# Novel EDAS Methodology Based on Single-Valued Neutrosophic Aczel-Alsina Aggregation Information and Their Application in Complex Decision-Making

Authors: Shahzaib Ashraf; Shakoor Ahmad; Muhammad Naeem; Muhammad Riaz; Md. Ashraful Alam
Journal: Complexity
 (2022)

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## Abstract

In this article, we proposed an extended EDAS (Evaluation based on Distance from Average Solution) method based on the single-valued neutrosophic (SVN) Aczel-Alsina aggregation information. The fundamental concept of a single-valued neutrosophic (SVN) set is a universal mathematical tool for effectively managing uncertain and imprecise information. To accomplish our goal, we first extend the Aczel-Alsina t-norm and t-conorm to SVN scenarios and introduce a few new SVN operations on which we construct novel SVN aggregation operators. Furthermore, a decision support strategy is built in the SVN framework using the EDAS methodology and the suggested Aczel-Alsina aggregation operators. This method computes the aggregated outcomes of each investigated alternative, as well as their score values. Finally, to demonstrate the functionality of the developed SVN- EDAS, an application has been made related to the role of commercial banks in providing loans to their customers, which has recently affected our world, and the results are compared with other existing methods. The results suggest that the proposed method may overcome the inadequacies of the existing decision method’s lack of decision flexibility by using SVN aggregation operators.


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## Body

## 1. Introduction

Multi-attribute group decision making (MAGDM) is the practice of utilizing expert evaluation information to analyze, rank and pick the best solutions using a given decision-making (DM) approach. Making a decision entails providing relevant information and making a choice amongst several DM approaches. Enhancing the MADM approach has become a hot topic in today’s DM area since it is difficult to predict the future and because experts’ expertise is limited. In the DM procedure, there is vagueness and uncertainty, and Zadeh’s theory [1] of fuzzy sets gives an extremely effective technique to cope with these challenges. Atanasov [2] devised the concept of intuitionistic fuzzy sets to reflect uncertainty in DM (IFS). IFS comprise membership functions and non-membership functions. Some decision-makers started using intuitionistic fuzzy numbers to express their preferences for alternatives in the DM dilemma [3–5]. Because of this, intuitionistic fuzzy information is becoming more popular among academics.Following the acquisition of the expert data, we must integrate it using various aggregating operators (Agop). Several Agops, such as the IF averaging operator, were created by Xu [6]. Jana and Pal [7] introduced the decision making EDAS method to tackle the uncertainty in decision support problems. Several Einstein Agops, such as IFE averaging/geometric operators (Aveg/GAOs), were introduced by Liu and Liu [8]. Xu and Xu [9] created a prioritized list of Agops and explained how they could be used to solve DM challenges (Dcmp). Under the linguistic IF strategy, Wang and Wang [10] produced various unique Agops and proposed an algorithm to address the difficult uncertain Dcmp. Jana et al. [11] presented the multi-attribute decision making method using power Dombi operators and MABAC method to tackle the uncertain information in decision support problems. IF Bonferroni means Agop, Xu and Yager [12] established the DM technique (DMA). The Group DMA was created by Arora and Garg [13] built on the prioritized IF Agop under the linguistic set of data. Zhao et al. [14] examined the uses of generalized IF Agops, such as the generalized IF averaging/geometric operators, to deal with uncertainty in Dcmp. Yu [15] demonstrated certain IF Agops depending on levels of confidence and handled challenging real-world Dcmps. Yu [16] created the IF Agop and discussed its usefulness in DM using Heronian mean. Jiang et al. [17] devised a DM strategy depends on the IF power Agop and the entropy measurement. Senapati et al. [18] developed various IF Aczel-Alsina (Acz-Als) Agops depending on the Acz-Als norm and used them in the IF multi-attribute DM process. Khan et al. [19] created the unique generalized IF soft evidence Agops and investigated their use in DM.Even though all of these approaches are beneficial for representing incomplete data, they are unable to deal with indeterminate (neutral) data and inconsistent data in actual practice. Cuong [20] consider the three type of uncertain situation at a time to developed the picture fuzzy set (PFS). It is clear that PFS interpretations of ambiguous data are more rational and accurate than those provided by the FS and IFS models. Many researchers began researching on PFS after its development. The synthesis of the achievement degree of criterion necessitates the collection of information. To understand the various uncertain data in Dcmp, Ashraf et al. [21] proposed the list of novel picture fuzzy (PF) algebraic Agop and decision support (D-S) strategy. Riaz et al. [22] introduced the decision making method under bipolar picture fuzzy operators and distance measures. Some Agops, such as PF geometic operators, were created by Garg [23]. Wei [24] compiled a list of PF Agops and discussed their use in DM challenges. Ashraf et al. [25] presented decision based application for Internet finance soft power evaluation under fuzzy information. Khan et al. [26] proposed and investigated the use of generalized PF soft details Agops in DM. Some Einstein Agops, such as PF Einstein (Aveg/GAOs), were introduced by Khan et al. [27]. Under algebraic norm and linguistic deta set, Qiyas et al. [28] created some PF averaging/geometric Agop. Jana et al. [29] investigated certain Dombi Agops in PF situations, such as PF Dombi averaging/geometric operators. Jana and Pal [30] presented the dynamical hybrid method using GRA technique to tackle decision support problems. PF Hamacher averaging/geometric operators employing Hamacher t-norm and s-norm were presented by Wei [31]. In a cubic PF setting, Ashraf et al. [32] devised a novel distance measure dependent on algebraic Agops. Khan et al. [33] created some logarithmic PF Agops and discussed how to use them in DM. Qiyas et al. [34] introduced the linguistic approach to tackle the decision problems using picture fuzzy Dombi information.The portrayal of fuzzy sets and their extensions allow greater flexibility for DM, although there are still certain limitations. Data discontinuity and inconsistency cannot be solved such that the NS emerges as required by time. In this study, the degree of truth, uncertainty, and falsity are all taken into account. This theory can assist decision-makers in expressing their ideas more accurately and in detail and addresses issues that the fuzzy sets are unable to address. The concept of neutrosophic sets was first introduced by Smarandache [35]. This epistemology is a mathematical model that describes not only the origins, nature, and scope of impartiality, but also the interactions between their many conceptualization ranges. In order to overcome DM difficulties in unclear contexts, such enhancements have been developed. Smarandache [36] presented the plithogeny, plithogenic set, logic as a novel neutrosophic information. Ye [37] developed an AgOs-depends strategy for DM, while Peng et al. [38] highlighted the power of AgOs in dealing with uncertainty in NS data and examined their usefulness in the context of uncertainty in data management. In order to deal with uncertain data in the form of neutrosophic numbers, Chen and Ye [39] defined the SVN information depends Dombi AgOs, while Liu et al. [40] established the generalized Hamacher AgOs. Al-Hamido [41] described the novel algebraic structure of neutrosophic set. Garg and Nancy [42] presented the novel methodology depends on SVNSs with linguistic terms. Ashraf et al. [43] proposed novel decision model for hydrogen power plant selection using SVN sine trigonometric AgOs. The flexible DM correlating to favoured rankings of alternatives was not studied thoroughly in the MADM process, despite the fact that these operators provide some motivation for solving MADM difficulties.The EDAS approach was initially defined by Keshavarz Ghorabaee et al. [44] and used it to multi-criteria inventory categorization issues. Kahraman et al. [45] developed a novel EDAS model for solid waste disposal site selection. Batool et al. [46] proposed the EDAS method with Pythagorean probabilistic hesitant information and discussed their applicability in decision making. Peng and Liu [47] proposed approaches for neutrosophic soft decision making depends on the EDAS algorithm using similarity measure. Karasan and Kahraman [48] developed the novel interval-valued neutrosophic EDAS approach. The EDAS method was used to develop the dynamic fuzzy approach for multi-criteria evaluation of subcontractors by Keshavarz-Ghorabaee et al. [49]. For more study, we refer to.In fuzzy sets and their extended fuzzy architectures, the t-norms and t-conorms are commonly acknowledged as important operations. The Acz-Als t-norm and t-conorm procedures, created by Aczel and Alsina, have the advantage of changeability by modifying a parameter [50]. Under the single valued neutrosophic framework, the Acz-Als t-norm and t-conorm procedures, [51–57] as well as a number of additional aggregation operators [58–60] (AOs), are proposed. In addition, to deal with uncertain data in complex real-life decision scenarios, an expanded EDAS (Evaluation based on Distance from Average Solution) method depends on the SVN Acz-Als aggregation operations is described.Our technique’s objectives are represented as follows:(1)
We devised certain Acz-Als operations for SVNNs in order to overcome the lack of algebraic, Einstein, and Hamacher processes and represent the relationship between the various SVNNs.(2)
In support of SVN data, we extended Acz-Als operators to SVN Acz-Als operators: SVN Aczel-Alsina weighted geometric (SV-NAWG) operator and SVN Aczel-Alsina order weighted geometric (SV-NAOWG) operator, which overcome the present operator’s drawbacks.(3)
Based on the suggested SVN Aczel-Alsina aggregation technique, we developed an expanded EDAS approach.(4)
Using SVN data, we developed an approach to deal with MAGDM difficulties.(5)
The suggested Aczel-Alsina aggregate operators and the EDAS technique are applied to the MAGDM issue in order to demonstrate their usability and reliability.(6)
The results suggest that the proposed process is more powerful and generates more authentic results when compared to existing methodologies.The remaining sections of the document are formatted in the following order: Under SVN information, Section2 provides some basic information on t-norms, SVNSs, and a few functional rules. Section 3 discusses the Aczel-Alsina working guidelines as well as the characteristics of SVNNs. In Section 4, we look at the various desirable qualities of several SVN Aczel-Alsina AOs. The next section develops an expanded EDAS method depends decision making algorithm using SVN Aczel-Alsina aggregation operations. In Section 6, we use an example to demonstrate the applicability of the suggested hybrid method. In Section 7, we examine at how a factor effects DM results. The comparison study for new and existing aggregating operators was developed in Section 8. Section 8 concludes the research work and elaborates future directions.


