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

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## Abstract

**:**

## 1. Introduction

#### 1.1. Uncertainty and Uncertainty Measure

- (i)
- The fact belongs to some certain interval
- (ii)
- The concept of the membership function is missing.

- (i)
- The conception of belongingness of the component has to be considered
- (ii)
- The concept of membership function is derived.

- (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

- (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

#### 1.2. Fuzzy Sets and Number

#### 1.3. Interval-Valued Fuzzy Numbers (IVFNs)

#### 1.4. Pentagonal Fuzzy Numbers

#### 1.5. Verbal Phrasesin Uncertainty Theory

**Example**

**1.**

**Example**

**2.**

#### 1.6. Ranking and Defuzzification

#### 1.7. Motivation

#### 1.8. Novelties

- (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.

#### 1.9. Structure of the Paper

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. PFN and Its Different Representations

**Definition**

**4.**

#### 3.1. Linear PFN with Symmetry

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**1.**

#### 3.2. Non-Linear Pentagonal Fuzzy Number with Symmetry

**Definition**

**7.**

**Definition**

**8.**

**Remark**

**2.**

## 4. Interval-Valued PFN

#### 4.1. Linear Interval-Valued PFN

#### 4.1.1. Linear Pentagonal Interval-Valued Fuzzy Number with Symmetry

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

**Remark**

**3.**

#### 4.1.3. Linear Pentagonal Interval-Valued Fuzzy Number with Asymmetry

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

#### 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

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

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

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

#### 4.3. Verbal Phrase for the PFN and Interval-Valued PFN

VL-Very Low, L-Low, M-Medium, H-High, M-Medium, VH-Very High

**Verbal Phrase for Interval-Valued Pentagonal Fuzzy Number**:

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

- (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.

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

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

#### 5.4. Defuzzification of Linear Symmetric PFN Based on Removalof Area Method for Linear Pentagonal Fuzzy Number

#### 5.5. Comparison ofthe Above Three Defuzzification Methods

**Remark**

**5.**

## 6. Ranking for the Pentagonal Fuzzy Number

#### 6.1. Basic Concept of Ranking Fuzzy Numbers

#### 6.2. Ranking of Pentagonal Fuzzy Numbers

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

#### 6.4. Numerical Computation

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

_{1},A

_{2},…,A

_{n}strategiesand B

_{1},B

_{2},…,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}.

#### 7.1. Operation for Solving a Game Problem Using Fuzzy

#### 7.2. Numerical Problems

**Example**

**1.**

_{1}are less than or equal to A

_{3}, here, A

_{3}dominates A

_{1}. So, according to the dominance properties, our table becomes Table 10:

_{3}are less than or equal to elements of B

_{1}, here B

_{1}dominates B

_{3}. So, according to the dominance properties our table becomes Table 11:

_{3}, B

_{4}(we take the average), that is, 3, 2, 4 is dominated by B

_{2}. Hence, we have Table 12,

_{3},A

_{4}(we take the average), that is, 2,4 is dominated by A

_{2}. Hence, we have Table 13,

**Example**

**2.**

**Example**

**3.**

**Remark**

**7.**

## 8. Conclusions and Scope of Future Research

- 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.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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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 |

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 |

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 |

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) |

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) |

No of Example | Value of k | $\mathbf{Value}\text{}\mathbf{of}\text{}{\mathit{n}}_{1},{\mathit{n}}_{2},{\mathit{m}}_{1},{\mathit{m}}_{2}$ | Defuzzification Value by Centroid Method | Defuzzification Value by Mean of Alpha-Cut Method | Defuzzification Value by Removal-Area Method |
---|---|---|---|---|---|

$\stackrel{\u02c7}{A}=\left(1,2,3,4,5\right)$ | 1 | 1,1,1,1 | 3.00 | 3.00 | 2.6 |

$\stackrel{\u02c7}{B}=\left(1,2,3,4,5\right)$ | 0.5 | 1,1,1,1 | 3.00 | 3.00 | 2.2 |

$\stackrel{\u02c7}{C}=\left(-2,1,0,2,4\right)$ | 0.3 | 2,2,2,2 | 1.0563 | 1.05 | 0.48 |

$\stackrel{\u02c7}{D}=\left(-2,-1,0,1,2\right)$ | 0.2 | 1,1,1,1 | 0.00 | 0.00 | 0.112 |

