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AccSamplingDesign: Acceptance Sampling Plan Design - R Package
In AccSamplingDesign: Acceptance Sampling Plans Design
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 7,
fig.height = 5
)
library(AccSamplingDesign)
1. Introduction
The AccSamplingDesign package provides tools for designing and evaluating Acceptance Sampling plans for quality control in manufacturing and inspection settings. It supports both attribute and variable sampling methods, applying nonlinear programming to minimize sample size while effectively controlling both producer’s and consumer’s risks.
Key features include:
- Attributes Sampling Plans — pass/fail decisions based on the proportion of nonconforming units.
- Variables Sampling Plans — support for normal and beta distributions, including compositional data.
- Operating Characteristic (OC) Curve Visualization — assess and compare plan performance.
- Risk-Based Optimization — minimize sample size while meeting Producer’s Risk (PR) and Consumer’s Risk (CR) conditions.
- Custom Plan Comparison — compare user-defined plans against optimized designs.
2. Installation
Install the stable release from CRAN:
install.packages("AccSamplingDesign")
Or install from GitHub
devtools::install_github("vietha/AccSamplingDesign")
Load package
library(AccSamplingDesign)
3. Attributes Sampling Plans
Note that we could use method optPlan() or optAttrPlan(), both work the same.
3.1 Create Attribute Plan
plan_attr <- optPlan(
PRQ = 0.01, # Acceptable Quality Level (1% defects)
CRQ = 0.05, # Rejectable Quality Level (5% defects)
alpha = 0.02, # Producer's risk
beta = 0.15, # Consumer's risk
distribution = "binomial"
)
3.2 Plan Summary
summary(plan_attr)
3.3 Acceptance Probability
# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)
3.4 OC Curve
plot(plan_attr)
3.5 Compare Attributes Optimal Plan vs Custom Plan
# Step1: Find an optimal Attributes Sampling plan
optimal_plan <- optPlan(PRQ = 0.01, CRQ = 0.05, alpha = 0.02, beta = 0.15,
distribution = "binomial") # could try "poisson" too
# Summarize the plan
summary(optimal_plan)
# Step2: Compare the optimal plan with two alternative plans
pd <- seq(0, 0.15, by = 0.001)
oc_opt <- OCdata(plan = optimal_plan, pd = pd)
oc_alt1 <- OCdata(n = optimal_plan$n, c = optimal_plan$c - 1,
distribution = "binomial", pd = pd)
oc_alt2 <- OCdata(n = optimal_plan$n, c = optimal_plan$c + 1,
distribution = "binomial", pd = pd)
# Step3: Visualize results
plot(pd, paccept(oc_opt), type = "l", col = "blue", lwd = 2,
xlab = "Proportion Defective", ylab = "Probability of Acceptance",
main = "Attributes Sampling - OC Curves Comparison",
xlim = c(0, 0.15), ylim = c(0, 1))
lines(pd, paccept(oc_alt1), col = "red", lwd = 2, lty = 2)
lines(pd, paccept(oc_alt2), col = "green", lwd = 2, lty = 3)
abline(v = c(0.01, 0.05), col = "gray50", lty = 2)
abline(h = c(1 - 0.02, 0.15), col = "gray50", lty = 2)
legend("topright", legend = c(sprintf("Optimal Plan (n = %d, c = %d)",
optimal_plan$n, optimal_plan$c),
sprintf("Alt 1 (c = %d)", optimal_plan$c - 1),
sprintf("Alt 2 (c = %d)", optimal_plan$c + 1)),
col = c("blue", "red", "green"),
lty = c(1, 2, 3), lwd = 2)
4. Variables Sampling Plans
Note that we could use method optPlan() or optVarPlan(), both work the same.
