Development of a Bayesian multi-state degradation model for up-to-date reliability estimations of working industrial components
Introduction
Multi-state degradation modeling is based on the discretization of the degradation process affecting an industrial equipment in three or more states, each one associated to a certain range of values of suitable degradation indicator variables (e.g., oxidized areas in gas turbine nozzle systems (Compare et al., 2015), electrical resistance values in electrical power switches [3], linear extent of wear in bearing shells [40], [41], performance levels (e.g., amount of power supplied by power generating systems [37] or symptoms (e.g., vibrational signals [4]. The main advantage of this modeling approach over the widely used binary model (i.e., considering only two operational states for the equipment, ‘good’ and ‘failed’) lies in its ability of more accurately describing the sequential phases of degradation, which can be even physically different [32], [36], [53].
The transition times from one state to another are often assumed to be Weibull-distributed (e.g., [2], [25], [41]). This choice is due to the flexibility of the Weibull distribution and the possibility it gives of keeping memory of the time spent in a degradation state, which influences the next stochastic transition time. This property gives rise to a semi-Markov model [33].
The estimation of the model parameters and the characterization of the corresponding uncertainties are fundamental to properly set and use multi-state degradation models. To do this, different approaches have been proposed in the literature, which mainly depend on the available knowledge, information and data. Namely, when a substantial amount of collected data is available, techniques from statistical analysis can be adopted, which may be purely analytical (e.g., [25], [50]) or numerical (e.g., Compare et al., 2015).
On the contrary, situations characterized by scarcity of data are common in industrial applications (e.g., very highly reliable components, new technology just introduced in the production system, etc.). In this case, expert opinion becomes a valuable source of information to be taken into account for developing semi-Markov degradation models. Different approaches have been proposed within probabilistic and non-probabilistic theoretical frameworks. Probabilistic approaches (e.g., [10], [11], [29]) are typically based on the elicitation of subjective expert judgements about the probabilities of occurrence of single events or about the probability distributions of uncertain quantities of interest. Often, the elicitation process is oriented to obtain a suitable prior distribution to be updated in light of the available data, according to the Bayesian paradigm (e.g., [14], [47], [19]). On the other side, within the non-probabilistic approaches, Possibility Theory has been used to tackle the situation in which the knowledge on each Weibull parameter is available in terms of a set of nested intervals with corresponding confidence levels provided by an expert [6], [7]. Similarly, Dempster-Shafer theory of Evidence has been applied to develop a semi-Markov degradation model in the situation in which an interval that is believed to contain the unknown parameter value is asked to each member of a team of experts [2]. Fuzzy logic has also been used with the same aim in [20].
In this work, we want to build a semi-Markov degradation model in an intermediate situation between that of having a sizeable amount of field data (which would justify the application of traditional statistical techniques in the frequentist probability framework) and the opposite, of no data available (which has been treated with non-probabilistic techniques that avoid information commitment). The objective is to fully exploit all the available sources of information in a coherent and solid way.
To do this, we resort to the Bayesian statistics framework, which allows combining the prior knowledge of experts with the evidence coming from field data to build a degradation model useful for maintenance applications (Compare & Zio, 2014). The proposed methodology allows the elicitation of the prior distributions of the model parameters, avoiding possible commitment of the information provided by the expert. Then, an adaptive Markov Chain Monte Carlo algorithm is developed to estimate the posterior distributions of the multi-state model parameters, which encode both the prior knowledge and the evidence brought by the available dataset. Finally, the developed stochastic model is used to derive the expected probabilities of occupying the degradation states over time for new components along with the corresponding credibility intervals, and to estimate the remaining useful life of a new component, which will be detected in a given degradation state after a certain working time.
The proposed procedure is applied to a case study in the biopharmaceutical industry, concerning the Ethylene Propylene Diene Monomer (EDPM) diaphragm installed within a production line of a company leader in that field.
The original contribution of the proposed method mainly lies in 1) the original combination of techniques, taken from the scientific literature of different contexts, which have been adapted to propose a comprehensive development pathway for building a semi-Markov degradation model based on expert knowledge and few field data, 2) the use of a Bayesian semi-Markov degradation model to support maintenance planning.
Furthermore, within the proposed framework for the construction of the Bayesian semi-Markov degradation model, we have developed a novel procedure to sample multiple parameters from their joint posterior distribution. The procedure is based on the combined use of i) an adaptive Markov Chain Monte Carlo (MCMC) algorithm to set possibly acceptable values for the entries of the covariance matrix and ii) the Normal-Random Walk Metropolis Hastings (N-RWMH).
An important property of the proposed framework for supporting maintenance planning is that it allows updating the posterior parameter distribution taking into account the outcome of the last inspection performed on the industrial component under observation. This additional updating is useful in those situations characterized by scarcity of data, where adding a single observation can significantly improve the parameter estimation.
The remainder of the paper is organized as follows. Section 2 introduces the case study motivating the development of the proposed framework. Section 3 defines the assumptions at the basis of the model development. Section 4 illustrates the development of the whole methodology, from the elicitation of the prior distribution to the posterior inference. This is applied to the EDPM diaphragms in Section 5, where the results of the study are also discussed. Some final considerations about the applicability of the proposed framework to other case studies are drawn in Section 6. Section 7 concludes the paper.
Section snippets
Case study motivating the framework development
The case study motivating the development of the proposed methodological framework concerns the gradual degradation of Ethylene Propylene Diene Monomer (EPDM) diaphragms used in the production line of a pharmaceutical company, where they are subject to harsh environmental conditions. The EDPM diaphragm is the sealing element in flow control valves whose function is to maintain the integrity of the hermetically sealed environments within production bioreactors. Severe degradation states of EPDM
Modeling assumptions
In this work, we frame the case study illustrated above as a general parameter estimation problem in which we want to exploit all the available knowledge, information and data [52]. For this, we first build a methodological framework and then we give some practical advices to readers interested in implementing the proposed procedure on other datasets.
We consider a three-state process (Fig. 1), where:
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State 1 refers to a new or mildly degraded component.
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State 2 refers to a fairly degraded
Prior elicitation
In the Bayesian analysis framework, the first important issue to address is the assignment of prior distributions to the model parameters. These distributions can be built starting either from data available from similar phenomena/experiments [17] or from the opinions of experts [8]. Otherwise, in complete absence of information, non-informative priors are used (e.g., [30]).
As mentioned in Section 2, we consider the case in which data for setting the priors are not available and, thus, we have
Case study
In order to develop the three-state model of the degradation process (Section 2), the prior distributions of the model parameters have been elicited from an expert using a questionnaire. The expert was asked to provide the 0.10, 0.50, 0.90 quantiles of the transition time and, additionally, he had the possibility of estimating for some proposed values the correspondent quantile orders. Expert answers are reported in Table 2. Finally, the expert indicated as Most Trustworthy Specification
Discussion
The aim of this Section is to comment on the applicability of the proposed methodology to other case studies.
From a wide perspective, it seems fair to say that when the objective of a research investigation is to exploit all the knowledge, information and data available to solve a general issue such as the estimation of the parameters of a multi-state model, then the peculiarities of the considered case study always compromise the generality of the developed solution.
In this work, the peculiar
Conclusions
In this paper, we have proposed a complete procedure for assessing the reliability of industrial equipment whose degradation process can be described through a three-state semi-Markov model. The method has been set within a Bayesian framework, that allows updating the prior distribution elicited from an expert with the available field data. In details, a Metropolis-Hastings algorithm, combined with numerical integration methods, has been proposed to sample from the complex posterior
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