Linear Programming, Sensitivity Analysis and Related Topics
By: Marie-France Derhy (author)Paperback
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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 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: Konig'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
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