4.1 Normal Distribution
4.1.1 Find an optimal plan and plot OC chart
# Predefine parameters
PRQ <- 0.025
CRQ <- 0.1
alpha <- 0.05
beta <- 0.1
norm_plan <- optPlan(
PRQ = PRQ, # Acceptable quality level (% nonconforming)
CRQ = CRQ, # Rejectable quality level (% nonconforming)
alpha = alpha, # Producer's risk
beta = beta, # Consumer's risk
distribution = "normal",
sigma_type = "known"
)
# Summary plan
summary(norm_plan)
# Probability of accepting 10% defective
accProb(norm_plan, 0.1)
# plot OC
plot(norm_plan)
4.1.2 Optimal Plan vs Custom Plan
# Setup a pd range to make sure all plans have use same pd range
pd <- seq(0, 0.2, by = 0.001)
# Generate OC curve data for designed plan
opt_pdata <- OCdata(norm_plan, pd = pd)
# Evaluated Plan 1: n + 6
eval1_pdata <- OCdata(n = norm_plan$n + 6, k = norm_plan$k,
distribution = "normal", pd = pd)
# Evaluated Plan 2: k + 0.1
eval2_pdata <- OCdata(n = norm_plan$n, k = norm_plan$k + 0.1,
distribution = "normal", pd = pd)
# Plot base
plot(100 * pd(opt_pdata), 100 * paccept(opt_pdata),
type = "l", lwd = 2, col = "blue",
xlab = "Percentage Nonconforming (%)",
ylab = "Probability of Acceptance (%)",
main = "Normal Variables Sampling - Designed Plan with Evaluated Plans")
# Add evaluated plan 1: n + 6
lines(100 * pd(eval1_pdata), 100 * paccept(eval1_pdata),
col = "red", lty = "longdash", lwd = 2)
# Add evaluated plan 2: k + 0.1
lines(100 * pd(eval2_pdata), 100 * paccept(eval2_pdata),
col = "forestgreen", lty = "dashed", lwd = 2)
# Add vertical dashed lines at PRQ and CRQ
abline(v = 100 * PRQ, col = "gray60", lty = "dashed")
abline(v = 100 * CRQ, col = "gray60", lty = "dashed")
# Add horizontal dashed lines at 1 - alpha and beta
abline(h = 100 * (1 - alpha), col = "gray60", lty = "dashed")
abline(h = 100 * beta, col = "gray60", lty = "dashed")
# Add legend
legend("topright",
legend = c(paste0("Designed Plan: n = ", norm_plan$sample_size, ", k = ", round(norm_plan$k, 2)),
"Evaluated Plan: n + 6",
"Evaluated Plan: k + 0.1"),
col = c("blue", "red", "forestgreen"),
lty = c("solid", "longdash", "dashed"),
lwd = 2,
bty = "n")
4.1.3 Compare known vs unknown sigma plans
p1 = 0.005
p2 = 0.03
alpha = 0.05
beta = 0.1
# known sigma plan
plan1 <- optPlan(
PRQ = p1, # Acceptable quality level (% nonconforming)
CRQ = p2, # Rejectable quality level (% nonconforming)
alpha = alpha, # Producer's risk
beta = beta, # Consumer's risk
distribution = "normal",
sigma_type = "know")
summary(plan1)
plot(plan1)
# unknown sigma plan
plan2 <- optPlan(
PRQ = p1, # Acceptable quality level (% nonconforming)
CRQ = p2, # Rejectable quality level (% nonconforming)
alpha = alpha, # Producer's risk
beta = beta, # Consumer's risk
distribution = "normal",
sigma_type = "unknow")
summary(plan2)
plot(plan2)
4.2 Beta Distribution
4.2.1 Find an Optimal Plan and Plot OC Chart
beta_plan <- optPlan(
PRQ = 0.05, # Target quality level (% nonconforming)
CRQ = 0.2, # Minimum quality level (% nonconforming)
alpha = 0.05, # Producer's risk
beta = 0.1, # Consumer's risk
distribution = "beta",
theta = 44000000,
theta_type = "known",
LSL = 0.00001
)
# Summary Beta plan
summary(beta_plan)
# Probability of accepting 5% defective
accProb(beta_plan, 0.05)
# Plot OC use plot function
plot(beta_plan)
4.2.2 Plot OC by Defective Rate and by The Mean
# plot use S3 method by default (defective rate)
plot(beta_plan)
# plot use S3 method by default by mean levels
plot(beta_plan, by = "mean")
5. Technical Specifications
5.1 Attribute Plan:
The Probability of Acceptance (( Pa )) is given by:
$$Pa(p) = \sum_{i=0}^c \binom{n}{i}p^i(1-p)^{n-i}$$
where:
- n is sample size
- c is acceptance number
- ( p ) is the quality level (non-conforming proportion)
5.2 Normal Variables Plan (Case of Known ( \sigma )):
The Probability of Acceptance (( Pa )) is given by:
[
Pa(p) = \Phi\left( -\sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right)
]
Or we could write:
[
Pa(p) = 1 - \Phi\left( \sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right)
]
where:
- ( \Phi(\cdot) ) is the CDF of the standard normal distribution.
