Gamma Degradation Process Research Articles

A reinforcement learning agent for maintenance of deteriorating systems with increasingly imperfect repairs

Efficient maintenance has always been essential for the successful application of engineering systems. However, the challenges to be overcome in the implementation of Industry 4.0 necessitate new paradigms of maintenance optimization. Machine learning techniques are becoming increasingly used in engineering and maintenance, with reinforcement learning being one of the most promising. In this paper, we propose a gamma degradation process together with a novel maintenance model in which repairs are increasingly imperfect, i.e., the beneficial effect of system repairs decreases as more repairs are performed, reflecting the degradational behavior of real-world systems. To generate maintenance policies for this system, we developed a reinforcement-learning-based agent using a Double Deep Q-Network architecture. This agent presents two important advantages: it works without a predefined preventive threshold, and it can operate in a continuous degradation state space. Our agent learns to behave in different scenarios, showing great flexibility. In addition, we performed an analysis of how changes in the main parameters of the environment affect the maintenance policy proposed by the agent. The proposed approach is demonstrated to be appropriate and to significatively improve long-run cost as compared with other common maintenance strategies.

Imperfect and worse than old maintenances for a gamma degradation process

AbstractThis article considers a condition‐based maintenance for a system subject to deterioration. The deterioration is modeled by a non‐homogeneous gamma process, more precisely the gamma process and the preventive maintenance are imperfect or worse than old. The corrective maintenance actions are as good as new. The maintenance efficiency or non‐efficiency parameters as well as the deterioration parameters are considered to be unknown. The monitoring data under consideration give indirect information on the maintenance parameters. Therefore, an expected maximum algorithm is applied for parameter estimation.

Non‐periodic inspection and replacement policy of system subject to non‐homogeneous gamma degradation process

AbstractThe non‐homogeneous gamma process can exhibit a convex average trend with a non‐linear shape parameter, providing significant flexibility in fitting an accelerated degradation process. However, few studies have been conducted on the maintenance modeling of systems undergoing a non‐homogeneous gamma degradation process despite its versatility. Meanwhile, most maintenance studies adopt a periodic inspection policy to reveal the degradation level. However, it is reasonable to inspect less frequently when the degradation level of the system is low and to perform inspections more frequently when the degradation level increases. Therefore, we conduct a study on the inspection/replacement policy of systems undergoing a non‐homogeneous gamma degradation process, where three non‐periodic inspection policies are investigated. We formulate the problem as a SMDP framework and present a value iteration algorithm to obtain the optimal action at each decision epoch that minimizes the long‐term average cost. A comprehensive numerical study is provided to illustrate the effectiveness of our proposed model.

Condition-based Maintenance Optimization for Gamma Deteriorating Systems under Performance-based Contracting

With the further development of service-oriented, performance-based contracting (PBC) has been widely adopted in industry and manufacturing. However, maintenance optimization problems under PBC have not received enough attention. To further extend the scope of PBC’s application in the field of maintenance optimization, we investigate the condition-based maintenance (CBM) optimization for gamma deteriorating systems under PBC. Considering the repairable single-component system subject to the gamma degradation process, this paper proposes a CBM optimization model to maximize the profit and improve system performance at a relatively low cost under PBC. In the proposed CBM model, the first inspection interval has been considered in order to reduce the inspection frequency and the cost rate. Then, a particle swarm algorithm (PSO) and related solution procedure are presented to solve the multiple decision variables in our proposed model. In the end, a numerical example is provided so as to demonstrate the superiority of the presented model. By comparing the proposed policy with the conventional ones, the superiority of our proposed policy is proved, which can bring more profits to providers and improve performance. Sensitivity analysis is conducted in order to research the effect of corrective maintenance cost and time required for corrective maintenance on optimization policy. A comparative study is given to illustrate the necessity of distinguishing the first inspection interval or not.

Imperfect condition-based maintenance for a gamma degradation process in presence of unknown parameters

A system subject to degradation is considered. The degradation is modelled by a gamma process. A condition-based maintenance policy with perfect corrective and an imperfect preventive actions is proposed. The maintenance cost is derived considering a Markov-renewal process. The statistical inference of the degradation and maintenance parameters by the maximum likelihood method is investigated. A sensibility analysis to different parameters is carried out and the perspectives are detailed.

