[abridged] We present a statistical exploration of the parameter space of the De Lucia and Blaizot version of the Munich semi-analytic model built upon the millennium dark matter simulation. This is achieved by applying a Monte Carlo Markov Chain method to constrain the 6 free parameters that define the stellar and black-hole mass functions at redshift zero. The model is tested against three different observational data sets, including the galaxy K-band luminosity function, B-V colours, and the black hole-bulge mass relation, separately and combined, to obtain mean values, confidence limits and likelihood contours for the best fit model. Using each observational data set independently, we discuss how the SA model parameters affect each galaxy property and to what extent the correlations between them can lead to improved understandings of the physics of galaxy formation. When all the observations are combined, we find reasonable agreement between the majority of the previously published parameter values and our confidence limits. However, the need to suppress dwarf galaxy formation requires the strength of the supernova feedback to be significantly higher in our best-fit solution than in previous work. To balance this, we require the feedback to become ineffective in halos of lower circular velocity than before, so as to permit the formation of sufficient high-luminosity galaxies: unfortunately, this leads to an excess of galaxies around L*. Although the best-fit is formally consistent with the data, there is no region of parameter space that reproduces the shape of galaxy luminosity function across the whole magnitude-range. We discuss modifications to the semi-analytic model that might simultaneously improve the fit to the observed luminosity function and reduce the reliance on excessive supernova feedback in small halos.
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