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Degradation-model-based remaining useful life (RUL) estimation is essential for effective prognostic and health management; this method can provide information for enabling effective maintenance decisions pertaining to degrading systems to avoid or mitigate loss due to impending failures. According to existing studies that estimate degradation-model-based RUL, stochastic-process-based methods are widely advocated by many researchers because they can capture stochastic dynamics within the degradation processes of systems. However, most existing studies primarily utilize the first passage time (FPT) to define the lifetime/RUL. The definition is generally conservative; thus, the lifetime/RUL estimation may be underestimated. This is particularly true for nonmonotonic stochastic degradation processes. In some extreme cases, with strong fluctuations within the degradation processes, the estimated lifetime/RUL under the FPT can be significantly less than the actual lifetime/RUL. To address this limitation, this study investigates prognostic issues, based on the stochastic degradation process, from the last exit time (LET) perspective. In contrast to the FPT, the lifetime/RUL of the degrading system is defined as the LET of its degradation process, i.e., the instant at which the degradation process passes the failure threshold for the last time. Given the new definition, we consider the most widely used degradation process model (i.e., Wiener-process-based model) as an example to demonstrate how the lifetime/RUL is estimated. Two general methods of lifetime estimation for the Wiener-process-based model are given, and some examples with associated exact and closed-form solutions are also provided for better illustration. Finally, numerical examples and a practical case study are presented to substantiate the theoretical results and illustrate the superiority of the proposed method. The results imply that the proposed method exhibits the potential to prevent premature maintenance and resource wastage because the lifetime/RUL estimation from the LET perspective can help avoid conservative results from being obtained. [ABSTRACT FROM AUTHOR]

For some design applications, the environment in actual field conditions does not remain constant but instead changes dynamically. In addition, stress levels might change sharply, and if the change happens for a long enough time and at a high enough rate, it could act as a shock to the product, although it is not actually a shock that occurs instantly. This situation is defined as an effective shock in this paper. An effective shock is caused by a rapid change in stress that has an analogous effect to an actual shock, where the stress change is immediate. Existing works on degradation modeling and lifetime prediction for products operating in a dynamic environment consider the effects of time-varying stress levels on the degradation rate of product performance. If shocks are considered at all, it is only those shocks that occur instantaneously. In this paper, a model that characterizes the degradation process of a product operating in a dynamic environment considering effective shocks is proposed to predict the residual lifetime of the product. This model is established based on a stress-dependent degradation rate and an effective shock function together with a Wiener process to model the stochastic degradation. A simulation study is presented as a demonstration of the application of this model, and a case study testing miniature light bulbs is presented to verify the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

Degradation data obtained through inspections have been widely used to estimate the remaining useful life (RUL) of deteriorating systems. At the same time, such inspections may introduce external stress or release interior stress on the degradation processes of systems. However, current studies have paid little attention to the influence of such inspections during degradation modeling and RUL estimation. In this paper, we incorporate the inspection influence into Wiener-process-based degradation modeling and develop an approach to estimate the RUL of deteriorating systems. In the proposed method, the lifetime and RUL distribution with the consideration of the inspection influence on degrading systems are derived under the concept of first hitting time. To achieve an adaptive estimation of the RUL when new observations arrive, we constructed a state-space model by augmenting the drift coefficient and the inspection influence as state variables and treating the degradation increments as the observation variables. To do so, a Kalman filter is employed to update the state estimation, and then, an analytical result of the estimated RUL distribution for periodic-inspected systems is achieved. To implement the proposed prognostic method, we apply the expectation maximization algorithm to estimate unknown parameters in the constructed state-space model based on the historical observations of inspections. Finally, the proposed approach is illustrated by a numerical example and demonstrated by a case study using the mechanical gyroscopes. [ABSTRACT FROM AUTHOR]

This paper presents a reliability modeling and analysis framework for load-sharing systems with identical components subject to continuous degradation. It is assumed that the components in the system suffer from degradation through an additive impact under increased workload caused by consecutive failures. A log-linear link function is used to describe the relationship between the degradation rate and load stress levels. By assuming that the component degradation is well modeled by a step-wise drifted Wiener process, we construct maximum likelihood estimates (MLEs) for unknown parameters and related reliability characteristics by combining analytical and numerical methods. Approximate initial guesses are proposed to lessen the computational burden in numerical estimation. The estimated distribution of MLE is given in the form of multivariate normal distribution with the aid of Fisher information. Alternative confidence intervals are provided by bootstrapping methods. A simulation study with various sample sizes and inspection intervals is presented to analyze the estimation accuracy. Finally, the proposed approach is illustrated by track degradation data from an application example. [ABSTRACT FROM AUTHOR]

