**1. Introduction**

Multiple criteria decision making (MCDM) is a hot research topic in the modern decision-making process to find the most suitable alternative(s) from the available ones. In this process, all the alternatives are to be evaluated under several attributes by both qualitatively and quantitatively [1,2]. Traditionally, the researchers offer his/her preference information towards the alternatives by using the crisp real numbers only. However, due to lack of knowledge, a time pressure, and other unavoidable factors, it is very difficult if not impossible to express the information precisely. Therefore, to handle the incomplete or incorrect information, the theory of fuzzy set (FS) also called as a type-1 fuzzy set (T1FS) [3] and its extensions as an intuitionistic FS (IFS) [4], type-2 FS (T2FS) [5] are widely used. Under these environments, authors have put forth the different techniques to solve the MCDM problems. For instance, geometric aggregation operators (AOs) for different intuitionistic fuzzy numbers (IFNs) are developed by Xu and Yager [6]. Garg [7,8] presented some Einstein norm based AOs for IFNs. Zhao et al. [9] presented some generalized AOs. Kaur and Garg [10] presented some generalized AOs using t-norm operations for cubic IFS information. However, apart from these, a comprehensive overview of the different approaches for solving the decision making (DM) problems by using aggregation operator (AOs) [11–21], information measures (IMs) [22–24] are summarized in these papers and their references.

In these existing works, authors have investigated the problem by taking quantitative environment to access the alternatives. However, not all the alternatives are accessed in terms of quantitative.

For this, there exists the concept of qualitative assessment in terms of linguistic variables/terms (LVs/LTs) [25,26]. By taking the advantages of LTs, Zhang [27] presented the linguistic IF (LIF) AOs to aggregate the LIF numbers. Chen et al. [28] presented an approach to solving the MCDM problem under LIFS environment. Garg and Kumar [29] presented AOs for LIF numbers (LIFNs) by using set pair analysis theory. Garg and Kumar [30] presented new possibility degree measure for LIFNs and an AO to aggregate the different LIFNs to solve MCDM problems. In many practical problems, it is not easy for any decision maker (DM) to discover an exact membership function of an FS corresponding to its element. To overthrow this limitation, type-2 fuzzy set (T2FS), an extension of T1FS, is applied to the model and is characterized by two functions: primary membership functions (PMF) and secondary membership function (SMF). Unfortunately, T2FSs are highly complex, it is troublesome for the DMs to implement it in the real situation; hence, their use is not ye<sup>t</sup> widespread. To reduce the computational complexity, Interval type-2 fuzzy (IT2F) sets (IT2FSs) [31] is the most widely used in T2FSs. In past decades, many methods have been developed to extend the theory of MCDM under IT2FS environment. Chen et al. [32] built up an expanded QUALIFLEX strategy for taking care of DM issues in view of IT2FSs and gave a contextual analysis of medicinal basic leadership. Chen [33] built up an ELECTRE-base outranking strategy for decision-making problems using IT2FSs. Wu and Mendel [34] proposed a linguistic weighted average AOs to deal with analytical hierarchical process (AHP) process under IT2F environment. Qin and Liu [35] investigated a family of type-2 fuzzy AOs in light of Frank triangular norm and built up another way to deal with MCDM problems under the IT2FSs setting. Gong et al. [36] extended the generalized Bonferroni mean (GBM) operator to the trapezoidal IT2F environment. Apart from these, some other studies under T2FS environment are conducted which are summarized in [35–48].

In all these above AOs, researchers have described the information by considering the independent of argumen<sup>t</sup> assumptions during the aggregation. However, the interaction between the multi-input parameters have commonly occurred and thus, it is necessary to add their features into the process. In that direction, Bonferroni mean (BM) and generalized BM (GBM)-based operators are proposed by the researchers [49,50]. But from them, it has been observed that they have considered only two or three multi-parameter at a single time. However, they are unable to analyze the effect of the multi-input argumen<sup>t</sup> into one analysis. Furthermore, in BM and GBM, there is a need for two and three parameters from the irrational set during the process which increases the computational complexity. An alternative to BM operators, Hamy mean (HM) [51] or Maclaurin symmetric mean (MSM) or Muirhead mean (MM) operator has advantages of capturing the inter-relationship among the multiple input arguments. Qin [46] make a correlation between the HM and the MSM and conclude that the MSM is an instance of HM [16,17]. Garg and Nancy [52] develop MCDM method by prioritized MM aggregation operators. Additionally, the HM operator involves the parameter, which can provide more flexibility and robustness during the aggregation operator. The existing - arithmetic and geometric mean- operators can be easily deduced from the HM by setting a particular value to its parameter. Be that as it may, the HM just accomplished a couple of research results on the hypothesis and application of inequality [53,54]. Therefore, it is a means to study the AOs using the HM operator.

It is noted from the above studies that T2FS or IT2FS are examined by considering only the membership degree (MD) of an element. But in practical problems, it is sometimes not possible for a DM to give their preferences in terms of MD only as there may be some amount of hesitation also. For discussing this, a type-2 IFS (T2IFS) [39] has been introduced which simultaneously considers the MDs, non-membership degrees (NMDs) and the footprint of uncertainties (FOU) between them. Later on, due to the high complexity of T2IFS, Garg and Singh [55] introduced the concept of triangular interval T2IFS (TIT2IFS) has introduced by considering the MDs and NMDs as a triangular fuzzy number.

Based on the above analysis, we can know that the decision-making problems have become more tedious these days. So in order to make a better decision in terms of selecting the best alternative(s) for the MCDM problems, it is necessary to consider the various factors such as MDs, NMDs, FOU between the alternatives. By keeping the advantages of both the AOs and the TIT2IFS, it is necessary to extend the Hamy mean AOs to process the TIT2IFNs by using linguistic features of MDs and NMDs and hence to develop some MCDM methods. Until now, we have not seen any work based on the AOs used to aggregate the TIT2IFS information. Thus, keeping in mind the advantages of T2IFS and the multiple input interaction between the argumen<sup>t</sup> of HM operator, this paper has presented the concept of the symmetric TIT2IFS and their desired properties. These considerations have led us to consider the main objectives of this paper:


To achieve the objective (1), we combine the T2IFSs and the symmetric triangular number to build a concept of the STIT2IFSs and studied their desired properties. To complete the objective (2), we presented the averaging AOs by using HM operations and named as symmetric triangular IT2IF HM averaging (STIT2IFHM) and weighted symmetric triangular IT2IF Hamy mean averaging (WSTIT2IFHM) operator for decision-making problems by keeping in mind the advantages of T2IFS and the multiple input interaction between the argumen<sup>t</sup> of HM operator. Several enviable properties and particular cases together with following different parameter values of this operator are calculated in detail. To cover the objective (3), we establish an MCDM method based on these proposed operators under the STIT2IFS environment where preferences related to each alternative is expressed in terms of linguistic STIT2IFNs. A numerical illustration is to given to exemplify the practicability of the proposed technique and a comparative analysis is analyzed in detail for fulfilling the Objective 4. Finally, the advantages of the proposed method in the state of the art are highlighted and discussed in detail.

The rest of the paper is organized as follows. In Section 2, some basic concepts on T2FS, IT2FS, T2IFS, and HM are reviewed briefly. In Section 3, we present the concept of the symmetric TIT2IF set and their desirable properties. Section 4 deals with new AOs based on HM operator to accommodate the STIT2IFN information and its special cases. In Section 5, we present an approach based on the WSTIT2IFHM operator to solve the MCDM problem. A practical example is discussed in Section 6 and some concluding remarks are summarized in Section 7.
