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A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims Karl D. Majeske The University of Michigan Business School 701 Tappan Street Ann Arbor, MI 48109-1234 Phone: 734-647-6978 E-mail: kdm@umich.edu ABSTRACT Automobile warranties and thus lifetimes are characterized in the two-dimensional space of time and mileage. This paper presents a non-homogenous Poisson Process (NHPP) predictive model for automobile warranty claims consisting of two components: a population size function and a failure or warranty claim rate. The population size function tracks the population in the time domain and accounts for mileage by removing vehicles from the population when they exceed the warranty mileage limitation. The model uses the intensity function of a NHPP - the instantaneous probability of failure - to model the occurrence of warranty claims. The approach was developed to support automobile manufacturers' process of using claims observed during the early portion (first seven months) of vehicle life to predict claims for the remainder of coverage, typically between three and five years. This paper uses manufacturer provided warranty data from a luxury car to demonstrate the NHPP model by predicting claims for three vehicle subsystems. Warranty predictions are then compared with the actual observed values. Keywords: Automobile warranty data, Two-dimensional lifetime, Non-homogeneous poisson process, Warranty claim predictions. Karl D. Ma eske Revised: 7-14-03 Page: 1

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 1. Introduction Manufacturers of consumer durable goods use many metrics to assess the quality and reliability of their products. One class of measures, referred to as field performance, represent how well a product has functioned during its life relative to customer expectations. Global automobile manufacturers track customer satisfaction with the Initial Quality Studyl, a third party measure of vehicle field performance. This survey primarily identifies items that annoy customers or vehicle features that fail to meet their expectations, and manufacturers use this information as an input into future designs. Warranty claims also measure the field performance of automobiles. Items repaired under warranty cause not only an inconvenience to the customer, but result in a direct deduction to the manufacturer's profit generated by the sale. These claims then represent an opportunity for cost reduction that manufacturers can glean through product and process design changes. To quantify the impact on customer satisfaction and cost of warranty, manufacturers use statistical techniques to model and predict the number and cost of warranty claims. Manufacturers use warranty predictions for a variey of purposes: to assess customer satisfaction, identify product and process design changes, and predict financial liability. This research focused on predicting the number of automobile warranty claims, which automobile manufacturers use as a measure of quality during the various stages of the product development life cycle. After completing product design, automobile manufactures make preproduction prototype vehicles. These vehicles are used in crash testing (to satisfy safety standards) and reliability testing (to establish field performance). Manufacturers also make warranty predictions based on the performance of these pre-production prototype vehicles. They use these predictions to motivate the need for, and justify the expense of, product design changes prior to the commencement of mass production. These early warranty predictions are also an 1 The Initial Quality Study, an across manufacturer comparative study of new vehicle quality, is published annually by J. D. Power and Associates, Agoura Hills, CA. Karl D. Majeske Revised: 7-14-03 Page: 2

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims input into product pricing and the viability of the vehicle in the marketplace. Sarawgi and Kurtz (1995) propose a method for predicting warranty claims using bench test data. Majeske and Herrin (1998) show an example where field performance dramatically differs from predictions using bench test data. Once a vehicle model reaches the marketplace, manufacturers predict warranty claims and costs based on data observed early in the product life (the first six or seven months of usage). Financial functions use these predictions to determine the amount of money to reserve to cover the future warranty liability. Engineering functions use these predictions for many purposes: identifying product design changes, identifying manufacturing or assembly process changes, cost justifying various changes, and evaluating the effectiveness of these changes once implemented. Wu and Meeker (2002) propose a method for early detection of reliability problems using warranty data. They argue the earlier problems are detected, the smaller the financial and goodwill costs that will result from the problems. Current trends in automobile manufacturing and marketing have increased basic warranty coverage from 12 months / 12,000 miles to as much as five years / 50,000 miles on the complete vehicle and ten years / 100,000 miles on the power-train. This increase in coverage has highlighted the lack of predictive validity of some automobile warranty forecasting techniques. The motivation for this research was the need to develop a technique that provided the manufacturer more accurate warranty predictions, especially in the later stages of warranty coverage. This paper contains the result of that work, a non-homogeneous Poisson Process (NHPP) predictive model for automobile warranty claims. This technique forecasts the number of warranty claims that a population of vehicles will experience based on the claims observed early in the vehicle lifetime. In addition to providing manufacturers a prediction of the total number of claims, the technique also forecasts the timing of claims during the vehicle lifetime. Karl D. Ma eske Revised: 7-14-03 Page: 3

