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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2023.028963

ARTICLE

Pythagorean Fuzzy Einstein Aggregation Operators with Z-Numbers: Application in Complex Decision Aid Systems

Shahzad Noor Abbasi1, Shahzaib Ashraf1,*, M. Shazib Hameed1 and Sayed M. Eldin2,*

1Institute of Mathematics, Khwaja Fareed University of Engineering & Information Technology, Rahim Yar Khan, 64200, Pakistan
2Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo, 11835, Egypt
*Corresponding Authors: Shahzaib Ashraf. Email: shahzaib.ashraf@kfueit.edu.pk; Sayed M. Eldin. Email: sayed.eldin22@fue.edu.eg
Received: 19 January 2023; Accepted: 16 March 2023; Published: 03 August 2023

Abstract: The primary goal of this research is to determine the optimal agricultural field selection that would most effectively support manufacturing producers in manufacturing production while accounting for unpredictability and reliability in their decision-making. The PFS is known to address the levels of participation and non-participation. To begin, we introduce the novel concept of a PFZN, which is a hybrid structure of Pythagorean fuzzy sets and the ZN. The PFZN is graded in terms of membership and non-membership, as well as reliability, which provides a strong advice in real-world decision support concerns. The PFZN is a useful tool for dealing with uncertainty in decision-aid problems. The PFZN is a practical way for dealing with such uncertainties in decision-aid problems. The list of aggregation operators: PFZN Einstein weighted averaging and PFZN Einstein weighted geometric, is established under the novel Pythagorean fuzzy ZNs. It is a more precise mathematical instrument for dealing with precision and uncertainty. The core of this research is to develop a numerical algorithm to tackle the uncertainty in real-life problems using PFZNs. To show the applicability and effectiveness of the proposed algorithm, we illustrate the numerical case study related to determining the optimal agricultural field. The main purpose of this work is to describe the extended EDAS approach, then compare the proposed methodology with many other methodologies now in use, and then demonstrate how the suggested methodology may be applied to real-world problems. In addition, the final ranking results that were obtained by the devised techniques were more efficient and dependable in comparison to the results provided by other methods presented in the literature.

Keywords: Pythagorean fuzzy Z-number Einstein weighted averaging; Pythagorean fuzzy Z-number Einstein weighted geometric; decision making

List of Abbreviations

FS | Fuzzy set
PFS | Pythagorean fuzzy set
IFS | Intuitionistic FS
CWW | Computing with words
PF | Pythagorean fuzzy
ZN | Z-number
FN | Fuzzy number
AO | Aggregation operator
PFZNE | PFZN Einstein
IVIFS | Interval-valued IFS
PFZNW | PFZN weighted
ASC | Appraisal score
CT | Conventional tillage
SPDA | Sum of PDA
SNDA | Sum of NDA
ZTB | Zero tillage with bed
AvS | Average solution
PFZNE | PFZN Einstein
RT | Reduce tillage
PFA | PF averaging
PFZN | Pythagorean fuzzy Z-number
MAGDM | Multi-attribute group decision-making
OWA | Ordered weighted average
MCDM | Multi-criteria decision-making
PFZNEWA | PFZN Einstein weighted averaging
PFZNEWGA | PFZNE weighted geometric averaging
PFN | Pythagorean fuzzy number
PFZNWG | PFZN weighted geometric
IVPFSs | Interval-valued PFSs
PFZNEOGA | PFZNE ordered weighting averaging
PFZNEGA | PFZNE geometric averaging
PFZNEWA | PFZNE weighted averaging
PFZNEOWA | PFZNE ordered weighted averaging
CTB | Conventional tillage with bed planting
RTB | Reduce tillage with bed planting
PFZNWGA | PFZN weighted geometric averaging
NDA | Negative distance from average
PFZNEOWGA | PFZNE ordered weighting geometric averaging
PFZNOWA | PFZN ordered weighted averaging
PDA | Positive distance from average

1  Introduction

The idea of FSs has been introduced since 1965 in a variety of ways and across many academic fields. Logic, computer science, medicine, decision theory, and robotics are a few fields where this theory has many applications. Mathematical innovations have reached a very high level and continue to be made today. MAGDM is a challenge in management, engineering, economics, and various other fields. People often think that the options for data access based on need and weight are given in real numbers. However, most desirable values are tainted by ambiguity, making it challenging for those in control of decision-making to identify the optimal alternative as the system gets more complicated every day. An informative assessment for the PFS has been developed in [1], together with a justification for its validity and a discussion of the performance of the anticipated information measure.

In [2], Zeng et al. provided an innovative IFS that prevents information loss from participation and non-participation degrees. Sen et al. [3] provided sustainable supplier selection from an IFS decision-making viewpoint in order to address the question of how to acquire supplier selection. They talked about applying for the classic FN and converting ZNs to conventional FNs, and they provided a supplier selection example to show how useful the suggested process is. In [4], Rahman et al. examined a number of fundamental and significant definitions of PFSs, a number of operations on PFSs, and a number of algebraic laws relating to PFSs. In [5], Wei et al. devised plenty of PF power AOs: e.g., the PF average operator, the PF power geometric operator, the PF power ordered weighted average operator, the PF power hybrid average operator, etc.

Ejegwa investigates the idea of PFSs and draws some conclusions about how the score and accuracy functions work in [6]. PFSs have several characteristics that have been described. Pythagorean fuzzy relation is a concept that is developed in a PFS setting using numerical examples to support the developed relation. In [7], a stochastic EDAS strategy is provided in order to cope with the situation when the performance of the alternative values for each criterion follows a normal distribution. Using the suggested methodology, they evaluated options and noted the ambiguity of the information used to make decisions by obtaining optimistic and pessimistic assessment ratings. Oz et al. provided risk assessment for the clearing and grading process of a natural gas pipeline project: An extended TOPSIS model with PFSs for prioritizing hazards in [8]. Yager et al. in [9] discussed the notions of PF subsets and Pythagorean participation grades, which are related concepts, and our attention was also drawn to the negation’s connection to the Pythagorean theorem. For the instance of PF subsets, they examined the fundamental set operations and complex numbers, and Pythagorean participation grades were shown to be related. In [10], Wang et al. investigated MCDM techniques using linguistic ZNs. In addition to defining and describing linguistic ZNs, this study also presented a comparison technique and a distance metric. Then they also introduced an expanded TODIM technique that relies on the Choquet integral for linguistic ZNs MCDM issues, taking into account the limited rationale of decision-making and the interactivity of criteria.

Tian et al. provided the procedure for calculating ZN relies on OWA weights and maximal entropy in [11], which is a simpler explanation of what ZN means. TOPSIS strategy simplifies MCDM situations, which rely on the idea of ZNs. Furthermore, Jia et al. [12] established a novel solution for ZNs based on complex FSs and its application in Decision-Making System.

Atanassov [13] added a second degree, known as the degree of non-participation, to the concept of the FS in 1986 to depict hesitancy and doubt on the degree of participation. For the first time, Aliev et al. in [14], present a broad strategy for building such functions that relies on the extension idea used with ZNs. The proposed method is useful for limiting the increase in uncertainty when computing the values of Z-valued functions, and it also takes into account a few ZN function characteristics.

In [15], Poleshchuk described a method to multicriteria decision making under Z-information. This method applied the anticipated utility paradigm to a standard economic decision-making issue. They also created an expanded TODIM strategy that relies on the Choquet integral for MCDM issues with linguistic ZNs. Jiang et al. [16] proffered a novel approach on the basis of Z-Network model based on Bayesian Network and ZN in which they expressed an application of the strategy to problems connected to cognitive and aesthetic concerns inherently defined by imperfect data, such as work satisfaction assessment and educational accomplishment of students appraisal. Kang et al. offered an environmental assessment under uncertainty using Dempster-Shafer theory and ZN in [17]. Internationally renowned tools, such as the Academic Motivation Scale, the Test of Attention (D2 Test), and Spielberger’s Anxiety Test completed by students, are used to measure psychological factors. These Articles take into account a multi-criteria supplier selection dilemma where all the alternative parts are characterized by Z-information. They employed utility theories to solve these difficulties, and after evaluating the options, they chose the best one [18]. Abiyev et al. furnished ZN based fuzzy system for control of omnidirectional robot in [19]. Pal et al. offered a thorough analysis of the ZN method for CWW in [20]. CWW simulations with ZNs, make a ZN-based operator for figuring out how much compliance is needed, and give an algorithm for CWW with ZNs. In particular, they presented a summary of what we know about the generic design philosophy, mechanism, and hurdles that underlie CWW in general. Finally, they discussed the benefits and drawbacks of ZNs and provided some recommendations for improving this technology. To address the linear goal programming with equally desired minima issue, Ding et al. [21] created an enhanced version of QUALIFLEX based on linguistic ZNs; they called it the linguistic Z-QUALIFLEX approach. Linguistic ZNs are first used in order to represent the judgements of the decision-makers, which may more accurately describe the views that are inherently held by the decision-makers. Kang et al. in [22] proposed a methodology for ZN-based supplier selection that necessitates the transformation of information. This paper was divided into two parts: the first part addresses the problem of converting a ZN to a traditional FN in accordance with a fuzzy expectation; the second part addresses the issue of obtaining the best priority weight for supplier selection using a genetic algorithm, which is a quick and convenient way to determine the priority weight of the judgement matrix. In [23], Ren et al. suggested an MCDM strategy that relies on generalized ZNs and the Dempster-Shafer theory. To do so, they increase the ZN to a larger version that is more influenced by mortal affirmation tendencies and intrigued by the concept of a hesitant fuzzy linguistic word set. Interval-valued PFSs ranking order has been proffered by Garg in [24], who has improved the score function. In [25], Garg introduced a new generalized improved score function of IVIFSs. The goal of that paper was split into two categories. First, by taking into account the concept of a weighted average of the level of uncertainty between their participation functions, a new generalized enhanced scoring function has been introduced from the perspective of IVIFSs. Second, an IVIFSs-based strategy was used to solve the MCDM problem. For aggregating uncertain data, in [26], Riaz et al. presented the Pythagorean m-polar fuzzy weighted averaging, Pythagorean m-polar fuzzy weighted geometric, and symmetric Pythagorean m-polar fuzzy weighted averaging and symmetric Pythagorean m-polar fuzzy weighted geometric operations. They created a class of non-standard PF subsets with participation grades (a,b) that satisfy the condition a2+b2≤1 by focusing on the Pythagorean complement. Garg developed several aggregation methods for PFS in [27]. In [28], Du et al. analyzed a strategy as a generalization of the ZN and the neutrosophic set. This study suggested the idea of a neutrosophic ZN set, which was a strategic platform of neutrosophic values with the neutrosophic measures of dependability. A significant amount of work on MCDM has been done in recent years by a variety of researchers using PFS, the picture fuzzy set, N-soft set in [29–35].

The following outline will be used to summarize this article. In Section 2, we define key terms and discuss major aspects of relevant ideologies in support of our primary arguments. For a complete description of the PFZN Einstein operational law, including its definition, properties, and related theorems, see Section 3. We build a PFZN with an Einstein weighted aggregation operator and discuss its formulation, properties, and related theorems in detail in Section 4. In Section 5, we define and characterize PFZNs with an Einstein weighted aggregation operator, and we prove and analyze the proofs of various related theorems. Section 6 defines PFZNs with Einstein ordered weighted averaging, Einstein weighted geometric averaging, and Einstein ordered weighted geometric averaging aggregation operators and describes their properties. It also proves various related theorems. We divide Section 7 into two subsections. In Subsection 7.1, we developed MCDM approach using the PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators, and in Subsection 7.2, we presented an example how to implemented these operators also provided comparison between these operators. In Section 8, we provided an EDAS method for the PFZNE operator and a numerical example for selecting agricultural fields. Finally, we provide a summary and some recommendations for future research in this field.

The extended EDAS approach is a novel concept that should be considered when looking for ways to deal with the truthness of membership and non-membership claims. The previously available approaches were incapable of managing this sort of data; hence, there was a vacuum in the market that needed to be addressed. As compared to the methods that were previously in use, the outcome obtained by this extended EDAS methodology yielded results that were more precise for the specific sort of data that was being examined.

The major goals of our study are:

1. To introduce a new approach for dealing with Pythagorean fuzzy Z-numbers using Einstein aggregation operators and an extended version of the EDAS method.

2. To demonstrate the effectiveness and robustness of the extended EDAS method in handling decision-making problems under uncertainty and imprecision.

3. To compare the proposed method with other existing operators and evaluate its superiority in terms of accuracy and performance.

4. To provide a comprehensive numerical example to illustrate the application of the proposed method in practical decision-making scenarios.

5. To contribute to the field of decision-making under uncertainty and imprecision by introducing a new method that can handle Pythagorean fuzzy Z-numbers effectively, and potentially be extended to other related areas of research.

Like any other model or approach, the proposed Pythagorean fuzzy Z-numbers with Einstein aggregation operators and extended EDAS method also has some limitations that need to be considered:

1. The proposed method assumes that the criteria weights are fixed and do not change over time or with changing conditions. However, in some cases, the weights may be dynamic, and the proposed method may not be suitable for such scenarios.

