- Email: [email protected]
Pentagonal fuzzy number, its properties and application in fuzzy equation
Accepted Manuscript Pentagonal fuzzy number its properties and application in fuzzy equation Sankar Prasad Mondal, Manimohan Mandal PII:
S2314-7288(17)30034-X
DOI:
10.1016/j.fcij.2017.09.001
Reference:
FCIJ 19
To appear in:
Future Computing and Informatics Journal
Received Date: 1 July 2017 Accepted Date: 14 September 2017
Please cite this article as: Mondal SP, Mandal M, Pentagonal fuzzy number its properties and application in fuzzy equation, Future Computing and Informatics Journal (2017), doi: 10.1016/ j.fcij.2017.09.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Pentagonal fuzzy number its properties and application in fuzzy equation Sankar Prasad Mondal1 and Manimohan Mandal2 1,2
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Corresponding author email: [email protected]
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Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore-721101, West Bengal, India
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Pentagonal fuzzy number its properties and application in fuzzy equation
Keywords: Pentagonal fuzzy number, fuzzy equation
1.1 Fuzzy sets and number
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1. Introduction:
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Abstract: The paper presents an adaptation of pentagonal fuzzy number. Different type of pentagonal fuzzy number is formed. The arithmetic operation of a particular type of pentagonal fuzzy number is addressed here. The difference between two pentagonal valued functions is also addressed here. Demonstration of pentagonal fuzzy solutions of fuzzy equation is carried out with the said numbers. Additionally, an illustrative example is also taken with the useful graph and table for usefulness for attained to the proposed concept.
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In 1965, Lotfi A. Zadeh [1], delibered new concept namely Fuzzy Sets theory. The theory of unsharp amounts has been applied with great success in many various fields. Chang and Zadeh [2] introduced the concept of fuzzy numbers. Differents mathematicians have been studying the theory (one-dimension or n-dimension fuzzy numbers, see for example [3,4]). With the various improvement of theories and applications of fuzzy sets theory the topic become a topic of great interest. 1.2 Pentagonal fuzzy number Many researcher take pentagonal fuzzy number with different types of membership function. In this subsection we study on some published work which is associated with pentagonal fuzzy number: Authors Information
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Panda and Pal [5]
Types of membership function Linear membership function of with symmetry
Anitha and Parvathi [6] Helen and Uma [7]
Linear membership function Linear membership function
Main contribution
Application Area
Define arithmetic operation and a exponent operation Find expected crisp value Find the parametric form of pentagonal fuzzy number
Fuzzy matrix theory
Inventory control problem Proof of all arithmetic operation using parametric form concept Find the ranking of pentagonal fuzzy number
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Linear membership and non membership function
Raj and Karthik [9] Dhanamand and Parimaldevi [10]
Linear membership function Linear membership function
Pathinathan and Ponnivalavan [11]
Reverse order linear membership function
Define all arithmetic operation Find the ranking of Intuitionistic fuzzy number Define all arithmetic operation Find the ranking of pentagonal fuzzy number using circumcenter of centroids and an index of modality Define arithmetic operation
Ponnivalavan and Pathinathan [12] Annie Christi and Kasthuri [13]
Linear membership and non membership function Linear membership and non membership function
Define arithmetic operation Define arithmetic operation and ranking
Application in network problem
Application in Neural network problem Apply in multi objective multi item inventory model
M AN U
SC
RI PT
Siji and Kumari [8]
Define different type of reverse order fuzzy number Find score and accuracy function Transportation problem
1.3 Motivation
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From the above literature survey we see that linear membership function with symmetry is only taken most of the cases. But what happen if we take non linear membership function or asymmetry on both ends or generalized case or their combinations ? Obviously the results are different. This article we propose to show all type of possibility.
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Fuzzy sets theory play an important role in uncertainty modeling. Now the question is if we wish to take a fuzzy number then how its geometrical representations are. What is its membership functions ? So if decision maker take a fuzzy number which can be graphically looks like a pentagon then how its membership function can be defined. From this point of view we try to define different type of pentagonal fuzzy number which can be a better choice of a decision maker in different situation.
1.4 Novelties
There is various articles where pentagonal fuzzy sets and number are introduced and apply to different fields. But there are so many scopes to work on that topic. We try to summarize the work done on pentagonal fuzzy number as follows: (i) Formation of different types of pentagonal fuzzy number in easier manner. i.e.,
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Symmetric linear pentagonal fuzzy number, asymmetric linear pentagonal fuzzy number, symmetric non linear pentagonal fuzzy number, asymmetric nonlinear pentagonal fuzzy number are defined.
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(ii) The parametric form of the said above types of number are defined. (iii) Arithmetic operation of symmetric linear fuzzy numbers is defined and how can we prove it is illustrated.
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(iii) The number is considered with equation i.e., pentagonal fuzzy equation are defined and solved.
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1.5 Structure of the paper
2. Preliminaries
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The paper is organized as follows. In Section 2, the basic concept on fuzzy number and fuzzy difference are discussed. In Section 3 we give a brief description on how we can choose a suitable membership function in different pentagonal form. In Section 4 we addressed some arithmetic operation on Linear pentagonal fuzzy number with symmetry. In Section 5, solution of fuzzy equation with pentagonal fuzzy number with example is discussed. The conclusions are written in Section 6.
Definition 2.1: Fuzzy Number: A fuzzy set , defined on the universal set of real number R , is said to be a fuzzy number if its possess at least the following properties:
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is convex. is normal i.e., ∃ ∈ such that = 1. is piecewise continuous. must be closed interval for every [0,1]. The support of , i.e., must be bounded.
