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

Abstract

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.

1. Introduction

1.1. 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 1 that shows how different type of PFN can be formed as below:

1.2. 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] discovered the main idea of fuzzy sets and numbers. Mathematicians have further studied the different result on that theory [3,6] and the impressive and considerable development of ideas and different application of FST have resulted in the topic gaining a great deal of attention.

1.3. Interval-Valued Fuzzy Numbers (IVFNs)

Several studies have been done on the topic of IVFNs. We reviewed some of these papers and then chose the topics we would cover. Our investigation is summarized below in

Table 1. Studies done on Interval-Valued Fuzzy numbers.

Authors Information	Membership Function’s Type	Main Contribution in Theatrical Improvement	Application Area
Guijun and Xiaoping [7]	General fuzzy cases	Define IVFN and interval distribution number	Generalized pseudo-probability metric spaces and pseudo metric spaces
Anitha and Parvathi [8]	Correlation coefficient of IVFN	Properties of IVFN	Information energy
Lin [9]	Triangular Linear	Ranking using signed distance methods	Job-shop scheduling problem
Wei and Chen [10]	Trapezoidal Linear	Found similarity measures between two IVFN	Fuzzy risk analysis
Kalaichelvi et al. [11]	Not take the membership function concept	Extended the concept of IVFN to Soft IVFN	Fuzzy matrix theory
Kumar and Singh [12]	Triangular linear	Found the signed distance	Fuzzy fault tree
Abirami and Dinagar [13]	LR type	Found distance function	Project network
Su [14]	Triangular linear	Arithmetic operation and comparison	Linear programming problem
Bhatia and Kumar [15]	Triangular linear	Ranking for solving LPP	Linear programming problem
Mondal [16]	Triangular linear and non-linear	Arithmetic operation	Fuzzy differential equation
Ebrahimnejad [17]	Generalized trapezoidal linear	Arithmetic operation on LPP	Linear programming problem
Dahooie et al. [18]	Trapezoidal and triangular linear	Defuzzification and fuzzy additive ratio assessment method	Oil and gas well drilling projects

.

1.4. Pentagonal Fuzzy Numbers

Many researchers have investigated PFN with various types of membership functions. In this part we study the available papers that are related to PFN (see below).

From our survey, we can say that membership functions are taken as linear with symmetry at both ends for the majority of cases. However, what is the scenario if the non-linear membership function or interval-valued membership functions concept is taken and there is asymmetry on both 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 of PFNs into account. For details we have to see

Table 2. Studiesdone on PFNs.

Authors Information	Membership Function’s Type	Main Contribution in Theatrical Improvement	Application Area
Panda and Pal [19]	Linear and symmetry on both end	exponent operation and arithmetic operation	Fuzzy matrix theory
Anitha and Parvathi [20]	Linear	Find expected crisp value	Inventory management
Helen and Uma [21]	Linear	Defining the parametric representation	Proofing the arithmetic operation and find the ranking
Siji and Kumari [22]	Linear membership, non-membership functions	Arithmetic operation and find the ranking	Networking problem
Raj and Karthik [23]	Linear	Arithmetic operation	Neural network problem
Dhanamand and Parimaldevi [24]	Linear	Find the ranking using circumcenter of centroids method	Multi item and multi -objective inventory management problem
Pathinathan and Ponnivalavan [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	Arithmetic operation and ranking	Transportation problem
Mondal and Mandal [28]	Linear and non-linear	Define symmetric and asymmetric PFN	Fuzzy equation

.

1.5. 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. Verbal phrase of different uncertain parameters.

Type of Uncertain Parameter	Verbal Phrase	Quantity Information
Interval Number	[Low, High]	Example 1	The height lies in the range [2000,3000] ft.
		Example 2	The car number is between the interval [1000,2000] cars
Triangular Fuzzy Number	[Low, Medium, High]	Example 1	The height lies in the fuzzy number set [2010,2050,2150] ft
		Example 2	The car number is in the fuzzy number set [1000,1050,1200] cars
Trapezoidal Fuzzy Number	[Low, Low Mean, High Mean, High]	Example 1	The height lies in the fuzzy number set [2000,2205,2705,2300] ft.
		Example 2	The car number is in the fuzzy number set [1000,1025,1075,1200] cars

.

1.6. Ranking and Defuzzification

The concept of ranking and the defuzzification methods for any fuzzy numbers are not novel among decision makers. But what is the basic concept of the ranking and the defuzzification methods and what are their inter-relations?

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.

1.7. 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.

1.8. 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.

1.9. Structure of the Paper

The article is structured as follows as shown in the Figure 3:

2. Preliminaries

Definition 1.

Fuzzy set: Let us take a set$\tilde{A}$, which is defined by$\tilde{A}=\left\{\left(x,{\mu }_{\tilde{A}}\left(x\right)\right):x\in A,{\mu }_{\tilde{A}}\left(x\right)\in [0,1]\right\}$. If in the pair$\left(x,{\mu }_{\tilde{A}}\left(x\right)\right)$, the first one,$x$belonsg to the classical set$A$and the second one${\mu }_{\tilde{A}}\left(x\right)$belongs to the interval$[0, 1]$, then set$\tilde{A}$is called a fuzzy set. Here${\mu }_{\tilde{A}}\left(x\right)$is called a Membership function.

Definition 2.

Interval-valued fuzzy set (IVFS): [29] An IVFS$\tilde{A}$on$ℛ$ is defined by

$${\tilde{A}}_n=\left[\left\{x,\left({\mu }_{{\tilde{A}}^U}\left(x\right),{\mu }_{{\tilde{A}}^L}\left(x\right)\right)\right\}:x\in ℛ\right]$$

where $x\in ℛ$ and ${\mu }_{{\tilde{A}}_n^U}\left(x\right),$ maps $ℛ$ into $[0,\lambda$], ${\mu }_{{\tilde{A}}_n^L}\left(x\right)$ maps $ℛ$ into $[0, \omega$] $\forall x\in R$, ${\mu }_{{\tilde{A}}_n^L}\left(x\right)\le {\mu }_{{\tilde{A}}_n^U}\left(x\right)$. (λ and $\omega$ are the maximum value of upper and lower membership function, respectively).

Definition 3.

Non-linear interval-valued fuzzy number (IVFN): [29] An IVFN is denoted by

$${\tilde{A}}_{nLIVFN}=[\left\{\left(a_1,b,c_1;\lambda \right),\left(a,b,c;\omega \right)\right\};n_1,n_2,n_3,n_4]$$

where$0<\omega \le \lambda \le 1$and$a_1<a<b<c<c_1$.

The upper and lower membership function of IVFN is defined by

$${\mu }_{{\tilde{A}}^U}\left(x\right)=\{\begin{array}{c}\begin{array}{c}\lambda {\left(\frac{x-a_1}{b-a_1}\right)}^{n_1},a_1\le x\le b\\ \lambda ,x=b\end{array}\\ \lambda {\left(\frac{c_1-x}{c_1-b}\right)}^{n_2},b\le x\le c_1\\ 0,\mathrm{otherwise}\end{array}$$

and

$${\mu }_{{\tilde{A}}^L}\left(x\right)=\{\begin{array}{c}\begin{array}{c}\omega {\left(\frac{x-a}{b-a}\right)}^{n_3},a\le x\le b\\ \omega ,x=b\end{array}\\ \omega {\left(\frac{c-x}{c-b}\right)}^{n_4},b\le x\le c\\ 0,\mathrm{otherwise}\end{array}$$

3. PFN and Its Different Representations

In this section we extend special types of PFNs in different viewpoints.

Definition 4.

