Linear Programming and Network Flows

Mokhtar S. Bazaraa, John J. Jarvis, Hanif D. Sherali

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781118211328
• Edition: 4
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.

SIX: DUALITY AND SENSITIVITY ANALYSIS

For every linear program we solve, there is another associated linear program that we happen to be simultaneously solving. This new linear program satisfies some very important properties. It may be used to obtain the solution to the original program, and its variables provide extremely useful information about the set of optimal solutions to the original linear program.

This leads to rich economic interpretations related to the original linear programming problem. In fact, the roots of this problem lie in the characterization of the optimality conditions for the original linear program. For the sake of expository reference, we shall call the original linear programming problem the primal (linear programming) problem, and we shall call this related linear program the dual (linear programming) problem. Although the term “dual” comes from linear algebra, the term “primal” was suggested as an appropriate Latin antonym by Dantzig’s father, Tobias Dantzig (who was a mathematician), to substitute for the cumbersome phrase, “the original problem of which this is the dual.” Actually, the terms primal and dual for this related pair of linear programming problems are only relative, because the dual of the “dual” is the “primal” itself.

We shall begin by formulating this new dual (linear programming) problem and proceed to develop some of its important properties. These properties will lead to two new algorithms, the dual simplex method and the primal–dual algorithm, for solving linear programs. Finally, we shall discuss the effect of variations in the data, that is, the cost coefficients, the right–hand–side coefficients, and the constraint coefficients, on the optimal solution to a linear program.

6.1 FORMULATION OF THE DUAL PROBLEM

Associated with each linear programming problem there is another linear programming problem called the dual. The dual linear program possesses many important properties relative to the original primal linear program. There are two important forms (definitions) of duality: the canonical form of duality and the standard form of duality. These two forms are completely equivalent. They arise respectively from the canonical and the standard representation of linear programming problems.

Canonical Form of Duality

Suppose that the primal linear program is given in the (canonical) form:

Then the dual linear program is defined by:

Note that there is exactly one dual variable for each primal constraint and exactly one dual constraint for each primal variable. We shall say more about this later.

Example 6.1

Consider the following linear program and its dual:

Before proceeding further, try solving both problems and comparing their optimal objective values. This will provide a hint of things to come.

In the canonical definition of duality it is important for Problem P to have a “minimization” objective with all “greater than or equal to” constraints and all “nonnegative” variables. In theory, to apply the canonical definition of duality we must first convert the primal linear program to the foregoing format. However, in practice it is possible to immediately write down the dual of any linear program. We shall discuss this shortly.

Standard Form of Duality

Another equivalent definition of duality may be given with the primal linear program stated in the following standard form:

Then the dual linear program is defined by:

Example 6.2

Consider the following linear program and its dual (compare this with Example 6.1 above):

Given one of the definitions, canonical or standard, it is easy to demonstrate that the other definition is valid. For example, suppose that we accept the standard form as a definition and wish to demonstrate that the canonical form is correct. By adding slack variables to the canonical form of a linear program, we may apply the standard form of duality to obtain the dual problem.

But since −wI ≤ 0 is the same as w ≥ 0, we obtain the canonical form of the dual problem.

Dual of the Dual

Since the dual linear program is itself a linear program, we may wonder what its dual might be. Consider the dual in canonical form:

Applying the transformation techniques of Cha...

Table of contents

• COVER
• Title Page
• Copyright
• Dedication
• Preface
• ONE: INTRODUCTION
• TWO: LINEAR ALGEBRA, CONVEX ANALYSIS, AND POLYHEDRAL SETS
• THREE: THE SIMPLEX METHOD
• FOUR: STARTING SOLUTION AND CONVERGENCE
• FIVE: SPECIAL SIMPLEX IMPLEMENTATIONS AND OPTIMALITY CONDITIONS
• SIX: DUALITY AND SENSITIVITY ANALYSIS
• SEVEN: THE DECOMPOSITION PRINCIPLE
• EIGHT: COMPLEXITY OF THE SIMPLEX ALGORITHM AND POLYNOMIAL–TIME ALGORITHMS
• NINE: MINIMAL COST NETWORK FLOWS
• TEN: THE TRANSPORTATION AND ASSIGNMENT PROBLEMS
• ELEVEN: THE OUT–OF–KILTER ALGORITHM
• TWELVE: MAXIMAL FLOW, SHORTEST PATH, MULTICOMMODITY FLOW, AND NETWORK SYNTHESIS PROBLEMS
• BIBLIOGRAPHY
• INDEX