Introducing robustness in multi-objective optimization

In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EMO) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.