Condition Based Maintenance (CBM) or predictive maintenance is based on observing an indicator of the degradation state of the equipment at different intervals of time in order to make an informed decision concerning the maintenance of this equipment. The objectives of this thesis are:​ (a) to determine the optimal replacement policy and optimal inspection interval for a piece of equipment when the degradation process is not outwardly visible, the indicator does not directly indicate the equipment state, and the inspections are costly;​ (b) to determine the Reliability Function (RF) and the Mean Residual Life (MRL) of such equipment at each observation moment;​ (c) to introduce a method for estimating the parameters of the models introduced in previous objectives.

Throughout this thesis, we assume that the equipment's unobservable degradation state transition follows a Markov Chain and we model it by a Hidden Markov Model. Bayes' rule is used to determine the probability of being in a certain degradation state at each observation moment. Cox's time-dependent Proportional Hazards Model (PHM) is considered to model the equipment's failure rate.

The first part of this thesis introduces a model to find the optimal inspection period for Condition Based Maintenance (CBM) of a system when the information obtained from the gathered data on the system does not reveal the system's exact degradation state and the collection of data is costly. By using dynamic programming, the system's optimal replacement policy and its total long run average operating maintenance cost are found. Based on the long run average cost, the optimal inspection interval and the corresponding replacement criterion are specified. A numerical example shows the behaviour of the CBM model when the inspection is costly, and finds the optimal inspection period and the maintenance cost.

In the second part of this thesis, a model to calculate the Reliability Function (RF) and the Mean Residual Life (MRL) of a piece of equipment when its degradation state is not directly observable is introduced. At each observation moment, an indicator of the underlying unobservable degradation state is observed and the monitoring information is collected. The conditional reliability is derived from the PHM and it is used to calculate the RF and the MRL. Two examples are presented. The MRL is calculated at all possible state probabilities for four observation moments. It is shown that the MRL can be used as a supplementary decision tool, in particular when the cost elements of preventive replacement are unknown, or when there are criteria other than the cost to respect.

The third part of this thesis proposes a method to estimate the parameters of the models that were introduced in the previous parts. The parameters of the PHM, the Markov process transition matrix, and the stochastic matrix of observations/states are estimated based on the Maximum Likelihood Estimation (MLE) method. By using a Monte Carlo simulation approach, it is shown that the method used gives estimation results that converge to the real values of the parameters as the sample size increases. In addition, the behavior of the method has been examined when censored data exist.

La maintenance conditionnelle (CBM), est une strategic d'intervention en maintenance basee sur l'observation a des intervalles reguliers d'elements indiquant l'etat de degradation d'un equipement. Le principal probleme est de prendre les meilleures decisions pour effectuer l'inspection de l'equipement, etablir le lien entre les elements observes lors de l'inspection et l'etat de degradation effectif de l'equipement et d'evaluer la fonction de fiabilite et la duree de vie residuelle comme criteres de decision. Plus particulierement, cette these propose une demarche coherente afin:​ (a) de determiner la politique de remplacement optimale ainsi que Pintervalle d'inspection optimal des equipements lorsque le processus de deterioration n'est pas directement observable (non visible), (b) de determiner la fonction de fiabilite (RF) ainsi que Fesperance de vie residuelle (MRL) de tel equipement a chaque periode d'observation afin d'evaluer le pouvoir de prediction du modele de remplacement propose, (c) d'introduire des methodes d'estimation des parametres du modele dans un contexte ou la relation entre les symptomes de degradation (indicateurs) et l'etat reel de l'equipement n'est pas deterministe.

Un modele taux de defaillance proportionnel de Cox (PHM) est utilise pour modeliser le taux de defaillance de l'equipement. Un Modele de Markov Cache (HMM) est propose pour modeliser la degradation non visible. Nous presentons un politique CBM optimale et un intervalle d'inspection optimal, RF et MRL de l'equipement en plus de l'estimation des parametres permettant d'adapter le modele en situation reelle, lorsqu'il existe une relation stochastique entre la degradation non visible de Pequipement et la valeur de l'indicateur d'inspection.

La programmation dynamique (DP), le Processus de decision markovien partiellement observable (POMDP) et les probabilites appliquees sont utilises afin de resoudre les problemes etudies dans cette these. Des exemples numeriques sont donnes pour illustrer les modeles proposes. Des simulations ont ete effectuees afin de tester la robustesse et la convergence des methodes d'estimation des parametres proposes.