Likelihood-free Hamiltonian Monte Carlo for modeling piping degradation and remaining useful life prediction using the mixed gamma process

The gamma process is a popular model to characterize the temporal uncertainty in degradation processes. According to this model, degradation occurs in tiny independent increments creating a monotonic non-decreasing trend. Mechanical components or units are often repetitively observed over time for measuring the extent of degradation; apparently, they degrade at different rates although operating in the same environment. The mixed gamma degradation process is suitable for capturing these unexplained differences. However, statistical estimation of its parameters and subsequent prediction of remaining useful life (RUL) turn challenging with limited and noisy measurements.While limited observations bring uncertainties to the model parameters, noisy data turns the likelihood function intractable. In such cases, approximate Bayesian computation (ABC) methods are widely applicable for their ability to bypass the likelihood evaluation and generate samples from an approximate target posterior distribution. The currently available Markov chain Monte Carlo (MCMC) based ABC method shows promising results, although it requires significant thinning due to poor mixing of samples. A new ABC method is derived in this paper which imitates the sampling process of the Hamiltonian Monte Carlo (HMC) method. The proposed ABC-HMC algorithm explores a target probability space more effectively — generating better mixing of samples and requiring less thinning than the MCMC based ABC method. Proof of convergence of ABC-HMC is shown by fulfilling the detailed balance condition. The application of the method is shown using real degradation data of piping components of the heat transport system of a nuclear reactor.

Degradation‐based reliability analysis of magnetic jack type control rod drive mechanism

AbstractThe magnetic jack type control rod drive mechanism (CRDM) is the key actuator in the nuclear power plant reactor. The latch assembly of the CRDM acts as the execution component to lift, insert, and release the control rod assembly to accomplish the reactor startup, power regulation, and shutdown, etc. The accuracy and the timeliness of CRDM latch assembly action will directly determine the safety operation of the reactor. Considering the time variation of the structural strength deterioration in the latch assembly, the time‐dependent reliability of CRDM's lift action is evaluated based on the Gamma degradation process. P‐S‐N curve is introduced to model the shape function and the scale parameter of the Gamma process. The stress subjected to the key structure of the latch assembly is calculated by a multi‐physics field coupling simulation scheme with utilization of MATLAB to transfer simulated results between the ANSYS and ADAMS software. Finally, the time‐dependent reliability is calculated based on the Monte Carlo simulation method. In this paper, the lift pole thread of the assembly is presented to demonstrate the effectiveness of the proposed method.

A new gamma degradation process with random effect and state-dependent measurement error

In this paper, a new gamma-based degradation process with random effect is proposed that allows to account for the presence of measurement error that depends in stochastic sense on the measured degradation level. This new model extends a perturbed gamma model recently suggested in the literature, by allowing for the presence of a unit to unit variability. As the original one, the extended model is not mathematically tractable. The main features of the proposed model are illustrated. Maximum likelihood estimation of its parameters from perturbed degradation measurements is addressed. The likelihood function is formulated. Hence, a new maximization procedure that combines a particle filter and an expectation-maximization algorithm is suggested that allows to overcome the numerical issues posed by its direct maximization. Moreover, a simple algorithm based on the same particle filter method is also described that allows to compute the cumulative distribution function of the remaining useful life and the conditional probability density function of the hidden degradation level, given the past noisy measurements. Finally, two numerical applications are developed where the model parameters are estimated from two sets of perturbed degradation measurements of carbon-film resistors and fuel cell membranes. In the first example the presence of random effect is statistically significant while in the second example it is not significant. In the applications, the presence of random effect is checked via appropriate statistical procedures. In both the examples, the influence of accounting for the presence of random effect on the estimates of the cumulative distribution function of the remaining useful life of the considered units is also discussed. Obtained results demonstrate the affordability of the proposed approach and the usefulness of the proposed model.

