Applications of n th Power Root Fuzzy Sets in Multicriteria Decision Making

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Introduction
Making a decision is the process of selecting the best option/ options from a set of possibilities.Humans make numerous decisions throughout the course of their daily lives.Tere is no need to make a decision if there is just one alternative, but it is benefcial when there are two or more options.Multicriteria decision-making (MCDM) is a type of operational research that deals with one kind of outcomes by evaluating viable alternatives against a set of criteria in decision-making that is inconsistent.It is a far-fetched assumption that perfect numerical data are necessary to replicate real-world decision-making methods, which are characterized by intrinsic ambiguity in human judgments.Terefore, to cope with imprecise data, Zadeh [1] introduced the notion of fuzzy sets, and several studies on generalizations of the concept of fuzzy set were conducted after that.Generalization of fuzzy sets begun by Atanassov [2] who described the intuitionistic fuzzy sets as a fascinating generalization of fuzzy sets and explored essential features.Intuitionistic fuzzy sets have a wide range of applications in diferent felds including reservoir food control operation, image fusion [3], pattern recognition [4], medical diagnosis, optimization issues [5], group theory [6,7], and decision-making [8,9].Ten, Yager [10] explored Pythagorean fuzzy sets (PFSs) as a model for dealing with imprecise data, and Zhang and Xu [11] introduced the concept of a Pythagorean fuzzy number.Garg [12] examined the use of PFSs in decision-making situations.Senapati and Yager [13] introduced Fermatean fuzzy sets and basic processes on them, as well as a Fermatean fuzzy TOPSIS technique for solving multiple criteria decision-making problems.To broaden the scope of membership and nonmembership degrees, Yager [14] put forward the idea of q-rung orthopair fuzzy sets (q-ROFSs), where q ≥ 1.
Recently, it has been suggested diferent approaches to deal with the input data inspired by the fact that the signifcance of membership and nonmembership degrees need not to be equal in general cases.Tese approaches are useable to describe some real-life issues and enlarge the spaces of data under study.In this regard, Ibrahim et al. [15] defned the (3,2)-fuzzy sets as another type of generalized Pythagorean fuzzy set.Al-Shami et al. [16] introduced a new type of fuzzy set known as the SR-fuzzy set and studied its features in depth.Ten, Al-Shami [17] displayed the idea of (2,1)-fuzzy sets and furnished its basic set of operations.At the beginning of the year 2023, Al-Shami and Mhemdi [18] ofered the concept of (m, n)-fuzzy sets as a generalized frame for these types of fuzzy sets.Tey introduced diferent types of operations and aggregation operators via the environment of (m, n)-fuzzy sets.Gao and Zhang [19] provided the concept of linear orthopair fuzzy sets to address some empirical problems of vagueness.
Multiple attribute decision-making (MADM) is a strategy that takes into account the best possible alternatives.To deal with the complications and complexity of MADM problems, a variety of helpful mathematical methods, such as soft sets and fuzzy sets, were improved.MADM is a procedure that may produce ranking outcomes for fnite alternatives based on their attribute values, and it is an important part of decision sciences.Te concept of intuitionistic fuzzy weighted averaging operators was proposed by Xu [20], and Xu and Yager [21] proposed geometric weighted and geometric hybrid operators in the context of intuitionistic fuzzy sets.To cope with Pythagorean fuzzy MCDM difculties, Yager [22] devised a useful decision technique based on Pythagorean fuzzy aggregation operators.Senapati and Yager [23] developed the Fermatean fuzzy weighted power average operator over Fermatean fuzzy sets, as well as their attributes.Al-Shami et al. [16] proposed the SR-fuzzy weighted power average operator and used it to choose the best university.Akram et al. [24] discussed a new approach to opt for the optimal alternative(s) using the 2tuple linguistic T-spherical fuzzy numbers.Ambrin et al. [25] studied TOPSIS method in the frame of picture hesitant fuzzy sets utilizing linguistic variables.Jana et al. [26] discussed multiple attribute decision-making methods under Pythagorean fuzzy information.
Te notion of n th power root fuzzy sets was created by Al-Shami et al. [27], and they are more likely to be employed in uncertain situations than other forms of fuzzy sets due of their larger range of displaying membership grades.Tey also looked into the idea of topology for n th power root fuzzy sets.In this context, we continue to investigate some concepts and notions inspired by this type of extension of fuzzy sets and show how this class of extension of fuzzy sets we can enable to evaluate the input data with diferent signifcance for grades of membership and nonmembership, which is appropriate for some real-life issues.
Te layout of this manuscript is as follows.In Section 2, we survey orthopairs in the light of fuzzy computing with an illustrative example.In Section 3, we introduce a series of operations for the n th power root fuzzy set and investigate their major characteristics.In Section 4, we display the concept of a weighted power average operator defned over the class of n th power root fuzzy set.Ten, we go into the MADM issues that can arise when using this operator and provide an empirical example.It can be seen that the primary advantage of n th power root fuzzy sets is that they can be applied to a wide variety of decision-making scenarios.In Section 5, we supply a comparison analysis of the proposed nPR-FWPA operator with other well-known operators and compared the current operator with SR-FWPA [16] and FFWPA operators [23].Finally, in Section 6, we summarize the paper's major accomplishments and suggest some future research.

