## 1 Introduction  
Universally, in the representation of data information there exists some uncertainty in the decision-making (DM) process. To overcome this drawback, first time [PERSON]'s [1] defined fuzzy set (FS). In FS, [PERSON]'s only show the positive membership grade of a number in the defined set, and applied to many fields i.e., fuzzy decision making problems [2, 3]. However, there are no discussion for the negative membership grade. Then, the fuzzy set theory failed to solve the complete uncertainty in the real life problem due to non-membership grade. So, [PERSON] [4] defined an intuitionistic fuzzy set (IFS) concept, which represents positive and negative membership grades; IFS has the advantage of two memberships which diminish the fuzziness. [PERSON] [5, 6] displayed generalized improved interactive aggregation operators for the solution of a decision-making problem with the IF set condition. [PERSON] and [PERSON] [7] introduced a cubic intuitionistic fuzzy set (CIFS) aggregation operator. Later, [PERSON] & [PERSON] [8] developed generalized aggregation operators with the cubic IF set information using the t-norm operations. Aside from them, different authors ([PERSON] & [PERSON] [9]; [PERSON] & [PERSON] [10]; [PERSON] & [PERSON] [11]; [PERSON], [PERSON], & [PERSON] [12]) incorporated the idea of aggregation process into the different applications and also give their related decision-making approaches with the IF set and their expansion. [PERSON] & [PERSON] [13] developed the power geometric aggregation operators using the concept of connection number of set pair analysis theory with intuitionistic fuzzy information. [PERSON] & [PERSON] [14] proposed a new generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm. [PERSON] & [PERSON] [15] introduced the generalized intuitionistic fuzzy soft power aggregation operator using the t-norm and its application in multi-criteria decision-making. There are some situations, where \(\mu+\
u\geq 1\), unlike the cases captured in IFSs. Then, the Pythagorean fuzzy set (PFS) is defined by [PERSON] [16; 17], which is characterized by positive and negative membership grades, and satisfies that the square of the positive and negative membership grades and their sum is less than or equal to 1. To handle these type of situation [PERSON] [17] gives an example: a decision maker provides his positive suggestion for an alternative is \(\frac{\sqrt{3}}{2}\) and his negative suggestion is \(\frac{1}{2}\). Now, their sum is greater than 1, then they are not possible for IF set, but they are possible for PFS, because \(\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}\leq 1\). This shows that PFS is more able than IF set to process the uncertainty in real life problems. In recent times, the notion of PF set invites the thinking of researchers, who have applied the PF set in many research areas. [PERSON] et al. [18] introduced the geometric aggregation operators for the group decision-making problem with interval valued Pythagorean fuzzy sets (IVPFSs) environment.Liang & [PERSON] [19] proposed the notion of hesitant PF sets, and applied The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for energy project selection model. [PERSON] et al. [20] introduced a new algorithm for the multi-attribute group decision making (MAGDM) problem, using induced IVPF Einstein aggregation operator. [PERSON] [21] defined a decision-making model with probabilistic information, utilizing the notion of immediate probabilities to aggregate the data with the PFS information. Utilizing the notion of the immediate probabilities, [PERSON] [22] introduced a series of the aggregation operators with the PF information. [PERSON] [23] developed the generalized PF geometric aggregation operators utilizing different norm operations for multi-attribute group decision-making problems. [PERSON] et al.