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Soft Computing (2022) 26:1101–1122

https://doi.org/10.1007/s00500-021-06616-1

FOUNDATION, ALGEBRAIC, AND ANALYTICAL METHODS IN SOFT

COMPUTING

Generalization and ranking of fuzzy numbers by relative preference

relation

Kavitha Koppula1·Babushri Srinivas Kedukodi1·Syam Prasad Kuncham1

Accepted: 25 November 2021 / Published online: 28 December 2021

© The Author(s) 2021

Abstract

We deﬁne 2n+1 and 2nfuzzy numbers, which generalize triangular and trapezoidal fuzzy numbers, respectively. Then, we

extend the fuzzy preference relation and relative preference relation to rank 2n+1 and 2nfuzzy numbers. When the data

is representable in terms of 2n+1 fuzzy number, we generalize the FMCDM (fuzzy multi-criteria decision making) model

constructed with TOPSIS and relative preference relation. Lastly, we give an example from telecommunications to present

the proposed FMCDM model and validate the results obtained.

Keywords Fuzzy number ·Fuzzy preference relation ·Decision making

1 Introduction

Zadeh (1965) introduced the concept of fuzzy set, and it is

widely used to characterize vague or imprecise settings (con-

ditions). Fuzzy sets have applications in automata theory,

systems theory, decision theory, switching theory, pattern

recognition, image thresholding, etc. [Lalotra and Singh

(2020); Singh et al. (2019); Singh and Sharma (2019); Singh

et al. (2020)]. Fuzzy numbers generalize real numbers and

are very useful to represent data corresponding to uncer-

tain situations. There are several methods to rank or order

fuzzy numbers. Lee and Li (1988) utilized the concept of

probability measure to determine the order of fuzzy num-

bers by considering the mean and dispersion of alternatives.

Choobineh and Li (1993) proposed an indexing method to

order or rank the fuzzy numbers. Dias (1993) proposed a

computational approach to rank the alternatives using fuzzy

numbers. Fortemps and Roubens (1996) presented a method

BBabushri Srinivas Kedukodi

babushrisrinivas.k@manipal.edu

Kavitha Koppula

kavitha.koppula@manipal.edu

Syam Prasad Kuncham

syamprasad.k@manipal.edu

1Department of Mathematics, Manipal Institute of Technology,

Manipal Academy of Higher Education (MAHE), Manipal,

Karnataka 576104, India

to compare fuzzy numbers using the area compensation pro-

cedure. Cheng (1998) proposed the distance method and

coefﬁcient of variation (CV) index method to rank the fuzzy

numbers. Chu and Tsao (2002) proposed a method using

the area between centroid point and original point of the

fuzzy numbers to facilitate ranking. Wang and Lee (2008)

later revised this method. Lee (2005b) introduced the ’com-

parable’ property for fuzzy preference relation and showed

that only O(n)comparisons of fuzzy numbers are sufﬁcient

if a fuzzy preference relation satisﬁes certain conditions.

Asady and Zendehnam (2007) proposed a ranking method

for the fuzzy numbers by obtaining the nearest point of sup-

port function with respect to fuzzy quantity. Wang (2015b)

proposed a fuzzy relation with membership function repre-

senting preference degree to compare two fuzzy numbers. A

relative preference relation was deﬁned using fuzzy prefer-

ence relation to compare a set of fuzzy numbers. The relative

preference relation expresses preference degrees of several

fuzzy numbers over average and facilitates easy and quick

ranking of fuzzy numbers.

Decision-making methods often apply fuzzy sets in their

computations. Jain (1976) presented a decision method that

represented uncertain quantities as fuzzy sets and subse-

quently obtained an optimal alternative. Jain (1977)also

developed a procedure for decision making using fuzzy sets

by assigning quantitative numbers to qualitative terms. Wang

(2014,2015a,2020a,b) proposed various methods using rel-

ative preference relation to solve FMCDM problems. In the

123

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1102 K. Koppula et al.

multi-granulation decision-theoretic rough set, Mandal and

Ranadive (2019) introduced the optimistic and pessimistic

fuzzy preference relation models.

