Decision Support System for Single-Valued Neutrosophic Aczel–Alsina Aggregation Operators Based on Known Weights

Latif, Sajid; Ullah, Kifayat; Hussain, Abrar; Awsar, Amrullah

doi:https://doi.org/10.1155/2024/4362151

Discrete Dynamics in Nature and Society

On this page

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

Special Issue

Multi-Criteria Decision Making (MCDM) Models in a Fuzzy and Rough Environment

View this Special Issue

Research Article | Open Access

Volume 2024 | Article ID 4362151 | https://doi.org/10.1155/2024/4362151

Decision Support System for Single-Valued Neutrosophic Aczel–Alsina Aggregation Operators Based on Known Weights

Sajid Latif,¹Kifayat Ullah,¹Abrar Hussain,¹and Amrullah Awsar²

Academic Editor: Ya Jia

Received18 Nov 2022

Revised01 May 2023

Accepted21 Jul 2023

Published20 Mar 2024

Abstract

Multiattribute decision making (MADM) approach is a well-known decision-making process utilized in a variety of fields such as solid waste management, renewable energy resources, air quality assurance, hotel location decision, sustainable supplier selection, partner recognition, green supplier enterprises, game theory, construction development authority, and weapon group target estimation. The aggregation operators (AOs) are essential components of the decision-making process and have a great capability to deal with ambiguous and unpredictable information in the different fields of fuzzy environments. In this article, we expressed the theory concepts of single-valued neutrosophic (SVN) sets (SVNS) and also characterized their basic operations. The power aggregation tools are allowed to input arguments to support each other among different arguments. Recently, Aczel–Alisna aggregation tools conquered great attention from several research scholars. We also exposed some reliable operations of Aczel–Alsina aggregation models under the consideration of SVN information. We established a series of new approaches, including the “single-valued neutrosophic Aczel–Alsina power weighted average” (SVNAAPWA) operator and “single-valued neutrosophic Aczel–Alsina power weighted geometric” (SVNAAPWG) operators. To show the effectiveness and compatibility of derived approaches, some prominent characteristics are also established. We constructed a MADM technique to solve an application of engineering and construction materials under consideration of our derived methodologies. An experimental case study is also presented to determine a suitable optimal option from a group of options. To find the flexibility of our proposed work, we provided a comparative study that compares the results of existing AOs with our proposed work. A comprehensive overview is also presented here.

1. Introduction

In 1965, Zadeh [1] invented the fuzzy set (FS) and introduced a truth value (TV) between and in place of the common crisp value of and . In the theory and science of decision-making, the fuzzy theory is a significant and engaging study issue, but FS lacks a falsity value (FV); instead, it is only identified by its TMV, which ranges from to . To address the shortcomings of FS, Atanassov [2] developed the idea of an intuitionistic fuzzy set (IFS), which is distinguished by its TV and FV between and . Yager [3] extended the theory of IFSs in the framework of Pythagorean FS (PyFS) with the sum of the square of TV and FV lies on the interval . An interval-valued IFS (IVIFS), which is distinguished by its interval TV and interval FV in the unit interval , was presented by Atanassov and Gargov [4] as a further generalization of an IFS. Smarandache [5, 6] created a neutrosophic set (NS) from a philosophical point of view to express indeterminate and inconsistent information because IFSs and IVIFSs cannot represent it. In an NS , represents TV, represents the abstinence value (AV), and represents FV of a set . The difficulties of practical applications may, therefore, be caused by the nonstandard interval . Consequently, based on the actual standard range [0, 1], Ye [7] introduced a simplified neutrosophic set (SNS), which includes the concepts of a SVNS given by Wang et al. [8] and an interval NS (INS) introduced by the Ye [9], which are the extension of IFS and IVIFS. An innovative idea of SVNNs based on weighted aggregated sum product valuation to solve a MADM technique was developed by Mishra et al. [10]. Seikh and Dutta [11] illustrated the theory of SVN information and developed advanced programming for the solution of matrix game theory. These concepts were first presented as classifications of NS to be easily used for real-time applications. Consequently, SVNS and INS are subclasses of NS, whereas SNS is a subclass of NS. NS is the generalization of FS and IFS; we also studied literature related to current research work seen in [12–14].

The AOs are appropriate mathematical tools to aggregate ambiguous and uncertain information. Recently, several research scientists worked on different AOs in the system of fuzziness such as AOs of IFS given by Xu [15], AOs of PyFSs developed by Rahman et al. [16], AOs of interval-valued PyFSs (IVPyFSs) presented by the Peng and Yang [17], AOs of IV T-spherical FSs given by the Ullah et al. [18], AOs of picture FSs (PFSs) presented by the Garg [19], AOs of q-rung orthopair FSs (q-ROFSs) developed by the Jana et al. [20], AOs of the bipolar valued hesitant fuzzy system given by the Khan et al. [21], and AOs of complex IFSs given by the Hussain et al. [22]. Fan et al. [23] provided some AOs of SVNSs by using operations of linguistic variables and also defined a MADM technique. Sodenkamp et al. [24] provided a list of new AOs to find the reliability of multicriteria decision-making problems under the system of SVNSs. Garg [25] introduced a fresh idea of neutrality AOs by using operations of sum and scalar multiplications based on SVNSs. Saha et al. [26] checked the flexibility and compatibility of hesitant FSs which is the advanced version of FSs and determined a series of new approaches based on Archimedean aggregation tools. A robust theory of Dombi Archimedean aggregation tools based on Hesitant fuzzy was reviewed by Liu et al. [27]. Saha et al. [28] presented the theory of Dombi Bonferroni mean aggregation expressions to evaluate real-life challenges.

