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Symmetry 2017, 9, 228; doi:10.3390/sym9100228
www.mdpi.com/journal/symmetry
Article
Decomposition and Intersection of Two Fuzzy
Numbers for Fuzzy Preference Relations
Hui-Chin Tang
Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences,
Kaohsiung 80778, Taiwan; tang@kuas.edu.tw
Received: 11 September 2017; Accepted: 9 October 2017; Published: 14 October 2017
Abstract: In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy
preference relation is the reasonable representation of preference relations by a fuzzy membership
function. This paper studies Nakamura’s and Kołodziejczyk’s preference relations. Eight cases, each
representing different levels of overlap between two triangular fuzzy numbers are considered. We
analyze the ranking behaviors of all possible combinations of the decomposition and intersection of
two fuzzy numbers through eight extensive test cases. The results indicate that decomposition and
intersection can affect the fuzzy preference relations, and thereby the final ranking of fuzzy
numbers.
Keywords: fuzzy number; ranking; preference relations
1. Introduction
For solving decision-making problems in a fuzzy environment, the overall utilities of a set of
alternatives are represented by fuzzy sets or fuzzy numbers. A fundamental problem of a decision-
making procedure involves ranking a set of fuzzy sets or fuzzy numbers. Ranking functions,
reference sets and preference relations are three categories with which to rank a set of fuzzy numbers.
For a detailed discussion, we refer the reader to surveys by Chen and Hwang [1] and Wang and Kerre
[2,3]. For ranking a set of fuzzy numbers, this paper concentrates on those fuzzy preference relations
that are able to represent preference relations in linguistic or fuzzy terms and to make pairwise
comparisons. To propose the fuzzy preference relation, Nakamura
[4] employed a fuzzy minimum
operation followed by the Hamming distance. Kołodziejczyk [5] considered the common part of two
membership functions and used the fuzzy maximum and Hamming distance. Yuan [6] compared the
fuzzy subtraction of two fuzzy numbers with real number zero and indicated that the desirable
properties of a fuzzy ranking method are the fuzzy preference presentation, rationality of fuzzy
ordering, distinguishability and robustness. Li [7] included the influence of levels of possibility of
dominance. Lee [8] presented a counterexample to Li’s method [7] and proposed an additional
comparable property. The methods of Wang et al. [9] and Asady [10] were based on deviation degree.
Zhang et al. [11] presented a fuzzy probabilistic preference relation. Zhu et al. [12] proposed hesitant
fuzzy preference relations. Wang [13] adopted the relative preference degrees of the fuzzy numbers
over average.
This paper evaluates and compares two fundamental fuzzy preference relations—one is
proposed by Nakamura [4] and the other by Kołodziejczyk [5]. The intersection of two membership
functions and the decomposition of two fuzzy numbers are main differences between these two
preference relations. Since the desirable criteria cannot easily be represented in mathematical forms,
their performance measures are often tested by using test examples and judged intuitively. To this
end, we consider eight complex cases that represent all the possible cases the way two fuzzy numbers
can overlap with each other. For Nakamura’s and Kołodziejczyk’s fuzzy preference relations, this
Symmetry 2017, 9, 228 2 of 17
paper analyzes and compares the ordering behaviors of the deco m p osition and in t ersecti o n
th r ough a group of extensive cases.
The organization of this paper is as follows—Section 2 briefly reviews the fuzzy sets and fuzzy
preference relations and presents the eight test cases. Section 3 analyzes Nakamura’s fuzzy preference
relation and presents an algorithm. Section 4 presents the behaviors of Kołodziejczyk’s fuzzy
preference relation. Section 5 analyzes the effect of the decomposition and intersection on fuzzy
preference relations. Finally, some concluding remarks and suggestions for future research are
presented.
2. Fuzzy Sets and Test Problems
We first review the basic notations of fuzzy sets and fuzzy preference relations. Consider a fuzzy
set A defined by a universal set of real numbers ℛ by the membership function (x), where
]1,0[:)( →ℜxA .
Definition 1. Let A be a fuzzy set. The support of A is the crisp set }0)({ >ℜ∈= xAxS A. A is called
normal when 1)(sup =
∈xA
A
Sx . An -cut of A is a crisp set })({
α
α
≥ℜ∈= xAxA . A is convex if, and
only if, each of its
α
-cut is a convex set.
