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

Ashraf, Shahzaib; Ahmad, Shakoor; Naeem, Muhammad; Riaz, Muhammad; Alam, Md. Ashraful

doi:https://doi.org/10.1155/2022/2394472

Complexity

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Ethical Approval Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Uncertainty Models in Complex Systems 2022

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 2394472 | https://doi.org/10.1155/2022/2394472

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

Shahzaib Ashraf,¹Shakoor Ahmad,²Muhammad Naeem,³Muhammad Riaz,⁴and Md. Ashraful Alam⁵

Academic Editor: Zeljko Stevic

Received27 Jun 2022

Revised24 Jul 2022

Accepted01 Sept 2022

Published10 Oct 2022

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.

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, Section 2 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 mapping is a Acz-Als t-norm ifwhere , is fixed and is extreme t-norm defined as

Definition 2 (see [50]). A mapping is a Acz-Als s-norm ifwhere , is positive constant and is drastic s-norm defined asFor every the t-norm and s-norm are dual to each other.

Definition 3 (see [35]). A neutrosophic set in a fixed set is defined aswhere positive grade, neutral grade and negative grade of the value to neutrosophic set satisfying for each .

Definition 4 (see [35]). A single valued neutrosophic set (SV-NS) in is defined aswhere positive grade, neutral grade and negative grade of the element to SV-NS satisfying for each .

Definition 5 (see [35]). Let be two single valued neutrosophic numbers (SVNNs), where Then:(1) iff and for all (2) if and (3)(4)(5)

Definition 6. (see [35]). Let be two SVNNs, where Then the operations about any two SVNNs are defined as follows:(1)(2)(3), (4)On the basis of Definition 6, Ashraf [6] derived several operations in the following ways:

Definition 7. (see [51]). Let be a collection of SVNNs, where and Then(1)(2)(3)(4)(5)(6)(7)

Definition 8. (see [43]). Let be SV-NN. Then the score and accuracy are given as follows:(1)(2)

Definition 9. (see [43]). Let be two SVNNs, where Then the comparison technique of SVNNs can be defined as:(1) implies that (2) and implies that (3) and implies that Ashraf et al. [6] developed the algebraic AO under SVNNs illustrate in the superseding definition.

Definition 10. (see [52]). Let be a collection of SVNNs, where A single valued neutrosophic weighted geometric (SV-NWG) AO of dimention is a mapping with weight vector such that and as

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. Let be two SVNNs, where and is fixed. Then Acz-Als norms depends operations for SVNNs are defined as follows:(1)(2)(3), (4)

Theorem 1. Let be a collection of SVNNs, where and Then:(1)(2)(3)(4)(5)(6)(7)

Proof. (1)Let be collection of SVNNs, where and Then by the Definition 11, it follows that(2)Utilizing the Definition 11, it follows that(3)Utilizing the Definition 11, it follows that(4)It is similar of the proof of (3).(5)Utilizing the Definition 11, it follows that (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. Let be collection of SVNNs, where A SVN Acz-Als weighted geometric (SVNAWG) AO of dimention is a mapping with weight vector such that and as

Theorem 2. Suppose be collection of SVNNs, where A SVN Acz-Als weighted geometric (SVNAWG) AO of dimention is a mapping with weight vector such that and is defined as:

Proof. Theorem’s 2 proof is derived by implemention of induction method as follows:

Step 1. for we haveUtilizing the Definition 11, it follows thatandThereforeThus, Theorem 2 is valid.
Now, we assume that, Theorem 2 is valid for that isNow we prove that Theorem 2 is valid for That is, we proveProve that, Theorem 2 is valid for all
Using the operator SVNAWG, we can efficiently illustrate the following characteristics.

Theorem 3. (Idempotency) Let be a collection of equivalent SVNNs, for each Then

Proof. SincePutwe haveThus, holds.

Theorem 4. (Boundedness) Let be collection of SVNNs. Let and Then,

Proof. Since it follows thatSimilarlyNow we haveTherefore

Theorem 5. Let and be two collections of SVNNs. If for Then,

Proof. The proof is straightforward.

Definition 13. Let be a collection of SVNNs, where An SVN Acz-Als ordered weighted geometric (SV-NAOWG) AO of dimention is a mapping with such that and aswhere are the permutation in such a way that.

