Gamma Degradation Process Research Articles

Quantitative assessments of performance and robustness of maintenance policies for stochastically deteriorating production systems

Over the last few decades, many efforts have been invested in improving the economic performances of maintenance policies for stochastically deteriorating production systems. However, with the development of complex production systems, maintenance managers are interested not only in cost saving, but also in how to trustworthily plan and allocate the required maintenance budget. In this context, the robustness of maintenance policies which is related to the maintenance cost variability from a renewal cycle to another plays a pivotal role. This research deals with a quantitative approach to jointly assess the economic performance and robustness of some representatives of two most well-known classes of maintenance policies: time-based and condition-based maintenance. To this end, we first propose a new cost criterion which combines the long-run expected cost rate and standard deviation of maintenance cost per renewal cycle. Then, we develop and compare the associated mathematical cost models of the considered maintenance policies on the basis of the Gamma degradation process and the theory of stochastic renewal processes. The comparison results under different situations of maintenance costs and system characteristics show that the optimal configuration of maintenance policies gives the best compromise between the performance and robustness, and is mostly affected by the system downtime. Under this aspect, the condition-based maintenance remains more profitable than the time-based maintenance. Still, maintenance managers could implement condition-based maintenance policies that efficiently control the downtime to maximise the maintenance effectiveness of production systems from both performance and robustness viewpoints.

Performance evaluation of anti-radiation based on the gamma degradation process

In the environment of space radiation, the high-energy charged particles or high-energy photons acting on a spacecraft can cause either temporary device degradation or permanent failure. The traditional probability model is difficult to obtain reliable estimation of unit radiation resistance performance with small samples. Considering that different products will change differently after high-energy particle radiation, we construct a model based on the gamma degradation process. This model can efficiently describe the law of unit radiation resistance variation with the total radiation dose levels under the effect of the total dose and displacement damage. Finally, the proposed model is used to assess the anti-radiation performance of the N-channel power MOSFET device STRH60N20FSY3 produced by STM to obtain average unit radiation resistance, survival probability, survival function, etc.

How heterogeneity influences condition-based maintenance for gamma degradation process

In many applications, units from the same population exhibit heterogeneity that they degrade with different rates due to random factors. This article studies how this heterogeneity in degradation influences condition-based maintenance (CBM) policy. Many CBM polices are developed based on gamma process because it is popularly used to characterise monotone degradation processes. In this study, we also model the unit’s degradation by gamma process. To account for the heterogeneity among units’ degradation, we incorporate a random effect parameter in the gamma process. Then the optimal policy for CBM is obtained through Markov decision process. We show that when heterogeneity exists, the transition probability of degradation state depends on both unit’s age and observed degradation level. And consequently, the optimal maintenance policy is a monotone control limit policy. We conduct extensive numerical experiments to validate and demonstrate our findings in depth.

Optimal Design for Step‐Stress Accelerated Test with Random Discrete Stress Elevating Times Based on Gamma Degradation Process

Recently, a step‐stress accelerated degradation test plan, in which the stress level is elevated when the degradation value of a product crosses a prespecified value, was proposed. The times of stress level elevating are random and vary from product to product. In this paper, we extend this model to a more economic plan. The proposed extended model has two economical advantages compared with the previous one. The first is that the times of stress level elevating in the new model are identical for all products, which enable us to use only one chamber (oven) for testing all test units. The second is that, the new method does not require continuous inspection and to elevate the stress level, it is not necessary for the experimenter to inspect the value of the degradation continually. The new method decreases the cost of measurement, and also there is no need to use electronic sensors to detect the first passage time of the degradation to the threshold value in the new method. We assume that the degradation path follows a gamma process. The stress level is elevated as soon as the measurement of the degradation of one of the test units, at one of the specified times, exceeds the threshold value. Under the constraint that the total experimental cost does not exceed a prespecified budget, the optimal settings including the optimal threshold value, sample size, measurement frequency, and termination time are obtained by minimizing the asymptotic variance of an estimated quantile of the lifetime distribution of the product. A case study is presented to illustrate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Life distribution analysis based on Lévy subordinators for degradation with random jumps

AbstractFor a component or a system subject to stochastic degradation with sporadic jumps that occur at random times and have random sizes, we propose to model the cumulative degradation with random jumps using a single stochastic process based on the characteristics of Lévy subordinators, the class of nondecreasing Lévy processes. Based on the inverse Fourier transform, we derive a new closed‐form reliability function and probability density function for lifetime, represented by Lévy measures. The reliability function derived using the traditional convolution approach for common stochastic models such as gamma degradation process with random jumps, is revealed to be a special case of our general model. Numerical experiments are used to demonstrate that our model performs well for different applications, when compared with the traditional convolution method. More importantly, it is a general and useful tool for life distribution analysis of stochastic degradation with random jumps in multidimensional cases. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 483–492, 2015

Optimum Accelerated Degradation Tests for the Gamma Degradation Process Case under the Constraint of Total Cost

An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of a practical test period. In this paper, statistical methods for optimal designing ADT plans are developed assuming that the degradation characteristic follows a gamma process (GP). The GP-based approach has an advantage that it can deal with more frequently encountered situations in which the degradation should always be nonnegative and strictly increasing over time. The optimal ADT plan is developed under the total experimental cost constraint by determining the optimal settings of variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example.

