Using accelerated life tests data to predict warranty cost under imperfect repair
Introduction
A warranty is a contractual agreement between buyers and sellers in which the manufacturer or resellers commit to providing post-sale services concerning product failures for a discounted price or free of charge (Blischke, 1995). Warranty management is a key factor that influences the market performance and reputation of manufacturers. Attractive warranty terms usually promote product sales by indicating high quality and reliability (Murthy & Blischke, 2006). Many companies would highlight their product warranties as a marketing tool. For example, computer hardware and electronic device manufacturers usually provide free repair warranties for a specified period for their personal computers, e.g., laptops and cell phones. In the literature of warranty policy analysis, two-dimensional warranty (Gupta et al., 2014, Huang et al., 2016, Jack et al., 2009) and extended warranty (Huang et al., 2016, Jung et al., 2015, Vahdani et al., 2013, Ye and Murthy, 2016) have received considerable attention recently. Servicing a warranty incurs additional costs to the manufacturer, making it a trade-off to balance the warranty cost and sales revenue with the objective of maximizing the profit. Manufacturers should consider the quality and failure mechanism of products (Liu, Wu, & Xie, 2015), customers use rates (Hong & Meeker, 2010), maintenance policy (Sahin and Polatoglu, 2012, Shang et al., 2016) and sales (Akbarov & Wu, 2013) simultaneously to design warranty policies.
Field failure data and historical warranty claims are important records to forecast warranty claims and cost (Akbarov and Wu, 2012, Gupta et al., 2014, Limon et al., 2015, Majeske, 2007, Tseng et al., 2016, Yang et al., 2016). Many previous studies focused on the analysis of warranty data, summarized in Wu, 2012, Wu, 2013. However, for new products that have not been put on the market, it is challenging to design warranty policies and predict warranty cost because there is no such failure record. Murthy and Djamaludin (2002) gave a literature review on new product warranty by considering marketing, logistics, and warranty management simultaneously. Huang, Liu, and Murthy (2007) proposed a joint optimization model that involved reliability, warranty, and price for new products. Xie and Ye (2016) considered stochastic sales to predict aggregated discounted warranty cost for new products. Nevertheless, as an essential procedure to predict the number of failures during the warranty coverage, the reliability prediction was not considered in these studies.
Accelerated life tests (ALT) could be conducted to predict the reliability of new products. In an ALT, the environmental stresses are elevated to accelerate the occurrences of failures. By observing failure times (either exact or censored) from an ALT, the accelerated life (AL) model parameters are estimated and test planners can use these estimators to make statistical inferences of the lifetime distribution of test products, which helps to predict the number of warranty claims. In Yang (2010), optimal 3-level compromise ALT plans were discussed to minimize the asymptotic variance of maximum likelihood estimator of the warranty cost. For an overview of accelerated reliability tests, one can be referred to Meeker and Escobar (2015).
By utilizing ALT experimental data, field reliability prediction could be made. Van Dorp and Mazzuchi (2004) proposed a Bayesian inference method to derive the posterior lifetime quantiles by assuming exponentially distributed lifetime. By considering the variation in field environment, Meeker, Escobar, and Hong (2009) described a reliability prediction model for newly designed products with two failure modes. Pan (2009) introduced a calibration factor to the field stress and employed a Bayesian framework to predict reliability from ALT and field failure data. In Jiang and Chen (2015), the physics of failure and effect analysis were taken into account to predict field reliability. To deal with ALT data with a failure-free life, Chen, Tang, and Ye (2016) proposed a quantile regression framework which was distribution-free and efficient to deal with heavily censored data. Nevertheless, to our knowledge, there is no research in literature that addresses the variability in warranty prediction based on ALT data.
We consider two sources of variability when ALT data are used to predict the warranty cost. First, the estimated parameters of acceleration regression model by ALT data have sampling distributions, which introduces variability in the predicted reliability under field condition. Second, the field environment is not constant in practice, thus the randomness of field stress among products should also be addressed when predicting the lifetime.
In this paper, we propose a warranty cost analysis framework based on ALT experimental data. The field stress is regarded as a random variable. Field lifetime distribution is predicted as a random variable as well and is used to calculate the warranty cost by evaluating the number of warranty claims in a given warranty period. Our goal is to predict the expected warranty cost and give confidence intervals. Furthermore, from the framework we can calculate the warranty cost risk, i.e., the probability that the expected cost exceeds a specific threshold. Repairs and replacements are generalized as imperfect repairs in this study, thus a g-renewal equation (Kaminskiy & Krivtsov, 2000) needs to be solved to obtain the expected cumulative number of warranty claims within the warranty period. Generally there is no close solution for g-renewal equations. Previous statistical research have proposed various methods to approximate the g-renewal equation (Yevkin & Krivtsov, 2012), and some were used to carry out reliability and warranty analysis (Tanwar et al., 2014, Wang and Yang, 2012, Yang et al., 2016).
The remainder of the paper is organized as follows. Section 2 presents the framework that models ALT data to predict the field reliability. In Section 3 warranty cost model is constructed and the approximation method to evaluate warranty claims is described. Numerical simulation methods to compute the predicted warranty cost and risk are presented in Section 4. In Section 5, an illustrative example along with sensitivity analysis is given. Finally, Section 6 concludes the paper and discusses areas for future research.
Section snippets
ALT data modeling
The lifetime of the product of interest is assumed to follow a Weibull distribution. The ALT planning under Weibull lifetime assumption has been intensely investigated in literature and widely applied in engineering problems (Sha & Pan, 2014). Similarly, lognormal and log-logistic distributions also belong to the log-location-scale distribution family (Meeker, 1984), which can be analyzed in an analogous way.
We consider an ALT experiment with one stress variable . By assuming a log-linear
Evaluation of number of warranty claims and cost model
A free-rectification warranty of period is assumed to be attached to the product of interest. The rectification type is imperfect repair with virtual age assumption (Tanwar et al., 2014), which generalizes the cases of perfect repair (or replacement) and minimal repair.
Simulation procedures to predict warranty cost with parameter and field variability
Since there is no close form for the distribution or density function for the random variable , we need to employ MC methods to draw samples to compute the three measures in Section 3.2. By assuming the distribution of and as a bivariate joint distribution (as in Eq. (8)) and approximating the number of warranty claims as in Eq. (11), the MC sampling procedure is simplified to one with only two variables: and . Recall that the density given in Eq. (8) cannot be necessarily
Numerical example
In this section, we use a numerical example to illustrate the proposed method. The ALT example from Meeker and Escobar (1998, p. 535) is modified for illustration. The example is merely used to describe the framework systematically. Thus the example is adjusted for better illustration by changing the time unit from one day to one hour, which makes the assumptions more reasonable in practice. In the example the lifetime of the new product was assumed to follow a Weibull distribution. The product
Concluding remarks
In this paper, we have proposed a novel framework to predict warranty cost under imperfect repair based on ALT data. Variability of ALT data and field stress is taken into account simultaneously. Large sample approximations are employed to derive the marginal distribution of Weibull parameters. We use Monte Carlo simulation methods based on Metropolis-Hastings sampling to compute the warranty cost measures. Compared with the original MC simulation, our methods give the results much more
Acknowledgement
The authors would like to thank the editor and the anonymous referees for valuable comments and constructive suggestions, which have considerably improved the quality of the paper. This work was supported by National Natural Science Foundation of China (Grants No. 71532008) and Theme-based Research Scheme of Research Grants Council (No. T32-101/15-R) at Hong Kong SAR.
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