Utilizing experimental degradation data for warranty cost optimization under imperfect repair
Introduction
Warranty has been widely used by manufacturers to promote their products. A warranty is a contractual agreement between buyers and sellers in which the manufacturer provides post-sale services concerning the product quality for a discounted price of free of charge [1], [2]. At the testing phase of a product before it is put on the market, manufacturers generally cannot precisely estimate its lifespan. However, maintenance and warranty policies need to be determined during this phase to get the product prepared for introduction and marketing. To achieve more accurate reliability prediction for new products without any field data, engineers usually resort to reliability tests. For the purpose of saving testing time and cost, many reliability tests are conducted under accelerating covariates, and this also helps to evaluate reliability characteristics under dynamic environments [3]. The results from reliability tests can be utilized to predict the warranty cost, thereby facilitating the decision-making concerning maintenance and warranty policies.
Degradation analysis has been proven to outperform failure time modeling approaches for numerous engineering systems of which quality characteristics can be measured over time, both in tests and under field usage [4], [5]. For products suffering from observable degradation, with the aid of frequent degradation measurements, degradation tests are more efficient than life tests [6]. On the other hand, the system repair policy, which is often included in a warranty contract, can yield various outcomes for degrading systems. One general and realistic assumption is that the repair conducted on degrading products reduces the degradation level. More often than not, the repairs are not able to make the product as-good-as-new. In such cases, the repairs are “imperfect” [7]. For example, tamping is carried out to repair tiny geometric degrading cracks on railway tracks, yet this operation cannot completely eliminate the geometric degradation. Manufacturers or maintenance providers may change the effect of imperfect maintenance within a controllable range, which can be subject to varying cost and time. Warranty cost prediction and optimization of maintenance policies under imperfect repairs are of interest to many manufacturers. Motivated by this need, this study proposes a warranty cost optimization method for continuously degrading systems.
Under the proposed framework, we consider several sources of variability for cost prediction to enable decision makers to quantitively understand the uncertainty in predicted cost. First, the parameter estimates are subject to data uncertainty. Second, the unit-to-unit field usage condition varies for the product of interest. Third, the impact of imperfect repair is random, as illustrated in Fig. 1. Considering these realistic assumptions, we choose the optimal repair policy to minimize the aggregate warranty cost for products under a free-rectification warranty with a fixed time length.
For products that have been sold on the market for a considerable time, warranty cost analysis can be directly conducted based on historical warranty claims. Wu [8], [9] gave comprehensive reviews on warranty data analysis. In recent literature, two-dimensional and extended warranty policies have drawn considerable attention [10], [11], [12] In Jack et al. [13], the repair-replace strategy is optimized for products with usage effect sold under two-dimensional warranty. Dai et al. [14] utilized an accelerated failure time model to study the effect of usage rate with two-dimensional warranty data. For maintenance models under warranty, one can be referred to Shafiee and Chukova [15] for an overview. Specifically, for imperfect repair models, Park and Pham [16] used a quasi-renewal process to model the failure process under exponential life assumption. Warranty prediction based on a general imperfect repair model [17] was proposed in Yang et al. [18]. Zhao and Xie [19] utilized accelerated life testing data to predict warranty cost for products under various usage condition. However, these studies are all based on failure time models rather than degradation models.
Warranty policies are closely associated with maintenance policies [20], [21]. Under different maintenance schemes, a warranty policy can lead to contrasting costs [22]. Luo and Wu [23] reported that warranty policies considerably influenced the total profit of manufacturers and collectively optimized the profit for a portfolio of different products via a mean-variance optimization approach. In recent literature, imperfect repair models were developed to characterize the effect of imperfect maintenance on system degradation [24], [25]. In Zhang et al. [26], a stochastic filtering method was investigated to plan condition-based maintenance under a random improvement factor model. Liu et al. [27] employed an improvement factor model of power-law form to optimize maintenance planning for systems suffering from both degradation and external shocks. In Chen et al. [28], both positive and negative effects of imperfect maintenance were discussed for degrading systems. As suggested in the literature [29], it is more realistic to model the improvement factor as random because the uncertainty in imperfect maintenance is inevitable for many systems. In many engineering cases, the improvement effect of imperfect repairs can be controlled and the cost usually varies with the effect [30].
