Handling the epistemic uncertainty in the selective maintenance problem
Introduction
The occurrence of failure events in both continuous (e.g. chemical and power plants) and discontinuous (e.g. aircrafts and ships) operating systems may lead to dangerous or even disastrous consequences. Therefore, the choice of optimal maintenance policies as well as the identification of critical components to be maintained are fundamental issues to be faced to ensure high system’s reliability levels (Ding & Kamaruddin, 2014).
The implementation of a whatever maintenance optimization procedure is grounded on the estimate of components’ failure probabilities. However, maintenance data sets available in real industrial contexts are often sparse and not well documented (Bell and Percy, 2012, Dubey et al., 2015). As a consequence, components’ reliability information is unavoidably uncertain. In the literature, uncertainty is traditionally classified into aleatory and epistemic (Gharaei et al., in press, Hoffman and Hammonds, 1994). The aleatory uncertainty arises from the intrinsic variability of the failure process and it is not reducible by further measurements. On the other hand, the epistemic uncertainty is due to the lack of knowledge about the stochastic model which characterizes the failure behaviour of system’s components. In traditional risk and reliability analysis, both kinds of uncertainty have been mainly managed by means of the probabilistic approach. Nevertheless, when referring to rare failure events or to components that have not worked long enough, then the application of a purely probabilistic-based approach is challenging because of the poor availability or even the absence of sufficient reliability data (Gharaei et al., in press, Gharaei et al., 2019, Markowski et al., 2009, Zio, 2018). In order to keep on applying the traditional probabilistic approach, risk and reliability analysts commonly refer to worldwide recognized databases where reliability data of similar components are reported. Despite that, such data disregard the operating conditions of components under investigation and how they are maintained. With these recognitions, a number of alternative frameworks has been introduced in the literature to deal with the uncertainty’s representation and treatment in risk and reliability assessment (Aven and Zio, 2011, Aven, 2011, Baraldi et al., 2010, Dubois, 2010, Hao et al., 2018, Kang et al., 2016). The main part of these alternative approaches recognizes the incorporation of subjective experts’ opinions as playing a pivotal role in uncertain environments, being experts more and more considered as the only available and reliable source of information (Hoseini Shekarabi et al., in press, Sayyadi and Awasthi, 2018, Siuta et al., 2013, Tsao, 2015). To date, experts’ judgments uncertainty about the probability of occurrence of failure events has been mainly dealt with three approaches. The first one characterizes failure events by an interval-valued probability of occurrence named imprecise probability (Coolen, Troffaes, & Augustin, 2011). The only available information about every failure event is that the true value of its probability of occurrence lies on the specified interval. Aiming at determining the probability of the system’s failure, imprecise probabilities related to system’s components are handled by applying the interval arithmetic on union (OR gates) and intersection (AND gates) operators. Nevertheless, if more than one expert is involved during the elicitation stage, the arithmetic of intervals does not provide any structured aggregation rule. On the basis of the Laplace Principle of Insufficient Reason, the second approach proposed in the literature to deal with the uncertainty of experts’ judgments consists in the assignment of a uniform distribution over the interval-valued probability of occurrence of failure events (Aven, Baraldi, Flage, & Zio, 2014). The latter means that the probability of occurrence of failure events is also considered as a stochastic variable. Afterwards, the Monte Carlo simulation is used to propagate uncertainties of components to the whole system whose failure event is finally characterized by a probability distribution as well. However, the assumption of a uniform distribution for modelling the uncertainty which affects the probability of occurrence of failure events can lead to results only in appearance exact (Helton et al., 2004, Sayyadi and Awasthi, in press). Actually, such assumption is not based on any further knowledge about the failure process. In addition, none combination rule is available to aggregate such distributions when several experts are involved. The third and most used approach to manage the epistemic uncertainty affecting reliability’s parameters is the Fuzzy Set Theory (FST) (Awasthi and Omrani, 2019, Chai et al., 2016, Ferdous et al., 2009, Kerk et al., 2017, Rabbani et al., in press, Rabbani et al., 2019, Zadeh, 1978). Within FST, the uncertainty on the probability of occurrence of an event is commonly represented by a fuzzy number whose membership function is often arbitrarily assumed without any further information to support its choice. However, whatever chosen membership function arises from a misrepresentation of the original information provided by the experts. A wide number of fuzzy aggregation methods (e.g. max-min, arithmetic averaging, quasi-arithmetic means, weighted average, fuzzy Delphi method, symmetric sum and t-norm, etc.) is available in the literature. Despite that, the formulation and resolution under a fuzzy environment of constrained optimization models increase the problems computational complexity. Actually, the difficulty of solving fuzzy programming models is well known also only for linear problems (Yu & Li, 2001).