2. The method relies on subjective inputs from decision-makers, such as the membership values of the Pythagorean fuzzy Z-numbers and the preference weights of the criteria. These subjective inputs may introduce bias and uncertainty in the decision-making process.

3. The proposed method may not be appropriate for situations where there are a large number of alternatives and criteria, as it can become computationally expensive and time-consuming.

4. The method assumes that the Pythagorean fuzzy Z-numbers are independent of each other. However, in some cases, the relationships between the Pythagorean fuzzy Z-numbers may be correlated, and the method may not accurately reflect these relationships.

5. The method does not explicitly consider the possibility of incomplete or inconsistent information, which may occur in some decision-making scenarios.

Overall, the proposed method provides a useful framework for decision-making under uncertainty and imprecision. However, it is important to acknowledge its limitations and carefully consider the appropriateness of the method in specific decision-making situations.

2  Preliminaries

The following description and symbols have been abbreviated for time considerations in this article.

Definition 2.1. [36] Suppose that X is a nonempty set, and X has a participation function that is A. Where μA:X→[0,1], the function defines the level of participation of the element, ℘∈X.

That is: In X, a FS A is an object of the following form: A={⟨℘,μA(℘)⟩|℘∈X}.

Definition 2.2. In 2011, Zadeh [37] was the first person to propose the notion of the ZN. The ZN is discussed as, taking order pair of FNs Z=(S,T), where S is a fuzzy limitation on the values of M, and T is the dependability for S, with M being a universal set.

Definition 2.3. [8] Assume that B is the PFS and here M is a universal set which described as

B={℘,μB(℘),νB(℘)| ℘∈M},

where the mapping μB(℘):M→[0,1] and νB(℘):M→[0,1] are the level of participation and the level of non-participation respectively, which satisfies the following requirement:

0≤(μB(℘))2+(νB(℘))2≤1.

To make things easier, Oz et al. [8] denoted a PFN by (μP(℘),νP(℘)), P=(μP,νP). Consider three PFNs α=(μ,ν), α1=(μ1,ν1), and α2=(μ2,ν2), Yager et al. [9] revealed the fundamental operations, which are:

(1)   α−=[ν,μ];

(2)   α1∨α2=[max{μ1,μ2},min{ν1,ν2}];

(3)   α1∧α2=[min{μ1,μ2},max{ν1,ν2}];

(4)   α1⊕α2=[μ21+μ22−μ21μ22,ν1ν2];

(5)   α1⊗α2=[μ1μ2,ν12+ν22−ν12ν22];

(6)   I~.α=[1−(1−μ2)I~,νI~,I~>0];

(7)   αI~=[μI~,1−(1−ν2)I~,I~>0].

3  Pythagorean Fuzzy Z-Number

Definition 3.1. Suggest the PFZN to be Gz, and M be the universal set:

Gz={℘,μ(S,T)(℘),ν(S,T)(℘)|℘∈M}

where the mapping μ(S,T)(℘):M→[0,1] and ν(S,T)(℘):M→[0,1] are constructed as follows:

Gz={(μ(S,T)),ν(S,T)}= {(μS,μT),(νS,νT)}.

It meets the following requirements:

0≤(μS(℘))2+(νS(℘))2≤1,

0≤(μT(℘))2+(νT(℘))2≤1.

The characteristics of PFZNs will now be discussed, which already derive in Definition 3.1.

Definition 3.2. Let Gz1={(μ1(S,T)),ν1(S,T)}= {(μS1,μT1),(νS1,νT1)} and Gz2={(μ2(S,T)),ν2(S,T))}= {(μS2,μT2),(νS2,νT2)} be two PFZNs, which satisfies the following characteristics:

(1) Gz1⊇Gz2 if and only if  μS1≥μS2, μT1≥μT2 and νS1≤νS2,νT1≤νT2.

(2) Gz1=Gz2 if and only if Gz1⊇Gz2 and Gz1⊆Gz2,

(3) Gz1∪Gz2={(μS1∨μS2,μT1∨μT2),(νS1∧νS2,νT1∧νT2)},

(4) Gz1∩Gz2={(μS1∧μS2,μT1∧μT2),(νS1∨νS2,νT1∨νT2)},

(5) (Gz1)c={(νS1,νT1),(μS1,μT1)},

(6) Gz1⊕Gz2={(μS12+μS22−μS12μS22,μT12+μT22−μT12μT22)(νS1νS2,νT1νT2)},

(7) Gz1⊗Gz2={(μS1μS2,μT1μT2),(νS12+νS22−νS12νS22,νT12+νT22−νT12νT22)},

(8) I~Gz1={(1−(1−μS12)I~,1−(1−μT12)I~),(νS1I~,νT1I~)},

(9) GI~z1={(μS1I~μT1I~),(1−(1−νS12)I~,1−(1−νT12)I~)}.

Definition 3.3. Let Gz1= {(μS1,μT1),(νS1,νT1)} and Gz2= {(μS2,μT2),(νS2,νT2)} ∈PFZNS. This leads to the following formulation of the scoring function:

J(Gzı`)=1+μSı`μTı`−νSı`νTı`2.(1)

where J(Gzı`)∈[0,1]. The ranking of Gz1≥Gz2, then there is J(Gz1)≥J(Gz2).

Definition 3.4. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)},(ı`=1,2,...,n~) be a catalogue that contains of PFZNs and I~ı` is the weight of I~ı`(ı`=1,2,...,n~) such that I~ı`∈[0,1] and ∑ı`=1n~I~ı`=1 then, a PFZNEWA mapping signified by the operator of dimension n PFZNEWA:ϖn~→ϖ, and

PFZNEWA(Gz1,Gz2,...,Gzn~)=I~1.ϵGz1⊕ϵI~2.ϵGz2⊕ϵ...⊕ϵI~n~.ϵGzn~(2)

where ϖ is the catalogue that contains of all PFNs. In instance, if I~ı`=1n~, ∀ı`, then PFZNEWA operator simplified to PF averaging operator.

PFA(Gz1,Gz2,...,Gzn~)=1n~.ϵ(Gz1⊕ϵGz2⊕ϵ...⊕ϵGzn~)

Example 1. Consider two PFZN as Gz1= {(0.6,0.8),(0.1,0.3)} and Gz2= {(0.5,0.7),(0.2,0.4)}. As a result, the following is the definition of the score function used to rank a given PFZN by using Eq. (1),

J(Gz1)=(1+(0.6×0.8)−(0.1×0.3)2)=0.725,J(Gz2)=(1+(0.5×0.7)−(0.2×0.4)2)=0.595.

Hence, the ranking of  Gz1≥Gz2, then there is J(Gz1)≥J(Gz2). Using the operation (6) and (8) in Definition 3.2, we give the PFZNWGA operator of PFZNs.

Definition 3.5. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)}(ı`=1,2,...,n~) be a group of PFZNs and

PFZNWA:ϖn~→ϖ. Then, we will be able to categorize the PFZNWA operator as

PFZNWA(Gz1,Gz2,...,Gzn~)=∑ı`=1n~I~ı`Gzı`,

where I~ı`(ı`=1,2,...,n~) is the weight vector with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Theorem 3.1. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)}(ı`=1,2,...,n~) be a group of PFZNs.

Then, the acquire value of the PFZNWA operator is a PFZN, which is deduced using this formula:

PFZNWA(Gz1,Gz2,...,Gzn~)=∑ı`=1n~I~ı`Gzı`={(1−Πı`=1n~(1−μS12)I~ı`,1−Πı`=1n~(1−μT12)I~ı`),(Πı`=1n~νS1I~ı`,Πı`=1n~νT1I~ı`)}

where I~ı`(ı`=1,2,...,n~) is the weight vector with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1. Using the operation (7) and (9) in Definition 3.2, we give the PFZNWGA operator of PFZNs.

Definition 3.6. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)}(ı`=1,2,...,n~) be a group of PFZNs. Then the PFZNWGA:ϖn~→ϖ operator is defined as

PFZNWGA(Gz1,Gz2,...,Gzn~)=Πı`=1n~GI~ı`zı`,

where I~ı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Theorem 3.2. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)}(ı`=1,2,...,n~) be a group of PFZNs. Then, the collected value of the PFZNWGA operator is a PFZN, which is obtained by the following formula:

PFZNWA(Gz1,Gz2,...,Gzn~)=Πı`=1n~GI~ı`zı`={(Πı`=1n~μSı`I~ı`,Πı`=1n~μTı`I~ı`),(1−Πı`=1n~(1−νSı`2)I~ı`,1−Πı`=1n~(1−νTı`2)I~ı`)}

where I~ı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Definition 3.7. Let αı`={(μSαı`,μTαı`)(νSαı`,νTαı`)}(ı`=1,2,...,n~) be a catalogue of PFZNs, then the PFZN order weighted averaging aggregation operator is defined as

PFZNOWA(α1,α2,...,αn~)=I~1ασ(1)⊕I~2ασ(2)⊕...⊕I~n~ασ(n~),

where I~ı`(ı`=1,2,...,n~) is the weighted vector of αı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Theorem 3.3. Let αı`={(μSαı`,μTαı`)(νSαı`,νTαı`)}(ı`=1,2,...,n~) be a collection of PFZNs, then their aggregated value by using PFZNOWA operators as

PFZNOWA(α1,α2,...,αn~)={(1−Πı`=1n~(1−μSασ(ı`)2)I~ı`,1−Πı`=1n~(1−μTασ(ı`)2)I~ı`),(Πı`=1n~νSασ(ı`)I~ı`,Πı`=1n~νTασ(ı`)I~ı`)}

where I~ı`(ı`=1,2,...,n~) is the weighted vector of αı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

4  Einstein Operational Law of PFZNs

In this part of the article, we will cover Einstein operations on PFNs and look at plenty of the benefits associated with using them. In [35], Deshrijver et al. defined the generalized intersection of PFNs H and K, symbolized by “∧”, and the generalized union of PFNs H and K, symbolized by “∨”.

H∨K={(℘,T(μH(℘),μK(℘)),S(μH(℘),μK(℘)))|℘ϵX}

H∧K={(℘,S(μH(℘),μK(℘)),T(μH(℘),μK(℘)))| ℘ϵX}

To express related intersections and unions, one can choose from a variety of t-norms and t-conorms groupings. They typically give nearly identical smooth estimation as the algebraic product and the algebraic sum, correspondingly, Einstein product ⊗∈ and Einstein sum ⊕∈ are two such families that make good alternatives which are specified as follows in the PF framework:

S∈(h,k)=h2+k21+h.ϵ2k2,T∈(h,k)=h.∈k1+(1−h2).∈(1−k2)

Definition 4.1. Assuming Gz1= {(μS1,μT1),(νS1,νT1)} and Gz2= {(μS2,μT2),(νS2,νT2)} be two PFZNs, which satisfies the following characteristics:

(1) Gz1⊇Gz2 if and only if  μS1≥μS2, μT1≥μT2 and νS1≤νS2,νT1≤νT2.

(2) Gz1=Gz2 if and only if Gz1⊇Gz2 and Gz1⊆Gz2,

(3) Gz1∪Gz2={(μS1∨μS2,μT1∨μT2),(νS1∧νS2,νT1∧νT2)},

(4) Gz1∩Gz2={(μS1∧μS2,μT1∧μT2),(νS1∨νS2,νT1∨νT2)},

(5) (Gz1)c={(νS1,νT1),(μS1,μT1)},

(6) Gz1⊕Gz2=[(μS12+μS221+μS12⋅∈μS22,μT12μT221+μT12⋅∈μT22),(νS12⋅∈νS221+(1−νS12)⋅∈(1−νS22),νT12.∈νT221+(1−νT12)⋅∈(1−νT22))]

(7) Gz1⊗Gz2=[(μS12.∈μS221+(1−μS12).∈(1−μS22),μT12.∈μT221+(1−μT12).∈(1−μT22)),(νS12+νS221+νS12.∈νS22,νT12+νT221+νT12.∈νT22)]

(8) I~.Gz1=[((1+μS12)I~−(1−μS22)I~(1+μS12)I~+(1−μS22)I~,(1+μT12)I~−(1−μT22)I~(1+μT12)I~+(1−μT22)I~),(2(νS1)I~(2−νS12)I~+(νS12)I~,2(νT1)I~(2−νT12)I~+(νST12)I~)]

(9) (Gz1)I~=[(2(μS1)I~(2−μS12)I~+(μS12)I~,2(μT1)I~(2−μT12)I~+(μST12)I~),((1+νS12)I~−(1−νS22)I~(1+νS12)I~+(1−νS22)I~,(1+νT12)I~−(1−νT22)I~(1+νT12)I~+(1−νT22)I~.)]

Theorem 4.1. Let Gz1= {(μS1,μT1),(νS1,νT1)}, Gz2= {(μS2,μT2),(νS2,νT2)}, and Gz={(μS,μT),(νS,νT)} be three PFZNs, then both Gz3=Gz1⊕ϵGz2 and Gz4=I~.ϵGz(I~>0) are also PFZNs.