Definition 2. A normal and convex fuzzy set whose membership function is piecewise continuous is called a
fuzzy number.
Definition 3. A triangular fuzzy number A, denoted
),,( cbaA =
, is a fuzzy number with membership
function given by:
≤≤
≤≤
=−
−
−
−
otherwise0
if
if
)( cxb
bxa
xA bc
xc
ab
ax
where −∞<≤≤<∞. The set of all triangular fuzzy numbers on ℛ is denoted by )(TF ℜ.
Definition 4. For a fuzzy number A, the upper boundary set ̅ of A and the lower boundary set of A are
respectively defined as:
()=sup
()
and:
() = sup
().
Definition 5. The Hamming distance between two fuzzy numbers A and B is defined by:
d(
,)= |
()−()|
=
()−()
()()+()−
()
()().
Definition 6. Let A and B be two fuzzy numbers and × be an operation onℛ, such as +, ─, *, ÷…. By
extension principle, the extended operation ⊗ on fuzzy numbers can be defined by:
Symmetry 2017, 9, 228 3 of 17
)}(),(min{sup)(
:,
yBxAz
yxzyx
BA ×=
⊗
=
μ
.
Definition 7. A fuzzy preference relation R is a fuzzy binary relation with membership function
),( BAR
indicating the degree of preference of fuzzy number A over fuzzy number B.
1. R is reciprocal if, and only if,
),(1),( ABRBAR −=
for all fuzzy numbers A and B.
2. R is transitive if, and only if,
5.0),( ≥BAR
and
5.0),( ≥CBR
implies
5.0),( ≥CAR
for
all fuzzy numbers A, B and C.
3. R is a fuzzy total ordering if, and only if, R is both reciprocal and transitive.
4. R is robust if, and only if, for any given fuzzy numbers , and ε>0, there exists δ>0 for
which |(,)−(,)|<, for all fuzzy number ′ and max
(|inf−inf
|,|sup−
sup|)<.
For simplicity, we denote
),(' BAR
for the degree of preference of fuzzy number B over fuzzy
number A.
The evaluation criteria for the comparison of two fuzzy numbers cannot easily be represented
in mathematical forms therefore it is often tested on a group of selected examples. The membership
functions of two fuzzy numbers can be overlapping/nonovelapping, convex/nonconvex, and
normal/non-normal. All the approaches proposed in the literature seem to suffer from some
questionable examples, especially for the portion of overlap between two membership functions.
Let (,,) and (,,) be two triangular fuzzy numbers. Figure 1 displays eight test
cases of representing all the possible cases the way two fuzzy numbers A and B can overlap with each
other. Table 1 shows the area of i-th region in each case. More precisely, the eight extensive test
cases are as follows:
Case 1. ≤
, ≤
, ≤
.
Case 2. ≤
, ≥
, ≤
.
Case 3. ≤
, ≤
, ≥
.
Case 4. ≤
, ≥
, ≥
.
Case 5. ≥
, ≤
, ≤
.
Case 6. ≥
, ≥
, ≤
.
Case 7. ≥
, ≤
, ≥
.
Case 8. ≥
, ≥
, ≥
.
Case 1. ≤
, ≤
, ≤
.
Symmetry 2017, 9, 228 4 of 17
Case 2. ≤
, ≥
, ≤
Case 3. ≤
, ≤
, ≥
.
Case 4. ≤
, ≥
, ≥
.
Case 5. ≥
, ≤
, ≤
.
Symmetry 2017, 9, 228 5 of 17
Case 6. ≥
, ≥
, ≤
.
Case 7. ≥
, ≤
, ≥
.
Case 8. ≥
, ≥
, ≥
.
Figure 1. Eight test cases for two fuzzy numbers (
,
,
) and (
,
,
).
Table 1. The area
of i-th region for eight cases.