In a multi-criteria decision-making problem with multiple

data points, few data points, referred to as fuzzy numbers,

are utilized to arrive at a decision. In this regard, different

fuzzy numbers, such as triangular, trapezoidal, pentagonal

and hexagonal fuzzy numbers, have been reported. These

fuzzy numbers consider only a few data points to arrive at a

decision. For example, the number of data points in triangu-

lar fuzzy numbers is 3, in trapezoidal fuzzy numbers it is 4,

and in hexagonal fuzzy numbers it is 6. However, using a few

data points to represent data leads to the loss of information.

To address this situation, we generalize fuzzy numbers that

encompass more data points to represent the data, thus mini-

mizing the loss of information. Practically, when the decision

problem is highly sensitive to the number of data points, it

is reasonable to choose a larger value of n. The ﬂexibility in

implementing this idea is apparent in the case of data repre-

sentation by 2n+1(or2n) fuzzy numbers due to the choice

for n. Thus, we get a natural advancement to the existing

FMCDM methods.

In this paper, we deﬁne 2nand 2n+1 fuzzy numbers.

Clearly, 2n+1 fuzzy numbers yield triangular and pentagonal

cases when n=1,2,and 2nfuzzy numbers coincide with

trapezoidal and hexagonal fuzzy numbers when n=2,3,

respectively. We extend the fuzzy preference relation and

relative preference relation given by Wang (2015b) to rank

2nand 2n+1 fuzzy numbers. Then, we compare the results

obtained by fuzzy preference relation and relative preference

relation with Wang and Lee (2008) method. Wang (2014)

developed the FMCDM model with TOPSIS under fuzzy

environment and relative preference relation on fuzzy num-

bers. We present an extension to the FMCDM model when

the given data is representable in terms of 2n+1 fuzzy num-

bers. We illustrate the suitability of the proposed method in

solving FMCDM problems using an example. Subsequently,

the proposed method results are validated and compared with

VIKOR, MOORA and ELECTRE methods.

The rest of the paper is organized as follows. In Sect. 2,

we present the basic deﬁnitions and related primary results.

In Sect. 3, we give the deﬁnition of 2n+1 fuzzy number

and the extension of fuzzy preference and relative preference

relations on 2n+1 fuzzy numbers. In Sect. 4, we deﬁne 2n

fuzzy number and the extension of fuzzy preference and rela-

tive preference relations on 2nfuzzy numbers. In Sect. 5,we

present the proposed FMCDM model along with a telecom-

munication example. In Sect. 6, we validate the proposed

method with popularly used multi-criteria decision-making

methods.

2 Deﬁnitions and preliminaries

For the following deﬁnitions, we refer (Zadeh 1965;Zim-

mermann 1987,1991).

Deﬁnition 2.1 A fuzzy subset Aon the universe Uis a set

deﬁned by a membership function μArepresenting a map-

ping μA:U−→ [ 0,1].

Deﬁnition 2.2 Aα={x|μA(x)≥α}is called an α-cut of

the fuzzy set A.

Deﬁnition 2.3 Let Xbe a fuzzy number. Then, XL

αand XU

α

are, respectively, deﬁned as

XL

α=inf

μX(z)≥α(z)and XU

α=sup

μX(z)≥α

(z).

Deﬁnition 2.4 (Lee 2005a,b;Epp1990) A fuzzy preference

relation Ris a fuzzy subset of R×Rwith membership

function μR(A,B)representing preference degree of fuzzy

numbers Aover B.

1. Ris reciprocal if and only if μR(A,B)=1−μR(B,A)

for all fuzzy numbers Aand B.

