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Library of Congress Cataloging-in-Publication Data
Badiru, Adedeji Bodunde, 1952-
Handbook of industrial engineering equations, formulas, and calculations / authors,
Adedeji B. Badiru, Olufemi A. Omitaomu.
p. cm. -- (Industrial innovation series)
“A CRC title.”
Includes bibliographical references and index.
ISBN 978-1-4200-7627-1 (hardcover : alk. paper)
1. Industrial engineering--Mathematics--Handbooks, manuals, etc. 2. Engineering
mathematics--Handbooks, manuals, etc. I. Omitaomu, Olufemi Abayomi. II. Title. III.
Series.
T57.B33 2011
620.001’51--dc22 2010028413
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7. To our families for their unflinching support of our intellectual pursuits.

8. Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
1 Computational Foundations of Industrial Engineering
Efficacy of Mathematical Modeling ...............................................................................1-1
Industrial Engineering and Computations....................................................................1-1
Definition and Applications ...........................................................................................1-5
Orientation to STEM.......................................................................................................1-6
IE Catchphrases ...............................................................................................................1-6
Span and Utility of IE......................................................................................................1-6
Heritage from Industrial Revolution..............................................................................1-7
Historical Accounts.........................................................................................................1-8
Chronology of Applications..........................................................................................1-10
Importance of IE Calculations..............................................................................1-14
Importance of Calculations Guide ...............................................................................1-16
Basic Queuing Equations ..............................................................................................1-17
Queuing Birth–Death Processes...................................................................................1-21
Laws of Motion of Queuing Birth and Death..............................................................1-21
Queuing Birth–Death Law 1 .................................................................................1-21
Queuing Birth–Death Law 2 .................................................................................1-21
Queuing Birth–Death Law 3 .................................................................................1-21
Data Types for Computational Analysis......................................................................1-22
Nominal Scale ........................................................................................................1-22
Ordinal Scale ..........................................................................................................1-22
Interval Scale ..........................................................................................................1-22
Ration Scale............................................................................................................1-23
Cardinal Scale.........................................................................................................1-23
References ......................................................................................................................1-23
ix

9. x Contents
2 Basic Mathematical Calculations
Quadratic Equation .........................................................................................................2-1
Overall Mean....................................................................................................................2-2
Chebyshev’s Theorem .....................................................................................................2-2
Permutations....................................................................................................................2-2
Combinations ..................................................................................................................2-2
Failure.......................................................................................................................2-3
Probability Distribution..................................................................................................2-3
Probability........................................................................................................................2-3
Distribution Function .....................................................................................................2-3
Expected Value ................................................................................................................2-4
Variance ...........................................................................................................................2-5
Binomial Distribution .....................................................................................................2-5
Poisson Distribution........................................................................................................2-5
Mean of a Binomial Distribution....................................................................................2-6
Normal Distribution........................................................................................................2-6
Cumulative Distribution Function.................................................................................2-6
Population Mean .............................................................................................................2-6
Standard Error of the Mean............................................................................................2-6
t-Distribution...................................................................................................................2-7
Chi-Squared Distribution ...............................................................................................2-7
Definition of Set and Notation .......................................................................................2-7
Set Terms and Symbols ...................................................................................................2-8
Venn Diagrams................................................................................................................2-8
Operations on Sets...........................................................................................................2-9
De Morgan’s Laws ...........................................................................................................2-9
Counting the Elements in a Set.....................................................................................2-10
Permutations..................................................................................................................2-10
Combinations ................................................................................................................2-11
Probability Terminology...............................................................................................2-11
Basic Probability Principles ..........................................................................................2-11
Random Variable...........................................................................................................2-12
Mean Value x̂ or Expected Value μ .............................................................................2-12
Series Expansions ..........................................................................................................2-12
Mathematical Signs and Symbols.................................................................................2-15
Greek Alphabets ............................................................................................................2-16
Algebra ...........................................................................................................................2-17
Laws of Algebraic Operations...............................................................................2-17
Special Products and Factors ................................................................................2-17
Powers and Roots...................................................................................................2-19
Proportion..............................................................................................................2-19
Sum of Arithmetic Progression to n Terms .........................................................2-20
Sum of Geometric Progression to n Terms..........................................................2-20

