Package 'ProcessCapabilityR'

Process Potential Index (Cp)

Description

$C_p = (USL - LSL) / (6\sigma)$.

Usage

cp(LSL, USL, sigma)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
sigma	Short-term (within-subgroup) standard deviation.

Value

Numeric value of $C_p$.

See Also

pci

Examples

cp(LSL = 57, USL = 63, sigma = 1)   # 1.0

---

Process Capability Index (Cpk)

Description

$C_{pk} = \min\bigl((USL-\mu)/(3\sigma),\;(\mu-LSL)/(3\sigma)\bigr)$.

Usage

cpk(LSL, USL, mu, sigma)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.

Value

Numeric value of $C_{pk}$.

See Also

pci

Examples

cpk(LSL = 57, USL = 63, mu = 60, sigma = 1)   # 1.0

---

Lower Capability Index (Cpl)

Description

$C_{pl} = (\mu - LSL) / (3\sigma)$.

Usage

cpl(LSL, mu, sigma)

Arguments

LSL	Lower specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.

Value

Numeric value of $C_{pl}$.

See Also

pci

Examples

cpl(LSL = 57, mu = 60, sigma = 1)   # 1.0

---

Taguchi Capability Index (Cpm)

Description

$C_{pm} = (USL - LSL) / \bigl(6\sqrt{\sigma^2 + (\mu - T)^2}\bigr)$.

Usage

cpm(LSL, USL, mu, sigma, target)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.
target	Target value $T$.

Value

Numeric value of $C_{pm}$.

See Also

pci

Examples

cpm(LSL = 57, USL = 63, mu = 60, sigma = 1, target = 60)   # 1.0

---

Modified Taguchi Capability Index (Cpmk)

Description

$C_{pmk} = \min\bigl((USL-\mu),\;(\mu-LSL)\bigr) / \bigl(3\sqrt{\sigma^2+(\mu-T)^2}\bigr)$.

Usage

cpmk(LSL, USL, mu, sigma, target)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.
target	Target value $T$.

Value

Numeric value of $C_{pmk}$.

See Also

pci

Examples

cpmk(LSL = 57, USL = 63, mu = 60, sigma = 1, target = 60)   # 1.0

---

Upper Capability Index (Cpu)

Description

$C_{pu} = (USL - \mu) / (3\sigma)$.

Usage

cpu(USL, mu, sigma)

Arguments

USL	Upper specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.

Value

Numeric value of $C_{pu}$.

See Also

pci

Examples

cpu(USL = 63, mu = 60, sigma = 1)   # 1.0

---

Generalized Process Capability Index (Cpy)

Description

$C_{py} = p / p_0 = \bigl(F(USL)-F(LSL)\bigr) / \bigl(F(UDL)-F(LDL)\bigr)$, where $F$ is the CDF from dist (Maiti, Saha & Nanda, 2010).

Usage

cpy(dist, LSL, USL, LDL = NULL, UDL = NULL, p0 = NULL)

Arguments

dist	A pci_dist object.
LSL	Lower specification limit.
USL	Upper specification limit.
LDL	Lower desirable limit (optional if p0 is given).
UDL	Upper desirable limit (optional if p0 is given).
p0	Desired yield (optional if LDL/UDL are given).

Value

Numeric value of $C_{py}$.

See Also

pci, pci_dist

Examples

d <- pci_dist_normal(mean = 60, sd = 1)
cpy(d, LSL = 57, USL = 63, LDL = 57, UDL = 63)   # 1.0

---

Compute a Process Capability Index

Description

Computes a single process capability or performance index from the supplied parameters.

Usage

pci(
  index,
  dist = NULL,
  LSL = NULL,
  USL = NULL,
  target = NULL,
  mu = NULL,
  sigma = NULL,
  xbar = NULL,
  s = NULL,
  LDL = NULL,
  UDL = NULL,
  p0 = NULL
)

Arguments

index	Character string: one of "Cp", "Cpk", "Cpu", "Cpl", "Cpm", "Cpmk", "Pp", "Ppk", "Ppu", "Ppl", "Z", or "Cpy".
dist	A pci_dist object (required for "Cpy"; optional otherwise).
LSL, USL	Lower and upper specification limits.
target	Target value (required for Cpm and Cpmk).
mu, sigma	Short-term (within-subgroup) mean and standard deviation.
xbar, s	Long-term (overall) mean and standard deviation.
LDL, UDL	Lower and upper desirable limits for Cpy.
p0	Desired yield for Cpy. If LDL/UDL are not supplied, they are computed from p0 using compute_desirable_limits.

