Reliability assessment and bi-objective optimization in repairable redundant retrial systems with working breakdown and Bernoulli feedback

This paper examines a finite-capacity machine repair system with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> parallel operating machines, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> </mml:math> warm standby machines, and C cold standby machines. It incorporates retrial and Bernoulli feedback queueing characteristics to investigate reliability and optimization issues. The system is equipped with an unreliable repairman responsible for repairing failed machines. During operation, the repairman may experience working breakdown, and provides repair services at a lower rate during the failure. Firstly, based on Markov process theory, matrix analysis method, and Cramer’s rule, transient and steady-state analyses are conducted on the system. By solving the balance equations in matrix form, the queueing and reliability indicators of the system are obtained. Furthermore, the closed form solution for the system’s transient probabilities is derived using the Laplace transform and the eigenvalue method. Subsequently, numerical experiments and numerical simulations are presented to illustrate the effects of system parameters on performance indicators, followed by a sensitivity analysis of reliability measures. Finally, from the decision-maker’s perspective, a bi-objective optimization model is formulated to maximize steady-state availability and minimize total cost, using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO) algorithms to seek the Pareto fronts. By using Bootstrap method combined with similarity measurement, the stability of the obtained Pareto solution sets are evaluated to ensure the robustness and feasibility of optimization results. The results indicate that both algorithms can achieve effective optimization outcomes, but NSGA-II outperforms MOPSO in terms of optimization performance and solution set stability, exhibiting superior robustness and reliability, thereby providing a theoretical reference for decision-making in practical maintenance systems.