The Spherical q-Linear Diophantine Fuzzy Multiple-Criteria Group Decision-Making Based on Differential Measure

1  Literature Review

Multiple criteria decision-making (GCDM) is a subfield of operations research and a sophisticated mathematical method that explicitly evaluates viable options to competing with many criteria in order to get the best answer [1]. Due to the growing complexity of decisions in these contexts, group decision-making (GDM) becomes important since it is typically difficult for a single person to fully analyze all the aspects of a situation [2]. Group decision-making (GDM) is a procedure in which several people interact at once to analyze issues, evaluate various alternatives characterized by several conflicting criteria, and select a suitable alternative [1]. Science, engineering, and medicine all depend on logic and crisp set theory. A crisp set indicates a digital element’s belongingness by indicating whether or not it is part of a set. However, in many real-life circumstances, partial belongingness becomes essential for describing ambiguity. We have to cope with circumstances where the crisp-set theory fails on mostly real-world issues. The choice of such a person is ambiguous; for instance, if a decision maker (GDM) has to choose the best instructor, a bright student, a gorgeous female, a young researcher, etc., what constitutes being the best, young, and youthful varies from person to person. A fuzzy set consists of partial grade gain values in the unit close interval [0, 1] to show the element’s partial belongingness to a set under some specific characteristics. The fuzzy set concept created by Zadeh [3] is a powerful set-theoretic concept for expressing ambiguous information. The fuzzy set caught the researchers’ attention due to its effectiveness and usefulness. It is used all around the globe in a variety of domains, including computational intelligence, information fusion, pattern recognition, engineering, medical diagnostics, artificial intelligence, machine learning, neural networks, and solving issues with (GCDM). The intuitionistic FS (IFS) concept by Atanassova et al. [4] is one of the well-known extended structures. Since the IFS contains two grades: truth-membership grade (TM-grade) and falsity-membership grade (FM-grade), which fulfil the requirement that the total of their grade values gained from [0,1] cannot be surpassed by one also 0≤T+F≤1, the selection is limited to the satisfaction and dissatisfaction classes. In the event that 00.7+0.5>1, it fails. Yager [5] was able to overcome this problem by creating the PyFS principle, which imposes the condition that the square sum of the TM and FM grades cannot be more than 0≤T2+F2≤1. Pythagorean fuzzy information measures and their applications in pattern recognition, cluster analysis, and medical diagnostics were proposed by Peng et al. [6]. The Pythagorean fuzzy idea proposed by Peng et al. [6], used in information measures, has applications in pattern recognition, cluster analysis, and medical diagnostics. Peng et al. [7] highlighted some noteworthy PFS features. In addition, Yager [8] created a powerful conceptual thought called the qROFS, which is an extension of both the PFS and the IFS. IFS and PFS are not enough for modeling when 0.712+0.722=1.0225>0. Wang et al. [9] put out a concept for q-ortho-pair based on the conventional cosine and cotangent similarity metrics. He defined ten similarity measures and ten weighted similarity measures between q-ROFS for (GADM) issues such as pattern identification and scheme selection. Also for Q-rung ortho-pair fuzzy sets with and without hesitation degree, Donyatalab et al. [10] provided a novel similarity measure that is expanded based on square root cosine similarity. For q-rung ortho-pair fuzzy sets, four distinct types of square root cosine similarity metrics (q-ROFSs) created. Jana et al. [11] utilized this article, in which they employed Dombi operations to produce a number of Pythagorean fuzzy Dombi aggregation operators to address the problems associated with multiple-attribute decision-making. Jana et al. [12] built a couple Q-rung orthopair fuzzy Dombi aggregation operators using Dombi operations to address the problems associated with multiple-attribute decision-making in this setting. Qiyas et al. [13] provided a collection of distances and similarity measures utilizing q-rung linear diophantine fuzzy (q-ROLDF). He proposed a Jaccard similarity measure, an exponential similarity measure, and cosine and cotangent function-based similarity measures for q-LDFSs. More recently, Kutlu Gündogdu et al. [14] established the concept of the spherical fuzzy set (SFS). With the eye-catching characteristic that, if one model does not work, the other will handle the uncertain circumstances, these models have the potential to deal with uncertainties in actual issues. With the constraint that the sum of the squares of the three indexes, membership grade, hesitation, and non-membership grade, is less than or equal to 1, an SFS is very significant since it may communicate hazy and ambiguous information. By utilising SFSs, Ahmmad et al. [15] created novel average aggregation operators. Kutlu Gündoğdu et al. [16] introduced numerous properties and arithmetic operational laws of SFSs and their scores and accuracy functions. Shanshan was impressed by the similarity measure in fuzzy sets, which is one of the most powerful and essential methods for assessing how similar two things are in comparison. Under a spherical fuzzy environment, Shishavan et al. [17] developed the Jaccard, exponential, and square root cosine similarity metrics. It has several uses in decision-making, data mining, medical diagnosis, and pattern detection. Rafiq et al. [18] discussed the family of unique cosine-based similarity measures for spherical fuzzy sets. He also evaluated them in light of all prior reflections. Khan et al. [19] investigated the distances and cosines with cotangent similarity measures in the selection of megaprojects in underdeveloped countries. Rayappan et al. [20] used the spherical fuzzy weighted cross entropy between the ideal alternative and an alternative to rank the alternatives according to the cross entropy values and choosed the most desired one(s). Mao et al. [21] used a simple designed aggregation operator to propose a neutrosophic data envelopment analysis model with undesired results in his article. Kouatli et al. [22] used fuzzy logic in this paper to identify technical analyses (TAs). For fuzzyification of these TA(s), he used the modular approach of fuzzy logic and “fuzzimetric sets” to achieve the “fuzzy spectrum” of forecasted price tolerances when making buying and selling decisions. As an innovation, Iqbal et al. [23] used the geometric core of a trapezoidal linear diophantine fuzzy number while ranking trapLDFNs using a centroid index. Hasan et al. [24] set up a series of picture fuzzy mean operators in the decision-making process to aggregate the picture fuzzy set, namely picture fuzzy harmonic mean, picture fuzzy weighted harmonic mean, picture fuzzy arithmetic mean, picture fuzzy weighted arithmetic mean (PFWAM), picture fuzzy geometric mean, and picture fuzzy weighted geometric mean. Wang et al. [25] presented some innovative dice similarity measures of SFSs and its generalization, GDSM. He indicates that the dice similarity measures and asymmetric measures (projection measures) are the specific instances of the GDSM in some parameter values. Wei et al. [26] applied the cosine function over SFS to find similarity measures in pattern recognition. Mahmood et al. [27] introduced SFS-based cosine similarity and data measures for pattern recognition and medical diagnosis. For linear Diophantine fuzzy sets, Mohammad et al. [28] proposed some novel distance and similarity measures, including the normalized distance measure, generalized hamming distance, euclidian distance measure, generalized distance measure, jaccard similarity measure, exponential similarity measure, and cosine and cotangent function similarity measures. In a spherical fuzzy environment, Donyatalab et al. [29] developed several innovative distances based on Minkowski and Minkowski-Hausdorff distances. He discussed innovative SF-similarity measures based on Minkowski and Minkowski-Hausdorff distances with some properties.

