Arabian Journal for Science and Engineering

, Volume 44, Issue 3, pp 2497–2502 | Cite as

Bayesian Estimation of Parameters of Reliability and Maintainability of a Component under Imperfect Repair and Maintenance

  • Ahmed Farouk Abdul Moneim
  • Mootaz GhazyEmail author
  • Amr Hassnien
Research Article - Systems Engineering


The determination of maintenance policies in companies depend on the reliability and maintainability (RAM) parameters. This work estimates the RAM parameters of a component under the condition of imperfect maintenance. Exact mathematical expressions using virtual age model of Kijima type \(\mathrm{I}\) are derived for the evaluation of probability distributions of these parameters. The mathematical derivations are based on Bayes formula of posterior probabilities. Bayes formula is applied directly without the need of application of approximate sampling techniques. Computations of probability distribution of parameters of reliability and maintainability have been elaborated using Visual Basic. The application of Bayes formula of posterior probabilities to estimate the parameters shows satisfactory results.


Bayesian estimation Imperfect maintenance Reliability and maintainability parameters 

List of symbols


Preventive maintenance


Corrective maintenance


Reliability and maintainability parameters

\(h\left( {t_i } \right) \)

Failure rate at time \(t_i \)

\(R\left( {t_i } \right) \)

Reliability function


Time interval between two successive maintenances

\(t_i \)

Time interval considered in ith interval (\({{\varvec{t}}}_{{\varvec{i}}} =0\)) at start of ith interval. This interval is ended by either maintenance (\({{\varvec{t}}}_{{\varvec{i}}} =T\)) or by failure (\({{\varvec{t}}}_{{\varvec{i}}}=\hbox {ttf}\))


Time to failure is random variable distributed by Weibull distribution

\(\mathrm{VA}_i \)

Virtual age of the component at start of ith interval


= 0


\(= \left\{ \begin{array}{ll} \mathrm{VA}_{i - 1} + \alpha _\mathrm{R}t_{i - 1} \hbox { In case of failure at time }\\ t_{i - 1} \hbox {during the } ( i - 1 )\hbox {th} \hbox { interval}\, ( {i \ge 2} )\\ \mathrm{VA}_{i - 1} + \alpha _\mathrm{m}T \hbox { In case of no failure during the }\\ ( {i - 1})\hbox {th}\,\hbox { interval and performing}\\ \qquad \hbox { preventive maintenance at the end of}\\ ( {i - 1} ){\mathrm{th}} \hbox { interval} ( {i \ge 2})\\ \end{array}\right. \)

\(\alpha _\mathrm{m},\alpha _\mathrm{R}\)

Coefficients of virtual age remained after PM and CM, respectively

\(\eta ,\beta \)

Scale and shape factors of Weibull distribution of time to failure


Likelihood function


Number of operational intervals under consideration

\(\nu _i\)

\(= \left\{ \begin{array}{ll} 1 &{}\quad \hbox {in case of having failure during the ith}\\ {} &{}\quad \hbox {interval (``CM'')} \\ 0&{}\quad \hbox {in case of having No failure in the ith}\\ {} &{}\quad \hbox {interval (``PM'')} \\ \end{array} \right. \)


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Industrial and Management Engineering Department, Arab Academy for Science and Technology and Maritime TransportAlexandriaEgypt
  2. 2.Egyptian Projects Operation and Maintenance “EPROM”AlexandriaEgypt

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