Proof. As I~ be an any positive real number and Gz be PFZNs, then 0≤μS≤1,0≤μT≤1,0≤νS≤1, 0≤νT≤1,0≤(μ(S)(℘))2+(ν(S)(℘))2≤1, and 0≤(μ(T)(℘))2+(ν(T)(℘))2≤1, then 1−(μ(T)(℘))2≥(ν(T)(℘))2≥0,1−(ν(T)(℘))2≥(μ(T)(℘))2≥0, and (1−(μ(T)(℘))2)I~≥((ν(T)(℘))2)I~,(1−(μ(S)(℘))2)I~≥((ν(S)(℘))2)I~ we get

(1+(μS1(℘))2)I~−(1−(μS1(℘))2)I~(1+(μS1(℘))2)I~+(1−(μS1(℘))2)I~≤(1+(μS1(℘))2)I~−((νS1(℘))2)I~(1+(μS1(℘))2)I~+((νS1(℘))2)I~,

(1+(μT1(℘))2)I~−(1−(μT1(℘))2)I~(1+(μT1(℘))2)I~+(1−(μT1(℘))2)I~≤(1+(μT1(℘))2)I~−((νT1(℘))2)I~(1+(μT1(℘))2)I~+((νT1(℘))2)I~.

and

2(νS1(℘))I~(2−(νS1(℘))2)I~+((νS1(℘))2)I~≤2(νS1(℘))I~(1+(μS1(℘))2)I~+((νS1(℘))2)I~,

2(νT1(℘))I~(2−(νT1(℘))2)I~+((νT1(℘))2)I~≤2(νT1(℘))I~(1+(μT1(℘))2)I~+((νT1(℘))2)I~.

Thus,

((1+(μS1(℘))2)I~−(1−(μS1(℘))2)I~(1+(μS1(℘))2)I~+(1−(μS1(℘))2)I~)2+(2(νS1(℘))I~(2−(νS1(℘))2)I~+((νS1(℘))2)I~)2≤1,

((1+(μT1(℘))2)I~−(1−(μT1(℘))2)I~(1+(μT1(℘))2)I~+(1−(μT1(℘))2)I~)2+(2(νT1(℘))I~(2−(νT1(℘))2)I~+((νT1(℘))2)I~)2≤1.

Furthermore,

((1+(μS1(℘))2)I~−(1−(μS1(℘))2)I~(1+(μS1(℘))2)I~+(1−(μS1(℘))2)I~)2+(2(νS1(℘))I~(2−(νS1(℘))2)I~+((νS1(℘))2)I~)2=0,

((1+(μT1(℘))2)I~−(1−(μT1(℘))2)I~(1+(μT1(℘))2)I~+(1−(μT1(℘))2)I~)2+(2(νT1(℘))I~(2−(νT1(℘))2)I~+((νT1(℘))2)I~)2=0.

if and only if μS1(℘)=νS1(℘)=0,μT1(℘)=νT1(℘)=0. and

((1+(μS1(℘))2)I~−(1−(μS1(℘))2)I~(1+(μS1(℘))2)I~+(1−(μS1(℘))2)I~)2+(2(νS1(℘))I~(2−(νS1(℘))2)I~+((νS1(℘))2)I~)2=1,

((1+(μT1(℘))2)I~−(1−(μT1(℘))2)I~(1+(μT1(℘))2)I~+(1−(μT1(℘))2)I~)2+(2(νT1(℘))I~(2−(νT1(℘))2)I~+((νT1(℘))2)I~)2=1.

if and only if (μS1(℘))2+(νS1(℘))2=1,(μT1(℘))2+(νT1(℘))2=1. Thus, Gz4=I~.ϵGz is PFZN for any positive real number I~.

Remark. Now let us looking into I~.ϵGz and GzI~ for a few particular situations of I~ and Gz in the following. (a) If Gz= {(μS,μT),(νS,νT)}=(1,0), that is μS=μT=1,νS=νT=0, then

I~.ϵGz=[((1+μS2)I~−(1−μS2)I~(1+μS2)I~+(1−μS2)I~,(1+μT2)I~−(1−μT2)I~(1+μT2)I~+(1−μT2)I~),(2(νS)I~(2−νS2)I~+(νS2)I~,2(νT)I~(2−νT2)I~+(νST2)I~)]=(1,0)(Gz1)I~=[(2(μS)I~(2−μS2)I~+(μS2)I~,2(μT)I~(2−μT2)I~+(μST2)I~),((1+νS2)I~−(1−νS2)I~(1+νS2)I~+(1−νS2)I~,(1+νT2)I~−(1−νT2)I~(1+νT2)I~+(1−νT2)I~)]=(1,0)

i.e., I~.ϵ(1,0)=(1,0) and (1,0)I~=(1,0)

(b) If Gz= {(μS,μT),(νS,νT)}=(0,1), then

I~.ϵGz=[((1+μS2)I~−(1−μS2)I~(1+μS2)I~+(1−μS2)I~,(1+μT2)I~−(1−μT2)I~(1+μT2)I~+(1−μT2)I~),(2(νS)I~(2−νS2)I~+(νS2)I~,2(νT)I~(2−νT2)I~+(νST2)I~)]=(0,1)(Gz1)I~=[(2(μS)I~(2−μS2)I~+(μS2)I~,2(μT)I~(2−μT2)I~+(μST2)I~),((1+νS2)I~−(1−νS2)I~(1+νS2)I~+(1−νS2)I~,(1+νT2)I~−(1−νT2)I~(1+νT2)I~+(1−νT2)I~)]=(0,1)

(c) If I~⟶0 and 0<μS,νS<1, then

I~.ϵGz=[((1+μS2)I~−(1−μS2)I~(1+μS2)I~+(1−μS2)I~,(1+μT2)I~−(1−μT2)I~(1+μT2)I~+(1−μT2)I~),(2(νS)I~(2−νS2)I~+(νS2)I~,2(νT)I~(2−νT2)I~+(νST2)I~)]⟶(0,1)(Gz1)I~=[(2(μS)I~(2−μS2)I~+(μS2)I~,2(μT)I~(2−μT2)I~+(μST2)I~),((1+νS2)I~−(1−νS2)I~(1+νS2)I~+(1−νS2)I~,(1+νT2)I~−(1−νT2)I~(1+νT2)I~+(1−νT2)I~)]⟶(1,0)

(d) If I~⟶+∞ and 0<μS,νS<1, then

I~.ϵGz=[((1+μS2)I~−(1−μS2)I~(1+μS2)I~+(1−μS2)I~,(1+μT2)I~−(1−μT2)I~(1+μT2)I~+(1−μT2)I~),(2(νS)I~(2−νS2)I~+(νS2)I~,2(νT)I~(2−νT2)I~+(νST2)I~)]⟶(0,1)=[((1)I~−(1−μS21+μS2)I~(1)I~+(1−μS21+μS2)I~,(1)I~−(1−μT21+μT2)I~(1)I~+(1−μT21+μT2)I~),(2(1)I~(2−νS2νS)I~+(1)I~,2(1)I~(2−νT2νT)I~+(1)I~)]⟶(0,1)

since 0≤νS<1,(0≤νT<1)⟺νS<2−νS⟺1<2−νSνS. Thus (2−νSνS)I~=+∞ as I~⟶∞

(Gz1)I~=[(2(μS)I~(2−μS2)I~+(μS2)I~,2(μT)I~(2−μT2)I~+(μST2)I~),((1+νS2)I~−(1−νS2)I~(1+νS2)I~+(1−νS2)I~,(1+νT2)I~−(1−νT2)I~(1+νT2)I~+(1−νT2)I~)]⟶(1,0)

(d) If I~=1, then

I~.ϵGz=[((1+μS2)I~−(1−μS2)I~(1+μS2)I~+(1−μS2)I~,(1+μT2)I~−(1−μT2)I~(1+μT2)I~+(1−μT2)I~),(2(νS)I~(2−νS2)I~+(νS2)I~,2(νT)I~(2−νT2)I~+(νST2)I~)]=[((1+μS2)−(1−μS2)(1+μS2)+(1−μS2),(1+μT2)−(1−μT2)(1+μT2)+(1−μT2)),(2(νS)(2−νS2)+(νS2),2(νT)(2−νT2)+(νST2))]={(μS,μT),(νS,νT)}

i.e., I~.ϵ=Gz

Theorem 4.2. Let I~,I~1,I~2≥0, then

(1) Gz1⊕ϵGz2=Gz2⊕ϵGz1

(2) Gz1⊗ϵGz2=Gz2⊗ϵGz1

(3) I~.ϵ(Gz1⊕ϵGz2)=I~.ϵGz2⊕ϵI~.ϵGz1

(4) (Gz1⊗ϵGz2)I~=(Gz2)I~⊗ϵ(Gz1)I~

(5) I~1.ϵGz⊕I~2.ϵGz=(I~1+I~2).ϵGz

(6) (Gz)I~1⊗(Gz)I~2=(Gz)I~1+I~2

Proof. We prove the part (1), (3), and (5) and hence similar for other.

(1)

Gz1⊕Gz2=[(μS12+μS221+μS12.∈μS22,μT12μT221+μT12.∈μT22),(νS12.∈νS221+(1−νS12).∈(1−νS22),νT12.∈νT221+(1−νT12).∈(1−νT22))]=[(μS22+μS121+μS22.∈μS12,μT22μT121+μT22.∈μT12),(νS22.∈νS121+(1−νS22).∈(1−νS12),νT22.∈νT121+(1−νT22).∈(1−νT12))]=Gz2⊕Gz1

(3)

Gz1⊕ϵGz2=[(μS12+μS221+μS12.∈μS22,μT12μT221+μT12.∈μT22),(νS12.∈νS221+(1−νS12).∈(1−νS22),νT12.∈νT221+(1−νT12).∈(1−νT22))]

is equivalent to

Gz1⊕ϵGz2=[((1+μS12)⋅ϵ(1+μS22)−(1−μS12)⋅ϵ(1−μS22)(1+μS12)⋅ϵ(1+μS22)+(1−μS12)⋅ϵ(1−μS22),(1+μT12)⋅ϵ(1+μT22)−(1−μT12)⋅ϵ(1−μT22)(1+μT12)⋅ϵ(1+μST22)+(1−μT12)⋅ϵ(1−μT22),),(2νS1.∈νS2(2−νS12).∈(2−νS22)+νS12.∈νS22,2νT1.∈νT2(2−νT12).∈(2−νT22)+νT12.∈νT22)]

Take a1=(1+μS12)⋅ϵ(1+μS22), b1=(1−μS12)⋅ϵ(1−μS22), a2=(1+μT12)⋅ϵ(1+μT22), b2=(1−μT12)⋅ϵ(1−μT22), c1=νS1.∈νS2, d1=(2−νS12).∈(2−νS22), c2=νT1.∈νT2, and d2=(2−νT12).∈(2−νT22), then

Gz1⊕ϵGz2=[(a1−b1a1+b1,a2−b2a2+b2),(2c1d1+c1,2c2d2+c2)]

It arises from the Einstein Pythagorean law that

I~.ϵ(Gz1⊕ϵGz2)=I~.ϵ[(a1−b1a1+b1,a2−b2a2+b2),(2c1d1+c1,2c2d2+c2)]=[((1+a1−b1a1+b1)I~−(1−a1−b1a1+b1)I~(1+a1−b1a1+b1)I~+(1−a1−b1a1+b1)I~,(1+a2−b2a2+b2)I~−(1−a2−b2a2+b2)I~(1+a2−b2a2+b2)I~+(1−a2−b2a2+b2)I~)((2.(2c1d1+c1)I~)(2−2c1d1+c1)I~+(2c1d1+c1)I~,(2.(2c2d2+c2)I~)(2−2c2d2+c2)I~+(2c2d2+c2)I~)]=[((a1)I~−(b1)I~(a1)I~+(b1)I~,(a2)I~−(b2)I~(a2)I~+(b2)I~),(2(c1)I~(d1)I~+(c1)I~,2(c2)I~(d2)I~+(c2)I~)]=[((1+μS12)I~⋅ϵ(1+μS22)I~−(1−μS12)I~⋅ϵ(1−μS22)I~(1+μS12)I~⋅ϵ(1+μS22)I~+(1−μS12)I~⋅ϵ(1−μS22)I~,(1+μT12)I~⋅ϵ(1+μT22)I~−(1−μT12)I~⋅ϵ(1−μT22)I~(1+μT12)I~⋅ϵ(1+μST22)I~+(1−μT12)I~⋅ϵ(1−μT22)I~,),(2(νS1I~.∈νS2I~)(2−νS12)I~.∈(2−νS22)I~+(νS12)I~.∈(νS22)I~,2(νT1I~.∈νT2I~)(2−νT12)I~.∈(2−νT22)I~+(νT12)I~.∈(νT22)I~)]

