An R package for designing and analyzing acceptance sampling plans. This package is now available on CRAN! 🎉
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.
# Install from CRAN
R> install.packages("AccSamplingDesign")# Install from GitHub
R> devtools::install_github("vietha/AccSamplingDesign")# Load package
R> library(AccSamplingDesign)Note that we could use method optPlan() or optAttrPlan(), both work the same.
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"
)summary(plan_attr)# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)plot(plan_attr)# 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)Note that we could use method optPlan() or optVarPlan(), both work the same.
# 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)# 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")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)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)# plot use S3 method by default (defective rate)
plot(oc_data)
# plot use S3 method by default by mean levels
plot(oc_data, by = "mean")The Probability of Acceptance (Pa) is:
\[ Pa(p) = \sum_{i=0}^c \binom{n}{i}p^i(1-p)^{n-i} \]
where: - \(n\) is sample size - \(c\) is acceptance number - \(p\) is the quality level (non-conforming proportion)
The Probability of Acceptance (Pa) is:
\[ Pa(p) = \Phi\left( -\sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right) \]
or:
\[ Pa(p) = 1 - \Phi\left( \sqrt{n_{\sigma}} \cdot (\Phi^{-1}(p) + k_{\sigma}) \right) \]
where: - \(\Phi(\cdot)\) is the CDF of the standard normal distribution. - \(\Phi^{-1}(p)\) is the standard normal quantile corresponding to \(p\). - \(n_{\sigma}\) is the sample size. - \(k_{\sigma}\) is the acceptability constant.
Sample size and acceptability constant:
\[ n_{\sigma} = \left( \frac{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)}{\Phi^{-1}(1 - PRQ) - \Phi^{-1}(1 - CRQ)} \right)^2 \]
\[ k_{\sigma} = \frac{\Phi^{-1}(1 - PRQ) \cdot \Phi^{-1}(1 - \beta) + \Phi^{-1}(1 - CRQ) \cdot \Phi^{-1}(1 - \alpha)}{\Phi^{-1}(1 - \alpha) + \Phi^{-1}(1 - \beta)} \]
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.
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. - \(n_s\) is the adjusted sample size:
\[ n_s = n_{\sigma} \times \left( 1 + \frac{k_s^2}{2} \right) \]
(Reference: Wilrich, P.T. (2004))
For Beta distributed data:
\[ f(x; a, b) = \frac{x^{a-1} (1 - x)^{b-1}}{B(a, b)} \]
where \(B(a, b)\) is the Beta function.
Reparameterized as:
\[ \mu = \frac{a}{a + b}, \quad \theta = a + b, \quad \sigma^2 \approx \frac{\mu(1 - \mu)}{\theta} \quad (\text{for large } \theta) \]
Probability of acceptance:
\[ Pa = P(\mu - k \sigma \geq L \mid \mu, \theta, m, k) \]
where: - \(L\) = lower specification limit - \(m\) = sample size - \(k\) = acceptability constant
Parameters \(m\) and \(k\) are found to satisfy:
\[ Pa(\mu_{PRQ}) = 1 - \alpha, \quad Pa(\mu_{CRQ}) = \beta \]
Implementation Note:
For a nonconforming proportion \(p\)
(e.g., PRQ or CRQ), the mean \(\mu\) is
derived by solving:
\[ P(X \leq L \mid \mu, \theta) = p \]
where \(X \sim \text{Beta}(\theta \mu, \theta (1-\mu))\).
Problem is solved using Non-linear programming.
For unknown \(\theta\), sample size is adjusted:
\[ m_s = \left(1 + 0.85k^2\right)m_\theta \]
where: - \(k\) remains the same.
This adjustment considers the variance ratio:
\[ R = \frac{\text{Var}(S)}{\text{Var}(\hat{\mu})} \]
Unlike the normal distribution where \(\text{Var}(S) \approx \frac{\sigma^2}{2n}\), in the Beta case, \(R\) depends on \(\mu\), \(\theta\), and sample size \(m\).