A novel complex fuzzy N-soft sets and their decision-making algorithm

Complex fuzzy N-soft set (CFN-SS) is an important technique to manage awkward and unreliable information in realistic decision-making problems. CFN-SS is a blend of two separate theories, called N-soft sets (N-SSs) and complex fuzzy sets (CFSs), which are the modiﬁed versions of soft sets (SSs) and fuzzy sets (FSs) to depict vague and uncertain information in daily life problems. In this manuscript, the novel concept of CFN-SS is explored and their fundamental laws are discussed. CFN-SS contains the grade of truth in the form of a complex number whose real and imaginary parts are limited to the unit interval. Besides, we examine some algebraic properties for CFN-SS like union, intersections and justify these properties with the help of some numerical examples. To examine the superiority and effectiveness of the proposed approaches, the special cases of the investigated approaches are also discussed. A decision-making procedure is developed by using the investigated ideas based on CFN-SSs. Further, some numerical examples are also illustrated with the help of explored ideas to ﬁnd the reliability and effectiveness of the proposed approaches. Finally, the comparative analysis of the investigated ideas with some existing ideas is also demonstrated to prove the quality of the proposed works. The graphical expressions of the obtained results are also discussed.


Introduction
With the development of the information age, the decisionmaking problems and decision-making environments are more and more complex.Thus, it becomes more and more difficult to express attribute values of alternatives.Based on this, Zadeh firstly developed the definition of fuzzy sets (FSs) [1], which can easily express fuzzy information for multi-attribute decision-making (MADM) problems and multi-attribute group decision-making (MAGDM) problems.The probability theory is also one of the most important and useful techniques to cope with awkward and complicated information in realistic issues, which were explored by numerous scholars [2,3].Later, FSs got a lot of attention B Tahir Mahmood tahirbakhat@iiu.edu.pkUbaid ur Rehman ubaid5@outlook.comZeeshan Ali zeeshanalinsr@gmail.com 1 Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan from researchers and got extensions such as interval-valued FSs (IvFSs) [4], intuitionistic FSs (IFSs) [5], interval-valued IFSs (IvIFSs) [6], etc. Historically, the set of real numbers was extended to the set of complex numbers, this extension motivated Ramot et al. [7] to introduce the notion of CFSs.Mahmood et al. [8] interpreted the concept of complex hesitant fuzzy sets.
The volume and intricacy of the gathered information in our advanced society are developing quickly.There regularly exist different sorts of vagueness in that information is identified with complex issues in different fields such as engineering, economics, social science, environmental science, and biology, etc.To portray and extract useful data covered up with uncertain information, researchers in computer science, mathematics, and related areas interpreted various theories like fuzzy set theory [1], vague set theory [9], probability theory, rough set theory [10][11][12][13][14], and interval mathematics [15].Moreover, the theory of SS was developed by Molodtsove [16] in 1999, as another mathematical instrument to cope with the vagueness which is liberated from the hurdles influencing existing methods.A wide scope of applications of SSs has been established in various fields, such as game theory, probability theory, smoothness off functions, Riemann integration, operations research, and perron integration [16,17].SS and its applications got a lot of attention from researchers in recent years.Ali et al. [18] defined some new operations such as union, intersection for SS.In SS theory the application in DM problem was discussed Maji et al. [19].Maji et a. [20] extended the SS, called Fuzzy SS (FSS).Roy and Maji [21] discussed a FSS theoretic approach to DM problems.Zhang and Zhan [22] described fuzzy soft β-covering based fuzzy rough sets and corresponding DM applications.The certain types of soft coverings based rough sets with applications were established by Zhan and Wang [23].Jiang et al. [24] defined the MADM approach to covering based variable precision fuzzy rough set.. Yang et al. [25] combined an interval-valued fuzzy set with a SS model to develop a new concept of interval-valued fuzzy soft sets (IvFSSs).Soft set relations and functions were developed by Babitha and Sunil [26].Feng et al. [27] combined SSs with FSs and roughs sets.Cagman and Enginoglu [28] interpreted a DM method namely, the uni-int DM method, and redefined SS operations to utilize effectively in DM problems.Zhan and Alcantud [29] presented a novel type of soft rough covering and tis application to multicriteria group decision-making.A survey of parameter reduction of SSs and corresponding algorithms were given by Zhan and Alcantud [30].Zhan et al. [31] presented covering based variable precision fuzzy rough sets with PROMETHEE-EDAS method.An application to rating problem by using TOPSIS-WAA method based on a covering-based fuzzy rough set is introduced in [32].Zhan and Xu [33] introduced a novel approach based on threeway decisions in the fuzzy information system.The notion of complex FSS was interpreted by Thirunavukarasu et al. [34].
