VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHES

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International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 8, Issue 5, September - October 2017, pp. 73–82, Article ID: IJARET_08_05_009
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ISSN Print: 0976-6480 and ISSN Online: 0976-6499
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VARIOUS FUZZY NUMBERS AND THEIR
VARIOUS RANKING APPROACHES
Lavanya P
Lecturer in Mathematics, Government Polytechnic College,
Arakandanallur, Villupuram, Tamil Nadu, India
ABSTRACT
A brief survey of this study is to identify the ranking formulas for various fuzzy
numbers derived from research papers published over the past few years. This paper
presents the latest results of fuzzy ranking applications very clearly and simply, as well
as highlighting key points in the use of fuzzy numbers. This paper discusses the
importance of pointing out the concepts of fuzzy numbers and their formulas for
ranking.
Key words: Fuzzy Set, Fuzzy Numbers, Various Ranking Methods
Cite this Article: Lavanya P, Various Fuzzy Numbers and their Various Ranking
Approaches, International Journal of Advanced Research in Engineering and
Technology, 8(5), 2017, pp 73–82.
https://iaeme.com/Home/issue/IJARET?Volume=8&Issue=5
1. INTRODUCTION
In 1965 Lotfi. A. Zadeh introduced the concept of Fuzziness. An approximation of analytical
functions by Dubois, Prade, and Yager (1978), involves the division of the membership
functions of algebraic operations into a left side and a representation of the right side by a simple
analytical form. Due to the ascent within the study of Fuzzy sets, we have different types of
Fuzzy numbers, namely Triangular, Trapezoidal, Pentagonal, Hexagonal, Heptagonal fuzzy
numbers. Well, S.H. Chen was studying operations on fuzzy numbers with real-valued
functions in 1985, as well as Klement. In 2004 Michael Hanss introduced the Triangular Fuzzy
number. In many applications, fuzzy numbers are used, such as control theory, signal
processing, and approximation theory.
We also discussed the basic definitions of different Fuzzy numbers and their numerous
fuzzy ranking approaches in this survey article. This survey alone would make it easy to
consider their review of the effects of the ranking methods and their future development in the
future.
2. FUZZY NUMBERS
2.1. Introduction
In 1975, Hutton, B [HU] & Rodabaugh, SE [Rod] introduced a fuzzy number. A fuzzy number
is the fundamental precept of the fuzzy set theory we typically use. It is chosen from the default
fuzzy set of all real numbers. Like standard numbers, fuzzy sets have been either positive or

Various Fuzzy Numbers and their Various Ranking Approaches
negative, where the whole space is symmetrically empty. The linguistic form is often selected
to address the fuzzy number, which includes slightly, quietly. Calculations with fuzzy numbers
allow parameters, properties, geometry, and initial conditions to be inserted into uncertainty. In
the literature on fuzzy sets, Zadeh (1965) notes that granulation plays a part in human cognition.
Membership functions are structured to represent individual and subjective human
experiences as part of a fuzzy set. A fuzzy set has several functions of membership, known as
operations from a well-defined universe. X with an interval between units, 0 to 1, as seen in the
following equation:
: [0,1]
A
X
 → (1)
The degree of notification for a vague class with an infinite set of range values between 0
and 1. The notification level for fuzzy numbers with an infinite set of range values between 0
and 1 is specified by the membership function. Fuzzy numbers play a crucial role in many fields
in computation, communications products engineering, scientific testing, decision-making,
approximate reasoning, and optimization.
Definition (Fuzzy Numbers): A fuzzy set N
 
is said to be a fuzzy number based on the real
number R
 
:
A fuzzy set N
 
is concave.
A fuzzy set N
 
is normal. i.e., 0 0
such that ( ) 1.
N
x R x
 
 
  =
( )
N
x
  , is piecewise continuous.
N
 
, must be closed interval for every [0,1].
 