Details

The twelve supported indices are:

Cp

$(USL - LSL) / (6\sigma)$

Cpk

$\min\bigl((USL-\mu)/(3\sigma),\;(\mu-LSL)/(3\sigma)\bigr)$

Cpu

$(USL - \mu) / (3\sigma)$

Cpl

$(\mu - LSL) / (3\sigma)$

Cpm

$(USL - LSL) / \bigl(6\sqrt{\sigma^2+(\mu-T)^2}\bigr)$

Cpmk

$\min\bigl((USL-\mu),\;(\mu-LSL)\bigr) / \bigl(3\sqrt{\sigma^2+(\mu-T)^2}\bigr)$

Pp

$(USL - LSL) / (6s)$

Ppk

$\min\bigl((USL-\bar x)/(3s),\;(\bar x-LSL)/(3s)\bigr)$

Ppu

$(USL - \bar x) / (3s)$

Ppl

$(\bar x - LSL) / (3s)$

Z

$\min\bigl((USL-\mu)/\sigma,\;(\mu-LSL)/\sigma\bigr)$

Cpy

$p / p_0 = \bigl(F(USL)-F(LSL)\bigr) / \bigl(F(UDL)-F(LDL)\bigr)$

Value

A single numeric value, or NA with a warning when a one-sided index is requested but the relevant specification limit is NULL.

See Also

pci_dist, pci_ci, pci_grid

Examples

pci("Cp",  LSL = 57, USL = 63, sigma = 1)            # 1.0
pci("Cpk", LSL = 57, USL = 63, mu = 60, sigma = 1)   # 1.0
pci("Z",   LSL = 57, USL = 63, mu = 60, sigma = 1)   # 3.0

# Generalized Cpy
d <- pci_dist_normal(60, 1)
pci("Cpy", dist = d, LSL = 57, USL = 63, LDL = 57, UDL = 63) # 1.0

---

Bootstrap Confidence Interval for a Process Capability Index

Description

Computes a confidence interval for a PCI via parametric or nonparametric bootstrap.

Usage

pci_ci(
  index,
  dist,
  n,
  data = NULL,
  LSL = NULL,
  USL = NULL,
  target = NULL,
  LDL = NULL,
  UDL = NULL,
  p0 = NULL,
  alpha = 0.05,
  B = 2000,
  method = c("percentile", "bca")
)

Arguments

index	Character: one of the 12 index names (see pci).
dist	A pci_dist object.
n	Integer sample size.
data	Optional numeric vector of observed data. If supplied, nonparametric bootstrap is used instead of parametric.
LSL, USL	Specification limits.
target	Target value (for Cpm/Cpmk).
LDL, UDL	Desirable limits (for Cpy).
p0	Desired yield (for Cpy).
alpha	Significance level; the interval covers $1 - \alpha$. Default 0.05.
B	Number of bootstrap replications. Default 2000.
method	Character: "percentile" (default) or "bca".

Details

**Parametric bootstrap** (when data = NULL): B samples of size n are drawn from dist via inverse-CDF sampling. For each sample the process mean, standard deviation (and, for $C_{py}$, the empirical CDF) are estimated, and the index is recomputed.

**Nonparametric bootstrap** (when data is supplied): B resamples of the same length are drawn with replacement from data.

Two interval methods are available:

percentile

Uses the alpha/2 and 1 - alpha/2 quantiles of the bootstrap distribution.

bca

Bias-corrected and accelerated interval using jackknife acceleration and bootstrap bias correction (Efron, 1987).

Value

An S3 object of class "pci_ci" with components:

index	The index name.
estimate	Point estimate.
lower	Lower CI bound.
upper	Upper CI bound.
alpha	The significance level used.
method	The interval method used.
boot_values	Numeric vector of all B bootstrap estimates.