Case Area
1
= −(
−−(−
+−))
(
)/(
)
=
−+(−−
+)
−
=
−−(−−
+
)
−
= (
−−(−
+−)
(
)/(
)
2
=
−+(−−
+)
(
)/(
)
= −(
−−(−+
−))
(
)/(
)
=
−+(−−+)
(
)/(
)−
=
−−(−
−+)
(
)/(
)−
Symmetry 2017, 9, 228 6 of 17
=
−−(−−
+
)
()/()
= (
−−(−+
−))
()/()
−
−
3
= −(
−−(−
+−))
()/()
=
−+(−−
+)
−
=
−
−(−−+
)
()/()−
=
−−(−
−+)
()/()
= (
−−(−
+−)
()/()
−
4
=
−+(−−
+)
()/()
= −(
−−(−+
−))
()/()
=
−+(−−+)
()/()−
=
−−(−
−+)
−
= (
−−(−+
−))
()/()
−
5
=
−+(−−+)
()/()
= −(
−−(−
+−))
()/()
=
−+(−−
+)
()/()−
=
−−(−−
+
)
−
= (
−−(−
+−)
()/()
−
6
= −(
−−(−+
−))
()/()
=
−+(−−+)
−
=
−−(−
−+)
()/()−
=
−−(−−
+
)
()/()
= (
−−(−+
−))
()/()
−
7
=
−+(−−+)
()/()
= −(
−−(−
+−))
()/()
=
−+(−−
+)
()/()−
Symmetry 2017, 9, 228 7 of 17
=
−
−(−−+
)
()/()−
=
−−(−
−+)
()/()
= (
−−(−
+−)
()/()
−
−
8
= −(
−−(−+
−))
()/()
=
−+(−−+)
−
=
−−(−
−+)
−
= (
−−(−+
−))
()/()
3. Nakamura’s Fuzzy Preference Relation
Using fuzzy minimum, fuzzy maximum, and Hamming distance, Nakamura’s fuzzy preference
relations [4] are defined as follows:
Definition 8. For two fuzzy numbers A and B, Nakamura [4] defines (,) and ′(,) as fuzzy
preference relations by the following membership functions:
(
,)=(
,
,)+(
,
,)
,+
,
and:
′(
,)=(
∩,0)+(A,
(
,B))
(
,0)+(,0)
respectively. Yuan [6] showed that N(,) is reciprocal and transitive, but not robust. Wang and Kerre [3]
derived that:
,
, = ,
,
,
, = (,
,)
,
,+
,
, =
,
,
,+
,
, =
,
and:
2(A∩B,0)+A,
(
,B)+B,
(
,B)=(
,0)+(,0).
It follows that: (
,)+(,
)=1
and: ′(
,)+′(,
)=1.
For two triangular fuzzy numbers (,,) and (,,), then:
=[,]=[
+(−),−(−
)]
=[,]=[
+(−),−(−)]
so:
(
,)=
,
,+
,
,
,+
,
Symmetry 2017, 9, 228 8 of 17
=
.
Define:
=
−
=
−+(−−+)
()
=
−
=
−+(−−
+)
()
=
−
=
−−(−
−+)
()
=−
=−
−(−−
+
)
() ,
then:
(
,)=
.
Let =
−, ℬ=
−
and =
−
. The steps for implementing the Nakamura’s
fuzzy preference relation (,) are as in Algorithm 1:
Algorithm 1. Nakamura’s fuzzy preference relation
If ≥0
If ≥0
If ℬ≥0, then (
,)=0 else (
,)=ℬ(ℬ)
(ℬ)(ℬ)(ℬ)(ℬ).
else if ℬ≥0, then (
,)=
(ℬ)(ℬ)ℬ else
(
,)=1−
(ℬ)(ℬ)ℬ.
else if ≥0
If ℬ≥0, then (
,)=
(ℬ)(ℬ)ℬ else
(
,)=1−
(ℬ)(ℬ)ℬ.
else if ℬ≥0, then (
,)=1− ℬ(ℬ)
(ℬ)(ℬ)(ℬ)(ℬ) else (
,)=1.
Table 2 shows the values of (,) and ′(,) for each test case. The first observation of this
table is that: (
,)+
(
,)=1
(
,)+
(
,)=1
(
,)+
(
,)=1
(
,)+
(
,)=1.
Secondly, comparing the values of (,) with that of ′(,), we have that 1−
(,)≥
(,) and 1−
(,)≤
(,). If +2+≥
−2
−
, we obtain that 1−
(,)≤
(,), 1−
(,)≥
(,), 1−
(,)≤
(,), 1−
(,)≥
(,), 1−
(,)≤
(,) and 1−
(,)≥
(,).
Table 2. (,) and ′(,) for eight cases.
