Ta-Chung Chu
1*
Chun-Chi Chen
2
Wei-Chun Hsu
3
1,2,3 Graduate Institute of Industrial Management’
Southern Taiwan University, Taiwan, R.O.C.
* Corresponding Professor. Email: tcchu@mail.stut.edu.tw
Abstract
The evaluation and selection of a distribution center has been an important issue for a company to upgrade its distribution efficiency and operation performance in order to fulfill the diverse demands of consumer. This paper suggests a maximizing set and minimizing based fuzzy MCDM model to resolve this problem, where ratings of alternatives versus qualitative criteria and the importance weights of all the criteria are assessed in linguistic values represented by fuzzy numbers. Ranking formulae and membership functions for the final fuzzy evaluation values can be clearly developed for better executing the decision making. A numerical is used to demonstrate the feasibility of the proposed approach.
Keywords: Fuzzy MCDM, Distribution center, Location selection, Ranking, Maximizing set and minimizing set.
1.
Introduction
Manufacturers, customers and suppliers are important members of a supply chain. To some extent, the success of a manufactory depends on its ability to link these members seamlessly. In the real logistics systems, it often requires some distribution centers to connect manufactories and their customers for the improvement of product flow [38].
Distribution center (DC) is viewed as the competence that links an enterprise with its customers and suppliers. A distribution center is usually supplied by the sources such as manufacturing factories, vendors, etc., and in turn it supplies the consumers or demand locations. In order to reduce transportation cost, enforce operation efficiency and logistics
1

performance, evaluating and selecting a DC location has become one of the most important decision issues for distribution industries. [7].
Evaluating a DC location, many conflicting criteria must be considered. These criteria can be classified into two categories: (1) objective – these criteria can be evaluated quantitatively, e.g.
investment cost, and (2) subjective – these criteria have qualitative definitions, e.g.
expansion possibility, closeness to demand market, etc. In addition, these criteria may have different importance [5,29-32]. Numerous precision-based methods for location selection problems have been investigated [2-3,11,14-16,19,22-23,26-28]. A review can be seen in [34]. All the above methods are based on the concept of accurate measure and crisp evaluation, i.e.
the measuring values must be numerical and exact. In [24], Perez et al. pointed out “Location problems concern a wide set of fields where it is usually assumed that exact data are known. However, in real applications, the location of facility considered can be full of linguistic vagueness, that can be appropriately modeled using networks with fuzzy values.” Moreover, the values for the qualitative criteria are often imprecisely defined for the decision-makers. Obviously, the precision-based methods are not adequate to resolve the DC location selection problem. To resolve the above problems, a fuzzy multiple criteria decision making (MCDM) model is needed.
Fuzzy set theory, initially proposed by Zadeh [41], has been extensively applied to objectively reflect the ambiguities in human judgment and effectively resolve the uncertainties in the available information in an ill-defined multiple criteria decision making environment. Numerous approaches have been proposed to solve fuzzy MCDM problems.
A review and comparison of many of these methods can be found in Carlsson and Fuller [6],
Chen and Hwang [9], Ribeiro [25], Triantaphyllou and Lin [33] and Deng [17]. Some recent applications on locations evaluation and selection can be found in [4,7,12]. Despite the merits, most of the above papers can not present membership functions for the final fuzzy evaluation values, nor can they clearly develop defuzzification formulae from the membership functions of the final fuzzy evaluation values, limiting the applicability of the fuzzy MCDM methods available. To resolve the these limitations, this work suggests a maximizing set and minimizing set based fuzzy MCDM approach for the evaluation and selection of distribution centers. Many fuzzy number ranking methods have been studied. A comparison of many of these ranking methods can be found in Cheng [10] and Wang and Kerre [36]. Some recent approaches can be seen in [1,13,21,37,39]. However, in spite of the merits, some of these methods are computational complex and difficult to implement and none of them can satisfactorily rank fuzzy numbers in all situations and cases. Herein, the ranking approach of
2

maximizing set and minimizing set [8] is applied for defuzzification due to its simplicity of implementation. Furthermore, defuzzification procedure can be clearly presented and formulae can be developed. Finally, a numerical example demonstrates the computational process of the proposed model .
The rest of this work is organized as follows. Section 2 briefly introduces fuzzy set theory. Section 3 introduces the suggested model. Meanwhile, an example is presented in
Section 4 to demonstrate the feasibility of the proposed model and conclusions are made in
Section 5
2.
Fuzzy set theory
2.1 Fuzzy Sets
A

  x , f
A
  
| x

U

, where U is the universe of discourse, x is an element in U , A is a fuzzy set in U , f
A
is the membership function of A at x [20]. The large f
A stronger the grade of membership for x in A .
, the
2.2 Fuzzy Numbers
A real fuzzy number A is described as any fuzzy subset of the real line R with membership function f
A
which possesses the following properties [18]:
(a)
(b)
(c)
(d)
(e)
(f) f
A
is a continuous mapping from R to [0,1]; f
A