Pentagonal fuzzy number (PFN): [28] A PFN$\tilde{A}=\left(a_1,a_2,a_3,a_4,a_5\right)$ should satisfy the following condition:

(1)${\mu }_{\tilde{A}}\left(x\right)$ is a continuous function in the interval [0,1]

(2)${\mu }_{\tilde{A}}\left(x\right)$is strictly non-decreasing continuous function on the intervals$[a_1,a_2]$and$[a_2,a_3]$

(3)${\mu }_{\tilde{A}}\left(x\right)$is strictly non-increasing continuous function on the intervals$[a_3,a_4]$and$[a_4,a_5]$

3.1. Linear PFN with Symmetry

Definition 5.

Linear PFN with symmetry: [28] A linear PFN is written as${\tilde{A}}_{LS}=\left(a_1,a_2,a_3,a_4,a_5;k\right)$ whose corresponding membership function is written as

$${\mu }_{{\tilde{A}}_{LS}}\left(x\right)=\{\begin{array}{c}\begin{array}{c}k\frac{x-a_1}{a_2-a_1}\mathrm{for}{a}_1\le x\le a_2\\ 1-\left(1-k\right)\frac{x-a_2}{a_3-a_2}\mathrm{for} a_2\le x\le a_3\end{array}\\ \begin{array}{c}1\mathrm{for} x=a_3\\ 1-\left(1-k\right)\frac{a_4-x}{a_4-a_3}\mathrm{for}{a}_3\le x\le a_4\end{array}\\ \begin{array}{c}k\frac{a_5-x}{a_5-a_4}\mathrm{for}{a}_4\le x\le a_5\\ 0\mathrm{for} x>a_5\end{array}\end{array}$$

Definition 6.

$\alpha$-cut or the parametric form of linear PFN with symmetry: [28] $\alpha$-cut or parametric form of LPFNS is written by the formulae

$$A_{\alpha }=\left\{x\in X|{\mu }_{{\tilde{A}}_{LS}}\left(x\right)\ge \alpha \right\}$$

$$=\{\begin{array}{c}{A}_{1L}\left(\alpha \right)=a_1+\frac{\alpha }{k}\left(a_2-a_1\right)\mathrm{for}\alpha \in [0,k]\\ \begin{array}{c}{A}_{2L}\left(\alpha \right)=a_2+\frac{1-\alpha }{1-k}\left(a_3-a_2\right)\mathrm{for} \alpha \in [k,1]\\ A_{2R}\left(\alpha \right)=a_4-\frac{1-\alpha }{1-k}\left(a_4-a_3\right)\mathrm{for} \alpha \in [k,1]\end{array}\\ A_{1R}\left(\alpha \right)=a_5-\frac{\alpha }{k}\left(a_5-a_4\right)\mathrm{for} \alpha \in [0,k]\end{array}$$

where$A_{1L}\left(\alpha \right)$, $A_{2L}\left(\alpha \right)$is the nondecreasing function with respect to$\alpha$and$A_{2R}\left(\alpha \right)$, $A_{1R}\left(\alpha \right)$is the decreasing function with respect to$\alpha$

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).

3.2. Non-Linear Pentagonal Fuzzy Number with Symmetry

Definition 7.

Non-Linear PFN with symmetry: A linear PFN is written as${\tilde{A}}_{LNS}={\left(a_1,a_2,a_3,a_4,a_5;k\right)}_{\left(n_1,n_2;m_1,m_2\right)}$ where the membership function is written as

$${\mu }_{{\tilde{A}}_{NS}}\left(x\right)=\{\begin{array}{c}k{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}\mathrm{for}{a}_1\le x\le a_2\\ 1-\left(1-k\right){\left(\frac{x-a_2}{a_3-a_2}\right)}^{n_2}\mathrm{for}{a}_2\le x\le a_3\\ 1\mathrm{for} x=a_3\\ 1-\left(1-k\right){\left(\frac{a_4-x}{a_4-a_3}\right)}^{m_1}\mathrm{for}{a}_3\le x\le a_4\\ k{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2}\mathrm{for}{a}_4\le x\le a_5\\ 0\mathrm{for} x>a_5\end{array}$$

Definition 8.

$\alpha$-cut or parametric form of non-linear PFN with symmetry:$\alpha$-cut or parametric form of LPFNS is written by the formulae

$$A_{\alpha }=\left\{x\in X|{\mu }_{{\tilde{A}}_{LS}}\left(x\right)\ge \alpha \right\}$$

$$=\{\begin{array}{c}{A}_{1L}\left(\alpha \right)=a_1+{\left(\frac{\alpha }{k}\right)}^{n_1}\left(a_2-a_1\right)\mathrm{for} \alpha \in [0,k]\\ \begin{array}{c}{A}_{2L}\left(\alpha \right)=a_2+{\left(\frac{1-\alpha }{1-k}\right)}^{n_2}\left(a_3-a_2\right)\mathrm{for} \alpha \in [k,1]\\ A_{2R}\left(\alpha \right)=a_4-{\left(\frac{1-\alpha }{1-k}\right)}^{m_1}\left(a_4-a_3\right)\mathrm{for} \alpha \in [k,1]\end{array}\\ A_{1R}\left(\alpha \right)=a_5-{\left(\frac{\alpha }{k}\right)}^{m_2}\left(a_5-a_4\right)\mathrm{for} \alpha \in [0,k]\end{array}$$

where$A_{1L}\left(\alpha \right)$, $A_{2L}\left(\alpha \right)$are increasing functions with respect to$\alpha$and$A_{2R}\left(\alpha \right)$, are decreasing functions with respect to$\alpha$.

Remark 2.

The basic idea of the mentioned number in Figure 5 is that the left picked value and right picked value are same but the boundary of the fuzzy area be supposed to not linear. The membership function can be formed as a non-linear function. So, we have to present the non-linearity on the membership function as an addition.

4. Interval-Valued PFN

4.1. Linear Interval-Valued PFN

4.1.1. Linear Pentagonal Interval-Valued Fuzzy Number with Symmetry

A linear pentagonal interval-valued fuzzy number is written as ${\tilde{A}}_{LPIS}=\left\{\left(a_1,a_2,c,a_4,a_5;k,p\right),\left(b_1,b_2,c,b_4,b_5;w,q\right)\right\}$ whose upper membership function and lower membership function are defined as follows

$${\mu }_{{\check{A}}_{LPIS}^U}\left(x\right)=\{\begin{array}{c}\begin{array}{c}p\frac{x-a_1}{a_2-a_1}\mathrm{for} a_1\le x\le a_2\\ k-\left(k-p\right)\frac{x-a_2}{c-a_2}\mathrm{for} a_2\le x\le c\end{array}\\ \begin{array}{c}k \mathrm{for} x=c\\ k-\left(k-p\right)\frac{a_4-x }{a_4-c}\mathrm{for}c\le x\le a_4\end{array}\\ \begin{array}{c}p\frac{a_5-x}{a_5-a_4}\mathrm{for}{a}_4\le x\le a_5\\ 0\mathrm{for}x>a_5\end{array}\end{array}$$

and

$${\mu }_{{\check{A}}_{LPIS}^L}\left(x\right)=\{\begin{array}{c}\begin{array}{c}q\frac{x-b_1}{b_2-b_1}\mathrm{for}{b}_1\le x\le b_2\\ w-\left(w-q\right)\frac{x-b_2}{b_3-b_2}\mathrm{for}{b}_2\le x\le c\end{array}\\ \begin{array}{c}w \mathrm{for} x=c\\ w-\left(w-q\right)\frac{b_4-x}{b_4-c}\mathrm{for}c\le x\le b_4\end{array}\\ \begin{array}{c}q\frac{b_5-x}{b_5-b_4}\mathrm{for} b_4\le x\le b_5\\ 0\mathrm{for} x>b_5\end{array}\end{array}$$

4.1.2. The $\mathsf{\alpha }$-Cut of a Linear Pentagonal Interval-Valued Fuzzy Number with Symmetry

Corresponding Alpha-cut ($\alpha -cut)$ are defined for the upper and lower membership functions as follows.

