A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic Environment

: In this paper, we advance the study of plithogenic hypersoft set (PHSS). We present four classiﬁcations of PHSS that are based on the number of attributes chosen for application and the nature of alternatives or that of attribute value degree of appurtenance. These four PHSS classiﬁcations cover most of the fuzzy and neutrosophic cases that can have neutrosophic applications in symmetry. We also make explanations with an illustrative example for demonstrating these four classiﬁcations. We then propose a novel multi-criteria decision making (MCDM) method that is based on PHSS, as an extension of the technique for order preference by similarity to an ideal solution (TOPSIS). A number of real MCDM problems are complicated with uncertainty that require each selection criteria or attribute to be further subdivided into attribute values and all alternatives to be evaluated separately against each attribute value. The proposed PHSS-based TOPSIS can be used in order to solve these real MCDM problems that are precisely modeled by the concept of PHSS, in which each attribute value has a neutrosophic degree of appurtenance corresponding to each alternative under consideration, in the light of some given criteria. For a real application, a parking spot choice problem is solved by the proposed PHSS-based TOPSIS under fuzzy neutrosophic environment and it is validated by considering two different sets of alternatives along with a comparison with fuzzy TOPSIS in each case. The results are highly encouraging and a MATLAB code of the algorithm of PHSS-based TOPSIS is also complied in order to extend the scope of the work to analyze time series and in developing algorithms for graph theory, machine learning, pattern recognition, and artiﬁcial intelligence.


Introduction
A strong mathematical tool is always needed in order to combat real world problems involving uncertainty in the data. This necessity has urged scholars to introduce different mathematical tools to facilitate the world for solving such problems. In 1965, the concept of fuzzy set was introduced by Zadeh [1], in which each element is assigned a membership degree in the form of a single crisp value in the interval [0, 1]. It has been studied extensively by the researchers and a number of real life problems have been solved by fuzzy sets [2][3][4][5]. However, in some practical situations, it is seen that this membership degree is hard to be defined by a single number. The uncertainty in the membership degree became the cause to introduce the concept of interval-valued fuzzy set in which the degree of membership is an interval value in [0, 1]. Later on, the concept of intuitionistic fuzzy set (IFS) was proposed by Atanassov [6] in 1986, which incorporates the non-membership degree. IFS had many applications [7][8][9][10]. the attributes may have an infinite number of attribute values. In such a case, every structure with non-Archimedean metrics can be dealt in depth [30,31].
Moreover, a soft set over U can be regarded as a parameterized family of the subsets of U . For an attribute α ∈ B, F B (α) is considered as the set of α-approximate elements of the soft set (F , B).
It may be noted that hypersoft set is a generalization of soft set.

Plithogenic Sets
A set X is called a plithogenic set if all of its members are characterized by the attributes under consideration and each attribute may have any number of attribute values [24]. Each attribute value possesses a corresponding appurtenance degree of the element x, to the set X, with respect to some given criteria. Moreover, a contradiction degree function is defined between each attribute value and the dominant attribute value of an attribute in order to obtain accuracy for aggregation operations on plithogenic sets. These degrees of appurtenance and contradiction may be fuzzy, intuitionistic fuzzy or neutrosophic degrees. Remark 1. Plithogenic set is regarded as a generalization of crisp, fuzzy, intuitionistic fuzzy. and neutrosophic sets, since the elements of later sets are characterized by a combined single attribute value (degree of appurtenance), which has only one value for crisp and fuzzy sets i.e., membership, two values in case of intuitionistic fuzzy set i.e., membership and non-membership, and three values for neutrosophic set i.e., membership, indeterminacy, and non-membership. In the case of plithogenic set, each element is separately characterized by all attribute values under consideration in terms of degree of appurtenance.