The triangular norms are powerful tools to overcome the influence of vague and impression information. Several researchers invented different TNM and TCM to aggregate information in the system of fuzziness. Firstly, the concepts of TNM and TCNM were developed by Klement [29] in 1982. There are such types of TNM and TCNM which include Lukasiewicz TNM and TCNM [30], Nilpotent TNM and TCNM [31], Drastic TNM and TCNM [32], Archimedean TNM and TCNM [33], Einstein TNM and TCNM [34], and Frank TNM and TCNM [35]. Many researchers explored the concepts of TNM and TCNM in the different fuzzy frameworks. Klement et al. [36] worked on a family of TNM and TCNM. Babu and Ahmed [37] invented several parametric TNMs based on the function generator. Seikh and Mandal [38] presented some new AOs of PFSs and their application based on the MADM technique. Seikh and Mandal [39] also elaborated on the theory of Dombi TNM and TCNM under the system of IFS. Seikh and Mahnaz [40] generalized the system of T-spherical FS by using the basic operational laws of frank TNM and TCNM.

A well-known and efficient aggregation model is known as the power average (PA) operator. The theory PA given by Yager [41] is utilized to express support among each other in the aggregation process. Xu and Yager [42] also elaborated on the theory of PA and presented a robust aggregation model of the power geometric (PG) operator. Liu [43] utilized the theoretic concepts of PA and integrated an application of a MADM problem. Jana and Pal [44] constructed some appropriate power aggregation methodologies based on SVNSs under consideration of Dombi operational laws. Ashraf et al. [45] illustrated some trigonometric aggregation models to solve a MADM technique for the selection of a hydrogen power plant. Senapati et al. [46] exposed a series of new aggregation approaches and tried to solve an application of sharing sustainable transportation enterprises.

Aczél and Alsina [47] gave an appropriate mathematical tool to overcome the effect of unreasonable and vague information in a fuzzy framework under condition . Alsina et al. [48] worked on probabilistic metric space and its basic properties. Senapati et al. [49] explored the idea of A-TNM and A-TCNM under the system of IFSs and also established a MADM to select a suitable selection for a multinational company. Senapati et al. [50] also explored the concept of A-TNM and A-TCNM in the framework of IVIFSs with an application under a MADM technique. Naeem et al. [51] developed a series of new AOs of PFSs based on Aczel–Alsina operations. Hussain et al. [52] extended the model of PyFSs by using Aczel–Alsina operations and studied a MADM technique for the selection process of a multinational company. The theory of Aczel–Alsina aggregation tools has been explored by numerous research scholars. Senapati et al. [53] generalized the theory of Aczel–Alsina aggregation tools and developed a class of new approaches in light of PyFS information. Senapati et al. [54] also illustrated the potential of cyclone disaster enterprises under consideration of Hesitant Fuzzy system. Farid et al. [55] constructed an algorithm for evaluation of robust green supplier management. A robust construction of interval-valued PyF information based on Aczel–Alsina aggregation tools was presented by Senapati et al. [56]. Sustainable green supplier enterprises were evaluated by the Riaz et al. [57]. Alcantud [58] published latest research work related to decision-making process and also developed the series of robust methodologies in a book.

The abovementioned fuzzy environments carried restricted information, and decision-makers face a lot of challenges due to insufficient information during the aggregation process. The SVNSs are a more reliable generalization of FSs, IFSs, and IVIFSs and provide freedom to decision-makers for the decision process. The power aggregation tools are well-known and provide a smooth approximation in the decision-making process. Recently, Aczel–Alsina aggregation expressions acquired a lot of extension from multiresearch scholars. We found a research gap in the environments of SVNSs under consideration of the power aggregation models. We extended research mythologies which are presented in [59] and developed some new appropriate aggregation approaches under consideration of power aggregation models. The major purpose of this article is particularized as follows.(a)To express the notion of SVNSs and their related appropriate aggregation operations.(b)To study well-known aggregation operators such as power average and power geometric operators under consideration of SVN environment.(c)We also expressed the theoretic concepts of Aczel–Alsina aggregation tools and some fundamental operational laws under consideration in SVN environments.(d)We also derived a series of new approaches including SVNAAPWA and SVNAAPWG operators with some special characteristics of our derived approaches.(e)We construct a MADM technique to solve an application of engineering and construction materials under consideration of our derived methodologies. An experimental case study is also presented to determine a suitable optimal option from a group of options.(f)To find the flexibility of our proposed work, we provide a comparative study that compares the results of existing AOs with those of our proposed work. A comprehensive overview is also presented here.

