Interval-Valued Pentagonal Fuzzy Numbers

International Journal of Pure and Applied Mathematics Volume 119 No. 9 2018, 177-187 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue

Interval-Valued Pentagonal Fuzzy Numbers Pathinathan T1 . and Ajay Minj2 1 P.G. Research Department of Mathematics, Loyola College, Chennai - 34, Tamil Nadu, India [email protected] 2 P.G. Research Department of Mathematics, Loyola College, Chennai - 34, Tamil Nadu, India [email protected] Abstract We define interval-valued pentagonal fuzzy number. Geometric representation is illustrated and some algebraic operations are defined. We establish a relation between the set of pentagonal intuitionistic fuzzy numbers and a subset of interval-valued pentagonal fuzzy numbers. We propose a direction for the application of this relation and intervalvalued pentagonal fuzzy number to multicriteria decision making.

AMS Subject Classification: 03E72 Keywords: interval-valued trapezoidal fuzzy number, intuitionis-

tic trapezoidal fuzzy number, interval-valued pentagonal fuzzy number, intuitionistic pentagonal fuzzy number.

1

Introduction

The concept of fuzzy number is one of the most used fuzzy tools in various applications. Fuzzy numbers are used to model imprecise and approximate quantities such as about five, close to five, below five, more or less five, nearly five and the like in a practical manner. It is very useful in representing and capturing the subjectivity and vagueness prevalent in

1

177

ijpam.eu

International Journal of Pure and Applied Mathematics

humanistic system. Fuzzy numbers have been extended to intuitionistic fuzzy numbers to express membership and non-membership in relation to a fuzzy set. It is also extended to interval-valued fuzzy numbers when 0 experts opinions are in the form of intervals. Different fuzzy numbers and interval-valued fuzzy numbers have been used depending on the nature of impreciseness and problems in various applications. Pentagonal fuzzy number for the first time was used by Raj Kumar and Pathinathan T., [9, 10] to study the fuzzy term poverty and thereby sieve out the poor in a more meaningful way. Comprehensive understanding of poverty is five dimensional i.e. Roti, Kapda, Makaan, Kaam and Samman. In cases when the experts give their opinions in intervals, we need to accommodate their opinions without generalizing or diluting. In this paper, we introduce interval-valued pentagonal fuzzy number and establish the relationship between pentagonal intuitionistic fuzzy number and interval-valued pentagonal fuzzy number. The paper is organized in the following way. In second section we present survey of literature. In the third section preliminary definitions are presented. We define interval-valued pentagonal fuzzy numbers in the fourth section with geometrical representation and some algebraic operations. In the fifth section we establish relation between intuitionistic pentagonal fuzzy numbers and interval-valued pentagonal fuzzy numbers. In the sixth section we propose direction as to how we can use interval-valued pentagonal fuzzy numbers in the decision making.

2

Fuzzy Numbers: Literature Survey

Lotfi Zadeh introduced fuzzy sets in 1965 [17]. It has been extended to Type-2 fuzzy sets [12], Type-n fuzzy sets [15], Intuitionistic fuzzy sets [3, 4], and Hesitant fuzzy sets [15]. Fuzzy number is basically a fuzzy set with properties normality, convexity and piece-wise continuity. The concept of fuzzy numbers and fuzzy arithmetic were introduced by Zadeh in the year 1975 [18, 19]. Trapezoidal fuzzy numbers are a popular form of fuzzy numbers [8]. Interval numbers and triangular fuzzy number [8] are the special cases of trapezoidal fuzzy number. Based on 0 Atanassov s Intuitionistic fuzzy sets, Bustine, H., and Burillo, P., (1994) proposed the definition of intuitionistic fuzzy number [5]. This idea was used to introduce intuitionistic triangular fuzzy numbers and intuitionistic trapezoidal fuzzy numbers [1, 11]. In 2015, Ponnivalavan, K., and Pathinathan, T., [14] introduced intuitionistic pentagonal fuzzy number. Based on generalized trapezoidal fuzzy number Chen and Chen [6]

2

178

Special Issue


presented the concept of generalized interval-valued trapezoidal fuzzy number. Deschrijver G. and E. Kerre [7] established the relationship between interval-valued fuzzy sets and intuitionistic fuzzy sets. Based on this result Adrian I. and Delia A. Tuse [2] established bijective mapping between trapezoidal/triangular intuitionistic fuzzy numbers and interval-valued trapezoidal/triangular fuzzy numbers. In this paper, we introduce interval-valued pentagonal fuzzy number and establish the relationship between pentagonal intuitionistic fuzzy number and intervalvalued pentagonal fuzzy number.

