Nuclear energy systems of the future must rise to the twin challenges of sustainable energy production and nuclear waste reduction. Sodium-cooled Fast Reactors (SFR) appear to be the technology that is the closest to industrial deployment in the coming decades. However, standard SFR's face the problem of positive void coefficients and significant production of Minor Actinides (MA) if transmutation is not performed. Different methods of MA transmutation using either uranium or thorium fuel cycles were performed for a SFR. These neutronics simulations used MURE (MCNP Utility for Reactors Evolution), a C++ object-oriented code that performs nuclear reactor time-evolution using successive calls to the widely-used particle transport code MCNP. For each strategy, the Sodium void coefficients and the radiotoxicities of the waste were determined. A scenario taking into account the radiotoxicity of waste produced each year and the radiotoxicity of the core (which becomes a waste when the SFR is shut down) is considered. A method is proposed to assess the advantages of the homogeneous transmutation and to compare different fuel cycle strategies for SFR, in terms of total waste production.