Modeling and Sequential Repairs of Systems Considering Aging and Repair Effects

Abstract

The reliability of a repairable system depends on the system age and the number of repairs it experienced. When these effects are considered, predicting the system reliability metrics, such as the cumulative number of failures and failure intensity in the future, becomes a challenging problem. Many existing models utilize Monte Carlo simulations to do prediction but this entails significant computational efforts. This chapter presents a modified Proportional Failure Intensity model to analyze repairable systems. By further modification (approximation) to the model, the system reliability metrics and the associated confidence bounds can be effectively predicted without conducting time-consuming simulations. Moreover, to make repair/replacement decisions, most research assumes the repair model of the system is available beforehand. In practice, however, the model needs to be estimated based upon failures and sequential repair/replacement decisions must be made based on the predicted system reliability metrics. The proposed model is utilized in this decision-making paradigm considering a short-run cost rate criterion. Unlike the widely used long-run cost rate, this criterion emphasizes the economic impact of a repair/replacement decision on the next fail-and-fix cycle of the system. Two benchmark data sets are analyzed to demonstrate the model in practical use.

Appendices

Appendix 1: Proof of Theorem 1

Suppose both the replacement time and the first failure time after the replacement follow exponential distributions with parameters $\lambda_{\xi }$ and $\lambda_{0}$, respectively. Then, the pdf of X is:

$$ \begin{aligned} g_{X} \left( x \right) & = \int\limits_{0}^{\infty } {\frac{{C_{R} }}{{x^{2} }}\xi \lambda_{\xi } e^{{ - \lambda_{\xi } \xi }} \lambda_{0} e^{{\frac{{\lambda_{0} \left( {C_{R} - x} \right)\xi }}{x}}} d\xi = \frac{{C_{R} \lambda_{0} \lambda_{\xi } }}{{x^{2} }}\int\limits_{0}^{\infty } {\xi e^{{ - \left( {\lambda_{\xi } + \frac{{\lambda_{0} C_{R} - \lambda_{0} x}}{x}} \right)\xi }} d\xi } } \\ & = \frac{{C_{R} \lambda_{0} \lambda_{\xi } }}{{x^{2} }}\left( {\frac{x}{{\left( {\lambda_{\xi } - \lambda_{0} } \right)x + \lambda_{0} C_{R} }}} \right)^{2} = \frac{{C_{R} \lambda_{0} \lambda_{\xi } }}{{\left( {\left( {\lambda_{\xi } - \lambda_{0} } \right)x + \lambda_{0} C_{R} } \right)^{2} }}, \\ \end{aligned} $$

(10.25)

which is in the form of Cauchy distribution. Since $X \in \left( {0,C_{R} } \right),$ X follows the Truncated Cauchy distribution.

Appendix 2: Proof of Theorem 2

Suppose the replacement time has the exponential distribution with parameter $\lambda_{\xi }$ and the hazard rate of the first failure time after the replacement is in the form of $\lambda \left( t \right) = \lambda_{0} \beta t^{\beta - 1} .$ Then, the pdf of $X$ is:

$$ \begin{aligned} g_{X} \left( x \right) & = \int\limits_{0}^{\infty } {\frac{{C_{R} }}{{x^{2} }}\xi \lambda_{\xi } e^{{ - \lambda_{\xi } \xi }} } \lambda_{0} \beta \left( {\frac{{\left( {C_{R} - x} \right)\xi }}{x}} \right)^{\beta - 1} e^{{ - \lambda_{0} \left( {\frac{{\left( {C_{R} - x} \right)\xi }}{x}} \right)^{\beta } }} d\xi \\ & = \frac{{\lambda_{0} \beta C_{R} \left( {C_{R} - x} \right)^{\beta - 1} }}{{x^{\beta + 1} }}\int\limits_{0}^{\infty } {\xi^{\beta } e^{{ - \left( {\frac{{\lambda_{0} \left( {C_{R} - x} \right)^{\beta } }}{{x^{\beta } }}} \right)\xi^{\beta } }} \lambda_{\xi } e^{{ - \lambda_{\xi } \xi }} d\xi } \\ & = \frac{{\lambda_{\xi } C_{R} }}{{x\left( {C_{R} - x} \right)}}\int\limits_{0}^{\infty } {\left( {\xi e^{{ - \lambda_{\xi } \xi }} } \right)d\left( {1 - \exp \left( { - \left( {\frac{{\sqrt[\beta ]{{\lambda_{0} }}\left( {C_{R} - x} \right)}}{x}} \right)^{\beta } } \right)} \right).} \\ \end{aligned} $$

(10.26)