is bounded.
Supp N
 
3. TRIANGULAR FUZZY NUMBER
Definition: ( )
, ,
N a b c
 
= be the fuzzy number is known as a triangular fuzzy number if its
membership functions :
  
0,1
R
 
→ are equal to
 
 
,
1
( )
,
0
N
x a
if x a b
b a
x b
x
c x
if x b c
c b
otherwise
 
−


 −

=

= 
−
 
 −


Where( )
, ,
a b c
, this fuzzy number is denoted by( )
, ,
a b c
.
Figure 1 Triangular Fuzzy Number

Lavanya P
3.1. Ranking Methods for Triangular Fuzzy Numbers
Many types of ranking procedures have 'Triangular fuzzy numbers.' All the rankings listed here
have been discovered over the past few years for 'Triangular fuzzy numbers' and compiled from
studying various research papers. Only the most important of them are listed here. They are,
“ − Cut”for Triangular Fuzzy Numbers[24]:
( ) ( ) ( )
 
L
a , ,
= − + − −
U
a b a a c c b
   
“Yeager’s ranking” method for Triangular Fuzzy Numbers[29]:
( )
1
( , , ) , ,
2
= 
L U
Y a b c a a d
  
( ) ( ) ( )
 
L
a , ,
= − + − −
U
where a b a a c c b
   
“Sub interval Average” method for Triangular Fuzzy Numbers[3]:
( )
( )
4
, ,
12
+ +
 
=  
 
a b c
R a b c
“Sub interval Addition” method for Triangular Fuzzy Numbers[1]:
4( )
( , , )
6
+ +
 
=  
 
a b c
R a b c
“Pascal Triangular Graded Mean” for Triangular Fuzzy Numbers[24]:
( )
2
, ,
4
+ +
 
=  
 
a b c
P a b c
“Magnitude Ranking” for Triangular Fuzzy Numbers[24]:
( )
1
0
1
( , , ) 3
2
= + −

Mag a b c c a b rdr
“Centroid approach” for Triangular Fuzzy Numbers[10]:
( )
2 14 2 7
, , ;
6 6
+ +
   
=    
   
a b c w
C a b c w
4. TRAPEZOIDAL FUZZY NUMBER
Definition. Let ( )
, , ,
 
=
N a b c d be a fuzzy number, is known as a trapezoidal fuzzy number if its
membership function
 
 
 
,
1 ,
( )
,
0
 
 −
 

 
 −
 

 
= 
−
 
 
 
 −
 


N
x a
if x a b
b a
x b c
x
d x
if x c d
d c
otherwise

The trapezoidal fuzzy number  
N is denoted by quadruplet ( )
, , ,
 
=
N a b c d and has the shape
of a trapezoid.

Figure 2 Trapezoidal Fuzzy Number
4.1. Ranking Methods for Trapezoidal Fuzzy Numbers
There are 'Trapezoidal fuzzy numbers' for several forms of ranking procedures. Even the most
important of them here Over the past few years, all the rankings listed here have been found for
'Trapezoidal fuzzy numbers' and collected by reviewing different research papers.
“ −
 Cut” for Trapezoidal Fuzzy Numbers[8]:
( ) ( ) ( )
 
L
a , ,
= − + − −
U
a b a a d d c
   
“Yeager’s ranking” method for Trapezoidal Fuzzy Numbers[30]:
1 2
1 4 2
( )
2 5 3
 
   
= + − +
 
   
   
 
L U
R A a a  
“Sub interval Average” method for Trapezoidal Fuzzy Numbers[3]:
( )
( )
5
, , ,
20
+ + +
 
=  
 
a b c d
R a b c d
“Sub interval Addition” method for Trapezoidal Fuzzy Numbers[1]:
5( )
( , , , )
10
+ + +
 
=  
 
a b c d
R a b c d
“Pascal Triangular Graded Mean” for Trapezoidal Fuzzy Numbers[15]:
( )
3 3
, , ,
8
+ + +
 
