Reliability HotWire

Issue 64, June 2006

Hot Topics

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.

The approach to analyzing repairable system data is generally determined by what our interests are and the type and amount of data we have. For example, a particular transmission, identifiable by serial number, may have many repairs and at some point the transmission may be overhauled. An overhaul is more than minimum repair and restores the transmission to a significantly better reliability, but perhaps not as good as new. This process may be repeated many times for a particular transmission. Each cycle begins at age zero and continues until the transmission is overhauled. A particular transmission, therefore, may generate many cycles over its operational life. During each cycle there may be many failures and minimum repairs of the transmission. The fleet of transmissions comprises all the transmissions being operated and repaired.

We may be interested in the reliability characteristics of the transmission as it operates during a cycle or we may be interested in the general reliability performance of the fleet of transmissions. If we have failure data for the transmissions during cycles, we can conduct a within-cycle repairable analysis using RGA. If we have overall fleet total failures versus total operating times, we can conduct a fleet analysis utilizing RGA. In this article, we will address the fleet analysis situation directed toward addressing warranty failures.


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 this case, we are interested in how to forecast future warranty failures for a fleet and we want to know if there is any trend in these warranty failures. In particular, we want to be able to accurately forecast the quantity of future failures (on a quarterly or monthly basis) based on the previous field performance data that we have collected.

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


Quarter Accumulated Hours per Quarter (Ti) Number of Failures (Ni)
Q1 125000 992
Q2 142000 1190
Q3 119000 981
Q4 138000 1096

Over these four quarters we have a total of 524500 warranty operating hours and 4259 warranty failures. For the Crow-AMSAA grouped data method, the end of each cumulative operating time interval is T1 = 125000, T2 = 267000, T3 = 386000 and T4 = 524000, and the number of warranty failures in each interval is N1 = 992, N2 = 1190, N3 = 981 and N4 = 1096. This is the way the data are entered into RGA. In this illustration there are K = 4 intervals.


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.


RGA Methods

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 t is the cumulative fleet warranty operating hours and zero is the beginning of our data set. The warranty failure intensity function has two parameters λ and
β. λ measures the scale or magnitude of the rate of warranty failures and β measures any trend in the rate of warranty failures over cumulative fleet operating hours. If the warranty failure intensity rate is stable over time then the estimated β should be close to one. If the warranty failure intensity rate is getting worse then the intensity function is increasing and β is greater than one. If, because of corrective actions or other reasons, warranty failure intensity rate is getting better then the failure intensity function is decreasing and β is less than one. Therefore, the β estimate is a very important parameter in our analysis. The equations for the estimates of the parameters are:


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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