Some Generalised Einstein Hybrid Aggregation Operators and Their Application to Group Decision-making Using Pythagorean Fuzzy Numbers

Pythagorean fuzzy set (PFS) is one of the prosperous extensions of the intuitionistic fuzzy set (IFS) for handling the fuzziness and uncertainties in the data. Under this environment, in this paper, we introduce the notion of two generalised Einstein hybrid operators namely, generalised Pythagorean fuzzy Einstein hybrid averaging (in short GPFEHA) operator and generalised Pythagorean fuzzy Einstein hybrid geometric (in short GPFEHG) operator along with their desirable properties, such as idempotency, boundedness and monotonicity. The main benefit of the proposed operators is that these operators deliver more general, more correct and precise results as compared to their existing methods. Generalised Einstein operators combine Einstein operators with some generalised operators using Pythagorean fuzzy values. Therefore these methods play a vital role in real world problems. Finally, the proposed operators have been applied to decision-making problems to show the validity, practicality and effectiveness of the new attitude.


Introduction
Multi-criteria group decision-making problems have importance in most kinds of fields such as economics, engineering and management. Generally, it has been assumed that the information which accesses the alternatives in term of criteria and weight are expressed in real numbers. But due to the complexity of the system day-by-day, it is difficult for the decision-makers to make a perfect decision, as most of the preferred value during the decision-making process imbued with uncertainty. In order to handle the uncertainties and fuzziness, intuitionistic fuzzy set [1] theory is one of the prosperous extensions of the fuzzy set theory [2], which is characterised by the degree of membership and degree of non-membership has been presented. Xu [3] developed some basic arithmetic operators, including the IFWA operator, the IFOWA operator, and the IFHA operator. Xu and Yager [4] defined some basic geometric operators, such as the IFWG operator, the IFOWG operator, CONTACT K. Rahman khaista355@yahoo.com to MAGDM using PFNs. In Section 5, we construct numerical example. In Section 6, we compare the proposed operators to others operators. In Section 7, we have conclusion.

Some Generalised Pythagorean Fuzzy Einstein Hybrid Operators
In this section, we investigate the generalised Einstein operators such as, generalised Pythagorean fuzzy Einstein hybrid averaging operator and generalised Pythagorean fuzzy Einstein hybrid geometric operator Definition 3.1: GPFEHG operator can be defined as: whereκ (t) is the largest of the WPFVsκ t (κ t = κ nλ t t , t = 1, 2, . . . , n). = ( 1 , 2 , . . . , n ) T andλ = (λ 1 ,λ 2 , . . . ,λ n ) T be the associated and weighted vector, respectively, and both have the same condition, such as both belong to the closed interval and their sum is equal to 1. And the positive number n is called the balancing coefficient, the parameter ∂ is ∂ 0. Theorem 3.1: Let κ t = Ω κ t , κ t be a family PFVs, then the following conditions hold:

Theorem 3.2:
Let κ t = Ω κ t , κ t be a family PFVs, then the following conditions hold: Proof: Straightforward.

Definition 3.2:
GPFEHA operator can be defined as: whereκ (t) is the largest of the WPFVsκ t (κ t = nλ t κ t , t = 1, 2, . . . , n). = ( 1 , 2 , . . . , n ) T andλ = (λ 1 ,λ 2 , . . . ,λ n ) T be the associated and weighted vector respectively, and both have the same condition, such as both belong to the closed interval and their sum is equal to 1 and n is the balancing coefficient, and the parameter ∂ and is ∂ 0.
Proof: For proof see Theorem 3.

An Application of the Proposed Aggregation Operators
This section deals with multiattribute decision-making (MADM) problems based on the above-mentioned operators using PFNs. To show the superiority and practicality of the above-mentioned operators in daily life problems an example is also given.
Step 5: Utilise the proposed operators to derive the overall preference values.
Step 6: Calculate the scores of all values.
Step 7: Select that option which has the highest score function.

