Two key steps in multi-criteria decision making (MCDM) are to quantify the considered criteria and to fuse the quantified criterion information to sort all alternatives. One of the most recent and important tools for the first step is linguistic interval-valued intuitionistic fuzzy number (LIVIFN) and one of the most common and effective ways for the second step is aggregation operator (AO). So far, a number of AOs of LIVIFNs have been presented. Each AO can work well under certain conditions. But there is not yet an AO of LIVIFNs that can deal with the situation where the weights of the considered criteria are unknown and the criteria are in different priority levels and concurrently provide satisfying generality and flexibility in the aggregation of criterion information. To this end, a linguistic interval-valued intuitionistic fuzzy Archimedean prioritised and (LIVIFAPA) operator and a linguistic interval-valued intuitionistic fuzzy Archimedean prioritised or (LIVIFAPO) operator, which have such capabilities, are presented in this paper. The formal definitions and generalised expressions of the two AOs are firstly provided. Then their properties are explored and proved and specific expressions are established. After that, a new method for solving the LIVIFNs based MCDM problems is proposed on the basis of the presented AOs. Finally, the proposed method is illustrated via an example about additive manufacturing machine selection and is evaluated via a comparison with existing methods. The major contribution of the paper is the development of the LIVIFAPA and LIVIFAPO operators for MCDM, which can make up for the above shortcoming of the existing AOs of LIVIFNs.