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Volume 2012, Year 2012 Article ID jfsva-00051, 9 pages
doi:10.5899/2012/jfsva-00051
Research Article
The Median Value of Fuzzy Numbers and its
Applications in Decision Making
Rahim Saneifard
1∗
, Rasoul Saneifard2
(1) Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh, Iran.
(2) Department of Engineering Technology, Texas Southern University, Houston, Texas, USA.
c Rahim Saneifard and Rasoul Saneifard. This is an open access article distributed under
Copyright 2011 ⃝
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Abstract
In this paper, the researchers proposed a new method to rank fuzzy numbers by median
value of fuzzy numbers. The median value can be used as a crisp approximation with
respect to a fuzzy quantity. Therefore, a method for ordering the fuzzy numbers is defined
accordingly. This method can effectively rank various fuzzy numbers, their images and
overcome the shortcomings of the previous techniques.
Keywords : Ranking; Fuzzy numbers; Defuzzification; Interval-Value; Central interval.
1
Introduction
In many applications, ranking of fuzzy numbers is an important component of the decision process. In addition to a fuzzy environment, ranking is a very important decision
making procedure. Since Jain [9, 10] employed the concept of maximizing set to order
the fuzzy numbers in 1976 (1978), many authors have investigated various ranking methods. Some of these ranking methods have been compared and reviewed by Bortolan and
Degani [2], and more recently by Chen and Hwang [4]. Other contributions in this field
include: an index for ordering fuzzy numbers defined by Choobineh and Li [6], ranking
alternatives using fuzzy numbers studied by Dias [7], automatic ranking of fuzzy numbers
using artificial neural networks proposed by Requena et al. [11], ranking fuzzy values with
satisfaction function investigated by Lee et al. [5], ranking and defuzzification methods
based on area compensation presented by Fortemps and Roubens [8], and ranking alternatives with fuzzy weights using maximizing set and minimizing set given by Raj and
∗
Corresponding author. Email address: [email protected] Tel:+989149737077
1
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Journal of Fuzzy Set Valued Analysis
Kumar [12]. However, some of these methods are computationally complex and difficult
to implement, and others are counterintuitive and not discriminating. Furthermore, many
of them produce different ranking outcomes for the same problem. In 1988, Lee and Li
[13], proposed a comparison of fuzzy numbers by considering the mean and dispersion
(standard deviation) based on the uniform and the proportional probability distributions.
Having reviewed the previous methods, this article proposes a method to use the concept
of median value, so as to find the order of fuzzy numbers. This method can distinguish
the alternatives clearly. The main purpose of this article is that, the median value can
be used as a crisp approximation of a fuzzy number. Therefore, by the means of this
defuzzification, this article aims to present a new method for ranking of fuzzy numbers.
In addition to its ranking features, this method removes the ambiguities resulted from
the comparison of previous ranking. In Section 2, we recall some fundamental results on
fuzzy numbers. In Section 3, a crisp approximation of a fuzzy number is obtained. In this
Section some theorems and remarks are proposed and illustrated. Proposed method for
ranking fuzzy numbers is in the Section 4.
2
Basic Definitions and Notations
The basic definitions of a fuzzy number are given in [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]
as follows:
Definition 2.1. A fuzzy number A is a mapping µA (x) : ℜ→ [0, 1] with the following
properties:
1. µA is an upper semi-continuous function on ℜ,
2. µA (x) = 0 outside of some interval [a1 , b2 ] ⊂ ℜ.
3. There are real numbers a2 , b1 such that a1 ≤ a2 ≤ b1 ≤ b2 and
3.1 µA (x) is a monotonic increasing function on [a1 , a2 ],
3.2 µA (x) is a monotonic decreasing function on [b1 , b2 ],
3.3 µA (x) = 1 for all x in [a2 , b1 ].
The set of all fuzzy numbers is denoted by F .
Definition 2.2. Let ℜ be the set of all real numbers. We assume a fuzzy number A that
can be expressed for all x ∈ ℜ in the form
(
)n

