# Handbook of Industrial Engineering Equations, Formulas, and Calculations | Adedeji Badiru and Olufemi A. Omitaomu

Handbook of Industrial Engineering Equations, Formulas, and Calculations

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
2 Basic Mathematical Calculations
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 ˆx 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
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 Cubic Equations ………………………………………………………………………2-21
Trigonometric Solution of the Cubic Equation ……………………………………………2-22
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
Rectangle ………………………………………………………………………………………………….2-27
Parallelogram ……………………………………………………………………………………………2-27
Rhombus ………………………………………………………………………………………………….2-28
Trapezoid………………………………………………………………………………………………….2-28
Theorem………………………………………………………………………………………………………….2-28
Regular Polygon of n Sides Each of Length b……………………………………………….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
Prolemy’s Theorem ………………………………………………………………………………………….2-29
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
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
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
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
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
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
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
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 ˆx 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
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
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
Hartley’s Test ……………………………………………………………………………………………4-10
Kruskal–Wallis Test…………………………………………………………………………………..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
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
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
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
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

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