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims The remainder of this paper is organized as follows. Section 2 defines the warranty lifetime of an automobile in the two-dimensional space defined by time and mileage, and introduces the usage function. Section 3 provides an overview of warranty claim predictive models with a focus on automobile warranty. Section 4 develops the NHPP predictive model that consists of two components. The first component is a time based population size that accounts for mileage by censoring vehicles from the population when they exceed the warranty mile age limitation. The second component, the failure rate or the warranty claim rate, is captured by the intensity function of a NHPP. The model then predicts claims by integrating the product of the population size and the intensity function over the time domain of the warranty coverage. Section 5 presents an application of the approach to three subsystems of a luxury automobile using manufacturer supplied warranty data. Section 6 contains the conclusion. 2. Automobile Lifetime Under a Two-Dimensional Warranty Manufacturers provide product warranties that can be described in a multitude of ways (Blischke and Murthy 1992). Basic coverage describes the warranty included in the purchase price of a product. This coverage represents a service contract that a customer must buy if they wish to purchase the product. Extended coverage represents coverage not included in the purchase price of the product (Kelly and Conant 1991). Moskowitz and Chun (1994) define three major types of warranties: pro rata, free replacement, and lump sum. The pro rata warranty defines a cost sharing function between the manufacturer and seller with the most common setup transferring the repair burden from the manufacturer to the customer linearly over the coverage (e.g., automobile batteries and tires). With free replacement coverage, the manufacturer agrees to replace or repair the product during the coverage at no cost to the customer (e.g., automobiles, computers). The lump sum warranty provides the customer with a fixed payment, determined at time of sale, if the product fails. Karl D. Majeske Revised: 7-14-03 Page: 4

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims Manufacturers express warranty coverage using a quantitative definition of product lifetime. When one variable quantifies product life (e.g., hours of use for a boat motor) the product carries a one-dimensional warranty (Blischke and Murthy 1994). Defining product life in a single metric like "time" has a great deal of intuitive appeal and facilitates warranty data modeling. When two metrics define product life, the product carries a two-dimensional warranty (Blischke and Murthy 1996, Moskowitz and Chun 1994, and Singpurwalla and Wilson 1993). Manufacturers sell automobiles with a two-dimensional basic coverage characterized by time and mileage that offers free replacement. The two-dimensional automobile warranty provides customers a flexible coverage based on their personal usage pattern. The two-dimensional automobile warranty also protects the manufacturer from replacing components on high mileage newer vehicles and limits long-term manufacturer liability on low usage vehicles. Figure 1 shows the two-dimensional lifetime space for automobiles characterized by time and mileage. The date of final assembly establishes the physical existence of an automobile but the warranty lifetime doesn't begin until a customer purchases the vehicle. Let the random variable L represent the sales lag, the time from final vehicle assembly to vehicle sale. Sales lag is an unknown value at time of assembly; however, all vehicles at risk for warranty claims have an observed value because they have been sold to a customer. Robinson and McDonald (1991) show that the larger the sales lag, the more likely a vehicle will observe warranty claims. The cutoff date is the calendar date of a warranty prediction. Let the random variable T represent time in service - the elapsed time from product sale to the cutoff date - which captures product exposure or the amount of time the vehicle was eligible for warranty claims. Due to the random nature of sales lag, a population of vehicles assembled together will have varying values for time in service at a given cutoff date. This phenomenon of varying values for time in service can complicate the analysis and prediction of automobile warranty claims. Karl D. Ma eske Revised: 7-14-03 Page: 5

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims U(t) for vehicle 1 e, U(t) for vehicle 2 mWL Warranty Coverage Region L: sales lag L: \ sas l T: time in service tw~ time Final Vehicle Cutoffme Assembly Sale Date Figure 1: Two Dimensional Automobile Lifetime Space Figure 1 shows the usage function U (t) - the non-decreasing path through the lifetime space that relates time to mileage - for two hypothetical vehicles. Defining the time limitation and the mileage limitation of the two-dimensional coverage as tWL and mWL, respectively, results in a rectangular coverage region. In this example, vehicle 1 leaves the warranty coverage region by exceeding the time limit while vehicle 2 leaves coverage by exceeding the mileage limit. Because vehicles may leave coverage by exceeding the mile age limit prior to the time limit U(t) > mWL for t < tWL, the duration of time that a vehicle will remain under warranty coverage U -1 (mWL) is an unknown random variable at time of sale. The length of warranty exposure varies within a population depending on the vehicles' usage function U (t), further complicating predictions. Vehicles repaired while inside the basic coverage region, U (t) such that (t < tWL n U(t) < mL ), (may) result in warranty claims. At a cutoff date, each vehicle is at an unknown point on their U (t) function. Automobile manufacturers track the sale of each vehicle and thus know vehicle location in the time domain. Karl D. Majeske Revised: 7-14-03 Page: 6