3 3.1

Basic Definitions Fuzzy Set

[17] A fuzzy set (FS) A˜ in a universal set X is characterized by a membership function µA˜ : X → [0, 1], and it is given by A˜ = {(x, µA(x) ˜ )/x ∈ X}, where µA ˜ (x) represents the grade of member˜ ship of an element x ∈ X in A.

3.2

Intuitionistic Fuzzy Set

[3]An Intuitionistic fuzzy set (IFS) on a universe X is an object of the form A˜I = {(x, µA˜I (x), νA˜I (x))/x ∈ X} where µA˜I (x) and νA˜I (x) are the degree of membership and non-membership respectively of an element x ∈ X in A˜ with the condition: 0 ≤ µA˜I (x) + νA˜I (x) ≤ 1; ∀x ∈ X

3.3

Trapezoidal Intuitionistic Fuzzy Numbers

A Trapezoidal intuitionistic fuzzy number (TIFN) 0 0 0 0 A˜T I =[(a1 , a2 , a3 , a4 ), (a1 , a2 , a3 , a4 )] is an intuitionsistic fuzzy set on R, with membership function µA˜T I and non-membership function νA˜T I defined as [2]  x < a1   0   (x−a1 )    (a2 −a1 ) a1 ≤ x ≤ a2 1 a2 ≤ x ≤ a3 µA˜T I (x) =   (a4 −x)    (a −a ) a3 ≤ x ≤ a4   4 3 0 x > a4

3

179

Special Issue


νA˜T I (x) =

  1    (a02 −x)     (a02 −a01 )

0

  (x−a03 )    (a04 −a03 )   

1

Special Issue

x < a01 a01 ≤ x ≤ a02 a02 ≤ x ≤ a03 a03 ≤ x ≤ a04 x > a04

where a1 ≤ a2 ≤ a3 ≤ a4 , a1 ≤ a2 ≤ a3 ≤ a4 , a01 ≤ a1 , a02 ≤ a2 , a3 ≤ a03 and a4 ≤ a04 0

3.4

0

0

0

Interval-valued Fuzzy Sets

L U [2]An interval-valued fuzzy set (IVFS) A˜IV = [A˜IV , A˜IV ] is defined as A˜IV =(x, [µA˜ L (x), µA˜ U (x)]); x ∈ X; where µA˜ L : X → [0, 1]; IV

νA˜

IV

3.5

U

IV

IV

L U : X → [0, 1] and A˜IV ⊆ A˜IV

Interval-valued Trapezoidal Fuzzy Numbers

An interval-valued trapezoidal fuzzy number (IVTFN) is an interval-valued fuzzy set on R. It is defined by [2] ˜ T = [AIV ˜ T L , AIV ˜ TU] AIV ˜ T L =(aL , aL , aL , aL ), AIV ˜ T U = (aU , aU , aU , aU ) are trapezoidal where AIV 1 2 3 4 1 2 3 4 L ˜ T ⊆ AIV ˜ TU fuzzy numbers such that AIV

3.6

Pentagonal Fuzzy Numbers

A Pentagonal Fuzzy Number (PFN) A˜P =(a1 , a2 , a3 , a4 , a5 ) is a fuzzy set with membership function µA˜P defined as [13]

µA˜P (x) =

                          

0

(x−a1 ) (a2 −a1 ) 1(x−a2 ) 2(a3 −a2 )

1

1(a4 −x) 2(a4 −a3 ) (a5 −x) (a5 −a4 )

0

4

180

x a5


3.7

Special Issue

Pentagonal Intuitionistic Fuzzy Numbers

A pentagonal intuitionistic fuzzy number (PIFN) 0 0 0 0 0 A˜IP =[(a1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] is an intuitionsistic fuzzy set on R, with membership function µA˜IP and non-membership function νA˜IP defined as [14]

