Linguistic intuitionistic fuzzy number (LIFN) is a special intuitionistic fuzzy number where the membership and non-membership are expressed by linguistic terms and can more easily describe the vagueness and uncertainty in the real world. The Heronian mean (HM) provides an aggregation operator which can consider the interrelationship among the aggregated arguments. Nevertheless, the traditional HM can only aggregate crisp numbers rather than any other types of arguments. In this paper, we firstly define some new operational rules of the LIFNs based on Einstein operations, then the HM operator is extended to the LIFNs and some linguistic intuitionistic fuzzy Heronian mean operators based on Einstein operations are proposed, such as linguistic intuitionistic fuzzy Einstein Heronian mean (LIFEHM) operator, weighted linguistic intuitionistic fuzzy Einstein Heronian mean (WLIFEHM) operator. Further, some desired properties of these operators and some special cases with respect to the different parameter values in these operators are discussed. Finally, a decision-making approach is developed for multiple-attribute group decision-making (MAGDM) problems with linguistic intuitionistic fuzzy information and an example is given to demonstrate the effectiveness and superiority of the proposed method.
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