2nd Edition

Operations Research Calculations Handbook

By Dennis Blumenfeld Copyright 2009
    256 Pages 50 B/W Illustrations
    by CRC Press

    256 Pages
    by CRC Press

    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.


    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


    Example of a Feasible Region

    Alternative Formulations

    Diet Problem


    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


    Rules for Matrix Calculations

    Inverses of Matrices

    Series of Matrices

    Derivatives of Matrices



    Finite Sums

    Infinite Sums

    Interest Formulas




    Dennis Blumenfeld works in research and development at General Motors.