0 ,
 x


 
, a ] ; f
A
is strictly increasing on f
A

1 , x

 
;
 
; f
A
is strictly decreasing on
 
; f
A

0 ,
 x

[ d ,

) ; where a
 b
 c
 d , A can be denoted as
 a , b , c , d

. The membership function f
A
of the fuzzy number A can also be expressed as: f
A





1


0 f f
L
A
R
A
 
 
,
, a
, b c


 x x x



, otherwise b c d
(1)
3

where f
L
A
and f
R
A
 
are left and right membership functions of A , respectively [20].
A fuzzy triangular number can be denoted as ( a , b , c ) [35].
2.3 α-cuts
The α-cuts of fuzzy number
A can be defined as A
 
 x | f
A
  
,
 
, where
A

is a non-empty bounded closed interval contained in R and can be denoted by
A



A l

, A
 u

, where A l

and A
 u
are its lower and upper bounds, respectively [20].
2.4 Arithmetic Operations on Fuzzy Numbers
B

Given fuzzy numbers A and B ,

B l

, B
 u
A , B

R

, the α-cuts of
A and B are A



A l

, A
 u

and
 
, respectively. By the interval arithmetic, some main operations of A and B can be expressed as follows [20]:

A

B
 

A B
 

A

B
 






A l

A l

A l




B l

B
 u
B l

, A
 u
,
,
A
 u
A
 u



B
 u
B
 u
B l





A B
 

A
 r
 





A l

A l

 r
B
 u
,
,
A
 u
A
 u
 r



B l

, r

R

(2)
(3)
(4)
(5)
(6)
2.5 Linguistic Values
A linguistic variable is a variable whose values are expressed in linguistic terms.
Linguistic variable is a very helpful concept for dealing with situations which are too complex or not well-defined to be reasonably described by traditional quantitative expressions [40].
For example, “importance” is a linguistic variable whose values include UI (unimportant), LI
(less important), I (important), MI (more important) and VI (very important). These linguistic values can be further represented by triangular fuzzy numbers such as UI=(0.0,0.0,0.25),
LI=(0.0,0.25,0.5), I=(0.25,0.5,0.75), MI=(0.50,0.75,1.00) and VI=(0.75,1.00,1.00).
4

3.
Model development
Suppose decision makers D t
, t

1, 2,..., , are responsible for evaluating alternatives i
,
 m under selected criteria, C j
, j
 n Criteria are categorized into three groups such as benefit qualitative criteria C j
, j
 g benefit quantitative criteria
C j
, j
 
1,..., , and cost quantitative criteria C j
, j h The proposed model is developed as the following steps.
3.1 Aggregate ratings of alternatives versus qualitative criteria
Assume x ijt

( a ijt
, b ijt
, c ijt
) , i

1,..., ,
 g t
 l x ij
 l
1

( x ij 1
 x ij 2

...
 x ijl
) , (7) where a ij
 l
1 t l 

1 a ijt
, b ij

1 l t l 

1 b ijt
, c ij
 l
1 t l 

1 c ijt
. x ijt
denotes ratings assigned by each decision maker for each alternative versus each qualitative criterion. x ij
denotes averaged rating of each alternative versus each qualitative criterion.
3.2 Normalize values of alternatives versus quantitative criteria
Herein, Chen’s [7] method is applied to normalize values of alternatives versus quantitative criteria, including benefit and cost, in order to make data dimensionless for calculation rationale. Benefit quantitative data has the characteristics: the larger the better; whereas cost quantitative data has the characteristics: the smaller the better. Suppose y ij

( o ij
, ij
, ij
) denotes evaluation value of alternative i versus benefit quantitative criteria
,
 
as well as cost quantitative criteria , h And x ij
denotes the normalized value of y ij
, x
 ij
( o ij q
* ij p ij
, q
* ij q ij
, q
* ij
), q
* ij
 max q ij
, j