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(i) (ii) (iii) (iv) (v)
Definition 2.2: Generalized Hukuhara difference: [14] The generalized Hukuhara difference of two fuzzy numbers , ∈ ℜℱ is defined as follows In terms of -cut set we have
⊖!" = # ⟺ %
= +# = + −1 #
ACCEPTED MANUSCRIPT [ ⊖!" ] = [min+, , - ., max+, , - .]
Where, , = , − - , - = - − , The conditions for which the existence of ⊖!" exists if
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Let 1 = ⊖!"
1, = , − - , 1- = - − , , 1,2 3 = ,2 3 − -2 3 and 1-2 3 = -2 3 − ,2 3 with 1, ,1-2 3 are increasing and 1- , 1,2 3 are decreasing function for all , 3 ∈ [0,1] and 1, ≤ 1- , 1-2 3 ≤ 1,2 3 .
3. Pentagonal fuzzy number and its variation:
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Remark 2.1: Throughout the paper, we assume that ⊖!" ∈ ℜℱ .
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In this section we develop different type of pentagonal fuzzy number in different viewpoint.
Definition 3.1: Pentagonal fuzzy number: A pentagonal fuzzy number = , , - , 5 , 6 , 7 should satisfy the following condition (1) is a continuous function in the interval [0,1]
(2) is strictly increasing and continuous function on [ , , - ] and [ - , 5 ]
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(3) is strictly decreasing and continuous function on [ 5 , 6 ] and [ 6 , 7 ] Definition 3.2: Equality of two Pentagonal fuzzy number: Two pentagonal fuzzy number = , , - , 5 , 6 , 7 and 8 = , , - , 5 , 6 , 7 are equal if
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, = , , - = - , 5 = 5 , 6 = 6 , 7 = 7
Now we try to define some new types of pentagonal fuzzy number in their different form.
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3.1 Linear pentagonal fuzzy number with symmetry Definition 3.3: Linear pentagonal fuzzy number with symmetry (LPFNS): A linear pentagonal fuzzy number is written as 9: = , , - , 5 , 6 , 7 ; whose membership function is written as
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− , BC , ≤ ≤ - A - − , ? ?1 − 1 − − - BC ≤ ≤ 5 ? 5 − ? 1 BC = 5 <= = − BC 5 ≤ ≤ 6 @ 1 − 1 − 6 6 − 5 ? ? 7 − ? BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
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Definition 3.4: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae = F ∈ G| <= I J
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A ,9 = , + - − , C ∈ [0, ] ? 1− ?-9 = - + 5 − - C ∈ [, 1] 1 − = @ = − 1 − − C ∈ [, 1] 6 5 1− 6 ? -K ?
> ,K = 7 − 7 − 6 C ∈ [0, ]
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 1: Linear pentagonal fuzzy number with symmetry Key point 3.1: The basic concept of the above number is the left picked point and right picked point are same (See Fig. 1 the picked point is ).
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3.2 Linear pentagonal fuzzy number with asymmetry Definition 3.5: Linear pentagonal fuzzy number with asymmetry: A linear pentagonal fuzzy number is written as 9 : = , , - , 5 , 6 , 7 ; , whose membership function is written as
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− , BC , ≤ ≤ - A - − , ? ?1 − 1 − − - BC ≤ ≤ 5 ? 5 − ? 1 BC = 5
Note: (1) If = the asymmetry pentagonal fuzzy number becomes symmetry pentagonal fuzzy number.
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(2) For asymmetry pentagonal fuzzy number may be < or D
Definition 3.6: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae
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= F ∈ G| <= I J
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A ,9 = , + - − , C ∈ [0, ] ? 1− ?-9 = - + − - C ∈ [, 1] 1− 5 = @ = − 1 − − C ∈ [, 1] 6 5 1− 6 ? -K ?
> ,K = 7 − 7 − 6 C ∈ [0, ]
Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 2: Linear pentagonal fuzzy number with asymmetry
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Key point 3.2: The basic concept of the above number is the left picked point and right picked point are not same (See Fig. 2 the left picked point is but right picked point is ).
3.3 Non Linear pentagonal fuzzy number with symmetry
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Definition 3.7: Non Linear pentagonal fuzzy number with symmetry: A linear pentagonal fuzzy number is written as 9N: = , , - , 5 , 6 , 7 ; OP,OQ ;RP,RQ whose membership function is written as
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− , OP T U BC , ≤ ≤ - A - − , ? − - OQ ? 1 1 − − T U BC - ≤ ≤ 5 ? 5 − ? 1 BC = 5
S= = R @ 1 − 1 − T 6 − U P BC 5 ≤ ≤ 6 6 − 5 ? ? 7 − RQ ? T U BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
Definition 3.8: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae
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OP = + V W - − , C ∈ [0, ] A ,9 , ? 1 − OQ ? ? -9 = - + T U 5 − - C ∈ [, 1] 1− = 1 − RP @ -K = 6 − T U 6 − 5 C ∈ [, 1] ? 1 − ? RQ ? = − V W 7 − 6 C ∈ [0, ] 7 > ,K
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 3: Non Linear pentagonal fuzzy number with symmetry
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Key point 3.3: The basic concept of the above number is the left picked point and right picked point are same but the boundary of the fuzzy area should not be linear always. It can be non linear also. That is the membership function can define as a non linear function. So we can give the non linearity on the membership function. (See Fig. 2 the left picked point is but right picked point is ).