From the most recent studies of SS, one can see that most of the researchers in SS theory worked on a binary evaluation {0, 1} or closed interval [0, 1] [35,36].However, in real-world problems, we mostly find information with a non-binary yet discrete structure.For instance, in social judgment frameworks, Alcantud and Laruelle [37] determined the ternary voting framework.Non-binary evaluations are likewise expected in ranking and rating positions.Inspired by these concerns, Fatimah et al. [38] proposed an extended SS model, namely N-SS and they described the significance of the ordered grades in real-world problems.Motivated by N-SS set al [39] interpreted the notion of fuzzy N-SS (FN-SS) which is the combination of FS theory with N-SS.They consider the fuzzy nature of parameterization of the universe.
Many of the features are jointly desirable but happen in separate formal models of knowledge.To overcome this drawback, in this article we combine complex fuzzy set theory with N-soft sets to introduce a new hybrid model called complex fuzzy N-soft sets.This model takes in the uncertainties concerning two aspects of data: what specific grades are given to objects when parameterizations attributes are graded, which can be assigned as a partial degree of membership.The proposed model provides complete information about the occurrence of ratings and uncertainty under periodic function.It is also useful to get optimistic and pessimistic responses by decision-makers.Therefore, for the purpose of the modulization of decision-making problems it provides more flexibility when hesitation and complexity in the parameterizations are involved.We make the model fully applicable by developing decision-making algorithms that appeal to methodologies that have been validated in related frameworks.
In CFN-SS hypothesis, membership degree is unpredictable esteemed and are addressed in polar directions.The abundancy term comparing the membership degree gives the degree of belonging of an item in a CFN-SS and the stage term related with membership degree gives the extra data, for the most part related to periodicity.Since, in the current FN-SSs hypothesis, it is noticed that there is just a single boundary to address the data which brings about data misfortune in certain examples.Nonetheless, in everyday life, we go over complex characteristic wonders where we need to add the second measurement to the statement of membership grade.By presenting this subsequent measurement, the total data can be projected in one set, and henceforth loss of data can be stayed away from.For example, assume a specific organization chooses to set up biometric-based participation gadgets (BBPGs) in the entirety of its workplaces spread everywhere in the country.For this, the organization counsels a specialist who gives the data with respect to the two-measurements specifically, models of BBPGs and their comparing creation dates of BBPGs.The errand of the organization is to choose the most ideal model of BBPGs with its creation date all the while.It is clearly seen that such sort of issues can't be displayed precisely by thinking about both the measurements at the same time utilizing the conventional FN-SS hypotheses.Along these lines, for such sorts of issue, there is a need to improve the current hypotheses and thus a CFN-SS climate gives us a proficient method to deal with the two-venture judgment situations in which an abundancy term might be utilized to give an organization's choice with respect to demonstrate of BBPGs and the stage terms might be utilized to address organization's choice as to creation date of BBPGs in the choice making measure.Likewise, some different sorts of models under CFN-SSs incorporate a lot of informational indexes that are created from clinical exploration, just as government data sets for biometric and facial acknowledgment, sound, and pictures, and so on Hereafter, a CFN-SS is a more summed up expansion of the current hypotheses, for example, FSs, FN-SSs, CFSs.Obviously, the upside of the CFN-SSs is that it can contain considerably more information to communicate the data.Hence, maintaining the benefits of this set and tak-

