We suggest a systematic method for obtaining the optimal compression ratio in the multi-stage closed-loop compression process of carbon dioxide. Instead of adopting the compression ratio of 3 to 4 by convention, we propose a novel approach based on mathematical analysis and simulation. The mathematical analysis prescribes that the geometric mean is a better initial value than the existing empirical value in identifying the optimal compression ratio. In addition, the optimization problem considers the initial installation cost as well as the energy required for the operation. We find that it is best to use the fifth stage in the general closed-loop type carbon dioxide multi-stage compression process.