Generalization and ranking of fuzzy numbers by relative preference relation

Abstract

We define $$2n+1$$ 2 n + 1 and 2 n fuzzy numbers, which generalize triangular and trapezoidal fuzzy numbers, respectively. Then, we extend the fuzzy preference relation and relative preference relation to rank $$2n+1$$ 2 n + 1 and 2 n fuzzy numbers. When the data is representable in terms of $$2n+1$$ 2 n + 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.

Full text

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)Letbe a binary relation on
fuzzy numbers defined by ABif 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
21
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.
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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
21
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
kx−a1
a2−a1;
a1≤x≤a2
Sn(x):= 2n−2
k+2n−2
kx−an
an+1−an;
an≤x≤an+1
T(n,r)(x):= 2n−2r
k+2n−2r
kan+2−x
an+2−an+1;
an+1≤x≤an+2
Qn(x):= 1
ka2n+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
kx−a1
a2−a1;a1≤x≤a2
1
ka3−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
kx−a1
a2−a1;a1≤x≤a2
1
k+1
kx−a2
a3−a2;a2≤x≤a3
1
k+1
ka4−x
a4−a3;a3≤x≤a4
1
ka5−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
kx−a1
a2−a1;a1≤x≤a2
1
k+1
kx−a2
a3−a2;a2≤x≤a3
2
k+2
kx−a3
a4−a3;a3≤x≤a4
2
k+2
ka5−x
a5−a4;a4≤x≤a5
1
k+1
ka6−x
a6−a5;a5≤x≤a6
1
ka7−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
kx−a1
a2−a1m
;
a1≤x≤a2
Sm
n(x):= 2n−2
k+2n−2
kx−an
an+1−anm
;
an≤x≤an+1
Tm
(n,r)(x):= 2n−2r
k+2n−2r
kan+2−x
an+2−an+1m
;
an+1≤x≤an+2
Qm
n(x):= 1
ka2n+1−x
a2n+1−a2nm
;
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
Tdα+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 X1X2if and only if μR(X1,X2)> 1
2,where
is a binary relation.
Result 3.11 Let X1=a1,a2,a3;1
kand X2=(b1,b2,b3;
1
kbe 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
21
k
0(X1−X2)α
L+(X1−X2)α
U
Tdα+1
=1
2(a1−b3)+2(a2−b2)+(a3−b1)
2kT+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)
2T
+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)]/(2kT)]+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
22
k
0(X1−X2)α
L+(X1−X2)α
U
Tdα+1
=1
2{[[(a1−b5)+2(a2−b4)+2(a3−b3)+2(a4−b2)
+(a5−b1)]/(2kT)]+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)]/(4T)]+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)]/(2kT)]+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+