In order to evaluate logistics AVs, Bonab et al. [30] created an enhanced MCDM framework based on the Choquet integral (CI) and a spherical fuzzy set (SFS). By employing spherical fuzzy sets and the spherical fuzzy weight correlation coefficient, Ma et al. [31] initiated a novel failure mode and effect analysis (FMEA) that enhanced the performance of FMEA. Ali et al. [32] introduced a novel approach for making decisions called MCDM-total area based on orthogonal vectors with a few innovative distance measurements made possible by matrix norms. The limitations of the current distance measurements are removed by this measure, which also meets all axiomatic requirements.

Das et al. [33] provided a way for resolving group decision-making issues using intuitionistic fuzzy parameterized intuitionistic multi fuzzy N-soft sets of dimension q by providing its induced IFP-hesitant N-soft set as an addition to the multi-fuzzy N-soft set-based method. Khan et al. [34] provided multi-attribute decision-making in a T-spherical fuzzy environment using the Archimedean aggregation operator for three-dimensional data with n powers.

Measures of similarity and distance are two sides of the same coin. The majority of similarity tests rely on distance measurements. A lot depends on both metrics when comparing two PFSs. Even though several studies suggested various distance measurements, the ones that are currently in use still have certain shortcomings [35]. They can first result in counterintuitive consequences [36]. Second, it is possible that they will not be able to calculate the maximum distance measurement value [35]. This will skew the ranking values of the alternatives and result in inaccurate findings. As a result, Shraf has solved the open problem of determining the distance between two PFSs in the Pythagorean fuzzy environment [36]. But there is still a problem with the idea, which is that it might not function if the raw data is in triplet form. To remove this flaw, we created a new fuzzy set to address this problem.

An integer-coefficiented linear Diophantine equation is included in the fuzzy membership function of Q-linear Diophantine fuzzy sets, a particular kind of fuzzy set. A solution to a q-linear Diophantine equation is the value that is specifically described as the membership function of a q-linear Diophantine fuzzy set. Applications that utilise such fuzzy sets in fuzzy set theory include decision-making, control, pattern recognition, and image processing. Overall, q-linear Diophantine fuzzy sets are a practical tool in fuzzy set theory for representing ambiguous and imprecise information and have many applications in a variety of domains. The human mind extends much beyond TM and FM. As a result, a new fuzzy set was required to take human thinking into account, and Ashraf et al. [37] proposed the concept of a spherical fuzzy set that provides a more adaptable and expressive framework for the ideas of classical (crisp) sets. In spherical fuzzy sets, an element of a set is given a range of membership values rather than a single membership value to represent the degree of uncertainty or ambiguity in that element’s categorization. This study’s objective is to identify the differential measure among three parameters using the Sq-LDFS specified fuzzy set to cover the three degrees of membership (membership, non-membership and indeterminacy). Razzaque et al. [38] proposed a collection of innovative Einstein aggregation operations by using sq-LDF data. Additionally, a better VIKOR method is presented to address the uncertainty in categorising viral hepatitis.

