Template:Example: Simple-Actuarial Example

Simple-Actuarial Example

A group of 55 units are put on a life test during which the units are evaluated every 50 hours, with the following results:

[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 for the simple actuarial method can be obtained by expanding the data table to include terms used in calculation of the reliability estimates from the simple actuarial 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]