=  
 
a b c d
P a b c d
“Magnitude Ranking” for Trapezoidal Fuzzy Numbers[16]:
5 5
( , , , )
12
+ + +
 
=  
 
a b c d
Mag a b c d
“Centroid approach” for Trapezoidal Fuzzy Numbers[10]:
( )
2 7 7 7
, , , ;
6 6
+ + +
   
= 
   
   
a b c d w
C a b c d w

Lavanya P
5. PENTAGONAL FUZZY NUMBER
Definition. A pentagonal fuzzy number ( )
, , , ,
 
=
p
N a b c d e , should satisfy the following condition
In the interval  
0,1 , ( )
 
p
N
x
 is a continuous function.
   
, and b,c
a b is a continuous function and ( )
 
p
N
x
 is strictly increasing.
   
, and d,e
c d is a continuous function and ( )
 
p
N
x
 is strictly decreasing.
Figure 3 Pentagonal Fuzzy number
5.1. Ranking Methods for Pentagonal Fuzzy Numbers
They are many 'Pentagonal Fuzzy Numbers' ranking procedures. All the rankings listed here
have been developed for 'Pentagonal Fuzzy Numbers' over the last few years and collected from
the consideration of different research papers. Just the most important of them are listed here.
They're,
“ −
 Cut” for Pentagonal Fuzzy Numbers[26]:
 1 2 1 2
[ ( ), ( )], [0,0.5]}{[ ( ), ( )], [0.5,1]
P
N P P for Q Q for
      
 
=  
( ) ( )  
( ) ( )  
2 , 2 for 0,0.5
2 2 ,2 2 for 0.5,1
P
b a a e d e
N
c b b c d c d c

  
  
 
 − + − − + 
 
 
= 
− + − − − + 
 
 

“Robust ranking” method for Pentagonal Fuzzy Numbers[4]:
 
1
0
1
( ) ( ) , ( )
2
 
= − + − −

p
R N b a a e e d d
  
“Sub interval Average” method for Pentagonal Fuzzy Numbers[3]:
( )
( )
6
, , , ,
30
+ + + +
 
=  
 
a b c d e
R a b c d e
“Sub interval Addition” method for Pentagonal Fuzzy Numbers[1]:
6( )
( , , , , )
15
+ + + +
 
=  
 
a b c d e
R a b c d e

“Pascal Triangular Graded Mean” for Pentagonal Fuzzy Numbers[23]:
( )
4 6 4
, , , ,
16
+ + + +
 
=  
 
a b c d e
P a b c d e
“Magnitude Ranking” for Pentagonal Fuzzy Numbers[28]:
1
0
1
( ( ) ( ) ) , [0,1]
6
 
= + + + + 

p L U
Mag N A A b c d A for
    
“Centroid approach” for Pentagonal Fuzzy Numbers[6]:
( )
3 4 3 6 2 4
, , , , ;
18 9
+ + + +
   
= 
   
   
a b c d e w
R a b c d e w
6. HEXAGONAL FUZZY NUMBER
Definition. Let N
 
is a hexagonal fuzzy number denoted by ( )
, , , , ,
Hex
N a b c d e f
 
= , and its
membership function ( )
Hex
N
x
  is given as,
1
2
1 1
2 2
1
( )
1
1
2
1
2
0 .
Hex
N
x a
for a x b
b a
x b
for b x c
c b
for c x d
x
x d
- for d x e
e d
f x
for e x f
f e
otherwise
 
 −
 
 
 
 −
 

 −
 
+  
  
−
 

  

= 
−
 
  
 