10. Contents xi
Arithmetic Mean of n Quantities, A.....................................................................2-20
Geometric Mean of n Quantities, G .....................................................................2-20
Harmonic Mean of n Quantities, H .............................................................................2-20
Generalized Mean..................................................................................................2-20
Solution of Quadratic Equations ..........................................................................2-21
Solution of Cubic Equations .................................................................................2-21
Trigonometric Solution of the Cubic Equation...................................................2-22
Solution of Quadratic Equations ..................................................................................2-23
Partial Fractions.............................................................................................................2-23
Nonrepeated Linear Factors..................................................................................2-23
Repeated Linear Factors ........................................................................................2-24
General terms.................................................................................................................2-24
Repeated Linear Factors ........................................................................................2-25
Factors of Higher Degree ..............................................................................................2-25
Geometry........................................................................................................................2-25
Triangles.................................................................................................................2-25
Right Triangle ........................................................................................................2-26
Equilateral Triangle ...............................................................................................2-26
General Triangle ....................................................................................................2-26
Menelaus’ Theorem.......................................................................................................2-27
Ceva’s Theorem .............................................................................................................2-27
Quadrilaterals ................................................................................................................2-27
Rectangle ................................................................................................................2-27
Parallelogram .........................................................................................................2-27
Rhombus ................................................................................................................2-28
Trapezoid................................................................................................................2-28
General Quadrilateral............................................................................................2-28
Theorem.........................................................................................................................2-28
Regular Polygon of n Sides Each of Length b.......................................................2-28
Circle of radius r ....................................................................................................2-28
Regular Polygon of n sides Inscribed in a Circle of Radius r ..............................2-29
Regular Polygon of n Sides Circumscribing a Circle of Radius r .......................2-29
Cyclic Quadrilateral.......................................................................................................2-29
Prolemy’s Theorem .......................................................................................................2-29
Cyclic-Inscriptable Quadrilateral .................................................................................2-30
Sector of Circle of Radius r ...................................................................................2-30
Radius of a Circle Inscribed in a Triangle of Sides a, b, and c ............................2-30
Radius of a Circle Circumscribing a Triangle of Sides a, b, and c ......................2-30
Segment of a Circle of Radius r.............................................................................2-30
Ellipse of Semimajor Axis a and Semiminor Axis b............................................2-31
Segment of a Parabola ...........................................................................................2-31
Planar Areas by Approximation ...........................................................................2-31
Solids Bounded By Planes.............................................................................................2-31
Cube........................................................................................................................2-32

11. xii Contents
Rectangular Parallelepiped (or Box).....................................................................2-32
Prism.......................................................................................................................2-32
Truncated Triangular Prism .................................................................................2-32
Pyramid ..................................................................................................................2-32
Frustum of a Pyramid............................................................................................2-33
Prismatoid ..............................................................................................................2-33
Regular Polyhedra..................................................................................................2-33
Sphere of Radius r..................................................................................................2-34
Right Circular Cylinder of Radius r and Height h...............................................2-34
Circular Cylinder of Radius r and Slant Height ........................................................2-35
Cylinder of Cross-Sectional Area A and Slant Height ......................................2-35
Right Circular Cone of Radius r and Height h ....................................................2-35
Spherical Cap of Radius r and Height h...............................................................2-35
Frustum of a Right Circular Cone of Radii a and b and Height h......................2-35
Zone and Segment of Two Bases ..........................................................................2-35
Lune ........................................................................................................................2-36
Spherical Sector......................................................................................................2-36
Spherical Triangle and Polygon............................................................................2-36
Spheroids........................................................................................................................2-36
Ellipsoid..................................................................................................................2-36
Oblate Spheroid .....................................................................................................2-36
Prolate Spheroid.....................................................................................................2-37
Circular Torus........................................................................................................2-37
Formulas from Plane Analytic Geometry....................................................................2-37
Distance d between Two Points............................................................................2-37
Slope m of Line Joining Two Points .....................................................................2-37
Equation of a Line Joining Two Points ................................................................2-37
Equation of a Line in Terms of x-intercept a = 0 and y-intercept b = 0 ..........2-38
Normal Form for Equation of a Line....................................................................2-38
General Equation of a Line....................................................................................2-38
Distance From a Point (x1, y1) to the Line Ax + By + C = 0..............................2-38
Angle ψ between Two Lines Having Slopes m1 and m2 .....................................2-38
Area of a Triangle with Vertices ...........................................................................2-39
Transformation of Coordinates Involving Pure Translation..............................2-39
Transformation of Coordinates Involving Pure Rotation ..................................2-39
Transformation of Coordinates Involving Translation and Rotation................2-39
Polar Coordinates (r, θ).........................................................................................2-40
Plane Curves...........................................................................................................2-40
Catenary, Hyperbolic Cosine................................................................................2-40
Cardioid .........................................................................................................................2-40
Circle.......................................................................................................................2-40
Cassinian Curves....................................................................................................2-41
Cotangent Curve............................................................................................2-41
Cubical Parabola............................................................................................2-41

12. Contents xiii
Cosecant Curve..............................................................................................2-41
Cosine Curve..................................................................................................2-41
Ellipse .............................................................................................................2-41
Gamma Function...........................................................................................2-41
Hyperbolic Functions....................................................................................2-41
Inverse Cosine Curve ....................................................................................2-42
Inverse Sine Curve.........................................................................................2-42
Inverse Tangent Curve ..................................................................................2-42
Logarithmic Curve.........................................................................................2-42
Parabola..........................................................................................................2-42
Cubical Parabola............................................................................................2-42
Tangent Curve ...............................................................................................2-42
Distance d between Two Points....................................................................2-43
Logarithmic Identities...................................................................................................2-46
Series Expansions ..........................................................................................................2-47
Limiting Values..............................................................................................................2-47
Inequalities.....................................................................................................................2-47
Polynomial Approximations ........................................................................................2-48
Exponential Function Series Expansion ......................................................................2-49
Fundamental Properties................................................................................................2-49
Definition of General Powers .......................................................................................2-49
Logarithmic and Exponential Functions .....................................................................2-50
Periodic Property...................................................................................................2-50
Polynomial Approximations.................................................................................2-50
Slopes......................................................................................................................2-53
Trigonometric ratios .....................................................................................................2-53
Sine Law..................................................................................................................2-55
Cosine Law .............................................................................................................2-56
Algebra ...........................................................................................................................2-56
Expanding ..............................................................................................................2-56
Factoring.................................................................................................................2-57
Roots of a Quadratic Equation .....................................................................................2-57
Law of exponents ...................................................................................................2-57
Logarithms.....................................................................................................................2-58
3 Statistical Distributions, Methods, and Applications
Discrete Distributions .....................................................................................................3-1
Bernoulli Distribution .............................................................................................3-1
Beta Binomial Distribution .....................................................................................3-1
Beta Pascal Distribution ..........................................................................................3-3
Binomial Distribution .............................................................................................3-3
Discrete Weibull Distribution.................................................................................3-3
Geometric Distribution...........................................................................................3-3