No. of Example | Components | $\mathit{R}\left(\stackrel{\u02c7}{{\mathit{F}}_{\mathit{H}}}\right)$ | Conclusion | |
---|---|---|---|---|

Example 1 | Set-1 | $\tilde{A}\left(1,2,3,4,5;0.8\right)$ | (3.0,0.45) | $\stackrel{\u02c7}{D}<\stackrel{\u02c7}{C}<\stackrel{\u02c7}{B}<\stackrel{\u02c7}{A}$ |

Set-2 | $\stackrel{\u02c7}{B}\left(-1,0,2,4,5;0.8\right)$ | (2.0,0.45) | ||

Set-3 | $\stackrel{\u02c7}{C}\left(-2,-1,0,2,4;0.8\right)$ | (0.54,0.45) | ||

Set-4 | $\stackrel{\u02c7}{D}\left(-2,-1,0,1,3;0.8\right)$ | (0.083,0.45) | ||

Example 2 | Set-1 | $\stackrel{\u02c7}{A}\left(-1,0,0.2,0.3,0.4;0.6\right)$ | (0.079,0.36) | $\stackrel{\u02c7}{B}<\stackrel{\u02c7}{C}<\stackrel{\u02c7}{D}<\stackrel{\u02c7}{A}$ |

Set-2 | $\stackrel{\u02c7}{B}\left(-1,-0.5,0,0.4,0.5;0.6\right)$ | (−0.79,0.36) | ||

Set-3 | $\stackrel{\u02c7}{C}\left(-1,-0.6,-0.3,0.2,0.5;0.6\right)$ | (−0.22,0.36) | ||

Set-4 | $\stackrel{\u02c7}{D}\left(-1,-0.2,-0.1,0.2,0.3;0.6\right)$ | (−0.067,0.36) |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | (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) |

A_{2} | (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) |

A_{3} | (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) |

A_{4} | (−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) |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | 2 | 1 | 4 | 0 |

A_{2} | 3 | 4 | 2 | 4 |

A_{3} | 4 | 2 | 4 | 0 |

A_{4} | 0 | 4 | 0 | 8 |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{2} | 3 | 4 | 2 | 4 |

A_{3} | 4 | 2 | 4 | 0 |

A_{4} | 0 | 4 | 0 | 8 |

Strategy | B_{2} | B_{3} | B_{4} |
---|---|---|---|

A_{2} | 4 | 2 | 4 |

A_{3} | 2 | 4 | 0 |

A_{4} | 4 | 0 | 8 |

Strategy | B_{3} | B_{4} |
---|---|---|

A_{2} | 2 | 4 |

A_{3} | 4 | 0 |

A_{4} | 0 | 8 |

Strategy | B_{3} | B_{4} |
---|---|---|

A_{3} | 4 | 0 |

A_{4} | 0 | 8 |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | (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) |

A_{2} | (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) |

A_{3} | (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) |

A_{4} | (−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) |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | 3 | 2 | 5 | 0 |

A_{2} | 4 | 5 | 3 | 5 |

A_{3} | 5 | 3 | 5 | 0 |

A_{4} | 0 | 4 | 0 | 9 |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | (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) |

A_{2} | (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) |

A_{3} | (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) |

A_{4} | (-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) |

Strategy | B_{1} | B_{2} | B_{3} | B_{4} |
---|---|---|---|---|

A_{1} | 4 | 3 | 6 | 0 |

A_{2} | 5 | 6 | 4 | 6 |

A_{3} | 6 | 4 | 6 | 0 |

A_{4} | 0 | 5 | 0 | 10 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chakraborty, A.; Mondal, S.P.; Alam, S.; Ahmadian, A.; Senu, N.; De, D.; Salahshour, S.
The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems. *Symmetry* **2019**, *11*, 248.
https://doi.org/10.3390/sym11020248

**AMA Style**

Chakraborty A, Mondal SP, Alam S, Ahmadian A, Senu N, De D, Salahshour S.
The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems. *Symmetry*. 2019; 11(2):248.
https://doi.org/10.3390/sym11020248

**Chicago/Turabian Style**

Chakraborty, Avishek, Sankar Prasad Mondal, Shariful Alam, Ali Ahmadian, Norazak Senu, Debashis De, and Soheil Salahshour.
2019. "The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems" *Symmetry* 11, no. 2: 248.
https://doi.org/10.3390/sym11020248