Case (,) ′(,)
1 0 1+ ()
()()
2 ()(()())
(()())()(()())() ()
()()
3 ()
()()()() 1+ ()
()()
Symmetry 2017, 9, 228 9 of 17
4 1− ()
()()()() ()
()()
5 ()
()()()() 1+ ()
()()
6 1− ()
()()()() ()
()()
7 1− ()()
(()())()(()())() 1+ ()
()()
8 1 ()
()()
4. Kołodziejczyk’s Fuzzy Preference Relation
By considering the common part of two membership functions, Kołodziejczyk’s method [5] is
based on fuzzy maximum and Hamming distance to propose the following fuzzy preference
relations:
Definition 9. For two fuzzy numbers A and B, Kołodziejczyk [5] defines 1′(,) and 2′(,) as fuzzy
preference relations by the following membership functions:
1′(
,)=
,
,+
,
,+(
∩,0)
,+
,+2(
∩,0)
and:
2′(
,)=
,max
,+
,max
,
,+
,
respectively. 1′(,) is reciprocal, transitive and robust [3,5]. Since:
2′(
,)=1−(
,)
the results of 2′(,) can be obtained from those of (,).
For two triangular fuzzy numbers (,,) and (,,), then:
=[,]=[
+(−),−(−
)]
=[,]=[
+(−),−(−)].
Define:
=
,
, = −
=
−
,=
+
=,, = −
=
−
,=
+
and: =(
∩,0)=−
−−
−−
.
Symmetry 2017, 9, 228 10 of 17
Then:
1′(
,)=++
++++2
and:
2′(
,)=
.
In Table 3, we display the values of 1′(,) and 2′(,) for each test case. An examination
of the table reveals that:
1
(
,)=1
(
,)=1
(
,)=1
(
,)
= 1 − (−)
(−+−)(−−
+)+2(−
)
and:
1
(
,)=1
(
,)=1
(
,)=1
(
,)=()
()()()().
If =
, we have:
1
(
,)=1− −
(−+−)=−
(−+−)
and:
1
(
,)=−
(−+−)
so:
1
(
,)=1
(
,)
and:
1
(
,)+1
(
,)=()
().
It follows that:
1
(
,)+1
(
,)=0 for
=
and =
.
and:
1
(
,)+1
(
,)=1 for
=
and −=
−.
Table 3. 1′(,) and 2′(,) for eight cases.
Case ′(,) ′(,)
1 1− ()
()()()1
2 ()
()()()()1− ()(()())
(()())()(()())()
3 1− ()
()()()1− ()
()()()()
4 ()
()()()()()
()()()()
5 1− ()
()()()1− ()
()()()()
6 ()
()()()() ()
()()()()
7 1− ()
()()()()()
(()())()(()())()
8 ()
()()()()0
5. Two Comparative Studies of Decomposition and Intersection of Two Fuzzy Numbers
If the fuzzy number A is less than the fuzzy number B, then the Hamming distance between A
and (,) is large. Two representations are adopted. One is d(,(,)). The other is
Symmetry 2017, 9, 228 11 of 17
,, +,, which decomposes A into and . To analyze the effect of
decomposition, we consider the following preference relations without decomposition:
1′(
,)=(
,max
(
,)) +(
∩,0)
(
,0)+(,0)
and:
2′(
,)=(
,max
(
,))
(
,)
which are the counterparts of the Kołodziejczyk’s preference relations 1′(,) and 2′(,).
Therefore, the preference relations 1′(,) and 2′(,) consider the decomposition of fuzzy
numbers, while 1′(,) and 2′(,) do not. The preference relations 1′(,) and 1′(,)
consider the intersection of two membership functions, while 2′(,) and 2′(,) do not. For
completeness, Table 4 displays the values of (,), ′(,), 1′(,), 2′(,), 1′(,) and
2′(,)of each test case in terms of the values of . The 1′(,) considers both decomposition
and intersection of two fuzzy numbers, while 2′(,) do not. From 1′(,) to 2′(,), two
representations are:
1′(
,)→2′(
,)→2′(
,)
and:
1′(
,)→1′(
,)→2′(
,).
