An R package for designing and analyzing acceptance sampling plans.
The AccSamplingDesign package provides tools for designing acceptance sampling plans for both attribute and variable data. Key features include:
```{r eval=FALSE} # Install from GitHub devtools::install_github(“vietha/AccSamplingDesign”)
library(AccSamplingDesign)
# 3. Attribute Sampling Plans
## 3.1 Create Attribute Plan
```{r}
plan_attr <- optAttrPlan(
PRQ = 0.01, # Acceptable Quality Level (1% defects)
CRQ = 0.05, # Rejectable Quality Level (5% defects)
alpha = 0.05, # Producer's risk
beta = 0.10 # Consumer's risk
)summary(plan_attr)# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)plot(plan_attr)norm_plan <- optVarPlan(
PRQ = 0.025, # Acceptable quality level (% nonconforming)
CRQ = 0.1, # Rejectable quality level (% nonconforming)
alpha = 0.05, # Producer's risk
beta = 0.1, # Consumer's risk
distribution = "normal",
sigma_type = "known"
)
summary(norm_plan)
# Generate OC curve data
oc_data_normal <- OCdata(norm_plan)
# show data of Proportion Nonconforming (x_p) vs Probability Acceptance (y)
#head(oc_data_normal, 15)
plot(norm_plan)p1 = 0.005
p2 = 0.03
alpha = 0.05
beta = 0.1
# known sigma plan
plan1 <- optVarPlan(
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 <- optVarPlan(
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)
# Generate OC curve data
oc_data1 <- OCdata(plan1)
oc_data2 <- OCdata(plan2)
# Plot the first OC curve (solid red line)
plot(oc_data1@pd, oc_data1@paccept, type = "l", col = "red", lwd = 2,
main = "Operating Characteristic (OC) Curve",
xlab = "Proportion Nonconforming",
ylab = "P(accept)")
# Add the second OC curve (dashed blue line)
lines(oc_data2@pd, oc_data2@paccept, col = "blue", lwd = 2, lty = 2)
abline(v = c(p1, p2), lty = 1, col = "gray")
abline(h = c(1 - alpha, beta), lty = 1, col = "gray")
legend1 = paste0("Known Sigma (n=", plan1$sample_size, ", k=", plan1$k, ")")
legend2 = paste0("Unknown Sigma (n=", plan2$sample_size, ", k=", plan2$k, ")")
# Add a legend to distinguish the two curves
legend("topright", legend = c(legend1, legend2),
col = c("red", "blue"), lwd = 2, lty = c(1, 2))
# Add a grid
grid()beta_plan <- optVarPlan(
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)
# 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)
#head(oc_data)
# plot use S3 method
plot(oc_data)
# Plot the OC curve with Mean Level (x_m) and Probability of Acceptance (y)
plot(oc_data@pd, oc_data@paccept, type = "l", col = "blue", lwd = 2,
main = "OC Curve by the mean levels (plot by data)", xlab = "Mean Levels",
ylab = "P(accept)")
grid()p1 = 0.005
p2 = 0.03
alpha = 0.05
beta = 0.1
spec_limit = 0.05 # use for Beta distribution
theta = 500
# My package for beta plan
beta_plan1 <- optVarPlan(
PRQ = p1, # Target quality level (% nonconforming)
CRQ = p2, # Minimum quality level (% nonconforming)
alpha = alpha, # Producer's risk
beta = beta, # Consumer's risk
distribution = "beta",
theta = theta,
theta_type = "known",
USL = spec_limit
)
summary(beta_plan1)
beta_plan2 <- optVarPlan(
PRQ = p1, # Target quality level (% nonconforming)
CRQ = p2, # Minimum quality level (% nonconforming)
alpha = alpha, # Producer's risk
beta = beta, # Consumer's risk
distribution = "beta",
theta = theta,
theta_type = "unknown",
USL = spec_limit
)
summary(beta_plan2)
# Generate OC curve data
oc_beta_data1 <- OCdata(beta_plan1)
oc_beta_data2 <- OCdata(beta_plan2)
# Plot the first OC curve (solid red line)
plot(oc_beta_data1@pd, oc_beta_data1@paccept, type = "l", col = "red", lwd = 2,
main = "Operating Characteristic (OC) Curve",
xlab = "Proportion Nonconforming",
ylab = "P(accept)")
# Add the second OC curve (dashed blue line)
lines(oc_beta_data2@pd, oc_beta_data2@paccept, col = "blue", lwd = 2, lty = 2)
abline(v = c(p1, p2), lty = 1, col = "gray")
abline(h = c(1 - alpha, beta), lty = 1, col = "gray")
legend1 = paste0("Known Theta (n=", beta_plan1$sample_size, ", k=", beta_plan1$k, ")")
legend2 = paste0("Unknown Theta (n=", beta_plan2$sample_size, ", k=", beta_plan2$k, ")")
# Add a legend to distinguish the two curves
legend("topright", legend = c(legend1, legend2),
col = c("red", "blue"), lwd = 2, lty = c(1, 2))
# Add a grid
grid()# Probability of accepting 10% defective
accProb(norm_plan, 0.1)
# Probability of accepting 5% defective
accProb(beta_plan, 0.05)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 acceptance constant.
Sample size and acceptance 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 acceptance 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 and Monte Carlo simulation.
For unknown \(\theta\), sample size is adjusted:
\[ m_s = \left(1 + 0.5k^2\right)m_\theta \]
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 \(m\).