Non-stationary Gamma Process Research Articles

Joint optimization of inspection and condition-based maintenance for a deteriorating product under extended warranty

Condition-based maintenance (CBM) has been widely applied in engineering practice, but it has been rarely integrated into extended warranty policies. When the inspection cost is high, it becomes important to schedule the inspection plan when making CBM decisions. This paper jointly optimizes inspection and CBM for a deteriorating product under extended warranty. The deterioration of the product is characterized as a non-stationary gamma process. At each inspection epoch, the decision maker needs to determine whether to repair the product and when to perform the next inspection. The objective is to simultaneously optimize the inspection policy and the maintenance policy that minimize the warranty servicing cost. The joint optimization problem is formulated by stochastic dynamic programming. The optimal policy is proven to be of a control-limit form. A backward induction algorithm integrating the obtained structural properties is proposed to find the optimization results. A numerical example is conducted to illustrate the effectiveness of the proposed approach. A graphical representation of the obtained policy is provided to support practical maintenance implementation. The comparison with a popular CBM policy confirms the superiority of the proposed policy.

Maintenance strategies and life-cycle cost analysis of inspection robots in metro tunnels

Inspection robots are rarely employed in metro tunnels. This is related to the fact that the various technical indicators of inspection robots are not balanced under the existing technical conditions. This study proposes a method for developing a maintenance strategy and assessing the life-cycle cost of inspection robots in metro tunnels to determine whether they should be used. A tunnel structure deterioration model based on a non-stationary gamma process was used. Based on this, a life-cycle costing model for a tunnel inspection strategy was designed. This model considers the accuracy of inspection subject, inspection cycle, cost of attendance, and maximum mileage and achieves a good assessment of the inspection subject. The study shows that the inspection cycle has a non-linear relationship with the life-cycle cost and that the inspection cycle has a minimum cost range. The detection accuracy of the inspection subject is negatively correlated with the life-cycle cost. However, when the detection accuracy reached 71%, the life-cycle cost was only 5% higher than the minimum cost, which is a good result for low operating costs. An evaluation model of the inspection indices under the combined effect of departure cost d, maximum mileage m, inspection accuracy pi and probability of failure pf of the inspection robot was established to guide its intervention indexes.

Investigation on inspection scheduling for the maintenance of tunnel with different degradation modes

Inspection scheduling is an important task in the maintenance of tunnels, which will obviously affect the lifetime maintenance cost and risk. Traditional periodic inspection with fixed inspection interval has low efficiency in the early age or high risk in the later age of tunnel. This study aims at developing a novel non-periodic inspection policy and investigating the adaptability of several inspection scheduling functions with different degradation modes. Three classical degradation modes (namely, accelerated, decelerated and stationary) are categorized to describe the potential trends of degradation, and they are modeled by non-stationary Gamma process in a unified form. The effects of three types of inspection scheduling functions on the tunnel maintenance are compared in the view of the expectation of lifetime maintenance cost. The results of the numerical analysis show that inspection schedule function f2(x) (in the form of a convex curve) always yields the lowest maintenance cost, regardless of the degradation mode of tunnel. The results indicate that the inspection scheduling function in the type of a convex function is the best and shortening the inspection interval is an effective measure to reduce maintenance cost and risk. The proposed method and the obtained results are helpful for practitioners to make more effective inspection scheduling and to reduce both risk and cost associated with the maintenance of tunnels.

Reliability estimation considering multi-stress monotonic degradation test data with non-constant scale parameter

The accelerated degradation testing (ADT) has been proven to be effective in quickly assessing the reliability of a highly reliable product, especially during product design and development stage. However, the modeling of ADT data to estimate reliability at normal operating conditions is still a challenging task. It becomes even more complex when there are more than one accelerating factors. In this study, a nonstationary gamma process is considered to model the degradation behavior assuming that the nature of product deterioration is strictly monotonic and non-negative. It further assumes that both gamma process parameters are stress-dependent while considering multiple stresses and their interaction effects. To deal with a problem of computational complexity caused by the consideration of multiple stresses and parameter dependency, a maximum likelihood method has been used for the model parameter estimation. The lifetime and reliability parameters under normal operating conditions are estimated using the acceleration models. A Monte Carlo simulation with the Bayesian updating method has been incorporated to update the gamma parameters when new degradation data become available. A carbon-film resistor degradation data is employed to demonstrate the efficacy of the proposed reliability assessment methodology.