13. xiv Contents
Hypergeometric Distribution .................................................................................3-3
Negative Binomial Distribution..............................................................................3-4
Poisson Distribution................................................................................................3-4
Rectangular (Discrete Uniform) Distribution .......................................................3-4
Continuous Distributions...............................................................................................3-4
Arcsin Distribution..................................................................................................3-4
Beta Distribution......................................................................................................3-5
Cauchy Distribution ................................................................................................3-5
Chi Distribution.......................................................................................................3-5
Chi-Square Distribution..........................................................................................3-5
Erlang Distribution..................................................................................................3-5
Exponential Distribution.........................................................................................3-6
Extreme-Value Distribution ...................................................................................3-6
F Distribution ..........................................................................................................3-6
Gamma Distribution ...............................................................................................3-7
Half-Normal Distribution.......................................................................................3-7
Laplace (Double Exponential) Distribution...........................................................3-7
Logistic Distribution................................................................................................3-7
Lognormal Distribution ..........................................................................................3-7
Noncentral Chi-Square Distribution......................................................................3-8
Noncentral F Distribution ......................................................................................3-8
Noncentral t-Distribution.......................................................................................3-8
Normal Distribution................................................................................................3-9
Pareto Distribution..................................................................................................3-9
Rayleigh Distribution ..............................................................................................3-9
t-Distribution...........................................................................................................3-9
Triangular Distribution.........................................................................................3-10
Uniform Distribution............................................................................................3-10
Weibull Distribution .............................................................................................3-10
Distribution Parameters........................................................................................3-10
Average...........................................................................................................3-10
Variance .........................................................................................................3-11
Standard Deviation........................................................................................3-11
Standard Error ...............................................................................................3-11
Skewness.........................................................................................................3-11
Standardized Skewness..................................................................................3-11
Kurtosis ..........................................................................................................3-11
Standardized Kurtosis ...................................................................................3-11
Weighted Average .........................................................................................3-11
Estimation and Testing..........................................................................................3-11
100(1−α)% Confidence Interval for Mean..................................................3-11
100(1 − α)% Confidence Interval for Variance ...........................................3-11
100(1 − α)% Confidence Interval for Difference in Means ........................3-12
100(1 − α)% Confidence Interval for Ratio of Variances ...........................3-12

14. Contents xv
Normal Probability Plot ........................................................................................3-12
Comparison of Poisson Rates ...............................................................................3-13
Distribution Functions—Parameter Estimation .................................................3-13
Bernoulli.........................................................................................................3-13
Binomial .........................................................................................................3-13
Discrete Uniform...........................................................................................3-13
Geometric.......................................................................................................3-13
Negative Binomial..................................................................................................3-13
Poisson....................................................................................................................3-13
Beta .........................................................................................................................3-14
Chi-Square..............................................................................................................3-14
Erlang......................................................................................................................3-14
Exponential ............................................................................................................3-14
F Distribution ........................................................................................................3-14
Gamma ...................................................................................................................3-14
Log–Normal...........................................................................................................3-15
Normal....................................................................................................................3-15
Student’s t...............................................................................................................3-15
Triangular...............................................................................................................3-15
Uniform..................................................................................................................3-15
Weibull ...................................................................................................................3-16
Chi-Square Test for Distribution Fitting..............................................................3-16
Kolmogorov–Smirnov Test...................................................................................3-16
ANOVA .........................................................................................................................3-16
Notation..................................................................................................................3-16
Standard Error (Internal)......................................................................................3-17
Standard Error (Pooled)........................................................................................3-17
Interval Estimates ..................................................................................................3-17
Tukey Interval........................................................................................................3-17
Scheffe Interval.......................................................................................................3-17
Cochran C-Test......................................................................................................3-17
Bartlett Test............................................................................................................3-18
Hartley’s Test .........................................................................................................3-18
Kruskal–Wallis Test...............................................................................................3-18
Adjustment for Ties...............................................................................................3-18
Freidman Test ........................................................................................................3-18
Regression ..............................................................................................................3-19
Notation .........................................................................................................3-19
Regression Statistics...............................................................................................3-19
Predictions..............................................................................................................3-20
Nonlinear Regression ............................................................................................3-21
Ridge Regression....................................................................................................3-21
Quality Control......................................................................................................3-22
For All Quality Control Formulas ................................................................3-22