A common random effect induced bivariate gamma degradation process with application to remaining useful life prediction

Due to the complex structures and the multi-functionality of modern products, there are usually two or more performance characteristics which can reflect a product’s degradation states. The degradation processes corresponding to these performance characteristics are dependent in general, which brings challenges to the degradation data analysis. In this paper, a gamma process based degradation model is developed for the bivariate dependent degradation data, where the dependency between the two degradation processes is captured by a common random effect naturally. The expectation maximization algorithm is employed to estimate the model parameters. Then, a real-time prediction method for a product’s remaining useful life is proposed using the Bayesian method. Finally, both the simulation study and the case study are provided for illustration, whose results demonstrate that the proposed model as well as the corresponding inference methods does work well.

A time-varying mechanical structure reliability analysis method based on performance degradation

In mechanical engineering, most mechanical structural systems are composed of many parts, so their structures are complex. When a mechanical structure fails, it is generally caused by various failure modes. When designers want to improve the quality of mechanical products in the design of mechanical products, reliability analysis can play an important role. It is indispensable in the design and manufacture of mechanical products, engineering use, repair and maintenance. Research on time-varying reliability of mechanical structures is of great significance in improving the quality of mechanical products and bringing more economic benefits to engineering. It has received extensive attention in both academic fields and practical engineering applications. This paper takes the time-varying reliability analysis of mechanical structures based on performance degradation as the research topic, focusing on the reliability analysis methods and techniques of mechanical structures such as Gamma degradation process, material P-S-N curve, mixed Copula function, and nested Copula function. The main research content is divided into three parts. (1) The Gamma degradation process and the P-S-N material probabilistic fatigue life curve are used when establishing the strength degradation random model of the mechanical system. It provides the theoretical basis of strength degradation level for time-varying reliability analysis of mechanical structure based on performance degradation. (2) Combine the mixed Copula function and the nested Copula function to establish a comprehensive failure correlation analysis model of the mechanical system. It provides a theoretical basis for failure correlation for time-varying reliability analysis of mechanical structures based on performance degradation. (3) The time-varying reliability analysis of the one-stage gear transmission system in the engineering example is carried out. Researched and analyzed the time-varying reliability of the large and small gears in the one-stage gear transmission system and the mechanical structure of the system itself under the correlation of strength degradation and failure.

Prognostics for lithium-ion batteries using a two-phase gamma degradation process model

To address the degradation of rechargeable batteries, this paper presents a two-phase gamma process model with a fixed change-point for modeling the voltage-discharge curves of battery cycle aging under a constant current. The model can be applied to estimate the state of charge (SOC) and the remaining useful discharge time (RUT) in a cycle with consideration of the effect of cycle aging, and can also be applied to estimate the state of life (SOL) and the remaining useful life (RUL) across cycles. The applications of the proposed model are demonstrated using the experimental cycle aging data of a lithium iron phosphate battery. A comparison shows that the proposed model generates a more accurate prediction than the conventional two-term exponential model with capacity data under a particle filter framework, and this reveals the superiority of modeling with voltage over modeling with capacity. The analytical expression of mean useful discharge time in a cycle (or mean time to failure) is developed with approximation by a Taylor expansion and the Birnbaum-Saunders distribution, and the result is shown to be in good agreement with the true mean of a gamma process.

Exploring the optimal condition-based maintenance policy under the gamma degradation process

As an industry-wide ongoing effort of the system health management (SHM), condition-based maintenance (CBM) utilizes the modern sensor technology for periodic inspections of a system, and the maintenance actions are made based on the inspection of working conditions of the system unlike traditional methods. It is an effective method to reduce unexpected failures as well as the operations and maintenance costs. Among various stochastic processes to describe degradation paths, the gamma process is characterized by a monotonic degradation evolution, suitable for modeling crack growth and wears for example. Its flexibility and mathematical tractability also ensure the analytical properties of the degradation model. In this work, we discuss the condition-based maintenance/replacement policy with the optimal inspection points under the gamma degradation process. A random effect parameter is used to account for the potential population/environmental heterogeneities and its distribution is continuously updated at each inspection epoch. The observed degradation level along with the system age is utilized for making the optimal maintenance decision. Aiming to minimize the total discounted operational costs, we investigate the structural properties of the optimal policy and determine the optimal inspection intervals. It was realized that the optimal maintenance policy that minimizes the total discounted operational costs is a monotone control limit policy, similar to the cases of Wiener process and inverse Gaussian process.