The $\alpha$-cut or parametric representation of LPIS is represented by the formulae

$$\begin{array}{ll}{A}_{LPIS\alpha }=& {{{\displaystyle \cup }}^{}}_{{\alpha }_2}[A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right)]\\ & ⊝{{{\displaystyle \cup }}^{}}_{{\alpha }_1}[A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)]\end{array}$$

where,

$$A_{1L}^U\left({\alpha }_2\right)=a_1+\frac{{\alpha }_2}{p}\left(a_2-a_1\right)\mathrm{for}{\alpha }_2\in [0,p]$$

$$A_{2L}^U\left({\alpha }_2\right)=a_2+\frac{k-{\alpha }_2}{k-p}\left(c-a_2\right)\mathrm{for}{\alpha }_2\in [p,k]$$

$$A_{1R}^U\left({\alpha }_2\right)=a_5-\frac{{\alpha }_2}{p}\left(a_5-a_4\right)\mathrm{for}{\alpha }_2\in [0,p]$$

$$A_{2R}^U\left({\alpha }_2\right)=a_4-\frac{k-{\alpha }_2}{k-p}\left(a_4-c\right)\mathrm{for}{\alpha }_2\in [p,k]$$

$$A_{1L}^L\left({\alpha }_1\right)=b_1+\frac{{\alpha }_1}{q}\left(b_2-b_1\right)\mathrm{for} {\alpha }_1\in [0,q]$$

$$A_{2L}^L\left({\alpha }_1\right)=b_2+\frac{w-{\alpha }_2}{w-q}\left(c-b_2\right)\mathrm{for}{\alpha }_1\in [q,w]$$

$$A_{1R}^L\left({\alpha }_1\right)=b_5-\frac{{\alpha }_1}{q}\left(b_5-b_4\right)\mathrm{for}{\alpha }_1\in [0,q]$$

$$A_{2R}^L\left({\alpha }_1\right)=b_4-\frac{w-{\alpha }_1}{w-q}\left(b_4-c\right)\mathrm{for}{\alpha }_1\in [q,w]$$

where $A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right)$ are increasing functions with respect to ${\alpha }_2,{\alpha }_1$, respectively, and $A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)$ are decreasing functions with respect to ${\alpha }_2,{\alpha }_1$, respectively.

Remark 3.

If in Figure 6$a_1=b_1,a_2=b_2,a_4=b_4,a_5=b_5,k=w,p=q$then, the interval-valued PFN becomes simply PFN.

4.1.3. Linear Pentagonal Interval-Valued Fuzzy Number with Asymmetry

A linear pentagonal interval-valued fuzzy number is written as ${\tilde{A}}_{LPIAS}=\left\{\left(a_1,a_2,c,a_4,a_5;k,p,r\right),\left(b_1,b_2,c,b_4,b_5;w,q,s\right)\right\}$ whose upper membership function and lower membership function are defined as follows

$${\mu }_{{\check{A}}_{LPIAS}^U}\left(x\right)=\{\begin{array}{c}\begin{array}{c}p\frac{x-a_1}{a_2-a_1}\mathrm{for} a_1\le x\le a_2\\ k-\left(k-p\right)\frac{x-a_2}{c-a_2}\mathrm{for}{a}_2\le x\le c\end{array}\\ \begin{array}{c}k \mathrm{if} x=c\\ k-\left(k-r\right)\frac{a_4-x}{a_4-c}\mathrm{for}c\le x\le a_4\end{array}\\ \begin{array}{c}r\frac{a_5-x}{a_5-a_4}\mathrm{for} a_4\le x\le a_5\\ 0\mathrm{for} x>a_5\end{array}\end{array}$$

and

$${\mu }_{{\check{A}}_{LPIAS}^L}\left(x\right)=\{\begin{array}{c}\begin{array}{c}q\frac{x-b_1}{b_2-b_1}\mathrm{for} b_1\le x\le b_2\\ w-\left(w-q\right)\frac{x-b_2}{b_3-b_2}\mathrm{for}{b}_2\le x\le c\end{array}\\ \begin{array}{c}w \mathrm{for}x=c\\ w-\left(w-s\right)\frac{b_4-x }{b_4-c}\mathrm{for}c\le x\le b_4\end{array}\\ \begin{array}{c}s\frac{b_5-x}{b_5-b_4}\mathrm{for} b_4\le x\le b_5\\ 0\mathrm{for} x>b_5\end{array}\end{array}.$$

4.1.4. The $\alpha$-Cut ofa Linear Pentagonal Interval-Valued Fuzzy Number with Asymmetry

Corresponding Alpha-cut ($\alpha -cut)$ are defined for the upper and lower membership functions as follows.

The $\alpha$-cut or parametric form of LPIS is represented by the formulae

$$\begin{array}{ll}{A}_{LPIAS\alpha }=& {{{\displaystyle \cup }}^{}}_{{\alpha }_2}[A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right)]\\ & ⊝{{{\displaystyle \cup }}^{}}_{{\alpha }_1}[A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)]\end{array}$$

where,

$$A_{1L}^U\left({\alpha }_2\right)=a_1+\frac{{\alpha }_2}{p}\left(a_2-a_1\right)\mathrm{for} {\alpha }_2\in [0,p]$$

$$A_{2L}^U\left({\alpha }_2\right)=a_2+\frac{k-{\alpha }_2}{k-p}\left(c-a_2\right)\mathrm{for}{\alpha }_2\in [p,k]$$

$$A_{1R}^U\left({\alpha }_2\right)=a_5-\frac{{\alpha }_2}{r}\left(a_5-a_4\right)\mathrm{for}{\alpha }_2\in [0,r]$$

$$A_{2R}^U\left({\alpha }_2\right)=a_4-\frac{k-{\alpha }_2}{k-r}\left(a_4-c\right)\mathrm{for}{\alpha }_2\in [r,k]$$

$$A_{1L}^L\left({\alpha }_1\right)=b_1+\frac{{\alpha }_1}{q}\left(b_2-b_1\right)\mathrm{for}{\alpha }_1\in [0,q]$$

$$A_{2L}^L\left({\alpha }_1\right)=b_2+\frac{w-{\alpha }_2}{w-q}\left(c-b_2\right)\mathrm{for} \alpha \in [q,w]$$

$$A_{1R}^L\left({\alpha }_1\right)=b_5-\frac{{\alpha }_1}{s}\left(b_5-b_4\right)\mathrm{for}{\alpha }_1\in [0,s]$$

$$A_{2R}^L\left({\alpha }_1\right)=b_4-\frac{w-{\alpha }_1}{w-s}\left(b_4-c\right)\mathrm{for}{\alpha }_1\in [s,w]$$

where $A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right)$ are increasing functions with respect to ${\alpha }_2,{\alpha }_1,$ respectively, and $A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)$ are decreasing functions with respect to ${\alpha }_2,{\alpha }_1$, respectively.

Remark 4.

If in Figure 7$p=r,q=s$ then, the above number becomes symmetric PFN.