- ( \Phi^{-1}(p) ) is the standard normal quantile corresponding to the quality level ( p ).
- ( n_{\sigma} ) is the sample size.
- ( k_{\sigma} ) is the acceptability constant.
The required sample size (( n_{\sigma} )) and acceptability constant (( k_{\sigma} )) are:
[
n_{\sigma} = \left( \frac{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)}{\Phi^{-1}(1 - PRQ) - \Phi^{-1}(1 - CRQ)} \right)^2
]
[
k_{\sigma} = \frac{\Phi^{-1}(1 - PRQ) \cdot \Phi^{-1}(1 - \beta) + \Phi^{-1}(1 - CRQ) \cdot \Phi^{-1}(1 - \alpha)}{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)}
]
where:
- ( \alpha ) and ( \beta ) are the producer's and consumer's risks, respectively.
- ( PRQ ) and ( CRQ ) are the Producer's Risk Quality and Consumer's Risk Quality, respectively.
5.3 Normal Variables Plan (Case of Unknown ( \sigma )):
The formula for the probability of acceptance (( Pa )) is:
[
Pa(p) = \Phi \left( \sqrt{\frac{n_s}{1 + \frac{k_s^2}{2}}} \left( \Phi^{-1}(1 - p) - k_s \right) \right)
]
where:
-
( k_s = k_{\sigma} ) is the acceptability constant.
-
( n_s ): This is the adjusted sample size when the sample standard deviation ( s ) (instead of population ( \sigma )) is used for estimation. It accounts for the additional variability due to estimation:
[
n_s = n_{\sigma} \times \left( 1 + \frac{k_s^2}{2} \right)
]
(See Wilrich, PT. (2004) for more detail about calculation used in sessions 6.2 and 6.3)
5.4 Beta Variables Plan (Case of Known ( \theta )):
Traditional acceptance sampling using normal distributions can be inadequate for compositional data bounded within [0,1]. Govindaraju and Kissling (2015) proposed Beta-based plans, where component proportions (e.g., protein content) follow ( X \sim \text{Beta}(a, b) ), with density:
[
f(x; a, b) = \frac{x^{a-1} (1 - x)^{b-1}}{B(a, b)},
]
where ( B(a, b) ) is the Beta function. The distribution is reparameterized via mean ( \mu ) and precision ( \theta ):
[
\mu = \frac{a}{a + b}, \quad \theta = a + b, \quad \sigma^2 \approx \frac{\mu(1 - \mu)}{\theta} \quad (\text{for large } \theta).
]
Higher ( \theta ) reduces variance, concentrating values around ( \mu ). The probability of acceptance (( Pa )) parallels normal-based plans:
[
Pa = P(\mu - k \sigma \geq L \mid \mu, \theta, m, k),
]
where ( L ) is the lower specification limit, ( m ) is the sample size, and ( k ) is the acceptability constant. Parameters ( m ) and ( k ) ensure:
[
Pa(\mu_{PRQ}) = 1 - \alpha, \quad Pa(\mu_{CRQ}) = \beta,
]
with ( \alpha ) (producer’s risk) and ( \beta ) (consumer’s risk) at specified quality levels (PRQ/CRQ).
Note that: this problem is solved to find ( m ) and ( k ) used Non-linear programming.