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims However, due to the uncertain usage patterns of customers, manufacturers do not know a vehicles location in the mileage domain. Because not all vehicles stay at risk for claims through tWL, the remaining coverage - in both time and mileage - is unknown for each vehicle when making a prediction. When vehicles have experienced one or more warranty claims prior to the cutoff date, the manufacturer has an observed mileage corresponding to the time of the repair(s). For vehicles without warranty claims, the manufacturer has no information regarding the mileage domain of the lifetime path. Due to this limited information on mileage, many automobile manufacturers make warranty predictions purely in the time domain. 3. Warranty Claim Predictive Models Murthy and Djamaludin (2002) provide a comprehensive review of literature related to the various aspects of new product warranty. This section presents some specific statistical models applied to warranty claims, with a focus on automobile warranty claims. This section is divided into three sub-sections based on the type of statistical model used: Poisson or discrete models, stochastic process models, and repair per thousand vehicle models. 3.1 Poisson Models Poisson prediction models have intuitive appeal to many reliability engineers. These models assume X, the number of claims for a product, follows a Poisson distribution with probability mass function p(x) = e x= 0,1,2,... x! For a one-dimensional warranty, reliability engineers allow the Poisson parameter lambda X to be a function of time in service t p(x)= e- x = 0,1,2,... t> 0. (1) x! Karl D. Ma eske Revised: 7-14-03 Page: 7

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims Equation (1) allows fitting a Poisson distribution to a population with varying values of time in service. Assuming X is independent of t, you can predict the expected claims through the warranty time limit tWL for one vehicle as?tWL, and for a population of N independent and identically distributed vehicles as NXtWL. Kalbfleisch, Lawless, and Robinson (1991) develop a Poisson model for predicting automobile warranty claims in the time domain similar to Equation (1) above. They note that their Poisson approach yields biased estimates due to vehicles leaving the warranty coverage region by exceeding the mileage limit. Moskowitz and Chun (1994) suggest a Poisson model for two-dimensional warranty coverage. They formulate the warranty problem as a Poisson regression by fitting the cumulative Poisson parameter ug with various functions of time t and mileage m. One can easily estimate the parameters of the linear model - = f31t + P2m using ordinary least squares. Chen and Papova (2002) fit the bi-variate Poisson model to twodimensional warranty data with maximum likelihood estimation. In application, the lack of data on the mileage domain limits the ability to use this bi-variate approach to model and predict automobile warranty data. 3.2 Stochastic Process Models Stochastic processes model a product or system that generates a series of (time) ordered observations. Applications of stochastic processes to field performance data appear in the literature as repairable systems reliability. The Poisson Process (Kingman 1993) assumes times between failures are independent and identically distributed (i.i.d.) exponential random variables. A renewal process relaxes the exponential assumption for time between failures but still requires theses times to be i.i.d. The non-homogeneous Poisson Process (NHPP) does not require i.i.d. time between failures and provides a great deal of flexibility when fitting observational data. The NHPP is characterized by the intensity function Karl D. Majeske Revised: 7-14-03 Page: 8

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims ) m Pr(observe a failure in interval t,t + t) (2) v (t)= lirm (2) At-O At that represents the instantaneous probability of a failure. The expected number of failures (warranty claims) for the NHPP is the integral of the intensity function. For a one-dimensional warranty, the expected number of warranty claims through the warranty limit tWL is E[C(twL)] = Jv (t)dt. (3) Nelson (1988) suggested a graphical technique for evaluating either the total cost or number of repairs for a population of products. This technique plots the mean cumulative cost mcc(t) or mean cumulative repair mcr(t) versus time t. Nelson noted the derivatives of both mcc(t) and mcr(t) become v (t), the intensity function of a NHPP. Wang, Suzuki, and Yamamoto (2002) fit a NHPP to warranty data for products with an unknown sales date or products that don't observe a value for sales lag. The power law process (Rigdon and Basu 1989a), a NHPP fit by Roberts and Mann (1993) to repairable system data, has the intensity function v (t) = (t). (4) Crow (1974) showed that the power law process has the property that T1, the time to first failure, follows a Weibull distribution (Blischke and Murthy 2000, Meeker and Escobar 1998) with the cumulative distribution function F(t)=l- e-() t> 0. (5) Authors have referred to the power law process as the Weibull Process (Crow 1982); however, only the time to first failure follows a Weibull distribution. Huang (2001) presents a Bayesian approach for fitting a power law process to repairable system data. Muralidharan (2001) extends the power law process to allow the failure or repair affect the failure rate of the product. Karl D. Ma eske Revised: 7-14-03 Page: 9