The first feature of Table 4 is that the differences between 1′(,) and 1′(,) and between
2′(,) and 2′(,) are . More precisely, the numerators and denominators of both 1′(,)
and 2′(,) include 2 for cases 1, 3, 5 and 7, the denominators of both 1′(,) and 2′(,)
include 2 for cases 2, 4, 6 and 8. Therefore, 2 represents the effect of the decomposition of fuzzy
numbers. The differences between 1′(,) and 2′(,) and between 1′(,) and 2′(,)
are . More precisely, the numerators and denominators of both 1′(,) and 1′(,) include
and 2, respectively. Therefore, represents the effect of the intersection of two membership
functions. After some computations, the characteristics of 1′(,), 2′(,), 1′(,) and
2′(,) are described as follows:
Table 4. (,), ′(,), 1′(,), 2′(,), 1′(,) and 2′(,) for eight cases.
Case (,) ′(,) ′(,) ′(,) ′(,) ′(,)
1 0
1
1
2
3
4
5
6
+
+++2 +
+2+++2
+2++ +
+++2
++
7
8 1
0
0
Theorem 1. Let 2′(,)=
.
(1) If ≤
, β≤2
+ or ≥
, β+2≤, then 1′(,)≤2′
(,). If ≤
, β≥
2+ or ≥
, β+2≥, then 1′(,)≥2′
(,).
(2) If ≤
, then 2′(,)≥2′
(,). If ≥
, then 2′(,)≤2′
(,).
(3) If α≥β, then 1′(,)≤2′
(,). If α≤β, then 1′(,)≥2′
(,).
(4) If ≤
, then 1′(,)≥1′
(,). If ≥
, then 1′(,)≤1′
(,).
(5) If ≤
, β(2+)≤
or ≥
, β≤(
+2), then 1′(,)≤2′
(,). If
≤
, β(2+)≥
or ≥
, β≥(
+2), then 1′(,)≥2′
(,).
Symmetry 2017, 9, 228 12 of 17
For each test case of two triangular fuzzy numbers (,,) and (,,), we analyze the
behaviors of 1′(,), 2′(,), 1′(,) and 2′(,) by applying Theorem 1 as follows:
Firstly, for ≤
, we have:
1′(
,)≤1′
(
,)≤2′
(
,)=2′(
,)=1
for case 1. For cases 3, 5 and 7, we have the following results.
(1) From 2+−β=
(), we have that if +2+≥
+2
+
, then
1′(,)≤2′
(,); if +2+≤
+2
+
, then 1′(,)≥2′
(,).
(2) 2′(,)≥2′
(,).
(3) From α−β=()()()()
(), it follows that if ++≥
+
+
,
then 1′(,)≤2′
(,); if ++≤
+
+, then 1′(,)≥2′
(,).
(4) 1′(,)≥1′
(,).
(5) If +2+≥
+2
+, then 1′(,)≥2′
(,). If +2+≤
+2
+
,
then 1′(,)≤2′
(,).
Therefore, for the cases 3, 5 and 7, if +2+≤
+2
+, then:
1′(
,)≥2′
(
,)≥2′(
,)
and:
1′(
,)≥1′
(
,)≥2′(
,).
Secondly, for ≥
, we have:
2′(
,)=2′
(
,)=0≤1′
(
,)≤1′
(
,)
for case 8. For cases 2, 4 and 6, we have the following results.
(1) From −2
−β=
(), we obtain if +2+≥
+2
+, then
1′(,)≤2′
(,); if +2+≤
+2
+
, then 1′(,)≥2′
(,).
(2) 2′(,)≤2′
(,).
(3) From α−β=
(−+−2
+2−
++()
), it follows that
if +2+≥
+2
+, then 1′(,)≤2′
(,); if +2+≤
+2
+
,
then 1′(,)≥2′
(,).
(4) 1′(,)≤1′
(,).
(5) If +2+≥
+2
+, then 1′(,)≤2′
(,). If +2+≤
+2
+
,
then 1′(,)≥2′
(,).
Therefore, for the cases 2, 4 and 6, if +2+≥
+2
+, then:
1′(
,)≤2′
(
,)≤2′(
,)
and:
1′(
,)≤1′
(
,)≤2′(
,).
For the two triangular fuzzy numbers (,,) and (,,), the second comparative
study is comprised of the five case studies shown in Figure 2, which compares the fuzzy preference
relations 1′(,), 2′(,), 1′(,) and 2′(,).
Case (a)
(,,) and (,,) with ≥
.
Symmetry 2017, 9, 228 13 of 17
Case (b)
(−,,+) and (−,,+).