4.2. Non-Linear Interval-Valued Fuzzy NumberInterval-Valued Linear PFN with Symmetry

4.2.1. Non-Linear Pentagonal Interval-Valued Fuzzy Number with Symmetry

A non-linear interval-valued pentagonal fuzzy number can be written as ${\tilde{A}}_{NPIFNS}=\{{\left(a_1,a_2,c,a_4,a_5;k,p\right)}_{\left(n_1,n_2;m_1,m_2\right)}$, ${\left(b_1,b_2,c,b_4,b_5;w,q\right)}_{\left(n_1,n_2;m_1,m_2\right)}\}$, whose upper and lower membership function can be written as follows

$${\mu }_{{\tilde{A}}_{NPIFNS}}^U\left(x\right)=\{\begin{array}{c}\begin{array}{c}p{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}if for a_1\le x\le a_2\\ k-\left(k-p\right){\left(\frac{x-a_2}{c-a_2}\right)}^{n_2}for a_2\le x\le c\end{array}\\ \begin{array}{c}k if x=c\\ k-\left(k-p\right){\left(\frac{a_4-x}{a_4-c}\right)}^{m_1} for c\le x\le a_4\end{array}\\ \begin{array}{c}p{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2} for a_4\le x\le a_5\\ 0for x>a_5\end{array}\end{array}$$

and

$${\mu }_{{\tilde{A}}_{NPIFNS}}^L\left(x\right)=\{\begin{array}{c}\begin{array}{c}q{\left(\frac{x-b_1}{b_2-b_1}\right)}^{n_1} for b_1\le x\le b_2\\ w-\left(w-q\right){\left(\frac{x-b_2}{c-b_2}\right)}^{n_2} for b_2\le x\le c\end{array}\\ \begin{array}{c}k if x=c\\ w-\left(w-q\right){\left(\frac{b_4-x}{b_4-c}\right)}^{m_1} for c\le x\le b_4\end{array}\\ \begin{array}{c}q{\left(\frac{b_5-x}{b_5-b_4}\right)}^{m_2} for b_4\le x\le b_5\\ 0for x>b_5\end{array}\end{array}$$

4.2.2. $\mathsf{\alpha }$-Cut ofNon-Linear Pentagonal Interval-Valued Fuzzy Number with Symmetry

Corresponding Alpha-cut ($\alpha -cut)$ are defined for the upper and lower membership functions as follows.

The $\alpha$-cut or parametric form of NPIFNS is represented by the formulae

$$\begin{array}{ll}{A}_{NLPIFNS\alpha }=& {{{\displaystyle \cup }}^{}}_{{\alpha }_2}[A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right)]\\ & ⊝{{{\displaystyle \cup }}^{}}_{{\alpha }_1}[A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)]\end{array}$$

where,

$$A_{1L}^U\left({\alpha }_2\right)=a_1+{\left(\frac{{\alpha }_2}{p}\right)}^{n_1}\left(a_2-a_1\right) for {\alpha }_2\in [0,p]$$

$$A_{2L}^U\left({\alpha }_2\right)=a_2+{\left(\frac{k-{\alpha }_2}{k-p}\right)}^{n_2}\left(c-a_2\right) for {\alpha }_2\in [p,k]$$

$$A_{1R}^U\left({\alpha }_2\right)=a_5-{\left(\frac{{\alpha }_2}{p}\right)}^{m_2}\left(a_5-a_4\right) for {\alpha }_2\in [0,p]$$

$$A_{2R}^U\left({\alpha }_2\right)=a_4-{\left(\frac{k-{\alpha }_2}{k-p}\right)}^{m_1}\left(a_4-c\right) for {\alpha }_2\in [p,k]$$

$$A_{1L}^L\left({\alpha }_1\right)=b_1+{\left(\frac{{\alpha }_1}{q}\right)}^{n_1}\left(b_2-b_1\right) for {\alpha }_1\in [0,q]$$

$$A_{2L}^L\left({\alpha }_1\right)=b_2+{\left(\frac{w-{\alpha }_1}{w-q}\right)}^{n_2}\left(c-b_2\right) for{\alpha }_1\in [q,w]$$

$$A_{1R}^L\left({\alpha }_1\right)=b_5-{\left(\frac{{\alpha }_1}{q}\right)}^{m_2}\left(b_5-b_4\right) for {\alpha }_1\in [0,q]$$

$$A_{2R}^L\left({\alpha }_1\right)=b_4-{\left(\frac{w-{\alpha }_2}{w-q}\right)}^{m_1}\left(b_4-c\right) for {\alpha }_1\in [q,w]$$

where $A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right)$ are increasing functionswith respect to ${\alpha }_2,{\alpha }_1$, respectively, and $A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)$ are decreasing functions with respect to ${\alpha }_2,{\alpha }_1$, respectively.

Several different types of figures are given below for different values of $n_1,n_2;m_1,m_2$.

The above Figure 8, Figure 9, Figure 10 and Figure 11 represents different verity of Non-linear interval-valued PFN with symmetry.

4.2.3. Non-Linear Pentagonal Interval-Valued Fuzzy Number with Asymmetry

A non-linear pentagonal fuzzy number can be written as ${\tilde{A}}_{NPIFNAS}=\{{\left(a_1,a_2,c,a_4,a_5;k,p,s\right)}_{\left(n_1,n_2;m_1,m_2\right)}$, ${\left(b_1,b_2,c,b_4,b_5;w,q,t\right)}_{\left(n_1,n_2;m_1,m_2\right)}\}$, whose upper and lower membership function can be written as follows

$${\mu }_{{\tilde{A}}_{NPIFNAS}}^U\left(x\right)=\{\begin{array}{c}\begin{array}{c}p{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1} if a_1\le x\le a_2\\ k-\left(k-p\right){\left(\frac{x-a_2}{c-a_2}\right)}^{n_2} if a_2\le x\le c\end{array}\\ \begin{array}{c}k if x=c\\ k-\left(k-s\right){\left(\frac{a_4-x}{a_4-c}\right)}^{m_1} if c\le x\le a_4\end{array}\\ \begin{array}{c}s{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2} if a_4\le x\le a_5\\ 0if x>a_5\end{array}\end{array}$$

and

$${\mu }_{{\tilde{A}}_{NPIFNAS}}^L\left(x\right)=\{\begin{array}{c}\begin{array}{c}q{\left(\frac{x-b_1}{b_2-b_1}\right)}^{n_1} if b_1\le x\le b_2\\ w-\left(w-q\right){\left(\frac{x-b_2}{c-b_2}\right)}^{n_2} if b_2\le x\le c\end{array}\\ \begin{array}{c}w if x=c\\ w-\left(w-t\right){\left(\frac{b_4-x}{b_4-c}\right)}^{m_1} if c\le x\le b_4\end{array}\\ \begin{array}{c}t{\left(\frac{b_5-x}{b_5-b_4}\right)}^{m_2} if b_4\le x\le b_5\\ 0if x>b_5\end{array}\end{array}$$

4.2.4. $\mathsf{\alpha }$-Cut ofNon-Linear Pentagonal Interval-Valued Fuzzy Number with Asymmetry

Corresponding Alpha-cut ($\alpha -cut)$ are defined for the upper and lower membership functions as follows.