The structure of this article is as follows: In Section 1, we recall the previous history of our research work, Section 2 presents the basic notions of TNM and TCNM with the help of an example; furthermore, authors also expose the notion of SVNSs and its fundamental operational laws based on SVNNs. Section 3 presents Aczel–Alsina operations based on SVNSs and we also gave a numerical example to support Aczel–Alsina operations. In Section 4, we develop AOs of SVNAAPWA operator and their characteristics such as idempotency, monotonicity, and boundedness, In Section 5, we also present SVNAAPWG operator based on Aczel–Alsina operations. In Section 6, we study a MADM technique to solve an application of engineering and construction materials with the help of a numerical example. In Section 7, we contrast the results of exiting AOs with the results of our proposed technique. In Section 8, we summarized our research work.

2. Preliminaries

We recall the notions of TNM and TCNM with some examples. Moreover, we also discuss the notion of A-TNM and A-TCNM for further development of this article. Firstly, the concepts of TNM and TCNM were given by Klement [29]. Symbols with their appropriate meanings are listed in Table 1.

Definition 1 (see [29]). A function is a TNM, if it satisfies conditions such as symmetry, monotonicity, associativity, and one identity element.(i)(ii) if (iii)(iv)

Example 1. Some examples of TNM are as follows.(i)Product t-norm: (ii)Minimum t-norm: (ii)Lukasiewicz t-norm: (iv)Drastic t-norm:for all

Definition 2 (see [29]). A function is a TCNM, if it satisfies conditions such as symmetry, monotonicity, associativity, and one identity element.(i)(ii) if (iii)(iv)for all

Example 2. Some of the examples are given as follows.(i)Probabilistic sum: (ii)Minimum t-conorm: (iii)Lukasiewicz t-conorm: (iv)Drastic t-norm:for all where is TNM and is TCNM. For all .

Definition 3 (see [47, 48]). A new category of TNM was presented by Aczel–Alsina in 1980.
The A-TNM is defined as follows:A-TCNM of is defined as follows:In limiting cases, . For every , the A-TNM and A-TCNM are dual for each other. The type of A-TNM and A-TCNM is strictly maximizing.
Now, we will study the notion of NS and SVNS on a universal set and also discuss some basic operations of SVNS. To compare different SVNNs, we explore the notions of score function and accuracy function.

Definition 4 (see [2]). Let be a nonempty set, and an IFS over is defined as follows:where and represent truth value (TV) and falsity value (FV), respectively. The IFS must satisfy the following axiom:The hesitancy value of an IFS is given by .

Definition 5 (see [60]). Let be a nonempty set, and a linear Diophantine FS (LDFS) over is characterized as follows:where , and represent truth value, falsity value, and reference parameters, respectively. The LDFS must satisfy the following axiom:The hesitancy value of an LDFS is given by .

Definition 6 (see [8]). Let be a nonempty set, and a SVNS over is defined as follows:where , and represent truth value, abstinence value and falsity value, respectively. The SVNS must satisfy the following axiom:Furthermore, a single-valued neutrosophic number (SVNN) is denoted by .

Definition 7 (see [61]). Let and be two SVNSs. Then, some basic operations of SVNSs such as union, intersection, and compliment are defined as follows.(i)(ii)(iii) and only if , , and (iv) and only if and (v)Now, we expressed some appropriate comparison rules, which defined how we compare different SVNNs.

Definition 8 (see [61]). Consider be a SVNN. Then, the score value is defined as follows:

Definition 9 (see [61]). Consider be a SVNN. Then, the accuracy value is defined as follows:

Remark 10. Let and be two SVNNs. Then, the score value of and is denoted by and , respectively. Similarly, the accuracy value of and is denoted by and , respectively. The relation between the score function and accuracy function is defined as follows.(i)If , then (ii)If , then (iii)If , then(a)If , then (b)If , then (c)If , then

Definition 11 (see [61]). Consider , and be three SVNNs. Then, we have(i)(ii)(iii)(iv)

Definition 12 (see [62]). Consider , be the set of SVNNs, with corresponding weights vector such that and . Then, the aggregated value of SVN weighted averaging (SVNWA) and SVN weighted geometric (SVNWG) operators is defined as follows:

Definition 13 (see [41]). Consider , be the set of SVNNs, then the power averaging (PA) operators are termed as follows:where .

Definition 14 (see [42]). Consider , be the set of SVNNs, then the power geometric (PG) operators are termed as follows:where .

3. Aczel–Alsina Operation-Based SVNNs

In this section, we will discuss the basic operations of A-TNM and A-TCNM based on SVNNs with some practical examples.

Consider and represent A-TNM and A-TCNM, respectively. We defined some operations of Aczel–Alsina product and Aczel–Alsina sum. Furthermore, we also defined some generalized unions and intersections over two SVNNs and as follows:

Definition 15. Let , and be the three SVNNs. Then, some basic operations of A-TNM and A-TCNM are defined as follows: for and ,(i)(ii)(iii)(iv)

Example 3. Let be three SVNNs. Then, by using the Aczel–Alsina operation, for and , we have(i), , (ii), , (iii), , (iv), ,

Definition 16. (see [59]). Let , be the set of SVNNs, with corresponding weight vectors of such that and . Then, the SVNAAWA operator is given by the following equation:

Definition 17 (see [59]). Let , be the set of SVNNs, with corresponding weight vector of such that and . Then, the SVNAAWG operator is given by the following equation:

Theorem 18. Let , and be three SVNNs. Then, we have(i)(ii)(iii)(iv)(v)(vi)

Proof. Let , and be three SVNNs and . By proving Theorem 18 and by using Definition 15, we get(i), , (ii), , (iii)Let Then, . Using this equation, we get the following equation:

4. Single-Value Neutrosophic Aczel–Alsina Power Average Aggregation Operators

Now, by using the basic operations of Aczel–Alsina aggregation tools, we derived appropriate methodologies such as SVNAAPWA operators with some reliable properties under consideration of SVNNs. We also utilized a degree of weighted support throughout this article by using the following equation: , where the support of is denoted by and associated weight vector of is , and .

