The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems

: In this paper, different measures of interval-valued pentagonal fuzzy numbers (IVPFN) associated with assorted membership functions (MF) were explored, considering significant exposure of multifarious interval-valued fuzzy numbers in neoteric studies.Also, the idea of MF is generalized somewhat to nonlinear membership functions for viewing the symmetries and asymmetries of the pentagonal fuzzy structures. Accordingly,the construction of level sets, for each case of linear and nonlinear MF was also carried out. Besides, defuzzification was undertaken using three methods and a ranking method, which were also the main features of this framework.The developed intellects were implemented in a game problem by taking the parameters as PFNs, ultimately resulting in a new direction for modeling real world problems and to comprehend the uncertainty of the parameters more precisely in the evaluation process.


Uncertainty and Uncertainty Measure
Uncertainty theory plays important role in modeling for engineering and science problems.Recently, huge developments have taken place in this area.Analysis based on uncertainty theory is regularly required in the following fields, although it is not limited to these areas; mathematical modeling, physics, chemistry, economics, artificial intelligence, legal fact-finding,medical science, business administration, psychology, decision sciences, etc. Various concepts have been formulated to measure uncertainty.Fuzzy logic involvesthe constructionof an approximate acumen mechanism to tackle the uncertainty related with human behavior.Many activities and approaches have been designed to counteract or curtail the uncertainty that is generatedby decision making, however, there are few comprehensive theories available in the literature.Researchers are currently striving to construct a common approach.
There are some important differences between different types of imprecise or uncertain parameters.
If we consider an Interval number, the following observations can be made: (i) The fact belongs to some certain interval (ii) The concept of the membership function is missing.
If we consider a Fuzzy number [1][2][3], the following observation can be made: (i) The conception of belongingness of the component has to be considered (ii) The concept of membership function is derived.
If we consider an Intuitionistic fuzzy number [4], the following observations can be made: (i) The conception of belongingness, or non-belongingness of the component has to be considered (ii) The concepts of the membership, or non-membership function are derived If we take a Neutrosophic fuzzy number [5], the following observations can be made: (i) The conception of truthfulness, falsity and indeterminacy of the component have to be considered (ii) The concept of membership function for truthfulness, falsity and indeterminacy are derived We can easily see the idea by using a Figure 1that shows how different type of PFN can be formed as below:

Fuzzy Sets and Number
Zadeh [1]invented a new idea, which is known as fuzzy sets theory (FST).The basic theory of uncertainty has been used with immense success in different fields.Chang and Zadeh [2] discoveredthe main idea of fuzzy sets and numbers.Mathematicians have further studied the different result on that theory [3,6] and the impressive and considerabledevelopment of ideas and different application of FST have resulted in the topic gaining a great deal of attention.ends and the concept of generalized fuzzy numbers is involved?Obviously, the formations are very different.Also, the results are different from previous cases.In this paper, we try to take all the possibilities for the formation ofPFNs into account.For details we have to see Table 2.  [25] Reverse order linear Arithmetic operation Defining reverse order fuzzy number Ponnivalavan and Pathinathan [26] Linear membership, non-membership function

Arithmetic operation
Find score and accuracy function Annie Christi and Kasthuri [27] Linear membership, non-membership function

Transportation problem
Mondal and Mandal [28] Linear and non-linear Define symmetric and asymmetric PFN Fuzzy equation

Verbal Phrasesin Uncertainty Theory
An important consideration is how can we relate the concept of uncertainty theory to practical examples and what is the verbal phrase for a particular type of uncertainty.
Example 1.Suppose some mountaineers want to calculate the approximate height of a mountain range.They have different points of view after looking the mountain from different angles and in different situations so they use different kinds of uncertain parameters.The parameters might be anything from an interval number, a triangular fuzzy number, a trapezoidal fuzzy number.etc. Example 2.Suppose some traffic sergeants wants to compute the traffic intensity in a congested crossing in a certain time domain.They have different points of view of the congestion in their mind so they use different kinds of uncertain parameters.As before, we can take any one of the uncertain parameters.
The verbal phrase for different types of uncertain numbers for the above problems are shown in Table 3. Ranking a fuzzy number involves measuring up to two fuzzy numbers, and defuzzification is a technique whereby the fuzzy number is renewed to an approximated crisp number.Just as the decision maker takes two concepts that are the same, similarly, for this problem we have to convert the fuzzy number to a corresponding crisp number and compare the number on the basis of crisp values.

