Multi attribute group decision making is a problem in several disciplines, such as management, engineering, and economics. For a very long time, it has been considered that such data which accesses the options in terms of requirement and weight is expressed in actual numbers. The majority of the desired values however are contaminated by uncertainty, which making a decision difficult for the decision-makers to correct judgment as the system becomes more complex every day. To deal with the uncertainties, Zadeh (Zadeh, 1965) introduced the fuzzy set theory notion in 1965, which described the degree of membership. It performed a very significant role in decision making (Ashraf et al., 2022a; Ashraf et al., 2022b; Zhang et al., 2022). In 2011, Zadeh (Zadeh, 2011) further proposed the idea of Z-numbers to better highlight the limitations and reliability of the evaluation by an ordered pair of fuzzy numbers in unpredictable conditions. Compared to the traditional fuzzy number, it is a more extensive concept that is intimately linked to reliability. Given this, the Z-number suggests that a pair of fuzzy numbers in the order of restriction and reliability could be better equipped to explain human knowledge and judgments. Theoretically, the Z-numbers reflect the numeric values of internal machine-mind processes that facilitate adapted real-world understanding (Banerjee and Pal, 2015). The four basic mathematical operations in linear programming addition, subtraction, multiplication, and division as well as algebraic operations like maximum, minimum, square, and square root of continuous Z-numbers were established in 2016 (Aliev et al., 2016). On the basis of the extension principle applied to Z-numbers, a technique for the creation of functions was proposed. This methodology is very helpful in the reduction of uncertainty, while computing the values of Z-valued functions. Z-numbers can be arranged using the proposed total utility of Z-number, which can also be used to make multi criteria judgments under uncertain conditions (Kang et al., 2018b). Ronald proposes a Dempster-Shafer such as belief structure representation of Z-values that incorporates type-2 fuzzy logic (Yager, 2012). Ronald R discusses mixing Z-number and the Dempster-Shafer (D-S) evidence theory, where the Z-number is used to simulate the fuzziness and dependability of sensor data and the D-S evidence theory is used to combine the complex information from Z-numbers (Jiang et al., 2016). Numerous scholars created the TODIM technique based on the Choquet integral for circumstances involving multi-criteria decision-making using linguistic Z-numbers (Wang et al., 2017). Using the Z-number application, the supplier selection problem’s idea was illustrated (Jabbarova, 2017). The BWM technique and the Z-number extension were studied in order to handle the informational uncertainty in a multi-criteria decision system (Aboutorab et al., 2018). He created the QUALIFLEX method’s linguistic Z-QUALIFLEX extension, which uses linguistic Z-numbers to solve the LGEDM problem (Ding et al., 2020). In (Kang et al., 2018a), reliable methods that can more accurately and flexibly simulate the process of human competition and cooperation are presented for study analysis based on the usefulness of the Z-number in evolutionary games. They present a generalized Z-number that is more in line with human expression tendencies and a multi-criteria decision-making approach based on the Dempster-Shafer (DS) theory and generalized Z-numbers (Ren et al., 2020). To demonstrate the suggested framework and demonstrate its effectiveness in environmental evaluations, the provides an environmental evaluation framework based on the Dempster-Shafer theory and Z-numbers (Kang et al., 2020).