Definition 19. Let , be the set of SVNNs, with corresponding weight vectors of such that and . Then, the SVNAAPWA operator is a function as follows:

Theorem 20. Let , be the set of SVNNs. Then, the aggregated value of the SVNAAPWA operator is also a SVNN as follows:

Proof. We will prove Theorem 20 with the help of a mathematical induction technique by using the following method.(i)For depending on Aczel–Alsina operations of SVNNs, we get the following equation: Using abovementioned Definition 19, we have the following equation: Hence, this is true for .(ii)Now, suppose that this will be true for Then, we have the following equation:Now, we have to show that it also holds for as follows:which is true for .

Example 4. Let , and be the four SVNNs with weight vector . Now, we investigate the value of the SVNAAPWA operator for and .
The degree of weighted support associated with SVNNs is as follows: .

Theorem 21 (idempotency). Consider , be the set of identical SVNNs, if for all then we have the following equation:

Proof. Since , be the set of SVNNs, we can get the following equation:Thus, we can say that holds.

Theorem 22 (boundedness). Consider , be the set of SVNNs if and , and then we have the following equation:

Proof. Consider , be the set of SVNNs. Let and . We have , , and . Hence, there we have the following subsequent inequalities:Therefore,

Theorem 23 (Monotonicity). Let and , be two sets of SVNNs, if for all Then, we have the following equation:

Now, we will represent SVN Aczel–Alsina ordered weighted averaging (SVNAAOPWA) operator by using the basic operations of Aczel–Alsina operations.

Definition 24. Consider , be the set of SVNNs, with corresponding weight vectors of . Then, an SVNAAOPWA operator of dimension is a mapping as follows:

Theorem 25. Consider , be the set of SVNNs, with the corresponding weight vector such that and . A SVNAAOPWA operator of dimension is a mapping as follows:

Theorem 26 (Idempotency). Consider , to be the set of identical SVNNs, for all Then, we have the following equation:

Theorem 27 (boundedness). Consider , to be the set of SVNNs. Let . Then, we have the following equation:

Theorem 28 (monotonicity). Let and , be any two sets of SVNNs, if for all Then, we have the following equation:

Theorem 29 (commutativity). Let and , be any two sets of SVNNs. Then, we have the following equation:where , is any permutation of

5. Single-Value Neutrosophic Aczel–Alsina Power Weighted Geometric Aggregation Operators

In this section, we will study new AOs of SVN weighted geometric aggregation operator by following the Aczel–Alsina operations.

Definition 30. Let , be the set of SVNNs, with corresponding weight vector of such that and . Then, a SVNAAPWG operator is a function as follows:

Theorem 31. Let be the set of SVNNs. Then, the aggregated value of the SVNAAPWG operator is also a SNNN as follows:

Example 5. Let , and be the four SVNNs, with weight vectors . We investigate the value of the SVNAAPWG operator for and :
The degree of weighted support of SVNNs is .We can state the following characteristics of the SVNAAPWG operator by using the Aczel–Alsina operations on SVNNs.

Theorem 32 (idempotency). Let , be the set of identical SVNNs, for all . Then, we have the following equation:

Theorem 33 (boundedness). Let , be the set of SVNNs. Let and . Then, we have the following equation:

Theorem 34 (Monotonicity). Let and , be any two sets of SVNNs, if for all . Then, we have the following equation:

We can prove all these theorems in the same way as explained in Section 4.

6. Model of MADM Techniques with SVNN Information

In this section, we solved a MADM technique by using SVNAAPWA and SVNAAPWG operators under the system of SVN information. Consider to be the set of alternatives and to be the set of attributes with corresponding weight vectors , and . A decision matrix containing the information under SVNNs given by decision-making is as follows:

In the decision matrix, each 3-tuple represents the value of SVNN and , , and such that . There are two types of attributes: cost factor and beneficial factor. If a cost factor is involved, then the decision matrix is transformed into the following normalized matrix:

We will follow the following steps of the algorithm to solve a given MADM technique by the decision maker. Step 1: the decision maker collects the information and arranged it in a decision matrix based on SVNNs. Step 2: if the cost factor is involved in the set of attributes, then we need to transform the decision matrix into a normalized matrix. Step 3: calculate support value as follows: where is the expression for distance. Step 4: compute weighted support of by using the weights of the characteristics , and : and investigate weights corresponding to the SVNNs of . where the associated weight vector of is , . Step 5: apply our derived approaches of the SVNAAPWA and SVNAAPWG operators on information depicted in the decision matrix. Step 6: compute the score value of , under consideration of SVNs and rearrange all acquired score values of alternative to choose a desirable optimal option.

