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Web End = Ranking p-norm generalised fuzzy numbers with different left height and right height using integral values

Rituparna Chutia Rekhamoni Gogoi

D. Datta

Received: 19 May 2014 / Accepted: 13 February 2015 / Published online: 4 March 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract This paper considers ranking of generalised fuzzy numbers with different left height and right height using integral values. With the advances in new type of fuzzy number (generalised fuzzy number with different left height and right height) methods should be developed to compare them. Keeping this in view a new modied method has been proposed.

Keywords Generalised fuzzy number Ranking

p-Norm Left height Right height

Introduction

Decision making in engineering, medical and any other real-life problems may be interpreted in terms of fuzzy. This demands ranking or ordering of fuzzy quantities to make a transparent decision. With the advances in fuzzy set theory, different ranking methods are developed. This concept was rst proposed by Jain [7]. Some of the literatures that describe different approach of ranking fuzzy quantities are [1, 3, 4, 10, 1315]. Recently, ranking of trapezoidal fuzzy numbers based on the shadow length has

been discussed by Pour et al. [12]. Also, ranking triangular fuzzy numbers by Pareto approach based on two dominance stages is discussed by Bahri et al. [2].

The literatures that are available on ranking fuzzy quantities based on the integral values are [69]. These type of methods of ranking fuzzy numbers are based on the convex combination of right and left integral values through an index of optimism found in Liou and Wang [11] and Kim and Park [8]. This concept was further generalised to rank non-normal p-norm trapezoidal fuzzy numbers [6]. However, this method was found insufcient to rank non-normal p-norm fuzzy numbers with different height; keeping this in mind, Kumar et al. [9] developed an approach to overcome those shortcomings. With the advances of generalised fuzzy numbers (GFNs) with different left height and right height [6], Kumars approach fails to rank them. Hence, Kumars approach is only sufcient for ranking fuzzy numbers or non-normal p-norm fuzzy numbers with different height, but the method is insufcient for ranking GFNs with different left height and right height.

Keeping this in view, Kumars approach has been modied in this paper to rank p-norm GFNs with different left height and right height. This modied method thus handle both normal and non-normal trapezoidal fuzzy number with different height. The modied method can also rank non-normal p-norm trapezoidal fuzzy numbers with different height.

The structure of the paper is as follows. In Sect. 2, some general concept of the GFN is put forwarded. Membership function of GFN is dened. Also the membership function of p-norm GFN with different left height and right height is dened. Section 3 starts with denitions of different integral values of p-norm GFN with different left height and right height, And nally, some properties related to them are discussed in this section. Section 4 describes the

R. Chutia (&)

Department of Chemistry, Indian Institute of Technology, Guwahati 781039, Assam, Indiae-mail: [email protected]

R. GogoiNorth Lakhimpur Girls Higher Secondary School, Lakhimpur 787001, Assam, India

D. DattaHealth Physics Division, Bhabha Atomic Research Division, Mumbai 400085, India

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2 Math Sci (2015) 9:19

proposed modied method along with some numerical examples. Finally, in Sect. 4, conclusions are made.

Denitions and notations

In this section, brief review of some concepts of generalised fuzzy number with different left height and right height are put forwarded.

Generalised fuzzy number

Let be represented by a; b; c; d; hL; hR on the real line

where l 11 : 0;hL ! a;b , l 12 : hL;hR orhR;hL ! b;c and l 13 : 0;hR ! c;d are continuous. The function

l 11x and l 13x are strictly increasing and strictly de

creasing, respectively. The function l 12x is strictly in

creasing when hL\hR and strictly decreasing when hL [hR. Let a; b; c; d; hL; hR be a trapezoidal GFN with

different left height and right height then the membership function is dened as

hLx a b a

; if a x b; hLc b hR hLx b

c b

; if b x c;

8 >

> hRx d c d

; if c x d; 0; otherwise:

such that 1\a b c d\1 is called a GFN with

different left height and right height which is bounded and convex. The values a; b; c and d are real, hL is called the left height of the GFN, hR is called the right height of the

GFN, hL 2 0; 1 and hR 2 0; 1 [5]. For now, let

Denition 2.1.1 A GFNp a; b; c; d; hL; hRp is said

to be a p-norm GFN with different left height and right height if its membership function is given by

l~Apx

> f L~Ap x hL 1

R be

the set of all GFNs with different left height and right height. If hL hR 1 then the GFN reduces to a standard

trapezoidal fuzzy number.