x−a

g(x)
=
when x ∈ [a, b),

b−a



1,
when x ∈ [b, c],
)n
(
A(x) =
(2.1)
d−x

,
when
x
∈
(c,
d],
h(x)
=


d−c


0
otherwise .
where a, b, c, d are real numbers such that a < b ≤ c < d and g is a real valued function
that is increasing and right continuous and h is a real valued function that is decreasing
and left continuous. A fuzzy number A with shape function g and h, where n > 0, will be
denoted by A = ⟨a, b, c, d⟩n . If n = 1, we simply write A = ⟨a, b, c, d⟩, which is known as
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Journal of Fuzzy Set Valued Analysis
a trapezoidal fuzzy number. If n ̸= 1, a fuzzy number A∗ = ⟨a, b, c, d⟩n is a concentration
of A. If 0 < n < 1, then A∗ is a dilation of A. Concentration of A by n = 2 is often
interpreted as the linguistic hedge ”very”. Dilation of A by n = 0.5 is often interpreted as
the linguistic hedge ”more or less”. More about linguistic hedges can be found in [22].
Another important notion connected with fuzzy number A is an cardinality of a fuzzy
number A.
Definition 2.3. [1]. Cardinality of a fuzzy number A described by (2.1) is the value of
the integral
∫ b
∫ 1
cardA =
A(x)dx =
(bα − aα )dα.
(2.2)
a
0
If A = ⟨a, b, c, d⟩n then
b−a
d−c
+ (c − d) +
.
n+1
n+1
In this paper we will always refer to fuzzy number A described by (2.1).
cardA =
3
(2.3)
Median value
The median value of a fuzzy set A was introduced as a possible scalar representative value
of A. The researcher will adopt the description of the median value from [20].
Definition 3.1. [1]. The median value of a fuzzy number A characterized by (2.1) is the
real number mA from the support of A such that
∫ mA
∫ d
A(x)dx,
(3.4)
A(x)dx =
a
mA
For practical purposes expression (3.4) can be rewritten as
∫ mA
A(x)dx = 0.5 cardA.
(3.5)
a
The article can classify fuzzy numbers with respect to the ”distribution” of their cardinality
as follows: a fuzzy number A is called
∫b
∫d
1. a fuzzy number with equally heavy tails if a A(x)dx = c A(x)dx,
}
{∫
∫d
∫d
b
2. a fuzzy number with light tails if max a A(x)dx, c A(x)dx ≤ 0.5 a A(x)dx,
3. a fuzzy number with heavy left tail if
∫b
4. a fuzzy number with heavy right tail if
a
A(x)dx > 0.5
∫d
c
∫d
A(x)dx > 0.5
a
A(x)dx,
∫d
a
A(x)dx.
Now the article will study location of the median value mA in the support of A. The
article will also identify the fuzziness of mA determined by its membership grade A(mA ).
Proposition 3.1. [1]. If A is a fuzzy number with light tails then
(∫ d
)
∫ b
b+c
mA =
+ 0.5
A(x)dx −
A(x)dx ,
2
c
a
and A(mA ) = 1.
(3.6)
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Journal of Fuzzy Set Valued Analysis
Corollary 3.1. If A has equally heavy tails then mA =
b+c
2
and A(mA ) = 1.
Obviously, if A has a heavy left (right) tail then mA is not located in the core of A but
it is shifted to the left (right) end of the support of A. For a trapezoidal fuzzy number A
and for its modifications by selected linguistic hedges the article will provide formulas for
the location of the median value.
Proposition 3.2. [1]. Let A = ⟨a, b, c, d⟩n . Then
(
mA = a +
(b − a)n
(n + 1) cardA
2
)
1
(n+1)
,
(3.7)
,
(3.8)
if A has a heavy left tail, and
