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# Operations Research Calculations Handbook

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## Book Description

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.

## Table of Contents

**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**

## Author(s)

### Biography

Dennis Blumenfeld works in research and development at General Motors.