Implementation Note:
For a nonconforming proportion ( p ) (e.g.,PRQ or CRQ), the mean ( \mu ) at a quality level (PRQ/CRQ) is derived by solving:
[
P(X \leq L \mid \mu, \theta) = p,
]
where ( X \sim \text{Beta}(\theta \mu, \theta (1 - \mu)) ). This links ( \mu ) to ( p ) via the Beta cumulative distribution function (CDF) at ( L ).
5.5 Beta Variables Plans (Case of Unknown ( \theta )):
For a beta distribution, the required sample size (m_s) (unknown (\theta)) is derived from the known-(\theta) sample size (m_\theta) using the formula:
[
m_s = \left(1 + 0.85k^2\right)m_\theta
]
where (k) is unchanged. This adjustment accounts for the variance ratio (R = \frac{\text{Var}(S)}{\text{Var}(\hat{\mu})}), which quantifies the relative variability of the sample standard deviation (S) compared to the estimator (\hat{\mu}). Unlike the normal distribution, where (\text{Var}(S) \approx \frac{\sigma^2}{2n}), for beta distribution’s, the ratio (R) depends on (\mu), (\theta), and sample size (m). The conservative factor (0.85k^2) approximates this ratio for practical use (see Govindaraju and Kissling (2015) )
\newpage
6. References
- Schilling, E.G., & Neubauer, D.V. (2017). Acceptance Sampling in Quality Control (3rd ed.). Chapman and Hall/CRC. https://doi.org/10.4324/9781315120744
- Wilrich, PT. (2004). Single Sampling Plans for Inspection by Variables under a Variance Component Situation. In: Lenz, HJ., Wilrich, PT. (eds) Frontiers in Statistical Quality Control 7. Frontiers in Statistical Quality Control, vol 7. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2674-6_4
- K. Govindaraju and R. Kissling (2015). Sampling plans for Beta-distributed compositional fractions.
Quality Engineering, vol. 27, no. 1, pp. 1-13. https://doi.org/10.1016/j.chemolab.2015.12.009
- ISO 2859-1:1999 - Sampling procedures for inspection by attributes
- ISO 3951-1:2013 - Sampling procedures for inspection by variables
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 5 ) library(AccSamplingDesign)
1. Introduction
The AccSamplingDesign package provides tools for designing and evaluating Acceptance Sampling plans for quality control in manufacturing and inspection settings. It supports both attribute and variable sampling methods, applying nonlinear programming to minimize sample size while effectively controlling both producer’s and consumer’s risks.
Key features include:
- Attributes Sampling Plans — pass/fail decisions based on the proportion of nonconforming units.
- Variables Sampling Plans — support for normal and beta distributions, including compositional data.
- Operating Characteristic (OC) Curve Visualization — assess and compare plan performance.
- Risk-Based Optimization — minimize sample size while meeting Producer’s Risk (PR) and Consumer’s Risk (CR) conditions.
- Custom Plan Comparison — compare user-defined plans against optimized designs.
2. Installation
Install the stable release from CRAN:
install.packages("AccSamplingDesign")
Or install from GitHub
devtools::install_github("vietha/AccSamplingDesign")
Load package
library(AccSamplingDesign)
3. Attributes Sampling Plans
Note that we could use method optPlan() or optAttrPlan(), both work the same.
