An R package for designing and analyzing acceptance sampling plans. This package is now available on CRAN! 🎉
The AccSamplingDesign package provides tools for designing Acceptance Sampling plans for both attributes and variables data. Key features include:
# 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, oc_opt@paccept, 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, oc_alt1@paccept, col = "red", lwd = 2, lty = 2)
lines(pd, oc_alt2@paccept, 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 * opt_pdata@pd, 100 * opt_pdata@paccept,
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 * eval1_pdata@pd, 100 * eval1_pdata@paccept,
col = "red", lty = "longdash", lwd = 2)
# Add evaluated plan 2: k + 0.1
lines(100 * eval2_pdata@pd, 100 * eval2_pdata@paccept,
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)# Generate OC data
p_seq <- seq(0.005, 0.5, by = 0.005)
oc_data <- OCdata(beta_plan, pd = p_seq)
# 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 \(m\).