 −
 

 
−
  
 
 −
 



Figure 4 Hexagonal Fuzzy Number

Lavanya P
6.1. Ranking Methods for Hexagonal Fuzzy Numbers
“Hexagonal Fuzzy Numbers” also have many types of ranking procedures. Over the past several
years, all ranking listed here have been found for 'Hexagonal Fuzzy Numbers' and collected by
reviewing different research papers. The most relevant of these have been here. They are the,
“ − Cut” for Hexagonal Fuzzy Numbers[14]:
( ) ( )  )
( ) ( )  )
2 , 2 ] 0,0.5
2 2 , 2 2 0.5,1
Hex
b a a f e f for
N
c b c b e d e d for

  
  
 
−
 − + − − + 


= 
− − + − − + − 
 
 

“Yeager’s ranking” method for Hexagonal Fuzzy Numbers[5]:
1
0
( ) (0.5) ( )
Hex U L
Y N A A d
 

 
= +

“Sub interval Average” method for Hexagonal Fuzzy Numbers[3]:
7( )
( )
42
Hex
a b c d e f
R N
  + + + + +
 
=  
 
“Sub interval Addition” method for Hexagonal Fuzzy Numbers[1]:
7( )
( )
21
Hex
a b c d e f
R N
  + + + + +
 
=  
 
“Pascal Triangular Graded Mean” for Hexagonal Fuzzy Numbers[21]:
5 10 10 5
( )
32
Hex
a b c d e f
P N
  + + + + +
 
=  
 
“Magnitude Ranking” for Hexagonal Fuzzy Numbers[19]:
2 3 4 4 3 2
18
Hex
a b c d e f
Mag N
  + + + + +
 
=  
 
“Centroid approach” for Hexagonal Fuzzy Numbers[18]:
2 3 4 4 3 2 ) 5
( )
18 18
Hex
a b c d e f w
R N
  + + + + +
   
= 
   
   
7. HEPTAGONAL FUZZY NUMBER
Definition. A Heptagonal fuzzy numbers ( )
, , , , , ,
Hep
N a b c d e f g
 
= , is a fuzzy number and its
membership function
( )
( )
( )
( )
2
1
2
1
2
( )
1
2
1
2
2
0
Hep
N
x a
a x b
b a
b x c
x d
c x d
d c
x d x
d x e
e d
e x f
g x
f x g
g f
otherwise
 
 
−
 
 
 
−
 

  

 
−
 +  
 
 
 −
 


=  
−
+  
 
 
−
 

  

 
−
  
 
 
 −
 




Figure 5 Heptagonal Fuzzy Number
7.1. Ranking Methods for Heptagonal Fuzzy Numbers
In so many types of rank procedures, there are many heptagonal fuzzy numbers. Many of the
rankings listed have been discovered for 'Heptagonal Fuzzy Numbers' over the past few years
and gathered by analyzing numerous research papers. Only the most significant of them are
listed here. They are,
“ − Cut” for Heptagonal Fuzzy Numbers[26]:
2 ( ), 2 ( ) [0,0.5]
2( 1)( ) , 2( 1)( ) [0.5,1]
Hep
a b a g g f
N
d c d d e d


  
  
 
−
+ − − − 

= 
− − + − − − 

“Robust ranking” method for Heptagonal Fuzzy Numbers[27]:
( )
1
0
1
( ) ,
2
L U
Hep Hep Hep
R N a a d
  
 
= 
“Sub interval Average” method for Heptagonal Fuzzy Numbers[3]:
8(
( )
56
Hep
a b c d e f g
R N
  + + + + + +
 
=  
 
“Sub interval Addition” method for Heptagonal Fuzzy Numbers[1]:
8(
( )
28
Hep
a b c d e f g
R N
  + + + + + +
 
=  
 
“Pascal Triangular Graded Mean” for Heptagonal Fuzzy Numbers[9]:
( ) 6 15 20 15 6
64
Hep
a b c d e f g
P N
  + + + + + +
 
=  
 
“Centroid approach” for Heptagonal Fuzzy Numbers[13]:
0 0
( ) ( , )
2 7 7 22 7 7 2 11
54 54
Hep
R N R x y
a b c d e f g w
 
