A hesitant fuzzy set, allowing the membership of an element to be a set of several possible values, is very useful to express people's hesitancy in daily life. In this paper, we define the distance and correlation measures for hesitant fuzzy information and then discuss their properties in detail. These measures are all defined under the assumption that the values in all hesitant fuzzy elements (the fundamental units of hesitant fuzzy sets) are arranged in an increasing order and two hesitant fuzzy elements have the same length when we compare them. We can find that the results, by using the developed distance measures, are the smallest ones among those when the values in two hesitant fuzzy elements are arranged in any permutations. In addition, the derived correlation coefficients are based on different linear relationships and may have different results.

We introduce the concepts of entropy and cross-entropy for hesitant fuzzy information, and discuss their desirable properties. Several measure formulas are further developed, and the relationships among the proposed entropy, cross-entropy, and similarity measures are analyzed, from which we can find that three measures are interchangeable under certain conditions. Then we develop two multiattribute decision-making methods in which the attribute values are given in the form of hesitant fuzzy sets reflecting humans' hesitant thinking comprehensively. In one method, the weight vector is determined by the hesitant fuzzy entropy measure, and the optimal alternative is obtained by comparing the hesitant fuzzy cross-entropies between the alternatives and the ideal solutions; in another method, the weight vector is derived from the maximizing deviation method and the optimal alternative is obtained by using the TOPSIS method. An actual example is provided to compare our methods with the existing ones.

Intuitionistic fuzzy set is a widely used tool to express the membership, nonmembership, and hesitancy information of an element to a set. To aggregate the intuitionistic fuzzy information, a lot of aggregation techniques have been developed, especially, the ones which reflect the correlations of the aggregated arguments are the hot research topics, among which Bonferroni mean (BM) is an important aggregation technique. However, the classical BM ignores some aggregation information and the weight vector of the aggregated arguments. In this paper, we introduce the generalized weighted BM and the generalized intuitionistic fuzzy weighted BM, both of which focus on the group opinion. Paying more attention to the individual opinions, we further define the generalized weighted Bonferroni geometric mean and the generalized intuitionistic fuzzy weighted Bonferroni geometric mean. Various families of the existing operators can be obtained when the parameters of the developed aggregation techniques are assigned different values. Finally, we propose an approach to multicriteria decision making on the basis of the proposed aggregation techniques and an example is also given to illustrate our results.

The Pythagorean fuzzy set as an extension of the intuitionistic fuzzy set characterized by membership and nonmembership degrees has been introduced recently. Accordingly, the square sum of the membership and nonmembership degrees is a maximum of one. The Pythagorean fuzzy set has been previously applied to multiattribute group decision-making. This study develops a few aggregation operators for fusing the Pythagorean fuzzy information, and a novel approach to decision-making is introduced based on the proposed operators. First, we extend the generalized Bonferroni mean to the Pythagorean fuzzy environment and introduce the generalized Pythagorean fuzzy Bonferroni mean and the generalized Pythagorean fuzzy Bonferroni geometric mean. Second, a new generalization of the Bonferroni mean, namely, the dual generalized Bonferroni mean, is proposed by considering the shortcomings of the generalized Bonferroni mean. Furthermore, we investigate the dual generalized Bonferroni mean in the Pythagorean fuzzy sets and introduce the dual generalized Pythagorean fuzzy Bonferroni mean and dual generalized Pythagorean fuzzy Bonferroni geometric mean. Third, a novel approach to multiattribute group decision-making based on proposed operators is proposed. Lastly, a numerical instance is provided to illustrate the validity of the new approach.