(
mA = d −
(d − c)n
(n + 1) cardA
2
)
1
(n+1)
if A has a heavy right tail.
Corollary 3.2. Let A = ⟨a, b, c, d⟩ be a trapezoidal fuzzy number. Then
mA = a +
√
(b − a) cardA,
(3.9)
(d − c) cardA,
(3.10)
if A has a heavy left tail, and
mA = d −
√
if A has a heavy right tail.
When the median value is located outside the core of a fuzzy number A, it belongs to A
with the membership grade less than 1 [1].
4
Comparison of fuzzy numbers using a median value
In the following, we present a new approach for ranking fuzzy numbers based on the
distance method. The method not only considers the median value of a fuzzy number, but
also considers the minimum crisp value of fuzzy numbers. For ranking fuzzy numbers, this
study firstly defines a minimum crisp value τmin to be the benchmark and its characteristic
function µτmin (x) is as follows:
{
1,
when x = τmin ,
µτmin (x) =
(4.11)
0,
when x ̸= τmin .
When ranking n fuzzy numbers A1 , A2 , · · · , An , the minimum crisp value τmin is defined
as:
τmin = min{x|x ∈ Domain(A1 , A2 , · · · , An )}.
(4.12)
Assume that there are n fuzzy numbers A1 , A2 , · · · , An . The proposed method for ranking
fuzzy numbers A1 , A2 , · · · , An is now presented as follows:
Use the point (mAj , 0) to calculate the ranking value MAj = dist(mAj , τmin ) of the fuzzy
numbers Aj , where Aj , where 1 ≤ j ≤ n, as follows:
dist(mAj , τmin ) = ∥mAj − τmin ∥
(4.13)
Journal of Fuzzy Set Valued Analysis
5
From formula (4.13), we can see that MAj = dist(mAj , τmin ) can be considered as the
Euclidean distance between the point (mAj , 0) and the point (τmin , 0). We can see that
the larger the value of MAj , the better the ranking of Aj , where 1 ≤ j ≤ n.
Definition 4.1. Let, there are n fuzzy numbers A1 , A2 , · · · , An ., and MAj are the median
value of their. Define the ranking of Aj by M(.) on F , i.e.
1. MAi < MAj if onlyif Ai ≺ Aj ,
2. MAi > MAj if onlyif Ai ≻ Aj ,
3. MAi = MAj if onlyif Ai ∼ Aj .
Then, this article formulates the order ≽ and ≽ as Ai ≽ Aj if and only if Ai ≻ Aj or
Ai ∼ Aj , Ai ≼ Aj if and only if Ai ≺ Aj or Ai ∼ Aj .
4.1
Using The Proposed Ranking Method In Selecting Army Equip
System
From experimental results, the proposed method with some advantages: (a) without normalizing process, (b) fit all kind of ranking fuzzy number, (c) correct Kerre’s concept.
Therefore we can apply median value of fuzzy ranking method in practical examples. In
the following, the algorithm of selecting equip systems is proposed, and then adopted to
ranking a army example.
4.1.1
An algorithm for selecting equip system
We summarize the algorithm for evaluating equip system as below:
Step 1: Construct a hierarchical structure model for equip system.
Step 2: Build a fuzzy performance matrix Ã. We compute the performance score of the
sub factor, which is represented by triangular fuzzy numbers based on expert’s ratings,
average all the scores corresponding to its criteria. Then, build a fuzzy performance matrix
Ã.
Step 3: Build a fuzzy weighting matrix W̃ . According to the attributes of the equip
systems, experts give the weight for each criterion by fuzzy numbers, and then form a