On the other hand,

I~.ϵGz1=[((1+μS12)I~−(1−μS12)I~(1+μS12)I~+(1−μS12)I~,(1+μT12)I~.−(1−μT12)I~(1+μT12)I~+(1−μT12)I~,),(2(νS1I~)(2−νS12)I~+(νS12)I~,2(νT1I~)(2−νT12)I~+(νT12)I~)]=[(a3−b3a3+b3,a4−b4a4+b4),(2c3d3+c3,2c4d4+c4)]

and

I~.ϵGz2=[((1+μS22)I~−(1−μS22)I~(1+μS22)I~+(1−μS22)I~,(1+μT22)I~.−(1−μT22)I~(1+μT22)I~+(1−μT22)I~,),(2(νS2I~)(2−νS22)I~+(νS22)I~,2(νT2I~)(2−νT22)I~+(νT22)I~)]=[(a5−b5a5+b5,a6−b6a6+b6),(2c5d5+c5,2c6d6+c6)],

where a3=(1+μS12)I~, b3=(1−μS12)I~, a4=(1+μT12)I~, b4=(1−μT12)I~ c3=(νS1)I~, d3=(2−νS12)I~, c4=(νT1I~), d4=(2−νT12)I~,a5=(1+μS22)I~, b5=(1−μS22)I~, a6=(1+μT22)I~, b6=(1−μT22)I~ c5=(νS2I~), d5=(2−νS22)I~, c6=(νT2I~), d6=(2−νT22)I~. Therefore, in accordance with the operational rule of Einstein’s addition, we get

(I~.ϵGz1)⊕ϵ(I~.ϵGz2)=[(a3−b3a3+b3,a4−b4a4+b4),(2c3d3+c3,2c4d4+c4)]⊕ϵ[(a5−b5a5+b5,a6−b6a6+b6),(2c5d5+c5,2c6d6+c6)]=[(a3−b3a3+b3+a5−b5a5+b51+a3−b3a3+b3.ϵa5−b5a5+b5,a4−b4a4+b4+a6−b6a6+b61+a4−b4a4+b4.ϵa6−b6a6+b6),(2c3⋅ϵc5(d3+c3)⋅ϵ(d5+c5)1+(1−2c3d3+c3)⋅ϵ(1−2c5d5+c5),2c4⋅ϵc6(d4+c6).ϵ(d4+c6)1+(1−2c4d4+c4)⋅ϵ(1−2c6d6+c6))]=[(a3.ϵa5−b3⋅ϵb5a3⋅ϵa5+b3⋅ϵb5,a4⋅ϵa6−b4⋅ϵb6a4⋅ϵa6+b4⋅ϵb6),(2c3⋅ϵc5d3⋅ϵd5+c3⋅ϵc5,2c4⋅ϵc6d4⋅ϵd6+c4⋅ϵc6)]=[((1+μS12)I~⋅ϵ(1+μS22)I~−(1−μS12)I~⋅ϵ(1−μS22)I~(1+μS12)I~⋅ϵ(1+μS22)I~+(1−μS12)I~⋅ϵ(1−μS22)I~,(1+μT12)I~⋅ϵ(1+μT22)I~−(1−μT12)I~⋅ϵ(1−μT22)I~(1+μT12)I~⋅ϵ(1+μST22)I~+(1−μT12)I~⋅ϵ(1−μT22)I~,),(2(νS1I~.∈νS2I~)(2−νS12)I~.∈(2−νS22)I~+(νS12)I~.∈(νS22)I~,2(νT1I~.∈νT2I~)(2−νT12)I~.∈(2−νT22)I~+(νT12)I~.∈(νT22)I~)]

Hence, I.ϵ~(Gz1⊕ϵGz2)=(I.ϵ~Gz1)⊕ϵ(I.ϵ~Gz2)

(5) For I~1>0,I~2>0

I~1.ϵGz1=[((1+μS12)I~1−(1−μS12)I~1(1+μS12)I~1+(1−μS12)I~1,(1+μT12)I~1.−(1−μT12)I~1(1+μT12)I~1+(1−μT12)I~1,),(2(νS1I~1)(2−νS12)I~1+(νS12)I~,2(νT1I~1)(2−νT12)I~1+(νT12)I~1)]=[(a3−b3a3+b3,a4−b4a4+b4),(2c3d3+c3,2c4d4+c4)]I~2.ϵGz1=[((1+μS12)I~2−(1−μS12)I~2(1+μS12)I~2+(1−μS12)I~2,(1+μT12)I~2.−(1−μT12)I~2(1+μT12)I~2+(1−μT12)I~2,),(2(νS1I~2)(2−νS12)I~2+(νS12)I~,2(νT1I~2)(2−νT12)I~2+(νT12)I~2)]=[(a5−b5a5+b5,a6−b6a6+b6),(2c5d5+c5,2c6d6+c6)]

where aı`=(1+μSı`2)I~ı`, bı`=(1−μSı`2)I~ı`, cı`=(νSı`)I~ı`, dı`=(2−νSı`2)I~ı` for ı`=3,5 and aı`=(1+μTı`2)I~ı`, bı`=(1−μTı`2)I~ı`, cı`=(νTı`)I~ı`, dı`=(2−νTı`2)I~ı` for ı`=4,6.

(I~1.ϵGz1)⊕ϵ(I~2.ϵGz1)=[(a3−b3a3+b3,a4−b4a4+b4),(2c3d3+c3,2c4d4+c4)]⊕ϵ[(a5−b5a5+b5,a6−b6a6+b6),(2c5d5+c5,2c6d6+c6)]=[(a3−b3a3+b3+a5−b5a5+b51+a3−b3a3+b3⋅ϵa5−b5a5+b5,a4−b4a4+b4+a6−b6a6+b61+a4−b4a4+b4⋅ϵa6−b6a6+b6),(2c3⋅ϵc5(d3+c3)⋅ϵ(d5+c5)1+(1−2c3d3+c3)⋅ϵ(1−2c5d5+c5),2c4.ϵc6(d4+c6)⋅ϵ(d4+c6)1+(1−2c4d4+c4)⋅ϵ(1−2c6d6+c6))]

=[(a3⋅ϵa5−b3⋅ϵb5a3⋅ϵa5+b3⋅ϵb5,a4⋅ϵa6−b4⋅ϵb6a4⋅ϵa6+b4⋅ϵb6),(2c3⋅ϵc5d3⋅ϵd5+c3⋅ϵc5,2c4⋅ϵc6d4⋅ϵd6+c4⋅ϵc6)]=[((1+μS12)I~1⋅ϵ(1+μS12)I~2−(1−μS12)I~1⋅ϵ(1−μS12)I~2(1+μS12)I~1⋅ϵ(1+μS12)I~2+(1−μS12)I~1⋅ϵ(1−μS12)I~2,(1+μT12)I~1⋅ϵ(1+μT12)I~2−(1−μT12)I~1⋅ϵ(1−μT12)I~2(1+μT12)I~1⋅ϵ(1+μST12)I~2+(1−μT12)I~1⋅ϵ(1−μT12)I~2,),(2(νS1I~1⋅∈νS1I~2)(2−νS12)I~1⋅∈(2−νS12)I~2+(νS12)I~1⋅∈(νS12)I~2,2(νT1I~1⋅∈νT1I~2)(2−νT12)I~1⋅∈(2−νT12)I~2+(νT12)I~1⋅∈(νT12)I~2)]=[((1+μS12)I~1+I~2−(1−μS12)I~1+I~2(1+μS12)I~1+I~2+(1−μS12)I~1+I~2,(1+μT12)I~1+I~2−(1−μT12)I~1+I~2(1+μT12)I~1+I~2+(1−μT12)I~1+I~2,),(2(νS1)I~1+I~2(2−νS12)I~1+I~2+(νS12)I~1+I~2,2(νT1)I~1+I~2(2−νT12)I~1+I~2+(νT12)I~1+I~2)]=(I~1+I~2).ϵGz1

Hence (I~1.ϵGz1)⊕ϵ(I~2⋅ϵGz1)=(I~1+I~2).ϵGz1

Theorem 4.3. Let Gz1= {(μS1,μT1),(νS1,νT1)}, and Gz2= {(μS2,μT2),(νS2,νT2)}, be two PFZNs, then

(1) Gz1c∧ϵGz2c=(Gz1∨ϵGz2)c

(2) Gz1c∨ϵGz2c=(Gz1∧ϵGz2)c

(3) Gz1c⊕ϵGz2c=(Gz1⊗ϵGz2)c

(4) Gz1c⊗ϵGz2c=(Gz1⊕ϵGz2)c

(5) (Gz1∨ϵGz2)⊕ϵ(Gz1∧ϵGz2)=Gz1⊕ϵGz2

(6) (Gz1∨ϵGz2)⊗ϵ(Gz1∧ϵGz2)=Gz1⊗ϵGz2

Proof. It is omitted here because the proof is trivial.

Theorem 4.4. Let Gz1= {(μS1,μT1),(νS1,νT1)},Gz2= {(μS2,μT2),(νS2,νT2)},

and Gz3= {(μS3,μT3),(νS3,νT3)} be three Pythagorean fuzzy ZNs, then

(1) (Gz1∨ϵGz2)∧ϵGz3=(Gz1∧ϵGz3)∨ϵ(Gz2∧ϵGz3)

(2) (Gz1∧ϵGz2)∨ϵGz3=(Gz1∨ϵGz3)∧ϵ(Gz2∨ϵGz3)

(3) (Gz1∨ϵGz2)⊕ϵGz3=(Gz1⊕ϵGz3)∨ϵ(Gz2⊕ϵGz3)

(4) (Gz1∧ϵGz2)⊕ϵGz3=(Gz1⊕ϵGz3)∧ϵ(Gz2⊕ϵGz3)

(5) (Gz1∨ϵGz2)⊗ϵGz3=(Gz1⊗ϵGz3)∨ϵ(Gz2⊗ϵGz3)

(6) (Gz1∧ϵGz2)⊗ϵGz3=(Gz1⊗ϵGz3)∧ϵ(Gz2⊗ϵGz3)

Proof. It is omitted here because the proof is trivial.

5  Pythagorean Fuzzy Z-Numbers Einstein Weighted Aggregation Operators

The PFZNs weighted aggregating operators will be looked at in this section using Einstein operations.

Lemma 5.1. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)},I~ı`>0 for (ı`=1,2,...,n~) and ∑ı`=1n~I~ı`=1, then

Πı`=1n~(Gzı`)I~ı`≤∑ı`=1n~I~ı`Gzı`

with equality is true  ⟺ Gz1=Gz2=...=Gzn~.

Theorem 5.1. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)},(ı`=1,2,...,n~) be a collection of PFZNs, then the aggregated value is a PFZN,

PFZNEWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`,Πı`=1n~(1+μTı`2)I~ı`.−Πı`=1n~(1−μTı`2)I~ı`Πı`=1n~(1+μTı`2)I~ı`+Πı`=1n~(1−μTı`2)I~ı`,),(2Πı`=1n~(νSı`I~ı`)Πı`=1n~(2−νSı`2)I~ı`+Πı`=1n~(νSı`2)I~ı`,2Πı`=1n~(νTı`I~ı`)Πı`=1n~(2−νTı`2)I~ı`+Πı`=1n~(νTı`2)I~ı`)](3)

where I~ı` is the weight of Gzı`(ı`=1,2,...,n~) such that I~ı`∈[0,1] and ∑ı`=1n~I~ı`=1.

Proof. By applying mathematical induction on n, we conclude Eq. (3), for this take the value n=2,

PFZNEWA(Gz1,Gz2)=I~1⋅ϵGz1⊕ϵI~2⋅ϵGz2.

According to Theorem 4.2-operational law (5), we can see that both I~1⋅ϵGz1 and I~2⋅ϵGz2 are PFZNs, and the value of I~1⋅ϵGz1⊕ϵI~2⋅ϵGz2 is a PFZN.

I~1⋅ϵGz1=[((1+μS12)I~1−(1−μS12)I~1(1+μS12)I~1+(1−μS12)I~1,(1+μT12)I~1.−(1−μT12)I~1(1+μT12)I~1+(1−μT12)I~1,),(2(νS1I~1)(2−νS12)I~1+(νS12)I~,2(νT1I~1)(2−νT12)I~1+(νT12)I~1)]I~2.ϵGz2=[((1+μS22)I~2−(1−μS22)I~2(1+μS22)I~2+(1−μS22)I~2,(1+μT22)I~2.−(1−μT22)I~2(1+μT22)I~2+(1−μT22)I~2,),(2(νS2I~2)(2−νS22)I~2+(νS22)I~2,2(νT2I~2)(2−νT22)I~2+(νT22)I~2)]

⇒ ˜PFZNEWA(Gz1,Gz2)=I~1.ϵGz1⊕ϵI~2.ϵGz2=[((1+μS12)I~1−(1−μS12)I~1(1+μS12)I~1+(1−μS12)I~1+(1+μS22)I~2−(1−μS22)I~2(1+μS22)I~2+(1−μS22)I~21+((1+μS12)I~1−(1−μS12)I~1(1+μS12)I~1+(1−μS12)I~1).ϵ(1+μS22)I~2−(1−μS22)I~2(1+μS22)I~2+(1−μS22)I~2,(1+μT12)I~1.−(1−μT12)I~1(1+μT12)I~1+(1−μT12)I~1+(1+μT22)I~2.−(1−μT22)I~2(1+μT22)I~2+(1−μT22)I~21+((1+μT12)I~1.−(1−μT12)I~1(1+μT12)I~1+(1−μT12)I~1).ϵ(1+μT22)I~2.−(1−μT22)I~2(1+μT22)I~2+(1−μT22)I~2),((2(νS1I~1)(2−νS12)I~1+(νS12)I~).ϵ(2(νS2I~2)(2−νS22)I~2+(νS22)I~2)1+(2(νS1I~1)(2−νS12)I~1+(νS12)I~1).ϵ(2(νS2I~2)(2−νS22)I~2+(νS22)I~2),(2(νT1I~1)(2−νT12)I~1+(νT12)I~1).ϵ(2(νT2I~2)(2−νT22)I~2+(νT22)I~2)1+(2(νT1I~1)(2−νT12)I~1+(νT12)I~1).ϵ(2(νT2I~2)(2−νT22)I~2+(νT22)I~2))]=[((1+μS12)I~1⋅ϵ(1+μS22)I~2−(1−μS12)I~1⋅ϵ(1−μS22)I~2(1+μS12)I~1⋅ϵ(1+μS22)I~2+(1−μS12)I~1⋅ϵ(1−μS22)I~2,(1+μT12)I~1⋅ϵ(1+μT22)I~2−(1−μT12)I~1⋅ϵ(1−μT22)I~2(1+μT12)I~1⋅ϵ(1+μT22)I~2+(1−μT12)I~1⋅ϵ(1−μT22)I~2),(2((νS12)I~1⋅ϵ(νS22)I~2)(2−νS12)I~1⋅ϵ(2−νS22)I~2+(νS12)I~1⋅ϵ(νS22)I~2,2((νT12)I~1⋅ϵ(νT22)I~2)(2−νT12)I~1⋅ϵ(2−νT22)I~2+(νT12)I~1⋅ϵ(νT22)I~2)]

Therefore, the result is convincing when n is equal to 2.

