Estimating reliability of a repairable system with imperfect coverage and fuzzy parameters using parametric non-linear programming approach

Abstract

In this paper, we consider the problem of evaluating system characteristics (MTTF, reliability) using Markov modeling approach, in which times to failure and times to repair of the operating units are, assumed to follow fuzzified exponential distribution. A method has been developed to construct a fuzzy set as an estimator for unknown parameters in the proposed statistical model. Using the α-cut approach the fuzzy repairable system is extracted from the conventional crisp intervals for the desired system characteristics, which are determined with a set of parametric nonlinear programs using their membership functions. With the proposed approach, explicit closed-form expressions of the system characteristics are obtained by inverting the interval limits of α-cuts of membership functions.

Appendices

Appendix 1. The Laplace transform equations fo a reliability model of a two-unit repairabel system

Assume that the process initially is in state 2, so P₂(0) = 1, P₁(0) = 0 and P₀(0) = 0. For the state transition diagram given in Fig. 1, the system differential equations using Laplace transforms are obtained in terms of λ, μ, and coverage-factor c are given by

$$ \begin{array}{c}\hfill s{\tilde{P}}_2(s)-1=-2\lambda {\tilde{P}}_2(s)+\mu {\tilde{P}}_1(s)\hfill \\ {}\hfill s{\tilde{P}}_1(s)=-\left(\lambda +\mu \right){\tilde{P}}_1(s)+2\lambda c{\tilde{P}}_2(s)\hfill \\ {}\hfill s{\tilde{P}}_0(s)=\lambda {\tilde{P}}_1(s)+2\lambda \left(1-c\right){\tilde{P}}_2(s)\hfill \end{array} $$

On solving this system of linear equations we obtain the Laplace transforms of P_i(t) for i = 0, 1, 2.

$$ \begin{array}{c}\hfill {\tilde{P}}_2(s)=\frac{\left(s+\lambda +\mu \right)}{\left(s+2\lambda \right)\left(s+\lambda +\mu \right)-2\lambda \mu c}\hfill \\ {}\hfill {\tilde{P}}_1(s)=\frac{2\lambda c}{\left(s+2\lambda \right)\left(s+\lambda +\mu \right)-2\lambda \mu c}\hfill \\ {}\hfill {\tilde{P}}_0(s)=\frac{2{\lambda}^2+2\lambda \left(1-c\right)\left(s+\mu \right)}{s\left(s+2\lambda \right)\left(s+\lambda +\mu \right)}\hfill \end{array} $$

Appendix 2. The steady-state equations of an availability model for a two-unit repairabel system

From the state transition diagram given in Figs. 2 and 3 the steady-state equations of the process are given by

$$ \begin{array}{c}\hfill \left(\lambda +\mu \right){P}_1=2\lambda c{P}_2\hfill \\ {}\hfill 2\lambda {P}_2=\mu {P}_1\hfill \\ {}\hfill {P}_0=\lambda {P}_1+2\lambda \left(1-c\right){P}_2\hfill \end{array} $$

On solving this system of linear equations the steady-state probabilities can be obtained as:

$$ \begin{array}{c}\hfill {P}_2=\frac{\mu^2}{2{\lambda}^2+2\lambda \mu \left(2-c\right)+{\mu}^2}\hfill \\ {}\hfill {P}_1=\frac{2\lambda \mu }{2{\lambda}^2+2\lambda \mu \left(2-c\right)+{\mu}^2}\hfill \\ {}\hfill {P}_0=\frac{2{\lambda}^2+2\lambda \mu \left(1-c\right)}{2{\lambda}^2+2\lambda \mu \left(2-c\right)+{\mu}^2}\hfill \end{array} $$