15. xvi Contents
Subgroup Statistics.................................................................................................3-22
X Bar Charts...........................................................................................................3-22
Capability Ratios....................................................................................................3-23
R Charts..................................................................................................................3-24
S Charts...................................................................................................................3-24
C Charts..................................................................................................................3-24
U Charts .................................................................................................................3-24
P Charts ..................................................................................................................3-24
NP Charts...............................................................................................................3-25
CuSum Chart for the Mean...................................................................................3-25
Multivariate Control Charts..................................................................................3-25
Time-Series Analysis .............................................................................................3-25
Notation .........................................................................................................3-25
Autocorrelation at Lag k........................................................................................3-26
Partial Autocorrelation at Lag k............................................................................3-26
Cross-Correlation at Lag k ....................................................................................3-26
Box-Cox..................................................................................................................3-26
Periodogram (computed using Fast Fourier Transform) ...................................3-27
Categorical Analysis ..............................................................................................3-27
Notation .........................................................................................................3-27
Totals ......................................................................................................................3-27
Chi-Square..............................................................................................................3-27
Fisher’s Exact Test .................................................................................................3-28
Lambda...................................................................................................................3-28
Uncertainty Coefficient.........................................................................................3-28
Somer’s D ...............................................................................................................3-29
Eta ...........................................................................................................................3-29
Contingency Coefficient........................................................................................3-30
Cramer’s V .............................................................................................................3-30
Conditional Gamma..............................................................................................3-30
Pearson’s R .............................................................................................................3-30
Kendall’s Tau b.......................................................................................................3-30
Tau C ......................................................................................................................3-30
Probability Terminology .......................................................................................3-30
Basic Probability Principles...................................................................................3-31
Random Variable...................................................................................................3-31
Mean Value x̂ or Expected Value μ......................................................................3-31
Discrete Distribution Formulas............................................................................3-32
Bernoulli Distribution ...........................................................................................3-32
Beta Binomial Distribution ...................................................................................3-32
Beta Pascal Distribution ........................................................................................3-32
Binomial Distribution ...........................................................................................3-32
Discrete Weibull Distribution...............................................................................3-32
Geometric Distribution.........................................................................................3-33

16. Contents xvii
Hypergeometric Distribution ...............................................................................3-33
Negative Binomial Distribution............................................................................3-33
Poisson Distribution..............................................................................................3-34
Rectangular (Discrete Uniform) Distribution .....................................................3-34
Continuous Distribution Formulas......................................................................3-34
Arcsin Distribution................................................................................................3-34
Beta Distribution....................................................................................................3-35
Cauchy Distribution ..............................................................................................3-35
Chi Distribution.....................................................................................................3-35
Chi-Square Distribution........................................................................................3-35
Erlang Distribution................................................................................................3-35
Exponential Distribution.......................................................................................3-35
Extreme-Value Distribution .................................................................................3-36
F Distribution ........................................................................................................3-36
Gamma Distribution .............................................................................................3-36
Half-Normal Distribution.....................................................................................3-36
Laplace (Double Exponential) Distribution.........................................................3-37
Logistic Distribution..............................................................................................3-37
Lognormal Distribution ........................................................................................3-37
Noncentral Chi-Square Distribution....................................................................3-37
Noncentral F Distribution ....................................................................................3-38
Noncentral t-Distribution.....................................................................................3-38
Normal Distribution..............................................................................................3-38
Pareto Distribution................................................................................................3-38
Rayleigh Distribution ............................................................................................3-39
t-Distribution.........................................................................................................3-39
Triangular Distribution.........................................................................................3-39
Uniform Distribution............................................................................................3-39
Weibull Distribution .............................................................................................3-40
Variate Generation Techniques....................................................................................3-40
Notation..................................................................................................................3-40
Variate Generation Algorithms....................................................................................3-40
References ......................................................................................................................3-42
4 Computations with Descriptive Statistics
Sample Average ...............................................................................................................4-1
Application Areas ....................................................................................................4-1
Sample calculations..................................................................................................4-1
Sample Variance..............................................................................................................4-1
Application Areas ....................................................................................................4-1
Sample Calculations.................................................................................................4-2
Sample Standard Deviation.............................................................................................4-2
Application Areas ....................................................................................................4-2
Sample Standard Error of the Mean...............................................................................4-3

17. xviii Contents
Application Areas ....................................................................................................4-3
Skewness...................................................................................................................4-3
Standardized Skewness............................................................................................4-3
Kurtosis.....................................................................................................................4-4
Standardized Kurtosis..............................................................................................4-4
Weighted Average....................................................................................................4-4
Estimation and Testing ...................................................................................................4-4
100(1 − α)% Confidence Interval for Mean...........................................................4-4
100(1 − α)% Confidence Interval for Variance .....................................................4-4
100(1 − α)% Confidence Interval for Difference
in Means...................................................................................................4-4
100(1 − α)% Confidence Interval for Ratio
of Variances..............................................................................................4-5
Normal Probability Plot ..........................................................................................4-5
Comparison of Poisson Rates .................................................................................4-5
Distribution Functions and Parameter Estimation.......................................................4-5
Bernoulli Distribution .............................................................................................4-5
Binomial Distribution .............................................................................................4-5
Discrete Uniform Distribution ...............................................................................4-6
Geometric Distribution...........................................................................................4-6
Negative Binomial Distribution..............................................................................4-6
Poisson Distribution................................................................................................4-6
Beta Distribution......................................................................................................4-6
Chi-Square Distribution..........................................................................................4-6
Erlang Distribution..................................................................................................4-6
Exponential Distribution.........................................................................................4-7
Application Areas ............................................................................................4-7
F Distribution ..........................................................................................................4-7
Gamma Distribution ...............................................................................................4-7
Log–Normal Distribution .......................................................................................4-7
Normal Distribution................................................................................................4-8
Triangular Distribution...........................................................................................4-8
Uniform Distribution..............................................................................................4-8
Weibull Distribution ...............................................................................................4-8
Chi-Square Test for Distribution Fitting................................................................4-8
Kolmogorov–Smirnov Test.....................................................................................4-9
ANOVA ...........................................................................................................................4-9
Notation....................................................................................................................4-9
Standard Error .........................................................................................................4-9
Interval Estimates ....................................................................................................4-9
Tukey Interval........................................................................................................4-10
Scheffe Interval.......................................................................................................4-10
Cochran C-test.......................................................................................................4-10
Bartlett Test............................................................................................................4-10

