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This study considers a repairable production system operated under an age-based preventive replacement policy that is subject to independent random production waits and failures. In addition to the age-based replacement policy, we propose a maintenance model that uses production waits to schedule preventive replacement. That is, a decision maker can preventively replace the system either during a production wait or at the age threshold, whereas if a failure occurs during production, the decision maker must decide whether to perform a minimal repair or a corrective replacement to restore the system. Under the above setting, we develop a semi-Markov decision process (SMDP) to obtain the optimal maintenance policy that minimizes the long-run average maintenance cost rate. We establish the existence of the optimal maintenance policy and provide an algorithm to numerically obtain the optimal action for each state. We further generalize the model to incorporate imperfect maintenance and the nonhomogeneous arrival of production waits. In the latter case, it is computationally intractable to optimize the SMDP using a value iteration algorithm due to the curse of dimensionality. To address this challenge, we further develop an approximate dynamic programming framework to generate high-quality solutions. An attractive feature of our model is its generality, such that the model includes many existing maintenance models as special cases. A real-world example from a steel factory is used to demonstrate the proposed model.

Most systems can fail in multiple ways, and the failure modes are usually positively correlated. This phenomenon complicates the reliability analysis and makes the corresponding maintenance planning challenging. This article proposes a condition-based maintenance policy for systems that are subject to both degradation-induced soft failure and sudden hard failure, where a higher degradation level leads to a higher hazard rate of the hard failure. The Wiener process is adopted for the degradation process, and the Weibull model is used to describe the baseline hazard rate of the hard failure. The degradation level is then treated as a time-varying covariate that affects the hazard rate of the hard failure, and the closed-form of the reliability function is derived by using the Brownian bridge theory. An inspection/replacement maintenance policy is employed, and the long-run cost rate is formulated based on the semiregenerative property of the system state. The optimal inspection interval and the preventive replacement threshold are then jointly determined by minimizing the long-run cost rate. A numerical study on a hydraulic sliding spool system is conducted to validate the derived reliability function and the maintenance policy.

The optimal plan of a time-censored accelerated life test (ALT) depends on unknown parameters in the ALT model. The general strategy for coping with this problem is to replace the unknown parameters with their a priori estimated values, after which we can design test plans that are robust to deviations of the estimated values. In an ALT with log-location-scale lifetime distribution and linear acceleration relation function, the slope parameter of the acceleration relation function usually has the highest uncertainty. Instead of prespecifying the slope parameter in the ALT design, in this article, we propose a new design criterion that minimizes the risk of misclassifying whether the product's p th quantile meets the design specification. Through a case study of a circular electric connector, we illustrate the proposed method, and compare it with other test criteria. The comparison result shows that the proposed plan exhibits good performance in both estimating and s -testing the product's p th quantile.

Components in a system work under the same dynamic operating conditions, and their lifetimes are generally positively correlated. Ignorance of the correlation may lead to a significant bias in the evaluation of system reliability. Using the cumulative exposure principle, we model the equivalent operating time of the components, resulting from the cumulative effects of the dynamic environments, as a monotone increasing stochastic time scale. Commonly-used models, such as the compound Poisson, gamma, and the inverse Gaussian processes, are adopted for the stochastic time scale. Based on the above settings, reliability models for multicomponent systems are developed. We investigate how the stochastic time scale influences the system reliability and the correlations between component lifetimes. Under the stochastic time scale, the component lifetimes are shown to be positively quadrant dependent. Overlook of the correlation would overestimate the reliability of a parallel system but underestimate the reliability of a series system. When the stochastic time scale degenerates to a deterministic function of the calendar time, on the other hand, the system reliability becomes the reliability of the system where components work independently. The proposed models are successfully applied to lifetime data of brake pads in the automobile braking system.