## 2. Preliminaries

We will look at some key topics in this section that will be significant in the creation of this article.Definition 1 (see [50]).
A mappingW^a℘℘∈0,∞ is a Acz-Als t-norm if(1)W^a℘Δ,ℏ=W^DΔ,ℏif℘=0minΔ,if℘=∞e−−ℓn'Δ℘+−ℓn'℘1/℘otherwise,where Δ,ð∈0,1, ℘ is +ve fixed and W^D is extreme t-norm defined as(2)W^DΔ,=ΔifΔ=1ifΔ=10otherwise..Definition 2 (see [50]).
A mappingβa℘℘∈0,∞ is a Acz-Als s-norm if(3)βa℘Δ,=βDΔ,if℘=0maxΔ,if℘=∞1−e−−ℓn′1−Δ℘+−ℓn′1−℘1/℘otherwise,where Δ,ð∈0,1, ℘ is positive constant and βD is drastic s-norm defined as(4)βDΔ,=ΔifΔ=0ifΔ=00otherwise..
For every℘∈0,∞, the t-norm W^a℘ and s-norm βa℘ are dual to each other.Definition 3 (see [35]).
A neutrosophic setΞ in a fixed set k is defined as(5)Ξ=O¯Ξ♭,ΔΞ♭,GΞ♭∈0−,1+|♭∈k,where O¯Ξ positive grade, ΔΞ neutral grade and GΞ negative grade of the value ♭ to neutrosophic set Ξ, satisfying 0−≤O¯Ξ+ΔΞ+GΞ≤3+, for each ♭∈k.Definition 4 (see [35]).
A single valued neutrosophic set (SV-NS)Ξ in k is defined as(6)Ξ=O¯Ξ♭,ΔΞ♭,GΞ♭∈0,1|♭∈k,where O¯Ξ positive grade, ΔΞ neutral grade and GΞ negative grade of the element ♭ to SV-NS Ξ, satisfying 0≤O¯Ξ+ΔΞ+GΞ≤3, for each ♭∈k.Definition 5 (see [35]).
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be two single valued neutrosophic numbers (SVNNs), where ℑ=1,2. Then:(1)
Ξ1⊆Ξ2 iff O¯Ξ1≤O¯Ξ2,ΔΞ1≤ΔΞ2 and GΞ1≥GΞ2 for all ♭∈k;(2)
Ξ1=Ξ2 if Ξ1⊆Ξ2 and Ξ2⊆Ξ1;(3)
Ξ1=Ξ2=minO¯Ξ1,O¯Ξ2,maxΔΞ1,ΔΞ2,maxGΞ1,GΞ2;(4)
Ξ1=Ξ2=maxO¯Ξ1,O¯Ξ2,minΔΞ1,ΔΞ2,minGΞ1,GΞ2;(5)
Ξ1c=GΞ1,ΔΞ1,O¯Ξ1.Definition 6.
(see [35]). Let Ξℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be two SVNNs, where ℑ=1,2. Then the operations about any two SVNNs are defined as follows:(1)
Ξ1⊕Ξ2=O¯Ξ1+O¯Ξ2−O¯Ξ1O¯Ξ2,ΔΞ1ΔΞ2,GΞ1GΞ2;(2)
Ξ1⊗Ξ2=O¯Ξ1O¯Ξ2,ΔΞ1+ΔΞ2−ΔΞ1.ΔΞ2,GΞ1+GΞ2−GΞ1.GΞ2;(3)
η⋅Ξ1=1−1−O¯Ξ1η,ΔΞ1η,GΞ1η,, η>0;(4)
Ξ1η=O¯Ξ1η,1−1−ΔΞ1η,1−1−GΞ1η,η>0.
On the basis of Definition6, Ashraf [6] derived several operations in the following ways:Definition 7.
(see [51]). Let Ξℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be a collection of SVNNs, where ℑ=1,2,...,n and η1,η1>0. Then(1)
Ξ1⊕Ξ2=Ξ2⊕Ξ1;(2)
Ξ1⊗Ξ2=Ξ2⊗Ξ1;(3)
η1Ξ1⊕Ξ2=η1Ξ1⊕η1Ξ2;(4)
Ξ1⊗Ξ2η1=Ξ1η1⊗Ξ2η1;(5)
η1Ξ1⊕η2Ξ1=η1+η2Ξ1;(6)
Ξ1η1⊗Ξ1η2=Ξ1η1+η2;(7)
Ξ1η1η2=Ξ1η1η2.Definition 8.
(see [43]). Let Ξ=O¯Ξ,ΔΞ,GΞ be SV-NN. Then the score QΞ and accuracy aΞ are given as follows:(1)
QΞ=O¯Ξ−ΔΞ−GΞ;(2)
aΞ=O¯Ξ+ΔΞ+GΞ.Definition 9.
(see [43]). Let Ξℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be two SVNNs, where ℑ=1,2. Then the comparison technique of SVNNs can be defined as:(1)
QΞ1>QΞ2 implies that Ξ1>Ξ2;(2)
QΞ1=QΞ2 and aΞ1>aΞ2 implies that Ξ1>Ξ2;(3)
QΞ1=QΞ2 and aΞ1=aΞ2 implies that Ξ1=Ξ2.
Ashraf et al. [6] developed the algebraic AO under SVNNs illustrate in the superseding definition.Definition 10.
(see [52]). Let Ξℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be a collection of SVNNs, where ℑ=1,2,...,ℓ. A single valued neutrosophic weighted geometric (SV-NWG) AO of dimention ℓ is a mapping ℑℓ⟶ℑ with weight vector ϖ=ϖ1,ϖ1,...,ϖℓT such that ϖℑ>0 and as(7)SV−NWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞℑϖℑ=∏ℑ=1ℓO¯Ξℑϖℑ,1−∏ℑ=1ℓ1−ΔΞℑϖℑ,1−∏ℑ=1ℓ1−GΞℑϖℑ.