This article introduces the idea of differential measure (DFM), derived from [39] as a novel method for contrasting spherical q-linear Diophantine fuzzy sets (Sq-LDFSs) [40]. When the uncertainty grade has equally in both sets, there is a preference connection between two Sq-LDFS because of their relative positions in the attribute space and the proximity of their membership and non-membership degrees. Due to the equal consideration of the membership and non-membership degrees, and uncertainty grade, although each direction has a distinct connotation, the existing Distance measures and similarity measures across various fuzzy sets, namely PFSs with a zero uncertainty grade have several shortcomings. To take into To account the effects of each element of an Sq-LDFSs evaluation, the DFM uses signed distance. According to the degree of superiority or inferiority, two Sq-LDFSs are shown as the same, equal, superior, or inferior to one another. Based on the newly established DFM, a novel MCGDM approach is certified by our proposed idea. Since Sq-LDFS are more adaptable than IFSs, PFSs, q-LDFSs and SFSs to deal with absurdity and ambiguity, as well as to deal with human evaluation information, it is vital to pay greater attention to collective decision-making in this context. Therefore,

1.   The proposal of DFM is proposed as a revolutionary technique for contrasting Sq-LDFSs. It is an association between two Sq-LDFSs based on the position in the attribute space. To cover the limitations of the current distance in discriminating, a DFM maintains the identity of the Sq-LDFSs’ parameters.

2.   A novel (GCGDM) technique is presented based on the proposed DFM.

3.   The experts’ weights in (GCGDM) issues are computed using a novel method that is put forth. Furthermore, the (GCGDM) problems may have a solution using this method.

The rest of the part is structured as follows: The most recent information regarding Sq-LDFSs is provided in the following sections: Literature review in Section 1, preliminary definitions in Section 2, and an explanation of the differential measure in Section 3. We provided the differential rules suggested for Sq-LDFSs with short numerical examples in Section 4. In Section 5, an enhanced framework for MCGDM is proposed to explain all phases of the technique and offer some counter-examples for previously proposed differential measures by our technique. In Subsection 5.1, a numerical example of diagnosing cancer patients using a variety of tests is provided to bolster the differential measure. In Sections 6 and 7, a discussion and a conclusion are offered.

2  Preliminaries

Definition 2.1. [40] A spherical q-linear Diophantine fuzzy set (Sq−LDFS)℘ over a fixed set K is defined as

℘={ℓ,⟨T℘(ℓ),I℘(ℓ),F℘(ℓ)⟩,⟨i℘(ℓ),j℘(ℓ),k℘(ℓ)⟩:∀ℓ∈K}(1)

where each membership grade T(ℓ),I(ℓ),F(ℓ):℘→[0,1], and each control parameter i(ℓ),j(ℓ),k(ℓ):℘→[0,1], for an element ℓ∈K with the limitations.

1.   0≤iqT℘(ℓ)+jqI℘(ℓ)+kqF℘(ℓ)≤1

2.   0≤i℘q(ℓ)+j℘q(ℓ)+k℘q(ℓ)≤1.

Definition 2.2. A compliment of Sq−LDFS℘ over a fixed set K is defined as

℘c=[ℓ,{F℘(ℓ),I℘(ℓ),T℘(ℓ)},{k℘(ℓ),j℘(ℓ),i℘(ℓ)}:∀ℓ∈K]

Definition 2.3. For a Sq-LDFS a score function (SF) Ξ is defined by the transformation Ξ:S´q−LDFS(e)→[−1,1], and given by:

Ξ(A)=12N{(1+T−I−F)+(1+iq−jq−kq)2};q≥1(2)

where S´q−LDFS(ℓ) is a collection on S´q−LDFNs over ℓ,∀ℓ∈K.

Definition 2.4. The hamming distance using Sq-LDFSs [38] can be defined as:

HD=16{|T13−T23|+|I13−I23|+|F13−F23|+|i13−i23|+|j13−j23|+|k13−k23|}(3)

Definition 2.5. The euclidean function using Sq-LDFSs can be defined as:

ED=1n[∑i=1n|T13−T23|+|I13−I23|+|F13−F23|+|i13−i23|+|j13−j23|+|k13−k23|](4)

