ON THE EXTENSION OF THE MEDIAN CENTER AND THE
MIN-MAX CENTER TO FUZZY DEMAND POINTS
Julio Rojas-Mora, Didier Josselin
UMR ESPACE 6012 CNRS, Universit
´
e d’Avignon (UAPV), Avignon, France
Marc Ciligot-Travain
LANLG, Universit
´
e d’Avignon (UAPV), Avignon, France
Keywords:
Location problem, Median center, Min-max center, Fuzzy sets, Distance.
Abstract:
A common research topic has been the search of an optimal center, according to some objective function
that considers the distance between the potential solutions and a given set of points. For crisp data, closed
form expressions obtained are the median center, for the Manhattan distance, and the min-max center, for
the Chebyshev distance. In this paper, we prove that these closed form expressions can be extended to fuzzy
sets by modeling data points with fuzzy numbers, obtaining centers that, through their membership function,
model the “appropriateness” of the final location.
1 INTRODUCTION
Finding an optimal center in space became a com-
mon process in planning, because it allows to affect
a set of demands to one or several locations that of-
fer dedicated facilities. For instance, a center col-
lecting wastes, a vehicle depot for logistic purpose
or a hospital complex, all require a relevant metric
to minimize cost or maximize access to them. Math-
ematicians, economists and geographers developed
methods which locate these centers according to ei-
ther equity (minimax) or versus efficiency (minisum)
objectives, following the work in k-facilities location
problems on networks (Hakimi, 1964), that respec-
tively correspond to the k-median and the k-center.
Indeed, there exist many mathematical problems and
formalisms for optimal location problems (Hansen
et al., 1987). More recently, we can see a larger
scope of the domain and sets where these issues ap-
pear (Chan, 2005). Other books complete the state-of-
the-art (Drezner and Hamacher, 2004; Griffith et al.,
1998; Nickel and Puerto, 2005) or focus on applica-
tions in transportation (Labb
´
e et al., 1995; Thomas,
2002) or health care (Brandeau et al., 2004).
Methodologies for optimal location can be applied
on continuous space, finite space or networks (graphs
or roads for instance). If k = 1, then the aim is to find a
single center. The choice of the metric p is also signif-
icant because it involves, on the one hand, the method
to set the distance separating the demands to the cen-
ter, and on the other, how to combine these distances
according to a given objective function. Thus, there
exist many ways to calculate a center for many points
of demand, even when reducing complexity by con-
sidering a continuous space, a unique center and the
Minkowski distance of L
p
norms. The first parame-
ter, p, defines the norm of the distance separating the
demand points to the center: rectilinear (p = 1), Eu-
clidean (p = 2) or Chebyshev’s (p → ∞). The second
parameter, p
0
, relates to the calculation of the center
itself. The sum of the distances is minimized when
p
0
= 1, the sum of the squared distances when p
0
= 2
and the maximum of the distances when (p
0
→ ∞).
Among all the possibilities crossing p and p
0
of
the L
p
norms, only three cases can be computed in
closed form: the median center, which minimizes the
sum of the rectilinear distances (p = p
0
= 1), the cen-
troid or barycenter, which minimizes the sum of the
squared Euclidean distances (p = p
0
= 2) and the min-
max center, which minimizes the maximum of the
maximum distances (p → ∞ and p
0
→ ∞).
Scientists and planners use to consider the final
location to be accurate and crisp, or, at least, as a fi-
nite set of possible predefined locations. However,
there might be uncertainty on the estimated distances,
due to uncertainty carried by the demand location it-
407
Rojas-Mora J., Josselin D. and Ciligot-Travain M..
ON THE EXTENSION OF THE MEDIAN CENTER AND THE MIN-MAX CENTER TO FUZZY DEMAND POINTS.
DOI: 10.5220/0003674604070416
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 407-416
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