Theorem 6. Suppose be a collection of SVNNs, where An SVN Acz-Als ordered weighted geometric (SV-NAOWG) AO of dimention is a mapping with weight vector such that and is defined as:where are the permutation in such a way as.

Using the operator SV-NAOWG, we can efficiently illustrate the following characteristics.

Theorem 7. (1)(Idempotency) Let be collection of equivalent SVNNs, for each Then(2)(Boundedness) Let be a collection of SVNNs. Let and Then(3)Let and be two collection of SVNNs. If for Then

Proof. Using Theorem 2, 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 of alternatives, and an acceptable rating by the attributes . Then, the usefulness of different attributes is specified by such that and .

Let for , , be the acceptable rating of every attribute for each alternative, where depicts the grade function the alternative ( ) satisfies . and designate the nutral grade function and grade function, respectively. The decision matrix of SVNNs can be obtained based from assessment data: .

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 attributes .

Step 2. Normalization is used to obtain the normalised decision matrix as:where refers to “if is a benefit criterion” and refers to “if 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.

Step 4. Determine the average solution () depends on all provided characteristics:Based on Definition 11, we get

Step 5. According to the results of , we can compute the positive distance from average (PDA_v) and negative distance from average (NDA_v) by using the following formula:We may adopt the score function of SVNNs described in Definition 8 to compute the PDA and NDA as follows:where represented the score value.

Step 6. Determine SPDA and SNDA, which represent the weighted average of PDA and NDA as follows:Where

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

Step 8. Compute the values of appraisal score (ASC) depends on each alternative’s as:

Step 9. Depending on the calculating values of ,alternatives are ranked in a decreasing order, and the bigger value of 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 ();(2)Customers’ guarantee requirements and regulations ();(3)Paperwork costs and customer packages ;(4)Procedural time and credit time ();(5)safe and secure banking ().

In this evaluation, experts were requested to utilize SV neutrosophic information to determine the best bank for business loans.

Table 1 summarizes the expert assessment data for SVNNs:

Cost type SVN information is given in and . Therefore the normalised expert evaluation decision information matrix is enclosed in Table 2:

Because there is just one expert in this case study, we do not need to calculate the cumulative decision matrix.

As per Table 2, we may use formula (3) to get the depends on all provided characteristics.

Then we can get the value of as

According to the results of average solution , we can compute the positive distance from average (PDA ) and negative distance from average (NDA ) by using formula (5) which are listed in Tables 3–5.

Calculate the values of and by (6) and attributes weighting vector , we can obtain the results as:

Normalize weighted sum of PDA and NDA are evaluated by (7) as follows:

Compute the values of appraisal score (ASC) depends on each alternative's NSPDA and NSNDA as:

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 Table 1.

Now we apply proposed single valued neutrosophic Acz-Als geometric AOs to adrress the uncertainty and chose the best alternative as follows in Table 6:

Tables 7–9 illustrate how existing techniques of AOs are used to aggregate the ambiguous information.

Now, according to collective data, the overall ranking of alternative is in the following Table 10.

Based 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.

Attribute analysis within a few previous techniques are as follows in Table 12.

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.

Data Availability

The data utilized in this article is hypothetical any one can be use just citing this article.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