Degradation modeling for reliability estimation of DC film capacitors subject to humidity acceleration

To investigate the lifetime of DC film capacitors subject to humidity environment, this paper proposes a degradation model to extract reliability characteristics from performance degradation measures. The percentage of capacitance loss of DC film capacitors is driven by a non-stationary gamma process and the component heterogeneity is characterized by random effects. Based on the model, the method of maximum likelihood for the model parameter estimation and the reliability characteristics including the probability density function of failure time and its confidence interval are developed. The effectiveness and superiority of the proposed method are illustrated and verified by comparing with the conventional failure time analysis method. The results indicate that the proposed method is capable of obtaining comparable lifetime prediction with reduced accelerated testing time.

STATE-ORIENTED MAINTENANCE STRATEGY FOR DETERIORATING SEGMENTAL LINING OF TUNNEL

The lining of tunnel is the principal part in ensuring its usage for transportation or utilities. Maintenance during operation is the way to keep structure in the state of high performance. To consider the efficiency of maintenance within limited resources (maintenance cost), state-oriented maintenance (SOM) approach is proposed. The SOM approach takes service state of the tunnel lining as a levelled target which should be maintained by different degrees of repairing measures. The evolution of service state of tunnel lining is modelled as a non-stationary Gamma process in this investigation, as this process can characteristic the variations of degradation rates. Two cases of numerical analysis with SOM approach give their lifetime maintenance costs. Case one (MS-1) sets SOM strategy with two-levelled service state. The other case (MS-2) adopts SOM strategy with multi-levelled service state. The numerical results demonstrate that the later one would reduce maintenance cost during the whole service life of tunnel lining.

Planning of step-stress accelerated degradation test based on non-stationary gamma process with random effects

Abstract This paper discusses the design problem of the step-stress accelerated degradation test (SSADT) based on the non-stationary gamma process with random effects. The cumulative exposure (CE) model is used to link the degradation paths of the SSADT under different stress levels. The expectation maximization (EM) algorithm is applied to estimate the model parameters. The purpose is to design an optimal experiment plan by minimizing the asymptotic variance of the estimated reliability of the product at a predesigned mission time of interest. Under the budget and boundary constraints, the design variables such as sample size, the measurement frequency at each stress level, and the number of measurements at each stress level are obtained. In the end, an example about the light emitting diode (LED) chip is used to illustrate the proposed model.

Health evaluation method for degrading systems subject to dependent competing risks

This paper proposes a health evaluation method for degrading systems subject to competing risks of dependent soft and hard failures. To characterize the time-varying degradation rate, the degradation process is determined by a non-stationary Gamma process and the soft failure is encountered when it exceeds a predefined critical level. For the hard failure, a Cox's proportional hazard model is applied to describe the hazard rate of the time to system failure. The dependent relationship is modeled by incorporating the degradation process as a time-varying covariate into the Cox's proportional hazard model. To facilitate the health characteristics evaluation, a discretization technique is applied both to the degradation process and the monitoring time. All health characteristics can be obtained in the explicit form using the transition probability matrix, which is computationally attractive for practical applications. Finally, a numerical analysis is carried out to show the effectiveness and the performance of the proposed health evaluation method.

Condition-based maintenance modeling for a two-stage deteriorating system with random changes based on stochastic process

PurposeThe purpose of this paper is to develop a condition-based maintenance (CBM) model for those systems subject to the two-stage deterioration including a deterioration pitting initiation process and a deterioration pitting growth process.Design/methodology/approachRegarding environmental changes as random shocks, the effect of environmental changes on the deterioration process is considered. Then, non-homogeneous Poison process and non-stationary gamma process are introduced to model the deterioration pitting initiation process and the deterioration pitting growth process, respectively. Finally, based on the deterioration model, a CBM policy is put forward to obtain the optimal inspection interval by minimizing the expected maintenance cost rate. Numerical simulations are given to optimize the performance of the deteriorating system. Meanwhile, comparisons between a single-stage deterioration model and a two-stage deterioration model are conducted to demonstrate the application of the proposed approach.FindingsThe result of simulation verifies that the deterioration rate is not constant in the life cycle and is affected by the environment. Furthermore, the result shows that the two-stage deterioration model proposed makes up for the shortage of single-stage deterioration models and can effectively reduce system failures and unreasonable maintenance caused by optimistic prediction using single-stage deterioration models.Practical implicationsIn practical situations, except for normal deterioration caused by internal factors, many systems are also greatly influenced by the random shocks during operation, which are probably caused by the environmental changes. What is more, most systems have self-protection ability in some extent that protects them to keep running as new ones for some time. Under such circumstances, the two-stage deterioration model proposed can effectively reduce system failures and unreasonable maintenance caused by optimistic prediction using single-stage deterioration models. In the combination with the bootstrap estimation, the paper obtains the life distributions with approximate 95 percent confidence intervals which can provide valuable information for practical system maintenance scheduling.Originality/valueThis paper presents a new CBM model for those systems subject to the two-stage deterioration including a deterioration pitting initiation process and a deterioration pitting growth process. Considering the effect of the environmental change on the system deterioration process, a two-stage deterioration model with environmental change factors is proposed to describe the system deterioration.