## 3. Aczel–Alsina Operation for SVNNs

We discussed Acz-Als operations in relation to SVNNs, taking into account the Acz-Als t-norm and Acz-Als t-conorm.Definition 11.
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be two SVNNs, where ℑ=1,2 and ℘ is +ve fixed. Then Acz-Als norms depends operations for SVNNs are defined as follows:(1)
Ξ1⊕Ξ2=1−e−−ℓn′1−O¯Ξ1℘+−ℓn′1−O¯Ξ2℘1/℘,e−−ℓn'ΔΞ1℘+−ℓn'ΔΞ2℘1/℘,e−−ℓn′GΞ1℘+−ℓn′GΞ2℘1/℘;(2)
Ξ1⊗Ξ2=e−−ℓn′O¯Ξ1℘+−ℓn′O¯Ξ2℘1/℘,1−e−−ℓn′1−ΔΞ1℘+−ℓn′1−ΔΞ2℘1/℘,1−e−−ℓn′1−GΞ1℘+−ℓn′1−GΞ2℘1/℘;(3)
η⋅Ξ1=1−e−η−ℓn′1−O¯Ξ1℘1/℘,e−η−ℓn'ΔΞ1℘1/℘,e−η−ℓn′GΞ1℘1/℘,, η>0;(4)
Ξ1η=e−η−ℓn′O¯Ξ1℘1/℘,1−e−η−ℓn′1−ΔΞ1℘1/℘,1−e−η−ℓn′1−GΞ1℘1/℘,η>0.Theorem 1.
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be a collection of SVNNs, where ℑ=1,2,...,n and η1,η1>0. Then:(1)
Ξ1⊕Ξ2=Ξ2⊕Ξ1;(2)
Ξ1⊗Ξ2=Ξ2⊗Ξ1;(3)
η1Ξ1⊕Ξ2=η1Ξ1⊕η1Ξ2;(4)
Ξ1⊗Ξ2η1=Ξ1η1⊗Ξ2η1;(5)
η1Ξ1⊕η2Ξ1=η1+η2Ξ1;(6)
Ξ1η1⊗Ξ1η2=Ξ1η1+η2;(7)
Ξ1η1η2=Ξ1η1η2.Proof.
(1)
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be collection of SVNNs, where ℑ=1,2,...,n and η1,η1>0. Then by the Definition 11, it follows that(8)Ξ1⊕Ξ2=1−e−−ℓn′1−O¯Ξ1℘+−ℓn′1−O¯Ξ2℘1/℘,e−−ℓn'ΔΞ1℘+−ℓn'ΔΞ2℘1/℘,e−−ℓn′GΞ1℘+−ℓn′GΞ2℘1/℘=1−e−−ℓn′1−O¯Ξ2℘+−ℓn′1−O¯Ξ1℘1/℘,e−−ℓn'ΔΞ2℘+−ℓn'ΔΞ1℘1/℘,e−−ℓn′GΞ2℘+−ℓn′GΞ1℘1/℘=Ξ2⊕Ξ1.(2)
Utilizing the Definition11, it follows that(9)Ξ1⊗Ξ2=e−−ℓn′O¯Ξ1℘+−ℓn′O¯Ξ2℘1/℘,1−e−−ℓn′1−ΔΞ1℘+−ℓn′1−ΔΞ2℘1/℘,1−e−−ℓn′1−GΞ1℘+−ℓn′1−GΞ2℘1/℘=e−−ℓn′O¯Ξ2℘+−ℓn′O¯Ξ1℘1/℘,1−e−−ℓn′1−ΔΞ2℘+−ℓn′1−ΔΞ1℘1/℘,1−e−−ℓn′1−GΞ2℘+−ℓn′1−GΞ1℘1/℘=Ξ2⊗Ξ1.(3)
Utilizing the Definition11, it follows that(10)η1Ξ1⊕Ξ2=η11−e−−ℓn′1−O¯Ξ1℘+−ℓn′1−O¯Ξ2℘1/℘,e−−ℓn'ΔΞ1℘+−ℓn'ΔΞ2℘1/℘,e−−ℓn′GΞ1℘+−ℓn′GΞ2℘1/℘=1−e−η1−ℓn′1−O¯Ξ1℘+η1−ℓn′1−O¯Ξ2℘1/℘,e−η1−ℓn'ΔΞ1℘+η1−ℓn'ΔΞ2℘1/℘,e−η1−ℓn′GΞ1℘+η1−ℓn′GΞ2℘1/℘=1−e−η1−ℓn′1−O¯Ξ1℘1/℘,e−η1−ℓn'ΔΞ1℘1/℘,e−η1−ℓn′GΞ1℘1/℘⊕1−e−η1−ℓn′1−O¯Ξ1℘1/℘,e−η1−ℓn'ΔΞ1℘1/℘,e−η1−ℓn′GΞ1℘1/℘=η1Ξ1⊕η1Ξ2.(4)
It is similar of the proof of (3).(5)
Utilizing the Definition11, it follows that(11)η1Ξ1⊕η2Ξ1=1−e−η1−ℓn′1−O¯Ξ1℘1/℘,e−η1−ℓn'ΔΞ1℘1/℘,e−η1−ℓn′GΞ1℘1/℘⊕1−e−η2−ℓn′1−O¯Ξ1℘1/℘,e−η2−ℓn'ΔΞ1℘1/℘,e−η2−ℓn′GΞ1℘1/℘=1−e−η1−ℓn′1−O¯Ξ1℘+η2−ℓn′1−O¯Ξ1℘1/℘,e−η1−ℓn'ΔΞ1℘+η2−ℓn'ΔΞ1℘1/℘,e−η1−ℓn′GΞ1℘+η2−ℓn′GΞ1℘1/℘=1−e−η1+η2−ℓn′1−O¯Ξ1℘1/℘,e−η1+η2−ℓn'ΔΞ1℘1/℘,e−η1+η2−ℓn′GΞ1℘1/℘=η1+η2Ξ1.(6) and (7) are similar to the proof of (5).


