Direct method for uncertain multi-objective optimization based on interval non-dominated sorting

The interval optimization methods which describe the uncertainty by intervals are widely used in uncertain multi-objective optimization. Commonly, the interval values of the objective functions are substituted by approximate values, which will cause the loss of accuracy. To deal with the problem, a direct method is presented in this paper. In the framework of the multi-objective genetic algorithm, the upper and lower boundaries of the objective and constraint functions are found using the Monte Carlo stochastic simulation method and the constraints are handled by the degree of interval constraint violation; and then every two individuals from the combination population of the evolutionary population and the external elite population are compared using the interval comparison method. Afterwards, an interval non-dominated sorting based on the interval dominance relations is performed to sort the combination population into different non-dominated level. Furthermore, a general crowding distance is calculated for each individual to make individuals from the same non-dominated level comparable and also to maintain the diversity. The efficiency and accuracy of the proposed method is demonstrated by three test functions, the multi-objective structural optimization of the ten-bar truss, and the multi-objective structural optimization of a vehicle body for crashworthiness and lightweight.