3.4 Non Linear pentagonal fuzzy number with asymmetry Definition 3.9: Non Linear pentagonal fuzzy number with asymmetry: A linear pentagonal fuzzy number is written as N : = , , - , 5 , 6 , 7 ; , OP,OQ ;RP,RQ whose membership function is written as
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− , OP T U BC , ≤ ≤ - A - − , ? − - OQ ?
1 − 1 − T U BC - ≤ ≤ 5 ? 5 − ? 1 BC = 5 SL= = RP @ 1 − 1 − T 6 − U BC 5 ≤ ≤ 6 6 − 5 ? ? 7 − RQ ? T U BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
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Note: (1) If X, = X- = Y, = Y- = 1 then non linear pentagonal fuzzy number becomes linear pentagonal fuzzy number.
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Definition 3.10: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae = F ∈ G| <= I J
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OP A ,9 = , + V W - − , C ∈ [0, ] ? 1 − OQ ? ?-9 = - + T U 5 − - C ∈ [, 1] 1 − = 1 − RP @ = 6 − T U 6 − 5 C ∈ [, 1] ? -K 1− ? RQ ? = 7 − V W 7 − 6 C ∈ [0, ] ,K >
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 4: Non Linear pentagonal fuzzy number with asymmetry Key point 3.4: The previous concept is apply here only the basic concept of the number is the left picked point and right picked point are not same.
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4. Arithmetic operation on Linear pentagonal fuzzy number with symmetry i.e., [ \] = ^_ , ^` , ^a , ^b , ^c ; d Z (1) Multiplication by crisp number
If # is a positive crisp number then #9: = # , , # - , # 5 , # 6 , # 7 ; and # is a negative crisp number then #9: = # 7 , # 6 , # 5 , # - , # , ;
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Proof: We can proof by interval arithmetic on the parametric method
= [, , - ] = [,9 , -9 ; -K , ,K ]
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We can make the above four component ,9 , -9 , -K , ,K into two interval as , = [,9 , -9 ] = e , + - − , , - + [-K , ,K ] = e 6 −
,g ,gi
f
,g ,gf
5 − - h and - =
6 − 5 , 7 − 7 − 6 h i
Note 4.1: The concept on parametric form of a normal fuzzy number is that it behave like a interval number for fixed value of the parameter. For the pentagonal fuzzy number we have to take two interval together. Case 1: When j D 0
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Now if we multiply by positive crisp number then
#, = [#,9 , #-9 ] and #- = [#-K , #,K ]
So the resultant interval is # = [#, , #- ] = [#,9 , #-9 ; #-K , #,K ]
So # = e# , + f # - − # , , # - + ,gf # 5 − # - ; # 6 − ,gi # 6 − # 5 , # 7 − i
# 7 − # 6 h
,g
,g
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Case 2: When k < 0
Now if we multiply by positive crisp number then
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That means # is the -cut of #9: = # , , # - , # 5 , # 6 , # 7 ;
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#, = [#-9 , #,9 ] and #- = [#,K , #-K ]
So the resultant interval is # = [#- , #, ] = [#,K , #-K ; #-9 , #,9 ] So # = e# 7 − i # 7 − # 6 , # 6 − ,gi # 6 − # 5 ; # - + ,gf # 5 − # - , # , + f
# - − # , h
,g
,g
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That means # is the -cut of #9: = # 7 , # 6 , # 5 , # - , # , ;
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If # is a negative crisp number then #9: = # 7 , # 6 , # 5 , # - , # , ; (2) Addition of two pentagonal fuzzy numbers:
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Consider two pentagonal fuzzy numbers 9: = , , - , 5 , 6 , 7 ; , and 8 9: = , , - , 5 , 6 , 7 ; - then the addition of the two numbers is given by l9: = , + , , - + - , 5 + 5 , 6 + 6 , 7 + 7 ;
Where = min+, , - .
(3) Subtraction of two pentagonal fuzzy numbers:
Consider two pentagonal fuzzy numbers 9: = , , - , 5 , 6 , 7 ; , and 8 9: = , , - , 5 , 6 , 7 ; - then the addition of the two numbers is given by
ACCEPTED MANUSCRIPT [9: = , − 7 , - − 6 , 5 − 5 , 6 − - , 7 − , ; m
Where = min+, , - .
5. Solution of fuzzy equation with pentagonal fuzzy number Consider the linear equation with linear pentagonal symmetric fuzzy number as
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n9: + 9: = õ9:
Solution: Taking -cut of the equation we get
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Note 5.1: The above equation is a linear pentagonal symmetric fuzzy equation since the coefficients and parameters are all linear pentagonal symmetric fuzzy number. If any one of the parameter are linear pentagonal symmetric fuzzy number then it is also known as linear pentagonal symmetric fuzzy equation.