## 4. Aczel–Alsina Geometric Aggregation Operators for SVNNs

Acz-Als norms depends list of novel AOs under single valued neutrosophic settings are construct in this section.Definition 12.
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be collection of SVNNs, where ℑ=1,2,...,ℓ. A SVN Acz-Als weighted geometric (SVNAWG) AO of dimention ℓ is a mapping with weight vector ϖ=ϖ1,ϖ1,...,ϖℓT such that ϖℑ>0 and ∑ℑ=1ℓϖℑ= as(12)SV−NAWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞℑϖℑ.Theorem 2.
SupposeΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be collection of SVNNs, where ℑ=1,2,...,ℓ. A SVN Acz-Als weighted geometric (SVNAWG) AO of dimention ℓ is a mapping with weight vector ϖ=ϖ1,ϖ1,...,ϖℓT such that ϖℑ>0 and ∑ℑ=1ℓϖℑ= is defined as:(13)SV−NAWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞℑϖℑ=e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−GΞℑ℘1/℘.Proof.
Theorem’s 2 proof is derived by implemention of induction method as follows:Step 1.
forℓ=2, we have(14)SV−NAWGΞ1,Ξ2=Ξ1ϖ1⊗Ξ2ϖ2.
Utilizing the Definition11, it follows that(15)Ξ1ϖ1=e−ϖ1−ℓn′O¯Ξ1℘1/℘,1−e−ϖ1−ℓn′1−ΔΞ1℘1/℘,1−e−ϖ1−ℓn′1−GΞ1℘1/℘,and(16)Ξ2ϖ2=e−ϖ2−ℓn′O¯Ξ2℘1/℘,1−e−ϖ2−ℓn′1−ΔΞ2℘1/℘,1−e−ϖ2−ℓn′1−GΞ2℘1/℘.
Therefore(17)SV−NAWGΞ1,Ξ2=e−ϖ1−ℓn′O¯Ξ1℘1/℘,1−e−ϖ1−ℓn′1−ΔΞ1℘1/℘,1−e−ϖ1−ℓn′1−GΞ1℘1/℘⊗e−ϖ2−ℓn′O¯Ξ2℘1/℘,1−e−ϖ2−ℓn′1−ΔΞ2℘1/℘,1−e−ϖ2−ℓn′1−GΞ2℘1/℘=e−ϖ1−ℓn′O¯Ξ1℘+ϖ2−ℓn′O¯Ξ2℘1/℘,1−e−ϖ1−ℓn′1−ΔΞ1℘+ϖ2−ℓn′1−ΔΞ2℘1/℘,1−e−ϖ1−ℓn′1−GΞ1℘+ϖ2−ℓn′1−GΞ2℘1/℘=e−∑ℑ=12ϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=12ϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=12ϖℑ−ℓn′1−GΞℑ℘1/℘.Thus, Theorem 2 is valid. ℓ=2.
Now, we assume that, Theorem2 is valid for ℓ=d, that is(18)SV−NAWGΞ1,Ξ2,…,Ξd=e−∑ℑ=1dϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−GΞℑ℘1/℘.
Now we prove that Theorem2 is valid for ℓ=d+1. That is, we prove(19)SV−NAWGΞ1,Ξ2,…,Ξd,Ξd+1=∏ℑ=1ℓΞℑϖℑ⊗Ξd+1ϖd+1∏ℑ=1ℓΞℑϖℑ⊗Ξd+1ϖd+1=e−∑ℑ=1dϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−GΞℑ℘1/℘⊗e−ϖd+1−ℓn′O¯Ξd+1℘1/℘,1−e−ϖd+1−ℓn′1−ΔΞd+1℘1/℘,1−e−ϖd+1−ℓn′1−GΞd+1℘1/℘=e−∑ℑ=1dϖℑ−ℓn′O¯Ξℑ℘+ϖd+1−ℓn′O¯Ξd+1℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−ΔΞℑ℘+ϖd+1−ℓn′1−ΔΞd+1℘1/℘,1−e−∑ℑ=1dϖℑ−ℓn′1−GΞℑ℘+ϖd+1−ℓn′1−GΞd+1℘1/℘=e−∑ℑ=1d+1ϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1d+1ϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1d+1ϖℑ−ℓn′1−GΞℑ℘1/℘.
Prove that, Theorem2 is valid for all ℓ.
Using the operator SVNAWG, we can efficiently illustrate the following characteristics.Theorem 3.
(Idempotency) LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑℑ=1,2,...,ℓ be a collection of equivalent SVNNs, Ξℑ=Ξ for each ℑ=1,2,...,ℓ. Then(20)SV−NAWGΞ1,Ξ2,…,Ξℓ=Ξ.Proof.
Since(21)SV−NAWGΞ1,Ξ2,…,Ξℓ=e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−GΞℑ℘1/℘.
Put(22)Ξℑ=O¯Ξℑ,ΔΞℑ,GΞℑ=Ξℑ=1,2,...,ℓ,
we have(23)SV−NAWGΞ1,Ξ2,…,Ξℓ=e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−GΞ℘1/℘=e−−ℓn′O¯Ξ℘1/℘,1−e−−ℓn′1−ΔΞ℘1/℘,1−e−−ℓn′1−GΞ℘1/℘=O¯Ξ,ΔΞ,GΞ=Ξ.Thus, SV−NAWGΞ1,Ξ2,...,Ξℓ=Ξ holds.Theorem 4.
(Boundedness) LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑℑ=1,2,...,ℓ be collection of SVNNs. Let Ξℑ−=minℑO¯Ξℑ,maxℑΔΞℑ,maxℑGΞℑ and Ξℑ+=maxℑO¯Ξℑ,minℑΔΞℑ,minℑGΞℑℑ=1,2,...,ℓ. Then,(24)Ξℑ−≤SV−NAWGΞ1,Ξ2,…,Ξℓ≤Ξℑ+.Proof.
SinceminℑO¯Ξℑ≤O¯Ξℑ≤maxℑO¯Ξℑ, it follows that(25)e−∑ℑ=1ℓϖℑ−ℓn′minO¯Ξℑ℘1/℘≤e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξℑ℘1/℘≤e−∑ℑ=1ℓϖℑ−ℓn′maxO¯Ξℑ℘1/℘.
Similarly(26)1−e−∑ℑ=1ℓϖℑ−ℓn′max1−ΔΞℑ℘1/℘≤1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞℑ℘1/℘≤1−e−∑ℑ=1ℓϖℑ−ℓn′min1−ΔΞℑ℘1/℘.
Now we have(27)1−e−∑ℑ=1dϖℑ−ℓn′max1−GΞℑ℘1/℘≤1−e−∑ℑ=1dϖℑ−ℓn′1−GΞℑ℘1/℘≤1−e−∑ℑ=1dϖℑ−ℓn′min1−GΞℑ℘1/℘.
Therefore(28)Ξℑ−≤SV−NAWGΞ1,Ξ2,…,Ξℓ≤Ξℑ+.Theorem 5.
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ and Ξℑ∗=O¯Ξℑ∗,ΔΞℑ∗,GΞℑ∗ℑ=1,2,...,ℓ be two collections of SVNNs. If Ξℑ≤Ξℑ∗ for ℑ=1,2,...,ℓ. Then,(29)SV−NAWGΞ1,Ξ2,…,Ξℓ≤SV−NAWGΞ1∗,Ξ2∗,…,Ξℓ∗.Proof.
The proof is straightforward.Definition 13.
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be a collection of SVNNs, where ℑ=1,2,...,ℓ. An SVN Acz-Als ordered weighted geometric (SV-NAOWG) AO of dimention ℓ is a mapping ℑℓ⟶ℑ with ϖ=ϖ1,ϖ1,...,ϖℓT such that ϖℑ>0 and ∑ℑ=1ℓϖℑ= as(30)SV−NAOWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞτℑϖℑ,where τ1,τ2,...,τℓ are the permutation in such a way that. Ξτℑ≤Ξτℑ−1.Theorem 6.
SupposeΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ be a collection of SVNNs, where ℑ=1,2,...,ℓ. An SVN Acz-Als ordered weighted geometric (SV-NAOWG) AO of dimention ℓ is a mapping ℑℓ⟶ℑ with weight vector ϖ=ϖ1,ϖ1,...,ϖℓT such that ϖℑ>0 and ∑ℑ=1ℓϖℑ= is defined as:(31)SV−NAOWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞτℑϖℑ=e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξτℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞτℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−GΞτℑ℘1/℘,where τ1,τ2,...,τℓ are the permutation in such a way as. Ξτℑ≤Ξτℑ−1.Using the operator SV-NAOWG, we can efficiently illustrate the following characteristics.Theorem 7.
(1)
(Idempotency) LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑℑ=1,2,...,ℓ be collection of equivalent SVNNs, Ξℑ=Ξ for each ℑ=1,2,...,ℓ. Then(32)SV−NAOWGΞ1,Ξ2,…,Ξℓ=Ξ.(2)
(Boundedness) LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑℑ=1,2,...,ℓ be a collection of SVNNs. Let Ξℑ−=minℑO¯Ξℑ,maxℑΔΞℑ,maxℑGΞℑ and Ξℑ+=maxℑO¯Ξℑ,minℑΔΞℑ,minℑGΞℑℑ=1,2,...,ℓ. Then(33)Ξℑ−≤SV−NAOWGΞ1,Ξ2,…,Ξℓ≤Ξℑ+.(3)
LetΞℑ=O¯Ξℑ,ΔΞℑ,GΞℑ and Ξℑ∗=O¯Ξℑ∗,ΔΞℑ∗,GΞℑ∗ℑ=1,2,...,ℓ be two collection of SVNNs. If Ξℑ≤Ξℑ∗ for ℑ=1,2,...,ℓ. Then(34)SV−NAOWGΞ1,Ξ2,…,Ξℓ≤SV−NAOWGΞ1∗,Ξ2∗,…,Ξℓ∗.Proof.
Using Theorem2, 3 and 4, the proof is straightforward.