A class of non-standard Pythagorean fuzzy subsets with pairs of membership grades (a, b) meeting the requirement 0 ≤ a ² + b ² ≤ 1 was introduced using the Pythagorean complement and alternative definitions of complement operations. They provided a number of aggregation techniques and investigated multicriteria decision making in the scenario for these Pythagorean fuzzy set (Yager, 2013). According to (Garg, 2016), some of the aggregator operators discussed include generalized Pythagorean fuzzy Einstein weighted averaging, generalized Pythagorean fuzzy Einstein ordered weighted averaging, and Pythagorean fuzzy Einstein weighted averaging. The Einstein sum, product, and exponentiation as well as geometric aggregation operators and the intuitionistic fuzzy Einstein weighted and ordered weighted geometric operators were only a few of the operations on intuitionistic fuzzy sets that were discussed (Wang and Liu, 2011). The geometric aggregation operators, the intuitionistic fuzzy Einstein weighted and ordered weighted geometric operators, as well as the Einstein sum, product, and exponentiation, were all discussed (Rahman et al., 2017). The Pythagorean Dombi fuzzy aggregation operators introduced by (Akram et al., 2019), ELECTRI-I approach for it by (Akram et al., 2020) and TOPSIS approach by (Akram et al., 2021b). Garg and Sharaf (Garg et al., 2022) presented the spherical fuzzy EDAS approach. Generalized aggregation operators are proposed in (Ashraf and Abdullah, 2019; Chinram et al., 2020; Ashraf et al., 2023). Many scholars worked on its hybrid structures such as cubic Pythagorean linguistic fuzzy numbers (Naeem et al., 2021), rough Pythagorean fuzzy bipolar soft information (Akram and Ali, 2020) and sine trigonometric Pythagorean fuzzy information (Ashraf et al., 2021). It performs better role than fuzzy sets in decision making (Khan et al., 2019; Akram et al., 2021a; Zeng et al., 2023). Saeed et al. (Saeed et al. (2023) presented the refined Pythagorean fuzzy sets and interval-valued complex Pythagorean fuzzy set based methodology is developed in (Yazbek et al., 2023). We refer some decision making techniques (Ashraf et al., 2022c; Garg and Sharaf, 2022) for more details.

We looked at the connection between Z-numbers and linguistic summaries, For instance Dempster-Shafer belief systems and type-2 fuzzy logic were used to describe Z values (Kang et al., 2020). Theoretical components of mathematical operations including addition, subtraction, multiplication, division, and computing the square root of a Z-number over discrete Z-numbers have been proposed (Aliev et al., 2015b). They suggested a Z-number-based computing with words (CWW) algorithm, defined a Z-number-based operator for assessing the degree of requirement satisfaction, described CWW simulation experiments using Z-numbers, examined the benefits and drawbacks of Z-numbers, and offered potential resolution approaches (Pal et al., 2013). The authors developed Z-TOPSIS, a novel variant of the TOPSIS approach that streamlines multi-criteria decision-making problems based on the idea of Z-numbers, in order to support the concept of ranking alternatives using Z-numbers. Additionally, the authors offered a link to some existing information in fuzzy sets (Yaakob and Gegov, 2016). The researchers advise researching the fully Z-number based LP (Z-LP) model and utilizing a method to address Z-LP problems that combines differential evolution optimization and Z-number arithmetic created by the authors in order to better fit real-world problems within the LP framework. Using a benchmark LP problem, the suggested model and solution approach for Z-LP are then shown (Aliev et al., 2015a). For replicating the effects of Pilates exercises on students’ motivation, concentration, anxiety, and academic success, they have presented an innovative way. Due to the ambiguity of data relating to cognitive assessment of psychological features and their partial reliability, the use of “Z-if … then rules” for modelling the considered relationship has been pushed for the first time (Aliev and Memmedova, 2015). The topic (Kang et al., 2016) is split into two parts: the first explains how to use the fuzzy expectation to convert a Z-number into a concept of fuzzy number, and the second explains how to use the genetic algorithm to determine the best priority weight for supplier selection. This method of calculating the priority weight of the judgment matrix is quick and flexible. The first recommendation in this study is an enhanced ranking approach for generalized fuzzy numbers that take into account the weight of centroid points, fuzziness levels, and fuzzy number spreads. This approach is particularly effective for evaluating symmetric fuzzy numbers and crisp numbers, among other things, and can alleviate some of the drawbacks of existing approaches (Jiang et al., 2017). For the first time, a multi-layer method of grading Z-numbers is proposed in this work. This method has two layers: fuzzy number ranking and Z number conversion (Bakar and Gegov, 2015).