µA˜IP (x) =

νA˜IP (x) =

4

                          

                                  

0

(x−a1 ) (a2 −a1 ) (x−a2 ) (a3 −a2 )

1

(a4 −x) (a4 −a3 ) (a5 −x) (a5 −a4 )

x < a1 a1 ≤ x ≤ a2 a2 ≤ x ≤ a3 x = a3 a3 ≤ x ≤ a4

0

a4 ≤ x ≤ a5 x > a5

1

x < a1

0

0 (a2 −x) 0 0 (a2 −a1 ) 0 (a3 −x) 0 0 (a3 −a2 )

0

0 (x−a3 ) 0 0 (a4 −a3 ) 0 (x−a4 ) 0 0 (a5 −a4 )

1

0

0

0

0

a1 ≤ x ≤ a2 a2 ≤ x ≤ a3 x = a3 0

0

0

0

0

a3 ≤ x ≤ a4 a4 ≤ x ≤ a5 x > a05

Interval-valued Pentagonal Fuzzy Number

Based on the concepts of interval-valued trapezoidal fuzzy numbers and intuitionistic pentagonal fuzzy numbers we define interval-valued pentagonal fuzzy number.

4.1

definition

˜ P is an An interval-valued pentagonal fuzzy number (IVPFN) AIV interval-valued fuzzy set on R defined by ˜ P = [AIV ˜ P L , AIV ˜ PU] AIV ˜ P L =(aL , aL , aL , aL , aL ), AIV ˜ P U = (aU , aU , aU , aU , aU ) are penwhere AIV 1 2 3 4 5 1 2 3 4 5 ˜ P L ⊆ AIV ˜ PU; tagonal fuzzy numbers such that AIV

5

181


L L L L U U U U U aL 1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 ; a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 are positive real U L L U numbers and a1 ≤ a1 ; a5 ≤ a5 ; The set of all IVPFNs are denoted by IVPFN(R)

4.2

Geometric Representation

˜ P = [AIV ˜ P , AIV ˜ P ] = [(aL , aL , aL , aL , aL ), (aU , aU , aU , aU , aU )] AIV 1 2 3 4 5 1 2 3 4 5 L

U

Figure 1: Interval-valued Pentagonal Fuzzy Number Example: ˜ P = [AIV ˜ P L , AIV ˜ P U ] = [(2, 4, 5, 6, 8, ), (1, 3, 5, 7, 9)] AIV

Figure 2: Interval-valued Pentagonal Fuzzy Number 6

182

Special Issue


4.3

Important Algebraic Operations

We adapt the operation of addition, difference and scalar multiplication on interval-valued fuzzy numbers to the case of interval-valued pentagonal fuzzy numbers as follows: Addition: ˜ P +BIV ˜ P = [(aL , aL , aL , aL , aL ), (aU , aU , aU , aU , aU )]+ AIV 1 2 3 4 5 1 2 3 4 5 L L L L U U U U U [(bL 1 , b2 , b3 , b4 , b5 ), (b1 , b2 , b3 , b4 , b5 )] L L L L L L L L L = [(aL 1 + (b1 , a2 + b2 , a3 + b3 , a4 + b4 , a5 + b5 ), U , aU + bU , aU + bU , aU + bU , aU + bU )] (aU + b 1 1 2 2 3 3 4 4 5 5 Example: ˜ P +BIV ˜P AIV = [(2, 4, 5, 6, 8), (1, 3, 5, 7, 9)+[(2.5, 4.5, 5.5, 6.5, 8.5), (1.5, 3.5, 5.5, 7.5, 9.5)] = [(4.5, 8.5, 10.5, 12.5, 16.5), (2.5, 6.5, 10.5, 14.5, 18.5)] Difference: ˜ P -BIV ˜ P = [(aL , aL , aL , aL , aL ), (aU , aU , aU , aU , aU )]AIV 1 2 3 4 5 1 2 3 4 5 L L L L U U U U U [(bL 1 , b2 , b3 , b4 , b5 ), (b1 , b2 , b3 , b4 , b5 )] L L L L L = [(max(aL 1 − b5 , 0), (max(a2 − b4 , 0), (max(a3 − b3 , 0), L L L L (max(a4 − b2 , 0)(max(a5 − b1 , 0) U U U U U (max(aU 1 − b5 , 0), (max(a2 − b4 , 0), (max(a3 − b3 , 0), U U U U (max(a4 − b2 , 0)(max(a5 − b1 , 0)] Eample: ˜ P -BIV ˜P AIV = [(2, 4, 5, 6, 8), (1, 3, 5, 7, 9)−[(2.5, 4.5, 5.5, 6.5, 8.5), (1.5, 3.5, 5.5, 7.5, 9.5)] = [(0, 0, 0, 1.5, 5.5), (0, 0, 0, 3.5, 7.5)] Scalar Multiplication: ˜ P =λ[(aL , aL , aL , aL , aL ), (aU , aU , aU , aU , aU )] λAIV 1 2 3 4 5 1 2 3 4 5 L L L L U U U U U =λ[(λaL 1 , λa2 , λa3 , λa4 , λa5 ), λ(a1 , λa2 , λa3 , λa4 , λa5 )];λ ≥ 0 Example: ˜ P =2[(2,4,5,6,8),(1,3,5,7,9)] λAIV = [(4,8,10,12,16),(2,6,10,14,18)]