Motivation
Fuzzy sets theory plays several significant roles in the theory of uncertainty for modeling.An important issue is that if anybody wants to take a PFN, then what should its pictorial representations (uncertainty quantification area) look like?How should we define the membership functions?From this viewpoint, we formulated different types of PFN that may be a good choice for a decision maker in a practical scenario.

Novelties
There are several published works where pentagonal fuzzy sets have been formulated and decision makers have applied these to various fields.However, there are still many important scopes to be worked on for PFN.A summary of the work we have done on PFN is as follows: (i) The development of different types of interval-valued PFN, i.e., symmetric linear PFN, asymmetric linear PFN, symmetric nonlinear PFN, and asymmetric nonlinear PFN were defined.(ii) The representation of the said PFNs in parametric form was defined.
(iii) The ranking and defuzzification of PFN were done.(iv) The number was applied to a fuzzy game theory problem.
We can easily see by the following Figure 2 for the difference and relation between defuzzification and ranking

Structure of the Paper
The article is structured as follows as shown in the Figure 3:  where 0 <  ≤  ≤ 1 and  <  <  <  <  The upper and lower membership function of IVFN is defined by

PFN and Its Different Representations
In this section we extend special types of PFNs in different viewpoints.

Linear PFN with Symmetry
Definition 5.Linear PFN with symmetry: [28] A linear PFN is written as  = ( ,  ,  ,  ,  ; )whose corresponding membership function is written as 0 for  >  Definition 6. -cut or the parametric form of linear PFN with symmetry: [28] -cut or parametric form of LPFNS is written by the formulae where  (),  () is the nondecreasing function with respect to  and  (),  () is the decreasing function with respect to  Remark 1.The main idea of the symmetric PFN is that the left picked point is equal to the right picked point (see in Figure 4 the same picked value is k).

Non-Linear Pentagonal Fuzzy Number with Symmetry
Definition 7.Non-Linear PFN with symmetry: A linear PFN is written as  = ( ,  ,  ,  ,  ; ) ( , ; , ) where the membership function is written as

Verbal Phrase for the PFN and Interval-Valued PFN
In the case of real-life problems such as the salary of laborers in a factory, everything depends on lots of parameters.For example, there are some laborers whose salaries are very low, some are moderately low, and some have low and moderate salaries while others have high and very high salaries owing to years of experience and their designations.To identify such distinct cases, we need the pentagonal fuzzy number in different forms.Thus, we use verbal phrases to identify its nature.

VL-Very Low, L-Low, M-Medium, H-High, M-Medium, VH-Very High
In this particular game problem, we use different kind of strategies for player A and B. Thus, the members of the payoff matrix are of the pentagonal fuzzy type, and we also use corresponding verbal phrases for different kinds of members.For the details about the problem see Tables 4, 5.In the case of interval-valued problems, there is a finite range for the interval whose income is low, medium or high.It is specified as very low, low, low mean, large mean, high, very high, etc.Within this finite range the membership function actually varies.Generally, it is denoted by

The Defuzzification Method
The defuzzification process is very important for a problem for two important reasons: (i) Those who are not familiar with the fuzzy concept can relate to the result or solution.
(ii) The crispified value of the fuzzy solutions is identified.

Defuzzification of Non-Linear Symmetric PFN Based on Centroid Method
We proposed a method to compute the defuzzification of non-linear symmetric PFN  where, Hence,

Defuzzification of Linear Symmetric PFN Based on Removalof Area Method for Linear Pentagonal Fuzzy Number
We consider different types of areas of the corresponding linear PFN as shown below.For k=1,   , 0 =

Comparison ofthe Above Three Defuzzification Methods
We already found the analytical result for the defuzzification value for the pentagonal fuzzy number.Then, we compared the two methods numerically as follows in the Table 6: Remark 5.In the numerical study above, we show that calculating the defuzzification value by the centroid method and mean of alpha-cut method gives almost the same result.
6. Ranking for the Pentagonal Fuzzy Number

Basic Concept of Ranking Fuzzy Numbers
The concept of ranking fuzzy numbers is very important for decision making problems.Researchers have presented many different reasons for finding the ranking of fuzzy numbers .In this section we find a new concept for finding the ranking of PFNs.

