Risk priority number (RPN) is the most commonly used failure risk ranking method among all risk assessment methods. However, the traditional RPN method not only cannot handle incomplete information and hesitation information (such as hesitation information between the s5=Low and s6=Moderate) provided by experts, but it also does not consider the objective weights of risk assessment factors. These reasons will lead to biased conclusions, causing decision makers to make wrong judgments. To address the limitations of the traditional RPN technique, the aim of this paper is to propose the flexible RPN assessment method under an uncertain environment. The flexible RPN assessment method is an extension of the traditional RPN technique. The flexible RPN method integrates the traditional RPN method and interval-valued 2-tuple weighted average method, and, simultaneously, considers subjective weights and objective weights. In the numerical verification, this study has adopted the example of stages of treatment planning for proton beam radiation therapy to verify the correctness and validity of the proposed flexible RPN technique. The numerical results show that the proposed flexible RPN technique not only handles the hesitation and incomplete information provided by experts but also considers the subjective and objective weights of risk assessment factors, providing more reasonable ranking results in the risk analysis.