self. This is particularly true when considering urban
sprawl, as it can generate non negligible variations on
the location of the town’s center, which in place might
affect the location of the optimal center. There is also
the case when subjective or vague information is used
to define the demand location. The result, then, can-
not possibly be a crisp point, and solutions that as-
sume crisp data when non is available, might be at
risk being far from optimal.
By modeling the demand points as bi-dimensional
fuzzy sets we prove in this paper that the results ob-
tained for crisp environments can be easily extended
to the fuzzy ones, attaining homologous closed form
expressions. As the solutions depend only on arith-
metic operations of fuzzy numbers, thus obtaining
fuzzy numbers as its coordinates, the approach fol-
lowed in this work deviates from the path trailed by
many fuzzy location papers, in which constraints are
fuzzy, but the solution is not (Darzentas, 1987; Can
´
os
et al., 1999; Chen, 2001; Moreno P
´
erez et al., 2004).
Fuzzy solutions also give some leeway to planners
which might be forced to select the final location of
the center away from the place with the highest mem-
bership value, but that can the measure the impact of
their decision and, thus, asses its “appropriateness”.
This paper is structured in the following way. In
Section 2, we introduce the closed form expressions
for centers usually used in the literature. Then, on
Section 3, the basic concepts of fuzzy sets and fuzzy
numbers used through our paper are defined. Sec-
tion 4 covers the demonstrations used to prove that
the closed form expressions found for some centers
in crisp environments can be extended to fuzzy points.
A small numerical example, joined by some figures in
which the results can be easily seen, is developed in
Section 5. Finally, Section 6 presents the conclusions
as well as the future work based on our results.
2 THE MEDIAN CENTER AND
THE MIN-MAX CENTER
A recurrent problem in geography is the need to find
the center of a set of demand points that minimizes a
given objective function. Without taking into consid-
eration the road network that links these points, i.e., in
an open space, there are two simple, but also widely
used methods to solve this problem, the median center
and the min-max center.
Definition 1. For a set P = {p
(i)
} of n points in
R
2
, i.e., p
(i)
= {p
(i,x)
, p
(i,y)
}, the median center m =
{m
(x)
, m
(y)
} is found by the median of their coordi-
nates in x and y:
m
(x)
= median

p
(i,x)

(1)
m
(y)
= median

p
(i,y)

. (2)
Definition 2. For a set P = {p
(i)
} of n points in
R
2
, i.e., p
(i)
= {p
(i,x)
, p
(i,y)
}, the min-max center z =
{c
(x)
, c
(y)
} is found by the average of the extremes in
x and y:
z
(x)
=
max
i=1,...,n

p
(i,x)

+ min
i=1,...,n

p
(i,x)

2
(3)
z
(y)
=
max
i=1,...,n

p
(i,x)

+ min
i=1,...,n

p
(i,x)

2
. (4)
The median center is affected by changes in the
middle points, but changes in extreme points affect
only the min-max center. The selection of the ap-
propriate method to find the center depends on which
points are most likely to change (Ciligot-Travain and
Josselin, 2009).
3 FUZZY SETS AND FUZZY
NUMBERS
When it is difficult to say that an object clearly be-
longs to a class, classical set theory loses its useful-
ness. The fuzzy sets theory (Zadeh, 1965) overcomes
this problem by assigning degrees of membership of
elements to a set. In this section we will recall the
concepts of the fuzzy set theory that will be used in
this paper.
3.1 Basic Definitions
Definition 3. A fuzzy subset A
e
is a set whose elements
do not follow the law of the excluded middle that rules
over Boolean logic, i.e., their membership function is
mapped as:
µ
A
e
: X → [0, 1]. (5)
In general, a fuzzy subset A
e
can be represented
by a set of pairs composed of the elements x of the
universal set X, and a grade of membership µ
A
e
(x):
A
e
=
n
x, µ
A
e
(x)

| x ∈ X , µ
A
e
(x) ∈ [0, 1]
o
. (6)
Definition 4. An α-cut of a fuzzy subset A
e
is defined
by:
A
α
= {x ∈ X : µ
A
e
(x) ≥ α}, (7)
i.e., the subset of all elements that belong to A
e
at least
in a degree α.
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
408

Definition 5. A fuzzy subset A
e
is convex, if and only
if:
λx
1
+ (1 − λx
2
) ∈ A
α
∀x
1
, x
2
∈ A
α
, α, λ ∈ [0, 1], (8)
i.e., all the points in [x
1
, x
2
] must belong to A
α
, for any
α.
Definition 6. A fuzzy subset A
e
is normal, if and only
if:
max
x∈X

µ
A
e
(x)