Modeling condition-based maintenance and replacement strategies for an imperfect production-inventory system

In this article, we consider an imperfect production-inventory system which produces a single type of product to meet the constant demand. The system deteriorates stochastically with usage and the deterioration process is modeled by a non-stationary gamma process. The production process is imperfect which means that the system produces some non-conforming items and the product quality depends on the degradation level of the production system. To prevent the system from deteriorating worse and improve the product quality, preventive maintenance is performed when the level of the system degradation reaches a certain threshold. However, the preventive maintenance is imperfect which cannot restore the system as good as new. Hence, the aging system will be replaced by a new one after some production cycles. The preventive maintenance cost, the replacement cost, the production cost, the inventory holding cost and the penalty cost of lost sales are considered in this article. The objective is to minimize the total cost per unit item which depends on two decision variables: the preventive maintenance threshold and the time at which the system is replaced. We derive the explicit expression of the total cost per unit item and the optimal joint policy can be obtained numerically. An illustrative example and sensitivity analysis are given to demonstrate the proposed model.

Maintenance strategies optimisation of metro tunnels in soft soil

This paper proposes a quantitative approach for selecting effective maintenance strategies for metro tunnels in order to reduce maintenance cost and ensure tunnel safety in the presence of structural degradation during the service life. A non-stationary gamma process is used to simulate the tunnel gradual degradation and to investigate the effects of different maintenance measures on the life cycle maintenance cost of metro tunnels in soft soil. Two commonly applied reinforcement methods, Aramid Fibre Reinforced Polymer and bonded steel plate reinforcement, are investigated in this study. The feasibility of the proposed methodology is demonstrated with maintenance strategies based on periodic and aperiodic inspection policies. Numerical results show that the optimisation of maintenance strategy can reduce the life cycle maintenance cost of metro tunnels in soft soil while ensuring sufficient level of structural safety, and the maintenance strategy with aperiodic inspection policy produces lower maintenance cost.

Condition-based maintenance policies for a combined wear and shock deterioration model with covariates

This paper proposes a deterioration model which considers two mechanisms of system deterioration: wear and shock, both affected by environmental conditions. The deterioration is modeled by the sum of wear and cumulative shocks, where the wear damage is modeled by a non-stationary Gamma process and the random shock damage is modeled by a generalized Pareto distribution satisfying Poisson arrivals. The effect of the environment on the mean deterioration rate is modeled by covariates via the Cox proportional hazards model. Both preventive and corrective replacement will be considered according to the results of system-state inspections. Based on partial inspection information, the so-called wear-oriented and shock-oriented maintenance policies are proposed specially for the different deterioration sources. The performance of these maintenance policies are studied through numerical implementations and compared with the policies which use complete inspection information. The optimal maintenance policy leading to the minimal cost criterion is obtained and limitation of the use of these policies is discussed.

Joint optimisation of buffer size and preventive maintenance for a deteriorating upstream machine

We consider a manufacturing system in which an upstream machine (UM) transfers a raw material to a downstream machine. The UM deteriorates stochastically with usage and may fail. The deteriorating process is described by a non-stationary gamma process. Preventive maintenance action is performed on the UM at its working age T. The preventive maintenance cannot restore the machine to as good as new state. The machine tends to deteriorate faster with the increasing number of maintenance. A buffer has been built between the two machines in order to cope with unexpected failure and planned preventive maintenance. The buffer size K and the working age T at which the preventive maintenance is initiated are the two decision variables of the model. Under the bivariate policy (K, T) we derive the average cost rate C(K, T). The optimal policy (K*, T*) such that C(K, T) is minimised can be obtained numerically by trading off the operational cost, the preventive maintenance cost, the replacement cost, the inventory holding cost and the shortage cost. We study, by means of numerical results, the influence of the variation of some parameters on the optimal policy.