## 5. EDAS Method Based on SVN Aczel–Alsina Aggregation Information

A novel extended EDAS approach is built here to handle the complex uncertain data in real-life D-S issues in order to validate the effectiveness of the SVN Acz-Als geometric AOs. The following are the particular measures to take.Assume, that there is a set ofℓ alternatives, and an acceptable rating by the attributes ℝ1,ℝ2,...,ℝm. Then, the usefulness of different attributes ℝıı=1,2,...,m is specified by ϖ=ϖ1,ϖ1,...,ϖmT such that ϖı>0 and ∑ı=1mϖı=.LetΞℑı=O¯Ξℑı,ΔΞℑı,GΞℑı for O¯Ξℑı, ΔΞℑı, GΞℑı∈0,1 be the acceptable rating of every attribute for each alternative, where O¯Ξℓm depicts the +ve grade function the alternative (ℑ=1,2,…,ℓ) satisfies ℝıı=1,2,...,m. ΔΞℓm and GΞℓm designate the nutral grade function and −ve grade function, respectively. The decision matrix of SVNNs can be obtained based from assessment data: Ξ=Ξℑıℓm.The procedure for determining the best alternative using EDAS methodology depends on SVN Acz-Als aggregation information is prsented as the following steps: Step-1: Select a series of attributes that are suitable for use in evaluating the problem under consideration:The prospective assessment characteristics are gathered via a study of the literature, and an expert DM committee is created to screen the attributes in order to develop an acceptable collection of evaluation attributesℝıı=1,2,...,m.(35)Dℑ×ı=R1R2⋮Rℓℝ1ℝ2ℝmO¯Ξ11,ΔΞ11,GΞ11O¯Ξ12,ΔΞ12,GΞ12...O¯Ξ1m,ΔΞ1m,GΞ1mO¯Ξ21,ΔΞ21,GΞ21O¯Ξ22,ΔΞ22,GΞ22...O¯Ξ2m,ΔΞ2m,GΞ2m⋮⋮⋱⋮O¯Ξℓ1,ΔΞℓ1,GΞℓ1O¯Ξℓ2,ΔΞℓ2,GΞℓ2...O¯Ξℓm,ΔΞℓm,GΞℓm.Step 2.
Normalization is used to obtain the normalised decision matrix as:(36)Nℑ×ı=O¯Ξℑı,ΔΞℑı,GΞℑıifCIGΞℑı,ΔΞℑı,O¯ΞℑıifCII,where CI refers to “if ℝıı=1,2,...,m is a benefit criterion” and CII refers to “if ℝıı=1,2,...,m is a cost criterion”.Step 3.
Aggregated Data: Established SVN Acz-Als operators are utilized to aggregate the specialists uncertain data of D-S problems.(37)SV−NAWGΞ1,Ξ2,…,Ξℓ=∏ℑ=1ℓΞℑϖℑ=e−∑ℑ=1ℓϖℑ−ℓn′O¯Ξℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−ΔΞℑ℘1/℘,1−e−∑ℑ=1ℓϖℑ−ℓn′1−GΞℑ℘1/℘.Step 4.
Determine the average solution (AVS) depends on all provided characteristics:(38)AVS=AVSı1×m=∑ℑ=1ℓΞℑıℓ1×m
Based on Definition11, we get(39)AVS=AVSı1×m=∑ℑ=1ℓΞℑıℓ1×m=1−e−∑ℑ=1ℓ1/ℓ−ℓn′1−O¯Ξℑı℘1/℘,e−∑ℑ=1ℓ1/ℓ−ℓn′ΔΞℑı℘1/℘,e−∑ℑ=1ℓ1/ℓ−ℓn′GΞℑı℘1/℘Step 5.
According to the results ofAVS, we can compute the positive distance from average (PDAv) and negative distance from average (NDAv) by using the following formula:(40)PDAvℑı=max0,Ξℑı−AVSıAVSı,NDAvℑı=max0,AVSı−ΞℑıAVSı.
We may adopt the score function of SVNNs described in Definition8 to compute the PDA and NDA as follows:(41)PDAvℑı=max0,QΞℑı−QAVSıQAVSı,NDAvℑı=max0,QAVSı−QΞℑıQAVSı,where Q represented the score value.Step 6.
Determine SPDA and SNDA, which represent the weighted average of PDA and NDA as follows:(42)SPDAℑ=∑ı=1mϖıPDAvℑı,SNDAℑ=∑ı=1mϖıNDAvℑı,Where ϖı∈0,1,∑ı=1mϖı=1.Step 7.
Normalize weighted sum of PDA and NDA is denoted and defined as respectively:(43)NSPDAℑ=SPDAℑmaxSPDAℑ,NSNDAℑ=SNDAℑmaxSNDAℑ.Step 8.
Compute the values of appraisal score (ASC) depends on each alternative’s as:(44)ASCℑ=12NSPDAℑ+1−NSNDAℑ.Step 9.
Depending on the calculating values ofASC,alternatives are ranked in a decreasing order, and the bigger value of ASCℑ is the best alternative selected will be.


