Hesitant fuzzy linguistic preference relation (HFLPR) is a new preference structure that the decision makers (DMs) are hesitant about several possible linguistic terms of preference information for pairwise comparison between alternatives each other. The purpose of this paper is to investigate the additive consistency of hesitant fuzzy linguistic preference relation (HFLPR) with a new expansion principle for hesitant fuzzy linguistic term sets (HFLTSs). In order to normalize HFLTSs with different numbers of linguistic terms, a least common multiple extended (LCME) principle is proposed. According to the LCME principle, the additive consistent index of a HFLPR is defined to measure the consistency level of the HFLPR. For improving the unacceptable additive consistency of a HFLPR, a pure integer programming model is constructed to derive an acceptable additive consistent HFLPR. The similarities between DMs are calculated based on their individual HFLPR with acceptable additive consistency. Subsequently, the confidence degrees of DMs are defined to derive DMs' weights, and some examples are analyzed to verify the effectiveness of the proposed method.