---
title: "Reliability Demonstration Test Planning"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Reliability Demonstration Test Planning}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

```{r setup}
library(ReliaGrowR)
```

## Overview

A **Reliability Demonstration Test (RDT)** provides statistical evidence that a
product meets its reliability requirement. The `rdt()` function computes either:

- the **required test time** given a fixed sample size, or
- the **required sample size** given a fixed test duration.

The calculation uses the Weibull distribution; setting `beta = 1` (the default)
gives the exponential special case.

## Test Time

For a zero-failure test plan (`f = 0`), the required cumulative test time $T$
for $n$ units satisfying reliability $R$ at mission time $t_m$ with confidence
$C$ is:

$$T = t_m \left(\frac{-\ln C}{-\ln R}\right)^{1/\beta}$$

When allowable failures $f > 0$, the chi-squared quantile replaces $-\ln C$:

$$T = \frac{t_m}{(-\ln R)^{1/\beta}} \cdot \left(\frac{\chi^2_{1-C,\,2(f+1)}}{2n}\right)^{1/\beta}$$

## Solving for Test Time

Suppose you need to demonstrate 90% reliability at 500 hours with 90%
confidence, and you have 20 units available for testing:

```{r solve-time}
plan <- rdt(
  target       = 0.90,   # 90% reliability
  mission_time = 500,    # hours
  conf_level   = 0.90,   # 90% confidence
  n            = 20      # 20 test units
)
print(plan)
```

Each unit must be tested for `Required_Test_Time` hours (or failure, whichever
comes first) with zero failures allowed.

## Solving for Sample Size

Alternatively, fix the test duration at 800 hours per unit:

```{r solve-n}
plan2 <- rdt(
  target       = 0.90,
  mission_time = 500,
  conf_level   = 0.90,
  test_time    = 800
)
print(plan2)
```

## Effect of Weibull Shape

For products with wear-out (`beta > 1`) or infant mortality (`beta < 1`), the
Weibull shape parameter changes the required test time. Here we compare three
scenarios:

```{r weibull-shape}
betas <- c(0.8, 1.0, 1.5)
for (b in betas) {
  p <- rdt(target = 0.90, mission_time = 500, conf_level = 0.90, n = 20, beta = b)
  cat(sprintf("beta = %.1f  ->  required test time = %.1f hours\n",
              b, p$Required_Test_Time))
}
```

Higher beta (wear-out) demands shorter tests because failures concentrate near
end-of-life; lower beta (infant mortality) demands longer tests.

## Allowing Failures

A zero-failure plan is conservative. Allowing a small number of failures can
substantially reduce the test burden:

```{r allow-failures}
for (f in 0:3) {
  p <- rdt(target = 0.90, mission_time = 500, conf_level = 0.90, n = 20, f = f)
  cat(sprintf("f = %d  ->  required test time = %.1f hours\n",
              f, p$Required_Test_Time))
}
```

## Summary

| Parameter | Role |
|---|---|
| `target` | Required reliability R(t_m) |
| `mission_time` | The time at which reliability is evaluated |
| `conf_level` | Statistical confidence in the demonstration |
| `beta` | Weibull shape (1 = exponential) |
| `f` | Allowable failures (0 = most conservative) |
| `n` | Sample size → solves for test time |
| `test_time` | Test duration per unit → solves for sample size |