B , x ij
 o
* ij o
* ij o
* ij
( , , ), q ij p ij o ij o ij
*  min o ij
, j

C .
For calculation convenience, assume x ij

( a ij
, b ij
, c ij
) , j
 g

1 ,..., n .
5
(8)
(9)

3.3 Average importance weights
Assume w jt

( d jt
, e jt
, f jt
), w jt

R

, j

1 ,..., n , t

1 ,..., l , w j

1
 l
( w j 1
 w j 2

...
 w jl
), (10) where d j
 l
1 t l 

1 d jt
, e j
 l
1 t l 

1 e jt
, f j
 l
1 t l 

1 f jt
.
w jt
represents the weight assigned by each decision maker for each criterion and w j
represents the average importance weight of each criterion.
3.4 Develop membership functions
The membership function of the final fuzzy evaluation value, ,

1,..., n , of each alternative can be developed as Eq. (11). In Eq. (1), the first two parts are additive weighted ratings under benefit criteria. The third part is under cost criteria but given a negative sign.
Therefore, the larger the G i
value, the better performance A i
will have.
G i
 g
 j

1 w j
 x ij
 g h


1 w i
 x ij
 n
 j h 1 w j
 x ij
,
The membership functions are developed as:
G i
  j g


1 w
 j
 x
 ij
 h
 j g 1 w
 j
 x
 ij
 n
 j h 1 w
 j
 x
 ij
, w
 j

[( e j
 d j
)
  d j
, ( e j
 f j
)
  f j
] , x ij
 
[( b ij
 a ij
)
  a ij
, ( b ij
 c ij
)
  c ij
] .
(11)
(12)
(13)
(14)
From Eqs. (13) and (14), we can develop Eq. (15) as follows:
 w
 j
 x
 ij
 
( e j
 d j
)( b ij
 a ij
)
 2  
( ij
( j
 d j
)
 j
( ij
 a ij
))
   a d j
,

( b ij
 c ij
)( e j
 f j
)
 2  
( ij
( j
 f j
)
 j
( ij
 c ij
))
   c f j
(15)
By applying Eq. (15) to Eq. (12), three equations are developed:
6

j g 

1 w
 j
 x ij
  g
[
 j

1
( e j g  j

1
( b ij
 d j
)( b ij
 c ij
)( e j

 a ij
)

2 f j
)

2 
 j g 

1
( a ij
( e j j g 

1
( c ij
( e j

 d j f j
)
)

 d j
( b ij f j
( b ij

 a ij
))
 c ij
))

 j g 

1 a ij d ij
,
(16)
 j g 

1 c ij f j
] . j
 h  g

1 w
 j
 x ij
 
[ j
 h  g

1
( e j j
 h  g

1
( b ij
 d j
)( b ij
 c ij
)( e j
 a ij
)
 2
 f j
)

2 
 j
 h  g

1
( a ij
( e j j
 h  g

1
( c ij
( e j

 d j
) f j
)
 f
 d j
( b ij j
( b ij
 a ij
))

 c ij
))
 
 j
 h  g

1 a ij d ij
,
(17) j
 h  g

1 c ij f j
] . j n 
 h

1 w
 j
 x ij
 
[ j n 
 h

1 n  j
 h

1
( b ij
( e j
 d j
)( b ij
 a ij
)

2 
 c ij
)( e j
 f j
)

2  j
 n  h

1 j n 
 h

1
( a ij
( e j
( c ij
( e j

 d j
)
 d j
( b ij f j
)
 f j
( b ij
 a ij
))

 c ij
))
 
