---
title: "AccSamplingDesign: Acceptance Sampling Plan Design - R Package"
author: "Ha Truong"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{AccSamplingDesign: Acceptance Sampling Plan Design - R Package}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)
library(AccSamplingDesign)
```

# 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** — determine sampling plans that meet target Producer’s Risk (PR) and Consumer’s Risk (CR).
- **Custom Plan Comparison** — compare user-defined plans against optimized designs.

# 2. Installation

Install the stable release from **CRAN**:

```r
install.packages("AccSamplingDesign")
```

Or install from GitHub
```{r eval=FALSE}
devtools::install_github("vietha/AccSamplingDesign")
```
Load package
```r
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
)
```

## 3.2 Plan Summary
```{r}
summary(plan_attr)
```

## 3.3 Acceptance Probability
```{r}
# Probability of accepting 3% defective lots
accProb(plan_attr, 0.03)
```

## 3.4 OC Curve
```{r}
plot(plan_attr)
```

# 4. Variables Sampling Plans

## 4.1 Normal Distribution 
### 4.1.1 Find an optimal plan and plot OC chart
```{r}
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)
```

### 4.1.2 Compare known vs unknown sigma plans
```{r}
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()
```

## 4.2 Beta Distribution

### 4.2.1 Find an optimal plan and plot OC chart
```{r}
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()
```

### 4.2.2 Compare known vs unknown theta plans
```{r}
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()
```

## 4.3 Variables Plan Acceptance Probability
```{r}
# Probability of accepting 10% defective
accProb(norm_plan, 0.1)

# Probability of accepting 5% defective
accProb(beta_plan, 0.05)
```

# 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 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 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. <br>
- \( PRQ \) and \( CRQ \) are the Producer's Risk Quality and Consumer's Risk Quality, respectively.


## 6.3 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 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 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 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 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:  
\[
m_s = \left(1 + 0.5k^2\right)m_\theta
\]  
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.

\newpage

# 7. References
1. Schilling, E.G., & Neubauer, D.V. (2017). Acceptance Sampling in Quality Control (3rd ed.). Chapman and Hall/CRC. https://doi.org/10.4324/9781315120744
2. 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
3. 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
4. ISO 2859-1:1999 - Sampling procedures for inspection by attributes  
5. ISO 3951-1:2013 - Sampling procedures for inspection by variables  


