Based on the spin-pumping theory and first-principles calculations, the spin-mixing conductance (SMC) is theoretically studied for Pt/Permalloy (Ni$_{81}$Fe$_{19}$, Py) junctions. We evaluate the SMC for ideally clean Pt/Py junctions and examine the effects of interface randomness. We find that the SMC is generally enhanced in the presence of interface roughness as compared to the ideally clean junctions. Our estimated SMC is in good quantitative agreement with the recent experiment for Pt/Py junctions. We propose possible routes to increase the SMC in Pt/Py junctions by depositing a foreign magnetic metal layer in Pt, offering guidelines for designing the future spintronic devices.