 j
 n  h

1 a ij d ij
,
(18) j n 
 h

1 c ij f j
] .
Assume:
A i 1
 j g 

1
( e j
 d j
)( b ij
 a ij
) , A i 2
 j h 
 g

1
( e j
 d j
)( b ij
 a ij
) , A i 3
 j n 
 h

1
( e j
 d j
)( b ij
 a ij
) ,
B i 1
 j g 

1
[ a ij
( e j
 d j
)
 d j
( b ij
 a ij
)] , B i 2
 j h 
 g

1
[ a ij
( e j
 d j
)
 d j
( b ij
 a ij
)] ,
B i 3
 j
 n 

1
[ a h ij
( e j
 d j
)
 d j
( b ij
 a ij
)] , C i 1
 j g 

1
( b ij
 c ij
)( e j
 f j
) ,
C i 2
 j h

 g

1
( b ij
 c ij
)( e j
 f j
) , C i 3
 j n 
 h

1
( b ij
 c ij
)( e j
 f j
) ,
D i 1
 j g 

1
[ c ij
( e j
 f j
)
 f j
( b ij
 c ij
)] , D i 2
 j h

 g

1
[ c ij
( e j
 f j
)
 f j
( b ij
 c ij
)] ,
D i 3
 j n 
 h

1
[ c ij
( e j
 f j
)
 f j
( b ij
 c ij
)] , O i 1
 j g 

1 a ij d j
, O i 2
 j h

 g

1 a ij d j
, O i 3
 j n 
 h

1 a ij d j
,
P i 1
 j g 

1 b ij e j
, P i 2
 j h

 g

1 b ij e j
, P i 3
 j n 
 h

1 b ij e j
, Q i 1
 g  j

1 c ij f j
, Q i 2
 j h

 g

1 c ij f j
, Q i 3
 j n 
 h

1 c ij f j
.
7

By applying the above assumption, Eqs. (16)-(18) can be arranged as: j g 

1 w
 j
 x ij
 
[ A i 1

2 
B i 1
 
O i 1
, C i 1

2 
D i 1
 
Q i 1
] ,
 j h  g

1 w
 j
 x ij
 
[ A i 2

2 
B i 2
 
O i 2
, C i 2

2 
D i 2
 
Q i 2
] , j n 
 h

1 w
 j
 x ij
 
[ A i 3

2 
B i 3
 
O i 3
, C i 3

2 
D i 3
 
Q i 3
] .
(19)
(20)
(21)
Applying Eqs. (19)-(21) to Eq. (12) to produce Eq. (22):
G i


[( A i 1

A i 2

C i 3
)

( C i 1

C i 2

A i 3
)
 2
2


( B i 1

B i 2

D i 3
)
 
( O i 1

O i 2

Q i 3
) ,
( D i 1

D i 2

B i 3
)
 
( Q i 1

Q i 2

O i 3
)] .
(22)
The right and left membership functions of G i
can be obtained as shown in Eq. (23) and
Eq. (24) as follows:
  f
L
G i
( )


( B i 1

B i 2

D i 3
)

[( B i 1

B i 2

D i 3
)
2 ( A i 1
2


A
4 ( A i 1
 i 2

C i 3
A i 2
)

C i 3
)( x

( O i 1

O i 2

Q i 3
))]
1
2
(23)
If O i 1

O i 2

Q i 3
 x

P i 1

P i 2

P i 3
;
  f
R
G i
( )


( D i 1

D i 2

B i 3
)

[( D i 1

D i 2

B i 3
2 ( C i 1
)
2


C
4 ( C i 1 i 2


A i 3
C i 2
)

A i 3
)( x

( Q i 1

Q i 2

O i 3
))]
1
2
(24)
If P i 1

P i 2

P i 3
 x

Q i 1

Q i 2

O i 3
.
3.5 Rank fuzzy numbers
In this research, Chen’s maximizing set and minimizing set [8] is applied to rank all the final fuzzy evaluation values. This method is one of the most commonly used approaches of ranking fuzzy numbers in fuzzy decision making.
8

The maximizing set M is defined as: f
M




(
0 , x x
R i max

 x x min min otherwise
) k
, x min
.
 x
R i
 x max
,
(25)
The minimizing set N is defined as: f
N




(
0 , x x
L i min

 x x max max otherwise
)
, k
, x min
 x
L i
 x max
,
(26) where x min
 inf x
S , x max
 sup x
S , S
  i n

1
S i
, S i

{ x f
A i
( x )

0 } , usually k is set to 1.
The right utility of A i
is defined as:
U
M
( A i
)
 sup ( x f
M
( x )
 f
A i
( x )) , i

1 ~ n . (27)
The left utility of A i
is defined as:
U
N
( A i
)
 sup ( x f
M
( x )
 f
A i
( x )) , i

1 ~ n .
The total utility of A i
is defined as:
(28)
U
T
( A i
)

1
2
( U
M
( A i
)