Numerical Example
Suppose, in Hazara University department of mathematics needs to hire a doctor for department.
For this resolution, the university constructs a committee of four decision-makers, whose weight vector is = (0.10, 0.20, 0.30, 0.40) T . After the first selection five doctors, A t (t = 1, 2, 3, 4, 5)are consider for more process. Committee must take a decision according to the following four attributes: C 1 : experience and subject knowledge, C 2 : teaching skill, C 3 : salary and other facilities, C 4 : research skill and publications, where C 1 , C 3 are cost type criteria and C 2 , C 4 are benefiting type criteria, whose weighted vector is = (0.40, 0.30, 0.20, 0.10) T .

Compare with the Other Methods
To show the practicality and effectiveness of the proposed methods and operators, we can compare the proposed methods with some existing methods. First, we compare the  propose methods with methods, such as Pythagorean fuzzy hybrid geometric aggregation operator and Pythagorean fuzzy hybrid averaging aggregation operator proposed by Rahman et al. [24,26], are based on algebraic operations, and those in this paper are based on the generalised Einstein operations. Because the generalised Einstein operations for Pythagorean fuzzy numbers are with parameter ∂, the methods proposed in this paper are more general and more flexible. Secondly, we can compare with Einstein operators, such as Pythagorean fuzzy Einstein hybrid averaging aggregation operator and Pythagorean fuzzy Einstein hybrid geometric aggregation operator proposed by Rahman et al. [31,32], they are only the special cases of the proposed operators in this paper. The proposed methods can be comparing the methods proposed by Garg [33,34], in which the author introduced several operators such as GPFEWA operator, GPFEWG operator GPFEOWA operator and GPFEOWG operator. Actually, GPFEWA operator and GPFEWG operator weigh only the Pythagorean fuzzy arguments, while GPFEOWA operator and GPFEOWG operator weigh only the ordered positions of the Pythagorean fuzzy arguments instead of weighing the Pythagorean fuzzy arguments themselves. To overcome these limitations, we introduce the concept of GPFEHA operator and GPFEHG operator which weigh both the given Pythagorean fuzzy value and its ordered position. Thus the proposed operators are the generalisation of the existing methods.

Benefit of the Proposed Operators
Generalised Einstein operators combine Einstein operators with some generalised operators using Pythagorean fuzzy values. The main benefit of the proposed operators is that these operators deliver more general, more correct and precise results as compared to their existing methods. Therefore these methods play a vital role in real world problems.

Conclusion
The objective of this paper is to investigate the generalised Einstein hybrid operators based on PFNs and their application for daily life problems. Firstly, we have investigated two generalised Einstein operators along with their properties, namely the generalised Pythagorean fuzzy Einstein hybrid averaging operator and the generalised Pythagorean fuzzy Einstein hybrid geometric operator by combining the parameter of the decision-making ∂ during the calculation process. Furthermore, we have industrialised a technique for multi-criteria decision-making based on these operators, and the operational procedures have proved in detail. The suggested methodology can be used for any type of selection problem involving any number of selection attributes. We ended the paper with an application of the new approach in a decision-making problem. Garg [33,34] introduced the notion of GPFEWA operator, GPFEOWA operator, GPFEWG operator, GPFEOWG operator. Actually, GPFEWA operator and GPFEWG operator weigh only the Pythagorean fuzzy arguments, GPFEOWA operator and GPFEOWG operator weigh only the ordered positions of the Pythagorean fuzzy arguments instead of weighing the Pythagorean fuzzy arguments themselves. To overcome these limitations, we introduce the concept of GPFEHA operator and GPFEHG operator which weigh both the given Pythagorean fuzzy value and its ordered position. Thus the proposed operators are the generalisation of the existing methods.
In the future, we will extend the proposed approach to the different environment and then will apply to the fields of the pattern recognition, Symmetric operator, Inducing variable, Logarithmic operator, Power operator, Hamacher operator, Dombi operator, Linguistic terms, Confidence levels, Interval valued etc.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
K. Rahman received M. Phil. and Ph.D degrees in mathematics from Hazara University, Mansehra, Pakistan. His main research interests are pure mathematics that is aggregation operators, fuzzy theory, fuzzy logic and decision making problems. He has published more than 50 papers in different national and international journals.