The AccSamplingDesign package provides tools for designing and evaluating Acceptance Sampling plans for quality control in manufacturing and inspection settings. It supports both attributes and variables sampling methods with a focus on minimizing producer’s and consumer’s risks.
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Load package
summary(plan_attr)
#> Attributes Acceptance Sampling Plan
#> -----------------------------------
#> Distribution: binomial
#> Sample Size (n): 132
#> Acceptance Number (c): 3
#> Producer's Risk (PR = 0.04425251 ) at PRQ = 0.01
#> Consumer's Risk (CR = 0.0992283 ) at CRQ = 0.05
#> ----------------------------------
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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal
#> Sample Size (n): 19
#> Acceptance Limit (k): 1.579
#> Population Standard Deviation: known
#> Producer's Risk (PR = 0.05 ) at PRQ = 0.025
#> Consumer's Risk (CR = 0.1 ) at CRQ = 0.1
#> ----------------------------------
# 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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal
#> Sample Size (n): 18
#> Acceptance Limit (k): 2.185
#> Population Standard Deviation: known
#> Producer's Risk (PR = 0.05 ) at PRQ = 0.005
#> Consumer's Risk (CR = 0.1 ) at CRQ = 0.03
#> ----------------------------------
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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal
#> Sample Size (n): 62
#> Acceptance Limit (k): 2.192
#> Population Standard Deviation: unknown
#> Producer's Risk (PR = 0.05 ) at PRQ = 0.005
#> Consumer's Risk (CR = 0.1 ) at CRQ = 0.03
#> ----------------------------------
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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: beta
#> Sample Size (n): 14
#> Acceptance Limit (k): 1.186
#> Population Precision Parameter (theta): known
#> Producer's Risk (PR = 0.04999926 ) at PRQ = 0.05
#> Consumer's Risk (CR = 0.09976476 ) at CRQ = 0.2
#> Lower Specification Limit: 1e-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)
#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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: beta
#> Sample Size (n): 16
#> Acceptance Limit (k): 2.484
#> Population Precision Parameter (theta): known
#> Producer's Risk (PR = 0.04999959 ) at PRQ = 0.005
#> Consumer's Risk (CR = 0.09980403 ) at CRQ = 0.03
#> Uper Specification Limit: 0.05
#> ----------------------------------
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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: beta
#> Sample Size (n): 66
#> Acceptance Limit (k): 2.484
#> Population Precision Parameter (theta): unknown
#> Producer's Risk (PR = 0.04653701 ) at PRQ = 0.005
#> Consumer's Risk (CR = 0.09489874 ) at CRQ = 0.03
#> Uper Specification Limit: 0.05
#> ----------------------------------
# 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()
The Probability of Acceptance (\(Pa\)) is given by: \[Pa(p) = \sum_{i=0}^c
\binom{n}{i}p^i(1-p)^{n-i}\]
where:
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:
The required sample size (\(n_{\sigma}\)) and acceptance 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:
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.
\(n_s\): 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)
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 and Monte Carlo simulation.
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 \(L\).
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:
\[
m_s = \left(1 + 0.5k^2\right)m_\theta
\]
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 \(0.5k^2\) approximates this ratio for
practical use.