# Multi-Phase - Mixed Data

*This example appears in the Reliability Growth and Repairable System Analysis Reference*.

A one-shot system underwent reliability growth development for a total of 20 trials. The test was performed as a combination of configuration in groups and individual trial-by-trial. The following table shows the obtained data set. The **Failures in Interval** column specifies the number of failures that occurred in each interval, and the **Cumulative Trials** column specifies the cumulative number of trials at the end of that interval.

Mixed Data
| ||||

Event | Failures in Interval | Cumulative Trials | Classification | Mode |
---|---|---|---|---|

F | 1 | 8 | BD | 1 |

F | 1 | 8 | BD | 2 |

F | 1 | 8 | BD | 3 |

F | 0 | 10 | A | |

F | 0 | 11 | A | |

F | 0 | 12 | A | |

F | 1 | 13 | BD | 2 |

F | 0 | 14 | A | |

F | 0 | 15 | A | |

I | 0 | 15 | BD | 3 |

F | 1 | 16 | BD | 4 |

F | 0 | 17 | A | |

I | 0 | 17 | BD | 4 |

F | 0 | 18 | A | |

F | 0 | 18 | A | |

F | 1 | 20 | BD | 5 |

The table also gives the classifications of the failure modes. There are 5 BD modes. Of these 5 modes, 2 are corrected during the test (BD3 and BD4) and 3 have not been corrected by time [math]T=20\,\![/math] (BD1, BD2 and BD5). Do the following:

- Calculate the parameters of the Crow Extended - Continuous Evaluation model.
- Calculated the demonstrated reliability at the end of the test.
- Calculate parameter [math]p\,\![/math].
- Calculate the unfixed BD mode failure probability.
- Calculate the nominal growth potential factor.
- Calculate the nominal average effectiveness factor.
- Calculate the discovery failure intensity function at the end of the test.
- Calculate the nominal projected reliability at the end of the test.
- Calculate the nominal growth potential reliability at the end of the test.

**Solution**

- The next figure shows the data entered in the RGA software.
The parameters [math]\beta \,\![/math] and [math]\lambda \,\![/math] are calculated as follows (the calculations for these parameters are presented in detail in the Crow-AMSAA (NHPP) chapter):

- [math]\widehat{\beta }=0.8572\,\![/math]

and:

- [math]\widehat{\lambda }=0.4602\,\![/math]

- The corresponding demonstrated unreliability is calculated as:
- [math]\begin{align} {{f}_{D}}=\lambda \beta {{T}^{\beta -1}},\text{with }T\gt 0,\text{ }\lambda \gt 0\text{ and }\beta \gt 0 \end{align}\,\![/math]

- [math]{{f}_{D}}(20)=0.8572\cdot 0.4602\cdot {{20}^{0.8572-1}}=0.2572\,\![/math]

- [math]\begin{align} {{R}_{D}}= & 1-{{f}_{D}} \\ = & 1-0.2572=0.7428 \end{align}\,\![/math]

- Assume that the following effectiveness factors are assigned to the unfixed BD modes:
Classification Mode Effectiveness Factor Implemented at End of Phase BD 1 0.65 Phase 1 BD 2 0.70 Phase 1 BD 5 0.75 Phase 1 The parameter [math]p\,\![/math] is the total number of distinct unfixed BD modes at time [math]T\,\![/math] divided by the total number of distinct BD (fixed and unfixed) modes.

In this example:

- [math]p=\frac{3}{5}=0.6\,\![/math]

- The unfixed BD mode failure probability at time [math]T\,\![/math] is the total number of unfixed BD failures (classified at time [math]T\,\![/math] ) divided by the total trials. Based on the table at the beginning of the example, we have:
- [math]{{f}_{BD, unfixed}}=\frac{4}{20}=0.2\,\![/math]

- The nominal growth potential factor is:
- [math]\begin{align} {{\lambda }_{NGPFactor}}= & \underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{i}} \right)\frac{{{N}_{i}}}{T} \\ = & \left( 1-{{d}_{1}} \right)\frac{{{N}_{1}}}{T}+\left( 1-{{d}_{2}} \right)\frac{{{N}_{2}}}{T}+\left( 1-{{d}_{5}} \right)\frac{{{N}_{5}}}{T} \\ = & \left( 1-0.65 \right)\frac{1}{20}+\left( 1-0.70 \right)\frac{2}{20}+\left( 1-0.75 \right)\frac{1}{20} \\ = & 0.06 \end{align}\,\![/math]

- The nominal average effectiveness factor is:
- [math]\begin{align} {{d}_{N}}= & \frac{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{Ni}}}{M} \\ = & \frac{0.65+0.70+0.75}{3} \\ = & 0.70 \end{align}\,\![/math]

- The discovery function at time [math]T\,\![/math] is calculated using all the first occurrences of all the BD modes, both fixed and unfixed. In our example, we calculate [math]{{\widehat{\beta }}_{BD}}\,\![/math] and [math]{{\widehat{\lambda }}_{BD}}\,\![/math] using only the unfixed BD modes and excluding the second occurrence of BD2, which results in the following:
- [math]{{\widehat{\beta }}_{BD}}=0.6602\,\![/math]

- [math]{{\widehat{\lambda }}_{BD}}=0.6920\,\![/math]

- [math]\begin{align} \widehat{h}(T|BD)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ = & 0.6920\cdot 0.6602\cdot {{20}^{0.6602-1}} \\ = & 0.16507 \end{align}\,\![/math]

- The nominal projected failure probability at time [math]T=20\,\![/math] is:
- [math]\begin{align} {{f}_{NP}}= & {{f}_{NGP}}+{{d}_{N}}h(T) \\ = & 0.0701+0.7\cdot 0.16507 \\ = & 0.1865 \end{align}\,\![/math]

- [math]\begin{align} {{R}_{P}}= & 1-0.1856= \\ = & 0.8135 \end{align}\,\![/math]

- The nominal growth potential unreliability is:
- [math]{{f}_{NGP}} = {{f}_{D}} - {{f}_{BD unfixed}} + {{\lambda}_{NGPFactor}} + {{d}_{N}}\cdot p \cdot h(T) - {{d}_{N}}h(T)\,\![/math]

- [math]\begin{align} {{f}_{NGP}}= & 0.2572-0.2+0.06+0.7\cdot 0.6\cdot 0.16507-0.7\cdot 0.16507 \\ = & 0.0709 \end{align}\,\![/math]

- [math]\begin{align} {{R}_{NGP}}= & 1-0.0709 \\ = & 0.9291 \end{align}\,\![/math]