Linear Programming and Network Flows

ISBN-10: 0470462728

ISBN-13: 9780470462720

Edition: 4th 2010

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Description: 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.

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Book details

List price: $179.00
Edition: 4th
Copyright year: 2010
Publisher: John Wiley & Sons Canada, Limited
Publication date: 12/14/2009
Binding: Hardcover
Pages: 768
Size: 6.50" wide x 9.75" long x 1.50" tall
Weight: 2.816

The Linear Programming Problem
Linear Programming Modeling and Examples
Geometric Solution
The Requirement Space
Notes and References
Linear Algebra, Convex Analysis, and Polyhedral Sets
Simultaneous Linear Equations
Convex Sets and Convex Functions
Polyhedral Sets and Polyhedral Cones
Extreme Points, Faces, Directions, and Extreme Directions of Polyhedral Sets: Geometric Insights
Representation of Polyhedral Sets
Notes and References
The Simplex Method
Extreme Points and Optimality
Basic Feasible Solutions
Key to the Simplex Method
Geometric Motivation of the Simplex Method
Algebra of the Simplex Method
Termination: Optimality and Unboundedness
The Simplex Method
The Simplex Method in Tableau Format
Block Pivoting
Notes and References
Starting Solution and Convergence
The Initial Basic Feasible Solution
The Two-Phase Method
The Big-M Method
How Big Should Big-M Be?
The Single Artificial Variable Technique
Degeneracy, Cycling, and Stalling
Validation of Cycling Prevention Rules
Notes and References
Special Simplex Implementations and Optimality Conditions
The Revised Simplex Method
The Simplex Method for Bounded Variables
Farkas' Lemma via the Simplex Method
The Karush-Kuhn-Tucker Optimality Conditions
Notes and References
Duality and Sensitivity Analysis
Formulation of the Dual Problem
Primal-Dual Relationships
Economic Interpretation of the Dual
The Dual Simplex Method
The Primal-Dual Method
Finding an Initial Dual Feasible Solution: The Artificial Constraint Technique
Sensitivity Analysis
Parametric Analysis
Notes and References
The Decomposition Principle
The Decomposition Algorithm
Numerical Example
Getting Started
The Case of an Unbounded Region X
Block Diagonal or Angular Structure
Duality and Relationships with other Decomposition Procedures
Notes and References
Complexity of the Simplex Algorithm and Polynomial-Time Algorithms
Polynomial Complexity Issues
Computational Complexity of the Simplex Algorithm
Khachian's Ellipsoid Algorithm
Karmarkar's Projective Algorithm
Analysis of Karmarkar's Algorithm: Convergence, Complexity, Sliding Objective Method, and Basic Optimal Solutions
Affine Scaling, Primal-Dual Path Following, and Predictor-Corrector Variants of Interior Point Methods
Notes and References
Minimal-Cost Network Flows
The Minimal Cost Network Flow Problem
Some Basic Definitions and Terminology from Graph Theory
Properties of the A Matrix
Representation of a Nonbasic Vector in Terms of the Basic Vectors
The Simplex Method for Network Flow Problems
An Example of the Network Simplex Method
Finding an Initial Basic Feasible Solution
Network Flows with Lower and Upper Bounds
The Simplex Tableau Associated with a Network Flow Problem
List Structures for Implementing the Network Simplex Algorithm
Degeneracy, Cycling, and Stalling
Generalized Network Problems
Notes and References
The Transportation and Assignment Problems
Definition of the Transportation Problem
Properties of the A Matrix
Representation of a Nonbasic Vector in Terms of the Basic Vectors
The Simplex Method for Transportation Problems
Illustrative Examples and a Note on Degeneracy
The Simplex Tableau Associated with a Transportation Tableau
The Assignment Problem: (Kuhn's) Hungarian Algorithm
Alternating Path Basis Algorithm for Assignment Problems
A Polynomial-Time Successive Shortest Path Approach for Assignment Problems
The Transshipment Problem
Notes and References
The Out-Of-Kilter Algorithm
The Out-of-Kilter Formulation of a Minimal Cost Network Flow Problem
Strategy of the Out-of-Kilter Algorithm
Summary of the Out-of-Kilter Algorithm
An Example of the Out-of-Kilter Algorithm
A Labeling Procedure for the Out-of-Kilter Algorithm
Insight into Changes in Primal and Dual Function Values
Relaxation Algorithms
Notes and References
Maximal Flow, Shortest Path, Multicommodity Flow, And Network Synthesis Problems
The Maximal Flow Problem
The Shortest Path Problem
Polynomial-Time Shortest Path Algorithms for Networks Having Arbitrary Costs
Multicommodity Flows
Characterization of a Basis for the Multicommodity Minimal-Cost Flow Problem
Synthesis of Multiterminal Flow Networks
Notes and References
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