18. Contents xix
Hartley’s Test .........................................................................................................4-10
Kruskal–Wallis Test...............................................................................................4-11
Adjustment for ties........................................................................................4-11
Freidman Test ........................................................................................................4-11
Regression ......................................................................................................................4-12
Notation..................................................................................................................4-12
Statistical Quality Control.............................................................................................4-13
Subgroup Statistics.................................................................................................4-13
X-Bar Charts ..........................................................................................................4-14
Capability Ratios....................................................................................................4-14
R Charts..................................................................................................................4-15
S Charts...................................................................................................................4-15
C Charts..................................................................................................................4-15
U Charts .................................................................................................................4-15
P Charts ..................................................................................................................4-15
NP Charts...............................................................................................................4-16
CuSum Chart for the Mean...................................................................................4-16
Time-Series Analysis .....................................................................................................4-16
Notation..................................................................................................................4-16
Autocorrelation at Lag k........................................................................................4-17
Partial Autocorrelation at Lag k............................................................................4-17
Cross-Correlation at Lag k ....................................................................................4-17
Box-Cox Computation ..........................................................................................4-17
Periodogram (Computed using Fast Fourier
Transform).............................................................................................4-18
Categorical Analysis ......................................................................................................4-18
Notation..................................................................................................................4-18
Totals ......................................................................................................................4-18
Chi-Square..............................................................................................................4-18
Lambda...................................................................................................................4-19
Uncertainty Coefficient.........................................................................................4-19
Somer’s D Measure................................................................................................4-20
Eta ...........................................................................................................................4-20
Contingency Coefficient........................................................................................4-21
Cramer’s V Measure..............................................................................................4-21
Conditional Gamma..............................................................................................4-21
Pearson’s R Measure..............................................................................................4-21
Kendall’s Tau b Measure .......................................................................................4-21
Tau C Measure.......................................................................................................4-22
Overall Mean..........................................................................................................4-22
Chebyshev’s Theorem ...........................................................................................4-22
Permutation ...........................................................................................................4-22
Combination ..........................................................................................................4-22
Failure.....................................................................................................................4-22

19. xx Contents
5 Computations for Economic Analysis
Fundamentals of Economic Analysis.............................................................................5-1
Simple Interest .........................................................................................................5-1
Future value......................................................................................................5-1
Compound Interest .................................................................................................5-2
Continuous Compound Interest ....................................................................5-2
Effective Rate....................................................................................................5-3
Present Value with Compound Interest.........................................................5-3
Annuities ..................................................................................................................5-4
Present value of annuity..................................................................................5-4
Future value of an annuity ..............................................................................5-4
Amortization of Loans.............................................................................................5-5
Interest and Equity Computations.................................................................................5-5
Equity Break-Even Formula ...........................................................................................5-8
Sinking Fund Payment ............................................................................................5-9
Internal Rate of Return....................................................................................5-9
Benefit–Cost Ratio...........................................................................................5-9
Simple Payback Period....................................................................................5-9
Discounted Payback Period ..........................................................................5-10
Economic Methods of Comparing Investment Alternatives......................................5-10
Present Value Analysis ..........................................................................................5-10
Annual Value Analysis ..........................................................................................5-10
Internal Rate of Return Analysis...........................................................................5-11
External Rate of Return Analysis..........................................................................5-11
Incremental Analysis.....................................................................................................5-11
Guidelines for Comparison of
Alternatives ............................................................................................5-12
Asset Replacement and Retention Analysis.................................................................5-12
Replacement Analysis Computation............................................................................5-14
Depreciation Methods...................................................................................................5-15
Depreciation Terminology............................................................................................5-15
Depreciation Methods...........................................................................................5-16
Straight-Line (SL) Method ....................................................................................5-16
Declining Balance (DB) Method...........................................................................5-16
Sums-of-Years’ Digits (SYD) Method..................................................................5-17
MACRS Method ....................................................................................................5-17
Effects of Inflation and Taxes................................................................................5-18
Foreign Exchange Rates.........................................................................................5-21
After-Tax Economic Analysis...............................................................................5-21
Cost and Value Computations .....................................................................................5-22
Actual Cost of Work Performed...........................................................................5-23
Applied Direct Cost...............................................................................................5-23
Budgeted Cost for Work Performed.....................................................................5-23

