Context. The metallicity distribution function (MDF) of the Galactic bulge is characterized by a multi-peak shape, with a metal-poor peak centered at [Fe/H] similar to -0.3 dex and a metal-rich peak centered at [Fe/H] similar to +0.3 dex. The bimodality of the MDF is also reflected in the [alpha/Fe] versus [Fe/H] abundance ratios, suggesting the presence of different stellar populations in the bulge. Aims. In this work we aim to reproduce the observed MDF of the Galactic bulge by testing a scenario in which the metal-poor component of the bulge is formed by stars formed in situ, during a strong burst of star formation, while the metal-rich population is formed by stars created in situ during a second burst of star formation and/or stars accreted from the innermost part of the Galactic disk as an effect of a growing bar. Methods. We adopted a chemical evolution model that is able to follow the evolution of several chemical species with detailed nucleosynthesis prescriptions. In particular, because of the importance of the production of Fe in constraining the MDF, close attention is paid to the production of this element in both Type Ia supernovae and massive stars. In particular, we included yields from rotating massive stars with different rotational velocity prescriptions. Our model also takes the infall and outflow of gas into account, as well as the effect of stellar migration. Results are compared to similar to 13 000 stars from the SDSS/APOGEE survey that belong to the region located at a Galactocentric distance RGC <= 3.5 kpc. Results. We successfully reproduce the observed double-peak shape of the bulge MDF as well as the abundance trends of the alpha elements relative to Fe by assuming both (i) a multi-burst star formation history with a quenching of the first burst of similar to 102 Myr and (ii) migration of stars from the innermost part of the Milky Way disk, as an effect of a growing bar. According to our results, the fraction of the stellar mass of the bulge-bar that belongs to the inner disk is similar to 40%. In terms of the nucleosynthesis, we conclude that models that assume either no rotation for massive stars or a distribution of rotational velocities that favors slow rotation at high metallicities best reproduce the observed MDF as well as the [alpha/Fe] and the [Ce/Fe] versus [Fe/H] abundance patterns.