Accurate remaining useful life prediction is meaningful for cutting tool usability evaluation. Over the years, experience-based models, data-driven models, and physics-based models have been used individually to predict cutting tool remaining useful lives. In order to improve prediction performances, different prognostics models can be combined to leverage their advantages. In this paper, a hybrid cutting tool remaining useful life prediction approach is proposed by combining a data-driven model and a physics-based model. By using force, vibration and acoustic emission signals, the data-driven model monitors cutting tool wear conditions based on empirical mode decomposition and back propagation neural network. On the basis of the Wiener process, the physics-based model builds a cutting tool condition degradation model to predict cutting tool remaining useful lives. Experimental study verifies the approach's effectiveness, accuracy, and robustness. Then, cutting tool remaining useful lives can be predicted more accurately during the machining process. [ABSTRACT FROM AUTHOR]

Modern products are usually designed with high reliability and have complex structures, and degradation analysis of the complex systems with two or multiple performance characteristics is still a challenge. In this paper, we propose a new bivariate degradation model based on the Wiener process. There are three main merits of the proposed model: it can describe the common factor affecting the degradation of the two performance characteristics and unit-to-unit variation simultaneously, the reliability functions of the system and the remaining useful life of the system have analytic forms, and the model parameters and the missing values can be estimated by the Bayesian method and data augmentation. The simulation study and data analysis show that the Bayesian method and the proposed model have satisfactory performance. [ABSTRACT FROM AUTHOR]

Degradation test is an effective method in assessing product reliability when measurements of degradation leading to failure can be observed. The accuracy of reliability inference in degradation analysis highly depends on the fitted model to the observed degradation data. Sometimes, observed degradation paths of the products exhibit multiphase pattern over the testing period. In this paper, we propose a change-point Wiener process with measurement errors (CPWPME) to fit two-phase degradation paths of organic light-emitting diodes (OLEDs). We assume the unit-specific parameters of the CPWPME model by using hierarchical Bayesian method. Based upon the proposed approach, the failure-time distribution and the remaining useful life distribution along with mean time to failure and mean residual life function are derived in closed form. A simulation study shows the utility of the proposed CPWPME model and the validity of the hierarchical Bayesian approach for the degradation data possessing two-phase degradation characteristics. In the analysis of OLED degradation data, the hierarchical Bayesian CPWPME model provides higher modeling flexibility and prediction power for future testing units than existing three degradation models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Parallel constant-stress accelerated degradation testing (PCSADT) is widely used to assess the reliability of highly reliable products in a timely manner when the products’ degradation can be measured. Under a time-censored PCSADT, several groups of units are tested simultaneously, but under different stress levels, until a prespecified censoring time is reached. At this time, degradation values from the censored units, and failure times of the failed units are obtained. When the degradation follows a Wiener process where the parameters depend on the stress level through a life-stress model containing an unknown nuisance parameter, estimating this parameter often biases the maximum likelihood and least-squares estimators of the lifetime parameters. In this paper, we propose a two-stage procedure to address this problem. In the first stage, we transform the data under the different stress levels of a PCSADT so that the resulting data can be considered to have been obtained under normal stress. In the second stage, we introduce a latent variable for the unobserved degradation after the failure time for each failed unit to obtain a pseudodegradation value at the censoring time. We then use all degradation values (pseudo or observed) at the censoring time to develop latent variable estimators for all model parameters. Unlike other existing estimators, the proposed estimators are shown to be s-consistent, have closed-form expressions, and are easy to interpret. We use a real example of light-emitting diodes to illustrate the proposed method. In addition to proving s-consistencies, we conduct a simulation study to demonstrate that the proposed estimators also perform well in finite samples. [ABSTRACT FROM PUBLISHER]