Case (c)
(,+,+2) and (+α,++α+β,+α+2+2β).
Case (d) A(,, +) and B(,+,+).
Case (e)
(+,b,1−+) and (,0.5,1− ).
Figure 2. Five case studies of A and B for 1′(,), 2′(,), 1′(,) and 2′(,).
Symmetry 2017, 9, 228 14 of 17
Case (a) (,,) and (,,) with ≥
.
It follows that:
1′(
,)=(+)+(+)+0
(+)+(+)+0=1
2(,) = (+)+(+)
(+)+(+)=1
1′(
,)=0+(+)
(+)=1
and:
2′(
,)=()
()=1. (1)
For this simple case, all the preference relations give the same degree of preference of B over A.
Case (b) (−,,+) and (−,,+).
We have:
1′(
,)=+0+
++2=1/2
2(,)=+0
+=1/2
1′(
,)=+
(++)+=1/2
and:
2′(
,)=
=1/2.
From the viewpoint of probability, the fuzzy numbers A and B have the same mean, but B has a
smaller standard deviation. The results indicate that the differences between the decomposition and
intersection of A and B cannot affect the degree of preference for B over A.
Case (c) (,+,+2) and (+α,++α+β,+α+2+2β).
For this case, the fuzzy number B is a right shift of A. Therefore, B should have a higher ranking
than A based on the intuition criterion. We obtain:
1′(
,)=(+)+(+)+
(+)+(+)+2=(2 + + 2)
2(+4+4+2+2)>1/2
2(,)=(+)+(+)
(+)+(+)=1
1′(
,)=+(+)
(+)+(+)=1−(−2+ )
2(2+)
and:
2′(
,)=
=1.
All methods prefer B, but 1′(,) is indecisive. More precisely,
If 2+ < , then 1′(,)<1/2, so >
If 2+ = , then 1′(,)=1/2, so =
If 2+ > , then 1′(,)>1/2, so <.
Hence, a conflicting ranking order of 1′(,) exists in this case.
Case (d) (,,+) and (,+,+) with ≥.
This case is more complex for the partial overlap of A and B. The membership function of B has
the right peak, B expands to the left of A for the left membership function, and A expands to the right
of B for the right membership function. We have:
Symmetry 2017, 9, 228 15 of 17
1′(,)=(+)+(+)+
(++)+(++)+2=0.5+ (−2++2)
+3+2+−2−2
2(,) = (+)+(+)
(++)+(++)=(−+ + )
2++2+2−2 −4
1′(,)=+(+)
(++)+(
++)=(+3−)(−++)
4
and:
2′(
,)=
=()
.
It follows that:
If −2+ +2 < 0, then 1′(,)<1/2 and 2(,) < 1/2, so >;
If −2+ +2 = 0, then 1′(,)=1/2 and 2(,) = 1/2, so =;
If −2+ +2 > 0, then 1′(,)>1/2 and 2(,) > 1/2, so <;
If <(1+√2)(−), then 1′(,)<1/2 and 2′(,)<1/2, so >;
If =(1+√2)(−), then 1′(,)=1/2 and 2′(,)=1/2, so =;
If >(1+√2)(−), then 1′(,)>1/2 and 2′(,)>1/2, so <.
Three special subcases are considered as follows:
(1) Subcase (d1) If =(1+√2)(−), then (,,2+√2− (1+√2)) and (,1+√2−
√2,(1+√2)− √2), therefore 1′(,)=2′
(,)=0.5, so =. However, 1′(,)=
√
, 2(, ) = 2/3 and <.
(2) Subcase (d2) If =2(−), then (,,3−2) and (,2−,2−), therefore 1′(,)=
7/16, 2′(,)=1/3, so >. However, 1′(,)=2
(,) = 0.5 and =.
(3) Subcase (d3) If (0.3,0.3,0.9) and (0.1,0.7,0.7), then 1′(,)= 0.5556, 2(, ) = 0.6667,
1′(,)= 0.5556，2′(,)= 0.6667, so <.
Therefore, if <2(−), then 1′(,)<1/2, 2(,) < 1/2 , 1′(,)<1/2 and
2′(,)<1/2, so >; if >(1+√2)(−) , then 1′(,)>1/2, 2(, ) > 1/2 ,
1′(,)>1/2 and 2′(,)>1/2, so <.
Case (e) (+,b,1−+) and (,0.5,1 −).
