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

List of Abbreviations

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,