1

U
N
( A i
)) , i

1 ~ n . (29)
The total utility U
T
( A i
) is applied to rank fuzzy numbers. The larger the U
T
( A i
) , the larger the fuzzy number A i
. Applying Eqs. (25)-(29) to Eqs. (23)-(24), the total utility of fuzzy number G i
can be obtained as:
9

U
T
( G i
)

1
2
( U
M
( G i
 
U
N
( G i
)), i
 n

1
2
[

( D i 1

D i 2

B i 3
)

[( D i 1

D i 2

B i 3
)
2
2 ( C i 1


4 ( C i 1
C i 2


C i 2
A i 3
)

A i 3
)( x
R
I

1


( B i 1

B i 2

D i 3
)

[( B i 1

B i 2

D i 3
)
2
2 ( A i 1

4 ( A i 1

A i 2

A i

C i 3
)
2

C i 3
)( x
L i

( Q i 1

Q i 2

O i 3
))] 2
1

( O i 1

O i 2

Q i 3
))] 2
1
] . (30) where x
R i
 
( 2 ( C i 1

C i 2

A i 3
) x min

( x min
 x max
)( D i 1

D i 2

B i 3
 x min
 x max
))

( x max
 x min
)[( D i 1

D i 2

B i 3
 x min
 x max
)
2

4 ( C i 1

C i 2

A i 3
)( x min

Q i 1

Q i 2

O i 3
)] 2
1
/ 2 ( C i 1

C i 2

A i 3
) .
(31) x
L i
 
( 2 ( A i 1

A i 2

C i 3
) x max

( x max
 x min
)( B i 1

B i 2

D i 3
 x max
 x min
))

( x min
 x max
)[( B i 1

B i 2

D i 3
 x max
 x min
)
2

4 ( A i 1

A i 2

C i 3
)( x max

O i 1

O i 2

Q i 3
)] 2
1
/ 2 ( A i 1

A i 2

C i 3
) . (32)
4.
Numerical example
Assume that a logistics company is looking for a suitable city to set up a new distribution center. Three decision makers, D
1
, D
2
and D
3
of this company are responsible for the evaluation of three distribution center candidates, A
1
, A
2
and A
3
. Four benefit qualitative criteria such as expandability ( C
1
), convenience to acquiring materials ( C
2
), closeness to market ( C
3
), human resources ( C
4
); one benefit quantitative criterion such as area size ( C
5
); and one cost quantitative criterion such as investment cost ( C
6
) are chosen for evaluating the distribution centers.
Further assume that linguistic values and their corresponding triangular fuzzy numbers shown in Table 1 are used to evaluate each distribution center candidate versus each qualitative criterion. Ratings of distribution center candidates versus qualitative criteria are given by decision makers as shown in Table 2. Through Eq. (7), averaged ratings of distribution center candidates versus qualitative criteria can be obtained as also displayed in
Table 2. In addition, suppose values of distribution center candidates versus quantitative
10

criteria are present as in Table 3. According to Eqs. (8) and (9), values of alternatives under benefit and cost quantitative criteria can be normalized as shown in Table 4. The linguistic values and its corresponding fuzzy numbers, shown in section 2.5, are used by decision makers to evaluate the importance of each criterion as displayed in Table 5. The average weight of each criterion can be obtained using Eq. (10) and can also be shown in Table 5.
Table 1 Linguistic values and fuzzy numbers for ratings
Very low(VL) /Very difficult(VD) /Very far(VF)
Low(L)/Difficult(D)/Far(F)
Medium(M)
High(H)/Easy(E)/Close(C)
Very high(VH)/Very easy(VE)/Very close(VC)
(0.00,015,0.30)
(0.15,0.30,0.50)
(0.30,0.50,0.70)
(0.50,0.70,0.85)
(0.70,0.85,1.00)
Table 2 Ratings of distribution center candidates versus qualitative criteria
Candidates Criteria
A
A
A
1
2
3
C
1
C
2
C
3
C
4
C
1
C
2
C
3
C
4
C
1
C
2
C
3
C
4
L
VE
M
L
VH
M
C
VH
D
1
VH
VE
C
M
L
E
M
M
VH
M
C
VH
D
2
H
E
VC
H
H
VE
C
H
H
E
VC
VH
D
3
VH
M
VC
H
Averaged Ratings
(0.63,0.80,0.95)
(0.50,0.68,0.85)
(0.63,0.80,0.95)
(0.43,0.63,0.80)
(0.63,0.80,0.95)
(0.37,0.57,0.75)
(0.57,0.75,0.90)
(0.70,0.85,1.00)
(0.27,0.43,0.62)
(0.63,0.80,0.95)
(0.37,0.57,0.75)
(0.32,0.50,0.68)
11