The membership function of GFN with different left height and right height is as given below

l1x; if a x b; l2x; if b x c;

> l3x; if c x d;

0; otherwise

; if a x b;

f M~A

8 >

x b

a b

h i

1p; if b x c;

< x hL hR hL 1

x c

b c

h i

1p; if c x d; 0; otherwise

> f R~Ap x hR 1

x c

d c

where p is a positive integer. The functions f L~Ap : a; b !

0; hL and f R~Ap : c; d ! 0; hR are both continuous as well as strictly increasing and strictly decreasing functions, respectively. The function f M~A

where l1 : a; b ! 0; hL , l2 : b; c ! hL; hR or hR; hL and l3 : c; d ! 0; hR are continuous. The

functions l1x and l3x are strictly increasing and strictly

decreasing, respectively. The function l2x is strictly in

creasing when hL\hR and strictly decreasing when hL [ hR. Then the inverse of lx is

l 1~Ay

l 11y; if 0 y hL;l 12y; if hL y hR; or hR y hL;

> l 13y; if 0 y hR;

0; otherwise

8 >

: b; c ! hL; hR or hR; hL is strictly increasing (decreasing) when hL\hRhR\hL).

When p is one, the p-norm GFN with different left height and right height reduces to trapezoidal GFN with different left height and right height as dened by Eq. (3).

Denition 2.1.2 The inverse function of lp x as given by membership function in (4) is given by

8 >

y hL

; if 0 y hL;

< gM~Ap y c b c 1

; if hL y hR; or hR y hL;

> gR~Ap y c d c 1

l 1~Apy

> gL~Ap y b a b 1

y hL

hR hL

; if 0 y hR;

y hR

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Math Sci (2015) 9:19 3

1. The left membership function f L~Ap x is continuous and strictly increasing function and its left integral value is

p bhL

a b

p hL

C 1p 1

The functions gL~Ap : 0; hL ! a; b and gR~Ap : 0; hR !

c; d are both continuous as well as strictly increasing and

strictly decreasing functions, respectively. The function gM~Ap :hL; hR or hR; hL ! b; c is strictly increasing (decreasing) when hL\hRhR\hL:

Total integral value

Convex combination of right and left integral values through an index of optimism is called the total integral value [8, 11]. The middle integral value is zero for normal and non-normal p-norm trapezoidal fuzzy numbers with different height. However, for a p-norm GFNs with different left height and right height this integral value has to be counted for a transparent decision. Keeping this in view, following denitions are put forwarded.

Denition 3.1 If is a fuzzy number with different left height and right height as dened by the membership function (1) and the inverse membership function given by (2) then the left integral value of is dened as

Z hL

0 l 11ydy

C 1p

where Cx is Eulers gamma function, dened by

R 1

0 yx 1e ydy.2. The right membership function f R~Ap x is continuous and strictly decreasing function and its right integral value is

p chR

d c

p hR

C 1p 1

C 2p 1

C 1p

3. The middle membership function f M~A

C 2p 1

x is continuous and strictly increasing and strictly decreasing when hL\hR and hR\hL, respectively. The middle integral value is given by

p hR hLc

b c

p hR hL

C 1p 1

C 1p

C 2p 1

Denition 3.2 If is a fuzzy number with different left height and right height as dened by the membership function (1) and the inverse membership function given by (2) then the right integral value of is dened as

Z hR

0 l 13ydy

4. The total integral value with optimism a is

IaT

p achR 1 abhL hR hLc

ahRd c 1 ahLa b

hR hLb cg

C 1p 1

C 1p

Denition 3.3 If is a fuzzy number with different left height and right height as dened by the membership function (1) and the inverse membership function given by (2) then the middle integral value of is dened as

Z hR

hL l 12ydy or IM

Proof Continuity of the left membership function f L~Ap x is trivial. Also, this function is strictly increasing and its integral values are inherited from [6]. Similarly, for the right membership function f L~Ap x.