Radiation-induced degradation of cyclohexanebutyric acid in aqueous solutions by gamma ray irradiation

Abstract The radiation-induced degradation of cyclohexanebutyric acid under gamma ray irradiation was investigated. Degradation experiments were performed with 100 mL sealed Pyrex glass vessels loaded with 80 mL of cyclohexanebutyric acid solutions at various initial concentrations of 10, 20, and 40 mg L −1 . The absorbed doses were controlled at 0, 0.65, 1.95, 3.25, 6.5, 9.75, and 13 kGy. The results showed that gamma ray irradiation could effectively degrade cyclohexanebutyric acid in aqueous solutions. The removal rate of cyclohexanebutyric acid increased significantly with the increase of absorbed dose and the decrease of its initial concentration. At the same time, the removal of chemical oxygen demand (COD) was as effective as that of cyclohexanebutyric acid. The kinetic studies showed that the degradation of cyclohexanebutyric acid followed pseudo first-order reaction. Above all, the proposed mechanism obtained when NaNO 2 , NaNO 3 and tert-butanol were added showed that the ∙ OH radical played a major role in the gamma degradation process of cyclohexanebutyric acid, while ∙ H and e aq − played a minor role in the gamma degradation process. The degradation products were identified by Fourier transform infrared spectroscopy (FTIR) and gas chromatography/mass spectrometry (GC/MS) during cyclohexanebutyric acid degradation.

Residual life estimation based on bivariate non-stationary gamma degradation process

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method.

Optimal Design for Degradation Tests Based on Gamma Processes With Random Effects

Degradation models are usually used to provide information about the reliability of highly reliable products that are not likely to fail within a reasonable period of time under traditional life tests, or even accelerated life tests. The gamma process is a natural model for describing degradation paths, which exhibit a monotone increasing pattern, while the commonly used Wiener process is not appropriate in such a case. We discuss the problem of optimal design for degradation tests based on a gamma degradation process with random effects. To conduct a degradation experiment efficiently, several decision variables (such as the sample size, inspection frequency, and measurement numbers) need to be determined carefully. These decision variables affect not only the experimental cost, but also the precision of the estimates of lifetime parameters of interest. Under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal decision variables are found by minimizing the asymptotic variance of the estimate of the 100p-th percentile of the lifetime distribution of the product. Laser data are used to illustrate the proposed method. Moreover, we assess analytically the effects of model mis-specification that occur when the random effects are not taken into consideration in the gamma degradation model. The numerical results of these effects reveal that the impact of model mis-specification on the accuracy and precision of the prediction of percentiles of the lifetimes of products are somewhat serious for the tail probabilities. A simulation study also shows that the simulated values are quite close to the asymptotic values.

Mis-specification analyses of gamma and Wiener degradation processes

Degradation models are widely used these days to assess the lifetime information of highly reliable products if there exist some quality characteristics (QC) whose degradation over time can be related to the reliability of the product. In this study, motivated by a laser data, we investigate the mis-specification effect on the prediction of product's MTTF (mean-time-to-failure) when the degradation model is wrongly fitted. More specifically, we derive an expression for the asymptotic distribution of quasi-MLE (QMLE) of the product's MTTF when the true model comes from gamma degradation process, but is wrongly assumed to be Wiener degradation process. The penalty for the model mis-specification can then be addressed sequentially. The result demonstrates that the effect on the accuracy of the product's MTTF prediction strongly depends on the ratio of critical value to the scale parameter of the gamma degradation process. The effects on the precision of the product's MTTF prediction are observed to be serious when the shape and scale parameters of the gamma degradation process are large. We then carry out a simulation study to evaluate the penalty of the model mis-specification, using which we show that the simulation results are quite close to the theoretical ones even when the sample size and termination time are not large. For the reverse mis-specification problem, i.e., when the true degradation is a Wiener process, but is wrongly assumed to be a gamma degradation process, we carry out a Monte Carlo simulation study to examine the effect of the corresponding model mis-specification. The obtained results reveal that the effect of this model mis-specification is negligible.

Optimal Burn-In Policy for Highly Reliable Products Using Gamma Degradation Process

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.

Optimal Step-Stress Accelerated Degradation Test Plan for Gamma Degradation Processes

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.