Plithogenic Hypersoft Set (PHSS)
Let X ⊆ U and C = A 1 × A 2 × . . . × A n , where n ≥ 1 and A i is the set of all attribute values of the attribute a i , i = 1, 2, 3, . . . , n. Each attribute value γ possesses a corresponding appurtenance degree d(x, γ) of the member x ∈ X, in accordance with some given condition or criteria. The attribute value degree of appurtenance is a function that is defined by such that d(x, γ) ∈ [0, 1] j , and P([0, 1] j ) is the power set of [0, 1] j , where j = 1, 2, 3 are for fuzzy, intuitionistic fuzzy, and neutrosophic degree of appurtenance, respectively.
Furthermore, the degree of contradiction (dissimilarity) between any two attribute values of the same attribute is a function given by For any two attribute values γ 1 and γ 2 of the same attribute, it is denoted by c(γ 1 , γ 2 ) and satisfies the following axioms: Subsequently, (X, A, C, d, c) is called a plithogenic hypersoft set (PHSS) [22]. For an n-tuple (γ 1 , γ 2 , . . . , γ n ) ∈ C, γ i ∈ A i , 1 ≤ i ≤ n, a plithogenic hypersoft set F : C → P(U ) is mathematically written as

Illustrative Example
Let U = {m 1 , m 2 , m 3 , . . . , m 10 } be a universe containing mobile phones. A person wants to buy a mobile phone for which the mobile phones under consideration (alternatives) are contained in X ⊆ U , given by The characteristics or attributes of the mobile phones belong to the set A = {a 1 , a 2 , a 3 , a 4 }, such that a 1 = Processor power, a 2 = RAM, a 3 = Front camera resolution, a 4 = Screen size in inches.

The Four Classifications of PHSS
In this section, we propose the four different classifications of PHSS that are based on the number of attributes chosen for application and the characteristics of alternatives under consideration or that of the attribute value degree of appurtenance function. The same example from Section 2 is considered to each classification for a practical understanding. Figure 1 shows a diagram for these classifications.

The First Classification
This classification is based on the number of attributes that are chosen by the decision makers for application.

Uni-Attribute Plithogenic Hypersoft Set
Let α ∈ A be an attribute required by the experts for application purpose and the attribute values of α belong to the finite discrete set Y = {y 1 , y 2 , . . . , y m }, m ≥ 1. Hence, the degree of appurtenance function is given by such that d(x, y) ⊆ [0, 1] j , where P([0, 1] j ) denotes the power set of [0, 1] j and j = 1, 2, 3 stands for fuzzy, intuitionistic fuzzy, or neutrosophic degree of appurtenance, respectively.
The contradiction degree function between any two attribute values of α, is given by For any two attribute values y 1 , y 2 ∈ Y, it is denoted by c(y 1 , y 2 ) and the following properties hold: c(y 1 , y 1 ) = 0, c(y 1 , y 2 ) = c(y 2 , y 1 ).
Subsequently, (X, α, Y, d, c) is termed as a uni-attribute plithogenic hypersoft set. For an attribute value y ∈ Y, a uni-attribute plithogenic hypersoft set F : Y → P(U ) is mathematically written as

Multi-Attribute Plithogenic Hypersoft Set
Consider a subset B of A, consisting of all attributes that were chosen by the experts, given by Let the attribute values of b 1 , b 2 , . . . , b m belong to the sets B 1 , B 2 , . . . , B m , respectively, and Afterwards, the appurtenance degree function is such that d(x, y) ⊆ [0, 1] j , j = 1, 2, 3. In this case, the contradiction degree function is given by The degree of contradiction between any two attribute values y 1 and y 2 , is denoted by c(y 1 , y 2 ) and it satisfies the following axioms: c(y 1 , y 1 ) = 0, c(y 1 , y 2 ) = c(y 2 , y 1 ).
is called a multi-attribute plithogenic hypersoft set. For an m-tuple (y 1 , y 2 , . . . , y m ) ∈ Y m , y i ∈ B i , 1 ≤ i ≤ m, a multi-attribute plithogenic hypersoft set F : Y m → P(U ) is mathematically written as The attribute values of a 1 , a 2 , a 3 , a 4 are contained in the sets A 1 , A 2 , A 3 , A 4 given below: 4.5, 5, 5.5, 6}.