5

Relations between Pentagonal intuitionistic fuzzy number and interval-valued Pentagonal fuzzy number

We establish the relation between pentagonal intuitionistic fuzzy number and interval-valued pentagonal fuzzy number making use of

7

183

Special Issue


relationship between interval-valued fuzzy sets and intuitionistic fuzzy sets [7] and bijective mapping between trapezoidal/triangular intuitionistic fuzzy numbers and interval-valued trapezoidal/triangular fuzzy numbers [2]. L L L L U U U U U We consider IV P F N ∗ (R) =[(aL 1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] U L L U in IV P F N (R) such that a2 ≤ a2 and a4 ≤ a4 Proposition: The mapping ϕ:P IF N (R)−→IV P F N ∗ (R) 0 0 0 0 0 ϕ[(a1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] L L L L U U U U U =[(aL 1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] 0 L U where ai = ai ; i ∈ 1, 2, 3, 4, 5 and ai = ai ; i ∈ 1, 2, 3, 4, 5 is bijection. Proof: Consider, 0 0 0 0 0 0 0 0 0 0 [(a1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] and [(b1 , b2 , b3 , b4 , b5 ), (b1 , b2 , b3 , b4 , b5 )] in P IF N (R) Let 0 0 0 0 0 0 0 0 0 0 ϕ[(a1 , a2 , a3 , a4 , a5 ),(a1 , a2 , a3 , a4 , a5 )]=ϕ[(b1 , b2 , b3 , b4 , b5 ),(b1 , b2 , b3 , b4 , b5 )] L L L L U U U U U ⇒ [[(aL 1 , a2 , a3 , a4 , a5 ), (a1 , a2 , a3 , a4 , a5 )] L L L L L U U U U U = [(b1 , b2 , b3 , b4 , b5 ), (b1 , b2 , b3 , b4 , b5 )] L U ⇒ aL aU i = bi ; ∀i and i 0 = bi ; ∀i 0 ⇒ ai = bi and ai = bi 0 0 0 0 0 0 0 0 0 0 ⇒ [(a1 , a2 , a3 , a4 , a5 ),(a1 , a2 , a3 , a4 , a5 )]=[(b1 , b2 , b3 , b4 , b5 ),(b1 , b2 , b3 , b4 , b5 )] So the mapping is injective. By definition, the mapping is surjective. Therefore, the mapping is bijective.

6

Direction for Application

Adrian I. Ban and Delia A. Tuse [2] confirmed that there is bijection between trapezoidal/triangular intuitionistic fuzzy numbers and intervalvalued trapezoidal/triangular fuzzy numbers have good properties with respect to arithmetic operations and parameters associated with trapezoidal/triangular intuitionistic fuzzy numbers and interval-valued trapezoidal/triangular fuzzy numbers. This could be established also in case of pentagonal intuitionistic fuzzy numbers and interval-valued pentagonal fuzzy numbers. Then on many problems with pentagonal intuitionistic fuzzy numbers data can be analyzed by using interval-valued pentagonal fuzzy numbers.