20. Contents xxi
Budgeted Cost for Work Scheduled .....................................................................5-23
Direct Cost .............................................................................................................5-23
Economies of Scale ................................................................................................5-23
Estimated Cost at Completion..............................................................................5-23
First Cost ................................................................................................................5-24
Fixed Cost...............................................................................................................5-24
Incremental Cost....................................................................................................5-24
Indirect Cost...........................................................................................................5-24
Life-Cycle Cost.......................................................................................................5-24
Maintenance Cost..................................................................................................5-24
Marginal Cost.........................................................................................................5-24
Operating Cost.......................................................................................................5-24
Opportunity Cost...................................................................................................5-25
Overhead Cost........................................................................................................5-25
Standard Cost.........................................................................................................5-25
Sunk Cost ...............................................................................................................5-25
Total Cost ...............................................................................................................5-25
Variable cost...........................................................................................................5-25
Cash-Flow Calculations ................................................................................................5-26
Calculations with Compound Amount Factor....................................................5-26
Calculations with Present Worth Factor..............................................................5-27
Calculations with Uniform Series Present Worth Factor ...................................5-27
Calculations with Uniform Series Capital Recovery Factor................................5-28
Calculations with Uniform Series Compound Amount Factor..........................5-29
Calculations with Uniform Series Sinking Fund Factor .....................................5-29
Calculations with Capitalized Cost Formula .......................................................5-30
Arithmetic Gradient Series....................................................................................5-31
Internal Rate of Return..........................................................................................5-32
Benefit-Cost Ratio Analysis...................................................................................5-33
Simple Payback Period ..........................................................................................5-33
Discounted Payback Period ..................................................................................5-34
Time Required to Double Investment..................................................................5-35
Effects of Inflation on Industrial Project Costing................................................5-36
Mild Inflation.........................................................................................................5-40
Moderate Inflation.................................................................................................5-40
Severe Inflation ......................................................................................................5-40
Hyperinflation........................................................................................................5-40
Break-Even Analysis..............................................................................................5-40
Profit Ratio Analysis......................................................................................................5-42
Project Cost Estimation ................................................................................................5-46
Optimistic and Pessimistic Cost Estimates ..........................................................5-47
Cost Performance Index ...............................................................................................5-47
Cost Control Limits.......................................................................................................5-48
Project Balance Computation.......................................................................................5-48

21. xxii Contents
6 Industrial Production Calculations
Learning Curve Models and Computations ..................................................................6-1
The Average Cost Model.........................................................................................6-1
Computational Example..........................................................................................6-2
Computational Example..........................................................................................6-3
The Unit Cost Model...............................................................................................6-4
Productivity Calculations Using Learning Curves........................................................6-5
Determining Average Unit Time............................................................................6-5
Calculation of the Learning Factor.........................................................................6-5
Calculating Total Time for a Job.............................................................................6-5
Time Required to Perform a Task for the nth Time..............................................6-5
The Improvement Ratio..................................................................................6-5
Computational Example..........................................................................................6-5
Solution ............................................................................................................6-5
Computation for Improvement Goal in Unit Time..............................................6-6
Computation of Machine Output Capacities.........................................................6-6
Machine Utilization Ratio: Determining How Often a
Machine Is Idle......................................................................................6-6
Calculating Number of Machines Needed to Meet Output..................................6-7
Alternate Forms for Machine Formula..........................................................6-7
Calculating Buffer Production to Allow for Defects .............................................6-8
Adjusting Machine Requirements to Allow for Rejects................................6-9
Output Computations for Sequence of Machines.........................................6-9
Calculating Forces on Cutting Tools....................................................................6-10
Calculating Pressure ......................................................................................6-11
Finding Required Horsepower .....................................................................6-11
Machine Accuracy .........................................................................................6-12
Calculating the Goal Dimension...........................................................................6-12
Drill Thrust ....................................................................................................6-13
Drill Torque ...................................................................................................6-14
Drill Horsepower...........................................................................................6-14
Calculating Speed and Rotation of Machines ......................................................6-14
Shaft Speed .....................................................................................................6-15
Surface Speed .................................................................................................6-15
Tool Feed per Revolution..............................................................................6-16
Tool Feed per Minute....................................................................................6-16
Tool Feed per Tooth......................................................................................6-17
Computation to Find the Volume of Material Removed....................................6-17
Time Rate for Material Removal in a Rectangular Cut...............................6-17
Calculation of Linear Feed for Rectangular Material Removal ..........................6-18
Takt Time for Production Planning.............................................................................6-18
Production Crew Work Rate Analysis.........................................................................6-19
Production Work Rate Example...........................................................................6-20

22. Contents xxiii
Case of Multiple Resources Working Together...........................................................6-20
Computational Examples......................................................................................6-21
Calculation of Personnel Requirements.......................................................................6-22
Calculation of Machine Requirements.........................................................................6-23
References ......................................................................................................................6-23
7 Forecasting Calculations
Forecasting Based on Averages.......................................................................................7-1
Simple Average Forecast .........................................................................................7-1
Period Moving Average Forecast............................................................................7-2
Weighted Average Forecast.....................................................................................7-2
Weighted T-Period Moving Average Forecast......................................................7-2
Exponential Smoothing Forecast............................................................................7-3
Regression Analysis.........................................................................................................7-3
Regression Relationships.........................................................................................7-3
Prediction.................................................................................................................7-4
Control .....................................................................................................................7-4
Procedure for Regression Analysis .........................................................................7-5
Coefficient of Determination..................................................................................7-5
Residual Analysis .....................................................................................................7-7
Time-Series Analysis .......................................................................................................7-9
Stationarity and Data Transformation.................................................................7-10
Moving Average Processes....................................................................................7-12
Autoregressive Processes.......................................................................................7-13
Forecasting for Inventory Control........................................................................7-15
Economic Order Quantity Model.........................................................................7-15
Quantity Discount .................................................................................................7-16
Calculation of Total Relevant Cost.......................................................................7-16
Evaluation of the Discount Option.......................................................................7-17
Sensitivity Analysis ................................................................................................7-19
Wagner–Whitin Algorithm ..................................................................................7-20
Notation and Variables..........................................................................................7-21
8 Six Sigma and Lean
Concept of Six Sigma.......................................................................................................8-1
Taguchi Loss Function....................................................................................................8-1
Identification and Elimination of Sources of Defects............................................8-2
Roles and Responsibilities for Six Sigma................................................................8-3
Statistical Techniques for Six Sigma...............................................................................8-3
Control Charts .................................................................................................8-4
X and R-Charts ........................................................................................................8-4
Data Collection Strategies .......................................................................................8-5
Subgroup Sample Size .....................................................................................8-5
Frequency of Sampling....................................................................................8-6