Few articles, to our knowledge, seek to explore the rationality and certainty of information in efficient and effective decision-making in a systematic and comprehensive manner. For handling uncertainty effectively in real-world applications, fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, hesitant fuzzy sets, etc. all are unable to deal with the reliability of their membership levels. While reliability or certainty is the core component to deal with making decisions efficiently and trustworthy. In order to eliminate this constraint, this research uses the strategic illustration, conceptual evolution map, and major route analysis to explain the development of the novel Pythagorean fuzzy Z-number (PyFZN) field comprehensively. We suggest a PyFZN by considering Pythagorean fuzziness in terms of both membership and non-membership degrees with reliability. We defined some fundamental properties and aggregation operators, which perform a very significant role in making decisions. Furthermore, we suggested a multicriteria decision-making (MDM) approach, which aims to reduce the overall uncertainty of the decision matrix to increase the credibility of the conclusions. To demonstrate the effectiveness of this novel concept we illustrated a real-life example based on it. Moreover, we also presented a comparative analysis with the existing studies to show its supremacy. The primary goals of this study are: 1. To investigate a number of distinct interactive averaging and geometric AOs concerning PyFZNs to circumvent the constraints. 2. Getting some key characteristics and unique cases of the newly defined AOs. 3. To create a novel MDM strategy using the interactive AOs in a fuzzy Pythagorean context that has been suggested. 4. To present the benefits and viability of the developed MDM strategy.

The main goals of this study are to address the aforementioned research topics and close a knowledge gap.

The rest of this article is organized as follows. Preliminary definitions and concepts are presented in Section 2 and are required to support our primary findings. The concept of Pythagorean fuzzy Z-number, their properties, and score function is explained in Section 3. A thorough investigation of this idea is done in Section 4 where we defined a Pythagorean fuzzy Z-number with arithmetic and geometric aggregation operators and also demonstrate some theorems related to these ideas. In Section 5 we defined a Pythagorean fuzzy Z-number ordered weighted arithmetic and geometric aggregation operators and also demonstrate some theorems related to these ideas. The MDM technique based on intended operators clarified in Section 6, which also provided a numerical example for selecting the riskiest companies to invest money as business partners. We provide an EDAS approach for PyFZNs in Section 7 and also illustrate it with the aid of an example. Finally, in Section 8, we presented a comparative analysis and in Section 9, the conclusion and future research directions for this paper are depicted.

Definition 1

(Zadeh, 1965) Let X be a nonempty set, a fuzzy set A in X is characterized by a membership function

where μ_A: X → [0, 1], the function defines the degree of membership of the element, x ∈ X.

That is: A fuzzy set A in X is an object having the form: A = x , μ A x | x ∈ X .

Definition 2

(Yager, 2013) Assume that B is the Pythagorean fuzzy set and here M is a universal set described as B = l , μ B l , ν B l | l ∈ M , where the function μ B l : M → 0,1 and ν B l : M → 0,1 are the degree of membership and the degree of nonmembership respectively, which satisfies the following requirement: 0 ≤ μ B l 2 + ν B l 2 ≤ 1 .

Definition 3

The concept of Z-number first introduced by zadeh in 2011 (Zadeh, 2011). The Z-number is discussed as, taking order pair of fuzzy numbers Z = S , T , where S is a fuzzy ristriction on the values of M and T gives the reliability for S , here M being a universal set.

Now we define Pythagorean fuzzy Z-number (PyFZN) as given below.

Definition 4

Assume that Gz is a Pythagorean fuzzy Z-number P y F Z N and M be the universal set: G z = l , μ S , T l , ν S , T l | l ∈ M

where the funtion μ S , T l : M → 0,1 and ν S , T l : M → 0,1 are constructed as follows: G z = μ S , T , ν S , T = μ S , μ T , ν S , ν T

It meets the following requirements: 0 ≤ μ S l 2 + ν S l 2 ≤ 1 0 ≤ μ T l 2 + ν T l 2 ≤ 1 .