Preliminaries
In this section, we review some basic notions of FSs, CFSs, SSs, N -SSs, and fuzzy N -SSs (FN-SSs).The fundamental properties of the existing methods are also discussed in detail.Throughout this manuscript, the symbols M ∅ and Q are denoted the finite set and set of parameters, respectively.
Definition 1 [1] A fuzzy set (FS) is designated and defined by: where μ : M → [0, 1] with a condition 0 ≤ μ(x) ≤ 1. Throughout, this manuscript the fuzzy numbers (FNs) is demonstrated by A (x, μ A (x)).When you are taking a single element x from a universal set M and assign it to a function the resultant values are restarted to unit interval is called the fuzzy number and the collection of all fuzzy numbers under the middle brackets is called fuzzy sets.Let A (x, μ A (x)) and B (x, μ B (x)) are two fuzzy numbers (FNs).Then Definition 2 [7] A complex fuzzy set (CFS) is designated and defined by: where μ μ e i2π ω μ with a condition 0 ≤ μ , ω μ ≤ 1. Throughout, this manuscript the complex fuzzy numbers (CFNs) is demonstrated by A ⎞ ⎠ are two complex fuzzy numbers (CFNs).Then. .
Definition 3 [16] A pair (F, D) represents a SS over M if F : Next, we discussed the modified operations of SS, which was developed by Ali et al. [18].The existing operations called restricted and extended unions, intersections are discussed below.
Definition 4 [18] For any two SSs (F 1 , D 1 ) and (F 2 , D 2 ) with D 1 ∩ D 2 ∅, then the restricted union and intersection are denoted and defined by: where H (d) Definition 5 [18] For any two SSs (F 1 , D 1 ) and (F 1 , D 2 ), then the extended union and intersection are designated and defined by: is called d -approximation elements of (F, D, N ).Next, the existing operations, called restricted and extended unions, intersections are discussed below.The existing notion and their operational laws are initiated by Fatimah et al. [38].
Definition 7 [38] For any two N-SSs (F, D 1 , N 1 ) and (F 2 , D 2 , N 2 ) with D 1 ∩ D 2 ∅, then the restricted union and intersection are denoted and defined by: where [38] For any two N-SSs (F 1 , D 1 , N 1 ) and (F 2 , D 2 , N 2 ), then the extended union and intersection are designated and defined by: where L(d) where J(d) .
Definition 9 [20] A pair (F , D) represents a fuzzy soft set (FSS) over M if F : D → P (M), D ⊆ Q, where P (M) contains the set of all fuzzy sets of M. if d ∈ D, then F (d) ⊆ P (M) is called d− approximation elements of (F , D). Next, the existing operations, called union and intersection, are discussed below.The existing notion and their operational laws are initiated by Maji et al. [20].
Definition 10 [20] For any two FSSs (F 1 , D 1 ) and (F 2 , D 2 ), then the union and intersection are designated and defined by: Definition 11 [34] A pair F , D represents a complex fuzzy soft set (CFSS) over M if F : D → P (U ), D ⊆ Q, where P (M) contains the set of all complex fuzzy sets of M.
Definition 12 [39] A pair μ , (F, A, N ) represents a fuzzy Next, the existing operations, called restricted and extended unions, intersections are discussed below.The existing notion and their operational laws are initiated by Akram et al. [39].
Definition 14 [39] For any two FN-SSs and μ 2 , (F 2 , D 2 , N 2 ) , then the extended union and intersection are denoted and defined by:

Complex fuzzy N-soft sets
In this section, we propose the notion of complex fuzzy Nsoft sets (CFN-SSs) and their functional representation.The fundamental properties of the proposed method are also discussed in detail.

The notion of complex fuzzy N-soft sets
Definition 15 Let M be a universe of objects, Q be the set of attributes, D ⊆ Q, andZ {0, 1, 2, . . ., N − 1} with N ∈ {2, 3, . ..}.A pair (μ, K) is said to be CFN-SS when As stated by Definition (15), with every attribute the mapping μ allocates a complex fuzzy set on the image of that attributes by the mappingF.Consequently, for every d ∈ D and m ∈ M there exists a unique (m, The accompanying example explains this formal definition.Furthermore, It presents a helpful tabular portrayal for CFN-SSs.
Example 1 a university wants to appoint a faculty member for the department of mathematics based on star ranking and ratings awarded by the selection board of the university including the president, vice president, director academic, and dean of the faculty.Let M {m 1 , m 2 , m 3 , m 4 , m 5 } be the universe set of applicants appearing in an interview and Q be set of attributes "evaluation of applicants by selection board".The subset We can get 5-SSs from Table 1, where.This graded evaluation by stars can undoubtedly be related to numbers as Z {0, 1, 2, 3, 4}, where.0 serves as "o", 1 serves as " * ", 2 serves as " * * ", 3 serves as " * * * ", 4 serves as " * * * * ", Table 1 presented the information obtained from associated data, and also presented the tabular representation of its related 5-soft set.
The above example, motivated by the interpretations.We guess that the information is given in Table 2, are capture in any CFN-SSs over a finite universal set of alternatives, the general form of the attributes denotes in Table 3.

