
Confidence Intervals on StressStrength Analysis in Weibull++
A product's probability of failure is equal to the probability that the stress experienced by that product will exceed its strength. In other words, given one probability distribution function for a product's stress and another for its strength, the probability of failure can be estimated by calculating the area of the overlap between the two distributions (i.e., stressstrength analysis). This overlapping region (shown next) may be also referred to as stressstress interference.
However, there will usually be some uncertainty in the estimated parameters for these distributions, which entails uncertainty in the estimated reliability. This article will discuss how reliability can be estimated using stressstrength analysis and how confidence bounds on the reliability can be calculated. The effect of sample size on the confidence bounds will also be illustrated.
Estimating Reliability
Mathematically, the expected probability of failure, F, can be calculated as:
The expected probability of success, or the expected reliability, R, is calculated as:
Clearly, the product is more reliable when the interference region is smaller. Therefore, there are two ways to increase reliability: (i) increase the difference between the mean stress and strength values, which will make the two pdfs move away from each other, or (ii) decrease the standard deviations of the distributions, which will make the distributions narrower.
Ideally, the estimates of all stresses and all strengths for each element of a product would be perfectly accurate, but in general this is too costly and resourceintensive to accomplish. For example, we could estimate the stress distribution from customer usage data (e.g., the mileage per year of a passenger car). However, the stress conditions depend on the way the product is used, and these vary greatly based on the customer profiles and environmental conditions. For example, not every vehicle is driven the same miles per year under the same environmental conditions and personal preferences. In addition, the strength distribution mainly depends on the material used in the product, its dimensions and the manufacturing process. All these sources of variation can introduce uncertainty in the estimated reliability.
Confidence Intervals on the Probability
We have explained briefly how to obtain the stress and strength distributions and some of the factors affecting them. An accurate estimation of the probability of failure mainly depends on the sources of the distributions. The more accurately we can define the stress and strength distributions, the more accurate the estimated probability value will be. Based on the source of the distribution, which could be based on actual user data or specified by engineers based on engineering knowledge or existing references, there are two types of variation associated with the calculated probability: variation in the model parameters and variation in the probability values. We will focus on variation in the model parameters in this article and show how to calculate the confidence intervals with a simple example in Weibull++ using stressstrength analysis.
Variation in Model Parameters
Some degree of variation of the probability calculated from the stressstrength analysis is inevitable due to the uncertainties associated with the estimated distribution parameters from data sets. These uncertainties can be used to estimate the confidence intervals on the calculated probability. The variance of the reliability can be approximated to get the confidence intervals via:
Variance of f_{1}(x) and R_{2}(x) can be estimated from the Fisher information matrix. For details, please see http://ReliaWiki.org/index.php/Confidence_Bounds. The twosided confidence intervals can be calculated using:
where:
 Z_{1α/2} is the 1  α/2 percentile of a standard normal distribution
 α is 1  the confidence level
If the upper bound (U) and lower bound (L) are not infinite (∞) and zero, respectively, then the calculated variance of R is adjusted using [1/F_{1}(U)  F_{1}(L))]^{2}.
Example
We will use stressstrength analysis in this example to estimate the reliability of a component used in a printer. The stress is the distribution of the number of pages printed by users, and the strength is the distribution of the number of pages printed before the component failed during inhouse testing. The warranty for the printer is one year, and the goal is to estimate the reliability of the component within that period, assess the confidence bounds on the reliability and investigate the effect of sample size on confidence bounds.
The numbers of pages printed per year from twenty different users are sorted and shown next. This information represents stress because it describes how much "work" the component will perform in a given year.
Stress: Number of Pages Printed  
17987  27274 
19292  28352 
19358  28434 
20874  30172 
22586  32456 
22994  33038 
23442  33856 
24074  35692 
25496  39162 
25896  40642 
The numbers of pages printed before a failure occurred (for twenty printers) during inhouse testing are sorted and shown next (strength data).
Strength: Number of Pages Printed to Failure  
27348  55948 
39584  57868 
39916  57904 
43348  61944 
46972  66712 
47388  67476 
47884  68712 
48192  72584 
49392  79924 
49948  83084 
Solution
We will estimate the stress and strength distributions using the data presented in the above tables. Each data set will be analyzed in a Weibull++ standard folio using the lognormal distribution and MLE analysis method. As shown next, the parameters of the stress distribution are estimated to be logmean = 10.196956 and logstd = 0.238039.
The parameters of the strength distribution are estimated to be logmean = 10.892812 and logstd = 0.270042.
Now we can compare the two data sheets using the stressstrength folio. The pdfs of the two data sets are shown next.
The estimated reliability for the printer is 97.34%. The associated confidence bounds are estimated to be 99.22% for the upper bound and 91.33% for the lower bound (with a 90% confidence level).
However, with larger samples for the stress and strength data, the width of the bounds can be narrowed. Therefore, with larger samples, we can be more certain about the reliability of the product, which can translate into more accurate estimations for customer returns during the printer's warranty period.
Let's assume that the collected user data and the inhouse testing data are doubled. The extended data sets are shown below. We will analyze each data set using the same analysis settings.
Stress: Number of Pages Printed  
17987  28352 
19292  28434 
19358  28486 
19786  30002 
20874  30172 
21222  31188 
21294  31278 
22586  32456 
22962  33038 
22994  33190 
23442  33856 
24074  35692 
24845  35702 
25294  36342 
25496  37242 
25787  39162 
25896  39262 
26482  40642 
27274  43079 
28046  44707 
Strength: Number of Pages Printed to Failure  
27348  55948 
30083  57868 
39584  57904 
39916  61543 
43348  61944 
43543  63655 
43908  63695 
46972  66712 
47388  67476 
47683  68139 
47884  68712 
48192  72584 
49392  73384 
49948  74224 
51670  75584 
52127  79843 
52673  79924 
53012  83084 
54332  87917 
54943  91393 
The parameters of the stress distribution in this case are estimated to be logmean = 10.244621 and logstd = 0.239873. The parameters of the strength distribution are estimated to be logmean = 10.940471 and logstd = 0.270892. Now we can compare the two extended data sets using the stressstrength folio.
As you can see in the results shown next, the estimated reliability for the printer, 97.28%, did not change significantly. On the other hand, the associated confidence bounds are narrowed down to 98.84% for the upper bound and 93.76% for the lower bound (with the same 90% confidence level), which represents a more precise reliability estimation.
Conclusion
In this article, we presented the theory and mathematical formulations behind stressstrength analysis and confidence bounds calculations with a basic example. The example illustrates that increasing the amount of data collected during testing can lead to a more precise estimate of product reliability via stressstrength analysis.
References
 http://www.ReliaWiki.org/index.php/StressStrength_Analysis
 http://www.ReliaSoft.com/newsletter/1q2002/usage.htm
 http://en.wikipedia.org/wiki/Stressstrength_analysis
 http://www.mtbf.us/PSIExamples/MechanicalReliabilityPrediction/MechanicalReliability.pdf
 li. Utkin, L.V., Kozine, I.O., StressStrength Reliability Models
Under Incomplete Information.
http://www.levvu.narod.ru/Papers/Stress_Final.pdf