1. Introduction

The AccSamplingDesign package provides tools for designing acceptance sampling plans for both attribute and variable data. Key features include:

Attribute sampling plans (pass/fail inspection based on nonconforming proportion)
Variable sampling plans for normal and beta distributions, focusing on the proportion of nonconforming units
Measurement error adjustments to account for variability in the data
OC curve visualization to evaluate sampling plan performance
Risk-based optimization to minimize producer’s and consumer’s risks while meeting specified quality levels of Producer’s Risk Quality (PRQ) and Consumer’s Risk Quality (CRQ)

2. Installation

# Install from GitHub
devtools::install_github("vietha/AccSamplingDesign")

# Load package
library(AccSamplingDesign)

3. Attribute Sampling Plans

3.1 Create Attribute Plan

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
)

3.2 Plan Summary

summary(plan_attr)
#> Attribute 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 
#> ---------------------------------

3.3 Acceptance Probability

# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)
#> [1] 0.4384354

3.4 OC Curve

plot(plan_attr)

4. Variable Sampling Plans

4.1 Normal Distribution

4.1.1 Find an optimal plan and plot OC chart

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)
#> Variable 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)

4.1.2 Compare known vs unknown sigma plans

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)
#> Variable 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)
#> Variable 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()

4.2 Beta Distribution

4.2.1 Find an optimal plan and plot OC chart

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)
#> Variable 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()

4.2.2 Compare known vs unknown theta plans

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)
#> Variable 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)
#> Variable 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()

4.3 Variable Plan Acceptance Probability

# Probability of accepting 10% defective
accProb(norm_plan, 0.1)
#> [1] 0.0997

# Probability of accepting 5% defective
accProb(beta_plan, 0.05)
#> [1] 0.9498741

6. Technical Specifications

6.1 Attribute Plan:

The Probability of Acceptance (\(Pa\)) is given by: \[Pa(p) = \sum_{i=0}^c \binom{n}{i}p^i(1-p)^{n-i}\]
where:

n is sample size
c is acceptance number
\(p\) is the quality level (non-conforming proportion)

6.2 Normal Variable Plan (Case of Known \(\sigma\)):

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:

\(\Phi(\cdot)\) is the CDF of the standard normal distribution.
\(\Phi^{-1}(p)\) is the standard normal quantile corresponding to the quality level \(p\).
\(n_{\sigma}\) is the sample size.
\(k_{\sigma}\) is the acceptance constant.

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:

\(\alpha\) 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, respectively.

6.3 Normal Variable Plan (Case of Unknown \(\sigma\)):

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)

6.4 Beta Variable Plan (Case of Known \(\theta\)):

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.

Implementation Note:

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 \(L\).

6.5 Beta Variable Plan (Case of Unknown \(\theta\)):

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.

7. References

Schilling, E.G., & Neubauer, D.V. (2017). Acceptance Sampling in Quality Control (3rd ed.). Chapman and Hall/CRC. https://doi.org/10.4324/9781315120744
Wilrich, PT. (2004). Single Sampling Plans for Inspection by Variables under a Variance Component Situation. In: Lenz, HJ., Wilrich, PT. (eds) Frontiers in Statistical Quality Control 7. Frontiers in Statistical Quality Control, vol 7. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2674-6_4
K. Govindaraju and R. Kissling (2015). Sampling plans for Beta-distributed compositional fractions. Quality Engineering, vol. 27, no. 1, pp. 1-13. https://doi.org/10.1016/j.chemolab.2015.12.009
ISO 2859-1:1999 - Sampling procedures for inspection by attributes
ISO 3951-1:2013 - Sampling procedures for inspection by variables

AccSamplingDesign: Acceptance Sampling Plan Design - R Package

Ha Truong

2025-04-29