**1. Introduction**

With growing advancements in economic, socio-cultural as well as technical aspects of the world, uncertainties have started playing a dominant part in decision-making (DM) processes. The nature of DM problems is becoming more and more complex and the data available for the evaluation of these problems is increasingly having uncertain pieces of unprocessed information [1,2]. Such data content leads to inaccurate results and increase the risks by many folds. To decrease the risks and to reach the accurate results, decision-making has attained the attention of a large number of researchers. In the complex decision-making systems, often large cost and computational efforts are required to address the information, to evaluate it to form accurate results. In such situations, the major aim of the decision makers remain to decrease the computational overheads and to reach the desired objective in less space of time.

Time-to-time such DM techniques are framed which captures the uncertain information in an efficient way and results are calculated in such a manner that they comply easily to the real-life situations. From the crisp set theory, an analysis was shifted towards the fuzzy sets (FSs) and further Atanassov [3] extended the FS theory given by Zadeh [4] to Intuitionistic FSs (IFSs) by acknowledging the measures of disagreeness along with measures of agreeness. Afterward, Atanassov and Gargov [5] extended the IFS to the Interval-valued intuitionistic fuzzy sets (IVIFSs) which contain the degrees of agreeness and disagreeness as interval values instead of single digits. As it is quite a common phenomenon that different attributes play a vital part during the selection of best alternative among the available ones, so suitable aggregation operators to evaluate the data are to be chosen carefully by the experts to address the nature of the DM problem. In these approaches, preferences are given as falsity and truth membership values in the crisp or interval number respectively such that the corresponding degrees altogether sum to be less than or equal to one. In above-stated environments, various researchers have constructed their methodologies for solving the DM problems focussing on information measures, aggregation operators etc. For instance, Xu [6] presented some weighted averaging aggregation operators (AOs) for intuitionistic fuzzy numbers (IFNs). Wang et al. [7] presented some AOs to aggregate various interval-valued intuitionistic fuzzy (IVIF) numbers (IVIFNs). Garg [8,9] presented some improved interactive AOs for IFNs. Wang and Liu [10] gave interval-valued intuitionistic fuzzy hybrid weighted AOs based on Einstein operations. Wang and Liu [11] presented some hybrid weighted AOs using Einstein norm operations. Garg [12] presented a generalized AOs using Einstein norm operations for Pythagorean fuzzy sets. Garg and Kumar [13] presented some new similarity measures for IVIFNs based on the connection number of the set pair analysis theory. However, apart from these, a comprehensive overview of the different approaches under the IFSs and/or IVIFSs to solve MCDM problems are summarized in [14–24]. In the above theories, it is difficult to capture cases where the preferences related to different objects are given in the form of the multiple numbers of possible membership entities. To handle it, Torra [25] came up with the idea of hesitant fuzzy sets (HFSs). Zhu et al. [26] enhanced it to the dual hesitant fuzzy sets (DHFSs) by assigning equal importance to the possible non-membership values as that of possible membership values in the HFSs. In the field of AOs, Xia and Xu [27] established different operators to aggregated their values. Garg and Arora [28] presented some AOs under the dual hesitant fuzzy soft set environment and applied them to solve the MCDM problems. Wei and Zhao [29] presented some induced hesitant AOs for IVIFNs. Apart from these, some other kinds of the algorithms for solving the decision-making problems are investigated by the authors [30–38] under the hesitant fuzzy environments.

Although, these approaches are able to capture the uncertainties in an efficient way, ye<sup>t</sup> these works are unable to model the situations in which the refusal of an expert in providing the decision plays a dominant role. For example, suppose a panel of 6 experts is approached to select the best candidate during the recruitment process and 2 of them refused to provide any decision. While evaluating the informational data using the existing approaches, the number of decision makers is considered to be 4 instead of 6 i.e., the refusal providing experts are completely ignored and the decision is framed using the preferences given by the 4 decision-providing experts only. This cause a significant loss of information and may lead to inadequate results. In order to address such refusal-oriented cases, Zhu and Xu [39] corroborated probabilistic hesitant fuzzy sets (PHFSs). Wu et al. [40] gave the notion of AOs on interval-valued PHFSs (IVPHFSs) whereas Zhang et al. [41] worked on preference relations based on IVPHFSs and accessed the findings by applying to real life decision scenarios. Hao et al. [42] corroborated the concept of PDHFSs. Later on, Li et al. [43] presented the concept of dominance degrees and presents a DM approach based on the best-worst method under the PHFFSs. Li and Wang [44] comprehensively expressed way to address their vague and uncertain information. Lin and Xu [45] determined various probabilistic linguistic distance measures. Apart from them, several researchers [46–52] have shown a keen interest in applying probabilistic hesitant fuzzy set environments to different decision making approaches. Based on these existing studies, the primary motivation of this paper is summarized as below:

(i) In the existing DHFSs, each and every membership value has equal probability. For instance, suppose a person has to buy a commodity *X*, and he is confused that either he is 10% sure or 20% sure to buy it, and is uncertain about 30% or 40% in not buying it. Thus, under DHFS environment, this information is captured as ({0.10, 0.20}, {0.30, 0.40}). Here, in dual hesitant fuzzy set, each hesitant value is assumed to have probability 0.5. So, mentioning the same probability value repeatedly is omitted in DHFSs. But, if the buyer is more confident about 10% agreeness than that of 20% i.e., suppose he is certain that his agreeness towards buying the

commodity is 70% towards 10% and 30% towards 20% and similarly, for the non-membership case, he is 60% favoring to the 40% rejection level and 40% favoring the 30% rejection level. Thus, probabilistic dual hesitant fuzzy set is formulated as - {0.10 0.70, 0.20 0.30}, {0.30 0.4, 0.40 0.6} . So, to address such cases, in which even the hesitation has a some preference over the another hesitant value, DHFS acts as an efficient tool to model them.


Motivated by the aforementioned points regarding shortcomings in the existing approaches, this paper focusses on eradicating them by developing a series of AOs. In order to do so, the supreme objectives are listed below:


To achieve the first objective and to provide more degrees of freedom to practitioners, in this article, we consider PDHFS environment to extract data. For achieving the second objective, two distance measures are proposed; one in which the size of two PDHFSs should be the same whereas in the second one the size may vary. For achieving the third objective, a non-linear model is solved to capture the weighted information. For achieving fourth objective average and geometric Einstein AOs are proposed. To attain the fifth and sixth objective a real-life based case-study is conducted and its comparative analysis with the prevailing environments is carried out.

The rest of this paper is organized as follows: Section 2 highlights the basic definitions related to DHFSs, PHFSs, and PDHFSs. Section 3 introduces the two distance measures for PDHFSs along with their desirable properties. Section 4 introduces some Einstein operational laws on PDHFSs with the investigation of some properties. In Section 5, some averaging and geometric weighted Einstein AOs are proposed. A non-linear programming model for weights determination is elicited in Section 6. In Section 7, an approach is constructed to address the DM problems and includes the real-life marketing problem including a comparative analysis with the existing ones. Finally, concluding remarks are given in Section 8.