See Also

pci, pci_grid

Examples

d <- pci_dist_normal(mean = 60, sd = 1)
ci <- pci_ci("Cp", dist = d, n = 30, LSL = 57, USL = 63,
             alpha = 0.05, B = 500)
print(ci)

---

Create a Distribution Specification for Process Capability Analysis

Description

Constructs an S3 object of class "pci_dist" that encapsulates the probability density function (PDF) and cumulative distribution function (CDF) of a quality characteristic. This object is passed to pci, pci_ci, and pci_grid when computing generalised indices such as $C_{py}$.

Usage

pci_dist(pdf, cdf = NULL, params = list(), support = c(-Inf, Inf))

Arguments

pdf	A function function(x, ...) returning the density at x. Additional distribution parameters are passed via params.
cdf	A function function(x, ...) returning the cumulative probability $P(Z \le x)$. If NULL, a numerical CDF is derived via integrate over pdf.
params	A named list of distribution parameters forwarded to pdf and cdf via do.call.
support	A length-2 numeric vector giving the lower and upper bounds of the distribution's support (default c(-Inf, Inf)).

Value

An S3 object of class "pci_dist" with elements:

pdf_fn	Closure that evaluates the PDF at a given point.
cdf_fn	Closure that evaluates the CDF at a given point.
params	The supplied parameter list.
support	The supplied support bounds.
cdf_is_numeric	Logical; TRUE if CDF was derived numerically.

Examples

# Normal distribution
d <- pci_dist(
  pdf = function(x, mean, sd) dnorm(x, mean, sd),
  cdf = function(x, mean, sd) pnorm(x, mean, sd),
  params = list(mean = 0, sd = 1)
)
d$cdf_fn(0)   # 0.5

# Weibull with only PDF (CDF derived numerically)
d2 <- pci_dist(
  pdf = function(x, shape, scale) dweibull(x, shape, scale),
  params = list(shape = 2, scale = 10),
  support = c(0, 60)
)

---

Normal Distribution Shortcut

Description

Convenience constructor for a pci_dist based on the normal distribution. Useful for sanity-checking that $C_{py}$ collapses to the classical indices when $F$ is the normal CDF.

Usage

pci_dist_normal(mean = 0, sd = 1)

Arguments

mean	Mean of the normal distribution.
sd	Standard deviation of the normal distribution.

Value

An S3 object of class "pci_dist" encapsulating the normal distribution PDF and CDF functions, parameter list, and support bounds.

Examples

d <- pci_dist_normal(mean = 60, sd = 1)
d$cdf_fn(60)   # 0.5
d$pdf_fn(60)   # ~ 0.3989

---

Sensitivity Grid of Process Capability Indices

Description

Computes a PCI across a grid of parameter values (short-term standard deviation sigma, long-term standard deviation s, and/or desired yield p0), optionally with bootstrap confidence intervals at multiple significance levels.

Usage

pci_grid(
  index,
  dist = NULL,
  LSL = NULL,
  USL = NULL,
  target = NULL,
  sigma_vals = NULL,
  s_vals = NULL,
  p0_vals = NULL,
  mu = NULL,
  xbar = NULL,
  alpha_vals = c(0.1, 0.05, 0.01),
  n = NULL,
  B = 2000
)

Arguments

index	Character: one of the 12 index names (see pci).
dist	A pci_dist object (required for "Cpy").
LSL, USL	Specification limits.
target	Target value (for Cpm/Cpmk).
sigma_vals	Numeric vector of $\sigma$ values to sweep (for capability indices).
s_vals	Numeric vector of $s$ values to sweep (for performance indices).
p0_vals	Numeric vector of $p_0$ values to sweep (for Cpy).
mu	Short-term process mean.
xbar	Long-term process mean.
alpha_vals	Numeric vector of significance levels. Default c(0.10, 0.05, 0.01).
n	Sample size for bootstrap CIs. If NULL, CIs are omitted.
B	Bootstrap replications. Default 2000.

Value

A data.frame (with additional class "pci_grid") in long format with columns: index, sigma, s, p0, alpha, estimate, lower, upper. Columns that do not apply to the chosen index contain NA.