If the score values of optimal options are the same, the decision maker faces some difficulties in the selection process and can categorize or classify desirable options from a group of options. For this purpose, the accuracy function is defined in equation (14). Figure 1 also covers the steps of an algorithm of the MADM problem.

6.1. Application

The construction sector is one of the modern agencies and organizations that has a big impact on the economy of any nation. All of the real estate and infrastructure surrounding us were constructed by various parts of the construction economy. The building sector is an essential kind that significantly affects a nation’s economic growth. The construction industry, which is driven by investment, is very important to the government. The government enters into contracts with the building trade to construct infrastructure for the health, transportation, and educational sectors. The prosperity of each nation depends on the construction industry.

6.2. Numerical Example

In this numerical example, we investigate suitable construction materials for the construction process of a building. There are several construction materials but we have selected suitable cement for the construction process of our building. Consider five different types of cement ; panel select suitable cement based on given characteristics (attributes) . Here, represents the color of cement, represents the materials, represents the quality of the cement, and represents the availability of cement in the market.

To evaluate and investigate the suitable cement decision maker provided a set of weight vectors for the selection of the best cement, we follow the following steps of the Algorithm 1.

	Step 1: the decision maker collects the information under the system of SVNNs as depicted in Table 2.
	Step 2: there is no need to perform the transformation of the decision matrix into a normalized matrix because there is no involved cost factor in the set of attributes.
	Step 3: investigate support by given information using where .
	Step 4: compute weighted supports based on weight vectors of criteria associated with the alternatives.
	And also investigate weights of associated with the SVNNs, acquired results shown in the following matrix.

	Step 5: We apply our proposed methodologies of SVNAAPWA and SVNAAPWG operators for the parametric value of . Table 3 covers the results of the SVNAAPWA and SVNAAPWG operators.

	And,
	It is worth noticing that the results obtained in Table 3 are well in accordance with the Definition #.
	Step 6: investigate score values by using the consequences of the SVNAAPWA and SVNAAPWG operators which are shown in Table 3. Obtained results of the score values are shown in Table 4.

After ranking and ordering of score values, we observed that and for SVNAAPWA and SVNAAPWG operators. We have seen is the best objective (cement), by both AOs of the SVNAAPWA and SVNAAPWG operators. We also present the score values of the SVNAAPWA and SVNAAPWG operators in the graphical representation of Figure 2.

6.3. Influence of Different Parametric Values of on Our Proposed Methodologies

To determine the effect of different parametric values on the results of our proposed methodologies, we evaluated our discussed AOs for various parametric values of and the results of SVNAAPWA and SVNAAPWG operators are shown in Tables 5 and 6, respectively. We observed that when parametric values of increase, the results of our proposed methodologies gradually increase and the ranking of score values remain unchanged. This obtained technique of the ranking of score values provides isotonicity property, enabling decision-makers to select the ideal value following their preferences.

We also studied the results of score values of SVNAAPWA and SVNAAPWG operators in a graphical representation in Figures 3 and 4, respectively.

7. Comparative Study

In this section, we compared the results of invented works with the results of existing AOs. We applied some existing AOs on a decision matrix given by the decision maker. To find stability and feasibility of our invented works, we applied different existing AOs such as SVN Dombi weighted AOs given by Chen and Ye [61], SVN weighted average and geometric AOs given by Peng et al. [62], AOs of SVNNs given by [59], SVN Einstein weighted AOs given by Ye et al. [63], AOs of interval-valued SVNSs (IVSVNNs) given by Aiwu et al. [64], AOs of IVSVNNs based on frank operations developed by the Zhou et al. [65], and the theory of SVNNs in the framework of complex SVNNs (CSVNNs) by utilizing the idea of prioritization methods explored by Mahmood and Ali [66]. The results of applied existing AOs [59, 61–66] are shown in Table 7.

We observe that existing AOs discussed in [49–51] failed to aggregate this information. The graphical representation of the results of the comparative study is shown in Figure 5, which are depicted in Table 7.

8. Conclusion

The MADM technique is a convenient mathematical process that is used in every field of life, such as computational and environmental science, green supplier enterprises, construction development, game theory, computer programming, and social decision-making. In our daily life, a MADM technique plays a vital role in performing a selection process for suitable objects. AOs are the useful and effective components of a decision-making or MADM technique. In this article, we expressed the theoretic concepts of SVNNs, having a large capacity to handle vague and imprecision information under different fuzzy environments. Another useful and well-known theoretic concept provides support for different input arguments. There are several types of algebraic sum and algebraic products generated by the numerous triangular norms, but Aczel–Alsina aggregation expressions provide an effective and flexible smooth approximation. Recently, numerous research scholars utilized theoretic concepts of Aczel–Alsina aggregation expressions for new creation and evaluated some appropriate real-life applications. By inspiring the effectiveness and reliability of Aczel–Alsina aggregation tools, we derived some new approaches, including SVNAAPWA and SVNAAPWG operators. Some appropriate properties of our derived approaches are also characterized. To solve a MADM technique, we presented an application based on the selection process under consideration of construction enterprises. To check the reliability and validity of our derived approaches, we gave a numerical example to evaluate a suitable construction material based on SVN information. We made a comprehensive comparative study to compare the results of the existing AOs with our proposed methodologies.

