Production, Manufacturing, Transportation and LogisticsOptimal warranty policy with inspection for heterogeneous, stochastically degrading items
Introduction
The impact of heterogeneity on reliability characteristics of manufactured items was extensively studied in the literature (see, e.g., Finkelstein, 2009; Finkelstein & Cha, 2013 and references therein). Heterogeneity is an intrinsic property that characterizes many real-life populations of technical and biological objects. On the other hand, homogeneity can be considered as some simplifying assumption relevant for a specific, e.g., laboratory environment. The origin of heterogeneity in practice is often in different quality of resources and components used in the production process, operation and maintenance history, random environment, human errors, etc.
In data analysis, heterogeneity is often modeled via the unobserved random variable(s) that is(are) usually called frailty(ies) (Vaupel et al., 1979). Frailty characterizes relative susceptibility to failures of homogeneous subpopulations comprising heterogeneous populations. Note that, heterogeneity can change the distributional properties of lifetimes that describe the corresponding homogeneous subpopulations, which should be taken into account in practice. For instance, the failure rate of items in heterogeneous population can exhibit a dramatically different shape than that in a homogeneous one with the same baseline lifetime distribution. A well-known example is when the Weibull distribution with the increasing failure rate becomes a distribution with the bell-shaped failure rate for the heterogeneous case, i.e., first increasing and then decreasing to 0 (see, e.g., Block et al., 2003). Thus, neglecting heterogeneity can result in significant errors in describing the performance quality of heterogeneous items.
To the best of our knowledge, warranty polices for heterogeneous products were addressed only in Lee et al. (2016), where the non-renewable warranty policy was considered for items from a heterogeneous population consisting of two subpopulations. In the current paper, we will consider a continuous case defined by the corresponding continuous (not binary) frailty, which is common in practice. Moreover, the approach and the model suggested here is different from that in Lee et al. (2016), where the necessary information for distinguishing between two subpopulations (weak and strong) was contained in the number of minimal repairs performed on a system prior to the decision time.
In the current paper, we are considering deteriorating items with degradation observed at inspection. This degradation in the baseline homogeneous subpopulation is modeled by the inverse-Gaussian (IG) process, which is becoming a popular tool in describing degradation in reliability applications (see Qin et al., 2013; Wang & Xu; 2010 and Ye & Chen, 2014 to name a few). Other degradation processes, e.g., the gamma process can be discussed in the same line; however, this needs specific analysis and justification. It is important to note that although the baseline homogeneous process is the IG process, the corresponding degradation for the whole heterogeneous population is described by the mixed IG process, which is not formally the IG process anymore, as it does not possess the property of independent increments. This makes the stochastic analyses in our paper quite challenging.
We will consider the renewable warranty policy (Bai & Pham, 2006; Chien, 2008; Chien et al., 2019; Zuo et al., 2000), whereas the nonrenewable policy can be discussed in a similar way. That is, under this warranty, a product that fails within its warranty period is replaced by a new one and the warranty is renewed at no charge to a customer.
The innovating feature of our approach is that, in accordance with the proposed model, an item's state, i.e., the level of its degradation, is observed at some intermediate time T, . Thus, the observed level of degradation at inspection will act as a necessary information that will enable to distinguish between the weaker and the stronger subpopulations. As far as we know, this approach has not been considered in the literature so far. As it will be shown, this additional information can make a difference and provide a cost-effective warranty policy, which makes it practically sound.
Thus, the weaker items should be replaced at T, whereas the stronger ones should continue operating. In order to justify and apply our reasoning, the corresponding methodology has to be properly developed. As a practical example, consider a drilling rig with the warranty period that is used for the well drilling. At some intermediate inspection time T, , the level of degradation is observed (occurrence and propagation of fatigue cracks). If, given this information, it follows that the probability for the drilling rig to successfully operate without failure during the rest of the warranty period is small, it is replaced by a new drilling rig. Then the new warranty period for this replaced item is applied. Another example is a gas turbine. If an inspection reveals unacceptable vibration level in rotating bearings, the probability of a catastrophic failure during the rest of warranty period is high. After replacement of bearings, the new warranty period can be applied. We can also think about contamination of reverse osmosis filters used in water desalination process. If during the inspection, the unacceptable level of contamination (manifested by the increased pressure that can cause damage and failure of pumps) is revealed, the filters are replaced and the new warranty period starts.
In principle, the case of more than one inspection can be also considered in a similar way (although much more cumbersome), however, due to practical reasons, a warranty with one additional inspection can be already sufficient and convenient for the significant decrease in the expected costs (see our illustartive results in Section 5). This topic needs further clarification and justification in the future research.
The proposed warranty policy in this paper has similar features with the warranty policy with preventive maintenance. However, our approach is not just age-based, as in conventional models, but is degradation and age-based. Thus, at inspection time, an item is not necessarily replaced, but this action is performed only if degradation exceeds a certain level.
The paper is organized as follows. In Section 2, we describe the corresponding degradation process, whereas the probabilistic analysis of the warranty policy is performed in Section 3. In Section 4, the optimization problem is formulated and the required expected costs are derived. Section 5 illustrates our findings with numerical examples. Finally, concluding remarks are given in Section 6.
Section snippets
Heterogeneous degradation process
The warrantied items are subject to stochastic degradation, which is the main cause of their failures. In order to proceed with the description of the mixed/heterogeneous degradation/wear processes, recall first, some relevant definitions for mixtures of lifetimes.
Denote the time to failure of an item from the subpopulation characterized by the realization of the mixing variable (frailty) by and by , the pdf of . The corresponding survival function (Sf), the cumulative
Probabilistic analysis of the warranty policy
We consider a renewable warranty policy. Under this policy, a product, which fails within its warranty period , is replaced by a new one and the warranty is renewed at no charge to the customer. Furthermore, assume that during the warranty period, an inspection is performed at time , and, depending on the observed information, an item is preventively replaced by a new one or no replacement is performed. More specifically, at time , an item is sold to the customer and starts
Cost-wise analysis
In the previous section, we showed superiority of the proposed warranty policy in probabilistic terms. However, in practice, the cost aspect is also of a paramount importance and, therefore, we will discuss now the corresponding cost-minimization problem. For this, we first must derive the expected cost for the proposed warranty policy. Note that, the cost optimization problem also makes sense in the case of the homogeneous IG process (although much simpler), whereas, as already mentioned, the
Numerical illustration
Let U = 15, d = 2.5, Cr=10,Cp=5,CI=0.1; η=0.2, Λ(t)=at, μ(z)=bz and . Similar results can be obtained for other relevant values of parameters and mixing distributions. We have performed various numerical experiments in this respect and the typical forms of the functions and graphs are presented in what follows.
Recall that U is the duration of warranty, d is the failure threshold (whereas, T is the time of inspection and is the screening out threshold to be optimized). Cr is the
Concluding remarks
We consider deteriorating items with degradation observed at an inspection. The inverse-Gaussian (IG) process models this degradation in the baseline homogeneous subpopulation. However, the whole population is heterogeneous and the mixed IG process, therefore, describes the corresponding degradation. The latter does not possess the independent increments property and, in fact, has not been studied sufficiently in the literature (e.g., Peng, 2015), which makes our task quite challenging.
We
Acknowledgments
The authors are grateful to the reviewers for helpful comments and advices. The work of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: 2019R1A6A1A11051177).
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