Table 3 Values of distribution center candidates versus quantitative criteria
Criteria
Distribution Center Candidates
C
C
5
6
A
1
100
2
A
2
80
5
A
3
90
10
Table 4 Normalization of quantitative criteria
Units hectare million
Distribution Center Candidates
Criteria
C
5
C
6
A
1
1
1
A
2
0.8
0.4
A
3
0.9
0.2
Table 5 Averaged weight of each criterion
C
1
C
2
C
3
C
4
C
5
C
6
D
1
MI
IM
LI
UI
MI
VI
D
2
VI
MI
LI
IM
VI
VI
D
3
IM
LI
VI
VI
IM
VI
Averaged weights
(0.50,0.75,0.92)
(0.25,0.50,0.75)
(0.25,0.53,0.67)
(0.33,0.50,0.67)
(0.50,0.75,0.92)
(0.75,1.00,1.00)
Apply Eqs. (11)-(22) and g

4 , h

5 , n

6 to the numerical example to produce A A A i 3,
B i 1 ,
B i 2 ,
B i 3 ,
C i 1 ,
C i 2 ,
C i 3 ,
D i 1 ,
D i 2 ,
D i 3 ,
O i 1 ,
O i 2 ,
O i 3 ,
P i 1 ,
P i 2 ,
P i 3 ,
Q i 1 ,
Q i 2 ,
Q i 3 for each candidate as displayed in Table 6. The calculation values for A i 1

A i 2

C i 3 ,
B i 1

B i 2

D i 3 ,
O i 1

O i 2

Q i 3 ,
Q i 1

Q i 2

O i 3
are shown in Table 7.
C i 1

C i 2

A i 3 ,
D i 1

D i 2

B i 3 ,
P i 1

P i 2

P i 3
,
12

Table 6 Values for A i1
, A i2
, A i 3,
B i1
, B i2
, B i3,
C i1
, C i2
, C i3,
D i1
, D i2
, D i3,
O i1
, O i2
, O i3,
P i1
, P i2
, P i3,
Q i1
, Q i2
, Q i3
A
1
A
2
A
3
O i2
O i3
P i1
P i2
A i1
A i2
A i3
B i1
B i2
B i3
C i1
C i2
C i3
D i1
D i2
D i3
O i1
P i3
Q i1
Q i2
Q i3
0.17
0.00
0.00
0.77
0.25
0.25
0.12
0.00
0.00
-1.11
-0.17
0.00
0.74
0.50
0.75
1.68
0.75
1.00
2.68
0.92
1.00
0.17
0.00
0.00
0.76
0.16
0.10
0.12
0.00
0.00
-1.11
-0.13
0.00
0.78
0.40
0.30
1.71
0.60
0.40
2.70
0.73
0.40
0.17
0.00
0.00
0.62
0.23
0.05
0.12
0.00
0.00
-1.08
-0.15
0.00
0.49
0.45
0.15
1.28
0.68
0.20
2.23
0.83
0.20
Table 7 Values for A i1
+ A i2
C i3
, B i1
+ B i2
D i3
, O i1
+ O i2
Q i3
, C i1
+ C i2
A i3
, D i1
+ D i2
B i3
, P i1
+ P i2
P i3
,
Q i1
+ Q i2
O i3
A i1
+A i2
-C i3
B i1
+B i2
-D i3
O i1
+O i2
-Q i3
C i1
+C i2
-A i3
D i1
+D i2
-B i3
P i1
+ P i2
P i3
Q i1
+Q i2
-O i3
A
1
0.17
1.02
0.24
0.12
-1.53
1.43
2.84
A
2
0.17
0.92
0.78
0.12
-1.34
1.91
3.13
A
3
0.17
0.84
0.74
0.12
-1.28
1.75
2.91
13

Through Eqs. (23) and (24), the left, f
L
G i
( ) , and right, f
R
G i
( ) , membership functions of the final fuzzy evaluation value, ,

1,..., n , of each distribution center candidate can be obtained and displayed in Table 8. f
L
G
1
( )
Table 8 Left and right membership functions of G i