During the useful life period of a costly system, it has been a common practice to perform imperfect preventive maintenances (PMs) with the purpose of failure prevention and useful life extension. Nevertheless, a system cannot be restored to an as-good-as-new state even though the imperfect PM actions are performed with a high frequency. This is known as the saturation effect and it is commonly overlooked in the existing literature. Motivated by a PM problem in a manufacturing company, this study proposes two PM models to capture the dynamics of the saturation effect. The first model divides the system deterioration into recoverable damage and irreversible intrinsic fatigue. The PMs are assumed to be effective only in healing the first type of damage. When the intrinsic fatigue for some complex systems cannot be well defined, we propose another model that generalizes the existing virtual age models by allowing the proportion of virtual age reduction to depend on the PM frequency. The long-run average costs of the two models are derived, and optimization of the cost models is investigated. The proposed PM models are then applied to two types of mechanical systems in the manufacturing company. The case study shows that ignorance of the saturation effect will make inferior maintenance policy that incurs substantial losses. The proposed models are also shown to be robust in the sense that the parameter estimation errors cannot significantly increase the system operational cost rate.

Condition-based maintenance (CBM) is proved to be effective in reducing the long-run operational cost for a system subject to degradation failure. Most existing research on CBM focuses on single-unit systems where the whole system is treated as a black box. However, a system usually consists of a number of components and each component has its failure behavior. When degradation of the components is observable, CBM can be applied to the component level to improve the maintenance efficiency. This paper aims to study the optimal inspection/replacement CBM strategy for a multi-unit system. Degradation of each component is assumed to follow a Wiener process and periodic inspection is considered. We cast the problem into a Markov decision framework and derive the optimal maintenance decisions that minimize the maintenance cost. To better illustrate the optimal maintenance strategy, we start from a 1-out-of-2: G system and show that the optimal maintenance policy is a two-dimensional control limit policy. The argument used in the 1-out-of-2: G system can be readily extended to general cases in a similar way. The value iteration algorithm is used to find the optimal control limits, and the optimal inspection interval is subsequently determined through a one-dimensional search. A numerical study and a comprehensive sensitivity analysis are provided to illustrate the optimal maintenance strategy.

In addition to conventional censoring schemes such as Type-I or Type-II censoring, progressively censoring is a useful method to reduce cost and obtain additional reliability information in accelerated life testing. Statistical inference for accelerated life testing (ALT) with Weibull distribution and progressively censoring is found to be difficult. This paper develops generalized fiducial inference techniques for the constant-stress ALT model with Weibull distribution and progressively Type-II censoring. Simulation studies reveal the good performance of the proposed inference methods. An example is given for illustration.

This study investigates modeling, estimation and optimization of destructive degradation tests (DDTs) for highly reliable products with random initial degradation values. It is common to observe that the degradation paths of distinct products start from different values specified by a random variable. The random initial value introduces additional uncertainties to the degradation of the product. In this study, Wiener-process-based degradation models are developed for products with random initial values. We first consider a DDT without stress acceleration. In a DDT, the measurement of the degradation destroys a test unit and, thus, only one measurement is available for each unit. Closed-form maximum likelihood (ML) estimators are derived. Then, an accelerated DDT (ADDT) is considered. Based on these results, we investigate optimal designs of both DDT and ADDT with the objective of minimizing the asymptotic variance of the estimated $p$ th-quantile of the failure time distribution under use conditions. The optimal test plans have to be obtained through a numerical approach. Optimality of the plans is verified by the general equivalence theorem. An adhesive bond example with real degradation data is analyzed to show the performance of the proposed methods.

We propose a quantile regression framework to model accelerated life tests (ALT) data. The quantile of the failure time distribution at the usage level can be easily estimated using quantile regression. Compared with traditional parametric regression methods, quantile regression is distribution-free, efficient in the presence of censoring, and more flexible in modeling ALT relations. More importantly, we show that it is able to handle ALT data with a failure-free life, which is a great challenge in the ALT literature. We use extensive simulation studies and two real ALT case studies to demonstrate the effectiveness of the proposed method.

In a load-sharing system, total workloads are shared by all components and failure of one component increases stress of the surviving ones. The time interval between two consecutive component failures reflects component reliability under different stress and is of interest to us. This study develops an iterative algorithm for analysis of such data from load-sharing systems. In each iteration, we first obtain the equivalent operating time of each component under a given stress by capitalizing on the cumulative exposure principle. Then the equivalent operating times, which are simply right-censored, are fitted to update the parameter estimates. The conversion has closed forms for most common distributions such as the log-location-scale family and the gamma distribution, and the subsequent fitting of right-censored data is straightforward. Therefore, the algorithm is easy to implement compared with existing methods such as the maximum likelihood estimation. Convergence properties of the algorithm are investigated theoretically and through extensive simulations. Three examples representing different types of real problems are used to demonstrate the proposed algorithm.

