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Operations Research Calculations Handbook, Second Edition / Edition 2

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Overview

A handbook in the truest sense of the word, the first edition of the Operations Research Calculations Handbook quickly became an indispensible resource. While other books available tend to give detailed information about specific topics, this one contains comprehensive information and results useful for real-world problem solving. Reflecting the breadth and depth of growth in the field, the scope of the second edition has been expanded to cover several additional topics. And as with the first edition, it focuses on presenting analytical results and formulas that allow quick calculations and provide understanding of system models.

See what’s in the Second Edition:

  • New chapters include Order Statistics, Traffic Flow and Delay, and Heuristic Search Methods
  • New sections include Distance Norms, Hyper-Exponential and Hypo-Exponential Distributions
  • Newly derived formulas and an expanded reference list

Like its predecessor, the new edition of this handbook presents the analytical results and formulas needed in the scientific applications of operations research and management. It continues to provide quick calculations and insight into system performance. Presenting practical results and formulas without derivations, the material is organized by topic and offered in a concise format that allows ready-access to a wide range of results in a single volume.

The field of operations research encompasses a growing number of technical areas, and uses analyses and techniques from a variety of branches of mathematics, statistics, and other scientific disciplines. And as the field continues to grow, there is an even greater need for key results to be summarized and easily accessible in one reference volume. Yet many of the important results and formulas are widely scattered among different textbooks and journals and are often hard to find in the midst of mathematical derivations. This book provides a one-stop resource for many important results and formulas needed in operations research and management science applications.

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Product Details

  • ISBN-13: 9781420052404
  • Publisher: CRC Press
  • Publication date: 12/18/2009
  • Series: Operations Research Series
  • Edition number: 2
  • Pages: 256
  • Product dimensions: 0.60 (w) x 0.90 (h) x 0.70 (d)

Meet the Author

Dennis Blumenfeld works in research and development at General Motors.

