Arabian Journal for Science and Engineering https://doi.org/10.1007/s13369-018-3206-2 RESEARCH ARTICLE - SYSTEMS ENGINEERING Bayesian Estimation of Parameters of Reliability and Maintainability of a Component under Imperfect Repair and Maintenance Ahmed Farouk Abdul Moneim 1 · Mootaz Ghazy 1 · Amr Hassnien 2 Received: 8 August 2016 / Accepted: 20 March 2018 © King Fahd University of Petroleum & Minerals 2018 Abstract 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 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. Keywords Bayesian estimation · Imperfect maintenance · Reliability and maintainability parameters List of symbols PM Preventive maintenance CM Corrective maintenance RAM Reliability and maintainability parameters h (t i ) Failure rate at time t i R (t i ) Reliability function T Time interval between two successive maintenances t i Time interval considered in i th interval (t i = 0) at start of ith interval. This interval is ended by either maintenance (t i = T ) or by failure (t i = ttf) ttf Time to failure is random variable distributed by Weibull distribution VA i Virtual age of the component at start of i th interval VA 1 =0 B Mootaz Ghazy mootaz.ghazy@aast.edu Ahmed Farouk Abdul Moneim mail@ahmedfarouk.org Amr Hassnien amrhassnien@yahoo.com 1 Industrial and Management Engineering Department, Arab Academy for Science and Technology and Maritime Transport, Abou Keer Campus, P.O. Box 1029, Alexandria, Egypt 2 Egyptian Projects Operation and Maintenance “EPROM”, Alexandria, Egypt VA i = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ VA i -1 + α R t i -1 In case of failure at time t i -1 during the (i - 1)th interval (i ≥ 2) VA i -1 + α m T In case of no failure during the (i - 1)th interval and performing preventive maintenance at the end of (i - 1)th interval(i ≥ 2) α m ,α R Coefficients of virtual age remained after PM and CM, respectively η, β Scale and shape factors of Weibull distribution of time to failure L(.) Likelihood function N Number of operational intervals under consideration ν i = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 in case of having failure during the ith interval (“CM”) 0 in case of having No failure in the ith interval (“PM”) 1 Introduction Design and development of engineering systems of different structures require serious considerations of their reliabilities within a pre-specified span of life time. Moreover, one of the most important prerequisites of rational designs is to render these systems maintainable and repairable perfectly and efficiently. In order to enable engineers to undertake such extremely difficult assignments, mathematical models 123