3.1 Create Attribute Plan
plan_attr <- optPlan( PRQ = 0.01, # Acceptable Quality Level (1% defects) CRQ = 0.05, # Rejectable Quality Level (5% defects) alpha = 0.02, # Producer's risk beta = 0.15, # Consumer's risk distribution = "binomial" )
3.2 Plan Summary
summary(plan_attr)
3.3 Acceptance Probability
# Probability of accepting 3% defective lots accProb(plan_attr, 0.03)
3.4 OC Curve
plot(plan_attr)
3.5 Compare Attributes Optimal Plan vs Custom Plan
# Step1: Find an optimal Attributes Sampling plan optimal_plan <- optPlan(PRQ = 0.01, CRQ = 0.05, alpha = 0.02, beta = 0.15, distribution = "binomial") # could try "poisson" too # Summarize the plan summary(optimal_plan) # Step2: Compare the optimal plan with two alternative plans pd <- seq(0, 0.15, by = 0.001) oc_opt <- OCdata(plan = optimal_plan, pd = pd) oc_alt1 <- OCdata(n = optimal_plan$n, c = optimal_plan$c - 1, distribution = "binomial", pd = pd) oc_alt2 <- OCdata(n = optimal_plan$n, c = optimal_plan$c + 1, distribution = "binomial", pd = pd) # Step3: Visualize results plot(pd, paccept(oc_opt), type = "l", col = "blue", lwd = 2, xlab = "Proportion Defective", ylab = "Probability of Acceptance", main = "Attributes Sampling - OC Curves Comparison", xlim = c(0, 0.15), ylim = c(0, 1)) lines(pd, paccept(oc_alt1), col = "red", lwd = 2, lty = 2) lines(pd, paccept(oc_alt2), col = "green", lwd = 2, lty = 3) abline(v = c(0.01, 0.05), col = "gray50", lty = 2) abline(h = c(1 - 0.02, 0.15), col = "gray50", lty = 2) legend("topright", legend = c(sprintf("Optimal Plan (n = %d, c = %d)", optimal_plan$n, optimal_plan$c), sprintf("Alt 1 (c = %d)", optimal_plan$c - 1), sprintf("Alt 2 (c = %d)", optimal_plan$c + 1)), col = c("blue", "red", "green"), lty = c(1, 2, 3), lwd = 2)
4. Variables Sampling Plans
Note that we could use method optPlan() or optVarPlan(), both work the same.
4.1 Normal Distribution
4.1.1 Find an optimal plan and plot OC chart
# Predefine parameters PRQ <- 0.025 CRQ <- 0.1 alpha <- 0.05 beta <- 0.1 norm_plan <- optPlan( PRQ = PRQ, # Acceptable quality level (% nonconforming) CRQ = CRQ, # Rejectable quality level (% nonconforming) alpha = alpha, # Producer's risk beta = beta, # Consumer's risk distribution = "normal", sigma_type = "known" ) # Summary plan summary(norm_plan) # Probability of accepting 10% defective accProb(norm_plan, 0.1) # plot OC plot(norm_plan)
4.1.2 Optimal Plan vs Custom Plan
# Setup a pd range to make sure all plans have use same pd range pd <- seq(0, 0.2, by = 0.001) # Generate OC curve data for designed plan opt_pdata <- OCdata(norm_plan, pd = pd) # Evaluated Plan 1: n + 6 eval1_pdata <- OCdata(n = norm_plan$n + 6, k = norm_plan$k, distribution = "normal", pd = pd) # Evaluated Plan 2: k + 0.1 eval2_pdata <- OCdata(n = norm_plan$n, k = norm_plan$k + 0.1, distribution = "normal", pd = pd) # Plot base plot(100 * pd(opt_pdata), 100 * paccept(opt_pdata), type = "l", lwd = 2, col = "blue", xlab = "Percentage Nonconforming (%)", ylab = "Probability of Acceptance (%)", main = "Normal Variables Sampling - Designed Plan with Evaluated Plans") # Add evaluated plan 1: n + 6 lines(100 * pd(eval1_pdata), 100 * paccept(eval1_pdata), col = "red", lty = "longdash", lwd = 2) # Add evaluated plan 2: k + 0.1 lines(100 * pd(eval2_pdata), 100 * paccept(eval2_pdata), col = "forestgreen", lty = "dashed", lwd = 2) # Add vertical dashed lines at PRQ and CRQ abline(v = 100 * PRQ, col = "gray60", lty = "dashed") abline(v = 100 * CRQ, col = "gray60", lty = "dashed") # Add horizontal dashed lines at 1 - alpha and beta abline(h = 100 * (1 - alpha), col = "gray60", lty = "dashed") abline(h = 100 * beta, col = "gray60", lty = "dashed") # Add legend legend("topright", legend = c(paste0("Designed Plan: n = ", norm_plan$sample_size, ", k = ", round(norm_plan$k, 2)), "Evaluated Plan: n + 6", "Evaluated Plan: k + 0.1"), col = c("blue", "red", "forestgreen"), lty = c("solid", "longdash", "dashed"), lwd = 2, bty = "n")
4.1.3 Compare known vs unknown sigma plans