3  The Differential Measure

Song et al. [41] presented the analysis of the distinction between psychological distance measure and classical distance measurement. The classical distance measurements’ surprising potential to ignore alternative background information and their competing interactions resolved by psychological distance. To illustrate, we use the SF to evaluate the three alternatives A1={0.8300,0.2100,0.4300,0.7200,0.1300,0.1400}, A2={0.8300,0.2100,0.4300,0.5100,0.1800,0.1700} and A3={0.9300, 0.2100, 0.4300, 0.7200, 0.2100, 0.1300} using Sq-LDF evaluation. We observe that A3 is the best alternative with large value of membership grade and control variable. Also A1 is better alternative than A2 with same degree of support and indeterminacy degree but less degree of opposition. But by using the hamming distances according to Sq-LDFS rules we found that DHm(A2,A3)=0.08>DHm(A1,A3)=0.04, this shows that A1 is closed to A3, i.e., A3 is preferable substitute than A2. Therefore, we can not rely on traditional distance measurements to accurately depict the preferences of different choices. The reader is recommended to take an eye view at Xiao et al. [36] and Huang et al. [35] for counter-examples that demonstrate the shortcomings of the existing distance metrics on pythagorean fuzzy sets. A3 is superior to A1 since it has more support and less opposition. Since A3 is superior to A1, we can also claim that A1 is inferior to A3. Comparing options A3 and A2, it is noticeable that A3 is preferable to A2 due to its equal support and lesser opposition and indeterminacy. While comparing options A3 and A1, A3 is superior to A2 regarding support, but A1 has superiority to A3 regarding the opposition with CFs. Even if each parameter has a different effect, an increase in the membership degree is treated in the traditional distance measures in the same way as an increase in the non-membership degree. Therefore, support, indeterminacy, and opposition should be measured using signed distance. A step toward membership is a positive step, whereas a step away from membership shows as a negative step. For the three parameters, the differential measures show by using an Sq-LDFSs. Support T-grade increases as the difference between membership degrees rises, but opposition F-grade increases as the difference between non-membership degrees gets bigger and vice versa. Consider about the alternatives A2 and A3 that have the Sq-LD’s fuzzy evaluations A2={⟨0.83,0.21,0.43⟩, ⟨0.51,0.18,0.17⟩}, and A3={⟨0.93, 0.21, 0.41⟩, ⟨0.72, 0.21,0.13⟩}. Utilizing the accuracy function (11), as well as score function (10) Sc(A2)=0.1284, Sc(A3)=0.1473 Utilizing the accuracy function (11), as well as score function (10) A3={⟨0.93, 0.21, 0.41⟩, ⟨0.72, 0.21, 0.13⟩} A2={⟨0.83,0.21,0.43⟩, ⟨0.51,0.18,0.17⟩} and Sc(A2)=0.1284, Sc(A3)=0.1473. Human thoughts are not limited to satisfaction and dissatisfaction. There are chances to refuse or remain neutral except to accept or not accept. The elements of an Sq-LDFS {⟨T,I,F⟩,⟨i, j, k⟩} have a physical interpretation like “who votes in favor” (satisfaction degree), “who votes against” (dissatisfaction degree) “who refuses to vote” (refusal grade) “who abstain” (uncertainty or abstinence grade). Hence for alternative A1, the vote for the resolution in favor, remains neutral, and against is equal amount. In evaluation, a tie occurs and Sq-LDF evaluations A1, A2, and A3 with their control factors have the same impact. The accuracy function, which takes the amount of information under consideration, is biased in favor of A1 in this situation. Naturally, A1 communicates more information than A2 and A3. However, in this situation, the Sq-LDFS’s information is significant since it makes one or two solutions appear superior. Either A1 is superior to A2 and A3 or tie between three and vice versa. That leads us to the fundamental idea behind a scoring function: better Sq-LDFSs have high values for T and low values for I and F with their control variables. This is simply the difference between support, neutral, and opposition is all that matters. In research of sharaf, he found the difference between support and opposition. This research [39] encourages us to find differences among three variables.

4  Rules for Differentiation

The differential measure between two Sq-linear Diophantine fuzzy sets A={⟨TA(ℓ), IA(ℓ), FA(ℓ)⟩, ⟨iA(ℓ), jA(ℓ), kA(ℓ)⟩} and B={⟨TB(ℓ), IB(ℓ), FB(ℓ)⟩, ⟨iB(ℓ),jB(ℓ),kB(ℓ)⟩}, ∀ ℓ∈K is determine as follows.

For triplet grades, membership determinacy and non-membership degrees, the datum is one. In case where A’s membership degree exceeds B’s, the difference between the grades is added to one, showing a positive step; in case where A’s membership degree is less than B’s, the difference between the grades is deducted from one, showing a negative step.

Definition 4.1. A differential measurement between two spherical q-linear Diophantine fuzzy sets (Sq-LDFSs) A={⟨TA(ℓ), IA(ℓ), FA(ℓ)⟩, ⟨iA(ℓ), jA(ℓ), kA(ℓ)⟩} and B={⟨TB(ℓ), IB(ℓ), FB(ℓ)⟩, ⟨iB(ℓ),jB(ℓ),kB(ℓ)⟩}, ∀ℓ∈K is a preference relationship based on the closeness of two Sq-LDFSs’ membership, indeterminacy, and non-membership degrees about their positions in the attribute space with cotrol parameters. It can be defined as

diff(A,B)={⟨1+(TA−TB)3+(TA+IA+FA)−(TB+IB+FB),1+(IA−IB)3+(TA+IA+FA)−(TB+IB+FB),1+(FA−FB)3+(TA+IA+FA)−(TB+IB+FB)⟩⟨1+(iA−iB)3+(iA+jA+kA)−(iB+jB+kB),1+(jA−jB)3+(iA+jA+kA)−(iB+jB+kB),1+(kA−kB)3+(iA+jA+kA)−(iB+jB+kB)⟩}(5)

Definition 4.2. Two Sq-LDFSs A={⟨TA(ℓ), IA(ℓ), FA(ℓ)⟩,⟨iA(ℓ), jA(ℓ), kA(ℓ)⟩} and B={⟨TB(ℓ), IB(ℓ), FB(ℓ)⟩, ⟨iB(ℓ),jB(ℓ),kB(ℓ)⟩}, ∀ℓ∈K can be categories using differential measurement (4.1) as below:

1. If diff(A,B)=(13,13,13), then A and B are equivalent A≅B, with a degee of superiority DoS(A,B)= DoS(B,A)=0, and a degree of inferiority DoN(A,B)= DoN(B,A)=0. If TA=TB, IA=IB, FA=FB, iA=iB, jA=jB, kA=kB, then “A” and “B” are the identical “A≡B.” 2. If diff(A,B)={⟨TD,ID,FD⟩,⟨iD,jD,kD⟩} with iDqTD>0.5, then A is superior to B. A≻B with the superior degree DoS(A,B)=12n[(1+T−I−F)+(1+iq−jq−kq)/2]>0. 3. If diff(A,B)={⟨TD,ID,FD⟩,⟨iD,jD,kD⟩} with kDqFD>0.5, then A is inferior to B. A≺B with the superior degree DoN(A,B)=12n[(1+T−I−F)+(1+iq−jq−kq)/2]<0.

Proposition 4.1. For Sq-LDFSs A={⟨TA,IA,FA⟩, ⟨iA,jA,kA⟩},B={⟨TB,IB,FB⟩, ⟨iB,jB,kB⟩} and C={⟨TC,IC,FC⟩,⟨iC,jC,kC⟩}

1.   If A≻B,B≻C⟹A≻C.

2.   If A≺B,B≺C⟹A≺C.

Proposition 4.2. For the Sq-LDFSs A={⟨TA,IA,FA⟩, ⟨iA,jA,kA⟩} and Ac={⟨FA,IA,TA⟩, ⟨kA,jA, iA⟩}diff(A,Ac)=diff(Ac,A), And if A is superior to Ac, then DoN(Ac,A)=−DoN(A,Ac), and vice versa. Proof. If we apply the definition of difference between the set A and its compliment Ac, we get

diff(A,Ac)={⟨1+(TA−FA)3+(TA+IA+FA)−(FA+IA+TA),1+(IA−IA)3+(T+IA+FA)−(FA+IA+TA),1+(FA−TA)3+(TA+IA+FA)−(FA+IA+TA)⟩⟨1+(iA−kA)3+(iA+jA+kA)−(kA+jA+iA),1+(jA−jA)3+(iA+jA+kA)−(kA+jA+iA),1+(kA−iA)3+(iA+jA+kA)−(kA+jA+iA)⟩}={⟨1+(TA−FA)3+(TA+IA+FA)−(FA+IA+TA),13+(TA+IA+FA)−(FA+IA+TA),1+(FA−TA)3+(TA+IA+FA)−(FA+IA+TA)⟩⟨1+(iA−kA)3+(iA+jA+kA)−(kA+jA+iA),13+(iA+jA+kA)−(kA+jA+iA),1+(kA−iA)3+(iA+jA+kA)−(kA+jA+iA)⟩}

diff(Ac,A)={⟨1+(FA−TA)3+(FA+IA+TA)−(TA+IA+FA),1+(IA−IA)3+(FA+IA+TA)−(TA+IA+FA),1+(TA−FB)3+(FA+IA+TA)−(TA+IA+FA)⟩⟨1+(kA−iA)3+(kA+jA+iA)−(iA+jA+kA),1+(jA−jA)3+(kA+jA+iA)−(iA+jA+kA),1+(iA−kA)3+(kA+jA+iA)−(iA+jA+kA)⟩}={⟨1+(FA−TA)3+(FA+IA+TA)−(TA+IA+FA),13+(FA+IA+TA)−(TA+IA+FA),1+(TA−FB)3+(FA+IA+TA)−(TA+IA+FA)⟩⟨1+(kA−iA)3+(kA+jA+iA)−(iA+jA+kA),13+(kA+jA+iA)−(iA+jA+kA),1+(iA−kA)3+(kA+jA+iA)−(iA+jA+kA)⟩}

We noticed that diff(A,Ac)=diffc(Ac,A). And if A is superior than Ac, then DON(Ac,A)=−DOS(A,Ac) and vice versa.

Numerical Examples In this subsection a few examples are provided to demonstrate the use of differential measures.

Example 1. Let A1={⟨1.0,0.0,0.0⟩,⟨1.,0.0,0.0⟩},A2={⟨1.0,0.0,0.0⟩,⟨1.,0.0,0.0⟩}, then

diff(A1,A2)={⟨1+(1.0−1.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(0.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0)⟩,⟨1+(1.0−1.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(0.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0)⟩}={⟨0.33,0.33,0.33⟩⟨0.33,0.33,0.33⟩}diff(A2,A1)={⟨1+(1.0−1.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(0.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0)⟩,⟨1+(1.0−1.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(0.0+0.0+0.0)−(1.0+0.0+0.0),1+(0.0−0.0)3+(1.0+0.0+0.0)−(1.0+0.0+0.0)⟩}={⟨0.33,0.33,0.33⟩⟨0.33,0.33,0.33⟩}