Future studies can further expand on the existing model by breaking down the true value into smaller “subtruths,” abstinence into smaller “subabstinences,” and falsity values into smaller “subfalsehoods.” Moreover, we will extend our proposed methodologies in the framework of PF Maclaurin symmetric mean [67] and bipolar soft sets [68]. Moreover, we also explored our invented approaches in the form of a complex bipolar fuzzy soft set [69] and a T-spherical fuzzy set [70]. Furthermore, Table 8 covers all abbreviations used in this article.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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
R. R. Yager, “Pythagorean fuzzy subsets,” in Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), pp. 57–61, Edmonton, Canada, June 2013.
View at: Publisher Site | Google Scholar
K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343–349, 1989.
View at: Publisher Site | Google Scholar
F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, DE, USA, 1999.
F. Smarandache, in Neutrosophic Set–A Generalization of the Intuitionistic Fuzzy Set, University of New Mexico, Albuquerque, NM, USA, 2002.
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
H. Wang, F. Smarandache, Y. Zhang, and R. Sunderraman, Single Valued Neutrosophic Sets, Infinite study, Pazhambalacode, India, 2010.
J. Ye, “Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making,” Journal of Intelligent and Fuzzy Systems, vol. 26, no. 1, pp. 165–172, 2014.
View at: Publisher Site | Google Scholar
A. R. Mishra, P. Rani, and R. S. Prajapati, “Multi-criteria weighted aggregated sum product assessment method for sustainable biomass crop selection problem using single-valued neutrosophic sets,” Applied Soft Computing, vol. 113, Article ID 108038, 2021.
View at: Publisher Site | Google Scholar
M. R. Seikh and S. Dutta, “A nonlinear programming model to solve matrix games with pay-offs of single-valued neutrosophic numbers,” Neutrosophic Sets Syst, vol. 47, pp. 366–383, 2021.
View at: Google Scholar
M. Saad, T. Mahmood, K. Ullah, and N. Jan, “Computing shortest path in a single valued neutrosophic hesitant fuzzy network,” The Nucleus, vol. 56, no. 3, 2019.
View at: Google Scholar
J. Ye, “Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 5, pp. 2453–2462, 2014.
View at: Publisher Site | Google Scholar
H.-L. Yang, C.-L. Zhang, Z.-L. Guo, Y.-L. Liu, and X. Liao, “A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model,” Soft Computing, vol. 21, no. 21, pp. 6253–6267, 2017.
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: Publisher Site | Google Scholar
K. Rahman, A. Ali, M. Shakeel, M. A. Khan, and M. Ullah, “Pythagorean fuzzy weighted averaging aggregation operator and its application to decision making theory,” The Nucleus, vol. 54, no. 3, pp. 190–196, 2017.
View at: Google Scholar
X. Peng and Y. Yang, “Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators,” International Journal of Intelligent Systems, vol. 31, no. 5, pp. 444–487, 2016.
View at: Publisher Site | Google Scholar
K. Ullah, N. Hassan, T. Mahmood, N. Jan, and M. Hassan, “Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators,” Symmetry, vol. 11, no. 3, p. 357, 2019.
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
C. Jana, G. Muhiuddin, and M. Pal, “Some Dombi aggregation of Q-rung orthopair fuzzy numbers in multiple-attribute decision making,” International Journal of Intelligent Systems, vol. 34, no. 12, pp. 3220–3240, 2019.
View at: Publisher Site | Google Scholar
M. U. Khan, T. Mahmood, K. Ullah, N. Jan, and I. Deli, “Some aggregation operators for bipolar-valued hesitant fuzzy information based on Einstein operational laws,” Journal of Engineering and Applied Science, vol. 36, no. 2, 2017.
View at: Google Scholar
A. Hussain, K. Ullah, J. Ahmad, H. Karamti, D. Pamucar, and H. Wang, “Applications of the multiattribute decision-making for the development of the tourism industry using complex intuitionistic fuzzy hamy mean operators,” Computational Intelligence and Neuroscience, vol. 2022, Article ID 8562390, 18 pages, 2022.
View at: Publisher Site | Google Scholar
C. Fan, E. Fan, and K. Hu, “New form of single valued neutrosophic uncertain linguistic variables aggregation operators for decision-making,” Cognitive Systems Research, vol. 52, pp. 1045–1055, 2018.
View at: Publisher Site | Google Scholar
M. A. Sodenkamp, M. Tavana, and D. Di Caprio, “An aggregation method for solving group multi-criteria decision-making problems with single-valued neutrosophic sets,” Applied Soft Computing, vol. 71, pp. 715–727, 2018.
View at: Publisher Site | Google Scholar
H. Garg, “Novel neutrality aggregation operator-based multiattribute group decision-making method for single-valued neutrosophic numbers,” Soft Computing, vol. 24, no. 14, pp. 10327–10349, 2020.
View at: Publisher Site | Google Scholar
A. Saha, D. Dutta, and S. Kar, “Some new hybrid hesitant fuzzy weighted aggregation operators based on Archimedean and Dombi operations for multi-attribute decision making,” Neural Computing & Applications, vol. 33, no. 14, pp. 8753–8776, 2021.