Remark 1
We have the following observations.

It isn't compulsory to set a scale for membership values
to pick a grade as we picked in Example (1), one can also take arbitrary membership values to select a grade.2. Grade 0 ∈ D in Definition ( 15), speaks about the lowest score.It doesn't imply that there is incomplete information or an absence of assessment.3. Any CFN-SS can be considered as complex fuzzy N + 1 SS.Generally, it tends to be considered as an N -SS with N > N .
For example, the CF5-SS in Example (1) can be considered as CF6-SS over the same fuzzy parameterizations and parameters.Motivated by Point 3 in Remark (1) we interpret the Definition ( 16) Example 3 Consider (μ, K) be the CF6-SS is given in the tabular form in Table 4.
The minimized CFV-SS of CF6-SS is CF5-SS with V 5, given in Table 5.
Remark 2 Any efficient CFN-SS coincides with its minimized CFN-SS.

Functional representation of CFN-SSs
The tabular form of CFN-SSs provides that we can use a function to express a more suitable representation of this concept.

Definition 18
The functional representation of a CFN-SS is a mapping : and C is the family of all complex fuzzy numbers.
Table 3 shows this general representation and relates it with the tabular representation described in portion 2.1.We can retrieve the original formulation of the CFN-SS presented in Definition (15) from functional representation.

Fundamental operations for CF-NSSs
Example 4 Consider CF5-SS of Example (1) then the week complement of CF5-SS is given in Table 6.
Ramot et al. [7] defined the complex fuzzy complement of the complex fuzzy set as μ   We now interpret the complex fuzzy weak complement.In a fuzzy complement, the complex fuzzy grades are complementary and the parameterizations of the universe remain the same.
Example 5 Consider CF5-SS of Example (1) then the complex fuzzy week complement of CF5-SS are given in Table 7. (μ, (F, D, N )) be a CFN-SS on M. The top complex fuzzy weak complement of (μ, (F, D, N )) is μ c , F T , D, N , where.

Definition 23 Let
The top complex fuzzy weak complement of the CF5-SS in Example (1) is provided in Table 8.

Example 7
The bottom complex fuzzy weak complement of the CF5-SS in Example (1) is provided in Table 9.
Example 9 Consider CF5-SS and CF6-SS given in Tables 10 and 11 of Example (8) then in this case the extended intersection (φ, K 1 ∩ E K 2 ) is presented in Table 13).
Definition 28 Let i is the functional representation of (μ i , K i ), i 1, 2. Then the functional representation of their extended intersection is : Example 10 Consider CF5-SS and CF6-SS given in Tables 10 and 11 of Example (8) then in this case the restricted union (δ, K 1 ∪ R K 2 ) is presented in Table 14).
Definition 30 Let i is the functional representation of (μ i , K i ), i 1, 2. Then the functional representation of their restricted intersection is : Then extended union of (μ 1 , K 1 ) and (μ 2 , K 2 ) is designated and defined as and m j ∈ M, and φ d k is presented by.
Definition 32 Let i is the functional representation of (μ i , K i ), i 1, 2. Then the functional representation of their extended intersection is :

Relationships
The notion of CFN-SS can be identified with both N-SSs and complex fuzzy SSs (CFSSs).In this portion, we are going to clarify these relationships.For that reason, let us fix the accompanying setting: M means the set of objects and (μ, K) is a CFN-SS, where K (F, D, N ) is an N-SS.

Applications
This segment clarifies the decision-making (DM) process that works on models that we have interpreted in previous segments.Consequently, we characterize particular algorithms for problems that are described by CFN-SSs.To demonstrate their significance also, achievability we apply them to real circumstances that are completely developed.We interpreted the following three algorithms of CFN-SSs for DM.

Selection of faculty member in a university
In an educational field, the selection of faculty members is the way toward persons in the right the right for a job is a for educational institutes.choices can only be made when is matching.Selecting best cant as a faculty member will provide quality education in the best way.Taking Example (1) in which a university wants to appoint a faculty member for the department of mathematics based on star ranking and ratings awarded by the selection board of to the applicants, represented the tabular form Table 2.