In degradation modeling, environmental covariates can be conveniently incorporated to study the reliability of products under heterogeneous conditions [31], [32]. Numerous studies on accelerated degradation tests showed that the reliability under various stress level could be evaluated by extrapolation [6]. For new products, experimental data play a major role in predicting field reliability thereby the warranty claims. Tseng et al. [33] jointly modeled the life testing data and field claims to predict the warranty cost. Nevertheless, to our knowledge, none of the studies investigated how to use degradation data with covariates to predict the warranty claims within a period, and this study intends to fill the gap.
In this paper, for the first time, we propose an optimal imperfect repair policy for systems subject to continuous degradation to minimize the warranty cost within a fixed coverage period. The system parameters are estimated from degradation tests with controllable covariates, and the field usage condition is random. The proposed methods can help decision makers to choose the optimal imperfect repair policy with random improvement factor. The expected cost incurred by a single product unit is evaluated to optimize the repair factor.
The remainder the paper is organized as follows: Section 2 presents the degradation data modeling methods with parameter estimation and describes the random imperfect maintenance model. Evaluation of predicted warranty cost and the optimization of maintenance policy are discussed in Section 3. In Section 4 we use simulated data sets with various sample sizes to illustrate the proposed methods. An application example is shown in Section 5. Finally, Section 6 concludes the paper.
Section snippets
Wiener degradation process and test data modeling
In this study, a univariate Wiener process is used to model the degradation path. Wiener process as a degradation model appeared in numerous literature concerning products with non-monotone deterioration properties [34], [35]. Let Y(t) be the Wiener degradation process and it can be expressed as a drifted Brownian motion:where η > 0 is the drift coefficient, σ is the diffusion coefficient, Λ(t) is a time-scale transformation function to describe the nonlinearity in
Evaluation of warranty claims
Manufacturers wish to know the expected number of warranty claims in a fixed period to forecast the future cost that will be induced to them. With given η, σ, u and v, suppose that the manufacturer provides a warranty coverage of length tW, the mass function of NW, i.e., the number of warranty claims up to tW, is given bywhere Tk and follow known IG distributions as advised in Propositions 1 and 2, of which the PDF and CDF can be conveniently
Comparison of computational efforts
It is interesting to see how the warranty claim prediction method utilizing Tk and ΔTk performs in comparison to MC simulation. We set the input parameters to be fixed as follows:
Afterward, with different combinations of NSim for MC simulation and warranty period tW, we calculate E(NW; η, σ, u, v) with both MC simulation and the proposed analytical method. The results and computing time is listed in Table 2. MATLAB is used to compute the result by an Intel Core CPU with
An application example
An electric corporation conducted a degradation test on a type of mechanical systems with a sample size of 104. Test units were placed under chambers with various temperature and humidity. The main quality characteristic is the corrosion which is regarded as a degradation measure. For confidential reasons, the time scale has been transformed. Under the normal environments, the system can take up to more than one year to fail. Thus, the temperature and humidity stresses are elevated for the test
Concluding remarks
The paper presents a warranty prediction and optimization framework which utilizes the experimental degradation data with covariates. The methods can be applied to new products that will be sold with one-dimensional warranty. To be more realistic, imperfect repairs with random improvement factors are assumed. With the aid of moment generating functions, the expected number of warranty claims within a given warranty length is derived analytically. The computational burden is significantly
Acknowledgements
We would like to thank the editor and anonymous reviewers for their constructive comments on the original manuscript which led to this improved version. This work was supported in part by the Research Grants Council of Hong Kong under a theme-based project Grant (T32-101/15-R) and a GRF (CityU 11203815) and in part by the National Natural Science Foundation of China with grant no. 71532008.
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