Linear Programming and Resource Allocation Modeling

ISBN-13: 9781119509448 / Angielski / Twarda / 2018 / 448 str.

Michael J. Panik

Spis treści

Dedication

Preference

Symbols and Abbreviations

Chapter 1. Introduction

Chapter 2. Mathematical Foundations

2.1. Matrix Algebra

2.2. Vector Algebra

2.3. Simultaneous Linear Equation Systems

2.4. Linear Dependence

2.5. Convex Sets and n–Dimensional Geometry

Chapter 3. Introduction to Linear Programming

3.1. Canonical and Standard Forms

3.2. A Graphical Solution to the Linear Programming Problem

3.3. Properties of the Feasible Region

3.4. Existence and Location of Optimal Solutions

3.5. Basic Feasible and Extreme Point Solutions

3.6. Solutions and Requirements Spaces

Chapter 4. Computational Aspects of Linear Programming

4.1. The Simplex Method

4.2. Improving a Basic Feasible Solution

4.3. Degenerate Basic Feasible Solutions

4.4. Summary of the Simplex Method

Chapter 5. Variations of the Standards Simplex Routine

5.1. The M–Penalty Method

5.2. Inconsistency and Redundancy

5.3. Minimizing the Objective Function

5.4. Unrestricted Variables

5.5. The Two–Phase Method

Chapter 6. Duality Theory

6.1. The Symmetric Dual

6.2. Unsymmetric Duals

6.3. Duality Theorems

6.4. Constructing the Dual Solution

6.5. Dual Simplex Method

6.6. Computational Aspects of the Dual Simplex Method

6.7. Summary of the Dual Simplex Method

Chapter 7. Linear Programming and the Theory of the Firm

7.1. The Technology of the Frim

7.2. The Single–Process Production Function

7.3. The Multi–Activity Production Function

7.4. The Single–Activity Profit Maximization Model

7.5. The Multi–Activity Profit Maximization Model

7.6. Profit Indifference Curves

7.7. Activity Levels Interpreted as Individual Product Levels

7.8. The Simplex Method as an Internal Resource Allocation Process

7.9. The Dual Simplex Method as an Internal Resource Allocation Process

7.10. A Generalized Multi–Activity Profit–Maximization Model

7.11. Factor Learning and the Optimum Product–Mix Model

7.12. Joint Production Processes

7.13. The Single–Process Product Transformation Function

7.14. The Multi–Activity Joint Production Model

7.15. Joint Production and Cost Minimization

7.16. Cost Indifference Curves

7.17. Activity Levels Interpreted as Individual Resource Levels

Chapter 8. Sensitivity Analysis

8.1. Introduction

8.2. Sensitivity Analysis

8.2.1 Changing an Objective Function Coefficient

8.2.2. Changing a Component of the Requirements Vector

8.2.3. Changing a component of the Coefficient Matrix

8.3. Summary of Sensitivity Effects

Chapter 9. Analyzing Structural Changes

9.1. Introduction

9.2. Addition of a New Variable

9.3. Addition of a New Structural Constraint

9.4. Deletion of a Variable

9.5. Deletion of a Structural Constraint

Chapter 10. Parametric Programming

10.1. Introduction

10.2. Parametric Analysis

10.2.1. Parametrizing the Objective Function

10.2.2. Parametrizing the Requirements Vector

10.2.3. Parametrizing an Activity Vector

Appendix 10. A. Updating the Basis Inverse

Chapter 11. Parametric Programming and the Theory of the Firm

11.1. The Supply Function for the Output of an Activity (or for an Individual Product)

11.2. The Demand Function for a Variable Input

11.3. The Marginal (Net) Revenue Productivity Function for an Input

11.4. The Marginal Cost Function for an Activity (or Individual Product)

11.5. Minimizing the Cost of Producing a Given Output

11.6. Determination of Marginal Productivity, Average Productivity, Marginal Cost and Average Cost Functions

Chapter 12. Duality Revisited

12.1. Introduction

12.2. A Reformulation of the Primal and Dual Problems

12.3. Lagrangian Saddle Points

12.4. Duality and Complementary Slackness Theorems

Chapter 13. Simplex–Based Methods of Optimization

13.1. Introduction

13.2. Quadratic Programming

13.3. Dual Quadratic Programs

13.4. Complementary Pivot Method

13.5. Quadratic Programming and Activity Analysis

13.6. Linear Fractional Functional Programming

13.7. Duality in Linear Fractional Functional Programming

13.8. Resource Allocation with a Fractional Objective

13.9. Game Theory and Linear Programming

13.9.1. Introduction

13.9.2. Matrix Games

13.9.3 Transformation of a Matrix Game to a Linear Program

Appendix 13.A. Quadratic Forms

Chapter 14. Data Envelopment Analysis (DEA)

14.1. Introduction

14.2. Set Theoretic Representation of a Production Technology

14.3. Output and Input Distance Functions

14.4. Technical and Allocative Efficiency

14.4.1. Measuring Technical Efficiency

14.4.2. Allocative, Cost, and Revenue Efficiency

14.5. Data Envelopment Analysis (DEA) Modeling

14.6. The Production Correspondence

14.7. Input–Oriented DEA Model Under Constant Returns to Scale (CRS)

14.8. Input and Output Slack Variables

14.9. Modeling Variable Returns to Scale (VRS)

14.9.1.1. The Basic BCC(1984) DEA Model

14.9.1.2. Solving the BCC (1984) Model

14.9.1.3. BCC (1984) Returns to Scale

14.10. Output–Oriented DEA Models

References and Suggested Reading

Index

O autorze

Michael J. Panik, PhD, is Professor Emeritus in the Department of Economics at the University of Hartford, CT. He has taught courses in economic and business statistics, quantitative decision methods, introductory and advanced quantitative methods, and econometrics. Dr. Panik is the author of several books, including Stochastic Differential Equations and Growth Curve Modeling: Theory and Applications, both published by Wiley. He is also a co–author of Introduction to Quantitative Methods in Business: With Applications Using Microsoft Office Excel.

Blurb

Guides in the application of linear programming to firm decision making, with the goal of giving decision–makers a better understanding of methods at their disposal

Useful as a main resource or as a supplement in an economics or management science course, this comprehensive book addresses the deficiencies of other texts when it comes to covering linear programming theory especially where data envelopment analysis (DEA) is concerned and provides the foundation for the development of DEA.

Linear Programming and Resource Allocation Modeling begins by introducing primal and dual problems via an optimum product mix problem, and reviews the rudiments of vector and matrix operations. It then goes on to cover: the canonical and standard forms of a linear programming problem; the computational aspects of linear programming; variations of the standard simplex theme; duality theory; single– and multiple– process production functions; sensitivity analysis of the optimal solution; structural changes; and parametric programming. The primal and dual problems are then reformulated and re–examined in the context of Lagrangian saddle points, and a host of duality and complementary slackness theorems are offered. The book also covers primal and dual quadratic programs, the complementary pivot method, primal and dual linear fractional functional programs, and (matrix) game theory solutions via linear programming, and data envelopment analysis (DEA). This book:

• Appeals to those wishing to solve linear optimization problems in areas such as economics, business administration and management, agriculture and energy, strategic planning, public decision making, and health care
• Fills the need for a linear programming applications component in a management science or economics course
• Provides a complete treatment of linear programming as applied to activity selection and usage
• Contains many detailed example problems as well as textual and graphical explanations

Linear Programming and Resource Allocation Modeling is an excellent resource for professionals looking to solve linear optimization problems, and advanced undergraduate to beginning graduate level management science or economics students.