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
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
12.3 Lagrangian Saddle Points 311
12.4 Duality and Complementary Slackness Theorems 315
13 Simplex-Based Methods of Optimization 321
13.1 Introduction 321
13.2 Quadratic Programming 321
13.3 Dual Quadratic Programs 325
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 Quadratic Forms 363
13.A.1 General Structure 363
13.A.2 Symmetric Quadratic Forms 366
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
References and Suggested Reading 405
Index 411