Ranking of Pentagonal Fuzzy Numbers
We proposed a new way to find the ranking of PFNs.In Figure 21the pentagon is divided into three triangles and one rectangle.We can find the ranking by using the centroid formula of triangles and rectangles. represents the centroid of the corresponding triangle whose vertices are ( , 0), ( , 0), ( , ).

Numerical Computation
to convert the pentagonal fuzzy number into a crisp number, then we have the modified pay-off matrix asTable 9:   Now, the convex combination of A3,A4 (we take the average), that is, 2,4 is dominated by A2.Hence, we haveTable 13, Hence, the optimal solution to the main problem is for A(0,0, , ) for B(0,0, , ) and the corresponding game value is .
Example 2.Let us consider another fuzzy game pay-off matrix where two different players are A and B for different pentagonal fuzzy numbers as a pay-off matrix memberTable 14.Hence, the optimal solution to the main problem is for A(0,0, , ) for B(0,0, , ) and the corresponding game value is .
Remark 7.In the above problem we take different three sets of the pay-off matrix with symmetric PFN and after utilizing the concept of defuzzification we solve the game problem.We can see that corresponding optimal strategies are changed, and also the game values are changed.Thus, we can conclude that for different types of pentagonal fuzzy numbers, there will be different kinds of optimal solutions with different types of game value result.

Conclusions and Scope of Future Research
In this paper, we extended the characteristics of pentagonal fuzzy numbers to interval-valued fuzzy numbers and pentagonal fuzzy number.Bearing in mind the symmetric and asymmetric structures of a pentagonal shape, the membership functions were comprehensively investigated for both cases.The linear and non-linear for symmetric and asymmetric membership functions along with three cuts were also added to the discussion.The defuzzification of the pentagonal fuzzy number was carried out by three methods, namely:the centroid method, mean of Alpha (α)-cut method, and the removal of area method.The ranking between two IVFNswas found by the centroid formula method.Efficaciously, the produced solutions exhibited IVPFN structures that contain all the possible outcomes of the governing model.Thus, the following conclusion were made:

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The expansion of the interval-valued pentagonal fuzzy numbers and functions, adds a new tool for modeling different aspects of science, engineering and other environmental studies.

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The classifications of linear and nonlinear membership functions provided an appropriate strategy for decision makers.

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Detailed illustrations of membership functions,  -cuts and ranking, and defuzzification provide all the required information in one platform to model any real-world problem.
In our forthcoming work, we will apply these definitions to other types of fuzzy numbers such as hexagonal fuzzy numbers, and diamond shape fuzzy numbers and use them to model different dynamics of applied sciences, such as multi-criterion decision making, optimization, networking problem, etc.

Figure 3 .
Figure 3. Structure of the paper.

Table 3 .
Verbal phrase of different uncertain parameters.

Table 4 .
Verbal Phrase for Pentagonal Fuzzy Number.
Working Rule to Find the Ranking of a Pentagonal Fuzzy Number () is a set of PFNs defined on the set of real numbers, and the ranking of PFNs is actually a function : () →  which maps each fuzzy number into a real line.

Table 9 .
Defuzzified game pay-off matrix Now all elements of A1 are less than or equal to A3,here, A3dominates A1.So, according to the dominance properties, our table becomesTable 10:

Table 10 .
First step for solving the problem

Table 11 .
Second step for solving the problem

Table 12 .
Third step for solving the problem

Table 13 .
Fourth step for solving the problem

Table 14 .
Fuzzy game pay-off matrix table

Table 15 .
Solution of the problem

Table 16 .
Fuzzy game pay-off matrix table

Table 17 .
Solution of the problem