Considering $\psi \left( a \right) = E\left[ {e^{a\xi } } \right]$ is the moment generating function of a random variable $\xi$ and $\psi^{\prime}\left( a \right) = E\left[ {\xi e^{a\xi } } \right]$ is the first derivative of the function with respect to a, the pdf of X becomes $g_{X} \left( x \right) = \frac{{\lambda_{\xi } C_{R} }}{{x\left( {C_{R} - x} \right)}}\psi^{\prime}\left( { - \lambda_{\xi } } \right),$ where $\psi^{\prime}\left( { - \lambda_{\xi } } \right)$ is the first derivative, with respect to $- \lambda_{\xi } ,$ of the moment generating function $\psi \left( { - \lambda_{\xi } } \right)$ of the two-parameter Weibull random variable with scale parameter $\frac{{\sqrt[\beta ]{{\lambda_{0} \left( {C_{R} - x} \right)}}}}{x}$ and shape parameter β. Furthermore, the integral in Eq. (10.26) exists because:

$$ \begin{aligned} & \xi \ge 0,\quad \lambda_{\xi } > 0 \Rightarrow e^{{ -^{{\lambda_{\xi } \xi }} }} \le 1 \Rightarrow \xi e^{{ - \lambda_{\xi } \xi }} \le \xi \Rightarrow \\ & \int\limits_{0}^{\infty } {(\xi e^{{ - \lambda_{\xi } \xi }} } d)\left( {1 - \exp \left( { - \left( {\frac{{\sqrt[\beta ]{{\lambda_{0} \left( {C_{R} - x} \right)}}}}{x}\xi } \right)^{\beta } } \right)} \right) < \int\limits_{0}^{\infty } {\xi d\left( {1 - \exp \left( { - \left( {\frac{{\sqrt[\beta ]{{\lambda_{0} }}\left( {C_{R} - x} \right)}}{x}\xi } \right)^{\beta } } \right)} \right),} \\ \end{aligned} $$

where the term on the right hand side is the expectation $\left( { < \infty } \right)$ of the Weibull random variable.

Appendix 3: Proof of Theorem 3

Suppose the repair time has the exponential distribution with parameter $\lambda_{\eta}$ and the failure intensity of the system follows the modified Proportional Failure Intensity model with the failure intensity $\lambda \left( t \right) = \lambda_{0} \beta t^{\beta - 1} e^{{\gamma N\left( {t - } \right)}}$. Then, the pdf of Y _i is:

$$ \begin{aligned} g_{{Y_{i} }} \left( {y_{i} } \right) & = \int\limits_{0}^{\infty } {\frac{{C_{r} }}{{y_{i}^{2} }}} \eta \lambda_{\eta } e^{{ - \lambda_{\eta } \eta }} \lambda_{0} \beta \left( {\frac{{\left( {C_{r} - y_{i} } \right)\eta }}{{y_{i} }} + t_{i} } \right)^{\beta - 1} e^{i\gamma } e^{{ - e^{i\gamma } \lambda_{0} \left( {\left( {\frac{{\left( {C_{r} - y_{i} } \right)\eta }}{{y_{i} }} + t_{i} } \right)^{\beta } - t^{\beta }_{i} } \right)}} d\eta \\ & = \frac{{\lambda_{0} \beta C_{r} e^{i\gamma } }}{{y_{i}^{2} }}e^{{\lambda_{0} e^{i\gamma } t_{i}^{\beta } }} \int\limits_{0}^{\infty } {\left( {\frac{{\left( {C_{r} - y_{i} } \right)\eta }}{{y_{i} }} + t_{i} } \right)^{\beta - 1} e^{{ - \lambda_{0} e^{{i\gamma \left( {\frac{{\left( {c_{r} - y_{i} } \right)\eta }}{{y_{i} }}} + t_{i}\right)^{\beta } }} }} \lambda_{\eta } \eta e^{{ - \lambda_{\eta } \eta }} d\eta } \\ & = \frac{{C_{r} \lambda_{\eta } e^{{\lambda_{0} e^{i\gamma } t_{i}^{\beta } }} }}{{y_{i} \left( {C_{r} - y_{i} } \right)}}\int\limits_{0}^{\infty } {\eta e^{{ - \lambda_{n} \eta }} d\left( {1 - \exp \left( { - \left( {\sqrt[\beta ]{{\lambda_{0} e^{i\gamma } \frac{{(C_{r} - y_{i} )}}{{y_{i} }}}}\left( {\eta - \frac{{ - y_{i} t_{i} }}{{C_{r} - y_{i} }}} \right)} \right)^{\beta } } \right)} \right),} \\ \end{aligned} $$

(10.27)

which is in the form of $\frac{{\lambda_{\eta } C_{r} e^{{\lambda_{0} e^{i\gamma } t_{i}^{\beta } }} }}{{y_{i} \left( {C_{r} - y_{i} } \right)}}\psi^{\prime}\left( { - \lambda_{\eta } } \right),$ where $\psi^{\prime}\left( { - \lambda_{\eta } } \right)$ is the first derivative, with respect to $- \lambda_{\eta }$, of the moment generating function $\psi \left( { - \lambda_{n} } \right)$ of the three-parameter Weibull random variable with location parameter $\frac{{\sqrt[\beta ]{{\lambda_{0} e^{i\gamma } \left( {C_{r} - y_{i} } \right)}}}}{{y_{i} }}$ scale parameter $\frac{{\sqrt[\beta ]{{\lambda_{0} e^{i\gamma } \left( {C_{r} - y_{i} } \right)}}}}{{y_{i} }}$ and shape parameter β.