= 1. (9)
Definition 7. The core of a fuzzy subset A
e
is defined
as:
N
A
e
=
n
x : µ
A
e
(x) = 1
o
. (10)
Definition 8. A fuzzy number A
e
is a normal, convex
fuzzy subset with domain in R for which:
1. ¯x := N
A
e
, card(¯x) = 1, and
2. µ
A
e
is at least piecewise continuous.
The mean value ¯x (Zimmermann, 2005), also called
maximum of presumption (Kaufmann and Gupta,
1985), identifies a fuzzy number in such a way that
the proposition “about 9” can be modeled with a fuzzy
number whose maximum of presumption is x = 9. As
Zimmermann explains, for computational simplicity
there is a tendency to call “fuzzy number” any nor-
mal, convex fuzzy subset whose membership function
is, at least, piecewise continuous, without taking into
consideration the uniqueness of the maximum of pre-
sumption. Thus, this definition will include “fuzzy
intervals”, fuzzy numbers in which ¯x covers an in-
terval
1
, and particularly trapezoidal fuzzy numbers
(TrFN).
Definition 9. A TrFN is defined by the membership
function:
µ
A
e
(x) =









1 −
x
2
−x
x
2
−x
1
, if x
1
≤ x < x
2
1, if x
2
≤ x ≤ x
3
1 −
x−x
3
x
4
−x
3
, if x
3
< x ≤ x
4
0 otherwise.
(11)
This kind of fuzzy interval represents the case
when the maximum of presumption, the modal value,
can not be identified in a single point, but only in an
interval between x
2
and x
3
, decreasing linearly to zero
at the worst case deviations x
1
and x
4
. The TrFN is
represented by a 4-tuple whose first and fourth el-
ements correspond to the extremes from where the
membership function begins to grow, and whose sec-
ond and third components are the limits of the in-
terval where the maximum certainty lies, i.e., A
e
=
(x
1
, x
2
, x
3
, x
4
).
1
As a matter of fact, they are also called “flat fuzzy num-
bers” (Dubois and Prade, 1979).
Definition 10. The image of a TrFN is defined as:
Im

A
e

= (−a
4
, −a
3
, −a
2
, −a
1
).
Definition 11. The addition and subtraction of two
TrFN A
e
and B
e
are defined as:
A
e
⊕ B
e
= (a
1
+ b
1
, a
2
+ b
2
, a
3
+ b
3
, a
4
+ b
4
) (12)
A
e
 B
e
= A
e
⊕ Im(B)
e
. (13)
3.2 Miscelaneous Definitions
Comparing fuzzy numbers is a task that can only be
achieved via defuzzification, i.e., by calculating its
expected value. For its simplicity, we have selected
the graded mean integrated representation (GMIR) of
a TrFN (Chen and Hsieh, 1999) as the method used in
this paper to defuzzify and compare TrFN.
Definition 12 (Chen and Hsieh, 1999). The GMIR of
non-normal TrFN is:
E

M
e

=
R
max

µ
M
e

0
µ
2

L
−1
M
e
(µ) + R
−1
M
e
(µ)

dµ
R
max

µ
M
e

0
µdµ
. (14)
Remark 1. For a normal TrFN as defined in (11), the
GMIR is:
E

A
e

=
a
1
+ 2a
2
+ 2a
3
+ a
4
6
. (15)
Remark 2. The GMIR is linear, i.e., E(A
e
⊕ B
e
) =
E(A
e
) + E(B)
e
and E(α · A
e
) = α · E(A
e
).
To calculate the distance between two TrFN, we
must first define the absolute value of a TrFN. We will
rely on the work of (Chen and Wang, 2009) for this.
Definition 13 (Chen and Wang, 2009). The absolute
value of a TrFN is defined as:


A
e


=





A
e
, if E (A
e
) > 0
0, if E (A
e
) = 0
Im (A
e
), if E (A
e
) < 0.
(16)
Proposition 1. For a TrFN A
e
, E(


A
e


) =


E(A
e
)


.
Proof. For E(A
e
) ≥ 0 the proof is trivial. For E(A
e
) < 0
we have:
E




A
e




= E

Im

A
e

=
−a
4
− 2a
3
− 2a
2
− a
1
6
= −E

A
e

=



E

A
e




.
ON THE EXTENSION OF THE MEDIAN CENTER AND THE MIN-MAX CENTER TO FUZZY DEMAND POINTS
409

Definition 14. The fuzzy Minkoswki family of dis-
tances between two fuzzy n-dimensional vectors A
e
and
B
e
composed of TrFN:
d
p
e

A
e
, B
e

=
n
∑
i=1




A
i
e
 B
i
e




p
!
1
p
. (17)
Remark 3. As with the crisp Minkowski family of dis-
tances, the fuzzy Manhattan distance is defined for
p = 1, the fuzzy Euclidean distance is defined for
p = 2, and the fuzzy Chebyshev distance is defined
for p = ∞.
Remark 4. In our proofs, we will use the form:
d
e
p