Observations on degradation performance are often used to analyze the underlying degradation process of highly reliable products. From the two-phase degradation path of the bearing performance observations, we observed that there exists an abrupt increase in degradation measurement at a change point. Then, the following degradation process started with the abrupt degradation measurement will degrade in a higher degradation rate. Here, a stochastic process-based degradation model is constructed to interpret the jump at the change point in the degradation process which is governed by the linear Wiener process. Meanwhile, the distribution of the first passage time over a prespecified threshold for the process is discussed. In addition, to get the estimates of the model parameter, the expectation-maximization algorithm is utilized since the change points are unobservable. Furthermore, to demonstrate the model's advantages over estimate, a comparison is made between the proposed and the existing known models from the literature. The results reveal that considering the jump in the degradation process can improve the accuracy of estimations in real applications. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Repeated degradation testing data have been widely used to assess lifetime of highly-reliable products or components with scarce failures. In this paper, we consider the problem of planning repeated degradation testing for products exhibiting three-source variability in their degradation characteristics, including the temporal variability, unit-to-unit variability, and measurement variability. The primary objective of this paper is centered on deciding the amount of units and the measurement schedule to achieve required estimation precision for some important statistics of interest. In such a planning process, the testing budget is limited and used as a constraint of an optimization model. To do so, a kind of Wiener degradation process, which has a random drift coefficient and a constant volatility coefficient, is used to model the repeated degradation testing data, where the measurement errors are considered and described as additive zero-mean random variables. Under the presented modeling framework, the lifetime distribution is formulated under the concept of the first passage time of the stochastic degradation process. Then, the large-sample approximate standard errors of the maximum likelihood estimations for the mean failure time and the quantile of the degradation distribution are derived, respectively. Furthermore, the constrained optimization model, which incorporates the cost of each degradation measurement, is proposed to plan the degradation testing by minimizing the testing cost under the condition of a maximum acceptable approximate standard error. Finally, an example is provided to illustrate the procedure and advantages of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, the degradation based reliability demonstration test (RDT) plan design problems for long life products under a small sample circumstance are studied. Fixed sample method, sequential probability ratio test (SPRT) method, and sequential Bayesian decision method are provided based on univariate degradation testing. The simulation examples show the superiority of degradation based RDT methods compared with the traditional failure based methods, and the sequential-type methods have more test power than their fixed sample counterparts. The test power can be further improved by combining the test data of a reliability indicator with the data of its marker, based on which the bivariate fixed sample method and the sequential Bayesian decision method are defined. The simulation study shows the benefit from the combination. The degradation based RDT plan optimization model, and the corresponding searching-based solution algorithm using some heuristic rules discovered in the paper, are also presented. The case study of Rubidium Atomic Frequency Standard with a RDT plan design demonstrates the effectiveness of our methods on overcoming the difficulties of small samples in reliability demonstration of long life products. [ABSTRACT FROM AUTHOR]

Hybrid deteriorating systems, which are made up of both linear and nonlinear degradation parts, are often encountered in engineering practice, such as gyroscopes which are frequently utilized in ships, aircraft, and weapon systems. However, little reported literature can be found addressing the degradation modeling for a system of this type. This paper proposes a general degradation modeling framework for hybrid deteriorating systems by employing an additive Wiener process model that consists of a linear degradation part and a nonlinear part. Furthermore, we derive the analytical solution of the remaining useful life distribution approximately for the presented model. For a specific system in service, the posterior estimates of the stochastic parameters in the model are updated recursively by using the condition monitoring observations based on a Bayesian framework with the consideration that the stochastic parameters in the linear and nonlinear deteriorating parts are correlated. Thereafter, the posterior distribution of stochastic parameters is used to update in real-time the distribution of the remaining useful life where the uncertainties in the estimated stochastic parameters are incorporated. Finally, a numerical example and a practical case study are provided to verify the effectiveness of the proposed method. Compared with two existing methods in literature, our proposed degradation modeling method increases the one-step prediction accuracy slightly in terms of mean squared error, but gains significant improvements in the estimated remaining useful life. [ABSTRACT FROM PUBLISHER]

The use of the observed degradation data of a system can help to estimate its remaining useful life (RUL). However, the degradation progression of the system is typically stochastic, and thus the RUL is also a random variable, resulting in the difficulty to estimate the RUL with certainty. In general, there are three sources of variability contributing to the uncertainty of the estimated RUL: 1) temporal variability, 2) unit-to-unit variability, and 3) measurement variability. In this paper, we present a relatively general degradation model based on a Wiener process. In the presented model, the above three-source variability is simultaneously characterized to incorporate the effect of three-source variability into RUL estimation. By constructing a state-space model, the posterior distributions of the underlying degradation state and random effect parameter, which are correlated, are estimated by employing the Kalman filtering technique. Further, the analytical forms of not only the probability distribution but also the mean and variance of the estimated RUL are derived, and can be real-time updated in line with the arrivals of new degradation observations. We also investigate the issues regarding the identifiability problem in parameter estimation of the presented model, and establish the according results. For verifying the presented approach, a case study for gyros in an inertial platform is provided, and the results indicate that considering three-source variability can improve the modeling fitting and the accuracy of the estimated RUL. [ABSTRACT FROM PUBLISHER]