Choice value (CV) of CF5-SSs
We can calculate the CV of CF5-SS of the applicant's selection as From CVs given in Table 20) we can see that the applicant m 3 has the highest grade given by the selection board of the university.So the university will appoint the applicant m 3 as a faculty member in the department of mathematics.We also note the ranking as

I-CV of CF5-SSs
It is additionally practical to estimate the objects in M from the information in (μ, K), where K (F, D, N ) is N -SS and a threshold I , by standard applications of CVs to the CFSS μ I , D .For this reason, we speak to it Q I j the CV at choice j of CFSS μ R , D and call Q I j the I -CV of (μ, K), where K (F, D, N ) is an N -SS at choice j. we get Table 21.
In Table 21, we supposed I 3 for DM so we get the 3-CV of CF5-SS.It can be observed from Table 21) that the applicant m 1 has the highest grade so m 1 is selected as a faculty member.

Comparison table of CF5-SSs
Definition 36 [40] it is a square table in which the number of rows and columns are the same, and both are tag by the name of objects of the universe such as m 1 , m 2 , m 3 , . . ., m p and e jk be the entries, where e jk the number of parameters for which the value of m j exceeds or equal to the value of m k . 2 are presented in tabular form in Table 22.

Membership grades of Table
The comparison table of membership grades is presented in Table 23.
The membership grades of each applicant are derived by subtracting the column sum from the row sum of Table 24.
From Table 24 we observe that the highest score is 8 which obtained by m 3 and m 1 but m 3 has the highest grade.So applicant m 3 is selected as a faculty member of the department of mathematics.

Selection of a player in the national German football team
The German football association [or Deutscher Fußball-Bound (DFB)] is a sport's national governing body.26, 000 football clubs in Germany are associated with DFB.These clubs have more than 170, 000 teams with over 2 million players.The selection committee of DFB does a selection of the players for the national German football team from these clubs.It is not an easy job to select a skillful and talented player for the national team from these clubs.

CV of CF4-SSs
The calculated CV of CF4-SS of the player's selection is presented in Table 27.

I-CV of CF4-SSs
The calculated 3-CV of CF4-SS of the player's selection is presented in Table 28.
Here we let I 3 as threshold and Table 28 gives the 3-CV of CF4-SS.It can be observed from Table 28 that the player 4 has the highest grade so the selection committee selected 4 for the national football team.

Comparison table of CF4-SSs
Membership grades of Table 26 are presented in tabular form in Table 29.
The comparison table of membership grades is presented in Table 30.
The membership grades of each football player are derived by subtracting the column sum from the row sum of Table 31.
From Table 31 we observe that the highest score is 2 which is obtained by two player 4 and 1 but 4 has the highest grade.So player 4 is selected for the national German football team.

Evaluation criteria of cleaner production for gold mines
In this section, we applied the proposed ranking approach to evaluation criteria system of cleaner production including five criteria is utilized according to the particular Characteristics of gold mines.For selecting the suitable cleaner production, we consider the group of decision-makers to perform the evaluation, between the five alternatives i.e. 1 , 2 , 3 and 4 .According to four criteria, the decisionmaker describes the cleaner production, which is follow as: This graded evaluation by stars can undoubtedly related to numbers as Z {0, 1, 2, 3, 4, 5}, where 0 serves as "o", 1 serves as " * ", 2 serves as " * * ", 3 serves as " * * * ", 4 serves as " * * * * ", 5 serves as " * * * * * ", Table 32 presented the information obtained from related data, and also presented the tabular representation of its related 6-SS.
The geometrical expressions of the gold mines are discussed in the form of Fig. 1.

Procedure of CF6-SSs
The calculated the best way for finding the gold mines of CF6-SS for the best procedure to find out the gold mines is presented in Table 34.
From Table 34 we can note that the cleaner production 1 has the highest grade to find the best way for examining the gold mines.So the cleaner production 1 will be the best procedure for finding the gold mines.We also note the ranking as 1 > 3 > 4 > 5 > 2 .

I-CV of CF6-SSs
The calculated 4-CV of CF4-SS of the player's selection is presented in Table 35.
Here we let I 4 as threshold and Table 35 gives the 4-CV of CF6-SS.It can be observed from Table 35 that the cleaner production 1 has the highest grade so the Group of decision-makers will select 1 as the best cleaner production of the year.