Example 2. Let A1={⟨0.95,0.24,0.38⟩,⟨41.,0.21,0.11⟩},A2={⟨0.85,0.24,0.45⟩,⟨0.25,0.34,0.18⟩}

diff(A1,A2)={⟨1+(0.95−0.85)3+(0.95+0.24+0.38)−(0.85+0.24+0.45),1+(0.24−0.24)3+(0.95+0.24+0.38)−(0.85+0.24+0.45),1+(0.38−0.45)3+(0.95+0.24+0.38)−(0.85+0.24+0.45)⟩,⟨1+(0.41−0.25)3+(0.41+0.21+0.11)−(0.25+0.34+0.18),1+(0.24−0.24)3+(0.41+0.21+0.11)−(0.25+0.34+0.18),1+(0.38−0.45)3+(0.41+0.21+0.11)−(0.25+0.34+0.18)⟩}={⟨0.45,0.50,0.55⟩⟨0.65,0.50,0.35⟩}diff(A2,A1)={⟨1+(0.85−0.95)3+(0.85+0.24+0.45)−(0.95+0.24+0.38),1+(0.24−0.24)3+(0.85+0.24+0.45)−(0.95+0.24+0.38),1+(0.45−0.38)3+(0.85+0.24+0.45)−(0.95+0.24+0.38)⟩,⟨1+(0.41−0.25)3+(0.41+0.21+0.11)−(0.25+0.34+0.18),1+(0.24−0.24)3+(0.41+0.21+0.11)−(0.25+0.34+0.18),1+(0.38−0.45)3+(0.41+0.21+0.11)−(0.25+0.34+0.18)⟩}={⟨0.55,0.50,0.45⟩,⟨0.35,0.50,0.65⟩}

Following that, either A2 is suppior than A1 with a degree of superiority DoS (A2,A1)=0.2, or A1 is inferior than A2 with a degree of inferiority DoN (A1,A2)=−0.2.

5  An Improved Proposed Framework for MCGDM

(1) Problem identification. Evaluation of the criteria {C1,C2,...,Cq} that experts attribute to alternatives {A1,A2,...,Ap}. Experts’ {E1,E2,...,Ek} formation of DM about each alternative in columns and each criterion in rows.

(2) The creation of decision matrix and weight vector assigned by experts as Sq-LDFSs.

A=[aij]t×s=(a11∂a12∂⋯a1t∂a21∂a22∂⋯a1t∂a31∂a32∂⋯a1t∂⋮⋮⋱⋮as1∂as2∂⋯ast∂)

and W={w1∂,w2∂,...,wt∂},

(3) Normalize the information in DM to get the best results.

Ast=1−ast−minastmaxast−minast(6)

(4) Create the aggregated decision matrix using the formula (7), and then determine the best rating for each criterion.

Mst=[ast],ast={(∑∂=1zw∂Tst∂,∑∂=1zw∂Ist∂,∑∂=1zw∂Fst∂),(∑∂=1zw∂ist∂,∑∂=1zw∂jst∂,∑∂=1zw∂kst∂)}(7)

(5) Determine the DFM between the ideal rating and the rating of each alternative for a criterion to create the differential matrix.

IRt={(maxsTst∂,minsIst∂,minsFst∂),maxsist∂,minsjst∂,minskst∂},Fst=[fst],fst=Diff(IRs,ast)(8)

(6) Calculate the collective differential measure (CDFM) for each option.

CDFM(fst)={(∑s=1zwsTfst,∑s=1zwsIfst,∑s=1zwsFfst),(∑s=1zwsifst,∑s=1zwsjfst,∑s=1zwskfst)}(9)

(7) Calculate the CDFM’s score function and rank. The crisp value of the CDFM of each A is estimated using the score formula. The total degree to which the ideal ratings for the evaluation criteria outperform the ratings of an alternative is the CDFM. The superiority of the alternative rating decreases with decreasing ideal rating superiority. The option with the lowest value is the best, and the alternatives are arranged in ascending order.

The expert weights evaluation technique: With the socioeconomic environment becoming more complicated, switching from a single expert to a group of experts became necessary since it is challenging for a single expert to manage all the relevant components of a complex problem [42]. A homogenous group of specialists with comparable attitudes, knowledge, and experience is virtually impossible when group decision-making. Consideration should be given to the credibility of expert opinions and how they could influence the choice made. Results may be inaccurate if the relative importance of the experts is ignored. Studies on calculating the weights of the criteria are abundant, but studies on calculating the weights of the experts are few [43]. Usually, the weights of the experts are chosen based on personal preference. They are decided upon by a manager or through peer reviews. Credibility is low for this method of allocating the experts’ weights. To give the experts’ opinions greater weight from a more objective standpoint, objective procedures that make use of quantitative methodologies are used [44]. Based on the similarity between the opinions of the individual experts and the group, Zhang [45] suggested a method to calculate the experts’ unknown weights. Experts are more significant and should be given more weight when their opinions are more in line with the group’s viewpoint. This study suggests a precise approach for calculating the weights of the unknown experts based on their agreement with the ranking of the alternatives. The weighted sum approach is employed to rank each expert (WSM). Spearman’s correlation coefficient is then used to determine the correlation between each pair of experts’ ranks. The correlation between an expert and other experts is added to determine the expert’s overall correlation. The final step is to divide the sum of the total correlations of each expert by that of the experts as a whole. First, each expert’s assessments of a potential replacement for the evaluation criteria are combined using the Pythagorean fuzzy weighted averaging operator (7).