fuzzy weighting matrix W̃ .
Step 4: Aggregate evaluation. To multiple fuzzy performance matrix and fuzzy weighting
matrix W̃ , then get fuzzy aggregative evaluation matrix R̃. (i.e. R̃ = Ã ⊗ W̃ t ).
Step 5: Determinate the best alternative. After step 4, we can get the fuzzy aggregative
performance for each alternative, and then rank fuzzy numbers by median value of fuzzy
numbers.
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Journal of Fuzzy Set Valued Analysis
Table 1
Linguistic values for the ratings.
Linguistic value
Very Poor(VP)
Poor
Slightly(SP)
Fair(F)
Slightly good(SG)
Good(G)
Very good(VG)
TFNs
(0,0,0.16)
(0,0.16,0.33)
(0.16,0.33,0.5)
(0.33,0.5,0.66)
(0.5,0.66.0.83)
(0.66,0.83,1)
(0.83,1,1)
Table 2
Basic performance data for five types of main battle Tanks.
Item
Type
Tank A
Armament
Ammunition
120 mm gun
15.2 mm MG
12.7 mm MG
40
1000
Tank B
Tank C
Tank D
Tank E
120 mm gun
15.2 mm MG
120 mm gun
15.2 mm MG
105 mm gun
15.2 mm MG
120mm gun
7.62mm MG
12.7 mm MG
Up to 50
4000
42
40
44
4750
4
1500
10000
2×6
2×5
2×8
None
2×9
26.2
19.2
27.2
19.0
27.5
67
56
72
60
71
480
1.21
60
2.74
Good
Good
Fair
Medium
450
1.07
60
2.43
Excellent
Fair
Fair
Medium
550
1.0
60
3.00
Good
Good
Fair
Medium
300
1.2
55
2.51
Fair
Fair
Poor
Medium
550
1.23
60
2.92
Excellent
Good
Fair
Good
11400
Smoke grenade
discharges
Power to weight
ratio(hp/t)
Max. road
speed(km/h)
Max. range(km)
Fording(m)
Gradient
Trench
Armor protection
Acclimatization
Communication
Scout
4.1.2
The selecting of best main battle tank
In [3], the authors have constructed a practical example for evaluating the best main battle tank, and they selected x1 = M1 A1 (USA), x2 = Challenger2 (UK), x3 = Leopard2
(Germany) as alternatives. In [3], the experts opinion were described by linguistic terms,
which can be repressed in triangular fuzzy numbers. The fuzzy Delphi method is adopted
to adjust the fuzzy rating of each expert to achieve the consensus condition. The evaluating criteria of main battle tank are a1 : attackcapability, a2 : mobilitycapability,
a3 : self − def encecapability and a4 : communicationandcontrolcapability. In this example, we adopted the hierarchical structure constructed in [3] for selection of five main
battle tanks, and the step-by-step illustrations based on Sec. 4.1.1s algorithm are described bellow:
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Journal of Fuzzy Set Valued Analysis
Step 1: Construct a hierarchical structure model for equip system.
Step 2: Build a fuzzy performance matrix Ã. The basic performance data for five types
of main battle tanks are summarized in Table 2. Then based on the linguistic values in
Table 1, the fuzzy preference of five tanks toward four criteria are collected and shown in
Table 5.
Step 3: Build a fuzzy weighting matrix W̃ . The aggregative fuzzy weights of four criteria,
according to the linguistic values of importance in Table 3, are shown in Table 4.
Step 4: Aggregate evaluation. To multiple fuzzy performance matrix à and fuzzy weighting matrix W̃ , then get fuzzy aggregative evaluation matrix R̃ = à ⊗ W̃ t . Therefore, from
Table 4 and 5, we have



