Algorithmic analysis of a complex reliability system subject to multiple events with a preventive maintenance strategy and a Bernoulli vacation policy through MMAPs

In this work, a single-unit multi-state system is considered. The system is subject to internal failures, as well as external shocks with multiple consequences. It also incorporates a preventive maintenance strategy and a Bernoulli vacation policy for the repairperson. It is algorithmically modeled in both continuous and discrete time using Marked Markovian Arrival Processes (MMAP). The system's operation/degradation level is divided into an indeterminate number of levels. Upon returning from a vacation period, the repair technician may initiate corrective repair, perform preventive maintenance, replace the unit, remain idle at the workplace, or begin a new vacation period. The decision in the latter two cases is made probabilistically based on the system's operational level. This methodology allows the model and its associated measures to be algorithmically derived in both transient and stationary regimes, presented in a matrix-algorithmic form. Analytical-matrix methods are used to obtain the system's steady-state behaviour as well as various performance measures. Costs and rewards are introduced to analyze when the system becomes profitable. Measures associated with costs over time and in the stationary regime are defined and considered for optimization studies. A numerical example demonstrates the versatility of the model by solving a probabilistic optimization problem using a multi-objective Pareto analysis approach and performing a comparative evaluation of multiple models. Genetic algorithm is applied to find the optimization results in the reduced solution space. All modeling and associated measures have been computationally implemented in Matlab.