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AccSamplingDesign: Acceptance Sampling Plan Design - R Package

Ha Truong

2025-05-25

1. Introduction

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.

Key features include:

Attributes Sampling Plans — pass/fail decisions based on the proportion of nonconforming units.
Variables Sampling Plans — support for normal and beta distributions, including compositional data.
Operating Characteristic (OC) Curve Visualization — assess and compare plan performance.
Risk-Based Optimization — minimize sample size while meeting Producer’s Risk (PR) and Consumer’s Risk (CR) conditions.
Custom Plan Comparison — compare user-defined plans against optimized designs.

2. Installation

Install the stable release from CRAN:

install.packages("AccSamplingDesign")

Or install from GitHub

devtools::install_github("vietha/AccSamplingDesign")

Load package

library(AccSamplingDesign)

3. Attributes Sampling Plans

Note that we could use method optPlan() or optAttrPlan(), both work the same.

3.1 Create Attribute Plan

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"
)

3.2 Plan Summary

summary(plan_attr)
#> Attributes Acceptance Sampling Plan
#> -----------------------------------
#> Distribution: binomial 
#> Sample Size (n): 144 
#> Acceptance Number (c): 4 
#> Producer's Risk (PR = 0.01534843 ) at PRQ = 0.01 
#> Consumer's Risk (CR = 0.1487162 ) at CRQ = 0.05 
#> ----------------------------------

3.3 Acceptance Probability

# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)
#> [1] 0.5656415

3.4 OC Curve

plot(plan_attr)

3.5 Compare Attributes Optimal Plan vs Custom Plan

# 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)
#> Attributes Acceptance Sampling Plan
#> -----------------------------------
#> Distribution: binomial 
#> Sample Size (n): 144 
#> Acceptance Number (c): 4 
#> Producer's Risk (PR = 0.01534843 ) at PRQ = 0.01 
#> Consumer's Risk (CR = 0.1487162 ) at CRQ = 0.05 
#> ----------------------------------

# 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)

4. Variables Sampling Plans

Note that we could use method optPlan() or optVarPlan(), both work the same.

4.1 Normal Distribution

4.1.1 Find an optimal plan and plot OC chart

# 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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal 
#> Sample Size (n): 19 
#> Acceptability Constant (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 
#> ----------------------------------

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

# plot OC 
plot(norm_plan)

4.1.2 Optimal Plan vs Custom 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")

4.1.3 Compare known vs unknown sigma plans

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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal 
#> Sample Size (n): 18 
#> Acceptability Constant (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 <- 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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: normal 
#> Sample Size (n): 62 
#> Acceptability Constant (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)

4.2 Beta Distribution

4.2.1 Find an Optimal Plan and Plot OC Chart

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)
#> Variables Acceptance Sampling Plan
#> ----------------------------------
#> Distribution: beta 
#> Sample Size (n): 14 
#> Acceptability Constant (k): 1.186 
#> Population Precision Parameter (theta): known 
#> Producer's Risk (PR = 0.04999589 ) at PRQ = 0.05 
#> Consumer's Risk (CR = 0.09947491 ) at CRQ = 0.2 
#> Lower Specification Limit (LSL): 1e-05 
#> ----------------------------------
# Probability of accepting 5% defective
accProb(beta_plan, 0.05)
#> [1] 0.9500041

# Plot OC use plot function
plot(beta_plan)

4.2.2 Plot OC by Defective Rate and by The Mean

# 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")

5. Technical Specifications

5.1 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)

5.2 Normal Variables 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 acceptability constant.

The required sample size (\(n_{\sigma}\)) and acceptability 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.

5.3 Normal Variables 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 acceptability 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)

5.4 Beta Variables 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.

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 \(L\).

5.5 Beta Variables Plans (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:
\[ m_s = \left(1 + 0.85k^2\right)m_\theta \]
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 \(0.85k^2\) approximates this ratio for practical use (see Govindaraju and Kissling (2015) )

6. 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

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