23. xxiv Contents
Stable Process...........................................................................................................8-6
Out-of-Control Patterns..........................................................................................8-6
Calculation of Control Limits .................................................................................8-7
Plotting Control Charts for Range and Average Charts........................................8-8
Plotting Control Charts for Moving Range and Individual
Control Charts .........................................................................................8-9
Case Example: Plotting of Control Chart...............................................................8-9
Calculations..............................................................................................................8-9
Trend Analysis .......................................................................................................8-14
Process Capability Analysis for Six Sigma ...................................................................8-18
Capable Process (Cp) .............................................................................................8-18
Capability Index (Cpk
) ...........................................................................................8-19
Process Capability Example ..................................................................................8-20
Possible Applications of Process Capability Index..............................................8-21
Potential Abuse of Cp and Cpk
.............................................................................8-22
Lean Principles and Applications.................................................................................8-22
Kaizen of a Process ................................................................................................8-23
Lean Task Value Rating System....................................................................................8-24
9 Risk Computations
Cost Uncertainties...........................................................................................................9-1
Schedule Uncertainties............................................................................................9-2
Performance Uncertainties .....................................................................................9-2
Decision Tables and Trees ..............................................................................................9-2
Reliability Calculations....................................................................................................9-7
General Reliability Definitions................................................................................9-8
Exponential Distribution Used as Reliability
Function ...................................................................................................9-9
Reliability Formulas...............................................................................................9-10
General Reliability Definitions..............................................................................9-10
Exponential Distribution Used as Reliability
Function .................................................................................................9-11
10 Computations for Project Analysis
Planning.......................................................................................................................10-1
Organizing ...................................................................................................................10-1
Scheduling....................................................................................................................10-1
Control.........................................................................................................................10-2
CPM Scheduling..........................................................................................................10-2
Advantages for Communication...........................................................................10-3
Advantages for Control .........................................................................................10-3
Advantages for Team Interaction.........................................................................10-3
Activity Precedence Relationships .............................................................................10-4
Network Notation..................................................................................................10-4

24. Contents xxv
Forward-Pass Calculations....................................................................................10-5
CPM Example ........................................................................................................10-6
Backward-Pass Calculations .................................................................................10-7
Calculation of Slacks ...................................................................................................10-8
Calculations for Subcritical Paths ..............................................................................10-9
Plotting of Gantt Charts............................................................................................10-10
Calculations for Project Crashing ............................................................................10-13
Calculations for Project Duration Diagnostics........................................................10-18
PERT Formulas .........................................................................................................10-19
Activity Time Distributions......................................................................................10-19
Project Duration Analysis.........................................................................................10-20
Simulation of Project Networks..........................................................................10-24
11 Product Shape and Geometrical Calculations
Equation of a Straight Line .........................................................................................11-1
Quadratic Equation.....................................................................................................11-2
Conic Sections .............................................................................................................11-2
Case 1 of Conic Section: e = 1 (Parabola)............................................................11-3
Case 2 of Conic Section: e 1 (Ellipse) ...............................................................11-3
Case 3 Hyperbola: e 1 ........................................................................................11-4
Case 4 e = 1 (Circle)..............................................................................................11-5
Conic Section Equation ..............................................................................................11-6
Quadric Surface (Sphere)............................................................................................11-6
Identities.................................................................................................................11-7
Trigonometry...............................................................................................................11-7
Law of Sines............................................................................................................11-7
Law of Cosines .......................................................................................................11-8
Identities.................................................................................................................11-8
Complex Numbers...............................................................................................11-10
Polar Coordinates......................................................................................................11-10
Euler’s Identity.....................................................................................................11-11
Roots.....................................................................................................................11-11
Matrices......................................................................................................................11-11
Matrix Multiplication..........................................................................................11-11
Matrix Addition...................................................................................................11-11
Identity Matrix.....................................................................................................11-11
Matrix Transpose.................................................................................................11-11
Matrix Inverse......................................................................................................11-12
Determinants .......................................................................................................11-12
Vectors .......................................................................................................................11-12
Gradient, Divergence, and Curl..........................................................................11-13
Identities...............................................................................................................11-13
Progressions and Series.............................................................................................11-14
Arithmetic Progression .......................................................................................11-14

