An aggregation operator of linguistic interval-valued intuitionistic fuzzy numbers (LIVIFNs) is an important tool for solving cognitively inspired decision-making problems with LIVIFNs. So far, many aggregation operators of LIVIFNs have been presented. Each of these operators works well in its specific context. But they are not always monotone because their operational rules are not always invariant and persistent. Dempster-Shafer evidence theory, a general framework for modelling epistemic uncertainty, was found to provide the capability for operational rules of fuzzy numbers to overcome these limitations. In this paper, a weighted averaging operator of LIVIFNs based on Dempster-Shafer evidence theory for cognitively inspired decision-making is proposed. Firstly, Dempster-Shafer evidence theory is introduced into linguistic interval-valued intuitionistic fuzzy environment and a definition of LIVIFNs under this theory is given. Based on this, four novel operational rules of LIVIFNs are developed and proved to be always invariant and persistent. Using the developed operational rules, a new weighted averaging operator of LIVIFNs is constructed and proved to be always monotone. Based on the constructed operator, a method for solving cognitively inspired decision-making problems with LIVIFNs is presented. The application of the presented method is illustrated via a numerical example. The effectiveness and advantage of the method are demonstrated via quantitative comparisons with several existing methods. For the numerical example, the best alternative determined by the presented method is exactly the same as that determined by other comparison methods. For some specific problems, only the presented method can generate intuitive ranking results. The demonstration results suggest that the presented method is effective in solving cognitively inspired decision-making problems with LIVIFNs. Furthermore, the method will not produce counterintuitive ranking results since its operational rules are always invariant and persistent and its aggregation operator is always monotone.