Operations Research Calculations Handbook
Features
Summary
The field of Operations Research (OR) covers a wide range of mathematical topics. Because it is so broad, results and formulas relevant to the field are widely scattered in different texts and journals and can be hard to find. As the field continues to grow, OR practitioners and students need a convenient, one-stop source for the results relevant to their work.
The Operations Research Calculations Handbook meets that need. It contains more than 300 results in a single, concise volume. Organized by topic and listed in a convenient summary format, it allows readers to have frequently used results at their fingertips. Although based on the author's experience in the manufacturing industry, many of the results are basic to system modeling. They carry over easily to applications in other areas of operations research and management science.
While modern software packages are useful for obtaining numerical results, formulas continue to play a significant role in systems modeling. They allow one to draw general conclusions about system behavior, reveal the underlying system model, and help provide an understanding of system performance. Whether you are a student, professor, or seasoned professional, the Operations Research Calculations Handbook offers not only a handy reference that will save time, but also a tool that will help build the intuitive understanding you need to apply systems models with confidence.
Table of Contents
INTRODUCTION
MEANS AND VARIANCES
Mean (Expectation) and Variance of a Random Variable
Covariance and Correlation Coefficient
Mean and Variance for Sum of Random Variables
Mean and Variance of a Product of Two Random Variables
Mean and Variance of a Quotient of Two Random Variables
Conditional Mean and Variance for Jointly Distributed Random Variables
Conditional Mean of a Random Variable in a Limited Range
Mean and Variance for Sum of a Random Number of Random Variables
Mean of a Function of a Random Variable
Approximations for the Mean and Variance of a Function of a Random Variable
Mean and Variance of the Maximum of Exponentially Distributed Random Variables
Mean and Variance of the Maximum of Normally Distributed Random Variables
Moments for the Greatest of a Finite Set of Random Variables
Moment Generating Functions
DISCRETE PROBABILITY DISTRIBUTIONS
Bernoulli Distribution
Binomial Distribution
Geometric Distribution
Negative Binomial Distribution
Poisson Distribution
Hypergeometric Distribution
Multinomial Distribution
CONTINUOUS PROBABILITY DISTRIBUTIONS
Uniform Distribution
Exponential Distribution
Erlang Distribution
Gamma Distribution
Beta Distribution
Normal Distribution
Lognormal Distribution
Weibull Distribution
Logistic Distribution
Gumbel (Extreme Value) Distribution
Pareto Distribution
Triangular Distribution
PROBABILITY FORMULAS
Distribution of the Sum of Independent Random Variables
Distribution of the Maximum and Minimum of Random Variables
Change of Variable in a Probability Distribution
Conditional Probability Distribution for a Constrained Random variable
Combination of Poisson and Gamma Distributions
Bayes' Formula
Central Limit Theorem
Probability Generating Function (z-Transform)
Moment Generating Function
Characteristic Function
Laplace Transform
STOCHASTIC PROCESSES
Poisson Process and Exponential Distribution
Renewal Process Results
Markov Chain Results
QUEUING THEORY RESULTS
Notation for Queue Types
Definitions of Queuing System Variables
Little's Law and General Queueing System Relationships
Extensions of Little's Law
Formulas for Average Queue Length
Formulas for Average Time in Queue
References for Formulas for Average Queue Length and Time in Queue
Pollaczek-Khintchine Formula for Average Time in Queue
Additional Formulas for Average Time in Queue
Other Formulas in Queueing Theory
PRODUCTION SYSTEMS MODELING
Definitions and Notation for Workstations
Basic Relationship Between Workstation Parameters
Distribution of the Time to Produce Fixed Lot Size at a Workstation
Throughput of a Serial Production Line with Failures
Throughput of a Two-Station Serial Production Line with Variable Processing Times
Throughput of an N-Station Serial Production Line with Variable Processing Times
INVENTORY CONTROL
Economic Order Quantity
Economic Production Quantity
"Newsboy Problem:" Optimal Inventory to Meet Uncertain Demand in a Single Period
Inventory Replenishment Policies
(s,Q) Policy: Estimates of Reorder Point (s) and Order Quantity (Q)
(s,S) Policy: Estimates of Reorder Point (s) and Order-Up-To Level (S)
(T,S) Policy: Estimates of Review Period (T) and Order-Up-To Level (S)
(T,s,S) Policy: Estimates of Review Period (T), Reorder Point (s), and Order-Up-To Level (S)
Summary of Results for Inventory Policies
Inventory in a Production/Distribution system
A Note of Cumulative Plots
DISTANCE FORMULAS FOR LOGISTICS ANALYSIS
"Traveling Salesman Problem" Tour Distance: Shortest Path Through a Set of Points in a Region
Distribution of Distance between Two Random Points in a Circle
Great Circle Distance
LINEAR PROGRAMMING FORMULATIONS
General Formulation
Terminology
Example of Feasible Region
Alternative Formulations
Diet Problem
Duality
Special Cases of Linear Programming Problems
Integer Linear Programming Formulations
Solution Methods
MATHEMATICAL FUNCTIONS
Gamma Function
Beta Function
Unit Impulse Function
Modified Bessel Functions
Stirling's Formula
CALCULUS RESULTS
Basic Rules for Differentiation
Integration by Parts
Fundamental Theorem of Calculus
Taylor Series
Maclaurin Series
L'Hôpital's Rule
Lagrange Multipliers
Differentiation Under the Integral Sign
Change of Variable in an Integral
Change of Variables in a Double Integral
Changing the Order of Integration in a Double Integral
Changing the Order of Summation in a Double Sum
Numerical Integration
MATRICES
Rules for Matrix Calculations
Inverses of Matrices
Series of Matrices
COMBINATORICS
SUMMATIONS
Finite Sums
Infinite Sums
INTEREST FORMULAS
REFERENCES
Editorial Reviews
"The layout is good and the writing concise … . … I can recommend this book to all those who are looking for a one-stop, handy formulae collection."
- Journal of the Operational Research Society, Vol. 55, No. 2, 2004
"[A] useful resource book … ."
- Zentralblatt MATH, 1042