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[ ,9 , -9 ; -K , ,K ]. [,9 , -9 ; -K , ,K ] + [ ,9 , -9 ; -K , ,K ] = [o,9 , o-9 ; o-K , o,K ]
Now the problem is converted to some interval equation. We have now follow the following statregy
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min+[ ,9 , -9 ]. [,9 , -9 ]. + ,9 = o,9
max+[ ,9 , -9 ]. [,9 , -9 ]. + -9 = o-9
min+[ -K , ,K ]. [-K , ,K ]. + -K = o-K
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max+[ -K , ,K ]. [-K , ,K ]. + ,K = o,K
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If we consider nrOsr is positive interval valued intuitionistic fuzzy number then using the concept of generalized characterization theorem we can write ,9 ,9 + ,9 = o,9
-9 -9 + -9 = o-9
-K -K + -K = o-K ,K ,K + ,K = o,K
Whose solution can be written as
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-9 = -K = ,K =
o,9 − ,9 ,9
o-9 − -9 -9
o-K − -K -K o,K − ,K ,K
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,9 =
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Note 5.2: Clearly [,9 , -9 ; -K , ,K ] is the -cut of the problems, but it is necessary to check whether the all component of the solution is maintains the pentagonal fuzzy rules or not. The solution is strong solution if
t,9 t-9 , D0 t t
i.e., ,9 , -9 are increasing function with respect to and
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t-K t,K , <0 t t
i.e., ,9 , -9 are decreasing function with respect to
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Numerical example 5.1: Consider the pentagonal fuzzy equation + 9: = õ9:
Where = 4 , 9: = 4,8,10,14,16; 0.6 and õ9: = 30,50,60,80,90; 0.6
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Solution: The -cut of the solution is like
n9: = [,9 , -9 ; -K , ,K ]
Where, ,9 = V26 + ,K = 6 V74 − ,
E 0
,
6 6 5
W
~_\ E 6.5000
| 5
W, -9 = 62 − 20 , -K = 26 + 40 and ,
,
6
~`\ E
6
E 0
~` E
~_ E 18.5000
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10.5000 11.0000 11.5000 12.0000 12.5000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
16.5000 15.5000 14.5000 13.5000 12.5000
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Table 7.1: Solution for different value of
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18.1667 17.8333 17.5000 17.1667 16.8333 16.5000
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7.1667 7.8333 8.5000 9.1667 9.8333 10.5000
Figure 7.1: Graph of solution
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ACCEPTED MANUSCRIPT Remarks 5.3: From the above graph and table we see that,9 is increasing for ∈ [0,0.6], -9 is increasing for ∈ [0.6,1], -K is decreasing for ∈ [0.6,1] and ,K is decreasing for ∈ [0,0.6]. Hence the solution is a strong solution.
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6. Conclusion
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In this paper the concept on different type of pentagonal fuzzy number is defined. The said number valued function is extended to its generalized Hukuhara difference concept, where it is applied to elucidate the pentagonal fuzzy solutions of the equation. Arithmetic operations of a particular pentagonal fuzzy number are also addressed. Further a numerical example is illustrated with pentagonal fuzzy number with fuzzy equation. Mainly the whole work reaches on the following conclusion: Demonstrating different type pentagonal fuzzy numbers enabled to meet the imprecise parameters as well, which is approvingly the advantageous for the decision makers to analyze the result in a more precise manner.
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By different situation the decision maker can take different type of pentagonal fuzzy number as per the problem definition.
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Acknowledgement
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Thus in future we are interested to use these concepts to find the solution of different problem with different type of pentagonal fuzzy numbers and we can apply this in various fields of engineering and sciences.
References
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The first author of the article wishes to convey his heartiest thanks to Miss. Gullu for inspiring him to write the article.
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[1] L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353. [2] S.S.L. Chang, L.A. Zadeh, On fuzzy mappings and control, IEEE Trans. Syst. Man Cyberne 2 (1972) 30-34. [3] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets,World Scientific, Singapore, 1994. [4] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978) 613-626. [5] Apurba Panda and Madhumangal Pal, A study on pentagonal fuzzy number and its corresponding matrices, Pacific Science Review B: Humanities and Social Sciences 1 (2015) 131-139. [6] P. Anitha and P. Parvathi , An Inventory Model with Stock Dependent Demand, two parameter Weibull Distribution Deterioration in a juzzy environment, 2016 Online International Conference on Green Engineering and Technologies (IC-GET), pp 1-8.
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[7] R.Helen and G.Uma, A new operation and ranking on pentagon fuzzy numbers, Int Jr. of Mathematical Sciences & Applications, Vol. 5, No. 2, 2015, pp 341-346. [8] S. Siji and K. Selva Kumari, An Approach for Solving Network Problem with Pentagonal
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Intuitionistic Fuzzy Numbers Using Ranking Technique, Middle-East Journal of Scientific Research 24 (9): 29772980, 2016. [9] A.Vigin Raj and S.Karthik, Application of Pentagonal Fuzzy Number in Neural Network, International Journal of Mathematics And its Applications, Volume 4, Issue 4 (2016), 149-154.
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[10] K. Dhanamand and M. Parimaldevi, Cost analysis on a probabilistic multi objective-multi item inventory model using pentagonal fuzzy number, The Global Journal of Applied Mathematics & Mathematical Sciences, Vol. 9, No. 2, 2016, pp 151-163. [11] T.Pathinathan and K.Ponnivalavan, Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numbers, Annals of Pure and Applied Mathematics, Vol. 9, No. 1, 2015, 107-117.
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[12] K. Ponnivalavan and T.Pathinathan, Intuitionistic pentagonal fuzzy number, ARPN Journal of Engineering and Applied Sciences, VOL. 10, NO. 12, 2015, pp 5446-5450. [13] M.S.Annie Christi and B.Kasthuri, Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Using Ranking Technique and Russell’s Method, Int. Journal of Engineering Research and Applications, Vol. 6, Issue 2, 2016, pp 82-86.
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[14] Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. (in press).