p1 = 0.005 p2 = 0.03 alpha = 0.05 beta = 0.1 # known sigma plan plan1 <- optPlan( PRQ = p1, # Acceptable quality level (% nonconforming) CRQ = p2, # Rejectable quality level (% nonconforming) alpha = alpha, # Producer's risk beta = beta, # Consumer's risk distribution = "normal", sigma_type = "know") summary(plan1) plot(plan1) # unknown sigma plan plan2 <- optPlan( PRQ = p1, # Acceptable quality level (% nonconforming) CRQ = p2, # Rejectable quality level (% nonconforming) alpha = alpha, # Producer's risk beta = beta, # Consumer's risk distribution = "normal", sigma_type = "unknow") summary(plan2) plot(plan2)
4.2 Beta Distribution
4.2.1 Find an Optimal Plan and Plot OC Chart
beta_plan <- optPlan( PRQ = 0.05, # Target quality level (% nonconforming) CRQ = 0.2, # Minimum quality level (% nonconforming) alpha = 0.05, # Producer's risk beta = 0.1, # Consumer's risk distribution = "beta", theta = 44000000, theta_type = "known", LSL = 0.00001 ) # Summary Beta plan summary(beta_plan) # Probability of accepting 5% defective accProb(beta_plan, 0.05) # Plot OC use plot function plot(beta_plan)
4.2.2 Plot OC by Defective Rate and by The Mean
# plot use S3 method by default (defective rate) plot(beta_plan) # plot use S3 method by default by mean levels plot(beta_plan, by = "mean")
5. Technical Specifications
5.1 Attribute Plan:
The Probability of Acceptance (( Pa )) is given by:
$$Pa(p) = \sum_{i=0}^c \binom{n}{i}p^i(1-p)^{n-i}$$
where:
- n is sample size
- c is acceptance number
- ( p ) is the quality level (non-conforming proportion)
5.2 Normal Variables Plan (Case of Known ( \sigma )):
The Probability of Acceptance (( Pa )) is given by:
[ Pa(p) = \Phi\left( -\sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right) ]
Or we could write:
[ Pa(p) = 1 - \Phi\left( \sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right) ]
where:
- ( \Phi(\cdot) ) is the CDF of the standard normal distribution.
- ( \Phi^{-1}(p) ) is the standard normal quantile corresponding to the quality level ( p ).
- ( n_{\sigma} ) is the sample size.
- ( k_{\sigma} ) is the acceptability constant.
The required sample size (( n_{\sigma} )) and acceptability constant (( k_{\sigma} )) are: [ n_{\sigma} = \left( \frac{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)}{\Phi^{-1}(1 - PRQ) - \Phi^{-1}(1 - CRQ)} \right)^2 ]
[ k_{\sigma} = \frac{\Phi^{-1}(1 - PRQ) \cdot \Phi^{-1}(1 - \beta) + \Phi^{-1}(1 - CRQ) \cdot \Phi^{-1}(1 - \alpha)}{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)} ] where:
- ( \alpha ) and ( \beta ) are the producer's and consumer's risks, respectively.
- ( PRQ ) and ( CRQ ) are the Producer's Risk Quality and Consumer's Risk Quality, respectively.
5.3 Normal Variables Plan (Case of Unknown ( \sigma )):
The formula for the probability of acceptance (( Pa )) is:
[ Pa(p) = \Phi \left( \sqrt{\frac{n_s}{1 + \frac{k_s^2}{2}}} \left( \Phi^{-1}(1 - p) - k_s \right) \right) ]
where:
-
( k_s = k_{\sigma} ) is the acceptability constant.