Uni-attribute plithogenic hypersoft set
Consider the most demanding feature of a mobile phone given by the attribute a 3 that stands for front camera resolution. The set of attribute values of a 3 is A 3 = {2MP, 5MP, 8MP, 16MP}. Then, the uni-attribute plithogenic hypersoft set F : A 3 → P(U ) is given by where d x (γ) denotes the degree of appurtenance of x ∈ X, to the set X, w.r.t. the attribute value γ ∈ A 3 . For an attribute value 16MP ∈ A 3 , we have

Multi-attribute plithogenic hypersoft set
Let B = {a 3 , a 4 } be the set of attributes required by the customer. Therefore, we need A 3 and A 4 given by Suppose that the customer is interested to buy a mobile phone with specific requirements of 16MP front camera with 5.5 inch screen size. In this case, we take (16MP, 5.5) ∈ A 3 × A 4 and a multi-attribute plithogenic hypersoft set F : where d m 5 (γ) stands for the degree of appurtenance of m 5 to the set X w.r.t. the attribute value γ ∈ (16MP, 5.5).

The Second Classification
This classification is based on the nature of the attribute value degree of appurtenance that may be crisp, fuzzy, intuitionistic fuzzy, or neutrosophic degree of appurtenance.

Plithogenic Intuitionistic Fuzzy Hypersoft Set
If the attribute value appurtenance degree d x (γ) of each x ∈ X, w.r.t. each attribute value, is intuitionistic fuzzy degree, then it is called the plithogenic intuitionistic fuzzy hypersoft set. Mathematically, it is written as d x (γ) ∈ P([0, 1] 2 ).

The Third Classification
This classification is based on the properties of attribute values and degree of appurtenance function.

Plithogenic Refined Hypersoft Set
Let (X, A, C, d, c) be a plithogenic hypersoft set and A denote the set of attribute values of an attribute a. If an attribute value γ ∈ A of the attribute a is subdivided or split into at least two or more attribute sub-values γ 1 , γ 2 , γ 3 , . . . ∈ A, such that the attribute sub-value degree of appurtenance function d(x, γ i ) ∈ P([0, 1] j ), for i = 1, 2, 3, . . . and j = 1, 2, 3 for fuzzy, intuitionistic fuzzy, neutrosophic degree of appurtenance, respectively, then X is called a refined plithogenic hypersoft set. It is represented as (X r , A, C, d, c).

Plithogenic Hypersoft Overset
If the degree of appurtenance of any element x ∈ X w.r.t. any attribute value γ ∈ A of an attribute a is greater than 1, i.e., d(x, γ) > 1, then X is called a plithogenic hypersoft overset. It is represented as (X o , A, C, d, c).

Plithogenic Hypersoft Underset
If the degree of appurtenance of any element x ∈ X w.r.t. any attribute value γ ∈ A of an attribute a less than 0, i.e., d(x, γ) < 0, then X is called a plithogenic hypersoft underset. It is represented as (X u , A, C, d, c).

Plithogenic Hypersoft Offset
A plithogenic hypersoft set (X, A, C, d, c) is called a plithogenic hypersoft offset if it is both an overset and an underset. Mathematically, if d(x 1 , γ 1 ) > 1 and d(x 2 , γ 2 ) < 0 for the same or different attribute values γ 1 , γ 2 ∈ A that correspond to the same or different members x 1 , x 2 ∈ X, then (X off , A, C, d, c) is a plithogenic hypersoft offset.

Plithogenic Hypersoft Multiset
If an element x ∈ X repeats itself into the set X with same plithogenic components given by or with different plithogenic components given by then (X n , A, C, d, c) is called a plithogenic hypersoft multiset.

Plithogenic Bipolar Hypersoft Set
If the attribute value appurtenance degree function is given by where j = 1, 2, 3, then, (X b , A, C, d, c) is called plithogenic bipolar hypersoft set. It may be noted that, for an attribute value γ, d(x, γ) allots a negative degree of appurtenance in [−1, 0] and a positive degree of appurtenance in [0, 1] to each element x ∈ X with respect to each attribute value γ.
Remark 3. The concept of plithogenic bipolar hypersoft set can be extended to plithogenic tripolar hypersoft set and so on up to plithogenic multipolar hypersoft set.

Plithogenic Complex Hypersoft Set
If for any x ∈ X, the attribute value appurtenance degree function, with respect to any attribute value γ, is given by  The attributes are a 1 , a 2 , a 3 , a 4 , whose attribute values are contained in the sets A 1 , A 2 , A 3 , A 4 .

