Actuarial Simple Example
A group of 55 units are put on a life test during which the units are evaluated every 50 hours. The results are:
[math]\displaystyle{ \begin{matrix}
Start & End & Number of & Number of \\
Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} \\
0 & 50 & 2 & 4 \\
50 & 100 & 0 & 5 \\
100 & 150 & 2 & 2 \\
150 & 200 & 3 & 5 \\
200 & 250 & 2 & 1 \\
250 & 300 & 1 & 2 \\
300 & 350 & 2 & 1 \\
350 & 400 & 3 & 3 \\
400 & 450 & 3 & 4 \\
450 & 500 & 1 & 2 \\
500 & 550 & 2 & 1 \\
550 & 600 & 1 & 0 \\
600 & 650 & 2 & 1 \\
\end{matrix} }[/math]
Solution
The reliability estimates can be obtained by expanding the data table to include the calculations used in the actuarial-simple method:
[math]\displaystyle{ \begin{matrix}
Start & End & Number of & Number of & Available & {} & {} \\
Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} & \prod\mathop{}_{}^{}1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \\
0 & 50 & 2 & 4 & 55 & 0.964 & 0.964 \\
50 & 100 & 0 & 5 & 49 & 1.000 & 0.964 \\
100 & 150 & 2 & 2 & 44 & 0.955 & 0.920 \\
150 & 200 & 3 & 5 & 40 & 0.925 & 0.851 \\
200 & 250 & 2 & 1 & 32 & 0.938 & 0.798 \\
250 & 300 & 1 & 2 & 29 & 0.966 & 0.770 \\
300 & 350 & 2 & 1 & 26 & 0.923 & 0.711 \\
350 & 400 & 3 & 3 & 23 & 0.870 & 0.618 \\
400 & 450 & 3 & 4 & 17 & 0.824 & 0.509 \\
450 & 500 & 1 & 2 & 10 & 0.900 & 0.458 \\
500 & 550 & 2 & 1 & 7 & 0.714 & 0.327 \\
550 & 600 & 1 & 0 & 4 & 0.750 & 0.245 \\
600 & 650 & 2 & 1 & 3 & 0.333 & 0.082 \\
\end{matrix} }[/math]
As can be determined from the preceding table, the reliability estimates for the failure times are:
[math]\displaystyle{ \begin{matrix}
Failure Period & Reliability \\
End Time & Estimate \\
50 & 96.4% \\
150 & 92.0% \\
200 & 85.1% \\
250 & 79.8% \\
300 & 77.0% \\
350 & 71.1% \\
400 & 61.8% \\
450 & 50.9% \\
500 & 45.8% \\
550 & 32.7% \\
600 & 24.5% \\
650 & 8.2% \\
\end{matrix} }[/math]