Preliminaries
In this section, we recall some relevant defnitions related to this paper.Defnition 1.Let W be the universal set and let ϖ ℏ , ω ℏ : W ⟶ [0, 1] be the functions that, respectively, determine the degrees of membership and nonmembership for every w ∈ W.
Defnition 3 (see [27]).Let N be a set of all natural numbers and W be a universal set.An n th power root fuzzy set (briefy, nPR-FS) ℏ which is a set of ordered pairs over W is defned as following: where For the sake of simplicity, we shall mention the symbol ℏ � (ϖ ℏ , ω ℏ ) for the nPR-FS ℏ � 〈w, ϖ ℏ (w), ω ℏ (w)〉:  w ∈ W}.
Te spaces of some kinds of nPR-fuzzy membership grades are displayed in Figure 1.

Some Operations via nPR-Fuzzy Sets
In this section, we propose various new operations on nPRfuzzy sets and discuss some of their features in detail.In the entire work, we employ only three decimal places for computations.
It is obvious that 0 ≤ ω ε ℏ ≤ 1, then we can obtain the following equation: Similarly, we can also obtain the following equation: Terefore, εℏ and ℏ ε are nPR-FSs.
Proof.From Defnitions 6 and 4, we have Proof.From Defnitions 4 and 6, we have And, (2) It can be proved similar to (1).

nPR-Fuzzy Weighted Power Average Inspired by the Class of nPR-Fuzzy Sets
In this section, we put forth the operator of nPR-fuzzy weighted power average and evince its main characterizations.In particular, we prove the properties of boundedness, monotonicity, and idempotency for this operator.Ten, we illustrate how nPR-FWPA operator is applied to evaluate options of an MCDM problem with nPR-fuzzy data.

Journal of Mathematics
Terefore, we have Tus, from (1) and ( 2) we obtain the following equation: Proof.For any Ten, the inequalities for membership value are Similarly, for nonmembership value (41) Proof.Since for all i we have ϖ ℏ i ≤ ϖ L i and We introduce the score and accuracy functions of the nPR-FS in order to rank nPR-FSs.

Proof
(1) For any nPR-FS ℏ, we have (2) Te proof is obvious.□ Note 1.For any nPR-FSs ℏ i � (ϖ ℏ i , ω ℏ i ), the comparison technique is supposed as follows: (1) if s(ℏ 1 ) < s(ℏ 2 ), then In what follows, we will use an nPR-FWPA operator to MCDM issues in order to evaluate options with nPR-fuzzy data.Te proposed method, in general, intertwines the following steps: Step 1.We formulate the nPR-fuzzy decision matrix R � (a ij ) m 2 ×m 1 for an MCDM problem with values of nPR-FSs, where the elements a ij (j � 1, 2, . . ., m 1 , i � 1, 2, . . ., m 2 ) are the appraisals of the alternative L i ∈ W regarding the criterion K j ∈ K Step 2. Convert the nPR-fuzzy decision matrix R � (a ij ) m 2 ×m 1 into the normalized nPR-fuzzy decision matrix Step 3. To compute alternative preference values with related weights, we use the proposed nPR-FWPA operator Step 4. Calculate the scores and accuracy of the nPR-FSs values obtained in Step 3 Step 5.By using Note 1, determine the best ranking order for the alternatives and identify the best option Step 6. End In order to exemplify the proposed method, we will show a realistic example of evaluating specifc locations using nPR-fuzzy data.
Example 4. Every family on the planet fantasizes about having their own home.It is assumed that a family wishes to build their home at a specifc location.Tey go to fve diferent places: L 1 , L 2 , L 3 , L 4 , and L 5 and establish the following fve criteria for selecting a house-building site: Accessibility and location (K 1 ): make an efort to learn the location's address as well as any other pertinent information.Is it possible to locate the site using Google Maps?Is it simple to get to?You will have a leg up on the competition if you can fnd answers to questions like these.Access to raw resources and utility services (K 2 ): any construction or building project must be carried out in an area with easy access to infrastructure and utilities in order to be successful.Water, electricity, shopping mall, a good waste disposal system, and healthcare, among other things, should all be available.Shape and size (K 3 ): both of these aspects must be taken into account.Knowing the shape and size of your home will help you fnd a layout that is ideal for you.Te site should be large enough to accommodate future expansion, and the shape should be even and free of sharp corners.Te nature of the neighborhood and security (K 4 ): in any residential area, the protection of lives and property is critical.As a result, this element should not be taken lightly.Before you start anything, conduct a thorough investigation of the security system in place at the location and its environs.