Taking into account the fact that the result is accurate for n=k,

PFZNEWA(Gz1,Gz2,...,Gzk)=[((Πı`=1k1+μSı`2)I~ı`−Πı`=1k(1−μSı`2)I~ı`Πı`=1k(1+μSı`2)I~ı`+Πı`=1k(1−μSı`2)I~ı`,Πı`=1k(1+μTı`2)I~ı`.−Πı`=1k(1−μTı`2)I~ı`Πı`=1k(1+μTı`2)I~ı`+Πı`=1k(1−μTı`2)I~ı`,),(2Πı`=1k(νSı`I~ı`)Πı`=1k(2−νSı`2)I~ı`+Πı`=1k(νSı`2)I~ı`,2Πı`=1k(νTı`I~ı`)Πı`=1k(2−νTı`2)I~ı`+Πı`=1k(νTı`2)I~ı`)]

Hence, because n~=k+1, we yield

PFZNEWA(Gz1,Gz2,...,Gzk+1)=PFZNEWA(Gz1,Gz2,...,Gzk)⊕I~k+1⋅ϵGzk+1=[((Πı`=1k1+μSı`2)I~ı`−Πı`=1k(1−μSı`2)I~ı`Πı`=1k(1+μSı`2)I~ı`+Πı`=1k(1−μSı`2)I~ı`,Πı`=1k(1+μTı`2)I~ı`.−Πı`=1k(1−μTı`2)I~ı`Πı`=1k(1+μTı`2)I~ı`+Πı`=1k(1−μTı`2)I~ı`,),(2Πı`=1k(νSı`I~ı`)Πı`=1k(2−νSı`2)I~ı`+Πı`=1k(νSı`2)I~ı`,2Πı`=1k(νTı`I~ı`)Πı`=1k(2−νTı`2)I~ı`+Πı`=1k(νTı`2)I~ı`)]⊕[((1+μSk+12)I~k+1−(1−μSk+12)I~k+1(1+μSk+12)I~k+1+(1−μSk+12)I~k+1,(1+μTk+12)I~k+1.−(1−μTk+12)I~k+1(1+μTk+12)I~k+1+(1−μTk+12)I~k+1,),(2(νSk+1I~k+1)(2−νSk+12)I~k+1+(νSk+12)I~k+1,2(νTk+1I~k+1)(2−νTk+12)I~ı`+(νTk+12)I~ı`)]=[((Πı`=1k+11+μSk+12)I~k+1−Πı`=1k+1(1−μSk+12)I~k+1Πı`=1k+1(1+μSk+12)I~k+1+Πı`=1k+1(1−μSk+12)I~k+1,Πı`=1k+1(1+μTk+12)I~k+1.−Πı`=1k+1(1−μTk+12)I~k+1Πı`=1k+1(1+μTk+12)I~k+1+Πı`=1k+1(1−μTk+12)I~k+1,),(2Πı`=1k+1(νSk+1I~k+1)Πı`=1k+1(2−νSk+12)I~k+1+Πı`=1k+1(νSk+12)I~k+1,2Πı`=1k+1(νTk+1I~k+1)Πı`=1k+1(2−νTk+12)I~ı`+Πı`=1k(νTk+12)I~ı`)]

This means that n~=k+1 also satisfies Eq. (3). This implies that Eq. (3) is valid for all values of n~. The proof is completed.

Theorem 5.2. If Gzı`= {(μSı`,μTı`),(νSı`,νTı`)} ∈ PFZNs,ı`=1,2,...,n~, then the aggregated value by using the PFZNEWA operator is again a PFZN, that is, PFZNEWA(Gz1,Gz2,...,Gzn~) ∈ PFZN.

Proof. Since Gzı`= {(μSı`,μTı`),(νSı`,νTı`)}∈PFZNs, ı`=1,2,...,n~, by definition of PFZNs, we have

0≤(μSı`(℘))2+(νSı`(℘))2≤1,0≤(μTı`(℘))2+(νTı`(℘))2≤1.

Therefore,

Πı`=1n~(1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`=1−2Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≤1−Πı`=1n~(1−μSı`2)I~ı`≤1

Also (1+μSı`2)≥(1−μSı`2)⇒Πı`=1n~(1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`≥0⇒

Πı`=1n~(1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≥0.

0≤μPFZNEWA≤1

On the other hand,

2Πı`=1n~(νSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`=2Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≤Πı`=1n~(1−μSı`2)I~ı`≤1

Also,

Πı`=1n~(νSı`2)I~ı`≥0⇔2Πı`=1n~(νSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≥0

Thus,

0≤νPFZNEWA≤1

Moreover,

μPFZNEWA2+νPFZNEWA2=Πı`=1n~(1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`+2Πı`=1n~(νSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`1−2Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`+2Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≤1

Similarly, 0≤(μTı`(℘))2+(νTı`(℘))2≤1. PFZNEWA∈[0,1] follows, as a result, the PFZNs that the PFZNEWA operator aggregated are once again PFZNs.

Corollary 1. The PFZNEWA and PFZNWA operators are related to one another as shown in:

PFZNEWA(Gz1,Gz2,...,Gzn~)≤PFZNWA(Gz1,Gz2,...,Gzn~)

where Gzı`(ı`=1,2,...,n~) be a collections of PFZNs and I~=(I~1,I~2,...,I~n~)T is the weight vector of Gzı` such that I~ı`∈[0,1],(ı`=1,2,...,n~) and ∑ı`=1n~I~ı`=1.

Proof. Let PFZNEWA(Gz1,Gz2,...,Gzn~)= ⟨(μSı`P,μTı`P),(νSı`P,νTı`P)⟩=GPz and

PFZNWA(Gz1,Gz2,...,Gzn~)=⟨(μSı`,μTı`),(νSı`,νTı`)⟩=Gz. Since Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≤∑ı`=1n~I~ı`(1+μSı`2)+∑ı`=1n~I~ı`(1−μSı`2)=2, we get

(Πı`=1n~1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`≤1−Πı`=1n~(1−μSı`2)I~ı`⇔μGzp≤μGz

where equality holds if and only if μS1=μS2=....=μSn~.

2(Πı`=1n~νSı`2)I~ı`Πı`=1n~(2−νSı`2)I~ı`+(Πı`=1n~νSı`2)I~ı`≥2Πı`=1n~(νSı`2)I~ı`∑ı`=1n~I~ı`(2−νSı`2)+(∑ı`=1n~I~ı`νSı`2)≥Πı`=1n~(νSı`2)I~ı`⇒2(Πı`=1n~νSı`)I~ı`Πı`=1n~(2−νSı`2)I~ı`+(Πı`=1n~νSı`2)I~ı`≥Πı`=1n~(νSı`)I~ı`⇒νGzp≥νGz

where equality holds if and only if νS1=νS2=....=νSn~. Thus,

S(GPz)=(μGzp)2−(νGzp)2≤(μGz)2−(νGz)2=S(Gz)

If S(GPz)<S(Gz), then by the score definition, for every I~, we have

PFZEWA(Gz1,Gz2,...,Gzn~)≤PFZNWA(Gz1,Gz2,...,Gzn~)

If S(GPz)=S(Gz), that is, (μGzp)2−(νGzp)2=(μGz)2−(νGz)2, then by the condition (μGzp)≤(μGz), and (νGzp)≥(νGz), we have (μGzp)=(μGz), and (νGzp)≥(νGz); thus the accuracy function H(GPz)=μGzp+νGzp=μGz+νGz=H(Gz). Relies on the score definition in this instance, we demonstrate that:

PFZNEWA(Gz1,Gz2,...,Gzn~)=PFZNWA(Gz1,Gz2,...,Gzn~)

Hence,

PFZNEWA(Gz1,Gz2,...,Gzn~)≤PFZNWA(Gz1,Gz2,...,Gzn~)

At which the equality is valid if and only if Gz1=Gz2=...=Gzn~. The proposed PFZNEWA operator illustrates a more optimistic decision maker behavior than the PFZNWA operator proposed by Yager and Abbasov [9].

Example 2. Let Gz1=(μS1,μT1),(νS1,νT1)={(06,0.4),(0.5,0.3)}, Gz2=(μS2,μT2),(νS2,νT2)={(0.7,0.1),(0.3,0.5)}, and Gz3=(μS3,μT3),(νS3,νT3)={(0.4,0.1),(0.8,0.2)} be three PFZNs and I~=(0.2,0.5,0.3)T be the weight vector of Gzı`(ı`=1,2,3)

PFZNEWA(Gz1,Gz2,Gz3)=[((Πı`=131+μSı`2)I~ı`−Πı`=13(1−μSı`2)I~ı`Πı`=13(1+μSı`2)I~ı`+Πı`=13(1−μSı`2)I~ı`,Πı`=13(1+μTı`2)I~ı`.−Πı`=13(1−μTı`2)I~ı`Πı`=13(1+μTı`2)I~ı`+Πı`=13(1−μTı`2)I~ı`,),(2Πı`=13(νSı`I~ı`)Πı`=13(2−νSı`2)I~ı`+Πı`=13(νSı`2)I~ı`,2Πı`=13(νTı`I~ı`)Πı`=13(2−νTı`2)I~ı`+Πı`=13(νTı`2)I~ı`)]=[(0.619876,0.200639),(0.8294563,0.221921)]

If we aggregate the data employing the Yager et al. [9] developed PFZNWA operator, PFZN Gzı`(ı`=1,2,3), then we have the following results:

PFZNWA(Gz1,Gz2,Gz3)=[(1−Πı`=13(1−μS12)I~ı`,1−Πı`=13(1−μT12)I~ı`),(Πı`=13νS1I~ı`,Πı`=13νT1I~ı`)]=[(0.266926,0.09527),(0.445945,0.34294)]

Here, we have a few PFZNEWA operator properties based on Theorem 5.1.

PROPERTY: Let Gzı`=(μSı`,μTı`),(νSı`,νTı`)(ı`=1,2,...,n~) be a collections of PFZNs and I~=(I~1,I~2,...,I~n~)T is the linked weighted vector of Gzı` such that I~ı`∈[0,1],(ı`=1,2,...,n~) and ∑ı`=1n~I~ı`=1; then, we have the following:

(1) Idempotency: If Gzı`=Gz0=⟨(μS0,μT0),(νS0,νT0)⟩ for all i, then

PFZNEWA(Gz1,Gz2,...,Gzn~)=Gz0.

(2) Boundedness: Let G−zı`=⟨minı`(μSı`,μTı`),maxı`(νSı`,νTı`)⟩, G+zı`=⟨maxı`(μSı`,μTı`)minı`(νSı`,νTı`)⟩ then

G−zı`≤PFZNEWA(Gz1,Gz2,...,Gzn~)≤G+zı`.

(3) Monotonicity: Gzı`≤Dzı`, for all i, then

PFZNEWA(Gz1,Gz2,...,Gzn~)≤PFZNEWA(Dz1,Dz2,...,Dzn~).