Variables acceptance reliability sampling plan based on degradation test

Until now, various acceptance reliability sampling plans have been developed based on different life tests. In most of the reliability sampling plans, the decision procedures are based on the lifetimes of the items observed on tests, or the number of failures observed during a pre-specified testing time. However, frequently, the items are subject to degradation phenomena and, in these cases, the observed degradation level of the item can be used as a decision statistic. In this paper, assuming the gamma degradation process, we develop a variables acceptance sampling plan based on the information on the degradation process of the items. It is shown that the developed sampling plan improves the reliability performance of the items conditional on the acceptance in the test and that the lifetimes of items after the reliability sampling test are stochastically larger than those before the test. A study comparing the proposed degradation-based sampling plan with the conventional sampling plan which is based on a life test is also performed.

Gamma Process-Based Models for Disease Progression

Classic chronic diseases progression models are built by gauging the movement from the disease free state, to the preclinical (asymptomatic) one, in which the disease is there but has not manifested itself through clinical symptoms, after spending an amount of time the case then progresses to the symptomatic state. The progression is modelled by assuming that the time spent in the disease free and the asymptomatic states are random variables following specified distributions. Estimating the parameters of these random variables leads to better planning of screening programs as well as allowing the correction of the lead time bias (apparent increase in survival observed purely due to early detection). However, as classical approaches have shown to be sensitive to the chosen distributions and the underlying assumptions, we propose a new approach in which we model disease progression as a gamma degradation process with random starting point (onset). We derive the probabilities of cases getting detected by screens and minimize the distance between observed and calculated distributions to get estimates of the parameters of the gamma process, screening sensitivity, sojourn time and lead time. We investigate the properties of the proposed model by simulations.

Designing an accelerated degradation test plan considering the gamma degradation process with multi-stress factors and interaction effects

ABSTRACT The effectiveness of accelerated degradation test and subsequent reliability assessment depends on the efficient test plan. Designing an accelerated test plan requires determining the optimal values of several decision variables. The problem of designing an accelerated degradation test becomes even more challenging when multiple stress factors and their interaction effects are considered. In this work, an optimal accelerated degradation test plan is presented considering the stochastic gamma process with constant stress loading. Both gamma parameters are treated as stress-dependent variables to capture the impact of stress levels. An optimization model is formulated considering multiple stress factors and their interaction effects for offering a more realistic test plan. The optimization criteria are set to minimizing the asymptotic variance of the maximum likelihood estimator of the lifetime at normal operating conditions within the total experimental cost constraints. The decision variables comprised in the optimization model are measurement time intervals, the frequency of measurements, stress levels, and sample allocation at each stress level. The sensitivity analysis is performed to investigate the model's robustness against the budget constraints and pre-estimates of model parameters. To demonstrate the proposed methodology, a case example of carbon film-resistors’ performance data is considered for designing the degradation test plan.

Copula-based reliability analysis of gamma degradation process and Weibull failure time

Purpose The purpose of this paper is the reliability analysis for systems with dependent gamma degradation process and Weibull failure time. Design/methodology/approach Consider a life testing experiment in which a sample of n devices starts to operate at t=0 and the data are available on failure time and failure-evolving process on each individual, called in some contents wear or degradation. Ignoring the between performance characteristics dependency structure may lead us to different reliability estimations, while the dependency justly exists. In previous research, dependency between the degradation process and hard failure time has been studied in limited detail (special closed form expression). Thereafter, the dependency between two degradation processes with the same structure (gamma process) in a system is considered using the copula function. Findings The results indicate that ignoring the dependency structure may lead us to different reliability estimations while the dependency justly exists. Originality/value This study gives some contributions that evaluate reliability metrics with more than one failure mechanism that may not be independent and possibly follow a different distribution function. The authors have used the copula function as a basis to develop a proposal model and analysis methods. In addition, the authors discussed the identifiability of the copula. Finally, simulation data were used to review the suggested approach.

