**1. Introduction**

The choices we make today determine our future, therefore, to choose the best alternative subject to certain attributes is an important problem. Multi-attribute group decision making (MAGDM) has established its importance by providing the optimal solution considering different attributes in many real life problems. For this purpose, many sets and models have been designed to express and comprehend the opinions of DMs. The classical set theory is too restrictive to express one's opinion, as some real life scenarios are too complicated and the vague data are often involved, therefore the DMs are unable to form a definite opinion. Fuzzy set theory is proposed as a remedy for such kind of real

life problems. Fuzzy set approaches are suitable to use when the modelling of human knowledge is necessary and when human evaluations are required. However, the usual fuzzy set theory is limited to the modelling in which the diversity of variants occurs at the same time.

To overcome such situation, different extensions of fuzzy set have been proposed to better model the real world, such as intuitionistic fuzzy set [1], hesitant fuzzy set [2], hesitant probabilistic fuzzy set [3], hesitant probabilistic multiplicative set [4], and necessary and possible hesitant fuzzy set [5]. Zadeh [6] suggested the concept of a linguistic variable that is more natural for humans to express there will in situations where data are imprecise. Thus far, linguistic environment has been extensively used to cope with the problems of decision making within [7]. Mergió et al. [8] used the Dempster–Shafer theory of evidence to construct an improved linguistic representation model for the sake of decision making process. Next, they introduced several linguistic aggregation operators. Zhu et al. [9] proposed a two-dimensional linguistic lattice implication algebra to determine implicitly and further the compilation of two-dimensional linguistic information decision in MAGDM dilemmas. Meng and Tang [10] generalized the 2-tuple linguistic aggregation operators and then used them in MAGDM dilemmas. Li and Dong [11] gave an introduction to the proportional 2-tuple linguistic form to make easy the solving of MAGDM dilemmas. Xu [12] introduced a dynamic linguistic weighted geometric operator to cumulate the linguistic information and then solved the problem of MAGDM when the judgment in different periods to change the linguistic information. Li [13] applied the concept of extended linguistic variables to construct an advanced way to cope with MAGDM dilemmas under linguistic environments. Agell et al. [14] used qualitative thinking approaches to perform and incorporate linguistic decision information and then applied it to MAGDM dilemmas.

Because of the uncertainty, vagueness and complexity of real world problems, it is troublesome for experts to gran<sup>t</sup> linguistic judgment using a single linguistic term. Torra [2] managed the situation where several membership values of a fuzzy set are possible by defining hesitant fuzzy set (HFS). Experts may hesitate among several possible linguistic terms. For this purpose, Rodriguez et al. [15] introduced the concept of hesitant fuzzy linguistic term sets (HFLTS) to improve the flexibility of linguistic information within hesitant situation. Zhu and Li [16] designed hesitant fuzzy linguistic aggregation operators based on the Hamacher t-norm and t-conorm. Cui and Ye [17] proposed multiple-attribute decision-making method using similarity measures of hesitant linguistic neutrosophic numbers regarding least common multiple cardinality. Liu et al. [18] defined new kind of similarity and distance measures based on a linguistic scale function. However, in some cases, the probabilities of these possible terms are not equal. Given this reality, Peng et al. [19] proposed the more generalized concept, called probabilistic linguistic term sets (PLTSs). PLTSs allow DMs to state more than one linguistic term, as an assessment for linguistic variable. This increases the flexibility and the fruitfulness of the expression of linguistic information and it is more reasonable for DMs to state their preference in terms of PLTSs because the PLTSs can reflect different probabilities for each possible assessment of a given object. Therefore, the research on the PLTSs is necessary. Thus, they used PLTSs in multi-attribute group decision making problem and construct an extended TOPSIS method as well as an aggregation-based method for MAGDM. Recently, in 2017, Lin et al. [20] extended the PLTSs to probabilistic uncertain linguistic term set, which is designed as some possible uncertain linguistic terms coupled with the corresponding probabilities, and developed an extended approach for preference to rank the alternatives.

Atanassov [1,21] presented the concept of the intuitionistic fuzzy set (IFS) which has three main parts, membership function, non-membership function and hesitancy function, and is better suited to handling uncertainty than the usual fuzzy set. Many researchers have been applying IFS for multi-attribute decision making under various different fuzzy environments. Up to now, the intuitionistic fuzzy set has been applied extensively to decision making problems [22–27]. Beg and Rashid [28] generalized the concept of HFLTS by hesitant intuitionistic fuzzy linguistic term set (HIFLTS) which is characterized by a membership and non-membership function that is more applicable for dealing with uncertainty than the HFLTS. HIFLTS collects possible membership and

non-membership linguistic values provided by the DMs. This approach is useful to model more complex real life scenarios.

In this article, we introduce the concept of PHILTS. The main idea is to facilitate DMs to provide their opinions about membership and non-membership linguistic terms more freely to cope with the vagueness and uncertainties of real life. To make meaningful decision making, the basic framework of PHILTS is developed. In this regard, normalization process for the purpose to equalize the length of PHILTSs, basic operations and their governing laws are presented. Furthermore, to deal with different scenarios, range of aggregation operators, i.e., probabilistic hesitant intuitionistic linguistic averaging operator, probabilistic hesitant intuitionistic linguistic weighted averaging operator, probabilistic hesitant intuitionistic linguistic geometric operator and probabilistic hesitant intuitionistic linguistic weighted geometric operator are proposed. The DM can choose the aggregation operator according to his preference. Lastly, for practical use of PHILTS in decision making, an extended TOPSIS method is derived, in which the DMs provide their opinions in PHILTSs which are further aggregated and processed according to the proposed mechanism of extended TOPSIS to find the best alternative.

This paper is organized as follows. In Section 2, we review some basic knowledge needed to understand our proposal. In Section 3, the concept of PHILTSs is firstly proposed and then some concepts concerning PHILTS, i.e., normalization process, deviation degree, score function, operations and comparison between probabilistic hesitant intuitionistic linguistic term elements (PHILTEs), are also discussed. In Section 4, aggregation operators, deviation degree between two PHILTEs and weight vector are derived. In Section 5, we propose an extended TOPSIS method and aggregation based method designed for MAGDM with probabilistic hesitant intuitionistic linguistic information. An example is provided in Section 6 to illustrate the usefulness and practicality of our methodology by ranking of alternatives. Section 7 is dedicated to highlighting the advantages of the proposed set and comparing proposed models with existing theory. Finally, some concluding remarks are given in Section 8.