25. xxvi Contents
Geometric Progression........................................................................................11-14
Properties of Series ..............................................................................................11-15
Power Series .........................................................................................................11-15
Taylor’s Series ......................................................................................................11-16
Differential Calculus .................................................................................................11-16
The Derivative......................................................................................................11-16
Test for a Maximum....................................................................................11-16
Test for a Minimum ....................................................................................11-16
Test for a Point of Inflection.......................................................................11-16
The Partial Derivative..................................................................................11-16
Curvature in Rectangular Coordinates ......................................................11-17
The Radius of Curvature.............................................................................11-17
L’Hospital’s Rule........................................................................................................11-17
Integral Calculus........................................................................................................11-18
Derivatives and Indefinite Integrals.........................................................................11-18
Derivatives .................................................................................................................11-18
Indefinite Integrals...............................................................................................11-20
Mensuration of Areas and Volumes ........................................................................11-21
Nomenclature ......................................................................................................11-21
Parabola................................................................................................................11-21
Ellipse....................................................................................................................11-21
Circular Segment .................................................................................................11-22
Circular Sector .....................................................................................................11-22
Sphere ...................................................................................................................11-22
Parallelogram .......................................................................................................11-22
Regular Polygon (n Equal Sides).........................................................................11-23
Prismoid ...............................................................................................................11-23
Right Circular Cone.............................................................................................11-23
Right Circular Cylinder.......................................................................................11-23
Paraboloid of Revolution.....................................................................................11-23
Centroids and Moments of Inertia...........................................................................11-24
Difference Equations.................................................................................................11-24
First-Order Linear Difference Equation.............................................................11-25
Second-Order Linear Difference Equation ........................................................11-25
Numerical Methods...................................................................................................11-25
Newton’s Method for Root Extraction...............................................................11-25
Newton’s Method of Minimization....................................................................11-25
Numerical Integration.........................................................................................11-26
Euler’s or Forward Rectangular Rule .........................................................11-26
Trapezoidal Rule..........................................................................................11-26
Simpson’s Rule/Parabolic Rule (n Must Be
an Even Integer)................................................................................11-27
Calculation of Best-Fit Circle ...................................................................................11-27

26. Contents xxvii
12 General Engineering Calculations
Six Simple Machines for Materials Handling............................................................12-1
Machine 1: The Lever ............................................................................................12-1
Machine 2: Wheel and Axle ..................................................................................12-1
Machine 3: The Pulley ...........................................................................................12-2
Machine 4: The Inclined Plane .............................................................................12-2
Machine 5: The Wedge..........................................................................................12-2
Machine 6: The Screw............................................................................................12-2
Mechanics: Kinematics ...............................................................................................12-3
Scalars and Vectors................................................................................................12-3
Distance and Displacement...........................................................................12-3
Acceleration ...................................................................................................12-3
Speed and Velocity ........................................................................................12-3
Frequency.......................................................................................................12-3
Period .............................................................................................................12-3
Angular Displacement...................................................................................12-3
Angular Velocity............................................................................................12-4
Angular Acceleration.....................................................................................12-4
Rotational Speed ............................................................................................12-4
Uniform Linear Motion ................................................................................12-4
Uniform Accelerated Linear Motion....................................................................12-4
Rotational Motion .........................................................................................12-5
Uniform Rotation and a Fixed Axis .....................................................................12-5
Uniform Accelerated Rotation about a Fixed Axis..............................................12-5
Simple Harmonic Motion...........................................................................................12-6
Pendulum ...............................................................................................................12-7
Free Fall ..................................................................................................................12-7
Vertical Project ......................................................................................................12-7
Angled Projections.................................................................................................12-8
Horizontal Projection (α = 0)...............................................................................12-8
Sliding Motion on an Inclined Plane....................................................................12-8
Rolling Motion on an Inclined Plane ...................................................................12-9
Mechanics: Dynamics ...............................................................................................12-10
Newton’s First Law of Motion ............................................................................12-10
Newton’s Second Law of Motion........................................................................12-10
Newton’s Third Law of Motion ..........................................................................12-10
Momentum of Force............................................................................................12-10
Impulse of Force ..................................................................................................12-11
Law of Conservation of Momentum ..................................................................12-11
Friction .................................................................................................................12-11
General Law of Gravity .............................................................................................12-11
Gravitational Force..............................................................................................12-12
Centrifugal Force .................................................................................................12-12
Centripetal Force .................................................................................................12-12

27. xxviii Contents
Torque ..................................................................................................................12-12
Work.....................................................................................................................12-13
Energy...................................................................................................................12-13
Conservation of Energy.......................................................................................12-14
Power....................................................................................................................12-14
Appendix A: Mathematical Patterns, Series, and Formulae
Number Sequence and Patterns ................................................................................A-1
Closed-Form Mathematical Expressions..................................................................A-3
Derivation of the Quadratic Formula........................................................................A-7
Appendix B: Measurement Units, Notation, and Constants
Common Notation ..................................................................................................... B-1
Scientific Constants.................................................................................................... B-3
Numbers and Prefixes ................................................................................................ B-3
Appendix C: Conversion Factors
Area Conversion Factors............................................................................................C-1
Volume Conversion Factors ......................................................................................C-2
Energy Conversion Factors........................................................................................C-2
Mass Conversion Factors...........................................................................................C-3
Temperature Conversion Factors..............................................................................C-3
Velocity Conversion Factors......................................................................................C-4
Pressure Conversion Factors .....................................................................................C-4
Distance Conversion Factors.....................................................................................C-5
Physical Science Equations ........................................................................................C-6
English and Metric Systems.......................................................................................C-7
Household Measurement Conversion ......................................................................C-7
Appendix D: Factors and Tables
Formulas for Interest Factor......................................................................................D-1
Summation Formulas for Closed-Form Expressions...............................................D-3
Interest Tables.............................................................................................................D-3
Appendix E: Greek Symbols and Roman Numerals
Greek Symbols ............................................................................................................ E-1
Roman Numerals........................................................................................................ E-2
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography-1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Index-1