# Linear Programming and Resource Allocation Modeling

## Books

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.

Preface xi

Symbols and Abbreviations xv

1 Introduction 1

2 Mathematical Foundations 13

2.1 Matrix Algebra 13

2.2 Vector Algebra 20

2.3 Simultaneous Linear Equation Systems 22

2.4 Linear Dependence 26

2.5 Convex Sets and n-Dimensional Geometry 29

3 Introduction to Linear Programming 35

3.1 Canonical and Standard Forms 35

3.2 A Graphical Solution to the Linear Programming Problem 37

3.3 Properties of the Feasible Region 38

3.4 Existence and Location of Optimal Solutions 38

3.5 Basic Feasible and Extreme Point Solutions 39

3.6 Solutions and Requirement Spaces 41

4 Computational Aspects of Linear Programming 43

4.1 The Simplex Method 43

4.2 Improving a Basic Feasible Solution 48

4.3 Degenerate Basic Feasible Solutions 66

4.4 Summary of the Simplex Method 69

5 Variations of the Standard Simplex Routine 71

5.1 The M-Penalty Method 71

5.2 Inconsistency and Redundancy 78

5.3 Minimization of the Objective Function 85

5.4 Unrestricted Variables 86

5.5 The Two-Phase Method 87

6 Duality Theory 95

6.1 The Symmetric Dual 95

6.2 Unsymmetric Duals 97

6.3 Duality Theorems 100

6.4 Constructing the Dual Solution 106

6.5 Dual Simplex Method 113

6.6 Computational Aspects of the Dual Simplex Method 114

6.7 Summary of the Dual Simplex Method 121

7 Linear Programming and the Theory of the Firm 123

7.1 The Technology of the Firm 123

7.2 The Single-Process Production Function 125

7.3 The Multiactivity Production Function 129

7.4 The Single-Activity Profit Maximization Model 139

7.5 The Multiactivity Profit Maximization Model 143

7.6 Profit Indifference Curves 146

7.7 Activity Levels Interpreted as Individual Product Levels 148

7.8 The Simplex Method as an Internal Resource Allocation Process 155

7.9 The Dual Simplex Method as an Internalized Resource Allocation Process 157

7.10 A Generalized Multiactivity Profit-Maximization Model 157

7.11 Factor Learning and the Optimum Product-Mix Model 161

7.12 Joint Production Processes 165

7.13 The Single-Process Product Transformation Function 167

7.14 The Multiactivity Joint-Production Model 171

7.15 Joint Production and Cost Minimization 180

7.16 Cost Indifference Curves 184

7.17 Activity Levels Interpreted as Individual Resource Levels 186

8 Sensitivity Analysis 195

8.1 Introduction 195

8.2 Sensitivity Analysis 195

8.2.1 Changing an Objective Function Coefficient 196

8.2.2 Changing a Component of the Requirements Vector 200

8.2.3 Changing a Component of the Coefficient Matrix 202

8.3 Summary of Sensitivity Effects 209

9 Analyzing Structural Changes 217

9.1 Introduction 217

9.2 Addition of a New Variable 217

9.3 Addition of a New Structural Constraint 219

9.4 Deletion of a Variable 223

9.5 Deletion of a Structural Constraint 223

10 Parametric Programming 227

10.1 Introduction 227

10.2 Parametric Analysis 227

10.2.1 Parametrizing the Objective Function 228

10.2.2 Parametrizing the Requirements Vector 236

10.2.3 Parametrizing an Activity Vector 245

10.A Updating the Basis Inverse 256

11 Parametric Programming and the Theory of the Firm 257

11.1 The Supply Function for the Output of an Activity (or for an Individual Product) 257

11.2 The Demand Function for a Variable Input 262

11.3 The Marginal (Net) Revenue Productivity Function for an Input 269

11.4 The Marginal Cost Function for an Activity (or Individual Product) 276

11.5 Minimizing the Cost of Producing a Given Output 284

11.6 Determination of Marginal Productivity, Average Productivity, Marginal Cost, and Average Cost Functions 286

12 Duality Revisited 297

12.1 Introduction 297

12.2 A Reformulation of the Primal and Dual Problems 297

12.4 Duality and Complementary Slackness Theorems 315

13 Simplex-Based Methods of Optimization 321

13.1 Introduction 321

13.4 Complementary Pivot Method 329

13.5 Quadratic Programming and Activity Analysis 335

13.6 Linear Fractional Functional Programming 338

13.7 Duality in Linear Fractional Functional Programming 347

13.8 Resource Allocation with a Fractional Objective 353

13.9 Game Theory and Linear Programming 356

13.9.1 Introduction 356

13.9.2 Matrix Games 357

13.9.3 Transformation of a Matrix Game to a Linear Program 361

13.A.1 General Structure 363

13.A.3 Classification of Quadratic Forms 367

13.A.4 Necessary Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 368

13.A.5 Necessary and Sufficient Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 369

14 Data Envelopment Analysis (DEA) 373

14.1 Introduction 373

14.2 Set Theoretic Representation of a Production Technology 374

14.3 Output and Input Distance Functions 377

14.4 Technical and Allocative Efficiency 379

14.4.1 Measuring Technical Efficiency 379

14.4.2 Allocative, Cost, and Revenue Efficiency 382

14.5 Data Envelopment Analysis (DEA) Modeling 385

14.6 The Production Correspondence 386

14.7 Input-Oriented DEA Model under CRS 387

14.8 Input and Output Slack Variables 390

14.9 Modeling VRS 398

14.9.1 The Basic BCC (1984) DEA Model 398

14.9.2 Solving the BCC (1984) Model 400

14.9.3 BCC (1984) Returns to Scale 401

14.10 Output-Oriented DEA Models 402

Index 411

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