The $\alpha$-cut or parametric form of NPIFNAS is represented by the formulae

$$\begin{array}{ll}{A}_{NLPIFNAS\alpha }=& {{{\displaystyle \cup }}^{}}_{{\alpha }_2}[A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right)]\\ & ⊝{{{\displaystyle \cup }}^{}}_{{\alpha }_1}[A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)]\end{array}$$

where,

$$A_{1L}^U\left({\alpha }_2\right)=a_1+{\left(\frac{{\alpha }_2}{p}\right)}^{n_1}\left(a_2-a_1\right) for {\alpha }_2\in [0,p]$$

$$A_{2L}^U\left({\alpha }_2\right)=a_2+{\left(\frac{k-{\alpha }_2}{k-p}\right)}^{n_2}\left(c-a_2\right) for {\alpha }_2\in [p,k]$$

$$A_{1R}^U\left({\alpha }_2\right)=a_5-{\left(\frac{{\alpha }_2}{s}\right)}^{m_2}\left(a_5-a_4\right) for {\alpha }_2\in [0,s]$$

$$A_{2R}^U\left({\alpha }_2\right)=a_4-{\left(\frac{k-{\alpha }_2}{k-s}\right)}^{m_1}\left(a_4-c\right) for {\alpha }_2\in [s,k]$$

$$A_{1L}^L\left({\alpha }_1\right)=b_1+{\left(\frac{{\alpha }_1}{q}\right)}^{n_1}\left(b_2-b_1\right) for {\alpha }_1\in [0,q]$$

$$A_{2L}^L\left({\alpha }_1\right)=b_2+{\left(\frac{w-{\alpha }_1}{w-q}\right)}^{n_2}\left(c-b_2\right) for {\alpha }_1\in [q,w]$$

$$A_{1R}^L\left({\alpha }_1\right)=b_5-{\left(\frac{{\alpha }_1}{t}\right)}^{m_2}\left(b_5-b_4\right) for {\alpha }_1\in [0,t]$$

$$A_{2R}^L\left({\alpha }_1\right)=b_4-{\left(\frac{w-{\alpha }_2}{w-t}\right)}^{m_1}\left(b_4-c\right) for {\alpha }_1\in [t,w]$$

where $A_{1L}^U\left({\alpha }_2\right),A_{2L}^U\left({\alpha }_2\right),A_{1L}^L\left({\alpha }_1\right),A_{2L}^L\left({\alpha }_1\right)$ are increasing functions with respect to ${\alpha }_2,{\alpha }_1,$ respectively, and $A_{1R}^U\left({\alpha }_2\right),A_{2R}^U\left({\alpha }_2\right),A_{1R}^L\left({\alpha }_1\right),A_{2R}^L\left({\alpha }_1\right)$ are decreasing functions with respect to ${\alpha }_2,{\alpha }_1$, respectively.

Several different types of figures are given below for different values of $n_1,n_2;m_1,m_2$.

The above Figure 12, Figure 13, Figure 14 and Figure 15 represents different verity of Non-linear interval-valued PFN with symmetry.

4.3. 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

Table 4. Verbal Phrase for Pentagonal Fuzzy Number.

Strategy	${\mathit{B}}_1$	${\mathit{B}}_2$	${\mathit{B}}_3$	${\mathit{B}}_4$
${\mathit{A}}_{\mathbf{1}}$	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)
${\mathit{A}}_{\mathbf{2}}$	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)
${\mathit{A}}_{\mathbf{3}}$	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)
${\mathit{A}}_{\mathbf{4}}$	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)	(VL,M,H,M,L)

and

Table 5. Verbal Phrase for Interval-Valued Pentagonal Fuzzy Number

Strategy	${\mathit{B}}_1$	${\mathit{B}}_2$	${\mathit{B}}_3$	${\mathit{B}}_4$
${\mathit{A}}_{\mathbf{1}}$	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)
${\mathit{A}}_{\mathbf{2}}$	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)
${\mathit{A}}_{\mathbf{3}}$	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)
${\mathit{A}}_{\mathbf{4}}$	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)	(VL,HM,VH,HM,L; L,LM,VH,H,VL)

.

Verbal Phrase for Interval-Valued Pentagonal Fuzzy Number:

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

VL-Very Low, L-Low, LM-Low Mean, HM-High Mean, H-High, VH-Very High;

VL-Very Low, L-Low, LM-Low Mean, HM-High Mean, H-High, VH-Very High

5. Defuzzification of Linear Symmetric PFN

5.1. 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 is a method for constructing an irrefutable outcome in fuzzy logic, given the fuzzy sets and the equivalent degree of membership functions. There are many defuzzification skills and some of the useful tools are as follows:

(1)

Centre of area (COA) [30]

(2)

Largest of maxima (LOM) [31]

(3)

Smallest of maxima (SOM) [32]

(4)

Bisector of area (BOA) [33]

(5)

Mean of maxima (MOM)

(6)

Regular weighted point (RWP)

(7)

Graded mean integration value (GMIV)

(8)

Centre of approximated interval (COAI).

5.2. Defuzzification of Non-Linear Symmetric PFN Based on Centroid Method

We proposed a method to compute the defuzzification of non-linear symmetric PFN as,

$$R=\frac{{{\displaystyle \int }}_{a_1}^{a_2}x.\mu \left(x\right)dx+{{\displaystyle \int }}_{a_2}^{a_3}x.\mu \left(x\right)dx+{{\displaystyle \int }}_{a_3}^{a_4}x.\mu \left(x\right)dx+{{\displaystyle \int }}_{a_4}^{a_5}x.\mu \left(x\right)dx}{{{\displaystyle \int }}_{a_1}^{a_2}\mu \left(x\right)dx+{{\displaystyle \int }}_{a_2}^{a_3}\mu \left(x\right)dx+{{\displaystyle \int }}_{a_3}^{a_4}\mu \left(x\right)dx+{{\displaystyle \int }}_{a_4}^{a_5}\mu \left(x\right)dx}$$

$$=\frac{{{\displaystyle \int }}_{a_1}^{a_2}x.\left\{k{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}\right\}dx+{{\displaystyle \int }}_{a_2}^{a_3}x.\left\{1-\left(1-k\right){\left(\frac{x-a_2}{a_3-a_2}\right)}^{n_2}\right\}dx+{{\displaystyle \int }}_{a_3}^{a_4}x.\left\{1-\left(1-k\right){\left(\frac{a_4-x}{a_4-a_3}\right)}^{m_1}\right\}dx+{{\displaystyle \int }}_{a_4}^{a_5}x.\left\{k{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2}\right\}dx}{{{\displaystyle \int }}_{a_1}^{a_2}\left\{k{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}\right\}dx+{{\displaystyle \int }}_{a_2}^{a_3}\left\{1-\left(1-k\right){\left(\frac{x-a_2}{a_3-a_2}\right)}^{n_2}\right\}dx+{{\displaystyle \int }}_{a_3}^{a_4}\left\{1-\left(1-k\right){\left(\frac{a_4-x}{a_4-a_3}\right)}^{m_1}\right\}dx+{{\displaystyle \int }}_{a_4}^{a_5}\left\{k{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2}\right\}dx}$$

$$=\frac{P}{Q}$$

where,

$$\begin{array}{l}P=\underset{a_1}{\overset{a_2}{{\displaystyle \int }}}\left\{k{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}\right\}dx\\ & +\underset{a_2}{\overset{a_3}{{\displaystyle \int }}}\left\{1-\left(1-k\right){\left(\frac{x-a_2}{a_3-a_2}\right)}^{n_2}\right\}dx\\ & +\underset{a_3}{\overset{a_4}{{\displaystyle \int }}}\left\{1-\left(1-k\right){\left(\frac{a_4-x}{a_4-a_3}\right)}^{m_1}\right\}dx+\underset{a_4}{\overset{a_5}{{\displaystyle \int }}}\left\{k{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2}\right\}dx\end{array}$$

$$\begin{array}{l}Q=\underset{a_1}{\overset{a_2}{{\displaystyle \int }}}\left\{k{\left(\frac{x-a_1}{a_2-a_1}\right)}^{n_1}\right\}dx\\ & +\underset{a_2}{\overset{a_3}{{\displaystyle \int }}}\left\{1-\left(1-k\right){\left(\frac{x-a_2}{a_3-a_2}\right)}^{n_2}\right\}dx\\ & +\underset{a_3}{\overset{a_4}{{\displaystyle \int }}}\left\{1-\left(1-k\right){\left(\frac{a_4-x}{a_4-a_3}\right)}^{m_1}\right\}dx+\underset{a_4}{\overset{a_5}{{\displaystyle \int }}}\left\{k{\left(\frac{a_5-x}{a_5-a_4}\right)}^{m_2}\right\}dx\end{array}$$