Degradation models are widely used to assess the lifetime information of highly reliable products possessing quality characteristics that both degrade over time and can be related to reliability. The performance of a degradation model largely depends on an appropriate model description of the product's degradation path. Conventionally, the random or mixed-effect model is one of the most well-known approaches presented in the literature in which the normal distribution is commonly adopted to represent unit-to-unit variability in the degradation model. However, this assumption may not appropriately signify accurate projections for practical applications. This paper is motivated by laser data wherein the normal distribution is relaxed with a skew-normal distribution that consequently provides greater flexibility as it can capture a broad range, non-normal, asymmetric behavior in unit-to-unit variability. Based on the proposed degradation model, we first derive analytical expressions for a product's lifetime distribution along with its corresponding mean-time-to-failure (MTTF). We then utilize the laser data to illustrate advantages gained by the proposed model. Finally, we address effects from the skewness parameter with regard to the accuracy of both a product's MTTF and its qth failure quantile; especially when the underlying skew-normal distribution is mis-specified as a normal distribution. The result demonstrates that effects from the skewness parameter on the tail probabilities of a product's lifetime distribution are not negligible when the random effect of the true degradation model follows a skew-normal distribution. [ABSTRACT FROM AUTHOR]

Burn-in test is a manufacturing process applied to products to eliminate latent failures or weak components in the factory before the products reach customers. The traditional burn-in test over a short period of time to collect time-to-failure or go/no-go data is rather inefficient. This decision problem can be solved if there exists a suitable quality characteristic (QC) whose degradation over time can be related to the lifetime of the product. Recently, optimal burn-in policies have been discussed in the literature assuming that the underlying degradation path follows a Wiener process. However, the degradation model of many materials (especially in the case of fatigue data) may be more appropriately modeled by a gamma process that exhibits a monotone-increasing pattern. Here, motivated by laser data, we first -propose a mixed gamma process to describe the degradation path of the product. Next, we present a decision rule for classifying a unit as typical or weak. A cost model is used to determine the optimal termination time of a burn-in test, and a motivating example is then presented to illustrate the proposed procedure. Finally, a simulation study is carried out to examine the effect of wrongly treating a mixed gamma process as a mixed Wiener process, and the obtained results reveal that the effect on the probabilities of misclassification is not negligible. [ABSTRACT FROM AUTHOR]

For highly reliable products with very few test units, a progressive-stress accelerated degradation test (PSADT) has been proposed in the literature to obtain timely information of the product's lifetime distribution. The results, however, are restricted to the case where the product's degradation path follows a Wiener process (Brownian motion) with a linear drift rate. But in practical applications, the product's mean degradation path may be non-linear. Hence, how to address the lifetime distribution in this situation is a worthy topic for reliability analysts. In this paper, a PSADT with a non-linear degradation path is constructed using the cumulative exposure model. Then the product's lifetime distribution can be analytically obtained by the first passage time of its degradation path. Furthermore, we derive an exact relationship between the lifetime distributions of the PSADT, and the conventional constant-stress degradation test (CSDT), which allows us to extrapolate the product's lifetime distribution under typical stress. Finally, the usage of the proposed model, and the efficiency of PSADT to reduce the product's life testing time are demonstrated in the example. [ABSTRACT FROM AUTHOR]

Step-stress accelerated degradation testing (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products (under a typical-use condition) when the available test items are very few. Recently, an optimal SSADT plan was proposed based on the assumption that the underlying degradation path follows a Wiener process. However, the degradation model of many materials (especially in the case of fatigue data) may be more appropriately modeled by a gamma process which exhibits a monotone increasing pattern. Hence, in practice, designing an efficient SSADT plan for a gamma degradation process is of great interest. In this paper, we first introduce the SSADT model when the degradation path follows a gamma process. Next, under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal settings such as sample size, measurement frequency, and termination time are obtained by minimizing the approximate variance of the estimated MTTF of the lifetime distribution of the product. Finally, an example is presented to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

Today, many products are designed to function for a long period of time before they fail. For such highly-reliable products, collecting accelerated degradation test (ADT) data can provide useful reliability information. However, it usually requires a moderate sample size to implement an ADT. Hence, ADT is not applicable for assessing the lifetime distribution of a newly developed or very expensive product which only has a few available test units on hand. Recently, a step-stress ADT (SSADT) has been suggested in the literature to overcome the above difficulty. However, in designing an efficient SSADT experiment, the issue about how to choose the optimal settings of variables was not discussed, such as sample size, measurement frequency, and termination time. In this study, we first use a stochastic diffusion process to model a typical SSADT problem. Next, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal settings of these variables are obtained by minimizing the asymptotic variance of the estimated 100pth percentile of the product's lifetime distribution. Finally, an example is used to illustrate the proposed method. [ABSTRACT FROM AUTHOR]