Most commercial products are sold with a warranty. Product repairs during the warranty period are often free of charge, and contribute significantly to the costs of a manufacturer. Estimation of the warranty costs is important for the manufacturer to prepare sufficient warranty reserves for future claims. Because products are often sold to customers intermittently, the total number of sold units under warranty varies over time, which presents difficulties in forecasting the warranty costs over time. In this study, we consider the stochastic sales process, and derive the expectation and variance of the aggregate discounted warranty costs within a given period of time. By taking the variation of the warranty costs into consideration, the result can be used in the preparation of a conservative warranty reserve for a new product. The discounted life-cycle warranty cost is a special case of our model, and it can also be determined from our result. Numerical results show the applicability of our model in estimating periodic discounted warranty costs, and preparing both short-term and long-term warranty reserves.

Motivated by an industrial problem affecting a water utility, we develop a model for a load sharing system where an operator dispatches work load to components in a manner that manages their degradation. We assume degradation is the dominant failure type, and that the system will not be subject to sudden failure due to a shock. By deriving the time to degradation failure of the system, estimates of system probability of failure are generated, and optimal designs can be obtained to minimize the long run average cost of a future system. The model can be used to support asset maintenance and design decisions. Our model is developed under a common set of core assumptions. That is, the operator allocates work to balance the level of the degradation condition of all components to achieve system performance. A system is assumed to be replaced when the cumulative work load reaches some random threshold. We adopt cumulative work load as the measure of total usage because it represents the primary cause of component degradation. We model the cumulative work load of the system as a monotone increasing and stationary stochastic process. The cumulative work load to degradation failure of a component is assumed to be inverse Gaussian distributed. An example, informed by an industry problem, is presented to illustrate the application of the model under different operating scenarios.

The IG process models have been shown to be an important family in degradation analysis. In this paper, we are interested in optimal constant-stress accelerated degradation tests (ADTs) planning when the underlying degradation follows the inverse Gaussian (IG) process. We first consider ADT planning for the IG process without random effects. Asymptotic variance of the estimate of a lower quantile is derived, and the objective of the planning is to minimize this variance by properly choosing the testing stresses, and the number of samples allocated to each stress. Next, ADT planning for a random-effects IG process model is considered. We then applied the IG process to fit the stress relaxation data of a component, and use the developed methods to help with the optimal ADT design.

Degradation signals that reflect a system's health state are important for diagnostics and health management of complex systems. However, degradation signals are often compounded and contaminated by measurement errors, making data analysis a difficult task. Motivated by the wear problem of magnetic heads used in hard disk drives (HDDs), this paper investigates Wiener processes with measurement errors. We explore the traditional Wiener process with positive drifts compounded with i.i.d. Gaussian noises, and improve its estimation efficiency compared with the existing inference procedure. Furthermore, to capture the possible heterogeneity in a population, we develop a mixed effects model with measurement errors. Statistical inferences of this model are discussed. The mixed effects model subsumes several existing Wiener processes as its limiting cases, and thus it is useful for suggesting an appropriate Wiener process model for a specific dataset. The developed methodologies are then applied to the wear problem of magnetic heads of HDDs, and a light intensity degradation problem of light-emitting diodes.

Degradation, and shock are two common mechanisms accounting for product failures. This paper presents a convenient means of capturing both shock and degradation in a single model when the extent of degradation and the magnitude of shocks are not observable, but only the failure times and the corresponding failure modes are recorded. We assume that the lifetime of a degradation-oriented failure, which is regarded as some initial random resource, belongs to some distribution family. Shocks arrive according to a non-homogeneous Poisson process, and the destructive probability depends on the transformed remaining resource of the system. Under these assumptions, we propose the single failure time model, and the recurrent event model. This study complements the well-known Brown-Proschan model. The single failure time model has successfully been applied to a real time data set. We also conduct a simulation study to examine the accuracy of our model.