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Table of Contents

Preface to the Second Edition xi

Preface to the First Edition xiii

Acknowledgments xv

Author xvii

1 Introduction 1

2 Means and Variances 3

2.1 Mean (Expectation) and Variance of a Random Variable 3

2.2 Covariance and Correlation Coefficient 5

2.3 Mean and Variance of the Sum of Random Variables 5

2.4 Mean and Variance of the Product of Two Random Variables 7

2.5 Mean and Variance of the Quotient of Two Random Variables 8

2.6 Conditional Mean and Variance for Jointly Distributed Random Variables 9

2.7 Conditional Mean of a Constrained Random Variable 9

2.8 Mean and Variance of the Sum of a Random Number of Random Variables 11

2.9 Mean of a Function of a Random Variable 11

2.10 Approximations for the Mean and Variance of a Function of a Random Variable 12

2.11 Mean and Variance of the Maximum of Exponentially Distributed Random Variables 13

2.12 Mean and Variance of the Maximum of Normally Distributed Random Variables 13

3 Discrete Probability Distributions 17

3.1 Bernoulli Distribution 17

3.2 Binomial Distribution 18

3.3 Geometric Distribution 20

3.4 Negative Binomial Distribution 22

3.5 Poisson Distribution 24

3.6 Hypergeometric Distribution 26

3.7 Multinomial Distribution 29

4 Continuous Probability Distributions 31

4.1 Uniform Distribution 31

4.2 Exponential Distribution 32

4.3 Erlang Distribution 34

4.4 Gamma Distribution 37

4.5 Beta Distribution 41

4.6 Normal Distribution 43

4.6.1 Sum of Normally Distributed Random Variables 44

4.6.2 Standard Normal Distribution 45

4.6.3 Partial Moments for the Normal Distribution 46

4.6.4 Approximations for the Cumulative Normal Distribution Function 47

4.7 Lognormal Distribution 48

4.8 Weibull Distribution 51

4.9 Logistic Distribution 53

4.10 Gumbel (Extreme Value) Distribution 54

4.11 Pareto Distribution 56

4.12 Triangular Distribution 58

4.13 Hyper-Exponential and Hypo-Exponential Distributions 60

4.13.1 Hyper-Exponential Distribution 60

4.13.2 Hypo-Exponential Distribution 63

4.13.3 Additional Comments on Hyper- and Hypo-Exponential Distributions 66

5 Probability Relationships 69

5.1 Distribution of the Sum of Independent Random Variables 69

5.2 Distribution of the Maximum and Minimum of Random Variables 69

5.2.1 Example for the Uniform Distribution 70

5.2.2 Example for the Exponential Distribution 71

5.3 Change of Variable in a Probability Distribution 72

5.4 Conditional Probability Distribution for a Constrained Random Variable 73

5.5 Combination of Poisson and Gamma Distributions 75

5.6 Bayes' Formula 76

5.7 Central Limit Theorem 77

5.8 Probability Generating Function (z-Transform) 78

5.9 Moment Generating Function 79

5.10 Characteristic Function 81

5.11 Laplace Transform 83

6 Stochastic Processes 85

6.1 Poisson Process and Exponential Distribution 85

6.1.1 Properties of the Poisson Process 85

6.1.2 "Lack of Memory" Property of the Exponential Distribution 85

6.1.3 Competing Exponentials 86

6.1.4 Superposition of Independent Poisson Processes 86

6.1.5 Splitting of a Poisson Process 86

6.1.6 Arrivals from a Poisson Process in a Fixed Interval 87

6.2 Renewal Process Results 87

6.2.1 Mean and Variance of the Number of Arrivals in a Renewal Process 87

6.2.2 Distribution of First Interval in a Renewal Process 87

6.3 Markov Chain Results 88

6.3.1 Discrete-Time Markov Chains 89

6.3.2 Continuous-Time Markov Chains 89

7 Queueing Theory Results 91

7.1 Notation for Queue Types 91

7.2 Definitions of Queueing System Variables 91

7.3 Little's Law and General Queueing System Relationships 92

7.4 Extension of Little's Law 93

7.5 Formulas for Average Queue Length, Lq 93

7.6 Formulas for Average Time in Queue, Wq 94

7.7 References for the Formulas for Average Queue Length and Time in Queue 95

7.8 Pollaczek-Khintchine Formula for Average Time in Queue, Wq 96

7.9 Additional Formulas for Average Time in Queue, Wq 96

7.10 Heavy Traffic Approximation for Distribution of Time in Queue 97

7.11 Queue Departure Process 98

7.12 Distribution Results for the Number of Customers in M/M/1 Queue 99

7.13 Distribution Results for Time in M/M/1 Queue 99

7.14 Other Formulas in Queueing Theory 100

8 Production Systems Modeling 101

8.1 Definitions and Notation for Workstations 101

8.2 Basic Relationships between Workstation Parameters 101

8.3 Distribution of the Time to Produce a Fixed Lot Size at a Workstation 102

8.4 Throughput of a Serial Production Line with Failures 104

8.4.1 Line without Buffers 104

8.4.2 Line with Buffers 105

8.5 Throughput of a Two-Station Serial Production Line with Variable Processing Times 106

8.5.1 Two Stations without a Buffer 106

8.5.2 Two Stations with a Buffer 107

8.6 Throughput of an N-Station Serial Production Line with Variable Processing Times 107

9 Inventory Control 109

9.1 Economic Order Quantity 109

9.2 Economic Production Quantity 111

9.3 "Newsboy Problem": Optimal Inventory to Meet Uncertain Demand in a Single Period 114

9.4 Inventory Replenishment Policies 116

9.5 (s, Q) Policy: Estimates of Reorder Point (s) and Order Quantity (Q) 119

9.6 (s, S) Policy: Estimates of Reorder Point (s) and Order-Up-To Level (S) 122

9.7 (T, S) Policy: Estimates of Review Period (T) and Order-Up-To Level (S) 124

9.8 (T, s, S) Policy: Estimates of Review Period (T), Reorder Point (s), and Order-Up-To Level (S) 126