S2314-7288(17)30034-X
DOI:
10.1016/j.fcij.2017.09.001
Reference:
FCIJ 19
To appear in:
Future Computing and Informatics Journal
Received Date: 1 July 2017 Accepted Date: 14 September 2017
Please cite this article as: Mondal SP, Mandal M, Pentagonal fuzzy number its properties and application in fuzzy equation, Future Computing and Informatics Journal (2017), doi: 10.1016/ j.fcij.2017.09.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Pentagonal fuzzy number its properties and application in fuzzy equation Sankar Prasad Mondal1 and Manimohan Mandal2 1,2
AC C
EP
TE D
M AN U
SC
Corresponding author email: [email protected]
RI PT
Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore-721101, West Bengal, India
ACCEPTED MANUSCRIPT
Pentagonal fuzzy number its properties and application in fuzzy equation
Keywords: Pentagonal fuzzy number, fuzzy equation
1.1 Fuzzy sets and number
M AN U
1. Introduction:
SC
RI PT
Abstract: The paper presents an adaptation of pentagonal fuzzy number. Different type of pentagonal fuzzy number is formed. The arithmetic operation of a particular type of pentagonal fuzzy number is addressed here. The difference between two pentagonal valued functions is also addressed here. Demonstration of pentagonal fuzzy solutions of fuzzy equation is carried out with the said numbers. Additionally, an illustrative example is also taken with the useful graph and table for usefulness for attained to the proposed concept.
EP
TE D
In 1965, Lotfi A. Zadeh [1], delibered new concept namely Fuzzy Sets theory. The theory of unsharp amounts has been applied with great success in many various fields. Chang and Zadeh [2] introduced the concept of fuzzy numbers. Differents mathematicians have been studying the theory (one-dimension or n-dimension fuzzy numbers, see for example [3,4]). With the various improvement of theories and applications of fuzzy sets theory the topic become a topic of great interest. 1.2 Pentagonal fuzzy number Many researcher take pentagonal fuzzy number with different types of membership function. In this subsection we study on some published work which is associated with pentagonal fuzzy number: Authors Information
AC C
Panda and Pal [5]
Types of membership function Linear membership function of with symmetry
Anitha and Parvathi [6] Helen and Uma [7]
Linear membership function Linear membership function
Main contribution
Application Area
Define arithmetic operation and a exponent operation Find expected crisp value Find the parametric form of pentagonal fuzzy number
Fuzzy matrix theory
Inventory control problem Proof of all arithmetic operation using parametric form concept Find the ranking of pentagonal fuzzy number
ACCEPTED MANUSCRIPT
Linear membership and non membership function
Raj and Karthik [9] Dhanamand and Parimaldevi [10]
Linear membership function Linear membership function
Pathinathan and Ponnivalavan [11]
Reverse order linear membership function
Define all arithmetic operation Find the ranking of Intuitionistic fuzzy number Define all arithmetic operation Find the ranking of pentagonal fuzzy number using circumcenter of centroids and an index of modality Define arithmetic operation
Ponnivalavan and Pathinathan [12] Annie Christi and Kasthuri [13]
Linear membership and non membership function Linear membership and non membership function
Define arithmetic operation Define arithmetic operation and ranking
Application in network problem
Application in Neural network problem Apply in multi objective multi item inventory model
M AN U
SC
RI PT
Siji and Kumari [8]
Define different type of reverse order fuzzy number Find score and accuracy function Transportation problem
1.3 Motivation
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From the above literature survey we see that linear membership function with symmetry is only taken most of the cases. But what happen if we take non linear membership function or asymmetry on both ends or generalized case or their combinations ? Obviously the results are different. This article we propose to show all type of possibility.
AC C
EP
Fuzzy sets theory play an important role in uncertainty modeling. Now the question is if we wish to take a fuzzy number then how its geometrical representations are. What is its membership functions ? So if decision maker take a fuzzy number which can be graphically looks like a pentagon then how its membership function can be defined. From this point of view we try to define different type of pentagonal fuzzy number which can be a better choice of a decision maker in different situation.
1.4 Novelties
There is various articles where pentagonal fuzzy sets and number are introduced and apply to different fields. But there are so many scopes to work on that topic. We try to summarize the work done on pentagonal fuzzy number as follows: (i) Formation of different types of pentagonal fuzzy number in easier manner. i.e.,
ACCEPTED MANUSCRIPT
Symmetric linear pentagonal fuzzy number, asymmetric linear pentagonal fuzzy number, symmetric non linear pentagonal fuzzy number, asymmetric nonlinear pentagonal fuzzy number are defined.
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(ii) The parametric form of the said above types of number are defined. (iii) Arithmetic operation of symmetric linear fuzzy numbers is defined and how can we prove it is illustrated.
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(iii) The number is considered with equation i.e., pentagonal fuzzy equation are defined and solved.
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1.5 Structure of the paper
2. Preliminaries
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The paper is organized as follows. In Section 2, the basic concept on fuzzy number and fuzzy difference are discussed. In Section 3 we give a brief description on how we can choose a suitable membership function in different pentagonal form. In Section 4 we addressed some arithmetic operation on Linear pentagonal fuzzy number with symmetry. In Section 5, solution of fuzzy equation with pentagonal fuzzy number with example is discussed. The conclusions are written in Section 6.
Definition 2.1: Fuzzy Number: A fuzzy set , defined on the universal set of real number R , is said to be a fuzzy number if its possess at least the following properties:
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is convex. is normal i.e., ∃ ∈ such that = 1. is piecewise continuous. must be closed interval for every [0,1]. The support of , i.e., must be bounded.