A
e
, B
e

=
n
∑
i=1




A
i
e
 B
i
e




p
, (18)
except for p = ∞ in which:
d
e
∞

A
e
, B
e

= arg



A
i
e
B
i
e



n
max
i=1
E




A
i
e
 B
i
e




. (19)
4 FUZZY MEDIAN CENTER AND
FUZZY MIN-MAX CENTER
We will prove that for a set of fuzzy points, the fuzzy
median center and the fuzzy min-max center are ex-
tensions of their respective counterparts in crisp set-
tings, i.e., that they can be obtained by the median or
the average of the maximum X and Y coordinates of
the fuzzy points, respectively.
Proposition 2. For two TrFN p
(1)
g
and p
(2)
g
, such that
E(p
(1)
g
) < E(p
(2)
g
), argmin
c
e
E(
∑
i∈{1,2}
d
1
e
(p
(i)
f
, c
e
)) =
{c
e
: E(c
e
) ∈ [E(p
(1)
g
), E(p
(2)
g
)]}.
Proof. Let p
(i)
f
= (p
(i)
1
, p
(i)
2
, p
(i)
3
, p
(i)
4
) and
c
e
= (c
1
, c
2
, c
3
, c
4
), hence:
d
1
e

p
(i)
f
, c
e

=



p
(i)
f
 c
e



.
By properties of the GMIR:
E

p
(i)
f
 c
e

= E

p
(i)
f

− E

c
e

.
If E(c
e
) ≤ E(p
(1)
g
) and by (16), then:
d
e
1

p
(1)
g
, c
e

= p
(1)
g
 c
e
, (20)
d
e
1

p
(2)
g
, c
e

= p
(2)
g
 c
e
. (21)
By (20) and (21):
argmin
c
e
E
∑
i∈
{
1,2
}

d
e
1

p
(i)
f
, c
e

!
= p
(1)
g
,
as p
(1)
g
c
e
= (0, 0,0, 0) and p
(2)
g
c
e
= p
(2)
g
 p
(1)
g
. For
any {c
e
: E(c
e
) < E(p
(1)
g
)}, E(p
(2)
g
 c
e
) > 0 and E(c
e

p
(1)
g
) > E(p
(2)
g
 p
(1)
g
).
Equivalently, if E(p
(2)
g
) ≤ E(c
e
) by (16), then:
d
e
1

p
(1)
g
, c
e

= c
e
 p
(1)
g
, (22)
d
e
1

p
(2)
g
, c
e

= c
e
 p
(2)
g
. (23)
By (22) and (23),
argmin
c
e
E
∑
i∈
{
1,2
}

d
e
1

p
(i)
f
, c
e

!
= p
(2)
g
,
as c
e
 p
(2)
g
= (0, 0,0, 0) and c
e
 p
(1)
g
= p
(2)
g
 p
(1)
g
. For
any {c
e
: E(p
(2)
g
) < E(c
e
)}, E(c
e
 p
(2)
g
) > 0 and E(c
e

p
(1)
g
) > E(p
(2)
g
 p
(1)
g
).
Given that E(p
(1)
g
) < E(c
e
) and by (16), then:
d
e
1

p
(1)
g
, c
e

= c
e
 p
(1)
g
=

c
1
−p
(1)
4
,c
2
−p
(1)
3
,c
3
−p
(1)
2
,c
1
−p
(1)
4

.(24)
Given that E(c
e
) < E(p
(2)
g
) and by (16), then:
d
e
1

p
(2)
g
, c
e

= p
(2)
g
 c
e
=

p
(2)
1
−c
4
,p
(2)
2
−c
3
,p
(2)
3
−c
2
,p
(2)
4
−c
1

.(25)
From (24) and (25):
∑
i∈
{
1,2
}

d
e
1

p
(i)
f
,c
e

=

c
1
−p
(1)
4
,c
2
−p
(1)
3
,c
3
−p
(1)
2
,c
4
−p
(1)
1

⊕

p
(2)
1
−c
4
,p
(2)
2
−c
3
,p
(2)
3
−c
2
,p
(2)
4
−c
1

=

p
(2)
1
−p
(1)
1
+c
1
−c
4
,p
(2)
2
−p
(1)
2
+c
2
−c
3
,
p
(2)
3
−p
(1)
3
+c
3
−c
2
,p
(2)
4
−p
(1)
4
+c
4
−c
1

