Abstract: To answer the small failure probabilities with high-dimensional correlated variables in the practical engineering, the subset simulation(SS) is combined together with importance sampling(IS) method. The samples from the probability density functions(PDF) of the importance sampling are used to construct the intermediate failure events, by which the small failure probabilities are turned into a Markov chain, which is a product of a series large failure probabilities or conditional failure probabilities(CFP) which are easily answered, on which the structural reliability sensitivity(RS) can be efficiently simulated by directly obtaining the correlated samples. Multi-objectives optimization models are established on minimizing the RS of failure probability with respect to the variable mean, variance(including the correlated coefficient between them) respectively and volume, and the collaborative optimization idea for multi-objectives is put forward, in the meantime, in view of the problem that it is difficult to converge for multi-objectives to be collaboratively optimized because of the errors when the RS is used as an objective function, the idea and method that utilize the errors are proposed. To accelerate the convergence of genetic algorithm(GA) and particle swarm optimization(PSO), the elite strategy that have elitist cloned and to take part in evolution simultaneously and the idea of similar mating are put forward. And the individuals from the modified GA are hybridized with those individuals from PSO to further improve their convergence. Finally, the 3 planet carriers of three-stage planetary reducers in shield machine are as illustrative examples to answer the mathematical models according to the algorithm above, the results show that ① the SS of the IS with correlated variables can highly simulate failure probability and its sensitivity. ② the convergent velocity of the collaborative algorithm of hybrid GA-PSO is superior to that of the GA and PSO, it can reduce the total volume of the planet carriers by 7.06% when the correlated coefficient is equal to 0.7, ③ it is confirmed that the proposed idea and method that utilize the errors are feasible and correct when the RS acts as objective function.

Key words: Collaborative optimization, Correlated variables, Hybrid genetic particle swarm optimization, Hybrid Markov chain, Importance sampling, Reliability sensitivity, Subset simulation