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

Acknowledgments

The author (Muhammad Naeem) would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4310396DSR26.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
F. Shen, X. Ma, Z. Li, and D. Cai, “An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation,” Information Sciences, vol. 428, pp. 105–119, 2018.
View at: Publisher Site | Google Scholar
M. C. Wu and T. Y. Chen, “The ELECTRE multicriteria analysis approach based on Atanassov's intuitionistic fuzzy sets,” Expert Systems with Applications, vol. 38, no. 10, pp. 12318–12327, 2011.
View at: Publisher Site | Google Scholar
Y. X. Xue, J. X. You, X. D. Liu, and H. C. Liu, “An interval-valued intuitionistic fuzzy MABAC approach for material selection with incomplete weight information,” Applied Soft Computing, vol. 38, pp. 703–713, 2016.
View at: Publisher Site | Google Scholar
Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 6, pp. 1179–1187, 2007.
View at: Google Scholar
C. Jana and M. Pal, “Extended bipolar fuzzy EDAS approach for multi-criteria group decision-making process,” Computational and Applied Mathematics, vol. 40, no. 1, pp. 1–15, 2021.
View at: Publisher Site | Google Scholar
W. Liu and X. Liu, “Intuitionistic fuzzy information aggregation using Einstein operations,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 5, pp. 923–938, 2012.
View at: Publisher Site | Google Scholar
X. Xu and Z. Xu, “Prioritized intuitionistic fuzzy aggregation operators,” Information Fusion, vol. 14, no. 1, pp. 108–116, 2013.
View at: Publisher Site | Google Scholar
P. Wang and P. Wang, “Some improved linguistic intuitionistic fuzzy aggregation operators and their applications to multiple-attribute decision making,” International Journal of Information Technology and Decision Making, vol. 16, no. 03, pp. 817–850, 2017.
View at: Publisher Site | Google Scholar
C. Jana, H. Garg, and M. Pal, “Multi-attribute decision making for power Dombi operators under Pythagorean fuzzy information with MABAC method,” Journal of Ambient Intelligence and Humanized Computing, pp. 1–18, 2022.
View at: Publisher Site | Google Scholar
Z. Xu and R. R. Yager, “Intuitionistic fuzzy Bonferroni means,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 41, no. 2, pp. 568–578, 2010.
View at: Google Scholar
R. Arora and H. Garg, “Group decision-making method based on prioritized linguistic intuitionistic fuzzy AOs and its fundamental properties,” Computational and Applied Mathematics, vol. 38, no. 2, pp. 1–32, 2019.
View at: Publisher Site | Google Scholar
H. Zhao, Z. Xu, M. Ni, and S. Liu, “Generalized aggregation operators for intuitionistic fuzzy sets,” International Journal of Intelligent Systems, vol. 25, no. 1, pp. 1–30, 2010.
View at: Publisher Site | Google Scholar
D. Yu, “Intuitionistic fuzzy information aggregation under confidence levels,” Applied Soft Computing, vol. 19, pp. 147–160, 2014.
View at: Publisher Site | Google Scholar
D. Yu, “Intuitionistic fuzzy geometric Heronian mean aggregation operators,” Applied Soft Computing, vol. 13, no. 2, pp. 1235–1246, 2013.
View at: Publisher Site | Google Scholar
W. Jiang, B. Wei, X. Liu, X. Zheng, and H. Zheng, “Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making,” International Journal of Intelligent Systems, vol. 33, no. 1, pp. 49–67, 2018.
View at: Publisher Site | Google Scholar
T. Senapati, G. Chen, and R. R. Yager, “Aczel–Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making,” International Journal of Intelligent Systems, vol. 324, p. 83, 2021.
View at: Publisher Site | Google Scholar
M. J. Khan, P. Kumam, P. Liu, W. Ashraf, and S. Ashraf, “A novel approach to generalized intuitionistic fuzzy soft sets and its application in decision support system,” Mathematics, vol. 7, no. 8, p. 742, 2019.
View at: Publisher Site | Google Scholar
B. C. Cuong, Picture Fuzzy Sets-First Results. Part 1, Seminar Neuro-Fuzzy Systems with Applications, Institute of Mathematics, Hanoi, 2013.
S. Ashraf, T. Mahmood, S. Khan, and Q. Khan, “Different approaches to multi-criteria group decision making problems for picture fuzzy environment,” Bulletin of the Brazilian Mathematical Society, New Series, vol. 50, no. 2, pp. 373–397, 2019.
View at: Publisher Site | Google Scholar
M. Riaz, H. Garg, H. M. A. Chinram, and R. Chinram, “Multi-criteria decision making based on bipolar picture fuzzy operators and new distance measures,” Computer Modeling in Engineering and Sciences, vol. 127, no. 2, pp. 771–800, 2021.
