Analyzing Warranty Data of Repairable Systems
[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]
Systems can be categorized into two basic types: one-time or nonrepairable systems and reusable or repairable systems. In the latter case, under continuous operation, the system is repaired but not replaced after each failure. For example, if a water pump in a vehicle fails, the water pump is replaced and the vehicle is repaired.
A repair of a complex system typically does not make a system as good as new. Generally a repair is just enough to get the system operational again. That is, the repair addresses only what failed and does not restore the system to its original condition when first put into service. When we repair a system doing only what is necessary to get the system operational again, this is called minimum repair. The repairable systems models and methods in RGA are appropriate for the common practical situation where the systems under consideration are not replaced when they fail, but are repaired under the minimum repair strategy and put back into service.
The system under study may be part of a larger system. For example, a transmission is a complex system that is part of a larger system such as an automobile, truck or helicopter. So, in terms of the RGA methods, both the fleet of transmissions and the fleet of vehicles can be analyzed.
Suppose we are interested in warranty failures at the overall, global fleet level for a complex system. In general, a warranty is valid over a time period so the systems in our data set should be only systems during their operation over this warranty period.
In the example below the warranty data are grouped according to quarters. In each quarter we note the total number of fleet operating hours for all systems that are operational and under warranty during that period. For these same systems that are under warranty we also note the total number of warranty failures.
Table 1 - Quarterly Warranty Returns Reports
The goal is to fit a model that describes the behavior of the systems in the field and to project warranty returns for the next year by quarters, assuming that all the systems in the field accumulate on average 130000 hr every quarter for the next year.
For a within-cycle warranty analysis we would use the RGA power law model and enter the appropriate data. However, for a global, fleet-level warranty analysis we should use the RGA Crow-AMSAA model. If we want to also analyze what failure modes are causing the warranty failures, we would use the RGA Crow Extended model. Both models are easy to use by simply inputting the failure and operating time data in groups (e.g., months or quarters). The Crow-AMSAA method is described here.
Crow-AMSAA Model For Fleet Warranty Data
The Crow-AMSAA model estimates the number of warranty failures per hour of warranty operation time over the intervals. This is called the intensity rate of warranty failures and is estimated by the intensity function:
where n is the total number of warranty failures in our data set.
Analysis of Example Data
Enter the data into RGA using the Grouped Failure Times data type.
Click the Calculate button to obtain the parameters. The estimated β and λ are:
β is close to 1, which indicates that the rate of warranty failures is constant and that wearout or any other negative trend (e.g., β > 1) is not significant by the end of the collected data period. The following calculation shows that on average, at the end of the year, the mean time between warranty replacements (repairs) is about 122.28 accumulated hours of operation, which is expected to remain almost constant because β is close to 1.
The following plot shows the cumulative number of failures over time, which can be used to forecast warranty returns for future periods.
The accumulated quarterly number of returns during a 5-year period can be estimated using the Quick Calculation Pad (QCP) or the Function Wizard. If you insert a general spreadsheet into the folio, you can use the Function Wizard to insert the estimated cumulative number of failures at different times (with confidence bounds) into the sheet. The 95% lower one-sided confidence bound on the number of failures at 524,000 hours is calculated as shown next.
The resulting forecasted cumulative failures, with two-sided 90% confidence bounds, are displayed in the following table.
Table 2 - Quarterly Forecasted Cumulative Warranty Returns
The forecasted quarterly failures, also with 90% confidence bounds, can be easily calculated using the above table, by taking the difference of each cell and the cell above it. The results are shown in the following table.
Table 3 - Quarterly Forecasted Warranty Returns
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