After computing we have

$$\begin{array}{ll}P=\frac{k{\left(a_2-a_1\right)}^2}{n_1+2}+& \frac{k{a}_1\left(a_2-a_1\right)}{n_1+1}+\frac{\left(a_3^2-a_2^2\right)}{2}-\frac{\left(1-k\right){\left(a_3-a_3\right)}^2}{n_2+2}\\ & -\frac{a_2\left(1-k\right)\left(a_3-a_3\right)}{n_2+1}+\frac{\left(a_4^2-a_3^2\right)}{2}+\frac{\left(1-k\right){\left(a_4-a_3\right)}^2}{m_1+2}\\ & -\frac{a_4\left(1-k\right)\left(a_4-a_3\right)}{m_1+1}-\frac{k{\left(a_5-a_4\right)}^2}{m_2+2}+\frac{k{a}_5\left(a_5-a_4\right)}{m_2+1}\end{array}$$

$$\mathrm{Q}=\frac{k\left(a_2-a_1\right)}{n_1+1}+k\left(a_3-a_2\right)-\frac{\left(1-k\right)\left(a_3-a_2\right)}{n_2+1}+k\left(a_4-a_3\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{m_1+1}+\frac{k\left(a_5-a_4\right)}{m_2+1}$$

Hence,

$$R=\frac{\mathrm{L}+\mathrm{M}}{\mathrm{N}}$$

where,

$$L=\frac{k{\left(a_2-a_1\right)}^2}{n_1+2}+\frac{k{a}_1\left(a_2-a_1\right)}{n_1+1}+\frac{\left(a_3^2-a_2^2\right)}{2}-\frac{\left(1-k\right){\left(a_3-a_3\right)}^2}{n_2+2}-\frac{a_2\left(1-k\right)\left(a_3-a_3\right)}{n_2+1},$$

$$\begin{array}{ll}M=\frac{\left(a_4^2-a_3^2\right)}{2}& +\frac{\left(1-k\right){\left(a_4-a_3\right)}^2}{m_1+2}-\frac{a_4\left(1-k\right)\left(a_4-a_3\right)}{m_1+1}-\frac{k{\left(a_5-a_4\right)}^2}{m_2+2}\\ & +\frac{k{a}_5\left(a_5-a_4\right)}{m_2+1}\end{array}$$

And

$$\begin{array}{ll}N=\frac{k\left(a_2-a_1\right)}{n_1+1}& +k\left(a_3-a_2\right)-\frac{\left(1-k\right)\left(a_3-a_2\right)}{n_2+1}+k\left(a_4-a_3\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{m_1+1}\\ & +\frac{k\left(a_5-a_4\right)}{m_2+1}\end{array}$$

For the linear pentagonal fuzzy number, we have $n_1=1, n_2=1,m_1=1,m_2=1$

$$R_{New}=\frac{\frac{k{\left(a_2-a_1\right)}^2}{3}+\frac{k{a}_1\left(a_2-a_1\right)}{2}+\frac{\left(a_3^2-a_2^2\right)}{2}-\frac{\left(1-k\right){\left(a_3-a_3\right)}^2}{3}-\frac{a_2\left(1-k\right)\left(a_3-a_3\right)}{2}+\frac{\left(a_4^2-a_3^2\right)}{2}+\frac{\left(1-k\right){\left(a_4-a_3\right)}^2}{3}-\frac{a_4\left(1-k\right)\left(a_4-a_3\right)}{2}-\frac{k{\left(a_5-a_4\right)}^2}{3}+\frac{k{a}_5\left(a_5-a_4\right)}{2}}{\frac{k\left(a_2-a_1\right)}{2}+k\left(a_3-a_2\right)-\frac{\left(1-k\right)\left(a_3-a_2\right)}{2}+k\left(a_4-a_3\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{2}+\frac{k\left(a_5-a_4\right)}{2}}$$

For $k=1,n_1=1,n_2=1,m_1=1,m_2=1$ we have

$$P^{\prime }=\frac{a_5^2+a_4^2+a_5{a}_4-a_1{a}_2-a_2^2-a_1^2}{6}$$

$$Q^{\prime }=\frac{a_5+a_4-a_2-a_1}{2}$$

So, $R^{\prime }=\frac{\left(a_5^2+a_4^2+a_5{a}_4-a_1{a}_2-a_2^2-a_1^2\right)}{3.\left(a_5+a_4-a_2-a_1\right)}$.

5.3. Defuzzification of Linear Symmetric PFN Based on Mean of Alpha (α)-Cut Method

The left and right α-cut of a non-linear heptagonal fuzzy number are

$$L^{-1}\left(\alpha \right)=a_1+{\left(\frac{\alpha }{k}\right)}^{n_1}\left(a_2-a_1\right) for \alpha \in [0,k]$$

$$R^{-1}\left(\alpha \right)=a_5-{\left(\frac{\alpha }{k}\right)}^{m_2}\left(a_5-a_4\right) for \alpha \in [0,k]$$

$$L^{-1}\left(\alpha \right)=a_2+{\left(\frac{1-\alpha }{1-k}\right)}^{n_2}\left(a_3-a_2\right) for \alpha \in [k,1]$$

$$R^{-1}\left(\alpha \right)=a_4-{\left(\frac{1-\alpha }{1-k}\right)}^{m_1}\left(a_4-a_3\right) for \alpha \in [k,1]$$

We proposed the mean of interval method for defuzzification as

$$\check{A}=\underset{\alpha =0}{\overset{1}{{\displaystyle \int }}}\frac{\left(L^{-1}\left(\alpha \right)+R^{-1}\left(\alpha \right)\right)d\alpha }{2}$$

$$={{\displaystyle \int }}_{\alpha =0}^k\frac{\left(L^{-1}\left(\alpha \right)+R^{-1}\left(\alpha \right)\right)d\alpha }{2}+{{\displaystyle \int }}_{\alpha =k}^1\frac{\left(L^{-1}\left(\alpha \right)+R^{-1}\left(\alpha \right)\right)d\alpha }{2}$$

$$={{\displaystyle \int }}_{\alpha =0}^k\frac{\left(\left\{a_1+{\left(\frac{\alpha }{k}\right)}^{n_1}\left(a_2-a_1\right)\right\}+\left\{a_5-{\left(\frac{\alpha }{k}\right)}^{m_2}\left(a_5-a_4\right)\right\}\right)d\alpha }{2}+{{\displaystyle \int }}_{\alpha =k}^1\frac{\left(\left\{a_2+{\left(\frac{1-\alpha }{1-k}\right)}^{n_2}\left(a_3-a_2\right)\right\}+\left\{a_4-{\left(\frac{1-\alpha }{1-k}\right)}^{m_1}\left(a_4-a_3\right)\right\}\right)d\alpha }{2}$$

$$=\frac{a_{1}k+\frac{k\left(a_2-a_1\right)}{n_1+1}+a_{5}k-\frac{k\left(a_5-a_4\right)}{m_2+1}}{2}+\frac{a_2\left(1-k\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{m_1+1}+a_4\left(1-k\right)+\frac{\left(1-k\right)\left(a_3-a_2\right)}{n_2+1}}{2}$$

For the linear pentagonal fuzzy number, we have $n_1=1, n_2=1,m_1=1,m_2=1$.