8

184

Special Issue


7

Conclusion

We have defined interval-valued pentagonal fuzzy number (IVPFN). We have also illustrated it with geometric representation and proposed some algebraic operations. We have established the relation of bijection between the set of pentagonal intuitionistic fuzzy numbers (PIFN) and a subset of interval-valued pentagonal fuzzy numbers.

References [1] Abdullah, L., Kwan Ismail, W.,Hamming Distance in Intuitionistic Fuzzy Sets and Interval-valued Intuitionistic Fuzzy Sets: A Comparative Analysis Hamming Distance in Intuitionistic Fuzzy Sets and Interval-valued Intuitionistic Fuzzy Sets: A Comparative Analysis, 1, No. 1 (2012), 7-11. [2] Adrian I. Ban and Delia A. Tuse, Trapezoidal/triangular intuitionistic fuzzy numbers versus interval-valued trapezoidal/triangular fuzzy numbers and applications to multicriteria decision making methods, Notes on Intuitionistic Fuzzy Sets, 20, No. 2 (2014), 4351. [3] Atanassov, Krassimir T., Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20 No. 1, (1986) 87-96. [4] Atanassov, K.T., Intuitionistic fuzzy sets: Theory and applications, Studies in fuzziness and soft computing, Heidelberg, New York (1999). [5] Burillo, P., H. Bustince, and V. Mohedano, Some definitions of intuitionistic fuzzy number. First properties, Proceedings of the 1st Workshop on Fuzzy Based Expert Systems, 1994. [6] Chen, J.H., Chen S.M., A new method for ranking generalized fuzzy numbers for handling fuzzy risk analysis problems, Proceedings of the Ninth Conference on Information Sciences, (2006), 1196-1199. [7] Deschrijver, G., Etienne E.K., On the relationship between some extensions of fuzzy set theory, Fuzzy sets and systems, 133, No. 2 (2003), 227-235. [8] Giachetti, R.E., Robert E.Y., Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal

9

185

Special Issue


fuzzy numbers and the development of a new approximation, Fuzzy Sets and Systems, 91, No.1 (1997), 1-13. [9] Kumar, R., Pathinathan, T., Sieving out the Poor using Fuzzy decision Making Tools, Indian Journal of Science and Technology, 8, No.22 (2015), 1-16. [10] Kumar, R., Pathinathan, T., Sieving out the poor using Fuzzy Decision Making Tools with reference to Nalanda District, Bihar, India, Interantional Conference on Convergence Technology, 5, No. 1 (2015), 890-891. [11] Li, J., Wenyi Z., and Ping G.,Interval-valued intuitionistic trapezoidal fuzzy number and its application, Systems, Man and Cybernetics (SMC), IEEE International Conference, San Diego, USA 2014. [12] Mendel, J.M., John, R.I.B., Type-2 Fuzzy Sets Made Simple, IEEE Transaction on Fuzzy Systems, bf 10, No. 2 (2002), 117-127. [13] Pathinathan, T., and Ponnivalavan, K., Pentagonal Fuzzy Numbers International Journal of Computing Algorithm, 3, (2014), 10031005. [14] Ponnivalavan, K., Pathinathan, T., Intuitiionistic Pentagonal Fuzzy Number, ARPN Journal of Engineering and applied Sciences, 10, No. 12 (2014), 5446-5450. [15] Torra, V.,Hesitant fuzzy sets, International Journal of Intelligent Systems, 25, No. 6 (2010), 529-539. [16] Wang, G., Li, X., The Applications of interval-valued fuzzy numbers and interval-distribution numbers, Fuzzy Sets and System, 98, (1998), 331-335. [17] Zadeh, L.A., Fuzzy Set, Information and Control, 8, (1965), 338353. [18] Zadeh, L.A., The concept of linguistic variable and its application to approximate reasoning-I, Information Sciences, Elsevier Science Publications,No. 3 (1975), 199-249. [19] Zadeh, L.A., The concept of linguistic variable and its application to approximate reasoning-II, Information Sciences, Elsevier Science Publications, 8, No. 4 (1975), 301-357.

10

186

Special Issue

187

188