Accurate predictions about the quantity of products that will be returned under warranty can provide huge benefits to manufacturing organizations. Among other advantages, better warranty data analysis allows an organization to make the most efficient allocation of resources for providing warranty services. It also allows the manufacturer to anticipate customer support needs and to take the necessary steps to ensure customer satisfaction. Warranty data analysis also can provide a valuable earlywarning signal to the manufacturer when there is a serious product reliability or quality problem in the field, which gives the organization time to mobilize its resources to meet the challenge before serious financial, legal or other problems develop. Warranty return predictions can also be used as a tool to verify whether actual returns fall in line with expectations. By monitoring warranty return data, the analyst (and the organization) can detect specific return periods and/or batches of sales or shipments that deviate from the assumed model by comparing actual returns to the previously predicted returns. If the actual returns number deviates significantly from the number that was predicted, a flag would be raised calling for further investigation or for actions to be taken to treat the problem. Using this methodology, namely monitoring warranty returns using Statistical Process Control (SPC) techniques, one can also detect seasonal effects. Additional analysis then may be required to understand and quantify these effects. In previous publications, warranty analysis techniques using the Weibull++ software have been discussed. These analyses have entailed fitting a lifetime distribution to the return data and using the conditional reliability metric to forecast failures in subsequent periods. In this article, we expand upon this topic by presenting an approach for detecting and then accounting for seasonal effects in the analysis so that more accurate return predictions can be made. The ProblemFor this example, suppose that the reliability team at ACME Corporation is using returns data to analyze a particular component in the company’s product line and make warranty return predictions so that the financial impact can be quantified. The data under consideration is for a particular region where returns seem to be higher than normal. Sales and returns data for the units will be analyzed and a forecast of the expected number of returns over the next two years will be calculated. Sales data from January 2006 to April 2007 and returns data from February 2006 to May 2007 are available. The warranty period for the product is 12 months. Life Data Analysis (LDA) ApproachFollowing the usual procedure, the sales and returns data sets are entered into a Weibull++ Warranty Analysis Folio so that a distribution can be fitted to the population and a forecast of the expected failures can be calculated [Ref. 1]. Figure 1 shows the sales data and the returns data entered in a "Nevada" chart format.
Figure 1: Sales and return data entered in a Weibull++ Warranty Analysis Folio After careful consideration, the team concludes that the data represent a mixed population and decides to fit a mixed Weibull distribution with 2 populations, as shown in Figure 2. Although mixed Weibull models are typically not recommended for extrapolations, the team determines that predictions are acceptable in this case given that no predictions will be made beyond the 12 months of warranty, which is within the 16 months of data available. Figure 3 shows the forecast of the expected failures over the next 12 months. (Please note that the decay in the number of failures is due to the population "running out" of units and not due to an increase in the reliability in the product.)
As an additional analysis, ChiSquared values are calculated to identify outliers in the data [Refs. 2, 3]. These values measure the difference between the predicted and observed failure quantities. Periods that display an upper bound probability of the ChiSquared distribution at or below the critical and caution values are tagged. For the data in this example, March, April and May are flagged as statistically different than the rest of the return periods (with Alpha 0.1) and are colored in yellow in Figure 4.
The next step is to consider the reason for this variation. Given the nature of the product, the engineers expect seasonal effects to influence the returns. Specifically, they expect higher stresses during the colder months due to both usage and temperature. However, no quantitative analysis has been done to investigate the effect of temperature on the life of the units. The next section describes an approach that applies accelerated life testing data analysis techniques to the warranty data in order to better understand and quantify the relationship between temperature and component life. Accelerated Life Testing Analysis (ALTA) ApproachThe typical accelerated life testing analysis model consists of an underlying life distribution (such as Weibull, lognormal or exponential) that describes the failure behavior of the product at different stress levels and a lifestress relationship that quantifies the manner in which the life distribution changes across different stress levels. For this example, the lifestress model to be chosen needs to account for the timedependent nature of the stress and its cumulative effect on the units. Such a model is commonly referred to as a cumulative damage or cumulative exposure model and this type of analysis can be performed in the ALTA PRO software. The cumulative damage model can take on different forms depending on the nature of the stress and is used when the stress is timedependent. Specifically, the Arrhenius relationship will be used for this example as it tends to work well when the stress is thermal. With a timedependent stress x(t), the Arrhenius lifestress relationship is given by:
Where b and C are constants. It can be reparameterized in a form that is consistent with the general loglinear (GLL) relationship and with ALTA PRO as follows:
Where the following reparameterization is used:
As a preliminary study, the team analyzes the warranty data along with a use profile (which describes the stress experienced by the units over time) in order to have a better understanding of the stress effect on the life of the units. Given the region where these units were put into service, the monthly average temperature for a year will be used as a stress profile for this analysis. Figure 5 shows this timedependent profile. (Please note that for more accurate results, a more rigorous approach should be employed to obtain a use profile that is more representative of the field stress.)