Thus,

$$\begin{array}{ll}\check{A}& =\frac{a_{1}k+\frac{k\left(a_2-a_1\right)}{2}+a_{5}k-\frac{k\left(a_5-a_4\right)}{2}}{2}\\ & +\frac{a_2\left(1-k\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{2}+a_4\left(1-k\right)+\frac{\left(1-k\right)\left(a_3-a_2\right)}{2}}{2}\end{array}$$

For $k=1,n_1=1,n_2=1,m_1=1,m_2=1$, we have

$$\check{A}=\frac{\left(a_5+a_4+a_2+a_1\right)}{4}$$

5.4. 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.

Then, we find the following.

$R_1\left(\check{A},0\right)=$ Area of shaded region for Figure 16 = $\frac{\left(a_1+a_2\right)}{2}.k$

$R_2\left(\check{A},0\right)=$ Area of shaded region for Figure 17 = $\frac{\left(a_2+a_3\right)}{2}.\left(1-k\right)$

$R_3\left(\check{A},0\right)=$ Area of shaded region for Figure 18 = $a_3.1=a_3$

$R_4\left(\check{A},0\right)=$ Area of shaded region for Figure 19 = $\left\{a_4.1-\frac{\left(a_4-a_3\right)}{2}.\left(1-k\right)\right\}$

$R_5\left(\check{A},0\right)=$ Area of shaded region for Figure 20 = $\frac{\left(a_5+a_4\right)}{2}.k$

Hence,

$$R\left(\check{D},0\right)=\frac{R_1\left(\check{A},0\right)+R_2\left(\check{A},0\right)+R_3\left(\check{A},0\right)+R_4\left(\check{A},0\right)+R_5\left(\check{A},0\right)}{5}$$

$$=\frac{\frac{\left(a_1+a_2\right)}{2}.k+\frac{\left(a_2+a_3\right)}{2}.\left(1-k\right)+a_3+a_4-\frac{\left(a_4-a_3\right)}{2}.\left(1-k\right)+\frac{\left(a_5+a_4\right)}{2}.k}{5}$$

For k=1, $R\left(\check{D},0\right)=\frac{a_1+a_2+2a_3+3a_4+a_5}{10}$.

5.5. 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. Numerical examples.

No of Example	Value of k	$\mathbf{Value}\mathbf{of}{\mathit{n}}_1,{\mathit{n}}_2,{\mathit{m}}_1,{\mathit{m}}_2$	Defuzzification Value by Centroid Method	Defuzzification Value by Mean of Alpha-Cut Method	Defuzzification Value by Removal-Area Method
$\check{A}=\left(1,2,3,4,5\right)$	1	1,1,1,1	3.00	3.00	2.6
$\check{B}=\left(1,2,3,4,5\right)$	0.5	1,1,1,1	3.00	3.00	2.2
$\check{C}=\left(-2,1,0,2,4\right)$	0.3	2,2,2,2	1.0563	1.05	0.48
$\check{D}=\left(-2,-1,0,1,2\right)$	0.2	1,1,1,1	0.00	0.00	0.112

:

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

6.1. 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 [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]. In this section we find a new concept for finding the ranking of PFNs.

6.2. Ranking of Pentagonal Fuzzy Numbers

We proposed a new way to find the ranking of PFNs. In Figure 21 the pentagon is divided into three triangles and one rectangle. We can find the ranking by using the centroid formula of triangles and rectangles.

Then, considering the above figure where $\left(a_1,0\right),\left(a_2,k\right),\left(a_3,1\right),\left(a_4,k\right),\left(a_5,0\right)$ are the vertices of the pentagonal fuzzy numbers and $0<k<1$:

$T_1$ represents the centroid of the corresponding triangle whose vertices are $\left(a_1,0\right),\left(a_2,0\right),\left(a_2,k\right).$

$T_2$ represents the centroid of the corresponding triangle whose vertices are $\left(a_4,0\right),\left(a_5,0\right),\left(a_4,k\right).$

$T_3$ represents the centroid of the corresponding triangle whose vertices are $\left(a_3,1\right),\left(a_2,k\right),\left(a_4,k\right)$.

$T_4$ represents the centroid of the corresponding triangle whose vertices are $\left(a_2,k\right),\left(a_2,0\right),\left(a_4,0\right),\left(a_4,k\right)$.

Now, the centroids are in the form:

$$T_1=\left(\frac{a_1+2a_2}{3},\frac{k}{3}\right)$$

$$T_3=\left(\frac{a_2+a_3+a_4}{3},\frac{1+2k}{3}\right)$$

$$T_2=\left(\frac{a_5+2a_4}{3},\frac{k}{3}\right)$$

$$T_4=\left(\frac{a_2+a_4}{2},\frac{k}{2}\right)$$

If we consider the average of these, we can obtain the new ranking as

$$\mathrm{R}\left(\check{F_H}\right)=\left(\frac{2a_1+9a_2+2a_3+9a_4+2a_5}{24},\frac{11k+2}{24}\right)$$

6.3. Working Rule to Find the Ranking of a Pentagonal Fuzzy Number

$F\left(R\right)$ is a set of PFNs defined on the set of real numbers, and the ranking of PFNs is actually a function $R: F\left(R\right) \to R$ which maps each fuzzy number into a real line.

Suppose we consider two different PFNs $\tilde{A}$ and $\tilde{B}$, which are to be ranked. Then using the above method and after finding the $R\left(A\right)$ and $R\left(B\right)$ by using the previous formula we can easily say that if

(1) $R(\tilde{A})>R(\tilde{B})$ then $\tilde{A}>\tilde{B}$

(2) $R(\tilde{A})<R(\tilde{B})$ then $\tilde{A}<\tilde{B}$

(3)$R(\tilde{A})=R(\tilde{B})$ then $\tilde{A}\approx \tilde{B}$

The flowchart of the proposed method is given below in Figure 22.

6.4. Numerical Computation

Remark 6.

By using the above concept of ranking PF’s, we can easily compare them (see

Table 7. Numerical example.

No. of Example		Components	$\mathit{R} \left(\check{{\mathit{F}}_{\mathit{H}}}\right)$	Conclusion
Example 1	Set-1	$\tilde{A}\left(1,2,3,4,5;0.8\right)$	(3.0,0.45)	$\check{D}<\check{C}<\check{B}<\check{A}$
	Set-2	$\check{B}\left(-1,0,2,4,5;0.8\right)$	(2.0,0.45)
	Set-3	$\check{C}\left(-2,-1,0,2,4;0.8\right)$	(0.54,0.45)
	Set-4	$\check{D}\left(-2,-1,0,1,3;0.8\right)$	(0.083,0.45)
Example 2	Set-1	$\check{A}\left(-1,0,0.2,0.3,0.4;0.6\right)$	(0.079,0.36)	$\check{B}<\check{C}<\check{D}<\check{A}$
	Set-2	$\check{B}\left(-1,-0.5,0,0.4,0.5;0.6\right)$	(−0.79,0.36)
	Set-3	$\check{C}\left(-1,-0.6,-0.3,0.2,0.5;0.6\right)$	(−0.22,0.36)
	Set-4	$\check{D}\left(-1,-0.2,-0.1,0.2,0.3;0.6\right)$	(−0.067,0.36)

).

7. Application of Pentagonal Fuzzy Number in a Game Problem in a Fuzzy Environment Using the Dominance Method

Let us consider two different players, A and B playing a game of carom, with A₁,A₂,…,A_n strategiesand B₁,B₂,…,B_m strategies, respectively. We also consider that each player can choose pure strategies. Let, a_pq be the elements of the pay-off matrix. The strategy of player A is denoted by A_p and the strategy of player B is denoted by B_q.

