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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 \(m\).
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.