## 1 Introduction  
Many scientists are currently researching problems related to vagueness [1]. Discovering effective knowledge from ambiguous data has become a major area of research [2], leading to the development of several techniques for identifying uncertain information, such as fuzzy set (FS) theory [3], quotient space theory [4], and rough set theory (RST) [5]. These theories aim to address issues based on ambiguity and uncertainty. The theory of FSs, created by [PERSON] [3], has been extended by many investigators according to their needs [6, 7, 8]. Every fuzzy set has a pair of components that include a function of membership providing a membership grade in the range of [0, 1]. In 1986, [PERSON] [9] proposed the concept of an intuitionistic fuzzy set (IFS) to express ambiguous and complex information with the aid of membership as well as non-membership grades, where the sum of both grades cannot exceed 1. Another approach to cope with vagueness was formulated by [PERSON] [10], who described a hesitant fuzzy set (HFS). An HFS agrees to the membership grade holding a set of possible values of the interval from 0 to 1 and is an expanded form of FS. The idea of HFS is widely applied in several complications. Most scholars have critically investigated HF data accumulation procedures and their effects in DM [11; 12; 13]. Recently, [PERSON] et al. [14] introduced the concept of an intuitionistic hesitant fuzzy set (IHFS), which is a fusion of IFS and HFS. In IHFS, the grades reflect the structure of a collection of possible values ranging from [0, 1]. IHFS has developed as a powerful tool for explaining the fuzziness of DM complexities. Aggregation operators play a vital role in fuzzy logic, as they combine multiple fuzzy sets into a single value that represents overall fuzzy information. [PERSON] [15; 16] proposed power average and power geometric aggregation operators, while [PERSON] et al. [17] presented Dombi power Heronian mean aggregation operators. [PERSON] et al. [18; 19] defined some novel geometric aggregation operators for IFSs, and [PERSON] [20] extended Bonferroni mean aggregation for a dual hesitant environment. [PERSON] et al. [21] explained the Hamacher mean operators for selecting the best option during decision making. [PERSON] et al. [14] established power aggregation operators for IHFSs for decision making. Triangular norms (T.N) and triangular co-norms (T.CN) are two types of binary operations used in fuzzy logic to combine fuzzy sets. These are based on triangular-shaped membership functions that represent uncertainty and vagueness. The T.N was introduced by [PERSON] [22], and new procedures were introduced by [PERSON] and [PERSON] [23] under the names Aczel-Alsina T.N and Aczel-Alsina T.CN. [PERSON] et al. [24] produced Aczel-Alsina aggregation operators for the IFR environment, while [PERSON] et al. [25] explored novel Aczel-Alsina operators under hesitant fuzzy information and applied them in cyclone disaster assessment. [PERSON] et al. [26] proposed single-valued neutrosophic Aczel-Alsina and utilized them in the decision-making process. [PERSON] et al. [27] worked on the Aczel-Alsina Hamy mean operators for T-spherical information. Other researchers have also explored the Aczel-Alsina aggregation operators in depth [28; 29; 30; 31]. The three-way decision (3 WD) model, an extension of the rough set theory (RST), is a valuable tool for uncertain classification problems [32; 33]. By using a set of thresholds, the 3 WD model divides the universe into three zones: acceptance, deferment, and rejection [34]. Thus, three-way decision theory has found numerous applications in solving complex problems in various fields [35; 36]. DTRSs, a more extensive version of RST, have played a significant role in improving three-way decisions by incorporating the Bayesian decision technique [37]. Proposed by [PERSON] et al. [38], DTRSs involve rational decision semantics that reflect relevant risks. To obtain 3 WD with DTRSs, the minimum total risk is calculated. [PERSON] et al.