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 3.3 Repairs per Thousand Vehicles The most common automobile warranty-data reporting scheme uses cumulative warranty claims per thousand vehicles. While the notation differs between manufacturers, this approach appears to be the industry standard. Majeske, Riches, and Annadi (2003) suggest the following model for R(t) - repairs (warranty claims) per thousand vehicles. Let Ni represent the number of vehicles in the population with at least i months in service (MIS). Notice that Ni can decrease as i increases due to varying values for sales lag and vehicles exceeding the mileage limit. Let fi represent the observed number or frequency of claims in time period i - manufacturers generally track claims by month - for the Ni vehicles. Then calculate t f R(t) = i 1000 i=o Ni One method used to predict R(t) values utilizes transformed data. By regressing log[ R(t)] on log( t), one can fit the simple regression model log(R(t)) = Po + P log(t)+ E,. (6) To predict a future R(t) value, denoted R(t), simply extend the fit line and transform back to the original units of R(t) using R(t) = exp( P0 + P log( t)). (7) Wasserman (1992) developed a dynamic linear predictive model for R(t) using data from multiple product years of the same model vehicle. He shows that incorporating data from previous model years can increase the accuracy of predictions. Singpurwalla and Wilson (1993) developed a bivariate failure model for automobile warranty data indexed by time t and mileage m. They derived the marginal failure distributions and presented a method for predicting R(t) using a log-log model similar to Equation (6). Karl D. Majeske Revised: 7-14-03 Page: 10

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 4. Automobile Warranty Predictive Model This section presents a non-homogenous Poisson Process (NHPP) predictive model for automobile warranty claims. The approach presented here makes predictions in the time domain but accounts for mileage by removing vehicles from the population when they exceed the mileage limit. The technique models the warranty claims for an automobile with a NHPP that provides a great deal of flexibility when fitting observational data. 4.1 Service Time based Population Size Automobile manufacturers form vehicle populations based on assembly date. This allows relating warranty claims and costs to time based events such as product design changes, component manufacturing process changes, assembly process changes and vendor issues. Given the variation in sales lag and the random nature of when vehicles leave warranty coverage, the population size varies as it moves through calendar time. Specifically, the population size would start at zero, and increase by one each time a vehicle is sold, and decrease by one each time a vehicle leaves warranty coverage. To place vehicles in a common metric for warranty predictions, automobile manufacturers characterize lifetime using time in service. As previously shown in Figure 1, vehicle sale begins the warranty coverage and defines the origin for time in service. Due to the random nature of sales lag, a population of vehicles assembled on the same day will have different time in service values when making a prediction. Let N(t) represent the population size or the number of vehicles at risk for warranty claims at time in service t. For a one-dimensional coverage in the time domain, the manufacturer would have complete information on N(t). When making a prediction of total claims for a population of size N, N t < tWL N(t)= WL (8) 0 t>t WL Karl D. Ma eske Revised: 7-14-03 Page: 11

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims When performing warranty data analysis, many automobile manufacturers use a population definition similar to Equation (8) due to the limited information on vehicle mileage. Using Equation (8) fails to remove vehicles from the population when they exceed the mileage limit thus over estimating the population size at higher time in service values. Define u as the rate a vehicle accumulates mileage, which is the derivative or slope of the usage function U (t). Assuming a constant usage, a vehicle is no longer in the warranty coverage region if the mileage exceeds the mileage limit or ut> mWL. (9) By letting ui represent the (constant) usage rate of the ith vehicle, the variable Il if ut < mWL = WL (10) 0 else indicates if a vehicle remains in the population; that is, Equation (10) indicates vehicles still in the warranty coverage region at time t. To determine the population size at time t, sum the indicator variables of Equation (10) to obtain (N ldi t <tWL N(t,m)L. (11) o t > tWL Equations (10) and (11) define a time in service based population size that removes vehicles when they exceed the mileage limit. N(t, m) characterizes the population size in the time domain while taking into account the effects of mileage on warranty coverage. Automobile manufacturers do not have complete information on the population size N(t, m) as they do for N(t), due to the lack of information on vehicle usage. To use N(t, m), the manufacturer must estimate usage for each vehicle in the population. Vehicles with a warranty claim have an observed value on the lifetime path that can be used to estimate u. To estimate usage for vehicles without a claim, identify a parametric model for u - such as a log Karl D. Majeske Revised: 7-14-03 Page: 12

