Multiple Attribute Decision Making Algorithm via Picture Fuzzy Nano Topological Spaces

Picture fuzzy nano topological spaces is an extension of intuitionistic fuzzy nano topological spaces. Every decision in life ends with an answer such as yes or no, or true or false, but we have an another component called abstain, which we have not yet considered. This work is a gateway to study such a problem. This paper motivates an enquiry of the third component—abstain—in practical problems. The aim of this paper is to investigate the contemporary notion of picture fuzzy nano topological spaces and explore some of its properties. The stated properties are quantified with numerical data. Furthermore, an algorithm for Multiple Attribute Decision-Making (MADM) with an application regarding the file selection of building material under uncertainty by using picture fuzzy nano topological spaces is developed. As a practical problem, a comparison table is presented to show the difference between the novel concept and the existing methods.


Introduction
Multiple Attribute Decision-Making (MADM) is a method that specifically considers the best possible alternatives. In medieval times, decisions were made without coping with data uncertainties, which could lead to a potential outcome. Inadequate outcomes had reallife organizational conditions. The results would be ambivalent, undefined, or wrong if we deduced the result of obtained data without hesitation. MADM played an important role in real life problem such as management, diagnosing diseases, economics and industries. Each time, hundreds of decisions are taken by each decision maker to execute the major part of his/her work but it should be a logical judgment. MADM is used to solve complex and complex problems with various parameters for this. In MADM, the issue must be identified by defining potential alternatives, evaluating each alternative on the basis of the criteria set by the decision-maker or community of decision-makers and, ultimately, choosing the best alternative. A variety of valuable mathematical methods, such as fuzzy sets, neutrosophic sets, and soft sets, were developed to tackle the complexities and complexity of MADM problems.
Fuzzy set theory was introduced by Zadeh [1]. MADM algorithm via rough fuzzy information was introduced and developed by Zafer et al. [2]. Among different generalized FSs, in sight of IFSs introduced by Atanassov [3], lacking a logical scheme to effectively process inconsistent and indeterminate knowledge embedded in practical situations, Smarandache [4] introduced the structure of neutrosophic logic, and NSs, which This model is the most common type and is used to collect large-scale data in Artificial Intelligence, Engineering and medical applications. Similar research can be easily replicated in the future with other methods and different forms of hybrid structure.
The scheme of this manuscript is organized as follows. Section 2, gives the preliminary definitions of the literature in NS are discussed. In Section 3, we introduced a novel idea of PFNTSs and established some of its operations such as interior and closure with the help of illustrations, and defined a score function. In Section 4, We proposed an algorithm and flowchart for MADM problem. In Section 5, As a numerical example, we established a method for the solution of MADM problem related to civil engineering (material selection) using PFNTS. We also presented the efficiency, advantage, consistency and validity of the algorithms proposed. With some current methodologies we provided a brief overview and comparative review of our proposed approach. The conclusion of this work is essentially summed up and future scope of research are presented in Section 6.

Preliminaries
The definitions from [12,20] are used in sequel.
The union of S 1 and S 2 is

2.
The intersection of S 1 and S 2 is

3.
The symmetric difference of S 1 and S 2 is Definition 3 ([12]). Let S be a family of PFS on U = ∅. Then (X, S) is called a picture fuzzy topological space if it satisfies the following: • 0 S and 1 S are member of S.

•
Arbitrary union of picture fuzzy set S in S if each S in S • Finite intersection of picture fuzzy set S in S if each S in S Definition 4 ([12]). Let R and S be the equivalence relation and picture fuzzy set, respectively defined on universe of discourse U. The membership m S , the abstinence a S and nonmembership n S are the components of S. The approximation space (U, R) has three components, namely lower PFL R (S), upper PFU R (S), and boundary approximation PFB R (S) where (i) The upper approximation of S with respect to R is denoted by PFU R (S), i.e., The lower approximation of S with respect to R is the set is denoted by PFL R (S), i.e., The boundary region of S with respect to R is the set of all objects which can be classified neither as S nor as not S with respect to R and is denoted by

Picture Fuzzy Nano Topological Spaces
Definition 5. Let the Universe be U, equivalence relation on the non-void set S ⊆ U be R and where A ⊆ S and τ R satisfies the following axioms:

Remark 1.
In PFNTS, the picture fuzzy nano border will be a non-void set. Since the symmetric difference between picture fuzzy nano major and picture fuzzy nano minor approximations is defined here as the maximum and minimum of the values in the picture fuzzy sets.

Proposition 1.
Let U be a non-void universe and A be a picture fuzzy set on U. Then the following statements hold: 1.
The collection τ R (A) = {0 p , 1 p }, is the in-discrete picture fuzzy nano topology on U.