=
+ + + + + +
   
= 
   
   

Lavanya P
8. CONCLUSION
We've considered fuzzy numbers throughout this article. We clarified quite well during this
survey, and thus, the various types of fuzzy numbers have developed in the previous few years
and their different types of fuzzy ranking systems. This survey paper has been compiled very
clearly for the possible purpose of developing fuzzy numbers and its ranking method for the
future numerical evaluation system.
REFERENCES
[1] D.S. Dinagar and S. Kamalanathan, “Sub interval addition method for ranking of fuzzy
numbers,”(International Journal of Pure and Applied Mathematics,2017), 115(9),pp. 159–169.
[2] D.S. Dinagar and S. Kamalanathan, “Solving fuzzy linear programming problem using new
ranking procedures of fuzzy numbers,” (International Journal of Applications of Fuzzy Sets and
Artificial Intelligence,2017), 7, pp. 281–292.
[3] S. Dinagar, R.Kamalanathan and N. Rameshan, “Sub Interval Average Method for Ranking of
Linear Fuzzy Numbers,” (International Journal of Pure and Applied Mathematics, 2017),
114(6), pp. 119–130.
[4] D.S. Dinagar and M.M. Jeyavuthin, “Distinct Methods for Solving Fully Fuzzy Linear
Programming Problems with Pentagonal Fuzzy Numbers,” (Journal of Computer and
Mathematical Sciences, 2019), 10(6), pp. 1253–1260.
[5] S. Dhanasekar, G. Kanimozhi and A. Manivannan, “Solving Fuzzy Assignment Problems with
Hexagonal Fuzzy Numbers by using Diagonal Optimal Algorithm,”(International Journal of
Innovative Technology and Exploring Engineering,2019), 9(1), pp. 54–57.
[6] K.P. Ghadle and P.A. Pathade, “Solving transportation problem with generalized hexagonal and
generalized octagonal fuzzy numbers by ranking method,” (Global Journal of Pure and Applied
Mathematics,2017), 13(9), pp. 6367–6376.
[7] R. Hassanzadeh, I. Mahdavi, N. Mahdavi-Amiri and A. Tajdin, “An α-cut approach for fuzzy
product and its use in computing solutions of fully fuzzy linear systems,” (International Journal
of Mathematics in Operational Research,2018), 12(2), pp.167–189.
[8] R. Jansi and K. Mohana, “A Heptagonal Fuzzy Number in Solving Fuzzy Sequencing
Problem,”(International Journal of Innovative Research in Technology,2018), 5(6), pp. 240–
243.
[9] B. Kalpana and N. Anusheela, “Analysis of Fm/Fm/I Fuzzy Priority Queues Based On New
Approach of Ranking Fuzzy Numbers Using Centroid of Centroids,” (International Journal of
Pure and Applied Mathematics,2018), 119(7), pp. 457–465.
[10] A.J. Kamble, “Some notes on Pentagonal fuzzy numbers”, (Int. J. Fuzzy Math Arch,2017), 13,
pp. 113–121.
[11] S.P. Mondal and M. Mandal,“Pentagonal fuzzy number, its properties and application in fuzzy
equation,” (Future Computing and Informatics Journal,2017), 2(2), pp. 110–117.
[12] N.I. Namarta, N. Thakur andU.C. Gupta, “Ranking of heptagonal fuzzy numbers using incentre
of centroids,” (Int. J. Adv. Technol. Eng. Sci., 2017), 5, pp. 248–255.
[13] M. Prabhavathi, A.Elavarasi, M. Leelavathi and K. Ajitha, “Some Operation on Hexagonal
Fuzzy Number,” (IJRAR-International Journal of Research and Analytical Reviews
(IJRAR),2020), 7(1), pp. 469–482.
[14] A. PraveenPrakash and M. GeethaLakshmi, “A Comparative Study-Optimal Path using Trident
and Sub-Trident Forms through Fuzzy Aggregation, Ranking and Distance Methods,”
(International Journal of Pure and Applied Mathematics,2016), 109(10), pp.139–150.

