Despite the availability of highly qualified research personnel, up-to-date research facilities and experience in developing applied research and innovation, many worldwide research institutions face challenges when managing contracted Research and Development (R&D) projects. These difficulties are mainly due to expectations from Industry (private sector), particularly regarding the applied development procedures, managerial processes and timing. Such difficulties have motivated funding agents to create evaluation processes to check whether the operational procedures of funded research institutions are sufficient to provide timely answers to demand innovation from industry, and also to identify aspects that require quality improvement in research development. For this purpose, several multiple criteria decision-making approaches can be applied. In this context, the research institutions are considered as alternatives for funding and their processes for research development as decision criteria. Among the available multiple criteria approaches, sorting methods are one prominent tool to evaluate operational capacity. However, the first difficulty that one may face when applying multiple criteria sorting methods is the need to hierarchically structure multiple criteria in order to represent the intended decision process. Additional challenges include the elicitation of the preference information and the definition of criteria evaluation, since these are frequently affected by some imprecision. In most approaches, all these critical points are neglected, or, at best, only partially considered. In this paper, a new sorting method is proposed to deal with all of those critical points simultaneously. To consider multiple levels for the decision criteria, the FlowSort method is extended to account for hierarchical criteria. To deal with imprecise data, FlowSort is integrated with fuzzy approaches. To yield solutions that consider fluctuations from imprecise weights, the Stochastic Multicriteria Acceptability Analysis (SMAA) is used. Finally, the proposed method is applied to the evaluation of research institutions, classifying them according to their operational maturity for the development of applied research.