Proof. (1) AS Gzı`=⟨(μS0,μT0),(νS0,νT0)⟩∈PFZNs for all i, then

PFZNEWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μS02)I~ı`−Πı`=1n~(1−μS02)I~ı`Πı`=1n~(1+μS02)I~ı`+Πı`=1n~(1−μS02)I~ı`,Πı`=1n~(1+μT02)I~ı`.−Πı`=1n~(1−μT02)I~ı`Πı`=1n~(1+μT02)I~ı`+Πı`=1n~(1−μT02)I~ı`,),(2Πı`=1n~(νS0I~ı`)Πı`=1n~(2−νS02)I~ı`+Πı`=1n~(νS02)I~ı`,2Πı`=1n~(νTı`I~ı`)Πı`=1n~(2−νT02)I~ı`+Πı`=1n~(νT02)I~ı`)]=[((1+μS02)∑ı`=1n~I~ı`−(1−μS02)∑ı`=1n~I~ı`(1+μS02)∑ı`=1n~I~ı`+(1−μS02)∑ı`=1n~I~ı`,(1+μT02)∑ı`=1n~I~ı`.−(1−μT02)∑ı`=1n~I~ı`(1+μT02)∑ı`=1n~I~ı`+(1−μT02)∑ı`=1n~I~ı`,),(2(νS0)∑ı`=1n~I~ı`(2−νS02)∑ı`=1n~I~ı`+(νS02)∑ı`=1n~I~ı`,2(νTı`)∑ı`=1n~I~ı`(2−νT02)∑ı`=1n~I~ı`+(νT02)∑ı`=1n~I~ı`)]=⟨(μS0,μT0),(νS0,νT0)⟩=Gz0

(2) Let f(℘)=1−℘1+℘,℘∈[0,1], then f′(c)=−2(1+℘)2<0; f(℘) is decreasing function. Since μSı`,min2≤μSı`2≤μSı`,max2, for all ı`=1,2,...,n~, then f(μSı`,max2)≤f(μSı`2)≤f(μSı`,min2) for all i, that is, 1−μSı`,max21+μSı`,max2≤1−μSı`21+μSı`2≤1−μSı`,min21+μSı`,min2, for all i. Let I~=(I~1,I~2,...,I~n~)T is the associated weighted vector of Gzı` such that I~ı`∈[0,1],(ı`=1,2,...,n~) and ∑ı`=1n~I~ı`=1; then for all i, we have

(1−μSı`,max21+μSı`,max2)I~ı`≤(1−μSı`21+μSı`2)I~ı`≤(1−μSı`,min21+μSı`,min2)I~ı`

Thus

Πı`=1n~(1−μSı`,max21+μSı`,max2)I~ı`≤Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤Πı`=1n~(1−μSı`,min21+μSı`,min2)I~ı`⇔(1−μSı`,max21+μSı`,max2)∑ı`=1n~I~ı`≤Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤(1−μSı`,min21+μSı`,min2)∑ı`=1n~I~ı`⇔(1−μSı`,max21+μSı`,max2)≤Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤(1−μSı`,min21+μSı`,min2)⇔(21+μSı`,max2)≤1+Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤(21+μSı`,min2)⇔(1+μSı`,min22)≤11+Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤(1+μSı`,max22)⇔1+μSı`,min2≤21+Πı`=1n~(1−μSı`21+μSı`2)I~ı`≤1+μSı`,max2⇔μSı`,min2≤21+Πı`=1n~(1−μSı`21+μSı`2)I~ı`−1≤μSı`,max2⇔μSı`,min2≤(Πı`=1n~1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`−1≤μSı`,max2⇔μSı`,min2≤(Πı`=1n~1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`−1≤μSı`,max2

Similarly,

μTı`,min2≤(Πı`=1n~1+μTı`2)I~ı`−Πı`=1n~(1−μTı`2)I~ı`Πı`=1n~(1+μTı`2)I~ı`+Πı`=1n~(1−μTı`2)I~ı`−1≤μTı`,max2

On the other hand, let g(y)=2−yy, y∈(0,1]; then g′(y)=−2y2<0, i.e., g(y) is a decreasing function on (0,1]. Since νSı`,min2≤νSı`2≤νSı`,max2, for all ı`=1,2,...,n~, then g(νSı`,max2)≤g(νSı`2)≤g(νSı`,min2) for all i, that is, 2−νSı`,max2νSı`,max2≤2−νSı`2νSı`2≤2−νSı`,min2νSı`,min2. Then, we have

(2−νSı`,max2νSı`,max2)I~ı`≤(2−νSı`2νSı`2)I~ı`≤(2−νSı`,min2νSı`,min2)I~ı`

Thus,

Πı`=1n~(2−νSı`,max2νSı`,max2)I~ı`≤Πı`=1n~(2−νSı`2νSı`2)I~ı`≤Πı`=1n~(2−νSı`,min2νSı`,min2)I~ı`⟹2−νSı`,max2νSı`,max2≤Πı`=1n~(2−νSı`2νSı`2)I~ı`≤2−νSı`,min2νSı`,min2⟹2νSı`,max2≤Πı`=1n~(2−νSı`2νSı`2)I~ı`+1≤2νSı`,min2⟹νSı`,min22≤1Πı`=1n~(2−νSı`2νSı`2)I~ı`+1≤νSı`,max22⟹νSı`,min2≤2Πı`=1n~(2−νSı`2νSı`2)I~ı`+1≤νSı`,max2⟹νSı`,min2≤2Πı`=1n~(νSı`)I~ı`Πı`=1n~(2−νSı`2)I~ı`+Πı`=1n~(νSı`2)I~ı`≤νSı`,max2

i.e,

⟹νSı`,min≤2Πı`=1n~(νSı`)I~ı`Πı`=1n~(2−νSı`2)I~ı`+Πı`=1n~(νSı`2)I~ı`≤νSı`,max

Similarly,

νTı`,min≤2Πı`=1n~(νTı`)I~ı`Πı`=1n~(2−νTı`2)I~ı`+Πı`=1n~(νTı`2)I~ı`≤νSı`,max

Let PFZNEWA(Gz1,Gz2,...,Gzn~)=⟨(μGZ,μGZ),(νGZ,νGZ)⟩, then we have the following results, respectively;

μSı`,min2≤μGZ≤μSı`,max2, μTı`,min2≤μGZ≤μTı`,max2,

and

νSı`,min2≤νGZ≤νSı`,max2, νTı`,min2≤νGZ≤νTı`,max2.

where μSı`,min=minı`(μSı`),μSı`,max=maxı`(μSı`),μTı`,min=minı`(μTı`),μTı`,max=maxı`(μTı`),νSı`,min=minı`(νSı`),νSı`,max=maxı`(νSı`), and νSı`,min=minı`(νSı`),νSı`,max=maxı`(νSı`). So, S(GZ)=μGZ2−νGZ2=μmax2−νmin2=S(GZ+) and S(GZ)=μGZ2−νGZ2=μmin2−νmax2=S(GZ−). If S(GZ)<S(GZ+) and S(GZ)>S(GZ−), then by order relation between two PFZNs, we have

GZ−≤PFZNEWA(Gz1,Gz2,...,Gzn~)≤GZ+

the proof (3) is similar as (2).

6  Einstein Ordered Weighted Averaging Operator, Einstein Weighted Geometric Averaging, and Order Weighted Geometric Averaging Aggregating Operators under Pythagorean Fuzzy Z-Numbers

This part is divided into two subsections: in Subsection 6.1 we offered the PFZNEOWA operator, and in Subsection 6.2 we established the PFZNEWGA and PFZNEOWGA.

6.1 Pythagorean Fuzzy Z-Number Einstein Ordered Weighted Averaging Operator

In this part, we present the PFZNEOWA operator, which combines the OWA concept with the PFZNEWA operator. The OWA concept is not taken into consideration by the PFZNEWA operator throughout the information fusion process. The PFZNEOWA operator is discussed in the sections that follow, first with a brief introduction and then with a numerical example.

Definition 6.1. Let Gzı`={(μSαı`,μTαı`)(νSαı`,νTαı`)}(ı`=1,2,...,n~) be a collection of PFZNs, and I~=(ı`=1,2,...,n~)T is the associated weighted vector of Gzı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1; then the PFZN Einstein order weighted averaging aggregation operator of dimension n~ is a mapping PFZNEOWA:ϖn~⟶ϖ, and

PFZNEOWA(Gz1,Gz2,...,Gzn~)=I~1⋅ϵGzσ(1)⊕ϵI~2⋅ϵGzσ(2)⊕ϵ...⊕ϵI~n~⋅ϵGzσ(n~)

where ϖ is the set of all PFZNs,(σ(1),σ(2),...,σ(n~)) is a permutation of (ı`=1,2,...,n~) such that Gzσ(ı`−1)≥Gzσ(ı`) for all i.

The PFZNEOWA operator can be changed into the following form employing the PFZNs’ Einstein operational rules.

Theorem 6.1. Let Gzı`={(μSαı`,μTαı`)(νSαı`,νTαı`)}(ı`=1,2,...,n~) be a collection of PFZNs, then their aggregated value by employing the PFZNEOWA operator is also a PFZN and

PFZNEOWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μSσ(ı`)2)I~ı`−Πı`=1n~(1−μSσ(ı`)2)I~ı`Πı`=1n~(1+μSσ(ı`)2)I~ı`+Πı`=1n~(1−μSσ(ı`)2)I~ı`,Πı`=1n~(1+μTσ(ı`)2)I~ı`.−Πı`=1n~(1−μTσ(ı`)2)I~ı`Πı`=1n~(1+μTσ(ı`)2)I~ı`+Πı`=1n~(1−μTσ(ı`)2)I~ı`,),(2Πı`=1n~(νSσ(ı`)I~ı`)Πı`=1n~(2−νSσ(ı`)2)I~ı`+Πı`=1n~(νSσ(ı`)2)I~ı`,2Πı`=1n~(νTσ(ı`)I~ı`)Πı`=1n~(2−νTσ(ı`)2)I~ı`+Πı`=1n~(νTσ(ı`)2)I~ı`)](4)

(σ(1),σ(2),...,σ(n~)) is a permutation of (ı`=1,2,...,n~) such that Gzσ(ı`−1)≥Gzσ(ı`) for all i, I~=(ı`=1,2,...,n~)T is the associated weighted vector of Gzı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Proof. This theorem is included in this discussion because the evidence for it is quite similar to the proof for Theorem 5.1.

Corollary 2. The PFZNEOWA operator and PFZNOWA operator has the underlying links:

PFZNEOWA(Gz1,Gz2,...,Gzn~)≤PFZNOWA(Gz1,Gz2,...,Gzn~)

where Gzı`={(μSαı`,μTαı`)(νSαı`,νTαı`)}(ı`=1,2,...,n~) be a collection of PFZNs, and I~=(ı`=1,2,...,n~)T is the associated weighted vector of Gzı`(ı`=1,2,...,n~) with 0≤I~ı`≤1 and ∑ı`=1n~I~ı`=1.

Proof. The concept of verification is quite similar to that of Corollary 1.

Example 3. Let Gz1=(μS1,μT1),(νS1,νT1)={(06,0.4),(0.5,0.3)}, Gz2=(μS2,μT2),(νS2,νT2)={(0.7,0.1),(0.3,0.5)}, and Gz3=(μS3,μT3),(νS3,νT3)={(0.4,0.1),(0.8,0.2)} be three PFZNs and I~=(0.2,0.5,0.3)T be the weight vector of Gzı`(ı`=1,2,3). Then

PFZNEOWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μSσ(ı`)2)I~ı`−Πı`=1n~(1−μSσ(ı`)2)I~ı`Πı`=1n~(1+μSσ(ı`)2)I~ı`+Πı`=1n~(1−μSσ(ı`)2)I~ı`,Πı`=1n~(1+μTσ(ı`)2)I~ı`.−Πı`=1n~(1−μTσ(ı`)2)I~ı`Πı`=1n~(1+μTσ(ı`)2)I~ı`+Πı`=1n~(1−μTσ(ı`)2)I~ı`,),(2Πı`=1n~(νSσ(ı`)I~ı`)Πı`=1n~(2−νSσ(ı`)2)I~ı`+Πı`=1n~(νSσ(ı`)2)I~ı`,2Πı`=1n~(νTσ(ı`)I~ı`)Πı`=1n~(2−νTσ(ı`)2)I~ı`+Πı`=1n~(νTσ(ı`)2)I~ı`)]=[(0.610681,0.200639),(0.294563,0.221921)]

If we employ the PFZNOWA operator to aggregate the PFZNs Gzı`s, then we obtain

PFZNOWA(α1,α2,...αn~)={(1−Πı`=1n~(1−μSασ(ı`)2)I~ı`,1−Πı`=1n~(1−μTασ(ı`)2)I~ı`),(Πı`=1n~νSασ(ı`)I~ı`,Πı`=1n~νTασ(ı`)I~ı`)}={(0.266926,0.09527),(0.445945,0.34294)}

Similarly the PFZNEWA operator, the PFZNEOWA operator has some properties as follows.

Proposition 6.1. Let Gzı`=(μSı`,μTı`),(νSı`,νTı`)(ı`=1,2,...,n~) be a collections of PFZNs and I~=(I~1,I~2,...,I~n~)T is the associated weighted vector of Gzı` such that I~ı`∈[0,1],(ı`=1,2,...,n~) and ∑ı`=1n~I~ı`=1; then, we have the following:

(1) Idempotency: If all Gzı`(ı`=1,2,...,n~) are equal that is Gzı`=Gz for all i, then

PFZNEOWA(Gz1,Gz2,...,Gzn~)=Gz

(2) Boundedness: Let G−zı`=⟨minı`(μSı`,μTı`),maxı`(νSı`,νTı`)⟩, G+zı`=⟨maxı`(μSı`,μTı`)minı`(νSı`,νTı`)⟩ then

G−zı`≤PFZNEOWA(Gz1,Gz2,...,Gzn~)≤G+zı`

(3) Monotonicity: Let Gzı`=(μSGı`,μTGı`),(νSGı`,νTGı`) and Dzı`=(μSDı`,μTDı`),(νSDı`,νTDı`) be collection of PFZNs and Gzı`≤Dzı`, for all i, then

PFZNEWA(Gz1,Gz2,...,Gzn~)≤PFZNEWA(Dz1,Dz2,...,Dzn~)

6.2 Pythagorean Fuzzy Z-Numbers Einstein Weighted Geometric and Order Weighted Geometric Aggregating Operators

With the use of Einstein operations, we will explore the PFZNs geometric and order geometric aggregating operators in this section.