Trivially, the function f M~A

p C

2 p

Z hL

hR l 12ydy

x is continuous. Now,

ddx f M~Apx

ddx hL hR hL 1

x c

b c

h i

Denition 3.4 If is a fuzzy number with different left height and right height as dened by the membership function (1), then the total integral value with index of optimism a is dened as

IaT

aIR

1 aIL

Proposition 3.1 Letp a; b; c; d; hL; hRp be a

p-norm GFN with different left height and right height with membership function (4), where p is a positive integer. Then:

h i

1p 1

hR hL

1p 1

x c

b c

p 1 1

b c

x c

b c

Since 0

x c

b c

p 1, it trivially follows that ddx f M~Apx 0 if hR hL 0 and

ddx f M~A

x 0 if hR hL 0. Hence the function f M~A

x is strictly increasing when hR hL and strictly decreasing when hR hL. Also,

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4 Math Sci (2015) 9:19

Z hRor hL

hLor hR

c b c 1

" #

The proposed method

In this section, a method of ranking p-norm GFNs with different left height and right height is presented. The method calculates total integral value on the basis of left integral value, right integral value and middle integral value. Ranking is done on the basis of these evaluated total integral values. Letp a; b; c; d; hL; hRp and

~Bp

q; r; s; t; h0L; h0Rp be p-norm GFNs with different left

height and right height. Then

1.p

~Bp if IaT

p [ IaT

~Bp,

2.p

~Bp if IaT

p\IaT

~Bp,

3.p

~Bp if IaT

p IaT

~Bp,The following are the steps involved in this ranking method:

Step 1. Find h1 min hL; h0L and h2 min hR; h0R.

Step 2. Find IL

p, IR

p, IM

p and IL

~Bp, IR

~Bp,

~Bp, such that IL

Z h1

0 gL~Apxdx; where

gL~Apx b a b 1

y hL

hR hL

C 1p

hR hLc

b c

p hR hL

C 1p 1

C 2p 1

Now, the total integral value with optimism a is

IaT

p afIR

p IM

pg 1 afIL

p IM

achR 1 abhL hR hLc

ahRd c 1 ahLa b

hR hLb cg

C 1p 1

C 1p

p C

2 p

Remark 3.1 For a pessimistic decision maker a 0, for

an optimistic decision maker a 1 and for a moderate

decision maker a 0:5.

Corollary 3.1 For an pessimistic decision maker a 0

the total integral IaT

, for optimistic

decision maker a 1 the total integral value IaT

and for moderate decision maker a 0:5 the total integral value IaT

;

x h1

C 1p

g:Note 1 The above total integral values for pessimistic decision maker, optimistic decision maker and moderate decision maker reduce to IaT

, IaT

and IaT

12 fIR

g, respectively, for a non-

normal p-norm trapezoidal fuzzy number. This is because IM

0 when hL hR.