Step1: Create each expert’s judgment vector. The pth expert’s evaluations for the ith alternative are combined utilizing (7). Bs∂=[∑j=1s℘jpA1jp∑j=1s℘jpA2jp⋮∑j=1s℘jpAtjp].

Step2: Calculate the total rating score for each option and place it in order. Bs∂=[Sc(B1∂)Sc(B2∂)⋮Sc(Bt∂)]. The choice with the highest score is considered to be the best one, with the alternatives sorted in descending order. Next, the pth expert is ranked.

Step3: Use Spearman’s correlation coefficient to calculate the correlation between one expert’s ranking and the other. Using Spearman’s correlation coefficient, the ranking of the pth expert and each of the other experts is correlated. C(E∂,Ek)=1−6∑d∂k2n(n2−1), for 1 ≤∂,k≤R,∂≠k where d∂k is the difference in the Sth alternative’s rank as ordered by the ∂th and kth experts, and (E∂,Ek), C(E∂,Ek) is the correlation between the rankings of the E∂ and Ek experts.

Step4: Calculate each expert’s overall correlation. The expert’s total correlation is the result of adding up all of the expert’s correlations. Assign a weight to the experts. C(E∂)=∑k=1k≠∂RC(E∂,Ek).

Step5: Each expert’s weight is determined by dividing his or her overall correlation by the sum of the total correlations of the other experts. w (E∂)=C(E∂)∑∂=1RC(E∂).

Fig. 1 depicts the prescribed MCGDM framework.

Figure 1: Improved technique for MCGDM

Practical examples: In decision making problem, one of the efficient way to identifying an eligible preferred alternative from a group of alternatives is its similarity degree to the ideal rating. The best alternative is chosen by the degree of similarity. The least differential measurement indicates the ideal option when evaluating the differential measure. In this part, two real-world applications utilizing Pythagorean fuzzy information are discussed in two practically applications. The first application is the evaluation of solid-state drives (SSDs) derived from Huang et al. [35]. The second application is the identification of the finest photovoltaic cell taken from Zhang [45].

Evaluation of SSDs: Flash memory is the primary storage technology used in computers and mobile devices today. Modern computing systems frequently substitute solid-state drives (SSDs) for magnetic hard disc drives as secondary memory (HDDs). HDD performance remained sluggish due to limits on the search time of actuator arms and the rotational speed of magnetic platters. SSDs, on the other side, do not contain complex mechanical parts. Compared to HDDs, this has a reduced failure rate and latency. In addition, SSDs have several advantages over HDDs, including more bandwidth, reduced power use, better random I/O performance, stronger shock tolerance, and increased system dependability [46]. An enterprise wishes to select one type of SSD from the five accessible types, symbolized by the letters “A1, A2, A3, A4, A5”. Performance (ℶ1), dependability (ℶ2), capacity (ℶ3), form factor and connections (ℶ4), battery life (ℶ5), speed (ℶ6), durability (ℶ7), and pricing (ℶ8) all weigh into the evaluation. Every criterion is a benefit criterion, except the eighth, which is the price, which is a cost criterion. The criteria have a weights vector {0.19, 0.09, 0.11, 0.12, 0.12, 0.13, 0.07, 0.17}. A business expert uses PFSs to assess the SSDs according to the specified criteria. The best rating for the Pythagorean fuzzy decision matrix is in Table 1.

Utilizing DM, the differential measurement between the ideal rating (IR) and the rating for each criterion is computed with the help of the differential measure formula, as shown in Table 2. To obtain its integrated DiFM, the differential measurements of each alternative are aggregated. After that, the CDiFMs are defuzzified using the score function (2.3), allowing for the evaluation of the total degree of the best ratings’ superiority over the ratings of the alternatives available for the evaluating criterion.

The options are ranked as shown in Table 3 in ascending order. According to Table 3, the ranking order is A3>A1>A2>A4>A5. The outcomes of other decision-making methodologies, including the TOPSIS method proposed by Zhang et al. [47], the TODIM approach developed by Ren et al. [48], the distance and similarity measure introduced by Zeng et al. [49], the fuzzy weighted and ordered weighted aggregation operators presented by Garg [50], and the differential measure for pythagorean fuzzy sets, are compared with the outcomes of the proposed methodology. Table 4 displays the rankings that these techniques produced [39]. Table 4 demonstrates that the suggested technique produced the same optimal alternative as the ones previously employed. The finest choice is the third substitute. The ranking results of our proposed method are the same as the Zeng et al. [49], Ren et al. [48], Garg [50], and Huang et al. [35] techniques.