Now we will discussing the properties of Pythagorean fuzzy Z-numbers which already discuss in Definition 4.

Definition 5

Let G z 1 = μ 1 S , T , ν 1 S , T = μ S 1 , μ T 1 , ν S 1 , ν T 1 and G z 2 = μ 2 S , T , ν 2 S , T = μ S 2 , μ T 2 , ν S 2 , ν T 2 be two Pythagorean fuzzy Z-numbers ( P _y FZN _S ) and L= >0 which satisfies the following characteristics:

(1) Gz ₁ ⊇ Gz ₂ if and only if μ S 1 ≥ μ S 2 , μ T 1 ≥ μ T 2 and ν S 1 ≤ ν S 2 , ν T 1 ≤ ν T 2 .

(2) Gz ₁ = Gz ₂ if and only if Gz ₁ ⊇ Gz ₂ and Gz ₁ ⊆ Gz ₂ ,

(3) G z 1 ∪ G z 2 = ( μ S 1 ∨ μ S 2 , μ T 1 ∨ μ T 2 ) , ν S 1 ∧ ν S 2 , ν T 1 ∧ ν T 2 ,

(4) G z 1 ∩ G z 2 = μ S 1 ∧ μ S 2 , μ T 1 ∧ μ T 2 , ν S 1 ∨ ν S 2 , ν T 1 ∨ ν T 2 ,

(5) G z 1 c = ν S 1 , ν T 1 , μ S 1 , μ T 1 ,

(6) G z 1 ⊕ G z 2 = { ( μ S 1 2 + μ S 2 2 − μ S 1 2 μ S 2 2 , μ T 1 2 + μ T 2 2 − μ T 1 2 μ T 2 2 ) ν S 1 ν S 2 , ν T 1 ν T 2 } ,

(7) G z 1 ⊗ G z 2 = { μ S 1 μ S 2 , μ T 1 μ T 2 , ( ν S 1 2 + ν S 2 2 − ν S 1 2 ν S 2 2 , ν T 1 2 + ν T 2 2 − ν T 1 2 ν T 2 2 ) } ,

(8) L = G z 1 = 1 − 1 − μ S 1 2 L = , 1 − 1 − μ T 1 2 L = , ν S 1 L = , ν T 1 L = ,

(9) G L = z 1 = μ S 1 L = μ T 1 L = , 1 − 1 − ν S 1 2 L = , 1 − 1 − ν T 1 2 L = .

Definition 6

Let Gz ₁ = μ S 1 , μ T 1 , ν S 1 , ν T 1 and Gz ₂ = μ S 2 , μ T 2 , ν S 2 , ν T 2 ∈ P y F Z N S . Then the score funtion is defined as J G z d = 1 + μ S d μ T d − ν S d ν T d 2 . (3.1)

where J ( G z d ) ∈ 0,1 . If the score of J (Gz ₁) ≥ J (Gz ₂), then Gz ₁ ≥ Gz ₂.

Example 1

Consider two pythagorean fuzzy z-number as Gz ₁ = 0.6 , 0.8 , 0.1 , 0.3 and Gz ₂ = 0.5 , 0.7 , 0.2 , 0.4 . Therefore, the base of score funtion the ranking of given pythagorean fuzzy z-number is defined as:Using Eq. 3.1 J G z 1 = 1 + 0.6 × 0.8 − 0.1 × 0.3 2 = 0.725 J G z 2 = 1 + 0.5 × 0.7 − 0.2 × 0.4 2 = 0.595 .

Hence, the score of J (Gz ₁) ≥ J (Gz ₂), then Gz ₁ ≥ Gz ₂ .

We may propose the weighted aggregation operators for PyFZNs in this part based on actions (6) to (9) in Definition 2.

On the basis of the operations (6)–(9) in Definition 4, we may suggest the weighted aggregation operators of PyFZNs in this section.