Arithmetic operations

The arithmetic of p-norm GFNs with different left height and right height are reviewed from Chen [5]. Letp

a; b; c; d; hL; hRp and

~Bp q; r; s; t; h0L; h0Rp be p-norm

GFNs with different left height and right height. Then

1.p

~Bp a q; b r; c s; d t;

min hL; h0L; min hR; h0Rp2.p

~Bp a t; b s; c r; d q; min hL; h0L;

min hR; h0Rp3.

fIR

bh1

a b

p h1

C 1p 1

C 2p 1

Z h2

0 gR~Apxdx; where

gR~Apx c d c 1

x h2

;

C 1p

ch2

d c

p h2

C 1p 1

C 2p 1

1 gM~Apxdx;

where gM~Apx c b c 1

x h1

h2 h1

Z h2or h

h1or h

;

ch2 h1

b c

p h2 h1

C 1p 1

C 1p

C 2p 1

~Bp

Z h1

0 gL~Bpxdx;

where gL~Bpx r q r 1

;

x h1

(

C 1p

ka; kb; kc; kd; hL; hRp k [ 0

kd; kc; kb; ka; hL; hRp k\0

rh1

q r

p h1

C 1p 1

C 2p 1

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Math Sci (2015) 9:19 5

~Bp

Z h2

0 gR~Bpxdx; where

gR~Bpx s t s 1

(iii) IM

~B, if b c 2e;

(iv) IM

;

[ IM

~B, if h1\h2 and b c [ 2e or

h1 [ h2 and b c\2e;

(v) IM

x h2

\IM

~B, if h1\h2 and b c\2e or

h1 [ h2 and b c [ 2e;

(vi) IaT

C 1p

sh2

t s

p h2

C 1p 1

[ IaT

~B if ah2c e 1 ah1b e h2 h1b c 2e [ 0;

(vii) IaT

C 2p 1

\IaT

~B if ah2c e 1 ah1b e h2 h1 b c 2e\0 and

(viii) IaT

~Bp

gM~Bpxdx; where

gM~Bpx s r s 1

x h1

h2 h1

Z h2or h1

h1or h2

IaT

~B if ah2c e 1 ah1b

e h2 h1 b c 2e 0:

Proof From Eqs. (13), (14), (15) and (19), on appropriate substitutions of the variables the following could be easily obtained:

h1a b

2 ; IR

;

C 1p

sh2 h1

r s

p h2 h1

C 1p 1

C 2p 1

h2d c

2 ;

Step 3. Find IaT

p and IaT

~Bp, which are given by

IaT

h2 h1b c

2 ; IL

h1a e

2 ;

p ach2 1 abh1 h2 h1c

ah2d c 1 ah1a b

h2 h1b cg

C 1p 1

h2e d

2 ; IM

~B eh2 h1;

IaT

C 1p

2 fah2c d 1 ah1a b

h2 h1c bg and IaT

;

p C

2 p

~Bp ash2 1 arh1 h2 h1s ah2t s 1 ah1q r

h2 h1r sg

C 1p 1

IaT

C 1p

p C

2 p

2 fah2e d 1 ah1a e 2eh2 h1g

where h1 minhAL; hB and h2 minhAR; hB.

Now IL

~B h12 b e 0 as b e c, hence

inequality (i) is deduced. Similarly, inequality (ii) could be deduced. Again, we have IM

~B h2 h12 b c 2e hence the inequalities (iii), (iv) and (v) follow

immediately. Also we have

IaT

Step 4. Check IaT

p [ IaT

~Bp or IaT

p\IaT

~Bp or

IaT

p IaT

~Bp.

2 fah2c e 1 ah1b e

h2 h1c b 2eg:Hence the inequalities (vi), (vii) and (viii) can be deduced easily. h

Corollary 4.1 [9] Let a; b; c; d; hA and

a; e; d; hB be non-normal trapezoidal and triangular fuzzy

numbers, respectively, where 1\a b e c d\1.

Then

(i) IL

~B,

(ii) IR

~B,

(iii) IaT

[ IaT

~B if e\ca 1 ab,

(iv) IaT

IaT

~B if e ca 1 ab and

(v) IaT

\IaT

~B if e [ ca 1 ab:

Case (i) If IaT

p [ IaT

~Bp then

~Bp.

Case (ii) If IaT

p\IaT

~Bp then

~Bp.

Case (iii) If IaT

p IaT

~Bp then

~Bp.