View at: Publisher Site | Google Scholar
H. Garg, “Some picture fuzzy aggregation operators and their applications to multicriteria decision-making,” Arabian Journal for Science and Engineering, vol. 42, no. 12, pp. 5275–5290, 2017.
View at: Publisher Site | Google Scholar
G. Wei, “Picture fuzzy aggregation operators and their application to multiple attribute decision making,” Journal of Intelligent and Fuzzy Systems, vol. 33, no. 2, pp. 713–724, 2017.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Khan, and S. Khan, “Fuzzy decision support modeling for internet finance soft power evaluation based on sine trigonometric Pythagorean fuzzy information,” Journal of Ambient Intelligence and Humanized Computing, vol. 12, no. 2, pp. 3101–3119, 2021.
View at: Publisher Site | Google Scholar
M. J. Khan, P. Kumam, S. Kumam, and W. Kumam, “Generalized picture fuzzy soft sets and their application in decision support systems,” Symmetry, vol. 11, no. 3, p. 415, 2019.
View at: Publisher Site | Google Scholar
S. Khan and S. Ashraf, “Picture fuzzy aggregation information based on Einstein operations and their application in decision making,” Mathematical Sciences, vol. 13, no. 3, pp. 213–229, 2019.
View at: Publisher Site | Google Scholar
M. Qiyas, S. Abdullah, S. Aslam, and M. Aslam, “Utilizing linguistic picture fuzzy aggregation operators for multiple-attribute decision-making problems,” International Journal of Fuzzy Systems, vol. 22, no. 1, pp. 310–320, 2020.
View at: Publisher Site | Google Scholar
C. Jana, T. Senapati, M. Yager, and R. R. Yager, “Picture fuzzy Dombi aggregation operators: application to MADM process,” Applied Soft Computing, vol. 74, pp. 99–109, 2019.
View at: Publisher Site | Google Scholar
C. Jana and M. Pal, “A dynamical hybrid method to design decision making process based on GRA approach for multiple attributes problem,” Engineering Applications of Artificial Intelligence, vol. 100, p. 104203, 2021.
View at: Publisher Site | Google Scholar
G. Wei, “Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making,” Fundamenta Informaticae, vol. 157, no. 3, pp. 271–320, 2018.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Abdullah, and M. Aslam, “Cleaner production evaluation in gold mines using novel distance measure method with cubic picture fuzzy numbers,” International Journal of Fuzzy Systems, vol. 21, no. 8, pp. 2448–2461, 2019.
View at: Publisher Site | Google Scholar
S. Khan, S. Abdullah, L. Ashraf, and S. Ashraf, “Logarithmic aggregation operators of picture fuzzy numbers for multi-attribute decision making problems,” Mathematics, vol. 7, no. 7, p. 608, 2019.
View at: Publisher Site | Google Scholar
M. Qiyas, S. Abdullah, S. Abdullah, and L. Abdullah, “Linguistic picture fuzzy Dombi aggregation operators and their application in multiple attribute group decision making problem,” Mathematics, vol. 7, no. 8, p. 764, 2019.
View at: Publisher Site | Google Scholar
F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, MI, USA, 1998.
F. Smarandache, “Plithogeny, plithogenic set, logic, probability and statistics: a short review,” Journal of Computational and Cognitive Engineering, pp. 47–50, 2022.
View at: Google Scholar
J. Ye, “A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets,” Journal of Intelligent and Fuzzy Systems, vol. 26, no. 5, pp. 2459–2466, 2014.
View at: Publisher Site | Google Scholar
J. J. Peng, J. Q. Wang, X. H. Wu, J. Chen, and X. H. Chen, “Multi-valued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems,” International Journal of Computational Intelligence Systems, vol. 8, no. 2, pp. 345–363, 2015.
View at: Publisher Site | Google Scholar
J. Chen and J. Ye, “Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making,” Symmetry, vol. 9, no. 6, p. 82, 2017.
View at: Publisher Site | Google Scholar
P. Liu, Y. Chu, Y. Li, and Y. Chen, “Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making,” International Journal of Fuzzy Systems, vol. 16, no. 2, pp. 242–255, 2014.
View at: Google Scholar
R. K. Al-Hamido, “A new neutrosophic algebraic structures,” Journal of Computational and Cognitive Engineering, p. 5, 2022.
View at: Publisher Site | Google Scholar
H. Garg and S. Nancy, “Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making,” Journal of Ambient Intelligence and Humanized Computing, vol. 9, no. 6, pp. 1975–1997, 2018.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Abdullah, S. Zeng, H. Ghani, and F. Ghani, “Fuzzy decision support modeling for hydrogen power plant selection based on single valued neutrosophic sine trigonometric aggregation operators,” Symmetry, vol. 12, no. 2, p. 298, 2020.
