This study compares the performances of two sampling-based strategies for the simultaneous estimation of the first- and total-order variance-based sensitivity indices (a.k.a. Sobol’ indices). The first strategy corresponds to the current approach employed by practitioners and recommended in the literature. The second one was only recently introduced by the first and last authors of the present article. Both strategies rely on different estimators of first- and total-order Sobol’ indices. The asymptotic normal variances of the two sets of estimators are established and their accuracies are compared theoretically and numerically. The results show that the new strategy outperforms the current one. The global sensitivity analysis of the radiative forcing model of sulfur aerosols is performed with the new strategy. The results confirm that in this model interactions are important and only one input variable is irrelevant.