For this case, the membership function B is symmetric with respect to =0.5. The membership
function of A is parallel translation of that of B except its peak. We have the following results:
(1) 1′(,)=()()
()()=
. If (1-2a-2b)(3+2a-2b-4c)<0,
then 1(,) < 1/2, so >. For simplicity, the other two conditions are omitted.
(2) 2(,) = ()()
()()=()
(). If 2 + 2 −1 > 0, then 2(,) < 1/2, so >
.
(3) 1′(,)=()
()()=2(1-b-c)(1-2c)-
2(1+2a-2b)(3+2a-2b-4c)(1-2c) and 2′(,)=
=
()()
()()(). If >−0.5++
(1− 2)(3 −4− 2), then 1′(,)<1/2 and
2′(,)<1/2, so >.
Four special subcases are considered as follows:
(1) Subcase (e1) If =−0.5++
(1− 2)(3 −4− 2), then 1′(,)=2′
(,)=0.5, so
=. However, 1′(,)>0.5, 2(,) > 0.5 and <.
(2) Subcase (e2) If 2 +2 − 1 = 0 , then 1′(,)=0.5− ()
()()<0.5, 2′(,)=
()
()<0.5, so >. However, 1′(,)=2
(,) = 0.5 and =.
(3) Subcase (e3) If A(0.3,0.4,0.9) and B(0.2,0.5,0.8), then 1′(,)= 0.4896, 2(,) = 0.4286,
1′(,)= 0.4896, 2′(,)= 0.4286, so >.
(4) Subcase (e4) If ≥0.5, then (1-2a-2b)(3+2a-2b-4c)<0, 2 + 2 −1 > 0 and >−0.5++
Symmetry 2017, 9, 228 16 of 17
(1− 2)(3 −4−2) , so 1(,) < 1/2 , 2(,) < 1/2 , 1′(,)<1/2 and
2′(,)<1/2, hence >.
6. Conclusions
This paper analyzes and compares two types of Nakamura’s fuzzy preference relations—
((,) and ′(,))—two types of Kołodziejczyk’s fuzzy preference relations—(1′(,) and
2′(,))—and the counterparts of the Kołodziejczyk’s fuzzy preference relations—(1′(,) and
2′(,))—on a group of eight selected cases, with all the possible levels of overlap between two
triangular fuzzy numbers (,,) and (,,). First, for (,) and ′(,) we obtain
that (,)+
(,)=1, = 1,2,3,4 . If +2+≥
−2
−
, we have that 1−
(,) ≥ (,) for = 1,3,5,7 and 1−(, ) ≤ (,) for = 2,4,6,8 . Secondly, for
1′(,) and 2′(,), we have that 1
(,)=1
(,) for = 3,5,7 and 1
(,)=
1
(,) for =4,6,8. Furthermore, 1
(,)+1
(,)=0for
=
and
=
and
1
(,)+1
(,)=1for
=
and
−=
−. Thirdly, for test case 1, 1′(,)≤
1′(,)≤2′
(,)=2′
(,)=1. For the test cases 3, 5 and 7, if +2+≤
+2
+,
then 1′(,)≥2′
(,)≥2′(,) and 1′(,)≥1′
(,)≥2′(,). For the test case 8, we
have 2′(,)=2′
(,)=0≤1′
(,)≤1′
(,). For the test cases 2, 4 and 6, if +2+
≥
+2
+
, then 1′(,)≤2′
(,)≤2′(,) and 1′(,)≤1′
(,)≤2′(,).
These results provide insights into the decomposition and intersection of fuzzy numbers. Among the
six fuzzy preference relations, the appropriate fuzzy preference relation can be chosen from the
decision-maker’s perspective. Given this fuzzy preference relation, the final ranking of a set of
alternatives is derived.
Worthy of future research is extending the analysis to other types of fuzzy numbers. First, the
analysis can be easily extended to the trapezoidal fuzzy numbers. Second, for the hesitant fuzzy set
lexicographical ordering method, Liu et al. [14] modified the method of Farhadinia [15] and this was
more reasonable in more general cases. Recently, Alcantud and Torra [16] provided the necessary
tools for the hesitant fuzzy preference relations. Thus, the analysis of hesitant fuzzy preference
relations is a subject of considerable ongoing research.
Conflicts of Interest: The author declares no conflict of interest.
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