1.02

(1.02)
2 
4(0.17)( x

0.24)
1
2
, 0.24
1.43
f
R
G
1
( )
1.53
( 1.53)
2 
4(0.12)( x

2.84)
1
2
, 1.43
2.84
f
L
G
2
( )

0.92

(0.92)
2 
4(0.17)( x

0.78)
1
2
, 0.78
1.91
f
R
G
2
( )
1.34
( 1.34)
2 
4(0.12)( x

3.13)
1
2
, 1.91
3.13
f
L
G
3
( )

0.84

(0.84)
2 
4(0.17)( x

0.74)
1
2
, 0.74
1.75
f
R
G
3
( )
1.28
( 1.28)
2 
4(0.12)( x

2.91)
1
2
, 1.75
2.91
By Eqs. (25)-(32), the total utilities, U
T
( G i
), x
R i
and x
L i
can be obtained and shown in
Table 9.
Alternatives
Table 9 Total utilities U
T
( G i
), x
R i
and x
L i
G
1
G
2
G
3 x i
1.97 2.26 2.12 x i
U
T
( G i
)
1.39
0.315
1.40
0.551
1.33
0.517
14

Then according to values in Table 9, candidate A
2
has the largest total utility,
U
T
( G
2
) =0.551. Therefore A
2
becomes the most suitable distribution center candidate for this company.
5.
Conclusions
A fuzzy MCDM model is proposed for the evaluation and selection of the locations of distribution centers, where ratings of alternatives versus qualitative criteria and the importance weights of all the criteria are assessed in linguistic values represented by fuzzy numbers. Membership functions of the final fuzzy evaluation values can be developed through interval arithmetic and α-cuts of fuzzy numbers. Chen’s maximizing set and minimizing set is applied to defuzzify the final fuzzy evaluation values in order to rank all the alternatives. Ranking formulae are clearly developed for better executing the decision making. Finally a numerical has demonstrated the computational procedure of the proposed approach.
15

References
[1] Abbasbandy, S. and Asady, B., Ranking of fuzzy numbers by sign distance, Information
Sciences 176 (2006) 2405-2416.
[2] Aikens, C.H., Facility location models for distribution planning, European J. Oper. Res .
22 (1985) 263-279.
[3] Alumur, S., Kara, B.Y., A new model for the hazardous waste location-routing problem,
Computers & Operations Research 34 (2007) 1406-1423.
[4] Anagnostopoulos, K. Doukas, H. and Psarras, J., A linguistic multicriteria analysis system combining fuzzy sets theory, ideal and anti-ideal points for location site selection, Expert
Systems with Applications 35 (2008) 2041-2048.
[5] Bowersox, D.J. and Closs, D.J., Logistical Management – The Integrated Supply Chain
Process, McGraw-Hill, Singapore (1996).
[6] Carlsson, C. and Fuller, R., Fuzzy multiple criteria decision making: Recent developments, Fuzzy Sets and Systems 78 (1996) 139-153.
[7] Chen, C.T., A fuzzy approach to select the location of the distribution center, Fuzzy Sets and Systems 118 (2001) 65-73.
[8] Chen, S.H., Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems 17 (1985) 113-129.
[9] Chen, S.J. and Hwang, C.L., Fuzzy Multiple Attribute Decision Making Methods and
Applications, Springer, Berlin (1992).
[10] Cheng, C.H., A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95 (1998) 307-317.
[11] Cheng, E.W.L., Li, H. and, Yu, L., The analytic network process approach to location selection : a shopping mall illustration, Construction Innovation 5 (2005) 83-97.
[12] Chou, C.C., A fuzzy MCDM method for solving marine transshipment container port selection problems, Applied Mathematics and Computation 186 (2007) 435-444.
[13] Chou, C.C., The representation of multiplication operation on fuzzy numbers and application to solving fuzzy multiple criteria decision making problems, Lecture Notes in
Artificial Intelligence 4099 (2006) 161-169.
[14] Colebrook, M. and Sicillia, J., Undesirable facility location problems on multicriteria networks, Computers & Operations Research 34 (2007) 1491-1514.
[15] Cram, S., Sommer, I., Morales, L., Oropeza, O., Carmona, E. and Gonzales-Medrano, F.,
Suitability of the vegetation types in Mexico’s Tamaulipas state for the siting of
16