Remark 4.1 For any two arbitrary generalised fuzzy numbers with different left height and right height,p and ~Bp, we have

IaT

~Bp IaT

p IaT

~Bp

Proposition 4.1 Let a; b; c; d; hAL; hAR and

a; e; d; hB be GFN with different left height and right

height and non-normal triangular fuzzy number, respectively, such that 1\a b e c d\1. Then(i) IL

~B;

(ii) IR

~B;

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6 Math Sci (2015) 9:19

Corollary 4.2 [9] Let a; b; c; d; hA and

~B2

These inequalities are particular case of the inequalities in the Proposition 4.1. These can be obtained by appropriate substitutions on the inequalities in the Proposition4.1.

Proposition 4.2 Let a; b; c; d; hAL; hAR and

~B2

a; b; c; d; hBL; hBR2 be GFN and 2-norm GFN with dif

ferent left height and right height, respectively. Then

(i) IL

~B2;(ii) IL

~B2;(iii) IM

~B2 if h1\ [ h2;

(iv) IaM

[ IaM

~B2 if ah2d c h11 aa

b h2 h1b c\0;

(v) IaM

\IaM

~B2 if ah2d c h11 aa

b h2 h1b c [ 0 and

(vi) IaM

IaM

~B2 if ah2d c h11 aa

b h2 h1b c 0:

Proof and ~B2 are GFN and 2-norm GFN with different left height and right height. Hence by the proposed method

, IR

, IM

, IaT

~B2, IR

~B2, IM

~B2 and

IaT

~B2 are obtained by using Eqs. (13), (14), (15) and (19) as:

h1a b

2 ; IR

a; b; c; d; hB be non-normal trapezoidal fuzzy number and

non-normal 2-norm trapezoidal fuzzy number, respectively, then

(i) IL

~B2,(ii) IL

~B2,(iii) IaT\IaT

~B if ad c 1 aa b [ 0,

(iv) IaT IaT

~B if ad c 1 aa b 0 and

(v) IaT [ IaT

~B if ad c 1 aa b\0.

These inequalities are particular case of the inequalities in the Proposition 4.2. These can be obtained by appropriate substitutions on the inequalities in the Proposition 4.2.

Proposition 4.3 Let a; a; a; a; 1; 1 and

b; b; b; b; 1; 1 be GFNs with height 1. Then(i) IL

~B, if a b,

(ii) IM

~B2 if a b and

(iii) I1T

[ \I1T

~B if a [ \b:The Proposition 4.3 validates that the proposed method can also be applied for real numbers.

Example 4.1 Let 5; 7; 8; 9; 0:5; 0:6 and

~B2

5; 7; 8; 9; 0:7; 0:62 be GFN and 2-norm GFN with dif

ferent left height and right height, which are depicted in Fig. 1. But, according to the proposed modied method h1 min0:5; 0:6 and h2 min0:7; 0:6. Also, I0:5T

4:8000 and I0:5T

~B2 4:7144. Thus, I0:5T

[ I0:5T

~B2.

Example 4.2 Let 0:1659; 0:2803; 0:7463; 1:154; 0:5;

0:6,

~B 0:1611; 0:2475; 0:5696; 0:8187; 0:4; 0:5 and

0:1645; 0:2445; 0:5869; 0:8894; 0:5; 0:6, are GFNs

with different left height and right height. Figure 2 depicts the membership function of the above fuzzy

numbers.

Here I0:5T

0:3335, I0:5T

~B 0:2553 and I0:5T

0:2670, and I1T

0:1406, I1T

~B 0:1226 and I1T

h2d c

2 ;

h2 h1b c

2 ; IL

~B2 bh1

a b

4 h1p;

~B2 ch2

d c

4 h2p; IM

~B2

b 3c

4 h2 h1p;

2 fah2c d 1 ah1a b

h2 h1c bg and IaT

~B2

IaT

14 h2 h1f4c pb cg 1 a

h1f4b pa bg ah2f4c pd cg

where h1 minhAL; hBL and h2 minhAR; hBR.