2. Ris transitive if and only if μR(A,B)≥1

2and

μR(B,C)≥1

2⇒μR(A,C)≥1

2for all fuzzy numbers

A,Band C.

3. Ris a fuzzy total ordering if and only if Ris both recip-

rocal and transitive.

Ais preferred to Bif and only if μR(A,B)> 1

2and Ais

equal to Bif and only if μR(A,B)=1

2.

Deﬁnition 2.5 (Wang 2015b)Letbe a binary relation on

fuzzy numbers deﬁned by ABif and only if Ais preferred

to B(That is, μR(A,B)> 1

2).

Wang revised the extended fuzzy preference relation

deﬁned by Lee (2005b) as follows.

Deﬁnition 2.6 (Wang 2015b)LetAand Bbe two fuzzy num-

bers, where Ais an interval [al,ar]and Bis an interval

[bl,br].A fuzzy preference relation Pis a subset of R×R

with membership function μP(A,B)representing prefer-

ence degree of Aover B.

Deﬁne

μP(A,B)=1

21

0(A−B)L

α+(A−B)U

α

||T|| +1,

where

||T|| = 1

0

((T+−T−)L

α+(T+−T−)U

α)dαif t+

l≥t−

r.

123

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Generalization and ranking of fuzzy numbers by relative… 1103

=1

0

((T+−T−)L

α+(T+−T−)U

α

+2(t−

r−t+

l))dαif t+

l<t−

r.

T+is an interval [t+

l,t+

r],T−is an interval [t−

l,t−

r],t+

l=

max{al,bl},t+

r=max{ar,br},t−

l=min{al,bl}and t−

r=

min{ar,br}.

Similarly, fuzzy preference relation on triangular and

trapezoidal fuzzy numbers have also been deﬁned.

For the examples on decision-making problems, we refer

(Koppula et al. 2019,2020;Riazetal.2020; Chen and Huang

2021).

Deﬁnition 2.7 (Wang 2015b)Let S={X1,X2,...,Xn}

denote a set composed of nfuzzy numbers. A fuzzy number

Xi=[xil,xir]belongs to the set S,where i=1,2,...,n.

Assume X=iXi

nderived by extension principle is average

of the nfuzzy numbers in S.A relative preference relation

P∗with membership function μP∗(Xi,X)represents pref-

erence degree of Xiover Xin S.

We deﬁne

μP∗(Xi,X)=1

21

0(Xi−X)L

α+(Xi−X)U

α

||Ts|| +1,

where

Ts=1

0

((T+

s−T−

s)L

α+(T+

s−T−

s)U

α)dαif t+

sl ≥t−

sr ;

=1

0

((T+

s−T−

s)L

α+(T+

s−T−

s)U

α

+2(t−

sr −t+

sl ))dαif t+

sl <t−

sr ;

T+

sis an interval [t+

sl ,t+

sr ],T−

sis an interval [t−

sl ,t−

sr ],t+

sl =

maxi{Xil},t+

sr =maxi{Xir},t−

sl =mini{Xil}and t−

sr =

mini{Xir}.

Clearly, 0 <μ

P∗(Xi,X)<1,where i=1,2,...,n.

μP∗(Xi,X)<1

2expresses that Xis preferred to Xi.On the

other hand, μP∗(Xi,X)>1

2expresses that Xiis preferred

to X.

Similarly, relative preference relation is deﬁned on trian-

gular and trapezoidal fuzzy numbers.

3 Generalized 2n+1 fuzzy number

3.1 Generalized linear 2n+1fuzzynumber

Let {a1,a2,a3,...,a2n+1}be real numbers such that a1<

a2<a3<··· <a2n+1,n=1,2,3, ... and nis ﬁnite,

k≥2n−1.Then, we denote

P(x):= 1

kx−a1

a2−a1;

a1≤x≤a2

Sn(x):= 2n−2

k+2n−2

kx−an

an+1−an;

an≤x≤an+1

T(n,r)(x):= 2n−2r

k+2n−2r

kan+2−x

an+2−an+1;

an+1≤x≤an+2

Qn(x):= 1

ka2n+1−x

a2n+1−a2n;

a2n≤x≤a2n+1.