hazardous waste treatment plants, Journal of Environmental Management 80 (2006)
13-24.
[16] Dahlberg, M.D. and May, J.H., Linear programming for sitting of energy facilities, J.
Energy Eng.
(1980) 5-14.
[17] Deng, H, Multicriteria analysis with fuzzy pairwise comparison, International Journal of
Approximate Reasoning 21 (1999) 215-231.
[18] Dubois, D. and Prade, H., Operations on fuzzy numbers, International Journal of Systems
Science 9 (1978) 613-626.
[19] Hodder, J.E. and Dincer, M.C., A multifactor model for international plant location and financing under uncertainty, Comput. & Oper. Res.
13 (5) (1986) 601-609.
[20] Kaufmann, A. and Gupta, M.M., Introduction to Fuzzy Arithmetic: Theory and
Application, Van Nostrand Reinhold, New York (1991).
[21] Liu, X.W. and Han, S.L., Ranking fuzzy numbers with preference weighting function expectations, Computers and Mathematics with Applications 49 (2005) 1731-1753
[22] Malczewski, J., GIS-based multicriteria decision analysis: a survey of the literature,
International Journal of Geographical Information Science 20 (2006) 703-726.
[23] Pavic’, I. and Babic’, Z., The use of the PROMETHEE method in the location choice of a production, Internat. J. Prod. Econom.
23 (1991) 165-174.
[24] Perez, J.A.M., Vega, J.M.M. and Verdegay, J.L., Fuzzy location problems on networks,
Fuzzy Sets and Systems 142 (2004) 393-405.
[25] Ribeiro, R.A., Fuzzy multiple attribute decision making: A review and new preference elicitation techniques, Fuzzy Sets and Systems 78 (1996) 155-181.
[26] Rietveld, Ouwersloot, P.H., Ordinal data in multicriteria decision making, a stochastic dominance approach to sitting nuclear power plants, European J. Oper. Res.
56 (1992)
249-262.
[27] Rodriguez, J.J.S., Garcia, C.G., Perez, J.M. and Casermeiro, E.M., A general model for the undesirable single facility location problem, Operations Research Letters 34 (2006)
427-436.
[28] Sener, B., Suzen, M.L. and Doyuran, V., Landfill site selection by using geographic information systems, Environmental Geology 49 (2006) 376-388.
[29] Spohrer, G.A. and Kmak, T.R., Qualitative analysis used in evaluating alternative plant location acenarios, Indust. Eng.
August (1984) 52-56.
[30] Stevenson, W.J., Operations Management, 9 th
Edition (2007), McGraw-Hill, Irwin.
[31] Sule, D.R., Manufacturing Facility; Location, Planning and Design, PWS Publishing,
17

Boston, 1994.
[32] Tompkins, J.A. and White, J.A., Facilities Planning, Wiley, New York, 1984.
[33] Triantaphyllou, E. and Lin, C.T., Development and evaluation of five fuzzy multi-attribute decision-making methods, International Journal of Approximate
Reasoning 14 (1996) 281-310.
[34] Tuzkaya, G., Onut, S., Tuzkaya, U.R., and Gulsun B., An analytic network process approach for locating undesirable facilities: An example from Istanbul, Turkey, Journal of Environmental Management 88 (2008) 970-983.
[35] Van Larrhoven, P.J.M. and Pedrycz, W., A Fuzzy Extension of Satty’s Priority Theory,
Fuzzy Sets and Systems 11 (1983) 229-241.
[36] Wang, X. and Kerre, E.E., Reasonable properties for the ordering of fuzzy quantities (I)
& (II), Fuzzy Sets and Systems 118 (2001) 375-385, 387-405.
[37] Wang, Y.J. and Lee, H.S., The revised method of ranking fuzzy numbers with an area between the centroid and original points, Computers & Mathematics with Applications
55 (2008) 2033-2042.
[38] Yang, L., Ji, X., Gao, Z., and, Li, K., Logistics distribution centers location problem and algorithm under fuzzy environment, Journal of Computational and Applied Mathematics
208 (2007) 303-315.
[39] Yong, D., Zhu, Z. and Liu, Q., Ranking fuzzy numbers with an area method using radius of gyration, Computers and Mathematics with Applications 51 (2006) 1127-1136.
[40] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci.
8 (1975) 199-249(I), 301-357(II).
[41] Zadeh, L.A., Fuzzy sets, Information and Control 8 (1965) 338-353.
18