Bi-Objective RRAP Optimization With Mixed Redundancy: An Importance Measure-Based Two-Stage Algorithm Framework

Mixed redundancy, which combines active and cold-standby redundancy, can improve reliability design flexibility but substantially increases computational complexity. Consequently, it is rarely used in multi-objective reliability-redundancy allocation problems (MRRAPs), as balancing conflicting objectives proves challenging. To address it, this paper formulates a bi-objective RRAP (BRRAP) with mixed redundancy, aiming to maximize system reliability while minimizing cost. The reliability of cold-standby and mixed redundant subsystems is precisely evaluated using continuous time Markov chain models. Although swarm intelligence algorithms are widely used for MRRAPs because of their global search capability and implementation simplicity, their stochastic updating mechanism often leads to weak local exploitation and premature convergence. To overcome this limitation, an importance measure (IM)-based two-stage BRRAP optimization framework is developed, which iteratively combines swarm intelligence-based global search with IM-guided local refinement. By adjusting Pareto solutions from both reliability and redundancy perspectives, the IM-based local optimization effectively pushes the Pareto front toward higher reliability and lower cost. Experiments based on four benchmarks demonstrate that the proposed framework improves solution quality, convergence, and diversity of Pareto fronts.