Plithogenic refined hypersoft set
Consider an attribute a 4 = screen size in inches whose attribute values belong to the set A 4 = {4, 4.5, 5, 5.5, 6}. A refinement of A 4 is given by Therefore, a plithogenic refined hypersoft set F r : A 4 → P(U ) is given by

Plithogenic hypersoft overset
Let each attribute value has a single-valued fuzzy degree of appurtenance to all the elements of X. Subsequently, for (octa-core, 8GB, 16MP, 5.5) ∈ C, a plithogenic hypersoft overset F o : C → P(U ) is given by It may be noted that d m 5 (16MP) > 1.

Plithogenic hypersoft underset
A plithogenic hypersoft underset defined by the function F u : C → P(U ) is given by It may be noted that d m 5 (16MP) < 0.

Plithogenic hypersoft offset
A plithogenic hypersoft offset is a function F off : C → P(U ), as given by

Plithogenic hypersoft multiset
A plithogenic hypersoft multiset F m : C → P(U ) is given by It should be noted that the element m 5 repeats itself with different plithogenic components.

Plithogenic bipolar hypersoft set
A plithogenic bipolar hypersoft set F 2 : C → P(U ) is given by

Plithogenic complex hypersoft set
A plithogenic complex hypersoft set F com : C → P(U ) is given by

The Fourth Classification
The attribute value degree of appurtenance may be a single crisp value in [0, 1], a finite discrete set or an interval value in [0, 1]. Therefore, we have the following classification of PHSS.

Single-Valued Plithogenic Hypersoft Set
A plithogenic hypersoft set is called a single-valued plithogenic hypersoft set if the attribute value appurtenance degree is a single number in [0, 1].

Hesitant Plithogenic Hypersoft Set
If the attribute value degree of appurtenance is a finite discrete set of the form {m 1 , m 2 , . . . , m i },   Each attribute value has an interval value degree of appurtenance in [0, 1] to each element m 5 and m 8 .

The Proposed PHSS-Based TOPSIS with Application to a Parking Problem
In this section, we use the concept of PHSS in order to construct a novel MCDM method, called PHSS-based TOPSIS, in which we extend TOPSIS based on PHSS under fuzzy neutrosophic environment. Moreover, a parking spot choice problem is constructed in order to employ the newly developed PHSS-based TOPSIS to prove its validity and efficiency. Two different sets of alternatives are considered for the application and a comparison is performed with fuzzy TOPSIS in both cases.

Proposed PHSS-Based TOPSIS Algorithm
Let U be a non-empty universal set, and let X ⊆ U be the set of alternatives under consideration, given by X = {x 1 , x 2 , . . . , x m }. Let C = A 1 × A 2 × . . . × A n , where n ≥ 1 and A i is the set of all attribute values of the attribute a i , i = 1, 2, 3, . . . , n. Each attribute value γ has a corresponding appurtenance degree d(x, γ) of the member x ∈ X, in accordance with some given condition or criteria. Our aim is to choose the best alternative out of the alternative set X. The construction steps for the proposed PHSS-based TOPSIS are as follows: S1: Choose an ordered tuple (γ 1 , γ 2 , . . . , γ n ) ∈ C and construct a matrix of order n × m, whose entries are the neutrosophic degree of appurtenance of each attribute value γ, with respect to each alternative x ∈ X under consideration. S2: Employ the newly developed plithogenic accuracy function A p , to each element of the matrix obtained in S1, in order to convert each element into a single crisp value, as follows: where T γ , I γ , F γ represent the membership, indeterminacy, and non-membership degrees of appurtenance of the attribute value γ to the set X, and T γ d , I γ d , F γ d stand for the membership, indeterminacy, and non-membership degrees of corresponding dominant attribute value, whereas c F (γ, γ d ) denotes the fuzzy degree of contradiction between an attribute value γ and its corresponding dominant attribute value γ d . This gives us the plithogenic accuracy matrix.
S3: Apply the transpose on the plithogenic accuracy matrix to obtain the plithogenic decision matrix M p = [m ij ] m×n of alternatives versus criteria.
S4: A plithogenic normalized decision matrix N p = [y ij ] m×n is constructed, which represents the relative performance of alternatives and whose elements are calculated as follows: , j = 1, 2, 3, . . . , n.