Now, [P(x), Q1(x)]gives fuzzy membership function of the

generalized triangular fuzzy number (a1,a2,a3;1

k).

That is,

f(a1,a2,a3;x)=⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

1

kx−a1

a2−a1;a1≤x≤a2

1

ka3−x

a3−a2;a2≤x≤a3.

If k=1 in the above, we get a triangular fuzzy number.

Now,

f(a1,a2,a3,...,a2n+1;x)

=[P(x), S2(x), S3(x), ··· ,Sn−1(x), Sn(x), T(n,1)(x),

T(n+1,2)(x), T(n+2,3)(x),...,T(2n−2,n−1)(x), Qn(x)]...(1)

gives fuzzy membership function of the generalized fuzzy

number (a1,a2,a3,...,a2n+1;2n−1

k), where n≥2 and k≥

2n−1.

In particular, if k=2n−1then (1) gives fuzzy membership

function of the fuzzy number (a1,a2,a3,...,a2n+1;1).

For example, substitute n=2 in (1), then [P(x), S2(x),

T(2,1)(x), Q2(x)]gives fuzzy membership function of the

generalized pentagonal fuzzy number (a1,a2,a3,a4,a5;2

k).

That is,

f(a1,a2,a3,a4,a5;x)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

1

kx−a1

a2−a1;a1≤x≤a2

1

k+1

kx−a2

a3−a2;a2≤x≤a3

1

k+1

ka4−x

a4−a3;a3≤x≤a4

1

ka5−x

a5−a4;a4≤x≤a5.

123

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1104 K. Koppula et al.

Fig. 1 Pentagonal and generalized pentagonal fuzzy number

If k=2 in the above, we get a pentagonal fuzzy number

(Fig. 1).

Similarly,substitute n=3 i n (1) then [P(x), S2(x), S3(x),

T(3,1)(x), T(4,2)(x), Q3(x)]gives fuzzy membership func-

tion of the generalized heptagonal fuzzy number (a1,a2,a3,

a4,a5,a6,a7;4

k).

That is,

f(a1,a2,a3,a4,a5,a6,a7;x)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

1

kx−a1

a2−a1;a1≤x≤a2

1

k+1

kx−a2

a3−a2;a2≤x≤a3

2

k+2

kx−a3

a4−a3;a3≤x≤a4

2

k+2

ka5−x

a5−a4;a4≤x≤a5

1

k+1

ka6−x

a6−a5;a5≤x≤a6

1

ka7−x

a7−a6;a6≤x≤a7.

If k=4 in the above, we get a heptagonal fuzzy number.

3.2 Generalized nonlinear 2n+1fuzzynumber

Let {a1,a2,a3,...,a2n+1}be real numbers such that a1<

a2<a3<··· <a2n+1,m,n=1,2,3, ... and m,nare

ﬁnite. Now, take k≥2n−1.Then, we denote

Pm(x):= 1

kx−a1

a2−a1m

;

a1≤x≤a2

Sm

n(x):= 2n−2

k+2n−2

kx−an

an+1−anm

;

an≤x≤an+1

Tm

(n,r)(x):= 2n−2r

k+2n−2r

kan+2−x

an+2−an+1m

;

an+1≤x≤an+2

Qm

n(x):= 1

ka2n+1−x

a2n+1−a2nm

;

a2n≤x≤a2n+1.

Then, [Pm(x), Qm

1(x)]gives fuzzy membership function of

the generalized nonlinear triangular fuzzy number (a1,a2,a3;

1

k).Now,

fm(a1,a2,a3,...,a2n+1;x)

=[Pm(x), Sm

2(x), Sm

3(x),...,Sm

n−1(x), Sm

n(x),

Tm

(n,1)(x), Tm

(n+1,2)(x), Tm

(n+2,3)(x), . . . ,

Tm

(2n−2,n−1)(x), Qm

n(x)]...(3)

gives fuzzy membership function of the generalized non-

linear fuzzy number (a1,a2,a3,...,a2n+1;2n−1

k), where

n≥2 and k≥2n−1.

