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.
Introduction
Means and Variances
Mean (Expectation) and Variance of a Random Variable
Covariance and Correlation Coefficient
Mean and Variance of the Sum of Random Variables
Mean and Variance of the Product of Two Random Variables
Mean and Variance of the Quotient of Two Random Variables
Conditional Mean and Variance for Jointly Distributed Random Variables
Conditional Mean of a Constrained Random Variable
Mean and Variance of the 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
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
Hyper-Exponential and Hypo-Exponential Distributions
Probability Relationships
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
Queueing Theory Results
Notation for Queue Types
Definitions of Queueing System Variables
Little’s Law and General Queueing System Relationships
Extension of Little’s Law
Formulas for Average Queue Length, Lq
Formulas for Average Time in Queue, Wq
References for the Formulas for Average Queue Length and Time in Queue
Pollaczek–Khintchine Formula for Average Time in Queue, Wq
Additional Formulas for Average Time in Queue, Wq
Heavy Traffi c Approximation for Distribution of Time in Queue
Queue Departure Process
Distribution Results for the Number of Customers in M/M/1 Queue
Distribution Results for Time in M/M/1 Queue
Other Formulas in Queueing Theory
Production Systems Modeling
Definitions and Notation for Workstations
Basic Relationships between Workstation Parameters
Distribution of the Time to Produce a 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
Note on Cumulative Plots
Distance Formulas for Logistics Analysis
Distance Norms
“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
Average Rectangular Grid Distance between Two Random Points in a Circle
Great Circle Distance
Traffic Flow and Delay
Traffic Flow Parameters
Traffic Speeds
Delay to Vehicle Merging with Traffi c Stream
Critical Flow on Minor Road
Delay to Traffi c Queue on Minor Road Waiting to Merge
Delay to Vehicle at Traffi c Signal
Linear Programming Formulations
General Formulation
Terminology
Example of a Feasible Region
Alternative Formulations
Diet Problem
Duality
Special Cases of Linear Programming Problems
Integer Linear Programming Formulations
Solution Methods
Heuristic Search Methods
Overview of Heuristics
Local Search Methods
Simulated Annealing
Tabu Search
Genetic Algorithms
Other Heuristics
Order Statistics
General Distribution Order Statistics
Uniform Distribution Order Statistics
Exponential Distribution Order Statistics
Mathematical Functions
Gamma Function
Incomplete Gamma Function
Beta Function
Incomplete 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 (Leibnitz’s Rule)
Change of a 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
Derivatives of Matrices
Combinatorics
Summations
Finite Sums
Infinite Sums
Interest Formulas
References
Index
Biography
Dennis Blumenfeld works in research and development at General Motors.