See Also

pci, pci_ci, plot.pci_grid

Examples

grid <- pci_grid("Cp", LSL = 57, USL = 63,
                 sigma_vals = c(0.5, 1.0, 1.5),
                 mu = 60, alpha_vals = c(0.05, 0.01))
print(grid)

---

Plot a Process Capability Sensitivity Grid

Description

Produces a ggplot2 line-and-ribbon plot showing the PCI estimate and confidence bands as a function of the chosen sweep variable, faceted by significance level.

Usage

## S3 method for class 'pci_grid'
plot(x, x_axis = c("sigma", "s", "p0"), facet_by = "alpha", ...)

Arguments

x	A pci_grid object.
x_axis	Character: which sweep variable to place on the x-axis. One of "sigma", "s", or "p0".
facet_by	Character: column name to facet by. Default "alpha".
...	Additional arguments (currently ignored).

Value

Invisibly returns a ggplot object representing the sensitivity plot. Called primarily for its side effect of rendering a line-and-ribbon plot to the active graphic device.

See Also

pci_grid

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
  grid <- pci_grid("Cp", LSL = 57, USL = 63,
                   sigma_vals = seq(0.5, 2, 0.1), mu = 60,
                   alpha_vals = c(0.05, 0.01),
                   n = 30, B = 200,
                   dist = pci_dist_normal(60, 1))
  plot(grid, x_axis = "sigma")
}

---

Performance Index (Pp)

Description

$P_p = (USL - LSL) / (6s)$.

Usage

pp(LSL, USL, s)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
s	Long-term (overall) standard deviation.

Value

Numeric value of $P_p$.

See Also

pci

Examples

pp(LSL = 57, USL = 63, s = 1.5)   # 0.6667

---

Performance Capability Index (Ppk)

Description

$P_{pk} = \min\bigl((USL-\bar{x})/(3s),\;(\bar{x}-LSL)/(3s)\bigr)$.

Usage

ppk(LSL, USL, xbar, s)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
xbar	Long-term (overall) mean.
s	Long-term standard deviation.

Value

Numeric value of $P_{pk}$.

See Also

pci

Examples

ppk(LSL = 57, USL = 63, xbar = 60, s = 1.5)   # 0.6667

---

Lower Performance Index (Ppl)

Description

$P_{pl} = (\bar{x} - LSL) / (3s)$.

Usage

ppl(LSL, xbar, s)

Arguments

LSL	Lower specification limit.
xbar	Long-term mean.
s	Long-term standard deviation.

Value

Numeric value of $P_{pl}$.

See Also

pci

Examples

ppl(LSL = 57, xbar = 60, s = 1.5)   # 0.6667

---

Upper Performance Index (Ppu)

Description

$P_{pu} = (USL - \bar{x}) / (3s)$.

Usage

ppu(USL, xbar, s)

Arguments

USL	Upper specification limit.
xbar	Long-term mean.
s	Long-term standard deviation.

Value

Numeric value of $P_{pu}$.

See Also

pci

Examples

ppu(USL = 63, xbar = 60, s = 1.5)   # 0.6667

---

Print method for pci_ci objects

Description

Print method for pci_ci objects

Usage

## S3 method for class 'pci_ci'
print(x, ...)

Arguments

x	A pci_ci object.
...	Additional arguments (ignored).

Value

Invisibly returns the input x of class "pci_ci". Called for side effect of printing formatted confidence interval details to the console.

---

Print method for pci_dist objects

Description

Print method for pci_dist objects

Usage

## S3 method for class 'pci_dist'
print(x, ...)

Arguments

x	A pci_dist object.
...	Additional arguments (ignored).

Value

Invisibly returns the input x of class "pci_dist". Called for side effect of printing distribution parameters and specification details to the console.

---

Sigma Level (Z)

Description

$Z = \min\bigl((USL-\mu)/\sigma,\;(\mu-LSL)/\sigma\bigr)$.

Usage

z_level(LSL, USL, mu, sigma)

Arguments

LSL	Lower specification limit.
USL	Upper specification limit.
mu	Short-term process mean.
sigma	Short-term standard deviation.

Value

Numeric value of $Z$.

See Also

pci

Examples

z_level(LSL = 57, USL = 63, mu = 60, sigma = 1)   # 3.0

---