S5: Construct a plithogenic weighted normalized decision matrix
is a row matrix of allocated weights w k assigned to the criteria a k , k = 1, 2, 3, . . . , n and ∑ w k = 1, k = 1, 2, . . . , n. Moreover, all of the selection criteria are assigned different weights by the decision maker, depending on their importance in the decision making process.
S7: Calculate plithogenic positive distance S + i and plithogenic negative distance S − i of each alternative from V + p and V − p , respectively, while using the following formulas: S8: Calculate the relative closeness coefficient C i of each alternative by the following expression: S9: The highest value from {C 1 , C 2 , . . . , C m } belongs to the most suitable alternative. Similarly, the lowest value gives us the worst alternative.

Parking Spot Choice Problem
Based on the proposed method, a parking spot choice problem is constructed. Parking a vehicle at some suitable parking spot is an interesting real life MCDM problem. A number of questions arises in mind, for instance, how much will the parking fee be, how far is it, will it be an open or covered area, how many traffic signals will be on the way, etc. Thus, it becomes a challenging task in the presence of so many considerable criteria. This task is formulated in the form of a mathematical model in order to apply the proposed technique to choose the most suitable parking spot. Consider a person at a particular location on the road, who wants to park his car at a suitable parking place. Keeping in mind the person's various preferences, a few nearby available parking spots are considered, having different specifications in terms of parking fee, distance between the person's location and each parking spot, the number of signals between the car and the parking spot, and traffic density on the way between the car and the parking spot. Figure 2 shows the location of car to be parked at a suitable parking spot. Let U be a plithogenic universe of discourse consisting of all parking spots in the surrounding area, where U = {P 1 , P 2 , P 3 , . . . , P 10 }.
The attributes of the parking spots, chosen for the decision, are a 1 , a 2 , a 3 , a 4 given below: a 1 = Parking fee, a 2 = Distance between car and parking spot, a 3 = Number of traffic signals between car and parking spot, a 4 = Traffic density on the way between car and parking spot.
The attribute values of a 1 , a 2 , a 3 , a 4 belong to the sets A 1 , A 2 , A 3 , A 4 , respectively.
Two different sets of alternatives are considered for the application of PHSS-based TOPSIS, along with a comparison with fuzzy TOPSIS in each case.

Case 1
In this case, the parking spots under consideration (alternatives) are contained in the set X ⊆ U , given by X = {P 1 , P 2 , P 3 , P 4 }.
The neutrosophic degree of appurtenance of each attribute value corresponding to each alternative P 1 , P 2 , P 3 , P 4 is given in Table 1.
and consider an element ( f 2 , r 1 , s 2 , d 1 ) ∈ C for which the corresponding matrix that was obtained from Table 1  This MCDM problem is solved by the proposed PHSS-based TOPSIS and fuzzy TOPSIS, as follows:

A. Application of PHSS-based TOPSIS for Case 1
Apply the plithogenic accuracy function (1) to the matrix (2) in order to obtain the plithogenic accuracy matrix given by:  Table 2, of alternatives versus criteria may also be drawn to see the situation in a clear way.
whereas the plithogenic weighted normalized decision matrix V p = [v ij ] 4×4 is given, as follows: The plithogenic distance of each alternative from the V + p and V − p , respectively, is determined as: The relative closeness coefficient C i , i = 1, 2, 3, 4, of each alternative is computed as: The highest value corresponds to the most suitable alternative. Since C 3 = 0.7494 is the maximum value and it corresponds to P 3 , therefore, the most suitable parking spot is P 3 . The Table 3 is constructed to rank all alternatives under consideration. A bar graph presented in Figure 3 is given, in which all alternatives P 1 , P 2 , P 3 , P 4 are ranked by PHSS-based TOPSIS. It is evident that the parking spot P 3 is the most suitable place to park the car while P 4 is not a good choice for parking based on the selection criteria.