## 6. Numerical Illustration of EDAS Method

In order to authenticate the effectiveness and appropriate of the developed technique, We call on Akram and Khan [53]’s example related to the role of commercial banks in providing loans to their customers to validate the MAGDM method, and perform a sensitivity analysis and comparative analysis with other existing methods. The problem is described below.


### 6.1. Case Study

Low pricing are offered to clients by commercial banks. Like wholesale corporations, clients purchase large quantities and then resell them at a lower price to other customers. Discounts include free checking, no fees for creating accounts, and reduced interest rates for real estate loans. A debit card, credit card, or both are all options available at commercial banks, as are investment accounts, commercial real estate loans, and mortgage plans. In addition to managing their checking and savings accounts, customers may use Internet banking to pay bills, move money between accounts, apply for short- and long-term loans, and more. Using a 24-hour ATM, a client may maintain control of their accounts even if their bank is closed. Long and short-term loans are both available from commercial banks. Long-term loans are available to help a variety of businesses get their start-up capital. Suppose a customer is looking for financing for his food business. He had a list of five banks and wanted to check out. He called a DM specialist to help him choose a bank that could lend him money, and the expert looked at the following criteria:(1)
Markup and penalty (ℝ1);(2)
Customers’ guarantee requirements and regulations (ℝ2);(3)
Paperwork costs and customer packagesℝ3;(4)
Procedural time and credit time (ℝ4);(5)
safe and secure banking (ℝ5).In this evaluation, experts were requested to utilize SV neutrosophic information to determine the best bank for business loans.Table1 summarizes the expert assessment data for SVNNs:Table 1
Expert evaluation information.
R1R2R3R4R5∅10.5,0.3,0.40.3,0.2,0.50.2,0.2,0.60.4,0.2,0.30.3,0.3,0.4∅20.7,0.1,0.30.3,0.2,0.70.6,0.3,0.20.2,0.4,0.60.7,0.1,0.2∅30.5,0.3,0.40.4,0.2,0.60.6,0.1,0.20.3,0.1,0.50.6,0.4,0.3∅40.7,0.3,0.20.2,0.2,0.70.4,0.5,0.20.2,0.2,0.50.4,0.5,0.4∅50.4,0.1,0.30.2,0.1,0.50.4,0.1,0.50.6,0.3,0.40.3,0.2,0.4Cost type SVN information is given inI2 and I4. Therefore the normalised expert evaluation decision information matrix is enclosed in Table 2:Table 2
Normalized expert data.
R1R2R3R4R5∅10.5,0.3,0.40.5,0.2,0.30.2,0.2,0.60.3,0.2,0.40.3,0.3,0.4∅20.7,0.1,0.30.7,0.2,0.30.6,0.3,0.20.6,0.4,0.20.7,0.1,0.2∅30.5,0.3,0.40.6,0.2,0.40.6,0.1,0.20.5,0.1,0.30.6,0.4,0.3∅40.7,0.3,0.20.7,0.2,0.20.4,0.5,0.20.5,0.2,0.20.4,0.5,0.4∅50.4,0.1,0.30.5,0.1,0.20.4,0.1,0.50.4,0.3,0.60.3,0.2,0.4Because there is just one expert in this case study, we do not need to calculate the cumulative decision matrix.As per Table2, we may use formula (3) to get the AVS depends on all provided characteristics.(45)AVS2=1−e−1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.551/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.151/5e−1/5∗−ℓog0.35+1/5∗−ℓog0.35+1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.251/5=0.447278,0.157468,0.245549.AVS3=1−e−1/5∗−ℓog1−0.25+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.45+1/5∗−ℓog1−0.451/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.15+1/5∗−ℓog0.55+1/5∗−ℓog0.151/5e−1/5∗−ℓog0.65+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.551/5=0.69715,0.141507,0.23343.AVS4=1−e−1/5∗−ℓog1−0.35+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.451/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.45+1/5∗−ℓog0.15+1/5∗−ℓog0.25+1/5∗−ℓog0.351/5e−1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.25+1/5∗−ℓog0.651/5=0.606536,0.168535,0.252183.AVS5=1−e−1/5∗−ℓog1−0.35+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.45+1/5∗−ℓog1−0.351/5e−1/5∗−ℓog0.35+1/5∗−ℓog0.15+1/5∗−ℓog0.45+1/5∗−ℓog0.55+1/5∗−ℓog0.251/5e−1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.45+1/5∗−ℓog0.451/5=0.642351,0.176204,0.28647.Then we can get the value ofAVSı as(46)AVSı=0.513968,0.143849,0.278161,0.447278,0.157468,0.245549,0.69715,0.141507,0.23343,0.606536,0.168535,0.252183,0.642351,0.176204,0.28647.According to the results of average solutionAVSı, we can compute the positive distance from average (PDA v ) and negative distance from average (NDA v ) by using formula (5) which are listed in Tables 3–5.Table 4
The valus ofPDAv.
R1R2R3R4R5∅10.00.00.00.00.0∅22.2623.5180.00.01.226∅30.00.00.00.00.0∅41.1745.7770.00.00.0∅50.03.5180.00.00.0Table 3
Score value ofZℑı and AVSı.
R1R2R3R4R5∅10.20−0.6−0.3−0.4∅20.30.20.100.4∅3−0.200.30.1−0.1∅40.20.3−0.30.1−0.5∅500.2−0.2−0.5−0.3AVSı0.09190.04420.32220.18580.1796Table 5
The valus ofNDAv.
R1R2R3R4R5∅13.1741.002.8622.6143.226∅20.000.000.6891.000.00∅33.1741.000.0680.4613.782∅40.000.001.9310.4613.782∅51.000.001.6203.6902.669Calculate the values ofSPDAℑ and SNDAℑ by (6) and attributes weighting vector ϖ=0.15,0.28,0.20,0.22,0.15, we can obtain the results as:(47)SPDA1=0.00SPDA2=1.50SPDA3=0.00SPDA4=1.79SPDA5=0.98SNDA1=2.38SNDA1=0.35SNDA1=1.10SNDA1=1.05SNDA1=1.68NSNDA1=2.16NSNDA2=0.323NSND3=1.00NSNDA4=0.95NSNDA5=1.526.Normalize weighted sum of PDA and NDA are evaluated by (7) as follows:(48)NSPDA1=0.00NSPDA2=1.53NSPDA3=0.00NSPDA4=1.82NSPDA5=1.00.Compute the values of appraisal score (ASC) depends on each alternative's NSPDA and NSNDA as:(49)ASC1=−0.580ASC2=1.103ASC3=0.000ASC4=0.933ASC5=0.236.Based on ASC results, we may rate all of the possibilities; the greater ASC, the better the alternative selected. Obviously, ranking results are and is the best alternative.