Photovoltaic cells: Photovoltaic (PV) thermoelectric technology has gained a great deal of attention as a result of the present global crisis caused by the shortage of natural fuels in the earth’s crust [51]. The most effective alternative to conventional sources of energy power is photovoltaic solar energy, which offers both sustainable and environmentally friendly energy sources [52]. Photovoltaic systems produce zero pollution as they directly harness solar energy to produce electricity [53]. They also provide many benefits, including noiseless operation, low maintenance requirements, and excellent dependability [54]. The advancement of new photovoltaic technologies has recently been the topic of extensive research focused on boosting the effectiveness and sustainability of these devices, integrating with using cheap materials and techniques [52]. These significant efforts combined with outstanding efforts to identify the best alternative and choice based on the requirements, using multi-criteria decision approaches, which have been kept parallel to this ongoing search [55]. A photovoltaic cell is an apparatus that uses the photovoltaic effect, a well-known physical and chemical phenomenon, to transform light energy into electricity. A generational division of several PV cell types in use. The first generation, which is the most widely used and conventional kind consisting of monocrystalline or polycrystalline silicon, controls more than 90% of the market for solar energy today. Due to the expensive manufacture of crystalline silicon, the experts made more efforts to create alternative materials with low-cost fabrication methods. The second generation was created of thin-film cells using less expensive techniques. Amorphous silicon (a-Si), nanocrystalline silicon (nc-Si), cadmium telluride (CdTe), and copper indium gallium selenide are employed the most frequently (CIGS). However, they do not perform much better than first-generation cells. The third generation was created to be highly efficient and inexpensive. They include a range of thin film technologies, some of which employ organic materials to produce energy while others rely on inorganic materials, such as dye-sensitized cells (DSSCs), quantum dot-sensitized cells (QDSSCs), organic solar cells, and hybrid perovskite cells [51]. Table 5 represents the aggregated decision matrix consists of photovoltaic cell data.

After selecting the ideal site for a photovoltaic solar plant installation, it is necessary to choose the kind of cell among the many PV cell options that best optimize the construction, for example, enhances production or efficiency, reduces costs, and provides the highest maturation and reliability [55].

The selection of photovoltaic cells: The selection of optimal photovoltaic is required from the listed alternatives: CdTe and CIGS (A1), amorphous silicon (A2), thin-film III-V with tracking systems (A3), organic and hybrid cells (A4), and crystalline silicon (A5). These parameters are used to evaluate these cells: manufacturing cost (ℑ1), energy conversion efficiency (ℑ2), market share (ℑ3), emission of greenhouse gas from production (ℑ4), and energy payback period (ℑ5). The evaluation involved in this procedure used weights given by three experts (0.3191, 0.3533, 0.3276). The criteria’s weighting vector is (0.2, 0.4, 0.1, 0.1, 0.2). PFSs present how the experts rated the potential solutions to the evaluation criteria. The decision matrices are created and normalized to consider the cost criterion. Table 6 presents the aggregated decision matrix using the PFWAY (7) and the ideal rating (IR) utilizing (8). Zhang [45] provided detailed information on Pythagorean fuzzy decision matrices. Then the diff. measure between IR and the alternative’s rating calculated for evaluation criterion by using (4.1) and creating a differential matrix as shown in Table 6. The total differential estimation can be obtained, and the differential measurements of each alternative are combined. The options are then rated accordingly to those shown in Table 7 using the scoring function (2.3). The ultimate ranking is A2>A5>A3>A4>A1, as shown in Table 7. As a result, A2 is the ideal solar cell (amorphous silicon). A technique for multi-criteria group decision-making (MCGDM) based on similarity measures and newly created aggregation operators was described by Zhang [45]. An approach for MCGDM using similarity measures based on point operators was also put out by Biswas et al. [56]. They created the Pythagorean fuzzy-dependent averaging operator and the Pythagorean fuzzy-dependent geometric operator using these similarity measurements as a starting point. These operators are used to combine the opinions of the experts. The ranking achieved using the methods of Zhang [45] and Biswas et al. [56] are compared to the results of the suggested technique. Table 8 provides a comparison. Thus according to Table 8, the ranking produced by the suggested strategy matches the ranking of Biswas et al. [56].

It is also obvious that the ranking of the suggested technique is virtually identical to Zhang’s [45] results. The two best technologies and the poorest technologies are unchanged, however, the moderately performing technologies have altered somewhat.

Propose weighting method application: The proposed technique may be used to calculate the weights of the professionals for the previous PV cell selection task. The illustrations of the stages are listed below.

Step 1: From the given weight vector by each decision professional, by using (2.3) WVP1=[(0.59,0.67)(0.73,0.39)(0.39,0.54)(0.57,0.54)(0.51,0.54)], WVP2=[(0.48,0.68)(0.67,0.32)(0.53,0.56)(0.54,0.59)(0.62,0.44)], WVP3=[(0.52,0.65)(0.65,0.33)(0.56,0.54)(0.39,0.64)(0.517,0.39)]

Step 2: Evaluate each alternative’s score function (2.3) using Table 10 to rank them. WV⌢P1=[−0.10080.3808−0.13950.1649−0.0315],  WV⌢P2=[−0.23200.3465−0.0327−0.05650.1908],  WV⌢P3=[−0.15210.32010.0220−0.25750.1728]  . Using the score function’s values, each expert’s rank is R(E1) = [A2A4A5A1A3],R(E2)=[A2A5A3A4A1],R(E3)=[A2A5A3A1A4]