.
(26)
Applying GMIR to (26) :
E

∑
i∈
{
1,2
}

d
e
1

p
(i)
f
,c
e


= E

p
(2)
g
− p
(1)
g

. (27)
Being that (27) is independent from c
e
:
argmin
c
e
E

∑
i∈
{
1,2
}

d
e
1

p
(i)
f
,c
e

=
n
c
e
:E

c
e

∈
h
E

p
(1)
f

,E

p
(2)
f
io
.
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
410

The result obtained in Proposition 2 shows than
any fuzzy point c
e
between two fuzzy points p
(1)
g
and
p
(2)
g
gives an equally good solution to the problem
of the minimization of distances. An arbitrary, but
frequently found solution to the crisp version of this
problem, is using the average of both points:
c
e
=
p
(1)
g
⊕ p
(2)
g
2
. (28)
In the following proposition we will see what hap-
pens for a set of n fuzzy points, but first, let us define
the notion of order statistic for fuzzy numbers.
Definition 15. For a set P = {p
(i)
f
}, ∀i = 1, . . . , n, of
TrFN, the k−th order statistic p
([k])
g
is defined as the
k−th point for which E(p
([k])
g
) ≤ E(p
([k+1])
^
).
Proposition 3 (Fuzzy median center in R). For a
set P = {p
(i)
f
}, i = 1, . . . , n, of TrFN, c
∗
e
is the point
for which arg min
c
e
E(
∑
n
i=1
d
e
1
(p
([i])
g
, c
e
)) = {c
e
: E(c
e
) ∈
[E(p
([
n
2
])
]
), E(p
([
n
2
+1])
^
)]}, if n is even, but if it is odd
argmin
c
e
E(
∑
n
i=1
(d
e
1
(p
([i])
g
, c
e
))) = p
([
n+1
2
])
^
.
Proof. Given that the k−th order statistic of the set P
is p
([k])
g
, we can apply iteratively the result in Proposi-
tion 2. In first place, it is known that:
argmin
c
e
E
∑
i∈
{
1,n
}
d
e

p
([i])
g
, c
e

!
=

c
e
: E

c
e

∈

E

p
([1])
g

, E

p
([n])
g

.
From Definition 15, it is also known that:

E

p
([2])
g

, E

p
([n−1])
^

∈
h
E

p
([1])
g

, E

p
([n])
g
i
,
so the solution is now:
argmin
c
e
E
∑
i∈
{
1,2,n−1,n
}
d
1
e

p
([i])
g
, c
e

!
=

c
e
: E

c
e

∈

E

p
([2])
g

, E

p
([n−1])
^

.
If we keep applying iteratively this logic, and n is
even, we get that
argmin
c
e
E
n
∑
i=1
d
1
e

p
([i])
g
, c
e

!
=

c
e
: E

c
e

∈

E

p
(
[
n
2
]
)
]

, E

p
(
[
n
2
+1]
)
^

.
If n is odd, we will have three points in the next-to-
last iteration,

p
(
[
n−1
2
]
)
^
, p
(
[
n+1
2
]
)
^
, p
(
[
n+3
2
]
)
^

. We can
present the problem as:
argmin
c
e
E
n
∑
i=1

d
1
e

p
([i])
g
, c
e

=
argmin
c
e
E


n−3
2
∑
i=
n−1
2
d
1
e

p
([i])
g
, c
e



= argmin
c
e
E


∑
i=
{
n−1
2
,
n−3
2
}
d
1
e

p
([i])
g
, c
e

+
d
1
e

p
([
n+1
2
])
^
, c
e

.
We know that:
argmin
c
e
E


∑
i=
{
n−1
2
,
n−3
2
}

d
1
e

p
(i)
g
, c
e



=
(
c
e
: E

c
e

∈
"
E p
(
[
n−1
2
]
)
^
!
, E p
(
[
n−3
2
]
)
^
!#)
.
Therefore, it is clear that:
argmin
c
e
E

d
1
e

p
(
[
n+1
2
]
)
^
, c
e

= p
(
[
n+1
2
]
)
^
.
So, given that c
e
= p
([
n+1
2
])
^
and that E(p
([
n+1
2
])
) ∈
[E(p
([
n−1
2
])
^
), E(p
([
n−3
2
])
^
)], for n even:
argmin
c
e
E
n
∑
i=1

d
1
e

p
(i)
f
, c
e

!
= p
(
[
n+1
2
]
)
^
.
Applying (28) to the result of Proposition 3 we get
the definition of the median for a set of TrFN.
Definition 16. The median of a set P = {p
(i)
f
}, ∀i =
1, . . ., n, of TrFN is defined as:
median(P) =









p
(
[
n
2
]
)
]
⊕p
(
[
n
2
+1]
)
^
2
, if n is odd,
p
(
[
n+1
2
]
)
^
, if n is even.
In an R
2
space, the solution is equivalent, as we
will see in the following proposition.
ON THE EXTENSION OF THE MEDIAN CENTER AND THE MIN-MAX CENTER TO FUZZY DEMAND POINTS
411

