Besides inherent failures under normal conditions studied by classical reliability engineering, systems may also be subject to abnormal external failures, whose probabilities are hard, if not impossible, to predict, such as failures caused by natural disasters (e.g. earthquakes, floods, and tornados) and malicious attacks. Thus, abnormal external failures need to be considered when designing a system, as in reliability redundancy allocation problem. To deal with the uncertainty of abnormal external failures and based on robust optimization, this research proposes a multi-objective optimization approach that can simultaneously maximize system reliability under both normal and the worst cases of abnormal external failures for redundancy allocation problem. Since the amount of abnormal external failures is usually unknown, the worst case for each possible amount of abnormal external failures needs to be identified first. For this purpose, a new importance measure that can quantify the importance of a subset of system components with arbitrary cardinality is proposed. We also revise a widely used multi-objective evolutionary algorithm, multi-objective probabilistic solution discovery algorithm, to deal with the complexity of redundancy allocation problem. The merits of the revised algorithm are demonstrated by extensive experiments.