In particular, if k=2n−1then (3) gives fuzzy membership

function of the nonlinear fuzzy number (a1,a2,a3,...,a2n+1;

1).

Note 3.3 If m=1 in the fuzzy membership function of the

generalized nonlinear 2n+1 fuzzy number, then we get a

fuzzy membership function of the generalized linear 2n+1

fuzzy number.

3.4 ˛-cut of a Generalized linear 2n+1fuzzy

number

Let {a1,a2,a3,...,a2n+1}be real numbers such that a1<

a2<a3<··· <a2n+1,n=1,2,3, ... and nis ﬁnite. Now,

take k≥2n−1.Then, we denote

E(α) := a1+kα(a2−a1);

α∈0,1

k

Fn(α) := an+αk

2n−2−1(an+1−an);

α∈2n−2

k,2n−1

k

G(n,r)(α) := an+2−αk

2n−2r−1(an+2−an+1);

α∈2n−2r

k,2n−2r+1

k

Hn(α) := a2n+1−kα(a2n+1−a2n);

α∈0,1

k.

123

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Generalization and ranking of fuzzy numbers by relative… 1105

Then, [E(α), H1(α)]gives α-cut of the generalized triangu-

lar fuzzy number (a1,a2,a3;1

k). That is,

h(a1,a2,a3;α)

=⎧

⎪

⎪

⎨

⎪

⎪

⎩

a1+kα(a2−a1);α∈0,1

k

a3−kα(a3−a2);α∈0,1

k.

If k=1 in the above, we get an α-cut of a triangular fuzzy

number.

Now,

h(a1,a2,a3,...,a2n+1;α)

=[E(α), F2(α), F3(α),...,Fn−1(α), Fn(α), G(n,1)(α),

G(n+1,2)(α), G(n+2,3)(α), . . . , G(2n−2,n−1)(α), Hn(α)]...(5)

gives α-cut of the generalized fuzzy number (a1,a2,

a3,...,a2n+1;2n−1

k), where n≥2 and k≥2n−1.

For example, if n=3 in (5) then [E(α), F2(α), F3(α),

G(3,1)(α), G(4,2)(α), H3(α)]gives α-cut of the generalized

heptagonal fuzzy number (a1,a2,a3,a4,a5,a6,a7;4

k). That

is,

h(a1,a2,a3,a4,a5,a6,a7;α)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

a1+kα(a2−a1);α∈0,1

k

a2+(αk−1)(a3−a2);α∈1

k,2

k

a3+αk

2−1(a4−a3);α∈2

k,4

k

a5−αk

2−1(a5−a4);α∈2

k,4

k

a6−(αk−1)(a6−a5);α∈1

k,2

k

a7−kα(a7−a6);α∈0,1

k.

If k=4 (That is k=2n−1=23−1)inthe

above, we get an α-cut of a heptagonal fuzzy number

(a1,a2,a3,a4,a5,a6,a7).

Inﬁmum and Supremum of α-cut of a 2n+1fuzzy num-

ber:

Let Bbe any fuzzy number and μB(x)is the fuzzy

membership function of B.Then BL

α=infμB(x)≥α(x)and

BU

α=supμB(x)≥α(x).

For example, if B=(a1,a2,a3;1

k)is a triangular fuzzy

number. Then, BL

α=E(α) and BU

α=H1(α).

For a fuzzy number C=(a1,a2,a3,...,a2n+1;2n−1

k),

CL

α=(E(α), F2(α), F3(α), . . . , Fn−1(α), Fn(α)) and

CU

α=(G(n,1)(α), G(n+1,2)(α), G(n+2,3)(α), . . . ,

G(2(n−1),n−1)(α), Hn(α)).

