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In this paper, several cross-entropy measures for linguistic hesitant intuitionistic fuzzy information have been developed which integrating cross-entropy measures of intuitionistic fuzzy sets and hesitant fuzzy linguistic sets. Some desirable properties of new cross-entropy measures have been studied. Two new multiple attribute decision making methods have been presented based on the new cross-entropy measures in which attribute values are given in the form of linguistic hesitant intuitionistic fuzzy values to reflect human hesitantation and fuzzy thinking comprehensively. We consider different attribute weight situations including attribute weights are completely known, partly known and completely unknown. An optimization model is established to determine attribute weights if they are partly known and a formula is given if attribute weights are completely unknown. Finally, a numerical example is presented to illustrate practical advantages and effectiveness of the proposed approaches.