**1. Introduction**

To handle the uncertainty in real life problems effectively, Zadeh proposed the concept "fuzzy set" [1]. Thereafter, some extensions of fuzzy sets were proposed, for example, interval-valued fuzzy sets proposed by Zadeh [2–4], intuitionistic fuzzy sets proposed by Atanassov [5], and interval-valued intuitionistic fuzzy sets proposed by Atanassov and Gargov [6]. Most recently, Torra and Narukawa introduced hesitant fuzzy sets (HFSs) to deal with hesitant situations, which were not well managed by the previous tools [7,8]. In HFSs, the membership is a union of several memberships of fuzzy sets. Practices show that HFS is a useful mathematical tool for dealing with this kind of uncertainty. Nowadays, lots of branches of HFSs have been studied, such as intuitionistic hesitant fuzzy sets (see reference [9]), dual hesitant fuzzy sets (see reference [10]), etc.

Distance and similarity measures are two important research objects in fuzzy set theory and they have attracted the attention of many scholars. Zwick, Carlstein and Budescu [11], Pappis and Karacapilidis [12] proposed a comparative analysis on similarity measures on fuzzy sets, respectively. Wang introduced two influential similarity measures on fuzzy sets [13]. As for HFSs, Xu and Xia proposed a series of classical distance measures on HFSs [14,15]. Thereafter, Peng et al. proposed a novel hesitant fuzzy weighted distance measure [16]. Then, Rodríguez et al. gave a clear perspective

of HFSs [17]. Li et al. pointed out that the existing distance and similarity measures fail to consider the cardinal numbers of HFEs [18]. Thereafter, Li et al. proposed a concept of hesitance degree of HFEs and HFSs to introduce a decision maker's hesitance situation. They also proposed a series of distance and similarity measures on HFSs, which take both the values and the cardinal numbers of HFEs into consideration. In addition, Tang et al. introduced some continuous hesitant fuzzy distance measures which also consider the element number of HFEs [19]. It is noteworthy that the distance measures on HFSs are important in decision-making. As for this application, Alcantud et al. summarized the latest related studies in their work [20].

The distance measures proposed by Li et al. are innovative [18]. In particular, they introduced the concept of hesitance degree on HFSs. This is a new beginning, where the proposed distance and similarity measures should be explored further with consideration of hesitant degree. The aims of this study are to proceed towards the direction where the distance and similarity measures should develop according to reference [18]. Specifically, this study proposes a series of novel distance measures on HFSs. The main characteristic of the proposed distance measures is that they contain three parameters, i.e., credibility factor, conservative factor, and a risk factor. These newly proposed distance measures handle the relationship between the cardinal number and the element values of hesitant fuzzy set well. When using these functions, decision makers with different risk preferences are allowed to give different values for the three parameters.

The remaining part of this study is arranged as follows: Section 1 reviewed some basic notions on HFSs and introduced some classical distance measures. Section 2 proposes a series of novel distance measures on HFSs. Section 3 provides two examples to show the validity of the novel distance measures. Finally, innovations of this study are concluded in Section 4.
