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:

$$\begin{matrix} Start & End & Number of & Number of \\ Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} \\ 0 & 50 & 2 & 4 \\   50 & 100 & 0 & 5  \\   100 & 150 & 2 & 2  \\   150 & 200 & 3 & 5  \\   200 & 250 & 2 & 1  \\   250 & 300 & 1 & 2  \\   300 & 350 & 2 & 1  \\   350 & 400 & 3 & 3  \\   400 & 450 & 3 & 4  \\   450 & 500 & 1 & 2  \\   500 & 550 & 2 & 1  \\   550 & 600 & 1 & 0  \\   600 & 650 & 2 & 1  \\ \end{matrix}$$

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 for Eqn. (simpact):

$$\begin{matrix} Start & End & Number of & Number of & Available & {} & {} \\ Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & 1-\tfrac & \mathop{}_{}^{}1-\tfrac \\ 0 & 50 & 2 & 4 & 55 & 0.964 & 0.964 \\   50 & 100 & 0 & 5 & 49 & 1.000 & 0.964  \\   100 & 150 & 2 & 2 & 44 & 0.955 & 0.920  \\   150 & 200 & 3 & 5 & 40 & 0.925 & 0.851  \\   200 & 250 & 2 & 1 & 32 & 0.938 & 0.798  \\   250 & 300 & 1 & 2 & 29 & 0.966 & 0.770  \\   300 & 350 & 2 & 1 & 26 & 0.923 & 0.711  \\   350 & 400 & 3 & 3 & 23 & 0.870 & 0.618  \\   400 & 450 & 3 & 4 & 17 & 0.824 & 0.509  \\   450 & 500 & 1 & 2 & 10 & 0.900 & 0.458  \\   500 & 550 & 2 & 1 & 7 & 0.714 & 0.327  \\   550 & 600 & 1 & 0 & 4 & 0.750 & 0.245  \\   600 & 650 & 2 & 1 & 3 & 0.333 & 0.082  \\ \end{matrix}$$

As can be determined from the preceding table, the reliability estimates for the failure times are:

$$\begin{matrix} Failure Period & Reliability \\ End Time & Estimate \\ 50 & 96.4% \\   150 & 92.0%  \\   200 & 85.1%  \\   250 & 79.8%  \\   300 & 77.0%  \\   350 & 71.1%  \\   400 & 61.8%  \\   450 & 50.9%  \\   500 & 45.8%  \\   550 & 32.7%  \\   600 & 24.5%  \\   650 & 8.2%  \\ \end{matrix}$$