Now, IL

~B2 2 p4 h1a b 0 and

~B2 2 p4 h2d c 0. Thus the desired

inequalities (i) and (ii) are obtained. b c2 p is

always greater than or equal to zero, thus IM

~B2 h2 h1 b c2 p4 0 if h1\ [ h2,

which prove the inequality (iii). For the inequalities (iv),(v) and (iv), we have

IaM

~B2

2 p

0:1234. Thus a moderate decision and a pessimistic decision maker rank them as [ [ ~B.

Example 4.3 Let2 2; 1; 0; 1; 0:5; 0:52,

~B2

1:5; 0:5; 0:5; 1:5; 0:5; 0:62 and

2 1; 1:5; 2; 2:5;

0:6; 0:52, are GFNs with different left height and right

height. The membership functions of the fuzzy numbers are depicted in Fig. 3. Here, I0:5T

2 0:2500, I0:5T

~B2

0:000 and I0:5T

2 0:6787, and I0T

2 0:8927,

I0T

~B2 0:6427 and I0T

2 0:5537. A moderate de

cision maker (a 0:5) ranks

, ~B and as [ ~B [ and also a pessimistic decision maker ranks them in the same order.

4 fah2d c

1 ah1a b h2 h1b cg:

Hence the inequalities (iv), (v) and (vi) follow immediately. h

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Math Sci (2015) 9:19 7

Fig. 1 Fuzzy numbers 5; 7; 8; 9; 0:5; 0:6 and

~B2 5; 7; 8; 9; 0:7; 0:62

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2

Fig. 2 Fuzzy numbers 0:1659; 0:2803; 0:7463; 1:154; 0:5; 0:6,

~B 0:1611; 0:2475; 0:5696; 0:8187; 0:4; 0:5 and

0:1645; 0:2445;

0:5869; 0:8894; 0:5; 0:6

0:2; 0:5; 0:5; 0:6; 1; 1.

Set C 0:2; 0:4; 0:4; 0:6; 0:9; 0:9,

~B 0:2; 0:4; 0:4; 0:6; 1; 1,

0:2; 0:4; 0:4; 0:6; 0:5; 0:5.

Set D 0:4; 0:5; 0:5; 0:6; 0:5; 0:5,

~B 0:2; 0:4; 0:6; 0:8; 0:6; 0:6.

Set E 0:4; 0:5; 0:5; 0:6; 0:6; 0:7,

~B 0:2; 0:4; 0:6; 0:8; 0:5; 0:6.

Set F 0:4; 0:5; 0:5; 0:6; 0:6; 0:7,

~B 0:2; 0:4; 0:6; 0:8; 0:5; 0:62.

Example 4.4 Let 1; 1; 1; 1; 1; 1 and

2; 2; 2; 2; 1; 1 be GFNs which are actually real numbers.

Now by Proposition 4.3 ~B [ trivially.

Example 4.5 Consider the following set of fuzzy numbers,

Set A 0:4; 0:5; 0:5; 0:6; 1; 1,

~B 0:2; 0:4; 0:6; 0:8; 1; 1.

Set B 0:2; 0:3; 0:3; 0:6; 1; 1,

~B 0:2; 0:4; 0:4; 0:6; 1; 1,

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8 Math Sci (2015) 9:19

Fig. 3 Fuzzy numbers2 2; 1; 0; 1; 0:5; 0:52,

~B2

1:5; 0:5; 0:5; 1:5; 0:5; 0:62

and

2 1; 1:5; 2; 2:5; 0:6; 0:52

Table 1 Comparison of the proposed method with Kim and Park [8] and Kumar et al. [9] methods