View at: Publisher Site | Google Scholar
P. Liu, A. Saha, D. Dutta, and S. Kar, “Multi-Attribute decision-making using hesitant fuzzy dombi–archimedean weighted aggregation operators,” International Journal of Computational Intelligence Systems, vol. 14, no. 1, pp. 386–411, 2020.
View at: Publisher Site | Google Scholar
A. Saha, H. Garg, and D. Dutta, “Probabilistic linguistic q-rung orthopair fuzzy Generalized Dombi and Bonferroni mean operators for group decision-making with unknown weights of experts,” International Journal of Intelligent Systems, vol. 36, no. 12, pp. 7770–7804, 2021.
View at: Publisher Site | Google Scholar
E. P. Klement, “Construction of Fuzzy σ-algebras using triangular norms,” Journal of Mathematical Analysis and Applications, vol. 85, no. 2, pp. 543–565, 1982.
View at: Publisher Site | Google Scholar
E. P. Klement and M. Navara, “A characterization of tribes with respect to the Łukasiewicz t-norm,” Czechoslovak Mathematical Journal, vol. 47, no. 4, pp. 689–700, 1997.
View at: Publisher Site | Google Scholar
A. Mesiarová, “Approximation of k-Lipschitz t-norms by strict and nilpotent k-Lipschitz t-norms,” International Journal of General Systems, vol. 36, no. 2, pp. 205–218, 2007.
View at: Publisher Site | Google Scholar
S. Wang, “A fuzzy logic for the revised drastic product t-norm,” Soft Computing, vol. 11, no. 6, pp. 585–590, 2007.
View at: Publisher Site | Google Scholar
H. T. Nguyen, V. Kreinovich, and P. Wojciechowski, “Strict Archimedean t-norms and t-conorms as universal approximators,” International Journal of Approximate Reasoning, vol. 18, no. 3–4, pp. 239–249, 1998.
View at: Publisher Site | Google Scholar
W. Wang and X. Liu, “Intuitionistic fuzzy geometric aggregation operators based on Einstein operations,” International Journal of Intelligent Systems, vol. 26, no. 11, pp. 1049–1075, 2011.
View at: Publisher Site | Google Scholar
B. De Baets and H. E. De Meyer, “The frank t-norm family in fuzzy similarity measurement,” in Proceedings of the 2nd International Conference in Fuzzy Logic and Technology, pp. 249–252, Leicester, UK, September 2001.
View at: Google Scholar
E. P. Klement, R. Mesiar, and E. Pap, “Families of t-norms,” in Triangular Norms, pp. 101–119, Springer, Berlin, Germany, 2000.
View at: Google Scholar
M. Shohel Babu and S. Ahmed, “Function as the generator of parametric T-norms,” American Journal of Applied Mathematics, vol. 5, no. 4, pp. 114–118, 2017.
View at: Publisher Site | Google Scholar
M. R. Seikh and U. Mandal, “Some picture fuzzy aggregation operators based on frank t-norm and t-conorm: application to MADM process,” Informatica, vol. 45, no. 3, 2021.
View at: Publisher Site | Google Scholar
M. R. Seikh and U. Mandal, “Intuitionistic fuzzy Dombi aggregation operators and their application to multiple attribute decision-making,” Granul. Comput., vol. 6, no. 3, pp. 473–488, 2021.
View at: Publisher Site | Google Scholar
S. Mahnaz, J. Ali, M. G. A. Malik, and Z. Bashir, “T-spherical fuzzy Frank aggregation operators and their application to decision making with unknown weight information,” IEEE Access, vol. 10, pp. 7408–7438, 2022.
View at: Publisher Site | Google Scholar
R. R. Yager, “The power average operator,” IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 31, no. 6, pp. 724–731, 2001.
View at: Publisher Site | Google Scholar
Z. Xu and R. R. Yager, “Power-geometric operators and their use in group decision making,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 94–105, 2010.
View at: Publisher Site | Google Scholar
P. Liu, “Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators,” Computers and Industrial Engineering, vol. 108, pp. 199–212, 2017.
View at: Publisher Site | Google Scholar
C. Jana and M. Pal, “Multi-criteria decision making process based on some single-valued neutrosophic Dombi power aggregation operators,” Soft Computing, vol. 25, no. 7, pp. 5055–5072, 2021.
View at: Publisher Site | Google Scholar
S. Ashraf, S. Abdullah, S. Zeng, H. Jin, 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
T. Senapati, V. Simic, A. Saha, M. Dobrodolac, Y. Rong, and E. B. Tirkolaee, “Intuitionistic fuzzy power Aczel-Alsina model for prioritization of sustainable transportation sharing practices,” Engineering Applications of Artificial Intelligence, vol. 119, Article ID 105716, 2023.
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: Publisher Site | Google Scholar
C. Alsina, B. Schweizer, and M. J. Frank, Associative Functions: Triangular Norms and Copulas, World Scientific, Singapore, 2006.
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. 37, no. 2, pp. 1529–1551, 2022.
View at: Publisher Site | Google Scholar
T. Senapati, R. Mesiar, V. Simic, A. Iampan, R. Chinram, and R. Ali, “Analysis of interval-valued intuitionistic fuzzy aczel–alsina geometric aggregation operators and their application to multiple attribute decision-making,” Axioms, vol. 11, no. 6, p. 258, 2022.