9.9 Summary of Results for Inventory Policies 128

9.10 Inventory in a Production/Distribution System 129

9.11 Note on Cumulative Plots 131

10 Distance Formulas for Logistics Analysis 133

10.1 Distance Norms 133

10.2 "Traveling Salesman Problem" Tour Distance: Shortest Path through a Set of Points in a Region 136

10.3 Distribution of Distance between Two Random Points in a Circle 136

10.4 Average Rectangular Grid Distance between Two Random Points in a Circle 139

10.5 Great Circle Distance 139

11 Traffic Flow and Delay 143

11.1 Traffic Flow Parameters 143

11.2 Traffic Speeds 143

11.3 Delay to Vehicle Merging with Traffic Stream 147

11.4 Critical Flow on Minor Road 148

11.5 Delay to Traffic Queue on Minor Road Waiting to Merge 149

11.6 Delay to Vehicle at Traffic Signal 150

12 Linear Programming Formulations 153

12.1 General Formulation 153

12.2 Terminology 154

12.3 Example of a Feasible Region 155

12.4 Alternative Formulations 156

12.4.1 Minimization vs. Maximization 156

12.4.2 Equality Constraints 156

12.4.3 Reversed Inequality Constraints 157

12.5 Diet Problem 157

12.6 Duality 158

12.7 Special Cases of Linear Programming Problems 160

12.7.1 Transportation Problem 160

12.7.2 Transshipment Problem 163

12.7.3 Assignment Problem 164

12.8 Integer Linear Programming Formulations 165

12.8.1 Knapsack Problem 166

12.8.2 Traveling Salesman Problem 167

12.9 Solution Methods 169

12.9.1 Simplex Method 169

12.9.2 Interior-Point Methods 170

12.9.3 Network Flow Methods 170

12.9.4 Cutting Planes 170

12.9.5 Branch and Bound 171

13 Heuristic Search Methods 173

13.1 Overview of Heuristics 173

13.2 Local Search Methods 175

13.3 Simulated Annealing 176

13.4 Tabu Search 178

13.5 Genetic Algorithms 179

13.6 Other Heuristics 180

14 Order Statistics 181

14.1 General Distribution Order Statistics 181

14.2 Uniform Distribution Order Statistics 182

14.3 Exponential Distribution Order Statistics 184

15 Mathematical Functions 187

15.1 Gamma Function 187

15.2 Incomplete Gamma Function 187

15.3 Beta Function 188

15.4 Incomplete Beta Function 188

15.5 Unit Impulse Function 189

15.6 Modified Bessel Functions 189

15.7 Stirling's Formula 190

16 Calculus Results 191

16.1 Basic Rules for Differentiation 191

16.2 Integration by Parts 192

16.3 Fundamental Theorem of Calculus 193

16.4 Taylor Series 193

16.5 Maclaurin Series 194

16.6 L'H?pital's Rule 195

16.7 Lagrange Multipliers 196

16.8 Differentiation under the Integral Sign (Leibnitz's Rule) 197

16.9 Change of a Variable in an Integral 198

16.10 Change of Variables in a Double Integral 199

16.11 Changing the Order of Integration in a Double Integral 201

16.12 Changing the Order of Summation in a Double Sum 203

16.13 Numerical Integration 204

17 Matrices 209

17.1 Rules for Matrix Calculations 209

17.2 Inverses of Matrices 210

17.2.1 Inverse of 2?2 Matrix 210

17.2.2 Inverse of 3?3 Matrix 210

17.3 Series of Matrices 211

17.4 Derivatives of Matrices 211

18 Combinatorics 213

19 Summations 215

19.1 Finite Sums 215

19.2 Infinite Sums 215

20 Interest Formulas 217

References 219

Index 231

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