Gradual Reliability Sensitivity Analysis of Mechanical Part Considering Preventive Maintenance

A lot of mechanical parts are subject to failure due to the deterioration. Usually the preventive maintenance is taken to ensure the safety and reliability. Therefore, it is very important to study the gradual reliability design of the mechanical part for improving the gradual reliability of the mechanical system under the condition of considering the preventive maintenance. Beta distribution is employed to describe the randomness of the mechanical part state after the preventive maintenance. The deterioration process of the mechanical part is modeled using the nonstationary Gamma process. The gradual reliability model before the first preventive maintenance is proposed according to the gradual failure principle and using the initial state distribution and the properties of Gamma process. Then, the gradual reliability model between any two times of preventive maintenance is also derived. Subsequently, the sensitivity equations of the proposed gradual reliability model to each parameter are given. The application process and practicality of the proposed approach are described by a numerical example. This work solved the problem where the maintenance has not been well considered in the reliability design of the mechanical part and contributed to the theory and method of improving the safety and reliability operation of the mechanical system.

Lifetime models for remaining useful life estimation with randomly distributed failure thresholds

In order to predict in advance and with the smallest possible uncertainty when a component needs to be fixed or replaced, lifetime models are developed based on the information of the component deterioration trend and its failure threshold to estimate the stochastic distribution of the hitting time (the first time the deterioration exceeds the failure threshold) and the remaining useful life. A primary issue is how to effectively handle the uncertainties related to the component deterioration trend and failure threshold. This problem is here investigated considering a non-stationary gamma process to model the component deterioration and a gamma-distributed failure threshold. Two lifetime models are proposed for comparison on an application concerning deterioration of choke valves used in offshore oil platforms.

Condition-based maintenance for a deterioration system with shock in dynamic environment

The aim of this paper is to discuss the problem of modelling and optimizing condition-based maintenance policies for a deteriorating system in a dynamical environment. The non-stationary Gamma process and the Generalized Pareto distribution with Poisson arrival are employed to model the wear deterioration and shock damage to the system within the dynamic environment represented by a Markov chain, say covariates. A model similar to the proportional hazards model is used to show the influence of covariates on the deterioration. In the framework of the system under consideration, an appropriate inspection/replacement policy which minimizes the expected average maintenance cost is derived. The average cost under different maintenance policies is analyzed through simulation experiments to compare the policies performances.

A time-discrete extended gamma process for time-dependent degradation phenomena

The non-stationary Gamma process is a widely used mathematical model to describe degradation phenomena whose growth rate at time t depends only on the current age of the item and not on the accumulated damage up to t. Nevertheless, the Gamma process is not a proper choice when there is empirical evidence that the variance-to-mean ratio of the process varies with time, because the Gamma process implies a constant variance-to-mean ratio. This paper proposes a generalization of the non-stationary Gamma process, which can be viewed as a time discretization of the extended Gamma process and allows one to describe time-dependent degradation phenomena whose variance varies with time t, not necessarily in proportion to the mean. A way to approximate the exact distribution of the degradation growth over a given time interval is given and a test for assessing whether the assumption of the Gamma process can be rejected or not is discussed. Finally, the proposed model is applied to a real dataset consisting of the sliding wear data of four metal alloy specimens.

Modelling and optimizing imperfect maintenance of coatings on steel structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. This paper considers the problem of finding the set of actions that minimizes the expected (discounted) maintenance costs over both a finite horizon and an infinite horizon. To this end the size of the area affected by corrosion is modelled by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, due to maintenance the parameters of the gamma deterioration process may also change. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with numerical examples.

A comparison of models for measurable deterioration: An application to coatings on steel structures

Steel structures like bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by organic coating systems. This paper focuses on modelling the deterioration of the organic coating layer that protects steel structures from corrosion. Only if there is sufficient knowledge of the condition of the coating on these structures, maintenance actions can be done in the most efficient way. Therefore the course of the deterioration of the coating system and its lifetime, which is also of importance for doing maintenance, have to be assessed accurately. In this paper, three different stochastic processes, viz. Brownian motion with non-linear drift, the non-stationary gamma process and a two-stage hit-and-grow physical process, are fitted to two real data sets. In this way we are the first who compare the three stochastic processes empirically on criteria such as goodness-of-fit, computational convenience and ease of implementation. The first data set is based on expert judgement; the second consists of inspection results. In the first case the model parameters are obtained by a least-squares approach, in the second case by the method of maximum likelihood. A meta-analysis is performed on the two-stage hit-and-grow model by means of fitting Brownian motion and gamma process to the outcomes of this model.