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims normal or gamma distribution - and estimate the distribution parameters using the observed data. Then estimate usage values for vehicles without claims according to the probability distribution fit to the sample data. Once each vehicle has a usage value (either observed or estimated) you can determine N(t, m) using Equations (10) and (11). 4.2 Time Based Failure Rate The second component of the predictive technique models the warranty claims associated with a vehicle. A non-homogeneous poisson process (NHPP) provides a flexible model to capture the varied warranty claim patterns observed in automobile warranty data. A NHPP is characterized by the intensity function (instantaneous probability of a failure) of Equation (2). To use the predictive approach in this paper, one must identify an intensity function that captures the time-based pattern of warranty claims. Once a manufacturer has identified the functional form of the intensity function, they must estimate the parameters. Rigdon (1989b) estimated intensity function parameters with multiple observations from a single system and developed confidence intervals on parameters. Hossain and Dahiya (1993) develop a NHPP model for software reliability with a data set that represented one system with 34 failures. These two techniques provide methods for fitting a NHPP to a system with many failures. Automobile manufacturers make predictions early in the vehicle life. At six or seven months in service, a relatively small percentage of vehicles will have experienced multiple warranty claims. This limits the ability to use the above techniques to directly fit a NHPP to warranty data. Thompson (1981) shows that for any NHPP, the intensity function and the hazard function of T7, the time to first failure, have the same functional form v (t) = h(t1). (12) This provides an alternate approach to characterizing the intensity function. First, estimate the distribution parameters for time to first warranty claim (failure). Then use these coefficients to Karl D. Ma eske Revised: 7-14-03 Page: 13

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims characterize a NHPP intensity function. Majeske (2003) describes how to use maximum likelihood estimation to fit the right-censored data encountered in warranty data analysis. By including non-failed vehicles as censored observations, this incorporates vehicles without a warranty claim when fitting the stochastic process. In some applications, manufacturers use a parametric model for time to failure, which they could use as a starting point for modeling warranty claims. To identify a parametric model for time to first warranty claim, one could use the hazard plot method suggested by Majeske, Lynch-Caris, and Herrin (1997). 4.3 Predicting Warranty Claims with the Model Let the function C(t) represent the cumulative number of warranty claims observed by the population through t time in service. Using this definition, C(t) is a strictly increasing step function that starts at 0. Let C(t) represent the predicted number of claims through time t as the expected value of the cumulative warranty claim function C(t) = E[C(t)]. Modeling a single vehicle with a NHPP, you can predict claims by integrating the intensity function as shown in Equation (3). To predict claims for a population assume homogeneity; that is, all vehicles in the population have the same intensity function or failure rate. For a one-dimensional warranty coverage - use the population size function N(t) of Equation (8) - the expected number of claims is the product of the population size and the integral of the intensity function tWL C(tw) = E[C(tw)] = N v (t)dt. (13) o For the two-dimensional warranty coverage of automobiles, one can predict the expected number of warranty claims by integrating the product of N(t, m) - the population size function of Equation (11) - and the intensity function Karl D. Majeske Revised: 7-14-03 Page: 14

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims tWL C(twL) = E[C(t)] = jN(t,m)v (t)dt. (14) 0 In some cases, manufacturers may want to relax the homogeneity assumption. Allowing the model parameters to be a function of X (covariates such as vehicle option content) would allow the manufacturer to make inference on how these variables affect warranty claim rates. Including covariates makes predicting claims somewhat more difficult because vehicles are at risk for different lengths of time. Letting ti represent the time the ith vehicle is at risk for warranty claims, the expected number of claims for the population of vehicles is N C(tWL) = E[C(t)] = v(t, Xi)dt. (15) i=l The complicating factor with this approach is that the ti values depend on the vehicle mileage, a random variable with limited or no information. This research suggests predicting warranty claims by vehicle subsystem (e.g., powertrain, wiring, fit and finish, etc.) and then aggregating subsystem predictions to obtain a prediction for the population of vehicles. To account for differences in option content (e.g., V6 versus V8 engine) the manufacturer can stratify the population for each subsystem and make a prediction with Equation (14) for each strata within each subsystem. Then, aggregate the predictions across strata (option content differences) and subsystem (portions of the vehicle) to arrive at a prediction for the population of vehicles. 5. Application of Predictive Technique This section assesses the predictive validity of the NHPP using a set of 9,168 luxury cars. To emulate the manufacturer's prediction process, this section makes predictions using the data available to the manufacturer at a cutoff date approximately seven months after vehicle assembly, termed the current date. Using the model developed in the previous section, warranty claim predictions are made through a cutoff approximately 45 months after assembly, called the future Karl D. Ma eske Revised: 7-14-03 Page: 15