Proposition 4. For a set P =
n
P
(i)
f
: P
(i)
f
=
n
p
(i, j)
g
o
, ∀i = 1, .. ., n, j ∈
{
x, y
}
o
,
where p
(i, j)
g
is a TrFN,
argmin
C
e
E

∑
n
i=1
d
e
1

P
(i)
f
,C
e

=
n
median

p
(i,x)
g

, median

p
(i,y)
g
o
.
Proof. Due to the linearity of the GMIR:
E

∑
n
i=1

d
e
1

P
(i)
f
,C
e

=
∑
n
i=1
∑
j∈{x,y}





E p
(i, j)
g
!
−E

c
j
e






=
∑
n
i=1





E p
(i,x)
g
!
−E

c
(x)
f






+
∑
n
i=1





E p
(i,y)
g
!
−E

c
(y)
f






. (29)
As both terms in (29) are independent from each
other:
min
c
( j)
∑
j∈{x,y}
n
∑
i=1




E

p
(i,x)
g

− E

c
(x)
f





!
=
∑
j∈{x,y}
min
c
( j)
n
∑
i=1




E

p
(i,x)
g

− E

c
(x)
f





.
The optimization problem is then reduced to ap-
plying independently for each j ∈
{
x, y
}
the result of
Proposition 3 with Definition 16. Thus:
argmin
C
e
E
n
∑
i=1
d
e
1

P
(i)
f
,C
e

!
=

median

p
(i,x)
g

, median

p
(i,y)
g

. (30)
Finally, we will address the subject of the fuzzy
min-max center, found using (19).
Proposition 5. For a set P =
n
p
(i)
f
o
, ∀i = 1, .. ., n, of
TrFN, max
n
i=1
(E(



p
(i)
f
 c
e



) =
1
2
· E(p
([n])
− p
([1])
).
Proof. Let E(c
e
) =
1
2
(E(p
([1])
) + E(p
([n])
g
)). Due
to the linearity of the GMIR and Proposition 1,
max
n
i=1
(E(



p
(i)
f
 c
e



) = max
n
i=1



E(p
(i)
f
) − E(c
e
)



.
So:
−
E p
([n])
g
!
−E
(
p
([1])
)
2
≤E p
([i])
f
!
−
E p
([n])
g
!
+E p
([1])
g
!
2
≤
E p
([n])
g
!
−E
(
p
([1])
)
2
then:










E p
([i])
f
!
−
E p
([n])
g
!
+E p
([1])
g
!
2










≤
E p
([n])
g
!
−E
(
p
([1])
)
2
max
n
i=1










E p
([i])
f
!
−
E p
([n])
g
!
+E p
([1])
g
!
2










≤
E p
([n])
g
!
−E
(
p
([1])
)
2
.
In fact:










E p
([n])
g
!
−
E p
([n])
g
!
+E p
([1])
g
!
2










=










E p
([n])
g
!
−E p
([1])
g
!
2










=










−
E p
([n])
g
!
−E p
([1])
g
!
2










=










E p
([1])
g
!
−
E p
([n])
g
!
+E p
([1])
g
!
2










≥










E p
([i])
f
!
−
E p
([n])
g
!
+E p
([1])
g
!
2










.
So:
max
n
i=1








E

p
([i])
f

−
E

p
([n])
g

+E

p
([1])
g

2








=
E

p
([n])
g

−E

p
([1])
g

2
,
i.e.:
n
max
i=1



E

p
([i])
g

− E

c
e




=
E

p
([n])
g

− E

p
([1])
g

2
.
Proposition 6. For a TrFN c
0
e
, such that
for every TrFN p
e
max
n
i=1
(E(



p
(i)
f
 p
e



) ≥
max
n
i=1
(E(



p
(i)
f
 c
0
e



)), then E(c
e
) = (c
0
e
).
Proof. Let E(c
e
) =
1
2
(E(p
([1])
) + E(p
([n])
g
)). Taking
p
e
= c
e
:
n
max
i=1