Comparison table of CF6-SSs
Membership grades of Table 33 are presented in tabular form in Table 36.
The comparison table of membership grades is presented in Table 37.
From Table 38 we observe that the highest score is 15 which obtained the best way to find the gold mines 1 with the highest grade.So the best procedure is 1 .

Comparison
In this section, we do a comparison of our proposed work with the existing work done by Akram et al. [39].39, where.
Four stars appear for 'Excellent', Three stars appear for 'Very Good', Two stars appear for 'Good',

CV of F6-SSs
We can calculate the CV of F6-SS by both algorithms defined by Akram et al. [39], and the proposed algorithm defined in this article, which is presented in Tables 41 and 42, respectively.
From CVs given in Tables 41 and 42 we can see that the car c 2 has the highest grade and highest membership grade given by the experts of the cars.So the family will purchase the car c 2 .We also note the ranking as c 2 > c 3 > c 4 > c 5 > c 1 .

I-CV of F5-SSs
We can calculate the 3-CV of F6-SS by both algorithms defined by Akram et al. [39], and the proposed algorithm defined in this article, which is presented in Tables 43 and  44, respectively.
In Tables 43 and 44 we supposed I 3 for DM so we get the 3-CV of F5-SS.It can be observed from Tables 43 and  44 that the car c 3 has the highest membership grade so the family will purchase a car c 3 .The membership grades of each car are derived by subtracting the column sum from the row sum of Table 47.From Table 47 we observe that the highest score is 8 which obtained by c 2 and c 2 has also highest grade So the family will get the car c 2 .

CV of CF6-SSs
We can calculate the CV of CF6-SS by the proposed algorithm defined in this article, which is presented in Table 49 and Akram et al. [39] cannot calculate CF6-SS.
From CVs given in Table 49 we can see that the car c 3 has the highest membership grade given by the experts of the cars.So the family will purchase the car c 3 .We also note the ranking as c 3 > c 5 > c 2 > c 4 > c 1 .

I-CV of CF5-SSs
We can calculate 3-CV of CF6-SS by the proposed algorithm defined in this article, which is presented in Table 50 and Akram et al. [39] cannot calculate 3-CV of CF6-SS.
From Table 53) we observe that the highest score is 8 which obtained by c 2 and c 2 has also highest grade So the family will get the car c 2 .

Conclusion
In this manuscript, we interpreted a new enlarged and appropriate version of SSs, called CFN-SS, which is the combination of CFSs with N-SSs to handle the complicated data in DM.Further, we interpreted some handy algebraic properties of CFN-SSs and establish their basic operations in this manuscript.Our novel approach along with its properties is explained with the help of examples.Additionally, we described the relationship of our novel approach with some existing methods such as complex fuzzy soft sets (CFSSs), FSSs, and SSs.Moreover, three DM procedures have been interpreted and clarified with the help of examples.Finally, we compared our model CFN-SS with the existing model FN-SS.Our novel approach spread out new directions for research.Within this novel concept, it is workable to study similarity indices, entropy, and the correlation between CFN-SSs as a mechanism to help in DM processes.
In the future, we aim to discuss N-soft sets to generalize the notions of complex dual hesitant fuzzy sets [41], complex
{ 1 , 2 , 3 , 4 } be the universe of players and D {d 1 , d 2 , d 3 , d 4 } be set of attributes.A selection committee allocates grades or ratings to the players depend on the per-20 The tabular form of the CV of CF5-SSs (μ, (F, D, 5))

Table 1
Information obtained from the associated data and tabular form of the related 5-SS

Table 2
The tabular form of the CF5-SS in Example 1

Table 3
The tabular form of the general CFN-SS

Table 4
The

Table 7
The complex fuzzy weak complement of CF5-SS

Table 8
The top complex fuzzy weak complement of the CF5-SS

Table 24
Membership score table.

Table 30
Comparison table of membership grades.
Information related to alternatives for cleaner assessment production in gold mines.

Table 31
Membership score table.

Table 36
The tabular form of membership grades

Table 37
The comparison table of membership grades.

Table 43
[39]tabular form of 3-CV of F5-SS was calculated by using the algorithm defined by Akram et al.[39]

Table 45 .
Table 40 are presented in tabular form in The comparison table of membership grades is presented in

Table 50
The tabular form of 3-CV of CF5-SS

Comparison table of CF5-SSs Membership
grades of Table 48 are presented in tabular form in Table 51.The comparison table of membership grades is presented in Table 52.