Numerical method for Weibull generalized renewal process and its applications in reliability analysis of NC machine tools

In generalized renewal process (GRP) reliability analysis for repairable systems, Monte Carlo (MC) simulation method instead of numerical method is often used to estimate model parameters because of the complexity and the difficulty of developing a mathematically tractable probabilistic model. In this paper, based on the conditional Weibull distribution for repairable systems, using negative log-likelihood as an objective function and adding inequality constraints to model parameters, a nonlinear programming approach is proposed to estimate restoration factor for the Kijima type GRP model I, as well as the model II. This method minimizes the negative log-likelihood directly, and avoids solving the complex system of equations. Three real and different types of field failure data sets with time truncation for NC machine tools are analyzed by the proposed numerical method. The sampling formulas of failure times for the GRP models I and II are derived and the effectiveness of the proposed method is validated with MC simulation method. The results show that the GRP model is superior to the ordinary renewal process (ORP) and the power law non-homogeneous Poisson process (PL-NHPP) model.

► Numerical method for the Kijima type generalized renewal process model I and II is proposed using a programming approach.
► The sampling formulas of failure times for the Weibull generalized renewal process models I and II are derived.
► Three real and different types of field failure data sets with time truncation for NC machine tools are analyzed.