Definition 6.2. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)},(ı`=1,2,...,n~) be a catalogue of PFZNs and I~ı` is the weight of I~ı`(ı`=1,2,...,n~) such that I~ı`∈[0,1] and ∑ı`=1n~I~ı`=1 then, a PFZNEGA operator is a mapping of dimension n PFZNEGA:ϖn~→ϖ, and

PFZNEGA(Gz1,Gz2,...,Gzn~)=Πı`=1n~GI~ı`zı`

where ϖ is the collection of all PFZNs.

Theorem 6.2. Let Gzı`= {(μSı`,μTı`),(νSı`,νTı`)},(ı`=1,2,...,n~) be a catalogue of PFZNs, then the aggregated value are determined by

PFZNEGA(Gz1,Gz2,...,Gzn~)=[(2Πı`=1n~(μSı`I~ı`)Πı`=1n~(2−μSı`2)I~ı`+Πı`=1n~(μSı`2)I~ı`,2Πı`=1n~(μTı`I~ı`)Πı`=1n~(2−μTı`2)I~ı`+Πı`=1n~(μTı`2)I~ı`),((Πı`=1n~1+νSı`2)I~ı`−Πı`=1n~(1−νSı`2)I~ı`Πı`=1n~(1+νSı`2)I~ı`+Πı`=1n~(1−νSı`2)I~ı`,Πı`=1n~(1+νTı`2)I~ı`.−Πı`=1n~(1−νTı`2)I~ı`Πı`=1n~(1+νTı`2)I~ı`+Πı`=1n~(1−νTı`2)I~ı`,)](5)

where I~ı` is the weight of Gzı`(ı`=1,2,...,n~) such that I~ı`∈[0,1] and ∑ı`=1n~I~ı`=1.

Proof. Verification is trivial, therefore it is valid here.

Theorem 6.3. Let Gzı`={(μSı`,μTı`),(νSı`,νTı`)},(ı`=1,2,...,n~) be a catalogue of PFZNs, then the aggregated value are determined by

PFZNEOGA(Gz1,Gz2,...,Gzn~)=[(2Πı`=1n~(μSσ(ı`)I~ı`)Πı`=1n~(2−μSσ(ı`)2)I~ı`+Πı`=1n~(μSσ(ı`)2)I~ı`,2Πı`=1n~(μTσ(ı`)I~ı`)Πı`=1n~(2−μTσ(ı`)2)I~ı`+Πı`=1n~(μTσ(ı`)2)I~ı`),((Πı`=1n~1+νSσ(ı`)2)I~ı`−Πı`=1n~(1−νSσ(ı`)2)I~ı`Πı`=1n~(1+νSσ(ı`)2)I~ı`+Πı`=1n~(1−νSσ(ı`)2)I~ı`,Πı`=1n~(1+νTσ(ı`)2)I~ı`.−Πı`=1n~(1−νTσ(ı`)2)I~ı`Πı`=1n~(1+νTσ(ı`)2)I~ı`+Πı`=1n~(1−νTσ(ı`)2)I~ı`,)](6)

where I~ı` is the weight of Gzı`(ı`=1,2,...,n~) such that I~ı`∈[0,1] and ∑ı`=1n~I~ı`=1.

Proof. It is omitted here because the proof is trivial.

7  Multiple Aggregated Operators with Example

We developed MCDM for the PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA aggregating operators. An example is given to demonstrate the usefulness of these aggregating operators, and a comparison of the these aggregating operators is included as well.

7.1 Multi-Criteria Decision-Making Approach Using the PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA Operators

MCDM is a method used to make decisions in situations where there are multiple conflicting criteria. The PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators are four types of fuzzy aggregation operators that can be used in MCDM. To use these operators in MCDM, the decision maker needs to first define the set of criteria, which can be represented as a set of Pythagorean fuzzy Z-numbers. The PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators can then be used to aggregate the criteria and produce a final decision. The choice of which operator to use depends on the decision-maker’s preferences and the characteristics of the criteria. In order to handle MCDM difficulties, this part develops an MCDM methodology using assessment data for both Pythagorean values and Pythagorean reliability measures. This approach relates to PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators and the score function. The weight I~ı`, are define as the weight vector I~ı`=(I~1,I~2,...,I~n~). Consider PFZNs Gzjd={(μSjd,μTjd),(νSjd,νTjd)}(jd=1,2,...,n~), where μSjd,μTjd∈[0,1] and νSjd,νTjd∈[0,1]. Now the decision matrix of PFZN is determined as Gzjd=(Gzjd)r×m. The decision process is defined in MCDM problem as following:

Step 1: Using PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators, we defined the PFZNs in Equations, i.e., Eqs. (3)–(6), respectively.

Step 2: Using Eq. (1), the score values of J(Gzi)(i=1,2,...,n) are calculated.

Step 3: The best option among the rated options is chosen based on the score values.

Step 4: End.

7.2 Numerical Illustration

To illustrate the relevance and efficacy of the suggested MCDM technique with PZN information, this section gives an example concerning the challenge of agriculture fields such as productivity profitability and energy use efficiency of Wheat crop in different tillage system. Energy inputs estimations were relies on the human labor requirement, use of different types of machinery and quantity of materials, energy calculation was completed using different input and output energy equivalent. Energy efficiency in agricultural production is becoming more popular due to rising fuel prices. The agriculture sector, like other sectors, relied on energy sources like electricity and fossil fuels to produce more food than the growing population demanded. The expert panel provides a set of 5 alternatives Q={Q1,Q2,Q3,Q4,Q5} from potential agriculture partner. The three criteria’s weight vector is written as (0.2,0.5,0.3) to denote their relative importance.

Seed energy: was calculated by the multiplication of quantity (Qs) of seed (kg ha−1) used at the time of sowing with the amount of energy stored (SE) in each seed unit.

Total fuel energy: was quantify using volumetric method such as tillage, sowing, harvesting and threshing operations it was calculated by the multiplication of quantity of diesel (Qd) with the energy present (TE) in each liter.

Human labor energy: is required for individually in each tillage, fertilizer, irrigation, chemicals, harvesting and threshing operations during crop growth period. It was calculated by multiplication of total hours (Hd) per day needed with the human energy used (HE) per hour.

Irrigation energy: which applied to field during crop growth period measured with water flow meter. It was calculated by multiplication of total quantity of water (Qw) to field with the water energy (IE) use during irrigations As a result, PFZN decision matrix can be used to create all PFZNs in Table 1.

By using PFZNEWA operator: Now, we can apply PFZNEWA on Table 1.

Step 1: To find PFZNs GZı`(ı`=1,2,3,4,5) using Eq. (3) is defined as

Gz1={(0.61068082,0.200638579),(0.294562982,0.221921072)},Gz2={(0.332111451,0.452762799),(0.217490322,0.12872272)},Gz3={(0.468903966,0.478663074),(0.350640262,0.115056262)},Gz4={(0.45416745,0.531936912),(0.198209476,0.116587122)},Gz5={(0.525153511,0.292377101),(0.306654337,0.130166267)}.

The calculation step for Table 1 PFZNEWA operator for the first step Gz1 can be calculated as

i.e.,

Πı`=13(1+μSı`2)I~ı`=(1.36).2×(1.49).5×(1.16).3=1.357182692Πı`=13(1−μSı`2)I~ı`=(0.64).2×(0.51).5×(0.84).3=0.619876067Πı`=13(1+μTı`2)I~ı`=(1.16).2×(1.01).5×(1.01).3=1.038361786Πı`=13(1+μTı`2)I~ı`=(0.84).2×(0.99).5×(0.99).3=0.9579967Πı`=13(νSı`2)I~ı`=(0.5).2×(0.3).5×(0.8).3=0.445945334Πı`=13(2−νSı`2)I~ı`=(1.75).2×(1.91).5×(1.36).3=1.695063475.Πı`=13(νTı`2)I~ı`=(0.3).2×(0.5).5×(0.2).3=0.342940086Πı`=13(2−νTı`2)I~ı`=(1.91).2×(1.75).5×(1.96).3=1.842478884

Thus

PFZNEWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μSı`2)I~ı`−Πı`=1n~(1−μSı`2)I~ı`Πı`=1n~(1+μSı`2)I~ı`+Πı`=1n~(1−μSı`2)I~ı`,Πı`=1n~(1+μTı`2)I~ı`.−Πı`=1n~(1−μTı`2)I~ı`Πı`=1n~(1+μTı`2)I~ı`+Πı`=1n~(1−μTı`2)I~ı`,),(2Πı`=1n~(νSı`I~ı`)Πı`=1n~(2−νSı`2)I~ı`+Πı`=1n~(νSı`2)I~ı`,2Πı`=1n~(νTı`I~ı`)Πı`=1n~(2−νTı`2)I~ı`+Πı`=1n~(νTı`2)I~ı`)],

={(0.61068082,0.200638579),(0.294562982,0.221921072)}.

Step 2: The score values J(Gzd) of PFZNEWA for the alternatives Qj={1,2,3,4,5} are given below:

J(Gz1)=0.5285782, J(Gz2)=0.561185882, J(Gz3)=0.592051828,J(Gz4)=0.609239879, J(Gz5)=0.556813405.

Step 3: According to the score values J(Gz4)≥J(Gz3)≥J(Gz2)≥J(Gz5)≥J(Gz1), the five alternatives are ranked as Q4≥Q3≥Q2≥Q5≥Q1. Hence the best supplier is Q4.

Here we first order the given matrix with the help of score function then the original matrix becomes: By using PFZNEOWA operator: Now, we can apply PFZNEOWA on Table 1.

Step 1: To find PFZNs GZı`(ı`=1,2,3,4,5) using Eq. (3) is defined as:

Gz1={(0.61068082,0.200638579),(0.294562982,0.221921072)},Gz2={(0.339494347,0.392551733),(0.245832943,0.135896766)},Gz3={(0.405806341,0.524471713),(0.350640262,0.115056262)},Gz4={(0.45416745,0.531936912),(0.198209476,0.116587122)},Gz5={(0.45416745,0.339784731),(0.179706884,0.18252723)}.

Step 2: The score values J(Gzd) of PFZNEOWA for the alternatives Qj={1,2,3,4,5} are given below:

J(Gz1)=0.5285782,J(Gz2)=0.549930596,J(Gz3)=0.586245294,J(Gz4)=0.609239879,J(Gz5)=0.560758883.

Step 3: According to the score values J(Gz4)≥J(Gz3)≥J(Gz5)≥J(Gz2)≥J(Gz1), the five alternatives are ranked as Q4≥Q3≥Q5≥Q2≥Q1. Hence the best supplier is Q4. Now we can apply PFZNEWGA, in MCDM problem can be solved using the invented MCDM strategy using the PFZNEWGA operator, which is illustrated by using the accompanying decision-making procedure:

By using PFZNEWGA operator: Now, we can apply PFZNEWGA on Table 1.

Step 1: The total number of PFZNGzj(j=1,2,3,4,5) are obtained as follows:

Gz1={(0.368418396,0.089242623),(0.559293884,0.39555748)},Gz2={(0.197536032,0.208601424),(0.469106778,0.288036181)},Gz3={(0.254931725,0.171976027),(0.567378715,0.184401829)},Gz4={(0.262715535,0.312701971),(0.434303643,0.247675851)}.Gz5={(0.31523819,0.133824743),(0.604477583,00.271952294)}.

Step 2: The score values J(Gzd) of PFZNEWGA for the corresponding Qı`={1,2,3,4,5} are given below:

J(Gz1)=0.405822872,J(Gz2)=0.453043286,J(Gz3)=0.469608236,J(Gz4)=0.487292571,J(Gz5)=0.438898802.

Step 3: According to the score values J(Gz4)≥J(Gz3)≥J(Gz2)≥J(Gz5)≥J(Gz1), the five alternatives are ranked as Q4≥Q3≥Q2≥Q5≥Q1. Hence the best supplier is Q4. Now we can apply PFZNEOWGA, in MCDM problem can be solved using the invented MCDM strategy using the PFZNEOWGA operator, which is illustrated by using the accompanying decision-making procedure: By using PFZNEOWGA operator: Now, we can apply PFZNEOWGA on Table 1.

Step 1: The total number of PFZN Gzj(j=1,2,3,4,5) are obtained as follows:

Gz1={(0.368418396,0.089242623),(0.499927467,0.39555748)},Gz2={(0.204980379,0.17209745),(0.494956732,0.315188962)},Gz3={(0.221479946,0.231744777),(0.534224391,0.184401829)},Gz4={(0.262715535,0.312701971),(0.456694776,0.247675851)}.Gz5={(0.262715535,0.173775546),(0.430236559,0.375195664)}.