E





p
(i)
f
 c
0
e





≤
n
max
i=1

E





p
(i)
f
 c
e





=
E

p
([n])
g

− E

p
([1])
g

2
,
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
412

so:
E





p
([1])
g
 c
0
e





≤
E

p
([n])
g

− E

p
([1])
g

2
E





p
([n])
g
 c
0
e





≤
E

p
([n])
g

− E

p
([1])
g

2
and:
E

c
0
e

≤
E

p
([n])
g

− E

p
([1])
g

2
+ E

p
([1])
g

=
E

p
([n])
g

+ E

p
([1])
g

2
E

c
0
e

≥ E

p
([n])
g

−
E

p
([n])
g

− E

p
([1])
g

2
=
E

p
([n])
g

+ E

p
([1])
g

2
.
So:
E

c
0
e

=
E

p
([n])
g

+ E

p
([1])
g

2
Proposition 7. For a set P =
n
p
(i)
f
o
, ∀i = 1, . . . , n,
of TrFN and a TrFN p
e
, max
n
i=1
(E(



p
(i)
f
 p
e



) =
max
n
i=1
(E(



p
(i)
f
 c
e



).
Proof. Let E(c
e
) =
1
2
(E(p
([1])
)+E(p
([n])
g
)). If E(p
e
) ≤
E

c
e

, E(p
([n])
)
^
− E(p
e
) ≥ E(p
([n])
)
^
− E(c
e
). So:
n
max
i=1





E(p
([i])
)
^
− E(p
e
)





≥





E(p
([n])
)
^
− E(p
e
)





≥
E

p
([n])
g

− E

p
([1])
g

2
=
n
max
i=1




E

p
([i])
g

− E

c
e





.
If E(p
e
) ≥ E

c
e

, E(p
e
) − E(p
([1])
)
^
≥ E(c
e
) −
E(p
([1])
)
^
. So:
n
max
i=1





E(p
([i])
)
^
− E(p
e
)





≥





E(p
e
) − E(p
([1])
)
^





≥
E

p
([n])
g

− E

p
([1])
g

2
=
n
max
i=1




E

p
([i])
g

− E

c
e





.
Again, due to the linearity of the GMIR,
max
n
i=1



E(p
([i])
g
) − E(c
e
)



= max
n
i=1
E



p
([i])
g
 c
e



Proposition 8. For a set P = {P
(i)
f
: P
(i)
f
=
{p
(i, j)
g
}}, ∀i ∈ 1, .. ., n, j ∈
{
x, y
}
, where p
(i, j)
g
is a TrFN, and the fuzzy center C
e
= {c
( j)
f
},
max
n
i=1
(d
∞
f
(P
(i)
f
, P
e
)) ≥ max
n
i=1
(d
∞
f
(P
(i)
f
,C
e
)).
Proof. Let E(c
( j)
f
) =
1
2
E(p
([1], j)
^
⊕ p
([n], j)
^
) and a the
fuzzy point P
e
= {p
( j)
g
}. Then:
max
n
i=1
d
∞
e
d
∞
P
(i)
f
,P
e
!!
=max
n
i=1
max
j∈{x,y}
E





p
(i, j)
g
p
( j)
f





!
=max
j∈{x,y}
max
n
i=1





E p
(i, j)
g
!
−E p
( j)
f
!





.
From Proposition 7, we will recall that:
max
n
i=1




E

p
(i, j)
g

−E

p
( j)
f





≥max
n
i=1




E

p
(i, j)
g

−E

c
( j)
f





,
so:
max
j∈{x,y}
max
n
i=1





E p
(i, j)
g
!
−E p
( j)
f
!





≥
max
j∈{x,y}
max
n
i=1





E p
(i, j)
g
!
−E c
( j)
f
!





=max
n
i=1
d
∞
e
P
(i)
f
,C
e
!
.
Proposition 9. For a set P =
n
P
(i)
f
: P
(i)
f
=
n
p
(i, j)
g
o
, ∀i ∈ 1, .. . , n, j ∈
{
x, y
}
o
,
where p
(i, j)
g
is a TrFN, the fuzzy min-max center
C
∗
f
=
n
c
( j)
f
o
is argmin
C
e
max
n
i=1
d
∞
f

P
(i)
f
,C
e

= {c
e
:
E(c
( j)
) =
1
2
E(p
([1], j)
^
⊕ p
([n], j)
^
)}.
Proof. Let the fuzzy point P
e
= {p
( j)
g
}, then:
max
n
i=1
E d
∞
e
P
(i)
f
,P
e
!!
=max
n
i=1
max
j∈{x,y}
E





p
(i, j)
g
p
( j)
f





!
=max
n
i=1
max
j∈{x,y}





E p
(i, j)
g
!
−E p
( j)
f
!





