 528
The Simplex Tableau Associated with a Transportation Tableau	p. 535
The Assignment Problem: (Kuhn's) Hungarian Algorithm	p. 535
Alternating Path Basis Algorithm for Assignment Problems	p. 544
A Polynomial-Time Successive Shortest Path Approach for Assignment Problems	p. 546
The Transshipment Problem	p. 551
Exercises	p. 552
Notes and References	p. 564
The Out-Of-Kilter Algorithm	p. 567
The Out-of-Kilter Formulation of a Minimal Cost Network Flow Problem	p. 567
Strategy of the Out-of-Kilter Algorithm	p. 573
Summary of the Out-of-Kilter Algorithm	p. 586
An Example of the Out-of-Kilter Algorithm	p. 587
A Labeling Procedure for the Out-of-Kilter Algorithm	p. 589
Insight into Changes in Primal and Dual Function Values	p. 591
Relaxation Algorithms	p. 593
Exercises	p. 595
Notes and References	p. 605
Maximal Flow, Shortest Path, Multicommodity Flow, And Network Synthesis Problems	p. 607
The Maximal Flow Problem	p. 607
The Shortest Path Problem	p. 619
Polynomial-Time Shortest Path Algorithms for Networks Having Arbitrary Costs	p. 635
Multicommodity Flows	p. 639
Characterization of a Basis for the Multicommodity Minimal-Cost Flow Problem	p. 649
Synthesis of Multiterminal Flow Networks	p. 654
Exercises	p. 663
Notes and References	p. 678
Bibliography	p. 681
Index	p. 733
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Linear Programming and Network Flows

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