2.
If PFL R = PFU R = PF R , then the picture fuzzy nano topology is If PFU R = PFB R , then the picture fuzzy nano topology is be any PFNTS with respect to picture fuzzy subset of U and let A be a picture fuzzy nano set in S. Then the picture fuzzy nano interior and picture fuzzy nano closure of A are defined as follows:

Remark 2. For any picture fuzzy nano set
A is a PFNCS if and only if A − = A.

4.
A is a PFNOS if and only if A o = A.

5.
A − is a PFNCS in U.

6.
A o is a PFNOS in U.
Theorem 1. Let (U; τ R )(S) be a picture fuzzy nano topological space with respect to S where S is a picture fuzzy subset of U . Let A 1 and A 2 be picture fuzzy subsets of U. Then the following statements hold: A is picture fuzzy nano closed if and only if Proof.

1.
By definition of picture fuzzy nano closure, If A is a picture fuzzy nano closed set, then A is the smallest picture fuzzy nano closed set containing itself and hence A − = A. Conversely, if A − = A, then A is the smallest picture fuzzy nano closed set containing itself and hence A is a picture fuzzy nano closed set. 3.
Since 0 p and 1 p are picture fuzzy nano closed sets in (U;

5.
Since PFN set A 1 is a subset of union of two PFN setsA 1 and A 2 and PFN set A 2 is a subset of union of two PFN sets A 1 and Then closure of PFN set A 1 is a subset of closure of union of two PFN sets A 1 and A 2 and closure of PFN set A 2 is a subset of closure of union of two PFN sets A 1 and A 2 . Therefore, union of closure of PFN sets Since A − is a picture fuzzy nano closed set, then ( Theorem 2. (U; τ R )(S) be a picture fuzzy nano topological space with respect to S where S is a picture fuzzy subset of U. Let A be a picture fuzzy subset of U. Then 1.
Theorem 3. Let (U; τ R )(S) be a picture fuzzy nano topological space with respect to S where S is a picture fuzzy subset of U . Let A 1 and A 2 be picture fuzzy subsets of U. Then the following statements hold:

A is picture fuzzy nano open if and only if
Proof.

1.
A If Since  a 1 , n 1 } and A 2 = {m 2 , a 2 , n 2 } be two PFSs, the following comparison rules are used 1. if

Picture Fuzzy Nano Topology in Multiple Attribute Decision-Making
MADM is a procedure for seeking a best solution that has the highest degree of satisfaction from a set of possible alternative solutions. These types of MADM problems arise in a many real-time situations, and they are characterized by multiple attributes.
The proposed algorithm deals with abstinence of an object other than a yes or no choice, while the other algorithm in fuzzy set theory and intuitionistic fuzzy set theory failed to handle these cases. This algorithm shows how picture fuzzy nano topology is influenced in decision making. The procedure we carried out in the algorithm is simple and is an elementary one to handle.
A novel picture fuzzy nano topological approach is presented in this section for decision-making problems with picture fuzzy information. A methodological procedure for selecting the right alternatives and attributes in the decision-making environment is proposed as the following necessary steps.

Proposed Algorithm and Flowchart
The flow chart of proposed Algorithm 1 for MADM is given in Figure 1.

Algorithm 1: Ideal decision making with PFTSs
Input part: Step-1: Consider the universe of discourse (set of objects) O, the set of alternatives E, the set of decision attributes D.
Step-2: Construct a picture fuzzy matrix of alternative verses objects and object verses decision attributes. Calculation part: Step-3: Construct the picture fuzzy topologies C * τ 1 η and C * τ 2 ξ .
Step-4: Find the score and accuracy values by Definition 2.3 of each of the entries of the PFNTS. Conclusion Part: Step-5: Organize the complex neutrosopic score values of the alternatives G 1 ≤ G 2 ≤ .. ≤ G β and the attributes H 1 ≤ H 2 ≤ .. ≤ H γ . Choose the attribute H γ for the alternative G 1 and H γ − 1 for the alternative G 2 etc. If β ≤ γ, then ignore H ξ , where ξ = 1, 2, β − γ.
Enter the attributes Os, Ds and alternatives Es start

Construct the PF matrices Es/Os and Os/Ds
Concoct the PFTs C * τ 1 η and C * The flow chart of proposed Algorithm for MADM is given in Figure 2.
Enter the attributes Os, Ds and alternatives Es start

Construct the PF matrices
Es/Os and Os/Ds Frame the in-discernibility relations R Concoct the PFNTs C * τ 1 η and C * Algorithm 2: Ideal decision making with PFNTSs Input part: Step-1: Consider the universe of discourse (set of objects) O, the set of alternatives E, the set of decision attributes D.
Step-2: Construct a picture fuzzy matrix of alternative verses objects and object verses decision attributes. Calculation part: Step-3: Frame the in-discernibility relation R on O.
Step-5: Find the score and accuracy values by Definition 2.3 of each of the entries of the PFNTS. Conclusion Part: Step-6: Organize the complex neutrosopic score values of the alternatives G 1 ≤ G 2 ≤ .. ≤ G β and the attributes H 1 ≤ H 2 ≤ .. ≤ H γ . Choose the attribute H γ for the alternative G 1 and H γ − 1 for the alternative G 2 etc. If β ≤ γ, then ignore H ξ , where ξ = 1, 2, β − γ.

