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Book Description

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.

Table of Contents

  1. Cover
  2. Preface
  3. 1 Introduction
  4. 2 Mathematical Foundations
    1. 2.1 Matrix Algebra
    2. 2.2 Vector Algebra
    3. 2.3 Simultaneous Linear Equation Systems
    4. 2.4 Linear Dependence
    5. 2.5 Convex Sets and n‐Dimensional Geometry‐Dimensional Geometry
  5. 3 Introduction to Linear Programming
    1. 3.1 Canonical and Standard Forms
    2. 3.2 A Graphical Solution to the Linear Programming Problem
    3. 3.3 Properties of the Feasible Region
    4. 3.4 Existence and Location of Optimal Solutions
    5. 3.5 Basic Feasible and Extreme Point Solutions
    6. 3.6 Solutions and Requirement Spaces
  6. 4 Computational Aspects of Linear Programming
    1. 4.1 The Simplex Method
    2. 4.2 Improving a Basic Feasible Solution
    3. 4.3 Degenerate Basic Feasible Solutions
    4. 4.4 Summary of the Simplex Method
  7. 5 Variations of the Standard Simplex Routine
    1. 5.1 The M‐Penalty Method‐Penalty Method
    2. 5.2 Inconsistency and Redundancy
    3. 5.3 Minimization of the Objective Function
    4. 5.4 Unrestricted Variables
    5. 5.5 The Two‐Phase Method
  8. 6 Duality Theory
    1. 6.1 The Symmetric Dual
    2. 6.2 Unsymmetric Duals
    3. 6.3 Duality Theorems
    4. 6.4 Constructing the Dual Solution
    5. 6.5 Dual Simplex Method (Lemke 1954)
    6. 6.6 Computational Aspects of the Dual Simplex Method
    7. 6.7 Summary of the Dual Simplex Method
  9. 7 Linear Programming and the Theory of the Firm1
    1. 7.1 The Technology of the Firm
    2. 7.2 The Single‐Process Production Function
    3. 7.3 The Multiactivity Production Function
    4. 7.4 The Single‐Activity Profit Maximization Model
    5. 7.5 The Multiactivity Profit Maximization Model
    6. 7.6 Profit Indifference Curves
    7. 7.7 Activity Levels Interpreted as Individual Product Levels
    8. 7.8 The Simplex Method as an Internal Resource Allocation Process
    9. 7.9 The Dual Simplex Method as an Internalized Resource Allocation Process
    10. 7.10 A Generalized Multiactivity Profit‐Maximization Model
    11. 7.11 Factor Learning and the Optimum Product‐Mix Model
    12. 7.12 Joint Production Processes
    13. 7.13 The Single‐Process Product Transformation Function
    14. 7.14 The Multiactivity Joint‐Production Model
    15. 7.15 Joint Production and Cost Minimization
    16. 7.16 Cost Indifference Curves
    17. 7.17 Activity Levels Interpreted as Individual Resource Levels
  10. 8 Sensitivity Analysis
    1. 8.1 Introduction
    2. 8.2 Sensitivity Analysis
    3. 8.3 Summary of Sensitivity Effects
  11. 9 Analyzing Structural Changes
    1. 9.1 Introduction
    2. 9.2 Addition of a New Variable
    3. 9.3 Addition of a New Structural Constraint
    4. 9.4 Deletion of a Variable
    5. 9.5 Deletion of a Structural Constraint
  12. 10 Parametric Programming
    1. 10.1 Introduction
    2. 10.2 Parametric Analysis
    3. 10.A Updating the Basis Inverse
  13. 11 Parametric Programming and the Theory of the Firm
    1. 11.1 The Supply Function for the Output of an Activity (or for an Individual Product)
    2. 11.2 The Demand Function for a Variable Input
    3. 11.3 The Marginal (Net) Revenue Productivity Function for an Input
    4. 11.4 The Marginal Cost Function for an Activity (or Individual Product)
    5. 11.5 Minimizing the Cost of Producing a Given Output
    6. 11.6 Determination of Marginal Productivity, Average Productivity, Marginal Cost, and Average Cost Functions
  14. 12 Duality Revisited
    1. 12.1 Introduction
    2. 12.2 A Reformulation of the Primal and Dual Problems
    3. 12.3 Lagrangian Saddle Points
    4. 12.4 Duality and Complementary Slackness Theorems
  15. 13 Simplex‐Based Methods of Optimization
    1. 13.1 Introduction
    2. 13.2 Quadratic Programming
    3. 13.3 Dual Quadratic Programs
    4. 13.4 Complementary Pivot Method
    5. 13.5 Quadratic Programming and Activity Analysis
    6. 13.6 Linear Fractional Functional Programming
    7. 13.7 Duality in Linear Fractional Functional Programming
    8. 13.8 Resource Allocation with a Fractional Objective
    9. 13.9 Game Theory and Linear Programming
    10. 13.A Quadratic Forms
  16. 14 Data Envelopment Analysis (DEA)
    1. 14.1 Introduction
    2. 14.2 Set Theoretic Representation of a Production Technology
    3. 14.3 Output and Input Distance Functions
    4. 14.4 Technical and Allocative Efficiency
    5. 14.5 Data Envelopment Analysis (DEA) Modeling
    6. 14.6 The Production Correspondence
    7. 14.7 Input‐Oriented DEA Model under CRS
    8. 14.8 Input and Output Slack Variables
    9. 14.9 Modeling VRS
    10. 14.10 Output‐Oriented DEA Models
  17. References and Suggested Reading
  18. Index
  19. End User License Agreement
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