-
( n_s ): This is the adjusted sample size when the sample standard deviation ( s ) (instead of population ( \sigma )) is used for estimation. It accounts for the additional variability due to estimation:
[ n_s = n_{\sigma} \times \left( 1 + \frac{k_s^2}{2} \right) ]
(See Wilrich, PT. (2004) for more detail about calculation used in sessions 6.2 and 6.3)
5.4 Beta Variables Plan (Case of Known ( \theta )):
Traditional acceptance sampling using normal distributions can be inadequate for compositional data bounded within [0,1]. Govindaraju and Kissling (2015) proposed Beta-based plans, where component proportions (e.g., protein content) follow ( X \sim \text{Beta}(a, b) ), with density:
[ f(x; a, b) = \frac{x^{a-1} (1 - x)^{b-1}}{B(a, b)}, ]
where ( B(a, b) ) is the Beta function. The distribution is reparameterized via mean ( \mu ) and precision ( \theta ):
[ \mu = \frac{a}{a + b}, \quad \theta = a + b, \quad \sigma^2 \approx \frac{\mu(1 - \mu)}{\theta} \quad (\text{for large } \theta). ]
Higher ( \theta ) reduces variance, concentrating values around ( \mu ). The probability of acceptance (( Pa )) parallels normal-based plans:
[ Pa = P(\mu - k \sigma \geq L \mid \mu, \theta, m, k), ]
where ( L ) is the lower specification limit, ( m ) is the sample size, and ( k ) is the acceptability constant. Parameters ( m ) and ( k ) ensure:
[ Pa(\mu_{PRQ}) = 1 - \alpha, \quad Pa(\mu_{CRQ}) = \beta, ]
with ( \alpha ) (producer’s risk) and ( \beta ) (consumer’s risk) at specified quality levels (PRQ/CRQ).
Note that: this problem is solved to find ( m ) and ( k ) used Non-linear programming.
Implementation Note:
For a nonconforming proportion ( p ) (e.g.,PRQ or CRQ), the mean ( \mu ) at a quality level (PRQ/CRQ) is derived by solving:
[ P(X \leq L \mid \mu, \theta) = p, ]
where ( X \sim \text{Beta}(\theta \mu, \theta (1 - \mu)) ). This links ( \mu ) to ( p ) via the Beta cumulative distribution function (CDF) at ( L ).
5.5 Beta Variables Plans (Case of Unknown ( \theta )):
For a beta distribution, the required sample size (m_s) (unknown (\theta)) is derived from the known-(\theta) sample size (m_\theta) using the formula:
[
m_s = \left(1 + 0.85k^2\right)m_\theta
]
where (k) is unchanged. This adjustment accounts for the variance ratio (R = \frac{\text{Var}(S)}{\text{Var}(\hat{\mu})}), which quantifies the relative variability of the sample standard deviation (S) compared to the estimator (\hat{\mu}). Unlike the normal distribution, where (\text{Var}(S) \approx \frac{\sigma^2}{2n}), for beta distribution’s, the ratio (R) depends on (\mu), (\theta), and sample size (m). The conservative factor (0.85k^2) approximates this ratio for practical use (see Govindaraju and Kissling (2015) )
\newpage
6. References
- Schilling, E.G., & Neubauer, D.V. (2017). Acceptance Sampling in Quality Control (3rd ed.). Chapman and Hall/CRC. https://doi.org/10.4324/9781315120744
- Wilrich, PT. (2004). Single Sampling Plans for Inspection by Variables under a Variance Component Situation. In: Lenz, HJ., Wilrich, PT. (eds) Frontiers in Statistical Quality Control 7. Frontiers in Statistical Quality Control, vol 7. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2674-6_4
- K. Govindaraju and R. Kissling (2015). Sampling plans for Beta-distributed compositional fractions. Quality Engineering, vol. 27, no. 1, pp. 1-13. https://doi.org/10.1016/j.chemolab.2015.12.009
- ISO 2859-1:1999 - Sampling procedures for inspection by attributes
- ISO 3951-1:2013 - Sampling procedures for inspection by variables
Try the AccSamplingDesign package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
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