In the authors’ opinion, the aforementioned approaches are deemed to be inadequate to solve the selective maintenance optimization problem. Actually:
- (a)
in real industrial contexts, experts are more likely able to provide the mean time to failure of components in an interval form. As a consequence, the assumption of whatever membership function or probability distribution on intervals elicited from experts does seem like a stretch.
- (b)
When input data are gathered from different and independent sources of information (e.g. several experts), structured aggregation rules are often lacking.
With these recognitions, the present paper proposes a Dempster-Shafer theory (DST) (Shafer, 1976) based approach to deal with the epistemic uncertainty of components’ reliability data in the selective maintenance problem. Referring to a system that has to function for a certain timeframe (e.g. mission time or time between turnarounds), ensuring a high reliability level at the same time, a constrained optimization model for the system’s reliability maximization is formulated. Therefore, components to be maintained only during the system’s downtimes need to be identified. The aleatory model which characterizes the time to failure of the system’s components is incomplete (i.e. affected by epistemic uncertainty), and missing information are elicited from experts in an interval form. Under the DST framework, experts’ judgements are converted into belief masses and opportunely aggregated by means of the Yager combination rule (Yager, 1987). Therefore, the initial selective maintenance model is reformulated under the DST framework, and aggregated values are used as input data of the model to determine the minimum and maximum reliability value of the whole system. To this aim, an exact algorithm is used to solve the model.
The remainder of the paper is organized as follows. The literature review is reported in Section 2, whereas the formulated maintenance optimization model is described in Section 3. A brief overview on DST is reported in Section 4 and the DST-based approach to the selective maintenance problem is described in Section 5. The exact algorithm used to solve the optimization model is detailed in Section 6 and the numerical application reported in Section 7. Conclusions are finally given in Section 8.
Section snippets
Literature review
Literature contributions on the traditional selective maintenance optimization problem are very numerous. In this regard, Rice, Cassady, and Nachlas (1998) formulate a mathematical programming model and provide a heuristic resolution approach. Later, Cassady, Pohl, and Murdock (2001) extend the previous model by including the maintenance cost as a further objective function. Referring to a system whose components are constituted by identical items arranged in parallel, Rajagopalan and Cassady
Model formulation
Let’s refer to a system that has to function for a given timeframe T before maintaining its components during the planned system downtime. To ensure the maximum system’s reliability level R at the end of the period T, the set of components I to be maintained during the system’s stop has to be identified. On the other hand, the time available to perform maintenance activities (i.e. is assumed as a problem constraint. The execution of a whatever maintenance action turns every component back
Basic concepts
The theory of evidence was firstly introduced by Arthur P. Dempster in 1967 and later extended by Glenn Shafer in 1976 (Shafer, 1976) as a mathematical framework for the representation of the epistemic uncertainty. Within the DST framework, the characterization of such uncertainty is grounded on three different measures, namely the Basic Probability Assignment (bpa) or belief mass, the Belief (Bel), and the Plausibility (Pl). Their definition is based on the so called Frame Of Discernment (FOD)
DST-based approach to the selective maintenance optimization problem
The proposed DST-based approach to the selective maintenance problem comprises the following steps.
- 1.
Elicitation of reliability data related to system’s components. This phase involves experts in the field.
- 2.
Conversion of reliability data elicited from experts into basic probability assignments.
- 3.
Aggregation of basic probability assignments of each component.
- 4.
Formulation of the mathematical programming model under the DST framework.
Such stages are detailed in the following.
Optimization algorithm
Let’s consider a series system constituted by blocks, some of which are single components whereas others are arranged in series-parallel. Referring to single components arranged in series, the algorithm matches the two possible maintenance states (performing or not performing the maintenance action) of a first single component along with the two possible states of a second component. Four possible combinations are hence obtained, and here named sequences. Let RS1 and RS2 be the reliability of
Numerical application
Aiming at verifying the ability of the proposed methodology to be applied also to complex systems in an uncertain environment, a simulated case is hereinafter reported. It refers to a system constituted by 80 components. The first 41 components are arranged in series-parallel whereas the remaining 39 are arranged in series (Fig. 1).
The value of the parameter β is set equal to 2 for all components. Considering a team of three experts, MTTF intervals are randomly generated for every component and
Conclusions
The present paper represents the first attempt in the literature to deal with the selective maintenance problem under the presence of epistemic uncertainty. In particular, the knowledge of the aleatory model which characterizes the time to failure of system’s components is incomplete. Therefore, missing information are gathered, in an interval form, from different and independent experts. As a consequence, components’ reliability is also expressed in an interval form and afterwards properly
CRediT authorship contribution statement
Giacomo Maria Galante: Supervision. Concetta Manuela La Fata: Conceptualization, Methodology, Software, Writing - original draft. Toni Lupo: Conceptualization, Methodology, Writing - review & editing. Gianfranco Passannanti: Supervision.
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