Inference for constant-stress accelerated degradation test based on Gamma process

• A Gamma accelerated degradation model based on the principle of degradation mechanism invariance is proposed. • The existence of the maximum likelihood estimate of the parameters of the proposed model is proved. • The approximate confidence interval for the shape parameter based on Cornish–Fisher expansion is derived. • The generalized confidence intervals for other model parameters and some important quantities are developed. This paper proposes a Gamma constant-stress accelerated degradation model based on the principle of the degradation mechanism invariance. The maximum likelihood estimators of the parameters of the proposed model are derived. Based on Cornish–Fisher expansion, the approximate confidence interval for the shape parameter of the Gamma degradation process is developed. Since it is difficult to obtain the exact confidence intervals for other model parameters and some quantities such as the mean degradation in unit time, the quantile and the reliability function of the lifetime at the normal stress level, the generalized confidence intervals for these quantities are proposed. The percentiles of the proposed generalized pivotal quantities can be obtained by the simulation. The performances of the proposed confidence intervals are evaluated by the Monte Carlo simulation method. In the simulation study, the proposed confidence intervals are compared with the Wald and the bootstrap- p confidence intervals. The simulation results show that the proposed confidence intervals outperform the Wald and the bootstrap- p confidence intervals in terms of the coverage percentage. Finally, a real example is used to illustrate the proposed procedures.

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

AbstractThis paper proposes and illustrates a new perturbed gamma degradation process where the measurement error is modeled as a non‐Gaussian random variable that depends stochastically on the actual degradation level. The expression of the likelihood function for a generic set of noisy degradation measurements is derived, and the expression of the remaining useful life distribution of a degrading unit that fails when its degradation level exceeds a given threshold limit is formulated. A particle filter method is suggested, which allows one to compute the likelihood function and to estimate the remaining useful life distribution in a quick yet efficient manner. In addition, a closed‐form approximation of the perturbed gamma process is proposed to use in the special, yet meaningful, case where the standard deviation of the measurement error depends linearly on the actual degradation level. Finally, an applicative example is discussed, where the parameters of the perturbed gamma process, the remaining useful life distribution, and the mean remaining useful life of the degrading units are estimated from a set of noisy real degradation data.

Bayesian estimation and prediction for the transformed gamma degradation process

AbstractVery recently, a new degradation process model, named the transformed gamma process, has been proposed to describe Markovian degradation processes whose increments over disjoint intervals are not independent, so that the degradation growth over a future time interval can depend both on the current age and the current state (degradation level) of the unit. This paper introduces a Bayesian estimation approach for such a process, based on prior information on physical characteristics of the observed degradation process. Several different prior distributions are then proposed, reflecting different degrees of knowledge of the analyst on the observed phenomenon. A Monte Carlo Markov Chain technique is adopted to estimate the transformed gamma parameters and some functions thereof, such as the residual reliability of a unit, as well as to predict future degradation growth and residual lifetime. Finally, the proposed approach is applied to a real dataset consisting of wear measures of the liners of the 8‐cylinder engine which equips a cargo ship.

Gamma Degradation Process and Accelerated Model Combined Reliability Analysis Method for Rubber O-Rings

The sealing performance of rubber O-rings plays an irreplaceable role in ensuring the high reliability and safety of mechanical systems. However, traditional statistical analysis may not consider the aging process of rubber O-rings at different temperatures and the time-varying characteristics of rubber material parameters. In view of these problems, an improved reliability analysis method based on the accelerated aging test of rubber O-rings is proposed in this paper. This method is featured by a combination with the traditional accelerated model and the Gamma stochastic process. This paper first adopts the stationary Gamma process to describe the degradation of compression set which is the performance degradation indicator of rubber O-rings. Then, the reliability model is derived from the Gamma degradation model. Next, the Arrhenius model is used to represent the shape parameter of Gamma degradation model and the approach of maximum likelihood estimation is adopted to solve the parameters. Finally, the rubber O-rings of gas steering engine were selected and a series of accelerated aging tests were conducted to obtain the reliability and the storage lifetime of rubber O-rings at different temperature stresses through our improved method. Besides, the comparison with the traditional statistical method is also conducted to prove the advantage of this method.