Package {ProcessCapabilityR}

Type:	Package
Title:	Classical and Generalized Process Capability Indices
Version:	0.1.0
Maintainer:	Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Description:	Computes classical process capability indices (Cp, Cpk, Cpu, Cpl, Cpm, Cpmk, Pp, Ppk, Ppu, Ppl, Z) and the generalized process capability index Cpy (Maiti, Saha & Nanda, 2010) <doi:10.1080/16843703.2010.11673233> for any continuous or discrete quality characteristic. Users supply the probability density function (PDF) and cumulative distribution function (CDF) of the characteristic, and the package returns point estimates, bootstrap confidence intervals (percentile and BCa), and sensitivity tables/plots across ranges of short-term standard deviation (sigma), long-term standard deviation (s), desired yield (p0), and significance levels. Classical indices are recoverable as special cases under the normal distribution. The package follows the theory and notation of Kane (1986) <doi:10.1080/00224065.1986.11978984>, Chan, Cheng & Spiring (1988) <doi:10.1080/00224065.1988.11979102>, Pearn, Kotz & Johnson (1992) <doi:10.1080/00224065.1992.11979403>, Kotz & Johnson (2002) <doi:10.1080/00224065.2002.11980119>, Montgomery (2020, ISBN:978-1-119-39930-8), Juran (1974, ISBN:978-0-07-033176-1), Harry & Schroeder (2000, ISBN:978-0-385-49437-2), and the AIAG SPC Reference Manual (2005, ISBN:978-1-60534-026-3).
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.3
Depends:	R (≥ 3.5.0)
Imports:	stats
Suggests:	ggplot2 (≥ 3.0.0), testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-07-23 17:54:01 UTC; 30017827
Author:	Shikhar Tyagi [aut, cre], Sumit Kumar [aut], Vrijesh Tripathi [aut]
Repository:	CRAN
Date/Publication:	2026-08-03 18:00:24 UTC

Compute symmetric desirable limits for Cpy

Description

Given a pci_dist and desired yield p0, computes LDL and UDL symmetric about the median of the distribution such that F(UDL) - F(LDL) = p0.

Usage

compute_desirable_limits(dist, p0)

Arguments

Value

Named numeric vector c(LDL = ..., UDL = ...).

Compute a PCI from sample statistics

Description

Compute a PCI from sample statistics

Usage

compute_index_from_sample(
  index,
  sample_data,
  LSL,
  USL,
  target,
  LDL,
  UDL,
  p0,
  dist
)

Value

A numeric value of the calculated process capability index, or NA_real_ on error/invalid input.

Process Potential Index (Cp)

Description

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

Usage

cp(LSL, USL, sigma)

Arguments

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

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

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

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

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

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

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

Draw random samples from a pci_dist via inverse-CDF sampling

Description

Generates samples by drawing U ~ Uniform(0,1) and inverting the CDF using uniroot.

Usage

inverse_cdf_sample(dist, n)

Arguments

Value

Numeric vector of length n.

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

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

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:

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

Value

An S3 object of class "pci_dist" with elements:

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

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

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

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

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

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

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

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

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

Value

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

Require a parameter to be non-NULL

Description

Require a parameter to be non-NULL

Usage

require_param(x, name, index)

Arguments

Value

Invisibly returns NULL. Called for side effects (input validation).

Validate significance level alpha

Description

Validate significance level alpha

Usage

validate_alpha(alpha)

Arguments

Value

Invisibly returns NULL. Called for side effects (input validation).

Validate desired yield p0

Description

Validate desired yield p0

Usage

validate_p0(p0)

Arguments

Value

Invisibly returns NULL. Called for side effects (input validation).

Validate that a value is positive

Description

Validate that a value is positive

Usage

validate_positive(x, name)

Arguments

Value

Invisibly returns NULL. Called for side effects (input validation).

Validate specification limits

Description

Validate specification limits

Usage

validate_spec_limits(LSL, USL)

Arguments

Value

Invisibly returns NULL. Called for side effects (input validation).

Sigma Level (Z)

Description

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

Usage

z_level(LSL, USL, mu, sigma)

Arguments

Value

Numeric value of Z.

See Also

pci

Examples

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