An adaptive consensus decision model based on individual cumulative consensus contribution is proposed for multi-attribute group decision making problems under hesitant fuzzy linguistic information. Firstly, based on the classical theory of information entropy and relative entropy, the hesitant fuzzy linguistic score function is used to construct an optimization model that determines attribute weights, comprehensively considering the information difference among different alternatives under the same attributes as well as the information difference between each alternative and the positive ideal alternative or the negative ideal alternative. Then, the measure of individual cumulative consensus contribution and the global consensus measure are developed. A new adaptive consensus process for hesitant fuzzy linguistic information is constructed, where the consensus level is controlled by using the global consensus degree and the expert weight is adaptively corrected according to the expert's cumulative consensus contribution. The characteristics of the process are as follows. 1) The penalties of expert weights are performed for the non-full consensus experts with less cooperation. 2) The updating of expert weights causes the adaptive updating of attribute weights, which in turn influences the accumulation of individual consensus contribution. Finally, an example of consensus decision-making for the location of emergency medical facilities is given to illustrate the feasibility and effectiveness of the proposed method.