View at: Publisher Site | Google Scholar
M. Naeem, Y. Khan, S. Ashraf, W. Weera, and B. Batool, “A novel picture fuzzy Aczel-Alsina geometric aggregation information: application to determining the factors affecting mango crops,” AIMS Mathematics, vol. 7, no. 7, pp. 12264–12288, 2022.
View at: Publisher Site | Google Scholar
A. Hussain, K. Ullah, M. N. Alshahrani, M.-S. Yang, and D. Pamucar, “Novel aczel–alsina operators for pythagorean fuzzy sets with application in multi-attribute decision making,” Symmetry, vol. 14, no. 5, p. 940, 2022.
View at: Publisher Site | Google Scholar
T. Senapati, G. Chen, R. Mesiar, and A. Saha, “Multiple attribute decision making based on Pythagorean fuzzy Aczel-Alsina average aggregation operators,” Journal of Ambient Intelligence and Humanized Computing, vol. 14, no. 8, pp. 10931–10945, 2022.
View at: Publisher Site | Google Scholar
T. Senapati, G. Chen, R. Mesiar, R. R. Yager, and A. Saha, “Novel Aczel–Alsina operations-based hesitant fuzzy aggregation operators and their applications in cyclone disaster assessment,” International Journal of General Systems, vol. 51, no. 5, pp. 511–546, 2022.
View at: Publisher Site | Google Scholar
H. M. A. Farid and M. Riaz, “q-rung orthopair fuzzy Aczel–Alsina aggregation operators with multi-criteria decision-making,” Engineering Applications of Artificial Intelligence, vol. 122, Article ID 106105, 2023.
View at: Publisher Site | Google Scholar
T. Senapati, A. R. Mishra, A. Saha, V. Simic, P. Rani, and R. Ali, “Construction of interval-valued Pythagorean fuzzy Aczel-Alsina aggregation operators for decision making: a case study in emerging IT software company selection,” Sādhanā, vol. 47, no. 4, p. 255, 2022.
View at: Publisher Site | Google Scholar
M. Riaz, H. M. Athar Farid, D. Pamucar, and S. Tanveer, “Spherical fuzzy information aggregation based on aczel–alsina operations and data analysis for supply chain,” Mathematical Problems in Engineering, vol. 2022, Article ID 9657703, 20 pages, 2022.
View at: Publisher Site | Google Scholar
J. C. R. Alcantud, Fuzzy Techniques for Decision Making, MDPI Books, Basel, Switzerland, 2008.
S. Ashraf, S. Ahmad, M. Naeem, M. Riaz, and M. A. Alam, “Novel EDAS methodology based on single-valued neutrosophic Aczel-Alsina aggregation information and their application in complex decision-making,” Complexity, vol. 2022, Article ID 2394472, 18 pages, 2022.
View at: Publisher Site | Google Scholar
M. Riaz and M. R. Hashmi, “Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems,” Journal of Intelligent and Fuzzy Systems, vol. 37, no. 4, pp. 5417–5439, 2019.
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
J. Peng, J. Wang, J. Wang, H. Zhang, and X. Chen, “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
J. Ye, E. Türkarslan, M. Ünver, and M. Olgun, “Algebraic and Einstein weighted operators of neutrosophic enthalpy values for multi-criteria decision making in neutrosophic multi-valued set settings,” Granul. Comput., vol. 7, no. 3, pp. 479–487, 2022.
View at: Publisher Site | Google Scholar
Z. Aiwu, D. Jianguo, and G. Hongjun, “Interval valued neutrosophic sets and multi-attribute decision-making based on generalized weighted aggregation operator,” Journal of Intelligent and Fuzzy Systems, vol. 29, no. 6, pp. 2697–2706, 2015.
View at: Publisher Site | Google Scholar
L.-P. Zhou, J.-Y. Dong, and S.-P. Wan, “Two new approaches for multi-attribute group decision-making with interval-valued neutrosophic Frank aggregation operators and incomplete weights,” IEEE Access, vol. 7, pp. 102727–102750, 2019.
View at: Publisher Site | Google Scholar
T. Mahmood and Z. Ali, “Prioritized Muirhead mean aggregation operators under the complex single-valued neutrosophic settings and their application in multi-attribute decision making,” Journal of Computational and Cognitive Engineering, vol. 104, 2022.
View at: Google Scholar
K. Ullah, “Picture fuzzy maclaurin symmetric mean operators and their applications in solving multiattribute decision-making problems,” Mathematical Problems in Engineering, vol. 2021, Article ID 1098631, 21 pages, 2021.
View at: Publisher Site | Google Scholar
T. Mahmood, “A novel approach towards bipolar soft sets and their applications,” Journal of Mathematics, vol. 2020, Article ID 4690808, 11 pages, 2020.
View at: Publisher Site | Google Scholar
T. Mahmood and U. Rehman, “Multi-attribute decision-making method based on bipolar complex fuzzy Maclaurin symmetric mean operators,” Computational and Applied Mathematics, vol. 41, no. 7, pp. 331–425, 2022.
View at: Publisher Site | Google Scholar
T. Mahmood, K. Ullah, Q. Khan, and N. Jan, “An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets,” Neural Computing and Applications, vol. 31, no. 11, pp. 7041–7053, 2019.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2024 Sajid Latif 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

94

Downloads

75

Citations