0.7

0.6

0.5

0.4

0.3

0.2

0.1

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3

Test sets Index Kim and Park Kumars method Proposed method

~B ~B ~B

Set A a 1:0 0.5714 0.75 0.55 0.70 0.55 0.70 a 0:5 0.5000 0.50 0.50 0.50 0.50 0.50

a 0:0 0.4286 0.25 0.45 0.30 0.45 0.30

Set B a 1:0 0.5714 0.6666 0.8000 0.45 0.50 0.55 0.45 0.50 0.55 a 0:5 0.3857 0.4999 0.6143 0.35 0.40 0.45 0.35 0.40 0.45 a 0:0 0.2000 0.3333 0.4286 0.25 0.30 0.35 0.25 0.30 0.35

Set C a 1:0 0.45 0.50 0.25 0.45 0.50 0.25 a 0:5 0.36 0.40 0.20 0.36 0.40 0.20 a 0:0 0.27 0.30 0.15 0.27 0.30 0.15

Set D a 1:0 0.275 0.35 0.275 0.35 a 0:5 0.250 0.25 0.250 0.25

a 0:0 0.225 0.15 0.225 0.15

Set E a 1:0 0.3800 0.470 a 0:5 0.3275 0.335

a 0:0 0.2750 0.200

Set F a 1:0 0.3800 0.4985 a 0:5 0.3275 0.3321

a 0:0 0.2750 0.1658

The results of the sets A, B, C, D and E are depicted in the Table 1. Sets A and B consist of normal fuzzy numbers, hence the ranking order by the three methods is same. Sets C and D consist of non-normal fuzzy numbers, Kim and Park [8] give no option for ranking such type of fuzzy number. The method of Kumar et al. [9] and the proposed methods ranking order are same for the sets C and D. However, sets E and F which consist of p-norm generalised fuzzy numbers with different left height and right height can be ranked only by the proposed method.

Validation of the proposed modied ranking method

For the validation of the proposed ranking method, the following reasonable axioms that Wang and Kerre [13] have proposed for fuzzy numbers ranking are considered. Let RM be an ordering method, S the set of fuzzy numbers for which the method RM can be applied, and A and A0 nite subsets of

S. The statements of two elementsp and ~Bp in A satisfy that

p has a higher ranking than ~Bp when RM is applied to the

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Math Sci (2015) 9:19 9

can only deal with non-normal p-norm trapezoidal fuzzy numbers. The proposed method can handle non-normal p-norm trapezoidal fuzzy numbers as well as p-norm GFNs with different left height and right height.

Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions which improved the paper form technical as well as clarity point of view. The author RC would like to thank Indian Institute of Technology, Guwahati for funding the research work.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

1. Abbasbandy, S., Hajjari, T.: A new approach for ranking of trapezoidal fuzzy numbers. Comput. Math. Appl. 57, 413419 (2009)2. Bahri, O., Amor, N.B., El-Ghazali, T.: New Pareto approach for ranking triangular fuzzy numbers. Inf. Process. Manag. Uncertain. Knowl. Based Syst. Commun. Comput. Inf. Sci. 443, 264273 (2014)

3. Chen, S.M., Chen, J.H.: Fuzzy risk analysis based on ranking generalised fuzzy numbers with different heights and different spreads. Expert Syst. Appl. 36, 68336842 (2009)

4. Cheng, C.H.: A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst. 95, 307317 (1998)

5. Chen, S.M., Munif, A., Chen, G.S., Liu, H.C., Kuo, B.C.: Fuzzy risk analysis based on ranking generalised fuzzy numbers with different left heights and right heights. Expert Syst. Appl. 39, 63206334 (2012)

6. Chen, C.C., Tang, H.C.: Ranking non-normal p-norm trapezoidal fuzzy number with integral value. Comput. Math. Appl. 56, 23402346 (2008)

7. Jain, R.: Decision-making in the presence of fuzzy variables. IEEE Trans. Syst. Man Cybern. 6, 698703 (1976)

8. Kim, K., Park, K.S.: Ranking fuzzy numbers with index of optimism. Fuzzy Sets Syst. 35, 143150 (1990)

9. Kumar, A., Singh, P., Kaur, A., Kaur, P.: A new approach for ranking nonnormal p-norm trapezoidal fuzzy numbers. Comput. Math. Appl. 61, 881887 (2011)