3.5 Extension of fuzzy preference relation on 2n+1

fuzzy numbers

We extend the fuzzy preference relation given by Wang

(2015b) to rank 2n+1 fuzzy numbers as follows.

Deﬁnition 3.6 Let X1and X2be two fuzzy numbers, where

X1=[a1,a2,...,a2n+1]and X2=[b1,b2,...,b2n+1].An

extended fuzzy preference relation Ris a subset of R×Rwith

membership function μR(X1,X2)representing preference

degree of X1over X2. Then,

μR(X1,X2)

=1

2⎛

⎝2n−1

k

0(X1−X2)α

L+(X1−X2)α

U

Tdα+1⎞

⎠

.....(I)

where

T= 2n−1

k

0

((T+−T−)α

L+(T+−T−)α

U)dα;

if t+

1≥t−

2n−1

2n−1

k

0

((T+−T−)α

L+(T+−T−)α

U+2(t−

2n+1−t+

1))dα;

if t+

1<t−

2n−1,

T+is an interval [t+

1,t+

2,...,t+

2n+1],T−is an interval

[t−

1,t−

2,...,t−

2n+1]and t+

1=max {a1,b1},t+

2=max {a2,b2},

...,t+

2n+1=max{a2n+1,b2n+1},t−

1=min {a1,b1},t−

2=min

{a2,b2},…,t−

2n+1=min {a2n+1,b2n+1}.

In (I), if 2n−1

k

0((T+−T−)α

L+(T+−T−)α

U)dα≥0,then

μR(X1,X2)≥1

2and if X1=X2,then μR(X1,X2)=1

2.

Lemma 3.7 The extended fuzzy preference relation Ris

reciprocal. That is, μR(X1,X2)=1−μR(X2,X1)for all

2n+1 fuzzy numbers X1and X2.

Lemma 3.8 The extended fuzzy preference relation Ris tran-

sitive. That is, if μR(X1,X2)≥1

2and μR(X2,X3)≥1

2then

μR(X1,X3)≥1

2,where X1,X2and X3are 2n+1 fuzzy

numbers.

From Lemmas 3.7 and 3.8, the extended fuzzy preference

relation Ris a total ordering relation (Epp (1990) and Lee

(2005b)).

123

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1106 K. Koppula et al.

Lemma 3.9 Let X1and X 2be two 2n+1fuzzy numbers. By

the extended fuzzy preference relation R, X1is preferred to

X2if and only if μR(X1,X2)> 1

2.

Lemma 3.10 X1X2if and only if μR(X1,X2)> 1

2,where

is a binary relation.

Result 3.11 Let X1=a1,a2,a3;1

kand X2=(b1,b2,b3;

1

kbe two generalized triangular fuzzy numbers. Then,

1

k

0[(X1−X2)L

α+(X1−X2)U

α]dα

=(a1−b3)+2(a2−b2)+(a3−b1)

2k.

Now,

μR(X1,X2)

=1

21

k

0(X1−X2)α

L+(X1−X2)α

U

Tdα+1

=1

2(a1−b3)+2(a2−b2)+(a3−b1)

2kT+1.

If k=1 in the above, we get

1

0

[(X1−X2)L

α+(X1−X2)U

α]dα

=(a1−b3)+2(a2−b2)+(a3−b1)

2.

Then, μR(X1,X2)=1

2((a1−b3)+2(a2−b2)+(a3−b1)

2T

+1)(Wang (2015b)).

Lemma 3.12 Let X1=(a1,a2,a3,a4,a5;2

k)and X2=

(b1,b2,b3,b4,b5;2

k)be two generalized pentagonal fuzzy

numbers. Then, μR(X1,X2)=1

2{[[(a1−b5)+2(a2−b4)+

2(a3−b3)+2(a4−b2)+(a5−b1)]/(2kT)]+1}.

