Operations Research Calculations Handbook

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.

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