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims cutoff. Because the vehicles have already observed the claims through the future cutoff, comparing predictions with actual claims allows assessing the predictive technique. 5.1 Estimating the Population To estimate the population function N(t, m) through the future cutoff date, a usage value u was calculated for each vehicle with a warranty claim prior to the current cutoff date. Using maximum likelihood estimation, a two parameter gamma distribution, with probability density function 1 (r) rle -r7e r> 0, (16) r(a) was fit to the data yielding the parameter estimates d = 3.548 and f = 16.629. Figure 2 shows a histogram of the observed values and the gamma distribution. To generate usage values for vehicles without a claim, assume that the vehicles with an observed usage value (56% of the population) are representative of the entire population. Usage rates were then simulated for vehicles without an observed value from a gamma distribution with a = 3.548 and y = 16.629 to match the observed data. Table 1 contains the N(t, m) values through the future cutoff date calculated using Equations (10) and (11). Table 1 also contains N(t), the population size ignoring the effect of mileage. Figure 3 plots the data of Table 1 to provide a graphical comparison of the mileage censored population size N (t, m) with the time only population size N(t). Karl D. Majeske Revised: 7-14-03 Page: 16

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 450 400 350 *~300-gm adsrbto 4? 250 0 "200 E 150 100 50 0 A rate: miles Iday in service Figure 2: Histogram of observed usage rates and best fit Gamma distribution Table 1: Population size by time in service (months) Time Time Time in Service in Service in Service J.Month.-, N(t~m) N(t) (Month.-, N(Lm N4tL J.Months) N(t-m) N(t) 0 9168 9168 16 8480 9168 32 4561 9168 1 9168 9168 17 8296 9168 33 4328 9168 2 9168 9168 18 8082 9168 34 4098 9168 3 9168 9168 19 7875 9168 35 3903 9168 4 9168 9168 20 7620 9168 36 3705 9168 5 9168 9168 21 7360 9168 37 3538 9168 6 9166 9168 22 7131 9168 38 3340 9168 7 9166 9168 23 6849 9168 39 3173 9168 8 9158 9168 24 6601 9168 40 2979 9052 9 9141 9168 25 6341 9168 41 2810 8895 10 9109 9168 26 6059 9168 42 2586 8691 11 9061 9168 27 5794 9168 43 2383 8450 12 8992 9168 28 5523 9168 44 2156 8165 13 8881 9168 29 5290 9168 45 1963 7813 14 8754 9168 30 5054 9168 15 8620 9168 31 4806 9168 _____________ 10000 T~ 9000 8000 7000 -*N 6000-.0 5000 -= 4000 -0;6 3000 -2000 -1000 0 N (t, m) - — 'k-IN~t 0 1 year 2 year 3 year t: Time in Service Figure 3: Estimated population size with and without adjusting for mileage Karl D. Ma eske Karl D. MajeskeR evised: 7-14-03 P g: 1 Page: 17

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 5.2 Estimating the NHPP Intensity Function This section characterizes the NHPP intensity function for three subsystems of the luxury cars. First, a functional form is identified for time to first claim using empirical hazard plots. The parameters are then estimated for each subsystem. A figure containing the empirical hazard function h(t) - calculated using life tables with an interval width of seven days (Lawless 1982) - and the fit line graphically shows the aptness of the model. Subsystem 1: Majeske (2003) derives the Weibull-uniform mixture model with cumulative distribution function p t q - e-^} 0 t<A F(t) A)+ (17) p+q (_e-^t) t> A to fit warranty claims from subsystem 1 of these same luxury cars. The distribution was developed to model warranty claims as a mixture of manufacturing defects and usage related failures using the data available about two years after vehicle assembly. The mixture distribution of Equation (17) defines a NHPP with intensity function p- + (1 - p ()[aC 3 (Ot)- le-(ot T* _ O<t<T* v (t)= 1-p +(, -(1p) - e (18) ap(at ot)- t> T To apply the prediction method, the mixture model was fit to the data available seven months after vehicle assembly. Table 2 contains the mixture distribution parameter estimates obtained via maximum likelihood estimation and the standard errors estimated with the observed Fisher information matrix as outlined in Majeske (2003). Figure 4 shows the empirical hazard function Karl D. Majeske Revised: 7-14-03 Page: 18