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(i) (ii) (iii) (iv) (v)
Definition 2.2: Generalized Hukuhara difference: [14] The generalized Hukuhara difference of two fuzzy numbers , ∈ ℜℱ is defined as follows In terms of -cut set we have
⊖!" = # ⟺ %
= +# = + −1 #
ACCEPTED MANUSCRIPT [ ⊖!" ] = [min+, , - ., max+, , - .]
Where, , = , − - , - = - − , The conditions for which the existence of ⊖!" exists if
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Let 1 = ⊖!"
1, = , − - , 1- = - − , , 1,2 3 = ,2 3 − -2 3 and 1-2 3 = -2 3 − ,2 3 with 1, ,1-2 3 are increasing and 1- , 1,2 3 are decreasing function for all , 3 ∈ [0,1] and 1, ≤ 1- , 1-2 3 ≤ 1,2 3 .
3. Pentagonal fuzzy number and its variation:
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Remark 2.1: Throughout the paper, we assume that ⊖!" ∈ ℜℱ .
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In this section we develop different type of pentagonal fuzzy number in different viewpoint.
Definition 3.1: Pentagonal fuzzy number: A pentagonal fuzzy number = , , - , 5 , 6 , 7 should satisfy the following condition (1) is a continuous function in the interval [0,1]
(2) is strictly increasing and continuous function on [ , , - ] and [ - , 5 ]
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(3) is strictly decreasing and continuous function on [ 5 , 6 ] and [ 6 , 7 ] Definition 3.2: Equality of two Pentagonal fuzzy number: Two pentagonal fuzzy number = , , - , 5 , 6 , 7 and 8 = , , - , 5 , 6 , 7 are equal if
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, = , , - = - , 5 = 5 , 6 = 6 , 7 = 7
Now we try to define some new types of pentagonal fuzzy number in their different form.
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3.1 Linear pentagonal fuzzy number with symmetry Definition 3.3: Linear pentagonal fuzzy number with symmetry (LPFNS): A linear pentagonal fuzzy number is written as 9: = , , - , 5 , 6 , 7 ; whose membership function is written as
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− , BC , ≤ ≤ - A - − , ? ?1 − 1 − − - BC ≤ ≤ 5 ? 5 − ? 1 BC = 5 <= = − BC 5 ≤ ≤ 6 @ 1 − 1 − 6 6 − 5 ? ? 7 − ? BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
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Definition 3.4: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae = F ∈ G| <= I J
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A ,9 = , + - − , C ∈ [0, ] ? 1− ?-9 = - + 5 − - C ∈ [, 1] 1 − = @ = − 1 − − C ∈ [, 1] 6 5 1− 6 ? -K ?
> ,K = 7 − 7 − 6 C ∈ [0, ]
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 1: Linear pentagonal fuzzy number with symmetry Key point 3.1: The basic concept of the above number is the left picked point and right picked point are same (See Fig. 1 the picked point is ).
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3.2 Linear pentagonal fuzzy number with asymmetry Definition 3.5: Linear pentagonal fuzzy number with asymmetry: A linear pentagonal fuzzy number is written as 9 : = , , - , 5 , 6 , 7 ; , whose membership function is written as
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− , BC , ≤ ≤ - A - − , ? ?1 − 1 − − - BC ≤ ≤ 5 ? 5 − ? 1 BC = 5
Note: (1) If = the asymmetry pentagonal fuzzy number becomes symmetry pentagonal fuzzy number.
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(2) For asymmetry pentagonal fuzzy number may be < or D
Definition 3.6: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae
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= F ∈ G| <= I J
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A ,9 = , + - − , C ∈ [0, ] ? 1− ?-9 = - + − - C ∈ [, 1] 1− 5 = @ = − 1 − − C ∈ [, 1] 6 5 1− 6 ? -K ?
> ,K = 7 − 7 − 6 C ∈ [0, ]
Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 2: Linear pentagonal fuzzy number with asymmetry
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Key point 3.2: The basic concept of the above number is the left picked point and right picked point are not same (See Fig. 2 the left picked point is but right picked point is ).
3.3 Non Linear pentagonal fuzzy number with symmetry
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Definition 3.7: Non Linear pentagonal fuzzy number with symmetry: A linear pentagonal fuzzy number is written as 9N: = , , - , 5 , 6 , 7 ; OP,OQ ;RP,RQ whose membership function is written as
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− , OP T U BC , ≤ ≤ - A - − , ? − - OQ ? 1 1 − − T U BC - ≤ ≤ 5 ? 5 − ? 1 BC = 5
S= = R @ 1 − 1 − T 6 − U P BC 5 ≤ ≤ 6 6 − 5 ? ? 7 − RQ ? T U BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
Definition 3.8: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae
ACCEPTED MANUSCRIPT = F ∈ G| <= I J
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OP = + V W - − , C ∈ [0, ] A ,9 , ? 1 − OQ ? ? -9 = - + T U 5 − - C ∈ [, 1] 1− = 1 − RP @ -K = 6 − T U 6 − 5 C ∈ [, 1] ? 1 − ? RQ ? = − V W 7 − 6 C ∈ [0, ] 7 > ,K
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 3: Non Linear pentagonal fuzzy number with symmetry
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Key point 3.3: The basic concept of the above number is the left picked point and right picked point are same but the boundary of the fuzzy area should not be linear always. It can be non linear also. That is the membership function can define as a non linear function. So we can give the non linearity on the membership function. (See Fig. 2 the left picked point is but right picked point is ).