View at: Publisher Site | Google Scholar
M. Keshavarz Ghorabaee, E. K. Zavadskas, L. Turskis, and Z. Turskis, “Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS),” Informatica, vol. 26, no. 3, pp. 435–451, 2015.
View at: Publisher Site | Google Scholar
C. Kahraman, M. Keshavarz Ghorabaee, E. K. Zavadskas, S. C. Cevik Onar, and B. Oztaysi, “Intuitionistic fuzzy EDAS method: an application to solid waste disposal site selection,” Journal of Environmental Engineering and Landscape Management, vol. 25, no. 1, pp. 1–12, 2017.
View at: Publisher Site | Google Scholar
B. Batool, S. S. Abosuliman, S. Abdullah, and S. Ashraf, “EDAS method for decision support modeling under the Pythagorean probabilistic hesitant fuzzy aggregation information,” Journal of Ambient Intelligence and Humanized Computing, pp. 1–14, 2021.
View at: Publisher Site | Google Scholar
X. D. Peng and C. Liu, “Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set,” Journal of Intelligent and Fuzzy Systems, vol. 32, no. 1, pp. 955–968, 2017.
View at: Publisher Site | Google Scholar
A. Karasan and C. Kahraman, “A novel interval-valued neutrosophic EDAS method: prioritization of the United Nations national sustainable development goals,” Soft Computing, vol. 22, pp. 4891–4906, 2018.
View at: Google Scholar
M. Keshavarz-Ghorabaee, M. Amiri, E. K. Zavadskas, Z. Antucheviciene, and J. Antucheviciene, “A dynamic fuzzy approach based on the EDAS method for multi-criteria subcontractor evaluation,” Information, vol. 9, no. 3, p. 68, 2018.
View at: Publisher Site | Google Scholar
J. Aczél and C. Alsina, “Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements,” Aequationes Mathematicae, vol. 25, no. 1, pp. 313–315, 1982.
View at: Google Scholar
J. Ye, “Subtraction and division operations of simplified neutrosophic sets,” Information, vol. 8, no. 2, p. 51, 2017.
View at: Publisher Site | Google Scholar
G. Wei and Z. Zhang, “Some single-valued neutrosophic Bonferroni power aggregation operators in multiple attribute decision making,” Journal of Ambient Intelligence and Humanized Computing, vol. 10, no. 3, pp. 863–882, 2019.
View at: Publisher Site | Google Scholar
M. Akram and A. Khan, “Complex Pythagorean Dombi fuzzy graphs for decision making,” Granular Computing, p. 34, 2020.
View at: Publisher Site | Google Scholar
J. J. Peng, J. Q. Wang, and J. Wang, “Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems,” International Journal of Systems Science, vol. 47, no. 10, pp. 2342–2358, 2016.
View at: Publisher Site | Google Scholar
D. Nancy and H. Garg, “Novel single-valued neutrosophic aggregated operators under Frank norm operation and its application to decision-making process,” International Journal for Uncertainty Quantification, vol. 6, no. 4, pp. 361–375, 2016.
View at: Publisher Site | Google Scholar
X. Li, Y. Ju, D. Ju, W. Zhang, P. Wang, and A. Wang, “Multi-attribute group decision making method based on EDAS under picture fuzzy environment,” IEEE Access, vol. 7, pp. 141179–141192, 2019.
View at: Publisher Site | Google Scholar
P. Liu, T. Khan, and Q. Khan, “Group decision making based on power Heronian aggregation operators under linguistic neutrosophic environment,” International Journal of Fuzzy Systems, vol. 20, no. 3, pp. 970–985, 2018.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Khan, and S. Khan, “Fuzzy decision support modeling for internet finance soft power evaluation based on sine trigonometric Pythagorean fuzzy information,” Journal of Ambient Intelligence and Humanized Computing, vol. 12, no. 2, pp. 3101–3119, 2021.
View at: Publisher Site | Google Scholar
A. O. Almagrabi, S. Abdullah, M. Shams, Y. D. Al-Otaibi, and S. Ashraf, “A new approach to q-linear Diophantine fuzzy emergency decision support system for COVID19,” Journal of Ambient Intelligence and Humanized Computing, vol. 13, no. 4, pp. 1687–1713, 2022.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Abdullah, and F. Muneeza, “Some novel aggregation operators for cubic picture fuzzy information: application in multi-attribute decision support problem,” Granular Computing, vol. 6, no. 3, pp. 603–618, 2021.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Shahzaib Ashraf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

367

Downloads

431

Citations