B. Application of Fuzzy TOPSIS for Case 1
In order to see the implementation of fuzzy TOPSIS [32][33][34] for the current scenario of the parking problem, we apply the average operator [27,35] to each element of the matrix 2 and take the transpose of the resulting matrix in order to obtain the decision matrix given by: Applying the fuzzy TOPSIS to the decision matrix M, along with the same weights given in matrix (3), we obtain the values of positive distance S + , negative distance S − , relative closeness C i and ranking of each alternative, as given in Table 4. A bar graph in Figure 4 is given in which all alternatives P 1 , P 2 , P 3 , P 4 are ranked by Fuzzy TOPSIS. A comparison is shown in Table 5, in which it can be seen that the result obtained by the proposed PHSS-based TOPSIS is aligned with that of fuzzy TOPSIS.  It is observed in Table 5 that the results obtained by both methods coincide in terms of the ranking of each alternative, but differ in the values of the relative closeness of each alternative. It is due to the nature of the MCDM problem in hand in which each alternative needs to be evaluated against each attribute value possessing a neutrosophic degree of appurtenance w.r.t. each alternative and a contradiction degree is defined between each attribute value and its corresponding dominant attribute value to be taken into consideration in the decision process. In such a case, the proposed PHSS-based TOPSIS produces a more reliable relative closeness of each alternative, as it can been seen in the parking spot choice problem that was chosen for the study. Therefore, it is worth noting that the proposed PHSS-based TOPSIS can be regarded as a generalization of fuzzy TOPSIS [32], because the fuzzy TOPSIS cannot be directly applied to MCDM problems in which the attribute values have a neutrosophic degree of appurtenance with respect to each alternative. In the case of the parking problem, fuzzy TOPSIS is applied after applying simple average operator to the neutrosophic elements of the matrix (2). However, it does not takes into account the degree of contradiction between the attribute values, which is the limitation of fuzzy TOPSIS. This concern is precisely addressed by the proposed PHSS-based TOPSIS.

Case 2
In this case, the set of parking spots under consideration is given by The neutrosophic degree of appurtenance of each attribute value that corresponds to each alternative of {P 1 , P 5 , P 6 , P 7 } is given in Table 6. Table 6. Degree of appurtenance of each attribute value w.r.t each alternative.

Sr.
Variables and consider an element ( f 2 , r 1 , s 2 , d 1 ) ∈ C for which the corresponding matrix obtained from Table 6, is given below: The plithogenic weighted normalized decision matrix V p is given, as follows: The plithogenic positive distance S + , plithogenic negative distance S − , relative closeness C i , and ranking of each alternative is shown in Table 7. A graphical representation of the ranking of all alternatives obtained by PHSS-based TOPSIS, is shown in Figure 5. It can be seen that the parking spot P 7 is the most suitable alternative in the light of chosen criteria.

B. Application of Fuzzy TOPSIS for Case 2
In this case, the decision matrix M for the implementation of fuzzy TOPSIS is given by  (3), the values of positive distance S + , negative distance S − , relative closeness C i , and ranking of each alternative are shown in Table 8. The ranking of all alternatives can also been visualized as a bar graph in Figure 6, in which all alternatives P 1 , P 5 , P 6 , P 7 are ranked by Fuzzy TOPSIS.
The most suitable parking spot obtained by fuzzy TOPSIS is also P 7 . A comparison of rankings obtained by PHSS-based TOPSIS and fuzzy TOPSIS is shown in Table 9 for case 2.
It may be noted that similar results are obtained in case 2, with the help of proposed PHSS-based TOPSIS and fuzzy TOPSIS with exactly same ranking of each alternative, but with a considerably different values of the relative closeness of each alternative as shown in Table 9. Therefore, it is accomplished that the results that were obtained by the PHSS-based TOPSIS are valid and more reliable and PHSS-based TOPSIS can be regarded as the generalization of fuzzy TOPSIS on the basis of the study conducted in the article.

Conclusions
It has always been a challenging task to deal with real life MCDM problems, due to the involvement of many complexities and uncertainties. In particular, some real life MCDM problems are designed in a way that the given attributes need to be further decomposed into two or more attribute values such that each alternative is then required to be evaluated against each attribute value in order to perform a detailed analysis to reach a fair conclusion. To deal with such situations, a novel PHSS-based TOPSIS is proposed in the present study, and it is applied to a MCDM parking problem with different choices of the set of alternatives and a comparison with fuzzy TOPSIS is done to prove the validity and efficiency of the proposed method. All of the results are quite promising and graphically depicted for a clear understanding. Moreover, the algorithm of the proposed method is produced in MATLAB in order to broaden the scope of the study to other research areas, including graph theory, machine learning, pattern recognition, etc.