Semi Markov Chain models and Cross Entropy approach : tools for industrial maintenance costs optimization

Importance sampling methods, particularly the Cross Entropy (CE) Approach, were originally used to give estimates of rare events probabilities; but recent literature highlights also the usefulness of the Cross-Entropy Method for optimizations. The (CE) approach has its benefits for noisy optimization problems, for which the objective function has no closed analytical form. In this thesis, we present an example of such a noisy optimization problem: the minimization of the expected total cost per time unit of industrial maintenance actions over a fixed horizon. After giving a brief description of different maintenance philosophies (time based versus condition based, we embed our maintenance problem into a Semi Markov Chain (SMC) framework that allows for simulating failure occurrences in industrial systems. In the theoretical part of this PhD thesis, we show that convergence of the maintenance optimization problem reduces to proving the asymptotic stability of the Cross Entropy method that boils down to a stochastic differential equation of Robbins Monro form. The numerical part consists in considering (truncated) Gaussian and Weibull distributions as candidate distributions belonging to the so-called natural exponential family, and evaluating their impacts on minimizing maintenance costs.