10. Lee, E.S., Li, R.J.: Comparison of fuzzy numbers based on the probability measure of fuzzy events. Comput. Math. Appl. 15, 887896 (1988)

11. Liou, T.S., Wang, M.J.J.: Ranking fuzzy numbers with integral value. Fuzzy Sets Syst. 50, 247255 (1992)

12. Pour, N.S., Moghaddam, R.T., Basiri, M.: A new method for trapezoidal fuzzy numbers ranking based on the shadow length and its application to managers risk taking. J. Intell. Fuzzy Syst. 26, 7789 (2014)

13. Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets Syst. 118, 375385 (2001)

14. Yager, R.R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24, 143161 (1981)

15. Yoon, K.P.: A probabilistic approach to rank complex fuzzy numbers. Fuzzy Sets Syst. 80, 167176 (1996)

fuzzy numbers in A will be written as

~Bp by RM on A.

~Bp by RM on A are similarly

interpreted. The following axioms show the reasonable properties of the ordering approach RM.

A1 Forp 2 A,

p by RM on A.

A2 For

p; ~Bp 2 A2,

~Bp and ~Bp

p by RM on A,

we should havep

~Bp by RM on A.

A3 For

p; ~Bp;p 2 A3,

~Bp and ~Bp

p by RM on

~Bp by RM on A, and

A, we should havep

p by RM on A.

A4 For

p; ~Bp 2 A2, inf supp

~Bp [ sup supp

p, we

should havep

~Bp by RM on A.

A04 For

p; ~Bp 2 A2, inf supp

~Bp [ sup supp

p, we

should havep

~Bp by RM on A.

A5 Let

p; ~Bp 2 A \ A02. We obtain the ranking order

~Bp by RM onA0 if and onlyif

~Bp byRM on A.

A6 Letp, ~Bp,p

p and ~Bp

p be elements of S. If

~Bp by RM on f

p; ~Bpg, then

~Bp

by RM on f

p; ~Bp

pg.

A06 Letp, ~Bp,p

p and ~Bp

p be elements of S. If

~Bp by RM on f

p; ~Bpg, then

~Bp

pg when

p 6 /.

Proposition 4.4 The proposed ranking method RM has the properties A1, A2, A3, A4, A04, A5, A6 and A06:

Proof It is easy to verify that properties A1A5 are hold. For the proof of A6, consider the generalised fuzzy numbers with different left height and right height as p a; b; c; d; hL; hRp,

~Bp q; r; s; t; h0L; h0Rp and

p l; m; n; o; ; h00L; h00Rp. Let

~Bp by RM, hence

by RM on f

p; ~Bp

IaT

p IaT

~Bp;

by adding IaT

IaT

p IaT

~Bp IaT

and by Remark 4.1

IaT

p IaT

~Bp

Therefore,p

~Bp

p. Similarly A06 also holds. h

Conclusions

In this paper, ranking of p-norm GFNs with different left height and right height is proposed. The proposed method is generalization of Kumars approach. Kumars approach

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Islamic Azad University 2015

Abstract

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This paper considers ranking of generalised fuzzy numbers with different left height and right height using integral values. With the advances in new type of fuzzy number (generalised fuzzy number with different left height and right height) methods should be developed to compare them. Keeping this in view a new modified method has been proposed.

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Title

Ranking p-norm generalised fuzzy numbers with different left height and right height using integral values

Author

Chutia, Rituparna; Gogoi, Rekhamoni; Datta, D

Pages

1-9

Publication year

2015

Publication date

Mar 2015

Publisher

Springer Nature B.V.

ISSN

20081359

e-ISSN

22517456

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/s40096-015-0142-0

ProQuest document ID

1679673570

Islamic Azad University 2015

Ranking p-norm generalised fuzzy numbers with different left height and right height using integral values

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