3.4 Non Linear pentagonal fuzzy number with asymmetry Definition 3.9: Non Linear pentagonal fuzzy number with asymmetry: A linear pentagonal fuzzy number is written as N : = , , - , 5 , 6 , 7 ; , OP,OQ ;RP,RQ whose membership function is written as
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− , OP T U BC , ≤ ≤ - A - − , ? − - OQ ?
1 − 1 − T U BC - ≤ ≤ 5 ? 5 − ? 1 BC = 5 SL= = RP @ 1 − 1 − T 6 − U BC 5 ≤ ≤ 6 6 − 5 ? ? 7 − RQ ? T U BC 6 ≤ ≤ 7 7 − 6 ? 0 BC D 7 >
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Note: (1) If X, = X- = Y, = Y- = 1 then non linear pentagonal fuzzy number becomes linear pentagonal fuzzy number.
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Definition 3.10: E-cut or parametric form of LPFNS: -cut or parametric form of LPFNS is represented by the formulae = F ∈ G| <= I J
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OP A ,9 = , + V W - − , C ∈ [0, ] ? 1 − OQ ? ?-9 = - + T U 5 − - C ∈ [, 1] 1 − = 1 − RP @ = 6 − T U 6 − 5 C ∈ [, 1] ? -K 1− ? RQ ? = 7 − V W 7 − 6 C ∈ [0, ] ,K >
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Where ,9 , -9 is increasing function with respect to and -K , ,K is decreasing function with respect to
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Figure 4: Non Linear pentagonal fuzzy number with asymmetry Key point 3.4: The previous concept is apply here only the basic concept of the number is the left picked point and right picked point are not same.
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4. Arithmetic operation on Linear pentagonal fuzzy number with symmetry i.e., [ \] = ^_ , ^` , ^a , ^b , ^c ; d Z (1) Multiplication by crisp number
If # is a positive crisp number then #9: = # , , # - , # 5 , # 6 , # 7 ; and # is a negative crisp number then #9: = # 7 , # 6 , # 5 , # - , # , ;
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Proof: We can proof by interval arithmetic on the parametric method
= [, , - ] = [,9 , -9 ; -K , ,K ]
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We can make the above four component ,9 , -9 , -K , ,K into two interval as , = [,9 , -9 ] = e , + - − , , - + [-K , ,K ] = e 6 −
,g ,gi
f
,g ,gf
5 − - h and - =
6 − 5 , 7 − 7 − 6 h i
Note 4.1: The concept on parametric form of a normal fuzzy number is that it behave like a interval number for fixed value of the parameter. For the pentagonal fuzzy number we have to take two interval together. Case 1: When j D 0
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Now if we multiply by positive crisp number then
#, = [#,9 , #-9 ] and #- = [#-K , #,K ]
So the resultant interval is # = [#, , #- ] = [#,9 , #-9 ; #-K , #,K ]
So # = e# , + f # - − # , , # - + ,gf # 5 − # - ; # 6 − ,gi # 6 − # 5 , # 7 − i
# 7 − # 6 h
,g
,g
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Case 2: When k < 0
Now if we multiply by positive crisp number then
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That means # is the -cut of #9: = # , , # - , # 5 , # 6 , # 7 ;
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#, = [#-9 , #,9 ] and #- = [#,K , #-K ]
So the resultant interval is # = [#- , #, ] = [#,K , #-K ; #-9 , #,9 ] So # = e# 7 − i # 7 − # 6 , # 6 − ,gi # 6 − # 5 ; # - + ,gf # 5 − # - , # , + f
# - − # , h
,g
,g
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That means # is the -cut of #9: = # 7 , # 6 , # 5 , # - , # , ;
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If # is a negative crisp number then #9: = # 7 , # 6 , # 5 , # - , # , ; (2) Addition of two pentagonal fuzzy numbers:
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Consider two pentagonal fuzzy numbers 9: = , , - , 5 , 6 , 7 ; , and 8 9: = , , - , 5 , 6 , 7 ; - then the addition of the two numbers is given by l9: = , + , , - + - , 5 + 5 , 6 + 6 , 7 + 7 ;
Where = min+, , - .
(3) Subtraction of two pentagonal fuzzy numbers:
Consider two pentagonal fuzzy numbers 9: = , , - , 5 , 6 , 7 ; , and 8 9: = , , - , 5 , 6 , 7 ; - then the addition of the two numbers is given by
ACCEPTED MANUSCRIPT [9: = , − 7 , - − 6 , 5 − 5 , 6 − - , 7 − , ; m
Where = min+, , - .
5. Solution of fuzzy equation with pentagonal fuzzy number Consider the linear equation with linear pentagonal symmetric fuzzy number as
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n9: + 9: = õ9:
Solution: Taking -cut of the equation we get
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Note 5.1: The above equation is a linear pentagonal symmetric fuzzy equation since the coefficients and parameters are all linear pentagonal symmetric fuzzy number. If any one of the parameter are linear pentagonal symmetric fuzzy number then it is also known as linear pentagonal symmetric fuzzy equation.