## 6.1. Case Study

Low pricing are offered to clients by commercial banks. Like wholesale corporations, clients purchase large quantities and then resell them at a lower price to other customers. Discounts include free checking, no fees for creating accounts, and reduced interest rates for real estate loans. A debit card, credit card, or both are all options available at commercial banks, as are investment accounts, commercial real estate loans, and mortgage plans. In addition to managing their checking and savings accounts, customers may use Internet banking to pay bills, move money between accounts, apply for short- and long-term loans, and more. Using a 24-hour ATM, a client may maintain control of their accounts even if their bank is closed. Long and short-term loans are both available from commercial banks. Long-term loans are available to help a variety of businesses get their start-up capital. Suppose a customer is looking for financing for his food business. He had a list of five banks and wanted to check out. He called a DM specialist to help him choose a bank that could lend him money, and the expert looked at the following criteria:(1)
Markup and penalty (ℝ1);(2)
Customers’ guarantee requirements and regulations (ℝ2);(3)
Paperwork costs and customer packagesℝ3;(4)
Procedural time and credit time (ℝ4);(5)
safe and secure banking (ℝ5).In this evaluation, experts were requested to utilize SV neutrosophic information to determine the best bank for business loans.Table1 summarizes the expert assessment data for SVNNs:Table 1
Expert evaluation information.
R1R2R3R4R5∅10.5,0.3,0.40.3,0.2,0.50.2,0.2,0.60.4,0.2,0.30.3,0.3,0.4∅20.7,0.1,0.30.3,0.2,0.70.6,0.3,0.20.2,0.4,0.60.7,0.1,0.2∅30.5,0.3,0.40.4,0.2,0.60.6,0.1,0.20.3,0.1,0.50.6,0.4,0.3∅40.7,0.3,0.20.2,0.2,0.70.4,0.5,0.20.2,0.2,0.50.4,0.5,0.4∅50.4,0.1,0.30.2,0.1,0.50.4,0.1,0.50.6,0.3,0.40.3,0.2,0.4Cost type SVN information is given inI2 and I4. Therefore the normalised expert evaluation decision information matrix is enclosed in Table 2:Table 2
Normalized expert data.
R1R2R3R4R5∅10.5,0.3,0.40.5,0.2,0.30.2,0.2,0.60.3,0.2,0.40.3,0.3,0.4∅20.7,0.1,0.30.7,0.2,0.30.6,0.3,0.20.6,0.4,0.20.7,0.1,0.2∅30.5,0.3,0.40.6,0.2,0.40.6,0.1,0.20.5,0.1,0.30.6,0.4,0.3∅40.7,0.3,0.20.7,0.2,0.20.4,0.5,0.20.5,0.2,0.20.4,0.5,0.4∅50.4,0.1,0.30.5,0.1,0.20.4,0.1,0.50.4,0.3,0.60.3,0.2,0.4Because there is just one expert in this case study, we do not need to calculate the cumulative decision matrix.As per Table2, we may use formula (3) to get the AVS depends on all provided characteristics.(45)AVS2=1−e−1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.551/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.151/5e−1/5∗−ℓog0.35+1/5∗−ℓog0.35+1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.251/5=0.447278,0.157468,0.245549.AVS3=1−e−1/5∗−ℓog1−0.25+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.45+1/5∗−ℓog1−0.451/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.15+1/5∗−ℓog0.55+1/5∗−ℓog0.151/5e−1/5∗−ℓog0.65+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.25+1/5∗−ℓog0.551/5=0.69715,0.141507,0.23343.AVS4=1−e−1/5∗−ℓog1−0.35+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.55+1/5∗−ℓog1−0.451/5e−1/5∗−ℓog0.25+1/5∗−ℓog0.45+1/5∗−ℓog0.15+1/5∗−ℓog0.25+1/5∗−ℓog0.351/5e−1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.25+1/5∗−ℓog0.651/5=0.606536,0.168535,0.252183.AVS5=1−e−1/5∗−ℓog1−0.35+1/5∗−ℓog1−0.75+1/5∗−ℓog1−0.65+1/5∗−ℓog1−0.45+1/5∗−ℓog1−0.351/5e−1/5∗−ℓog0.35+1/5∗−ℓog0.15+1/5∗−ℓog0.45+1/5∗−ℓog0.55+1/5∗−ℓog0.251/5e−1/5∗−ℓog0.45+1/5∗−ℓog0.25+1/5∗−ℓog0.35+1/5∗−ℓog0.45+1/5∗−ℓog0.451/5=0.642351,0.176204,0.28647.Then we can get the value ofAVSı as(46)AVSı=0.513968,0.143849,0.278161,0.447278,0.157468,0.245549,0.69715,0.141507,0.23343,0.606536,0.168535,0.252183,0.642351,0.176204,0.28647.According to the results of average solutionAVSı, we can compute the positive distance from average (PDA v ) and negative distance from average (NDA v ) by using formula (5) which are listed in Tables 3–5.Table 4
The valus ofPDAv.
R1R2R3R4R5∅10.00.00.00.00.0∅22.2623.5180.00.01.226∅30.00.00.00.00.0∅41.1745.7770.00.00.0∅50.03.5180.00.00.0Table 3
Score value ofZℑı and AVSı.
R1R2R3R4R5∅10.20−0.6−0.3−0.4∅20.30.20.100.4∅3−0.200.30.1−0.1∅40.20.3−0.30.1−0.5∅500.2−0.2−0.5−0.3AVSı0.09190.04420.32220.18580.1796Table 5
The valus ofNDAv.
R1R2R3R4R5∅13.1741.002.8622.6143.226∅20.000.000.6891.000.00∅33.1741.000.0680.4613.782∅40.000.001.9310.4613.782∅51.000.001.6203.6902.669Calculate the values ofSPDAℑ and SNDAℑ by (6) and attributes weighting vector ϖ=0.15,0.28,0.20,0.22,0.15, we can obtain the results as:(47)SPDA1=0.00SPDA2=1.50SPDA3=0.00SPDA4=1.79SPDA5=0.98SNDA1=2.38SNDA1=0.35SNDA1=1.10SNDA1=1.05SNDA1=1.68NSNDA1=2.16NSNDA2=0.323NSND3=1.00NSNDA4=0.95NSNDA5=1.526.Normalize weighted sum of PDA and NDA are evaluated by (7) as follows:(48)NSPDA1=0.00NSPDA2=1.53NSPDA3=0.00NSPDA4=1.82NSPDA5=1.00.Compute the values of appraisal score (ASC) depends on each alternative's NSPDA and NSNDA as:(49)ASC1=−0.580ASC2=1.103ASC3=0.000ASC4=0.933ASC5=0.236.Based on ASC results, we may rate all of the possibilities; the greater ASC, the better the alternative selected. Obviously, ranking results are and is the best alternative.