R̃ = 


(0.7, 0.9, 1.0) (0.7, 0.8, 1.0)
(0.4, 0.6, 0.7) (0.4, 0.6, 0.8)
(0.6, 0.8, 0.9) (0.7, 0.9, 0.96)
(0.2, 0.3, 0.5) (0.3, 0.5, 0.6)
(0.7, 0.8, 0.9) (0.8, 0.9, 1.0)



=


(0.5, 0.7, 0.8)
(0.5, 0.7, 0.8)
(0.4, 0.6, 0.8)
(0.3, 0.5, 0.7)
(0.6, 0.8, 0.9)
(0.6, 0.8, 0.9)
(0.6, 0.8, 0.9)
(0.6, 0.8, 0.9)
(0.4, 0.6, 0.8)
(0.7, 0.8, 1.0)
(0.38, 0.61, 0.82)
(0.27, 0.48, 0.69)
(0.36, 0.58, 0.77)
(0.18, 0.34, 0.55)
(0.40, 0.64, 0.84)


(0.7, 0.9, 1.0)

  (0.8, 0.9, 1.0)
⊗
  (0.6, 0.7, 0.8)

(0.3, 0.5, 0.7)







.


Step 5: Determinate the best alternative. According to Eq. (4.13), we can get the median value
of fuzzy numbers of Tanks A-E, which are equal to 0.234, 0.423, 0.236, 0.323and0.289, respectively.
Therefore, we find that the ordering of median value is T ank A < T ank C < T ank F < T ank D <
T ank B. So, the best type of main battle Tank is Tank F .
Table 3
Linguistic values of the importance weights.
Linguistic value
TFNs
Very low(VL)
(0,0,0.167)
Low(L)
(0,0.167,0.333)
Slightly
(0.167,0.333,0.5)
Medium(M)
(0.333,0.5,0.667)
Slightly high(SH)
(0.5,0.667,0.833)
High(H)
(0.667,0.833,1)
Very High(VH)
(0.833,1,1)
Table 4
The importance weights of linguistic criteria and
Criteria
Experts
D1
Attack (W̃1 )
VH
Mobility (W̃2 )
VH
Self-defence (W̃3 )
M
Communication and command (W̃4 ) M
its mean.
Mean of TFNs
D2
H
H
VH
M
D3
H
VH
SH
M
(0.72,0.89,1)
(0.78,0.94,1)
(0.56,0.72,0.83)
(0.33,0.5,0.67)
8
Journal of Fuzzy Set Valued Analysis
Table 5
Basic performance data for five types of main battle Tanks.
Criteria
Type
Tank A
Tank B
Tank C
Attack
Armament
G
SG
SG
Ammunition
VG
SG
SG
Smoke grenade
G
SP
VG
dischargers
Mean
(0.7,0.8,1)
(0.3,0.5,0.7) (0.6,0.7,0.8)
Mobility
Power to weight
G
F
G
ratio
Max.
G
F
VG
road speed
Max. range
G
SG
VG
Fording/Gradient G
SG
SG
Trench
Mean
(0.6,0.8,1)
(0.4,0.5,0.7) (0.7,0.8,0.9)
Self-defence
Armor
SG
G
F
protection
Acclimatization
SG
F
SG
Mean
(0.5,0.6,0.8) (0.5,0.6,0.8) (0.4,0.5,0.7)
Communication
and command
Communication
G
G
G
Scout
SG
SG
SG
Mean
(0.5,0.7,0.9) (0.5,0.7,0.9) (0.5,0.7,0.9)
5
Tank D
Tank E
F
F
VP
SG
G
VG
(0.2,0.3,0.5)
(0.6,0.8,0.9)
F
G
SG
VG
P
F
VG
G
(0.2,0.4,0.6)
(0.7,0.9,1)
F
G
F
(0.3,0.5,0.6)
G
(0.5,0.7,0.9)
F
SG
(0.4,0.5,0.7)
G
G
(0.6,0.8,1)
Conclusion
In this paper, the researchers proposed a median value method to rank fuzzy numbers. This
method use a crisp set approximation of a fuzzy number. The method can effectively rank various
fuzzy numbers and their images (normal/nonnormal/trapezoidal/general ). The calculations of
this method are simpler than the other approaches.
References
[1] S. Bodjanova, Median value and median interval of a fuzzy number, Infomation Sciences 172
(2005) 73-89.
http://dx.doi.org/10.1016/j.ins.2004.07.018.
[2] G. Bortolan and R. Degani, A review of some methods for ranking fuzzy numbers, Fuzzy Sets
and Systems 15 (1985) 1-19.
http://dx.doi.org/10.1016/0165-0114(85)90012-0.
[3] CH. Cheng, Y. Lin, Evaluating weapon system by analitic hierarchy process based on fuzzy
scales, Fuzzy Sets and Systems 63 (2002) 1-10.
