Template:Example: Stress-Strength Analysis with Parameter Uncertainty

Stress-Strength Analysis with Parameter Uncertainty

Assume we are going to use stress-strength analysis to estimate the reliability of a component used in a vehicle. The stress is the usage mileage distribution and the strength is the miles-to-failure distribution of the component. The warranty is 1 year or 15,000 miles, whichever is earlier. This table gives the data for the mileage distribution per year (stress):

Stress: Usage Mileage Distribution
10096	12405
10469	12527
10955	12536
11183	12595
11391	12657
11486	13777
11534	13862
11919	13971
12105	14032
12141	14138

This table gives the data for the miles-to-failure distribution (strength.

Strength: Failure Mileage Distribution
13507	16125
13793	16320
13943	16327
14017	16349
14147	16406
14351	16501
14376	16611
14595	16625
14746	16670
14810	16749
14940	16793
14951	16862
15104	16930
15218	16948
15303	17024
15311	17041
15480	17263
15496	17347
15522	17430
15547	17805
15570	17884
15975	18549
16003	18575
16018	18813
16052	18944

The goal is to estimate the reliability of the component within the warranty period (1 year/15,000 miles).

Solution

First, we need to estimate the stress and strength distributions using the above data. The estimated stress distribution is:

The estimated strength distribution is:

Add one Stress-Strength folio and choose the stress and strength distributions:

The stress-strength tool shows the pdfs of the selected data folios. The pdf curves are shown next:

Since the warranty is 1 year/15,000 miles, all the vehicles with mileage larger than 15,000 should not be considered in the calculation. Therefore, the conditional stress distribution conditioning on the mileage less than 15,000 should be used. This is done by using the following settings in the tool:

The calculated results are:

The estimated reliability for vehicles less than 15,000 miles per year is 98.84%. The associated confidence bounds are estimated from the variance of the distribution parameters. With larger samples for the stress and strength data, the width of the bounds will be narrower.