Journal of Mathematics 11
Knowing the area crime rate allows you to make informed decisions and take preventative measures to safeguard yourself, your family, employees, and property.
Te neighborhood's nature is also highly important.Are your neighbors pleasant?Is there a lot of toxins in the environment?Is there any kind of contamination at the location that could endanger people's health?Recognize the soil type (K 5 ): on a given site, various varieties of soil can be found.As a result, you must pay close attention to the soil available on your site and assess whether it is suitable for construction.

It is assumed that
is a set of criteria for the selection of places.Table 1 shows how to build the nPR-fuzzy set decision-making matrices, could a chance to be demonstrated that the degree to which the area L i fulflls those criteria K i is ϖ L i and the level with which the area L i dissatisfes those criteria . Te weight vector of the criteria was established by the family as follows: η � (0.09, 0.24, 0.17, 0.31, 0.19) T .Tey place a lower priority on K 1 and a higher priority on K 4 .Now, using weight vectors η � (0.09, 0.24, 0.17, 0.31, 0.19) T and n � 3, 4, 5, 10, 15, 20, we apply the nPR-FWPA operator as follows in Table 2. Now, as shown in Table 3, we calculate the score value of each choice as well as their ranking.
We used diferent values of n to rank the options to explain the efect of the parameter n on MADM end fndings.Table 3 shows the results of the ranking order of the alternatives based on the nPR-FWPA operator.When n � 3, 4, 5, 10, 15, 20, we obtained a rank of alternatives as L 3 ≻L 5 ≻L 2 ≻L 1 ≻L 4 , here, L 3 is the best choice.

. Comparison Analysis
Tis section gives the comparison analysis of the proposed nPR-FWPA operator under nPR-fuzzy numbers with other well-known operators.We compared the results of nPR-FWPA operator with SR-FWPA [16] and FFWPA operators [23].Te following is a summary of the fndings, which can be found in Table 4.  Te most ideal ranking order of the fve areas is L 3 ≻L 5 ≻L 2 ≻L 1 ≻L 4 , if we utilize SR-fuzzy weighted power average (SR-FWPA) and Fermatean fuzzy weighted power average (FFWPA) operators for aggregating the distinctive options.Along these lines, the best option is L 3 , which is same as that of the suggested operator.As a result, our proposed method is more adaptable than other methods already in use.

Conclusions
In this study, a set of operations via the class of nPR-fuzzy sets have been studied and their relationship have been illustrated with the assistance of suitable examples.Ten, we have presented a new weighted aggregated operator over nPR-fuzzy sets and discussed their properties in details.In addition, with one fully practical example, we have demonstrated this procedure.Finally, the fndings of the nPR-FWPA operator have been compared to the outcomes of other well-known operators.
On the one side, the proposed type of fuzzy sets enables us to evaluate the input data with diferent signifcance for grades of membership and nonmembership, which is appropriate for some real-life issues.In contrast, the diferent values estimated for the nonmembership and membership spaces require a comprehensive realization of the situations by the experts they are in charge of to evaluate the inputs of the case under study.Tis procedure is not required in the previous types of extensions of IFSs inspired by the same values of nonmembership and membership spaces.
In future works, it is possible that other uses of nPRfuzzy sets will be investigated, for example, construct abstract structures like those given in [28].Furthermore, over nPR-FSs, we will try to provide several diferent types of weighted aggregated operators and study novel MCDM methods depending on these operators.

Figure 1 :
Figure 1: Grades spaces of some kinds of nPR-fuzzy sets.

Figure 2 :
Figure 2: Some comparisons between some kinds of nPR-fuzzy sets and other generalizations of IFSs.