Numerical Example
The proposed algorithms helps the builder to find the suitable building material. The method of classifying various sets of features of the material for flooring under a single form is very critical and complicated. In certain realistic circumstances, each dimension has the possibility within a form of the picture fuzzy sets. Therefore, further abstinence is involved in the medical diagnosis. Complicated situations are addressed by picture fuzzy nano topologies. This strategy is generally more versatile when it comes to less places of abstinence, and easier to use. With a score function between builder versus feature requirement and features versus material type, the proposed algorithms of picture fuzzy topological spaces and picture fuzzy nano topological spaces has the right choice of selection of material in picture fuzzy milieu.
First, we solve the material problem by using first PFTS-MADM method as given in Algorithm 1.
The key feature of this suggested method is that it appraises the factual participation, specific indeterminate and misrepresentation of each dimension in the form of a picture fuzzy set.
Our work is to analyze the builder's choice and decide on the flooring type of material suitable for them in a picture fuzzy environment.
Step-2: Frame the matrix of picture fuzzy system of relationship between builders and features and the matrix of picture fuzzy system of relationship between features and flooring material are given in Tables 1 and 2 respectively. Table 1. The picture fuzzy system of relationship between builders and features.  Table 2. The picture fuzzy system of relationship between features and flooring material. Step-5: Computation of picture fuzzy score functions for the builders and flooring materials are done as in step-5 of the algorithm are as follows: Score values for the builders are Score values for the flooring materials are As the score functions are not equal, the accuracy function does not need to be calculated.
Step-6: Arrange the picture fuzzy score values for the alternatives b 1 , b 2 , b 3 , b 4 and the attributes f 1 , f 2 , f 3 , f 4 in run-up order. We consider the sequences below Step 1 and Step 2 are same as in Algorithm 1.
Step-4: Construct the picture fuzzy nano topological spaces for each builder and each flooring material with respect to the features as follows: Picture fuzzy nano topologies for builders are C * As the score functions are not equal, the accuracy function does not need to be calculated.
Step-6: Arrange the picture fuzzy score values for the alternatives b 1 , b 2 , b 3 , b 4 and the attributes f 1 , f 2 , f 3 , f 4 in run-up order. We consider the sequences below b 3  We propose two algorithms for MADM of the real world problems. The first two steps and last two steps of Algorithm 1 and Algorithm 2 are the same. In Step 3 of Algorithm 1 we find PFTS while in step 3 and step 4 of Algorithm 2 we find in-discernibility relations and PFNTSs. Both algorithms give approximately same ranks; this difference does not mean incomplete information. This is only because both algorithms have different formulae. The constructed algorithms are valid and practical. Finally both algorithms gives the same final decision.
The comparison Table 3 shows the difference between novel picture fuzzy nano topological space with existing work.

Conclusions and Future Work
The application of the rough picture fuzzy set gained attention among researchers. However, the boundary of the region was not studied further by Nguyen [12]. In this paper, we introduced boundary of a region on picture fuzzy set along with upper and lower approximation. It is our opinion that picture fuzzy information can be best dealt with by unclear, vague, indeterminate, contradictory, and incomplete periodic / redundant information work. This paper aimed at bringing out the picture fuzzy nano topology which is more versatile and adaptable to real-time issues than rest of the types of general fluffy sets. Definitions of nano topology in picture fuzzy sets were identified, followed by the closure and interior operations. A new form of MADM technique in the picture fuzzy set has been introduced and applied to a building material selection process. To show the advantages and applicability, a comparison was made between the proposed method and the existing methods. The results are critical for enriching the picture fuzzy set awareness provided for decision making applications. Future research plans are to use the MADM technique for more practical applications and advance the practical interval valued complex picture fuzzy nano topological logic method for prediction of forecasting problems.
Author Contributions: All authors have contributed equally to this paper. The individual responsibilities and contributions of all authors can be described as follows: the idea of this whole paper was put forward by P.M. and C.O., I.A. and H.G. completed the preparatory work of the paper. C.O. and H.G. analyzed the existing work. The revision and submission of this paper was completed by P.M. and I.A. All authors have read and agreed to the published version of the manuscript.