Multi-objective Bayesian global optimization is an optimization framework that primarily relies on the Kriging surrogate model to evolve a design toward the discovery of global Pareto front. In this paper, our objective is to gain a computational insight regarding the impact and the efficiency of various multi-objective Bayesian optimization strategies for solving aerodynamic optimization problems. Three methods were investigated, that is, Euclidean-based expected improvement (EEI), expected hypervolume improvement (EHVI), and ParEGO. For this purpose, computational tests on three inviscid airfoil optimization problems and preliminary test on turbomachinery case were performed. Inviscid solver was used for airfoil problem due to its cheap computational cost, which allowed us to perform statistical analysis of the results. From the results, we observe that the EHVI method is able to provide higher quality solutions that are close to and well distributed on the Pareto front. The latter is especially one powerful aspect of the EHVI method as compared to the other two algorithms, in which EEI and ParEGO found it difficult to find a diverse set of solutions. In the light of these results, we suggest that EHVI is a potential method to be explored further for solving expensive aerodynamic design optimization problems.