Using a lognormal distribution (best fitting distribution) and the cumulative damage model in conjunction with the given stress profile, the analysts obtain the following parameters from ALTA:
Where α_{0} and α_{1} are the calculated lifestress relationship parameters and σ^{'}_{T} is the logstandard deviation of the lognormal distribution. The Reliability vs. Time plot shown in Figure 6 displays a much better fit of the data with the ALTA approach than with the LDA approach used initially. A cyclic effect of the temperature profile also can be detected in this plot, where the units accumulate damage at different rates during different months of the year.
For this example, then, by using a timedependent stress profile that describes the field stress, the analysts are able to make more accurate predictions of the component’s reliability over time. With the LDA approach, a mixed Weibull distribution was used because different populations could be observed but it was impossible to identify the cause of this difference. In fact, the difference is not due to differences in the populations themselves but in the damage accumulated and the varying rate at which it was accumulated. With the ALTA approach, a forecast of the expected failures is calculated in a manner similar to the LDA approach, by using the conditional reliability given the age of the remaining units. The reliability for the cumulative damage Arrheniuslognormal under a single stress is given by:
Where:
And:
And the conditional probability of failure is given by:
As an example, let's calculate the expected number of failures in June 2007 for units that were put in service in June 2006 (t = 1 since there is data up to May 2007). There are 914 units that have survived twelve months (T = 12). Therefore, the expected number of failures is calculated as shown next and other months are analyzed in a similar manner.
(Note that in the ALTA approach, units are assumed to be put in service at time zero, which corresponds to the lowest temperature in the profile. The effect of when the units were put in service is assumed to be small, which may not be the case in reality. A more correct and more laborintensive approach (not covered here) is to take into account at what point in the profile units are put into service and associate each of them with its corresponding profile. Alternatively, and possibly more beneficially, inhouse tests can be conducted to obtain a lifestress relationship that can be used in conjunction with any profile and would apply to other regions as well.) Figure 7 shows a comparison of the failure predictions between the analysis done with the LDA approach vs. the analysis done with the ALTA approach. The constant stress assumption in the LDA approach leads to different predictions as it does not account for the timedependent nature of the stress. This difference is expected to grow as we predict failures further out in time, both due to the constant stress assumption and due to moving away from the range of data in the mixed Weibull model.
Additional AnalysisOnce the analysts have obtained a distribution and lifestress relationship along with a stress profile representative of the field stresses, a number of interesting applications are possible. For example, assuming that this component will be part of a larger repairable system that experiences a similar stress profile, one could study the system using the reliability phase diagram technique supported by the BlockSim 7 software. Specifically, each month or season could be represented by a "phase" in the phase diagram and different components in the system would have different life characteristics during a given phase (estimated using the ALTA analysis described previously). Figure 8 shows the phase diagram for this example along with the reliability characteristics defined for January and July. The logstandard deviation is given directly by the cumulative damage model (σ^{'}_{T} = 1.1786) while the logmean has to be calculated for each phase or stress (using Eqn. (2)). Table 1 shows the logmeans for each period.
Once a phase diagram has been set up in BlockSim, the component reliability can be estimated through simulation. Figure 9 shows the Reliability vs. Time plot generated for this analysis compared with the results for the ALTA model. This is the first step to a number of analyses that can be performed via simulation (such as system reliability, maintainability and availability analysis, resource allocation, throughput, etc.).
ConclusionWhen analyzing warranty data where seasonality is a factor, studying the effects of stress on the life of the units might be beneficial in order to obtain more accurate predictions. In this article, one possible approach was shown where warranty data were analyzed in conjunction with a stress profile using accelerated life test data analysis techniques. It was considered in this case that accounting for the effect of the stress on the life of the units was valuable as it took into account the physics of failure of such units, leading to more accurate results. Once a model has been obtained, a wide variety of ancillary analyses can be performed with the use of reliability phase diagrams and simulation. References[1] ReliaSoft Corporation, "Predicting Warranty Returns," Reliability Edge, Volume 2, Issue 1, Tucson, AZ, 2001. [2] ReliaSoft Corporation, "Monitoring Warranty Returns Using Statistical Process Control (SPC)," Reliability HotWire, Issue 58, Tucson, AZ, 2005. [3] ReliaSoft Corporation, Life Data Analysis Reference, ReliaSoft Publishing, Tucson, AZ, 2007. [4] ReliaSoft Corporation, Accelerated Life Testing Reference, ReliaSoft Publishing, Tucson, AZ, 2007.