Step 2: The score values J(Gzd) of PFZNEOWGA for the alternatives Qj={1,2,3,4,5} are given below:

J(Gz1)=0.417564287,J(Gz2)=0.439635851,J(Gz3)=0.476407433,J(Gz4)=0.484519699,J(Gz5)=0.442115322.

Step 3: According to the score values J(Gz4)≥J(Gz3)≥J(Gz5)≥J(Gz2)≥J(Gz1), the five alternatives are ranked as Q4≥Q3≥Q5≥Q2≥Q1. Hence the best supplier is Q4. The four sorts of ranking orders indicated above for the five possibilities and the best option are the same, according to the developed MCDM technique, which uses the PFZNEWA, PFZNEOWA, PFZNEWGA, and PFZNEOWGA operators as well as the score function. The established MCDM technique thus functions.

8  Improved EDAS Method Based on Pythagorean Fuzzy Z-Number Einstein Aggregation Operators

MCDM is beneficial in locating solutions to a wide range of decision-making challenges that arise in the real world. Evaluation based on the distance from the average solution (EDAS) is an innovative and viable tool for MCDM tactic. The needed concerns, such as the average response being determined using weighted mean in this manner, may be effectively addressed by the EDAS technique. For the purpose of assessing the efficacy of the PFZN weighted geometric AOs, an original extended EDAS strategy is created to handle the complicated uncertain data in real-world DS scenarios. Assume there are a number of “alternatives” {Q1,Q2,...,Ql}, and a satisfactory rating {R1,R2,...,Rm} for each. Then, I~d=(I~1,I~2,...,I~m)T specifies the usefulness of various characteristics Rd(d=1,2,...,m), such that I~d>0 and Σd=1mI~d=1. Let Gzjd={(USjd,UTjd), (VSjd,VTjd)} where 0≤(USjd)2+(UTjd)2≤1, 0≤(VSjd)2+(VTjd)2≤1 be the permissible rating for each attribute for each option.

Step 1: Select a set of qualities that can be used to evaluate the problem: Prospective assessment characteristics are acquired by a review of the literature, and an expert decision-making committee is established to filter the characteristics for the purpose to generate a legitimate set of evaluation criteria. Rd(d=1,2,...,m).

Gzjd=(R1R2...Rm((US11,UT11),(VS11,(VT11))((US12,UT12),(VS12,VT12))...((US1m,UT1m),(VS1m,VT1m))((US21,UT21),(VS21,(VT21))((US22,UT22),(VS22,VT22))...((US2m,UT2m),(VS2m,VT2m))............((USr1,UTr1),(VSr1,(VTr1))((USr2,UTr2),(VSr2,VTr2))...((USrm,UTrm),(VSrm,VTrm)))

Step 2: Using normalization, the following steps are taken to construct the normalized decision matrix:

Gzjd={((USjd,UTjd), (VSjd,VTjd)),ifCI,((VSjd,VTjd),(USjd,UTjd)),ifCII.

if Rd(d=1,2,...,m) is a benefit criterion, the statement use CI, if Rd(d=1,2,...,m) is a cost criterion, the statement CII use.

Step 3: Aggregated Data: The skilled uncertain data of required situations are aggregated using established PFZNWGA operators.

PFZEGA(Gz1,Gz2,...,Gzn~)=Πı`=1n~GI~ı`zı`=[(2Πı`=1n~(μSı`I~ı`)Πı`=1n~(2−μSı`2)I~ı`+Πı`=1n~(μSı`2)I~ı`,2Πı`=1n~(μTı`I~ı`)Πı`=1n~(2−μTı`2)I~ı`+Πı`=1n~(μTı`2)I~ı`),((Πı`=1n~1+νSı`2)I~ı`−Πı`=1n~(1−νSı`2)I~ı`Πı`=1n~(1+νSı`2)I~ı`+Πı`=1n~(1−νSı`2)I~ı`,Πı`=1n~(1+νTı`2)I~ı`.−Πı`=1n~(1−νTı`2)I~ı`Πı`=1n~(1+νTı`2)I~ı`+Πı`=1n~(1−νTı`2)I~ı`).]

Step 4: Verify the average solution (AVS), which is relies on all the criteria given. AVS=[AVSd]1×m={Σj=1n´Gzjdn´}1×m Using Definition 3, we obtain AVS=[AVSd]1×m={Σj=1n´Gzjdn´}1×m

PFZNEWA(Gz1,Gz2,...,Gzn~)=[((Πı`=1n~1+μS02)I~ı`−Πı`=1n~(1−μS02)I~ı`Πı`=1n~(1+μS02)I~ı`+Πı`=1n~(1−μS02)I~ı`,Πı`=1n~(1+μT02)I~ı`.−Πı`=1n~(1−μT02)I~ı`Πı`=1n~(1+μT02)I~ı`+Πı`=1n~(1−μT02)I~ı`,),(2Πı`=1n~(νS0I~ı`)Πı`=1n~(2−νS02)I~ı`+Πı`=1n~(νS02)I~ı`,2Πı`=1n~(νTı`I~ı`)Πı`=1n~(2−νT02)I~ı`+Πı`=1n~(νT02)I~ı`)]

Step 5: The positive (PDAv) and negative (NDAv) distances from average must be determined from the AVS values:

PDAv=max(0,(Gzjd−AVS))AVS, and NDAv=max(0,(AVS−Gzjd))AVS.

The score function of PFZNs specified in Definition 3 can be employed as follows to assess the PDA and NDA:

PDAv=max(0,(J(Gzjd)−J(AVS))J(AVS), and n´DAv=max(0,(J(AVS)−J(Gzjd))J(AVS).

where W shows the score value.

Step 6: Calculate SPDA and SNDA, which represent for PDA and NDA′s weighted average, respectively:

SPDA=∑d=1mI~dPDAd, SNDA=∑d=1mI~dNDAd, I~d∈[0,1] and Σd=1mI~d=1.

Step 7: Normalize weighted sum of PDA and NDA is defined as, respectively

NSPDA=SPDAmax(SPDA), and NSPDA=SNDAmax(SNDA)

Step 8: Compute the values of ASC depends on each alternative’s as

ASC=12((NSPDA+1−NSNDA).

Step 9: Depending on the ASC calculations, alternatives are sorted in decreasing order, and the higher the ASC number, the better options will be.

The flow chart of EDAS method is given in Fig. 1.

Figure 1: Flow chart of EDAS method

8.1 Numerical Illustration by Using EDAS Method

Step 1: The expert decision information is given in Table 2.

Step 2: The normalized decision matrix is created using normalization as follows:

Gzjd={((USjd,UTjd),(VSjd,VTjd)),ifCI,((VSjd,VTjd),(USjd,UTjd)),ifCII.

if Rd(d=1,2,...,m) is a benefit criterion, the statement use CI, if Rd(d=1,2,...,m) is a cost criterion, the statement CII use. Here the given system is benefit type, so we not need to normalized. Step 3: Now that PFZNWGA is available, we can use it to address MCDM problems using the newly created MCDM method, as shown by the following decision-making procedure. The overall collected PFZN Gzj(j=1,2,3,4,5) are obtained as follows:

Gz1={(0.353445079,0.107280055),(0.593185747,0.357392481)},Gz2={(0.186477202,0.189131059),(0.424938759,0.343341288)},Gz3={(0.216371437,0.23958899),(0.5437384,0.173133908)},Gz4={(0.286909561,0.306199937),(0.480953768,0.265109265)},Gz5={(0.286909561,0.166765999),(0.545908753,0.318371496)}.

Step 4: The score values J(Gzd) of PFZNEWGA for the alternatives Qj={1,2,3,4,5} are given below:

J(Gz1)=0.412958741,J(Gz2)=0.444684805,J(Gz3)=0.47885033,J(Gz4)=0.480173195,J(Gz5)=0.437022487.

And verify the average solution (AVS) as

AV(Gz1)=0.266022568,AV(Gz2)=0.201793208,AV(Gz3)=0.517745085,AV(Gz4)=0.291469687.

Score function of average solution (AVS) we have

J[AV(Gz)]=0.451387275

Step 5: Using the AVS values, the positive distance from average (PDAv) and the negative distance from average (NDAv) is calculated in Tables 3 and 4, respectively.

By using the score function of PFZNs mentioned in Definition 3, the PDA and NDA is calculated in Tables 5 and 6, respectively:

Step 6: Determined SPDA and SNDA in Tables 7 and 8, respectively, which reflect the weighted average for PDA and NDA’s, and attributes weighting vector I~=(0.333,0.333,0.333,0.333,0.333), we may acquire the results as follows:

Step 7: The total PDA and NDA is presented in Tables 9 and 10, respectively, normalized by weight are characterized as:

Step 8: Values of the ASC are determined in Table 11:

Step 9: Ranking of EDAS method is given in Table 12 by using weighted geometric aggregation operator:

⇒Q4≥Q3≥Q5≥Q2≥Q1. Hence the best supplier is Q4.

Validation and Sensitivity Analysis

This section provides an example concerning the challenges that are faced in the agricultural fields, such as the productivity, profitability, and energy use efficiency of wheat crops grown under a variety of tillage systems. The purpose of this section is to illustrate the relevance and efficacy of the developed MCDM technique with PFZN information. Energy input estimates were based on the amount of human labor needed, the use of various kinds of equipment, and the amount of materials. Energy calculations were carried out using various input and output energy equivalents. Since the price of fuel continues to rise, there is a growing interest in improving agricultural production’s energy efficiency. To meet the needs of a more populous world, other industries, like agriculture, had to rely on non-renewable energy sources like electricity and fossil fuels to manufacture more food than was required.

Here, sensitivity analysis is performed to adjust the behavior of the offered methodologies. In Table 2, we provide five sets of possibilities. One criterion has a higher weight than the others for each set, as shown in the table. By following this technique, a large enough space of criterion weights has been constructed to test the method’s sensitivity to changes in the weighting of criteria. In Table 13 and Fig. 2, we provide the results of a sensitivity study that ranks different operators based on an extended EDAS method amenity alternative and a variety of criterion weight.

Figure 2: Ranking results

The graphical representation of ranking result is shown with the help of a Fig. 2.

8.2 Comparison Analysis

In this section we discussed the ranking results with the existing studies in the literature. The Tables 14 and 15 are given as follows:

9  Conclusion

The wheat crop had a distinct set of energy inputs compared to the other crops. The primary reason for the disparity in wheat energy inputs was the use of different fuels. There is a need for human labor, irrigation, and seed energy. Different types of tillage required different types of energy expenditure, one of which was higher in CTB. CT, RTB, RT, and ZTB all help to reduce fuel energy use when applied to wheat crops. The performance values of alternatives in MCDM circumstances are prone to being distributed arbitrarily, despite the fact that these values are essential in a wide variety of scientific, technical, and managerial disciplines. The optimistic assessment score and the pessimistic assessment score were used together to make the enhanced evaluation score, which was then used to make the final evaluation of the different options. We have defined favorable and unfavorable values for a number of components of the recommended method in order to account for the unpredictability of the data across the entire analytic strategy. This is done to work through numerical examples and demonstrate how the proposed method functions, as well as showcase the situations where it may be useful. We use the second example to compare the ranking results of the recommended technique with a few other ways, and the first example gives a step-by-step discussion of the EDAS strategy. In addition, the decisive rating outcomes of the competitions that were recommended were better aligned with the real decision-making techniques that were used.

Future Scope and Direction

There are several future research directions that can be explored based on the proposed method of handling Pythagorean fuzzy Z-numbers with Einstein aggregation operators and extended EDAS method. Some of these directions include:

1. Developing hybrid methods: The proposed method can be combined with other existing methods, such as TOPSIS, AHP, or PROMETHEE, to enhance its performance and accuracy.

2. Generalizing the proposed method: The proposed method can be generalized to handle other types of fuzzy numbers, such as interval-valued fuzzy numbers, type-2 fuzzy sets, and hesitant fuzzy sets.

3. Applications in diverse fields: The proposed method can be applied to various real-life problems, such as supplier selection, project evaluation, medical diagnosis, financial analysis, and environmental management.

4. Extending the EDAS method: The EDAS method can be further extended by incorporating additional distance metrics or modifying the weights calculation to improve its performance.

5. Development of software tools: Software tools can be developed to facilitate the implementation of the proposed method in decision-making problems.

6. The proposed method can be applicable to several real-life problems that involve decision-making under uncertainty and imprecision. For instance, in supplier selection, the proposed method can be used to evaluate and rank potential suppliers based on multiple criteria, such as quality, price, and delivery time. In financial analysis, the proposed method can be used to evaluate investment opportunities based on factors such as risk, return, and liquidity. In medical diagnosis, the proposed method can be used to analyze patient data and provide a diagnosis based on multiple symptoms and test results. Overall, the proposed method has the potential to be widely applied in various fields and can contribute to making more accurate and informed decisions in uncertain and imprecise situations.

Acknowledgement: The authors would like to thank the editors and the anonymous evaluators for their insightful and useful comments and suggestions, which significantly improved this article.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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