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims h(t) and the mixture hazard function using the parameter estimates from Table 2. The mixture model appears to provide a good fit to the data available at 7 months in service. Table 2: Parameter Estimates for Time to First Warranty Claim. Subsystem 1 Subsystem 2 Subsystem 3 Mixture Distribution Weibull Distribution Linear Hazard Parameter Estimate Std Error Parameter Estimate Std Error Parameter Estimate Std Error 0.00027 0.000021 o 0.00324 0.00006 Po 0.0001558 0.0000379 P 1.006 0.0339 P 1.057 0.0154 Pi 0.0000174 0.0000022 Up 0.047 0.0048 0.232 0.0295___ 0.0009 ~ 0.0008 = 0.0007 - 0 | 0.0006 o 0.0005 g 0.0004 0.0003 - 0.0002 0.0001 0 U U 5 l 13 l * Observed Values Fit Line U 131 E3 a [ I l I I II I I 0 5 10 15 20 Time in Service (Weeks) 25 30 Figure 4: Subsystem 1 - Empirical and mixture model hazard functions Subsystem 2: Reliability engineers often use the Weibull distribution of Equation (5) to model field failure and bench test data for subsystem 2. Modeling time to first failure with a Weibull distribution results in a special case of the NHPP, the power law process, with intensity function of Equation (4). The Weibull distribution was fit to the time to first subsystem 2 claim data available at the current cutoff. The parameter estimates and associated standard errors appear in Karl D. Ma eske Revised: 7-14-03 Page: 19

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims Table 2. Figure 5 plots the Weibull hazard with the observed values for subsystem 2. Notice that the Weibull does not capture the apparent decreasing hazard in weeks 15 through 30. 0.006 0.005 - X 0.004 - 0.003 Observed Values 0.001 - Fit Line 0 l I I I I I 0 5 10 15 20 25 30 Time in Service (Weeks) Figure 5: Subsystem 2 - Empirical and Weibull hazard functions Subsystem 3: The manufacturer did not use a parametric model for subsystem 3 claims; rather, they used the log-log R(t) approach of Equations (6) and (7). After evaluating the empirical hazard plot shown in Figure 6, the linear hazard function h(t) = o + A1t (19) was identified for the Subsystem 3 data. The linear hazard of Equation (19) defines a NHPP with intensity function v (t) = o + P1t. (20) The parameters of the linear hazard model were estimated using ordinary least squares by fitting a line to the empirical hazard function. The parameter estimates and associated standard errors appear in Table 2. Figure 6 shows the empirical hazard along with the linear hazard, which appears to provide a good fit to the data. Karl D. Ma eske Revised: 7-14-03 Page: 20

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 0.001 0.0009 *- I Observed Values 0.0008 - Fit Line 0.0007 - E v 0.0006 t 0.0005 I 0.0004- m ' 0.0003 2 = 0.0002 - - d E 0.0001 0 I I I I I I 0 5 10 15 20 25 30 Time in Service (Weeks) Figure 6: Subsystem 3 - Empirical and linear hazard functions 5.3 Predicting Warranty Claims This section predicts warranty claims through 45 months in service using the population size N(t, m) and NHPP intensify functions developed in sections 5.1 and 5.2 respectively. Because months in service don't correspond to calendar time, predictions were made for 30 day (month) intervals. To make predictions, Equation (14) was evaluated for each monthly interval Table 3 contains the observed claims by month at the current cutoff, the predicted claims through the future cutoff, and the observed claims at the future cutoff for each of the three subsystems. Comparing the total claims through 45 months, the NHPP approach predicted within 7% of observed claims for all three subsystems. Figures 7, 8 and 9 graphically depict the data in Table 3 by comparing the observed and predicted claims by month for subsystems 1, 2, and 3 respectively. Evaluating Figure 7, the Weibull portion of the mixture (t > 4 MIS) over predicts in the period 9 - 22 MIS and under predicts in the period 28 - 45 MIS. While the model has some correlation structure in the residuals, it does provide a good fit to the subsystem 1 warranty claims. From Figure 8, the power law process under predicts the subsystem 2 infant mortality failures and does not predict the downward trend at 10 MIS. However, the model very accurately predicts claims from 30 to Karl D. Ma eske Revised: 7-14-03 Page: 21