## 7. Comparison of EDAS Method with Some Aggregation Operators under SVNNs

In order to evaluate the established EDAS approach's potential and effectiveness and to compare it with new discoveries, we've included some applicable instances below.The expert evaluation information under SVNNs is given in Table1.Now we apply proposed single valued neutrosophic Acz-Als geometric AOs to adrress the uncertainty and chose the best alternative as follows in Table6:Table 6
SVN Acz-A1s geometric AOSVNAWG.
∅10.354965,0.228894,0.413376∅20.658326,0.226143,0.24175∅30.562678,0.194736,0.321488∅40.54847,0.316751,0.228029∅50.410712,0.152477,0.392657Tables7–9 illustrate how existing techniques of AOs are used to aggregate the ambiguous information.Table 7
Exisiting Aggregated SVN information.
SVNWA [54]SVNOWA [54]NWA [55]SVNFWA [42]∅10.37,0.22,0.400.38,0.24,0.400.37,0.23,0.420.37,0.22,0.40∅20.66,0.20,0.230.66,0.18,0.240.66,0.24,0.240.66,0.20,0.23∅30.56,0.17,0.310.55,0.18,0.310.56,0.21,0.320.56,0.17,0.31∅40.57,0.29,0.220.57,0.31,0.220.57,0.33,0.230.56,0.29,0.22∅50.41,0.14,0.360.39,0.13,0.360.41,0.16,0.410.41,0.14,0.36Table 8
Existing aggregated SVN information.
SVNHWA [40]SVNHWA [40]L−SVNWA [55]L−SVNOWA [55]∅10.37,0.22,0.400.36,0.22,0.400.31,0.17,0.350.32,0.19,0.36∅20.66,0.20,0.230.66,0.20,0.230.64,0.19,0.230.65,0.17,0.23∅30.56,0.17,0.310.56,0.30,0.220.49,0.17,0.330.48,0.18,0.33∅40.56,0.29,0.220.56,0.30,0.220.55,0.27,0.190.55,0.29,0.19∅50.41,0.14,0.360.41,0.14,0.370.28,0.12,0.370.24,0.12,0.38Table 9
Existing aggregated SVN information.
ST−SVNWA [43]ST−SVNWG [43]ST−SVNOWA [43]ST−SVNOWG [43]∅10.56,0.02,0.070.50,0.02,0.080.56,0.02,0.080.51,0.03,0.08∅20.86,0.02,0.020.85,0.03,0.030.86,0.01,0.020.85,0.02,0.03∅30.77,0.01,0.040.76,0.02,0.050.77,0.01,0.040.76,0.03,0.05∅40.78,0.04,0.020.73,0.06,0.020.78,0.04,0.020.72,0.06,0.02∅50.60,0.09,0.060.59,0.01,0.080.58,0.09,0.060.57,0.01,0.08Now, according to collective data, the overall ranking of alternative is in the following Table10.Table 10
Overall ranking of the alternatives.
Exsisting operationsRankingBest alternativeNWA [55]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNWA [54]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNOWA [54]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNFWA [42]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNHWA for γ=2 [40]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNHWA for γ=3 [40]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2L−SVNWA [55]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2L−SVNOWA [55]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2ST−SVNWA [43]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2ST−SVNWG [43]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2ST−SVNOWA [43]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2ST−SVNOWG [43]Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2SVNAWG (proposed)Q∅2>Q∅3>Q∅1>Q∅5>Q∅4∅2Based on the outcomes of the recommended operators and the previously employed methodologies, we may deduce that the ranking lists are identical. Based on Acz-Als aggregation operations depends EDAS technique is generalized and new way to dealing with uncertainty in DM situations. It is more flexible and economical in real-world issues to use Acz-Als norm-depends aggregate operators in a single valued NS context.


### 7.1. Sensitivity Analysis

Using the given SVN Acz-Als aggregation approaches, we change the parameter℘ value from 0 to 30 in this section to investigate the different patterns of scores and ranking of the alternatives. The findings from suggested SVNAWG operator is listed in Table 11. The obtained results illustrate to decision-makers that they can obtain the best option depends on their preferences.Table 11
Sensitivity analysis of parameter℘.
ϒOperatorsScoreRanking best alternativeQ∅1Q∅2Q∅3Q∅4Q∅5⟶0.2SVNAWG−0.2740.2030.0600.025−0.113∅2⟶1SVNAWG−0.3090.1680.022−0.034−0.168∅2⟶2SVNAWG−0.3520.129−0.022−0.107−0.232∅2⟶5SVNAWG−0.4640.051−0.116−0.265−0.357∅2⟶10SVNAWG−0.562−0.010−0.189−0.372−0.450∅2⟶15SVNAWG−0.606−0.038−0.223−0.413−0.494∅2⟶30SVNAWG−0.652−0.068−0.260−0.456−0.546∅2Attribute analysis within a few previous techniques are as follows in Table12.Table 12
Comparision.
TechniquesWhether utilized fuzzy dataWhether make a data aggregationthrough parameterPeng et al. [54]YesNoNancy and Gar [55]YesNoDeveloped methodYesYes


## 7.1. Sensitivity Analysis

Using the given SVN Acz-Als aggregation approaches, we change the parameter℘ value from 0 to 30 in this section to investigate the different patterns of scores and ranking of the alternatives. The findings from suggested SVNAWG operator is listed in Table 11. The obtained results illustrate to decision-makers that they can obtain the best option depends on their preferences.Table 11
Sensitivity analysis of parameter℘.
ϒOperatorsScoreRanking best alternativeQ∅1Q∅2Q∅3Q∅4Q∅5⟶0.2SVNAWG−0.2740.2030.0600.025−0.113∅2⟶1SVNAWG−0.3090.1680.022−0.034−0.168∅2⟶2SVNAWG−0.3520.129−0.022−0.107−0.232∅2⟶5SVNAWG−0.4640.051−0.116−0.265−0.357∅2⟶10SVNAWG−0.562−0.010−0.189−0.372−0.450∅2⟶15SVNAWG−0.606−0.038−0.223−0.413−0.494∅2⟶30SVNAWG−0.652−0.068−0.260−0.456−0.546∅2Attribute analysis within a few previous techniques are as follows in Table12.Table 12
Comparision.
TechniquesWhether utilized fuzzy dataWhether make a data aggregationthrough parameterPeng et al. [54]YesNoNancy and Gar [55]YesNoDeveloped methodYesYes


## 8. Conclusion

In this study, we extended Acz-Als t-norm and t-conorm to single-valued neutrosophic scenarios, suggested a few novel working rules for SVNNs, and studied their properties and linkages in this research. This led to the development of an additional set of aggregate operators, including an Acz-Als depends approach for dealing with conflicts in SVNNs, which were introduced at that time. Various attractive qualities and special circumstances of these operators, as well as the connections between these operators, have recently been investigated in more depth. The suggested operators and decision making methodology, along with SVN data, were placed on MAGDM problems, and to demonstrate the DM procedure, a mathematical formulation was offered. The influence of parameter℘ on the results of DM has been investigated. Operators may be used to find the best solution by adjusting a parameter ℘. This means that decision-makers now have a more flexible way for addressing the problems of SVN-MAGDM. The aggregation process is more clearly observable when a parameter is provided, making it easier to represent ambiguous information than other existing methodologies. The existing AOs [54, 55], on the other hand, do not make data aggregation more flexible. It is because of this that our suggested AOs in SVN data DM are more complex and trustworthy.We will investigate the use of Acz-Als weighted AOs of SVNNs in additional domains, such as intelligent manufacturing, machine learning, and data mining, in future research.



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*Source: 2394472-2022-10-10.xml*