.
ON THE EXTENSION OF THE MEDIAN CENTER AND THE MIN-MAX CENTER TO FUZZY DEMAND POINTS
413

By the result of Proposition 8:
E

c
( j)
f

=
E

p
(i, j)
g

+ E

p
( j)
g

2
.
Then:
n
max
i=1
d
∞
f

P
(i)
f
, P
e

≥
n
max
i=1
d
∞
f

P
(i)
f
,C
e

.
Given that the solution of the fuzzy min-max cen-
ter is a set of fuzzy points, we will extend the result
for crisp values with the following definition.
Definition 17 (Fuzzy min-max center in R
2
). For a
set P =
n
P
(i)
f
: P
(i)
f
=
n
p
(i, j)
g
o
, ∀i ∈ 1, . . . ,n, j ∈
{
x, y
}
o
,
where p
(i, j)
g
is a TrFN, the fuzzy min-max center C
e
=
n
c
( j)
f
o
is defined as:
c
( j)
f
:=
p
([1], j)
^
⊕ p
([n], j)
^
2
. (31)
5 NUMERICAL EXAMPLE
In the following numerical example we will see how
the three centers are found and how much they differ
from each other. Lets suppose there are three fuzzy
demand points:
P
(1)
g
=
n
p
(1,x)
]
, p
(1,y)
]
o
p
(1,x)
]
= (18, 35,37, 40)
p
(1,y)
]
= (31, 49,49, 68)
P
(2)
g
=
n
p
(2,x)
]
, p
(2,y)
]
o
p
(2,x)
]
= (58, 75,75, 94)
p
(2,y)
]
= (87, 103, 105, 121)
P
(3)
g
=
n
p
(3,x)
]
, p
(3,y)
]
o
p
(3,x)
]
= (73, 83, 86, 107)
p
(3,y)
]
= (10, 20,21, 29)
20 40 60 80 100 120
20 40 60 80 100 120
x
y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Figure 1: Fuzzy median center.
20 40 60 80 100 120
20 40 60 80 100 120
x
y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Figure 2: Fuzzy min-max center.
The expected values for these three points would be:
E

p
(1,x)
]

= 33.667
E

p
(1,y)
]

= 49.167
E

p
(2,x)
]

= 75.333
E

p
(2,y)
]

= 104
E

p
(3,x)
]

= 86.333
E

p
(3,y)
]

= 20.167
For these points, the fuzzy median center (see Fig-
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
414

ure 1) would be:
M
e
=
n
m
(x)
g
, m
(y)
g
o
m
(x)
g
= median
^
i=1,...,3

p
(i,x)
g

= (58, 75, 75, 94)
m
(y)
g
= median
^
i=1,...,3

p
(i,y)
g

= (31,49, 49,68).
And the min-max center (see Figure 2) would be:
Z
e
=
n
z
(x)
f
, z
(y)
f
o
z
(x)
f
=
1
2
∑
i∈
{
1,3
}
p
([i],x)
^
= (49.667, 64.333, 66, 80.333)
z
(y)
f
=
1
2
∑
i∈
{
1,3
}
p
([i],y)
^
= (42.667, 57.333, 58.333, 72.667) .
As we can see from the figures, the option of using
fuzzy numbers to model the demand points is much
more closer to what in reality geographers and plan-
ners face. The results obtained will give them flexi-
bility in the final location of the center, according to
constraints not easily modeled otherwise.
6 CONCLUSIONS
In this paper we have shown that the results found for
the solution of the median center and the min-max
center can be extended to fuzzy environments, where
both the demand points and the center are modeled
with fuzzy numbers. The use of fuzzy numbers is due
to the need to reflect the uncertainty about available
information on demand. Not only the data might be
vague or subjective, but it could also involve disagree-
ments or lack of confidence in the methodology used
in its collection. Therefore, it is necessary to have a
solution that, while simply obtained, incorporates this
uncertainty.
Fuzzy solutions can also give flexibility to plan-
ners on the final location of the center, according to
constraints that are not easily modeled. The selected
center will have a membership value that reflects its
“appropriateness” according to the data.
Future work deriving from this methodology will
follow solving the fuzzy 1-median problem as well
as the barycenter, when the solution is modeled with
fuzzy numbers. We would like to use the results found
for multicriteria analysis, creating a fuzzy Pareto front
by intersecting the solutions found for different values
of p.
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