Saddle point: In a game problem, the max-min for A and min-max for B should be equal, therefore, the corresponding game is assumed to have a saddle point or an equilibrium point. The gain at the saddle point, which is the arrangement in the associated matrix where the maximum of the row minima correspond with the minimum of the column maxima, is said to be the value of the game.

We represent the max-min value of the game by gamma(γ), and the min-max value of the game by $\overline{\gamma }$. The game is said to be fair. If γ = 0 = $\overline{\gamma }$.

The game is supposed to be strictly ascertainable. If γ = γ = $\overline{\gamma }$.

7.1. Operation for Solving a Game Problem Using Fuzzy

Step1: First, we have to examine whether a saddle point will exist or not. If it exists in the given problem, then we need to find it in a direct way. However, if it doesnot exist, then we need to follow the second step.

Step2: Compare column strategies.

(1) In the given pay-off matrix, if the elements of Column A ≤ elements of Column B, that is, Column A strategy will fully dominate over column B strategy, then, according to the rule we need to delete column B strategy from the given pay-off matrix.

(2) In the given pay-off matrix, we tally each column strategy with all other column strategies and omit high strategies as far as possible.

Step3: Compare row strategies.

(i) In the given pay-off matrix, if the elements of Row A ≥ elements of Row B, Row A strategy will dominate over Row B strategy. So, omit Row B strategy from the given payoff matrix.

(ii) In the given pay-off matrix, we tally each row strategy with all possible row strategies and omit low strategies as far as possible.

(iii) The game may decrease to a single cell giving an order about the value of the game and optimal master plan of the players. If not, then moves to step 4.

Step4: The dominance rule should not be based on the dominance of pure strategies only. A given approach can be dominated if it gives us a poor result to an usual of two or more other pure master plan.

7.2. Numerical Problems

Example 1.

Let us consider a fuzzy game pay-off matrix where two different players are A and B (

Table 8. Fuzzy game pay-off matrix table.

Strategy	B1	B2	B3	B4
A1	(0,1,2,3,4;0.5)	(−1,0,1,2,3;0.5)	(2,3,4,5,6;0.5)	(−2,−1,0,1,2;0.5)
A2	(1,2,3,4,5;0.5)	(2,3,4,5,6;0.5)	(0,1,2,3,4;0.5)	(2,3,4,5,6;0.5)
A3	(2,3,4,5,6;0.5)	(0,1,2,3,4;0.5)	(2,3,4,5,6;0.5)	(−2,−1,0,1,2;0.5)
A4	(−2,−1,0,1,2;0.5)	(2,3,4,5,6;0.5)	(−2,−1,0,1,2;0.5)	(6,7,8,9,10;0.5)

):

We apply the rule of defuzzification using Alpha ($\alpha$)-cut method,

$$\begin{array}{ll}\check{A}=& \frac{a_{1}k+\frac{k\left(a_2-a_1\right)}{2}+a_{5}k-\frac{k\left(a_5-a_4\right)}{2}}{2}\\ & +\frac{a_2\left(1-k\right)-\frac{\left(1-k\right)\left(a_4-a_3\right)}{2}+a_4\left(1-k\right)+\frac{\left(1-k\right)\left(a_3-a_2\right)}{2}}{2}\end{array}$$

to convert the pentagonal fuzzy number into a crisp number, then we have the modified pay-off matrix as

Table 9. Defuzzified game pay-off matrix.

Strategy	B1	B2	B3	B4
A1	2	1	4	0
A2	3	4	2	4
A3	4	2	4	0
A4	0	4	0	8

:

Now all elements of A₁ are less than or equal to A₃, here, A₃dominates A₁. So, according to the dominance properties, our table becomes

Table 10. First step for solving the problem.

Strategy	B1	B2	B3	B4
A2	3	4	2	4
A3	4	2	4	0
A4	0	4	0	8

:

Now all elements of B₃ are less than or equal to elements of B₁, here B₁dominates B₃. So, according to the dominance properties our table becomes

Table 11. Second step for solving the problem.

Strategy	B2	B3	B4
A2	4	2	4
A3	2	4	0
A4	4	0	8

:

Now, the convex combination of B₃, B₄ (we take the average), that is, 3, 2, 4 is dominated by B₂. Hence, we have

Table 12. Third step for solving the problem.

Strategy	B3	B4
A2	2	4
A3	4	0
A4	0	8

,

Now, the convex combination of A₃,A₄ (we take the average), that is, 2,4 is dominated by A₂. Hence, we have

Table 13. Fourth step for solving the problem.

Strategy	B3	B4
A3	4	0
A4	0	8

,

Hence, the optimal solution to the main problem is for A(0,0,$\frac{2}{3}$,$\frac{1}{3}$) for B(0,0,$\frac{2}{3}$,$\frac{1}{3}$) and the corresponding game value is $\frac{8}{3}$.

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 member

Table 14. Fuzzy game pay-off matrix table.

Strategy	B1	B2	B3	B4
A1	(1,2,3,4,5;0.5)	(0,1,2,3,4;0.5)	(3,4,5,6,7;0.5)	(−2,−1,0,1,2;0.5)
A2	(2,3,4,5,6;0.5)	(3,4,5,6,7;0.5)	(1,2,3,4,5;0.5)	(3,4,5,6,7;0.5)
A3	(3,4,5,6,7;0.5)	(1,2,3,4,5;0.5)	(3,4,5,6,7;0.5)	(−2,−1,0,1,2;0.5)
A4	(−2,−1,0,1,2;0.5)	(2,3,4,5,6;0.5)	(−2,−1,0,1,2;0.5)	(7,8,9,10,11;0.5)

.

After crispification we have a pay-off matrix as

Table 15. Solution of the problem.

Strategy	B1	B2	B3	B4
A1	3	2	5	0
A2	4	5	3	5
A3	5	3	5	0
A4	0	4	0	9

.

Hence, the optimal solution to the main problem is for A(0,0,$\frac{9}{14}$,$\frac{5}{14}$) for B(0,0,$\frac{9}{14}$,$\frac{5}{14}$) and the corresponding game value is $\frac{45}{14}$.

Example 3.

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 member (

Table 16. Fuzzy game pay-off matrix table.

Strategy	B1	B2	B3	B4
A1	(2,3,4,5,6;0.5)	(2,3,4,5,6;0.5)	(4,5,6,7,8;0.5)	(-2,-1,0,1,2;0.5)
A2	(3,4,5,6,7;0.5)	(4,5,6,7,8;0.5)	(3,4,5,6,7;0.5)	(4,5,6,7,8;0.5)
A3	(4,5,6,7,8;0.5)	(2,3,4,5,6;0.5)	(4,5,6,7,8;0.5)	(-2,-1,0,1,2;0.5)
A4	(-2,-1,0,1,2;0.5)	(4,5,6,7,8;0.5)	(-2,-1,0,1,2;0.5)	(8,9,10,11,12;0.5)

).

After crispification we have a pay-off matrix as

Table 17. Solution of the problem.

Strategy	B1	B2	B3	B4
A1	4	3	6	0
A2	5	6	4	6
A3	6	4	6	0
A4	0	5	0	10

.

Hence, the optimal solution to the main problem is for A(0,0,$\frac{5}{8}$,$\frac{3}{8}$) for B(0,0,$\frac{5}{8}$,$\frac{3}{8}$) and the corresponding game value is $\frac{15}{4}$.

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.

8. 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:

• 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.
• The classifications of linear and nonlinear membership functions provided an appropriate strategy for decision makers.
• Detailed illustrations of membership functions, $\alpha$-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.