[15] Ponnialagan,Dhanasekaran, JeevarajSelvaraj and LakshmanaGomathiNayagamVelu, “A
complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision
making,” (Neural Computing and Applications 30, no. 11, 2018), pp. 3303–3315.
[16] P. Rajarajeswari,“Ranking of hexagonal fuzzy numbers for solving multi objective fuzzy linear
programming problem,”(International Journal of Computer Applications, 2013), 84(8), pp. 14–
19.
[17] P. Rajarajeswari and A.S. Sudha,“Ordering generalized hexagonal fuzzy numbers using rank,
mode, divergence and spread,” (IOSR Journal of Mathematics,2014), 10(3), pp. 15–22.
[18] C. Rajendran andM. Ananthanarayanan, “Fuzzy Criticalpath Method with Hexagonal and
Generalised Hexagonal Fuzzy Numbers Using Ranking Method,” (International Journal of
Applied Engineering Research, 2018), 13(15), pp. 11877–11882.
[19] M.S. Ramya and M.B.J. Presitha, “Solving an Unbalanced Fuzzy Transportation Problem using
a Heptagonal Fuzzy Numbers by Robust Ranking Method,” (Journal of Analysis and
Computation, 2019), XII(I), pp. 1–13.
[20] K. Selvakumari and G. Sowmiya, “Fuzzy network problems using job sequencing technique in
hexagonal fuzzy numbers,” (International Journal of Advance Engineering and Research
Development,2017), 4(9).
[21] P. Selvam, A. Rajkumar and J.S. Easwari, “Ranking of pentagonal fuzzy numbers applying
incentre of centroids,” (International Journal of Pure and Applied Mathematics,2017), 117(13),
165–174.
[22] K. Selvakumari and S. Santhi, “A Pentagonal Fuzzy Number Solving Fuzzy Sequencing
Problem,” (International Journal of Mathematics and its Application,2018), 6(2-B), pp. 207–
211.
[23] R. Srinivasan, G. Saveetha and T. Nakkeeran, “Comparative study of fuzzy assignment problem
with various ranking,” (Malaya Journal of Matematik (MJM), 2020), (1), pp. 431–434.
[24] R. Srinivasan, T. Nakkeeran and G. Saveetha, “Evaluation of fuzzy non-preemptive priority
queues in intuitionistic pentagonal fuzzy numbers using centroidal approach,” (Malaya Journal
of Matematik (MJM),2020), (1), pp. 427–430.
[25] A.M. Shapique, “Arithmetic Operations on Heptagonal Fuzzy Numbers,” (Asian Research
Journal of Mathematics,2017), pp. 1–25.
[26] A.S. Sudha and S. Karunambigai, “Solving a transportation problem using a Heptagonal fuzzy
number,” (International Journal of Advanced Research in Science, Engineering and
Technology, 2017), 4(1).
[27] SyedaNaziyaIkram and A. Rajkumar, “Pentagon Fuzzy Number and Its Application to Find
Fuzzy Critical Path,”(International Journal of Pure and Applied Mathematics, 2017), 114(5),
pp. 183–185.
[28] G. Uthra, K. Thangavelu and B. Amutha, “An Approach of Solving Fuzzy Assignment Problem
using Symmetric Triangular Fuzzy Number,” (International Journal of Pure and Applied
Mathematics,2017), 113(7), 16–24.
[29] D.U. Wutsq and N. Insani, “Yager’s ranking method for solving the trapezoidal fuzzy number
linear programming,”(IOP Publishing, In Jou2rnal of Physics: Conference Series, March, 2018),
Vol. 983, No. 1, p. 012135.

VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHES

VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHES

VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHES

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