Matrix-augmentation approach for machine repair problem with generally distributed repair times during working breakdown periods

In the manufacturing industry, production facilities for product manufacturing often contain unreliable operating machines and standbys. The machines can break down and result in productivity loss, and have to be repaired by the repairmen. To evaluate the system performance in operations and production management accurately, a general mathematical model is proposed to investigate the machine repair problem with an unreliable repairman, working breakdowns, and multiple vacations. A novel matrix-augmentation approach is introduced to simplify the analysis process and to derive the stationary distribution of the number of failed machines in the system when the classical supplementary variable technique cannot evolve the steady-state probability recursively. Furthermore, explicit formulas of various performance metrics are developed and numerically computed corresponding to various repair time distributions during working breakdown periods. Finally, an optimization problem with multiple objective functions is formulated with two different objective functions: the expected cost and machine availability. The NSGA-II algorithm is applied to perform numerical experiments and to provide Pareto-efficient solutions for managers and decision-makers.