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Linear Programming, Sensitivity Analysis & Related Topics

Linear Programming, Sensitivity Analysis & Related Topics

Marie-France Derhy

Feb 2010, Paperback, 464 pages
ISBN13: 9780273723387
ISBN10: 0273723383
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This book covers all aspects of linear programming from the two-dimensional LPs and their extension to higher dimensional LPs, through duality and sensitivity analysis and finally to the examination of commented software outputs.
The book is organised into three distinct parts: the first part studies the concepts of linear programming and presents its founding theorems complete with proofs and applications; the second part presents linear programming in the diversity of its variants (Integer Programming, Game Theory, Transportation Problem, Assignment Model), and highlights the modelling problems that are involved in network optimisation; the final part furthers the discussion on selected topics and presents an opening to nonlinear programming through quadratic programming.

Preface

Acknowledgements

1 Introduction

1.1. Modelling using Linear Programming

1.2. Solving linear programmes

1.2.1. The graphical solution and the importance of visual displays.

1.2.2. The Simplex Method and its main variants

1.2.3. Computer software packages

1.2.4. Complementary information and Sensitivity Analysis

1.3. Linear Programming: the Approach par excellence for understanding modelling

1.3.1. The variants of Linear Programming

1.3.2. LP’s related topics

1.4. The Approach of the book

Part I Linear Programming and Sensitivity Analysis

2 The Geometric Approach

2.1. The founding concepts of Linear Programming

2.2. The Maximization Form

Application # 1: An advertising campaign [Aurel 2D]

