On the use of jump-diffusion process for maintenance decision-making: A first step
Résumé
A major concern for engineers and managers nowadays is to make high quality products and highly reliable systems. In this context, reliability analysis and efficient maintenance decision-making are strongly required. Reliability analysis needs collecting, modeling and analysis different types of measure especially for deteriorating systems. In presence of information data on system deterioration level or health indicator, the development of an accurate mathematical model describing the system behavior is indispensable. Stochastic processes are suitable tools to model the system behavior by taking into account its dynamic evolution and the possible influencing random phenomenon. The stochastic processes are largely used in reliability modeling and prognostics e.g. Markov chains, birth processes, counting processes, Gamma processes, and Wiener processes. In the case of gradually deteriorating systems, mainly Gamma and Wiener processes are considered to model the evolution of the deterioration in time respectively for monotonically deteriorating systems and for non-monotone phenomenon. A more general framework that encompasses these latter models is the diffusion process. In this framework the process is defined as the solution of a stochastic differential equation which means that each infinitesimal increment of degradation is described by the sum of drift and a random perturbation. Sudden events and relatively large fluctuations that may appear in real data variation can also be represented by allowing the stochastic process to include “jumps factors”. One important type of stochastic process that has the ability to describe the general behavior of data evolution as well as the particular large fluctuations is the jump-diffusion process. In this paper, based on a case study related to real collected data, the aim is to predict as precise as possible the failure time presented by the first crossing time of a given failure threshold by the deterioration level. In this purpose, a jump-diffusion process is used to model the data evolution and predict the future behavior of the system under consideration. The given dataset consists in several independent nonmonotone trajectories considered as the deterioration trajectories of different items. The variance behavior is found to be similar for the different independent trajectories. According to the variance behavior a division of the time do main into three different states is proposed. The transition between these states is supposed to be controlled by an explanatory variable, called covariate, following a Markov chain. At each state, data evolution is considered to follow the jump-diffusion process with Log-Uniform Jump Amplitude (log-normal distributed increments with jumps). The data fitting and parameters estimations are detailed in the rest of this paper. Eventually, the resulting model is used in forecasting and risk prediction. Giving an estimation of the future evolution of system within a certain confidence interval, this model permits to schedule the maintenance actions.