At a time when interest in improving the resilience of the distribution system is increasing, this paper proposes an optimal method for reinforcing the distribution system with regards to resilience. In this study, resilience is derived from the empirical distribution function, which is used as a constraint in the distribution facility reinforcement plan. The resilience calculation process utilizes Monte Carlo simulation and considers various uncertainties such as the number of facility failures, location, and recovery time for post-contingency. The case study identifies the minimum investment cost and optimal location for the facility reinforcement considering resilience for the modified 13 node test feeder. The results of this study demonstrate that the proposed method is more effective than the existing facility reinforcement plans based on several fault scenarios, because the calculated results for multiple unpredictable resilience indices are used as constraints for the reinforcement plan.