## 1 Introduction  
Fuzzy set theory was initially introduced by [PERSON] [1] in 1965 as an extension of the classical set theory. In classical set theory, an element either belongs to or does not belong to a set. In fuzzy set theory, a gradual assessment of elements of a set is described with a membership function that is in \([0,1]\), and it is successfully used in decision-making problems [2]. After this popular extension of set theory, several extensions and generalizations of fuzzy sets have been introduced. These extensions have been successfully used in several practical applications of both real life and scientific problems. Various applications of these extensions can be found in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics [3; 4; 5; 6]. [PERSON] felt that there is still a need to introduce one more extension of fuzzy set theory--the hesitant fuzzy set (HFS) theory [7]. This extension permits several possible membership degrees of an element in unit closed interval \([0,1]\). HFS provides a much better description than the other extensions of fuzzy sets when the difficulty of establishing the membership degree is tremendous and a specific set of possible values, instead of a single value, is chosen as possible membership values. Many studies on HFS have been conducted, see [8; 9; 10; 11; 12; 13; 14]. In the quest to model the real world more accurately, it was noted that the probabilities of occurrence for different membership values in HFS must be different. HFS does not take into account the occurrence probabilities, which can play a crucial role in decision-making. To overcome this flaw, [PERSON] and [PERSON] proposed the concept of a hesitant probabilistic fuzzy set [15], an extension of HFS. Furthermore, they investigated several aggregation operators with properties for hesitant probabilistic fuzzy set. Additionally, a novel algorithm was developed to handle the multi-criteria group decision making problems for the hesitant probabilistic fuzzy set [16]. This concept was further extended to the continuous form of hesitant probabilistic fuzzy set [17]. While [PERSON] et al. introduced the idea of a probabilistic linguistic term set [18], allowing a decision-maker (DM) to provide not only his opinion in linguistic terms but also the occurrence probabilities, in an attempt to comprehend the complexities of reality, [PERSON] et al. [16] proposed the notion of a probabilistic linguistic vector-term set with a novel algorithm for group decision-making with multi-granular linguistic information. This research work confirms the vital role of occurrence probabilities. Apart from interval \([0,1]\), the [PERSON]'s scale 1-9 [19; 20] is quite important and useful for expressing the opinion of the decision-maker. [PERSON] and [PERSON] [21] defined the hesitant multiplicative set (HMS) in order to express membership degrees in [PERSON]'s scale. However, the HMS has the same deficiency as HFS, as the occurrence probabilities are not studied. Therefore, it is an important research problem to study HMS in a hesitant probabilistic environment. In this paper, we extend HMS as a hesitant probabilistic multiplicative set (HPMS), by assigning the probabilities to membership degrees expressed in [PERSON]'s scale. For practical group decision-making with the proposed HPMS, our choice is the preference relations. The preference relations are quite elegant in making practical decision-making flawless.