Hybrid maintenance policy optimization of <i>k</i> -out-of-2 <i>n</i> : <i>F</i> balanced systems with common-mode degradation

To address the operation and maintenance challenges of rail grids in small-radius curved sections, this study proposes a k -out-of-2 n : F balanced system considering common-mode degradation. The system’s components are symmetrically arranged on both sides of a line, with two components in symmetric slots forming a single group. A hierarchical copula function is employed to quantitatively characterize the dependence structure among the degradation increments of these components. The system is defined to fail if at least k out of the 2 n components fail; additionally, a group incurs penalty costs when the state difference between its two components exceeds a preset threshold. Given the degradation heterogeneity across components, a hybrid maintenance strategy is put forward, integrating state difference elimination and component reallocation. To obtain the optimal hybrid maintenance policy minimizing total costs over a finite planning horizon, a multi-stage Markov decision process model is constructed. Numerical examples and comparative analyses are provided to verify the model’s feasibility and demonstrate its superiority.