2.2.1. The mathematical formulation

2.2.2. The graphical solution and the fundamental theorem of LP

- Basic vs. non-basic variables

- Basic solution vs. basic feasible solution

- The Fundamental Theorem of Linear Programming

2.2.3. Interpreting the slack and surplus variables

2.2.4. Shadow prices

Application # 2: Computer games

2.3. The Minimization Form

Application # 3: A Portfolio selection

Chapter 2 Exercises and applications

3 The Simplex Method

3.1. The Maximization Form

3.1.1. The Standard Form

3.1.2. The simplex algorithm (using tableaux)

3.1.3. Shadow prices and reduced costs.

3.1.4. The algorithm (using matrix algebra)

3.1.5. Introduction of artificial variables

3.1.6. The remarkable features of the simplex algorithm

3.2. The Minimization Form

3.3. The Revised Simplex Method

3.3.1. The revised simplex algorithm

3.3.2. Using artificial variables: adjusting the algorithm

Chapter 3 Exercises and applications

4 Understanding Special Cases and Mixed Function Problems

4.1. Identifying special cases: graphical and simplex approaches

4.1.1. Alternative optimal solutions

4.1.2. Unboundedness

4.1.3. Infeasibility versus point solution

4.1.4. Degenerate solutions

4.1.5. Special types of constraints

4.2. The mixed function problem

Chapter 4 Exercises and applications

5 Duality

5.1. Theorems of duality and relationships

5.1.1. The theorems of duality

5.1.2. Primal / dual relationships

5.1.3. Formulating duals using the general primal formats

5.1.4. Primal / dual interrelationships: the Complementary Slackness theorem

5.2. The Dual Simplex Method

5.3. Particular Cases:

5.3.1. Unrestricted-in-sign variables (free variables)

5.3.2. Revisiting the special cases: study of the behaviour of their duals

Chapter 5 Exercises and applications

6 Sensitivity Analysis

6.A A visual approach to Sensitivity Analysis

6.1. The Maximization Form

6.1.1. The range of optimality: separate and simultaneous changes

6.1.2. The range of feasibility

6.1.3. Note on a few specific cases

6.2. The Minimization Form

6.B Sensitivity Analysis under the Simplex Method using Matrix Algebra

6.3. The Maximization Form

6.3.1. Ranges of optimality: simple changes

6.3.2. Optimality ranges: simultaneous changes and restoring optimality

6.3.3. Simple and simultaneous ranges of feasibility

6.3.4. Restoring feasibility

6.3.5. The 100% Rule: optimality and feasibility tests

6.4. Introduction of a new variable or of a new constraint

6.5. Note on the Minimization Form

6.6. Embedded modifications

6.C Revisiting mixed function problems

6.7. Discussion on optimality ranges: simplex and graphical approaches

Chapter 6 Exercises and applications

7 Understanding Computer Outputs and LP Applications

7.A &absp; Highlighting outputs

7.1. Using software packages for solving LP problems

7.1.1. Lindo: How to take advantage of its functions

7.1.2. The Management Scientist

7.1.3. Solving LP's with Excel

7.2. Study of outputs with respect to Chapters 3 and 6: the Simplex Method and Sensitivity Analysis

7.3. Commented outputs with respect to Chapter 4 and 5: special cases and duality

7.B The Various Fields of Application

7.4. Production and make-or-buy

7.5. Purchase plans

7.6. Finance

7.7. Advertising

7.8. Staff scheduling

7.9. Blending and nutrition

7.10. Efficiency problems

Chapter Applications

Part II Variants and Related Topics

8 The Variants of Linear Programmes

8.1. Integer Programming

8.1.1. Pure and Mixed Integer Programming: the graphical insight

8.1.2. Binary Integer Programming

8.1.3. Formulating logical constraints

8.2. Game Theory

8.2.1. Strictly determined Games and the theorems of GT

8.2.2. Non-Strictly determined games and solution approach by LP

8.3. The Transportation Problem

8.3.1. The balanced problem: solution approach through simplex multipliers

8.3.2. The unbalanced problem

8.3.3. Comment on the LP formulation of unbalanced problems: the feasibility requirement

8.3.4. Special Transportation Problems: LP formulation and solution approach

8.3.5. Maximization problems

8.4. The Assignment Model

8.4.1. The Solution Approach: König's Algorithm

- The Balanced Problem

- Alternative Assignments

8.4.2. The Maximization Problem (Example also displaying an unbalance)

- Multiple Tableaux

8.4.3. Note on the LP Formulation and on Sensitivity Analysis: identifying alternative assignments

Chapter 8 Exercises and applications

9 Related Topics: Graphs and Networks

9.1. The main building concepts of Graph Theory.

9.1.1. Definitions and examples.

9.1.2. From the graph to the matrix: adjacency and incidence matrices

9.1.3. Directed graphs.

9.2. Flow networks

9.2.1. The LP Formulation and solution

9.2.2. Solving the capacitated network graphically

9.2.3. The Max-Flow Min-Cut Theorem (Ford-Fulkerson)

9.2.3. Transshipments

9.3. The Shortest Path

9.3.1. The LP formulation and solution

9.3.2. The graphical solution (Dijkstra’s Algorithm)

9.3.3. Floyd’s Algorithm

9.4. The Minimal Spanning Tree

9.4.1. The graphical solution: Kruskal's and Prim’s Algorithms

9.4.2. The limits of the LP Formulation

Chapter 9 Exercises and applications

Part III Mathematical Corner and Note on Nonlinear Programming

10 Mathematical Corner

10.1. Coping with infeasibility

10.1.1. Graphical insights

10.1.2. Discussion on changes

10.2. Flow networks:

10.2.1. Highlighting the cut on outputs

10.2.2. The cut revisited by duality

10.3. The Shortest Route Algorithm: discussion on Sensitivity Analysis

- Highlighting the leeway

- Reading alternative optimal solutions on outputs

10.4. The Minimal Spanning Tree

10.4.1. Minimal Spanning Trees and hierarchical clustering schemes

10.4.2. The LP formulation of Minimal Spanning Trees: a heuristic approach

Note on Sensitivity Analysis

Chapter 10 Exercises and applications

11 Note on Nonlinear Programming

11.1. Quadratic Programming: definition

11.2. Illustrations and graphical displays: solution methltipliers

11.3. Formulating the quadratic programme

11.4. Comment on shadow prices and on “RHS ranges”.

Chapter 11 Exercises

Basic Review Chapter

R.1. Basic Matrix Algebra

R.1.1on, subtraction and multiplication

R.1.2. Matrices: types, addition and subtraction, multiplication and inverses

R.1.3. Finding the inverse of mxm matrices by the Gauss-Jordan method

R.2. Derivatives and local extrema

R.2.1. Brief review of derivatives in the single-variable case

Comments on limits, continuity and differentiability.

R.2.2. Partial derivatives

Answers to Selected Problems and Applications

Study Applications

Index

  • A circular approach looks at same applications through various solution methods (geometric, simplex, revised simplex and dual simplex methods, duality, study of software outputs, sensitivity analysis), enabling the book to be structured to individual courses
  • The highly graphical approach aids understanding and visualisation
  • Each chapter includes a set of objectives, problems and applications.