Proof Now,

X1−X2=((a1−b5), (a2−b4), (a3−b3), (a4−b2), (a5−b1)).

Then,

2

k

0

(X1−X2)L

αdα

=1

k

0

((a1−b5)+kα((a2−b4)−(a1−b5)))dα

+2

k

1

k

((a2−b4)+(kα−1)((a3−b3)−(a2−b4)))dα

=(a1−b5)

k+(a2−b4)−(a1−b5)

2k

+2(a2−b4)

k+3((a3−b3)−(a2−b4))

2k−(a3−b3)

k.

Similarly,

2

k

0

(X1−X2)U

αdα

=1

k

0

((a5−b1)−kα((a5−b1)−(a4−b2))dα)

+2

k

1

k

((a4−b2)−(kα−1)((a4−b2)−(a3−b3))dα)

=(a5−b1)

k−(a5−b1)−(a4−b2)

2k+2(a4−b2)

k

−3((a4−b2)−(a3−b3))

2k−(a3−b3)

k.

Then,

2

k

0

(X1−X2)L

αdα+2

k

0

(X1−X2)U

αdα

=1

2k[(a1−b5)+2(a2−b4)

+2(a3−b3)+2(a4−b2)

+(a5−b1)].

Now,

μR(X1,X2)

=1

22

k

0(X1−X2)α

L+(X1−X2)α

U

Tdα+1

=1

2{[[(a1−b5)+2(a2−b4)+2(a3−b3)+2(a4−b2)

+(a5−b1)]/(2kT)]+1}.

Note 3.13 If k=2 in Lemma 3.12, then μR(X1,X2)=

1

2{[[(a1−b5)+2(a2−b4)+2(a3−b3)+2(a4−b2)+

(a5−b1)]/(4T)]+1}.

Lemma 3.14 Let X1=(a1,a2,...,an,an+1,an+2,...,a2n,

a2n+1;2n−1

k)and X2=(b1,b2,...,bn,bn+1,...,b2n,

b2n+1;2n−1

k)be two fuzzy numbers, where n≥3.

Then,

μR(X1,X2)=1

2{[[(a1−b2n+1)+2(a2−b2n)

+3[(a3−b2n−1)+2(a4−b2n−2)+4(a5−b2n−3)

+···+ 2n−3(an−bn+2)]+2n−1(an+1−bn+1)

+3[2n−3(an+2−bn)+2n−4(an+3−bn−1)

+...+2(a2n−2−b4)+(a2n−1−b3)]+2(a2n−b2)

+(a2n+1−b1)]/(2kT)]+1},

123

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Generalization and ranking of fuzzy numbers by relative… 1107

where

T= [(t+

1−t−

2n+1)+2(t+

2−t−

2n)+3[(t+

3−t−

2n−1)

+2(t+

4−t−

2n−2)+4(t+

5−t−

2n−3)+··· +2n−3(t+

n−t−

n+2)]

+2n−1(t+

n+1−t−

n+1)+3[2n−3(t+

n+2−t−

n)

+2n−4(t+

n+3−t−

n−1)+...+2(t+

2n−2−t−

4)+(t+

2n−1−t−

3)]

+2(t+

2n−t−

2)+(t+

2n+1−t−

1)]/2kif (t+

1−t−

2n+1)≥0.

={[(t+

1−t−

2n+1)+2(t+

2−t−

2n)+3[(t+

3−t−

2n−1)

+2(t+

4−t−

2n−2)+4(t+

5−t−

2n−3)+...+2n−3(t+

n−t−

n+2)]

+2n−1(t+

n+1−t−

n+1)+3[2n−3(t+

n+2−t−

n)+2n−4(t+

n+3−t−

n−1)

+...+2(t+

2n−2−t−

4)

+(t+

2n−1−t−

3)]+2(t+

2n−t−

2)+(t+