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[ ,9 , -9 ; -K , ,K ]. [,9 , -9 ; -K , ,K ] + [ ,9 , -9 ; -K , ,K ] = [o,9 , o-9 ; o-K , o,K ]
Now the problem is converted to some interval equation. We have now follow the following statregy
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min+[ ,9 , -9 ]. [,9 , -9 ]. + ,9 = o,9
max+[ ,9 , -9 ]. [,9 , -9 ]. + -9 = o-9
min+[ -K , ,K ]. [-K , ,K ]. + -K = o-K
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max+[ -K , ,K ]. [-K , ,K ]. + ,K = o,K
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If we consider nrOsr is positive interval valued intuitionistic fuzzy number then using the concept of generalized characterization theorem we can write ,9 ,9 + ,9 = o,9
-9 -9 + -9 = o-9
-K -K + -K = o-K ,K ,K + ,K = o,K
Whose solution can be written as
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-9 = -K = ,K =
o,9 − ,9 ,9
o-9 − -9 -9
o-K − -K -K o,K − ,K ,K
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,9 =
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Note 5.2: Clearly [,9 , -9 ; -K , ,K ] is the -cut of the problems, but it is necessary to check whether the all component of the solution is maintains the pentagonal fuzzy rules or not. The solution is strong solution if
t,9 t-9 , D0 t t
i.e., ,9 , -9 are increasing function with respect to and
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t-K t,K , <0 t t
i.e., ,9 , -9 are decreasing function with respect to
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Numerical example 5.1: Consider the pentagonal fuzzy equation + 9: = õ9:
Where = 4 , 9: = 4,8,10,14,16; 0.6 and õ9: = 30,50,60,80,90; 0.6
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Solution: The -cut of the solution is like
n9: = [,9 , -9 ; -K , ,K ]
Where, ,9 = V26 + ,K = 6 V74 − ,
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,
6 6 5
W
~_\ E 6.5000
| 5
W, -9 = 62 − 20 , -K = 26 + 40 and ,
,
6
~`\ E
6
E 0
~` E
~_ E 18.5000
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10.5000 11.0000 11.5000 12.0000 12.5000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
16.5000 15.5000 14.5000 13.5000 12.5000
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Table 7.1: Solution for different value of
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18.1667 17.8333 17.5000 17.1667 16.8333 16.5000
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7.1667 7.8333 8.5000 9.1667 9.8333 10.5000
Figure 7.1: Graph of solution
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ACCEPTED MANUSCRIPT Remarks 5.3: From the above graph and table we see that,9 is increasing for ∈ [0,0.6], -9 is increasing for ∈ [0.6,1], -K is decreasing for ∈ [0.6,1] and ,K is decreasing for ∈ [0,0.6]. Hence the solution is a strong solution.
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6. Conclusion
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In this paper the concept on different type of pentagonal fuzzy number is defined. The said number valued function is extended to its generalized Hukuhara difference concept, where it is applied to elucidate the pentagonal fuzzy solutions of the equation. Arithmetic operations of a particular pentagonal fuzzy number are also addressed. Further a numerical example is illustrated with pentagonal fuzzy number with fuzzy equation. Mainly the whole work reaches on the following conclusion: Demonstrating different type pentagonal fuzzy numbers enabled to meet the imprecise parameters as well, which is approvingly the advantageous for the decision makers to analyze the result in a more precise manner.
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By different situation the decision maker can take different type of pentagonal fuzzy number as per the problem definition.
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Acknowledgement
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Thus in future we are interested to use these concepts to find the solution of different problem with different type of pentagonal fuzzy numbers and we can apply this in various fields of engineering and sciences.
References
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The first author of the article wishes to convey his heartiest thanks to Miss. Gullu for inspiring him to write the article.
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[1] L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353. [2] S.S.L. Chang, L.A. Zadeh, On fuzzy mappings and control, IEEE Trans. Syst. Man Cyberne 2 (1972) 30-34. [3] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets,World Scientific, Singapore, 1994. [4] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978) 613-626. [5] Apurba Panda and Madhumangal Pal, A study on pentagonal fuzzy number and its corresponding matrices, Pacific Science Review B: Humanities and Social Sciences 1 (2015) 131-139. [6] P. Anitha and P. Parvathi , An Inventory Model with Stock Dependent Demand, two parameter Weibull Distribution Deterioration in a juzzy environment, 2016 Online International Conference on Green Engineering and Technologies (IC-GET), pp 1-8.
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[7] R.Helen and G.Uma, A new operation and ranking on pentagon fuzzy numbers, Int Jr. of Mathematical Sciences & Applications, Vol. 5, No. 2, 2015, pp 341-346. [8] S. Siji and K. Selva Kumari, An Approach for Solving Network Problem with Pentagonal
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Intuitionistic Fuzzy Numbers Using Ranking Technique, Middle-East Journal of Scientific Research 24 (9): 29772980, 2016. [9] A.Vigin Raj and S.Karthik, Application of Pentagonal Fuzzy Number in Neural Network, International Journal of Mathematics And its Applications, Volume 4, Issue 4 (2016), 149-154.
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[10] K. Dhanamand and M. Parimaldevi, Cost analysis on a probabilistic multi objective-multi item inventory model using pentagonal fuzzy number, The Global Journal of Applied Mathematics & Mathematical Sciences, Vol. 9, No. 2, 2016, pp 151-163. [11] T.Pathinathan and K.Ponnivalavan, Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numbers, Annals of Pure and Applied Mathematics, Vol. 9, No. 1, 2015, 107-117.
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[12] K. Ponnivalavan and T.Pathinathan, Intuitionistic pentagonal fuzzy number, ARPN Journal of Engineering and Applied Sciences, VOL. 10, NO. 12, 2015, pp 5446-5450. [13] M.S.Annie Christi and B.Kasthuri, Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Using Ranking Technique and Russell’s Method, Int. Journal of Engineering Research and Applications, Vol. 6, Issue 2, 2016, pp 82-86.
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[14] Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. (in press).