Optimization Using Linear Programming
- Length: 350 pages
- Edition: I
- Language: English
- Publisher: Mercury Learning and Information
- Publication Date: 2019-04-02
- ISBN-10: 1683923472
- ISBN-13: 9781683923473
- Sales Rank: #4325423 (See Top 100 Books)
Designedfor engineers, mathematicians, computer scientists, financial analysts, andanyone interested in using numerical linear algebra, matrix theory, and gametheory concepts to maximize efficiency in solving applied problems. The book emphasizes the solution of various typesof linear programming problems by using different types of software, butincludes the necessary definitions and theorems to master theoretical aspectsof the topics presented.
Features:
- Emphasizes the solution of various types of linear programming problems by using different kinds of software, e.g., MS-Excel, solutions of LPPs by Mathematica, MATLAB, WinQSB, and LINDO
- Provides definitions, theorems, and procedures for solving problems and all cases related to various linear programming topics
- Includes numerous application examples and exercises, e.g., transportation, assignment, and maximization
- Presents numerous topics that can be used tosolve problems involving systems of linear equations, matrices, vectors, gametheory, simplex method, and more.
Cover Page Title Copyright Contents Preface 1 Basics of Linear Algebra Using Ms-Excel 1.1. Vectors 1.2. Linear Independence and Dependence of Vectors 1.3. Solution to a System of Simultaneous Linear Equations 1.4. The Gauss-Jordan Method for Solving Systems of Linear Equations 1.5. Convex Sets Exercises 2 Introduction to LPPs and the Graphical Method 2.1. Introduction 2.2. Assumptions in a Linear Programming Problem 2.3. Theorems on Extreme Points 2.4. Areas of Application of LPPs 2.5. Formulation of Linear Programming Models 2.6. Graphical Method 2.6.1. Extreme Point Approach 2.6.2. ISO-Profit (cost) Function Line Approach 2.7. Solution of LPPs by the Graphical Method Using MS-Excel 2.8. Special Cases 2.8.1. Problem with Multiple Solutions 2.8.2. The Problem with Unbounded Solutions 2.8.3. The Problem with Inconsistent Constraints 2.8.4. The Problem with Redundant Constraint Equations Exercises 3 Simplex Method-I 3.1. Standard and Canonical Form of the General Linear Programming Problem 3.2. Slack and Surplus Variables 3.3. Algebraic Simplex Method 3.4. Relationship between the Simplex and Graphical Methods 3.5. Simplex Method in Tabular Form 3.6. Use of Solver in MS-Excel for Solving a Linear Programming Problem 3.7. Use of Jensen Add-Ins for a Linear Programming Problem Exercises 4 Simplex Method-II 4.1. Introduction 4.2. Big M Method (Penalty Method) 4.3. Two-Phase Method 4.4. Degeneracy in Linear Programming Problems 4.4.1. Perturbation Method for the Resolution of Degeneracy Problems in LPPs 4.5. Solving a System of Linear Equations Using the Simplex Method 4.6. Solution of a System of Linear Equations by Using Solver 4.7. Inverse of a Matrix Using the Simplex Method 4.8. Special Cases 4.8.1. The Problem with Alternative or Multiple Solutions 4.8.2. Unbounded Solutions 4.8.3. The Problem with Inconsistent Constraint Equations Exercises 5 Duality 5.1. Introduction 5.2. Rules for Finding the Dual of a Given Linear Programming Problem 5.3. Finding the Optimal Dual Solution from the Optimal Table of the Primal Problem 5.4. Use of the Graphical Method for Finding the Optimal Dual Solution 5.5. Construction of a Simplex Table 5.6. Duality Theorems 5.7. Economic Interpretation of Duality 5.8. Dual Simplex Method Exercises 6 Sensitivity Analysis 6.1. Introduction 6.2. Changes in the RHS Vector bi 6.2.1. Range of bi’s 6.2.2. Simultaneous Changes in bi’s 6.3. Addition of a New Constraint 6.3.1. When the Current Solution Satisfies the New Constraint 6.3.2. When the Current Solution Fails to Satisfy the New Constraint 6.4. Adding a New Activity or a Variable 6.5. Changes in the Objective Function Coefficients 6.5.1. Changes in the Objective Function Coefficients of Non-Basic Variables 6.5.2. Changes in Objective Function Coefficients of Basic Variables 6.5.3. 100% Rule for Making Simultaneous Changes in the Objective Function Coefficients 6.6. Changes in the aij Coefficients 6.6.1. Changes in aij Coefficients of Non-Basic Variables 6.6.2. Changes in aij- Coefficients of Basic Variables 6.7. Deletion of a Variable 6.8. Deletion of a Constraint 6.9. Sensitivity by Using Excel Solver Exercises 7 Transportation and Transshipment Problems 7.1. Introduction 7.2. Formulation of a Transportation Problem 7.3. Various Methods for Finding the Initial Basic Feasible Solution 7.3.1. North-West (N-W) Corner Method 7.3.2. Row Minima Method 7.3.3. Column Minima Method 7.3.4. Least-Cost or Matrix Minima Method 7.3.5. Vogel’s Approximation Method (VAM) 7.4. Closed Path or Loop in Transportation Problems 7.5. Moving Toward the Optimal Solution 7.5.1. Stepping-Stone Method 7.5.2. The Modified (MODI) Distribution or u-v Method 7.6. Solution of Transportation Problems in Excel 7.7. Some Special Cases in Transportation Problems 7.7.1. Unbalanced Transportation Problems 7.7.2. Restricted Entry 7.7.3. Maximization Problems 7.7.4. Multiple Solutions in Transportation Problems 7.7.5. Degeneracy in Transportation Problems 7.8. Transshipment Problems Exercises 8. Assignment Problems 8.1. Introduction 8.2. Mathematical Formulation 8.3. Assignment Problems as a Special Case of Transportation Problems 8.4. Hungarian Method 8.5. Special Cases in Assignment Methods 8.5.1. Maximization Problems 8.5.2. Restricted Entry 8.6. Solution of Assignment Problems Using Excel Solver Exercises 9 Game Theory 9.1. Introduction 9.2. Zero-Sum Games 9.3. Maximin and Minimax Principle 9.4. Game with a Saddle Point 9.5. Game without a Saddle Point 9.6. Graphical Solution of 2 × n and m × 2 Games 9.7. Method of Dominance 9.8. Solution of a Game Using the Simplex Method 9.9. Solution of a Game Using Gambit Exercises Appendix: Use of Mathematica, Matlab, Lindo, and WinQSB to Solve Linear Programming Models References Index
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