By detailed first-principles calculations we show that the Fermi energy and the Rashba splitting in disordered ternary surface alloys Bi_xPb_ySb_1−x−y/Ag(111) can be independently tuned by choosing the concentrations x and y of Bi and Pb, respectively. The findings are explained by three fundamental mechanisms, namely the relaxation of the adatoms, the strength of the atomic spin-orbit coupling, and band filling. By mapping the Rashba characteristics, i.e. the splitting k_R and the Rashba energy E_R, and the Fermi energy of the surface states in the complete range of concentrations, we find that these quantities depend monotonically on x and y, with a very few exceptions. Our results suggest that we should investigate experimentally effects which rely on the Rashba spin-orbit coupling depending on spin-orbit splitting and band filling.