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 45 MIS and provides an accurate prediction of total claims. Figure 9 shows the linear intensity model lends a good fit to the subsystem 3 data, yet slightly over predicts claims after 30 MIS. 6. Conclusion Automobile manufacturers rely on predictions for the number, timing, and cost of future warranty claims. Omitting the mileage data contained on automobile warranty claims, and making predictions purely in the time domain, ignores vehicle s leaving coverage prior to the warranty time limit. Incorporating a two-dimensional aspect will result in a predictive model that better represents the process being modeled. This paper develops a NHPP predictive modelthat has a parametric component (time to first failure) and provides a great deal of flexibility in application. The NHPP approach also allows incorporating past experience when identifying the failure process and bases predictions on early field performance of the products. Using the NHPP will result in more accurate predictions of warranty claims and support decision-making when implementing engineering design and manufacturing process changes. More accurate predictions will also assist the manufacturer in allocating reserves to pay for repairs covered under warranty. Karl D. Majeske Revised: 7-14-03 Page: 22

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims Table 3 Observed and Predicted Warranty Claims by Month in Service ISubsystemi J Subsystem 2 Subsystem 3 Time Observed Predicted Observed Observed Predicted Observed Observed Predicted Observed in Service Claims at Claims through Claims at Claims at Claims through Claims at Claims at Claims through Claims at (Months) I6MIS 45 MIS 45 MIS 6 MIS 45 MIS 45 MIS I6MIS 45 MIS 45 MIS 0 2 3 4 5 6 7 8 9 1 0 1 1 12 13 14 15 1 6 1 7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 8 1 100 81 188 155 189 199 158 200 164 160 170 185 162 190 93 96 98 41 73 51 50 73 74 73 84 73 64 73 60 73 58 72 57 71 61 70 40 69 43 68 41 67 49 65 32 63 41 61 43 59 45 58 39 55 56 53 42 51 43 49 52 47 38 45 47 43 50 41 46 39 43 37 54 35 62 33 46 32 51 30 43 29 48 27 35 26 33 24 36 23 36 21 40 19 33 17 19 16 21 414 1010 1039 1241 1576 1059 578 469 414 414 786 1022 844 1080 869 1301 886 1664 899 1143 909 723 917 775 924 903 929 854 931 881 932 856 929 694 922 587 913 605 903 554 891 522 875 521 855 522 836 468 811 534 786 530 764 529 735 507 710 461 684 437 655 491 628 502 600 501 576 526 551 567 525 618 499 603 474 561 450 522 429 530 408 476 390 436 369 399 351 384 330 351 312 397 287 374 265 388 240 373 219 332 90 90 90 66 53 69 71 74 79 103 94 119 122 115 151 133 135 159 141 156 182 138 176 230 197 264 217 247 236 258 256 291 274 252 290 238 306 263 320 263 334 322 345 307 355 420 363 389 369 391 372 350 377 407 377 393 378 345 378 348 374 329 371 360 366 359 363 371 358 427 351 420 343 395 335 325 327 290 320 290 312 261 306 212 296 166 289 174 278 149 268 181 253 204 238 154 220 150 205 131 I____ 1001 2786 2784 J 7386 30415 28418 1 864 12810 12175 Karl D. Ma eske Karl D. MajeskeR evised: 7-14-03 P g: 2 Page: 23

A Non-Homogeneous Poisson Process Predictive Model for Automobile Warranty Claims 250 T 200 -.a 150 -, 100 - 50 - * o 0 Observed Claims at 45 months 0 0 Prediction made at 6 months 00 ao0~ 00100000 0 10 20 30 Time in Service (Months) 40 50 Figure 7: Subsystem 1 - Mixture Model Prediction with Observed Claims 180T 1600'1400'"'1200'*f. 1000'0 w 800'E600'400 ' 200' o Observed Claims at 45 months Prediction made at 6 months 0 0 0 p0 0 00 00000000 00a000 00000 0 000 L0 0 10 20 30 40 50 Time in Service (months) Figure 8: Subsystem 2 - Power Law Process Prediction with Observed Claims 450 400 350 -300 50 0 1 50 100 50.0 0 0 Oseve Cais a 4 mnts Prdcinmd0t6mnh 0 10 20 30 40 50 Time in Service (months) Figure 9: Subsystem 3 - Linear Intensity Function Model Prediction with Observed Claims Karl D. Ma eske Karl D. MajeskeR evised: 7-14-03 P g: 2 Page: 24

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