Failure Mode and Effect Analysis (FMEA) is a useful risk assessment tool used to identify, evaluate, and eliminate potential failure modes in numerous fields to improve security and reliability. Risk evaluation is a crucial step in FMEA and the Risk Priority Number (RPN) is a classical method for risk evaluation. However, the traditional RPN method has deficiencies in evaluation information, risk factor weights, robustness of results, etc. To overcome these shortcomings, this paper aims to develop a new risk evaluation in FMEA method. First, this paper converts linguistic evaluation information into corresponding interval-valued intuitionistic fuzzy numbers (IVIFNs) to effectively address the uncertainty and vagueness of the information. Next, different priorities are assigned to experts using the interval-valued intuitionistic fuzzy priority weight average (IVIFPWA) operator to solve the problem of expert weight. Then, the weights of risk factors are subjectively and objectively determined using the expert evaluation method and the deviation maximization model method. Finally, the paper innovatively introduces the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator, Tchebycheff Metric distance, and the interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator into the ratio system, the reference point method, and the full multiplication form of MULTIMOORA sub-methods to optimize the information aggregation process of FMEA. The extended IVIF-MULTIMOORA method is proposed to obtain the risk ranking order of failure modes, which will help in obtaining more reasonable and practical results and in improving the robustness of results. The case of the Middle Route of the South-to-North Water Diversion Project&rsquo / s operation risk is used to demonstrate the application and effectiveness of the proposed FMEA framework.