[4] S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision Making, Springer- Verlag,
Berlin, 1972.
Journal of Fuzzy Set Valued Analysis
9
[5] K. Lee, H. Chu, H. Lee-Kwang, Ranking fuzzy values with satisfaction function, Fuzzy Sets
and Systems 64 (1994) 295-309.
http://dx.doi.org/10.1016/0165-0114(94)90153-8.
[6] F. Choobineh, H. Li, An index for ordering fuzzy numbers, Fuzzy Sets and Systems 54 (1993)
287-294.
http://dx.doi.org/10.1016/0165-0114(93)90374-Q.
[7] O. Dias, Ranking alternatives using fuzzy numbers: A computational approach, Fuzzy Sets
and Systems 56 (1993) 247-252.
http://dx.doi.org/10.1016/0165-0114(93)90151-7.
[8] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation,
Fuzzy Sets and Systems 82 (1996) 319-330.
http://dx.doi.org/10.1016/0165-0114(95)00273-1.
[9] R. Jain, Decision-making in the presence of fuzzy variable, IEEE Trans. Systems Man and
Cybernet.SMC. 6 (1976) 698-703.
http://dx.doi.org/10.1109/TSMC.1976.4309421.
[10] R. Jain, A procedure for multi-aspect decision making using fuzzy sets, Internat. J. Systems
Sci. 8 (1978) 1-7.
http://dx.doi.org/10.1080/00207727708942017.
[11] I. Requena, M. Delgado and J. I. Verdegay, Automatic ranking of fuzzy numbers with the
criterion of decision maker leant by an artificial neural network, Fuzzy Sets and Systems 64
(1994) 1-9.
http://dx.doi.org/10.1016/0165-0114(94)90002-7.
[12] P. A. Raj, D. N. Kumar, Ranking alternatives with fuzzy weights using maximizing set and
minimizing set, Fuzzy Sets and Systems 105 (1999) 365-375.
http://dx.doi.org/10.1016/S0165-0114(97)00243-1.
[13] R. Saneifard, Defuzzification method for solving fuzzy linear systems, International Journal
of Industrial Mathematics 4 (2009) 321-331.
[14] R. Saneifard, Ranking L-R fuzzy numbers with weighted averaging based on levels, International Journal of Industrial Mathematics 2 (2009) 163-173.
[15] R. Saneifard, T. Allahviranloo, F. Hosseinzadeh, N. Mikaeilvand, Euclidean ranking DMUs
with fuzzy data in DEA, Applied Mathematical Sciences 6 (2007) 2989-2998.
[16] R. Saneifard, A method for defuzzification by weighted distance, International Journal of
Industrial Mathematics 3 (2009) 209-217.
[17] R. Saneifard, R. Ezatti, Defuzzification Through a Bi Symmetrical weighted Function, Aust.
J. Basic Appl. Sci. 10 (2010) 4976-4984.
[18] Rahim Saneifard and Rasoul Saneifard, A modified method for defuzzification by probability
density function, Journal of Applied Sciences Research 2 (2011) 102-110.
[19] R. Saneifard and A. Asgari, A Method For Defuzzification Based on Probability Density
Function (II), Applied Mathematical Sciences 5 (2011) 1357-1365.
[20] Rahim Saneifard and Rasoul Saneifard, Evaluation of Fuzzy Linear Regression Models by
Parametric Distance, Australian Journal of Basic and Applied Sciences 5 (2011) 261-267.
[21] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353.
http://dx.doi.org/10.1016/S0019-9958(65)90241-X.
[22] L. A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, Journal of Cybernetics
2 (1972) 4-34.
http://dx.doi.org/10.1080/01969727208542910.