A Maximizing Set and Minimizing Set Based Fuzzy
MCDM Approach for the Evaluation and Selection of
the Distribution Centers
Advisor:Prof. Chu, Ta-Chung
Student: Chen, Chun Chi
1
Outline
• Introduction
• Fuzzy set theory
• Model development
• Numerical example
• Conclusion
2
Introduction
• Properly selecting a location for establishing a distribution center is
very important for an enterprise to effectively control channels,
upgrade operation performance, service level and sufficiently
allocate resources, and so on.
• Selecting an improper location of a distribution center may cause
losses for an enterprise.
• Therefore, an enterprise will always conduct evaluation and
selection study of possible locations before determining
distribution center.
3
Introduction (cont.)
• Evaluating a DC location, many conflicting criteria must be
considered:
1. objective – these criteria can be evaluated quantitatively, e.g.
investment cost.
2. subjective – these criteria have qualitative definitions, e.g.
expansion possibility, closeness to demand market, etc.
• Perez et al. pointed out “Location problems concern a wide set
of fields where it is usually assumed that exact data are known,
but in real applications is full of linguistic vagueness.
4
Introduction (cont.)
• Fuzzy set theory, initially proposed by Zadeh, and it can
effectively resolve the uncertainties in an ill-defined multiple
criteria decision making environment.
• Some recent applications on locations evaluation and selection
can be found, but despite the merits, most of the above papers
can not present membership functions for the final fuzzy
evaluation values and defuzzification formulae from the
membership functions.
• To resolve the these limitations, this work suggests a
maximizing set and minimizing set based fuzzy MCDM
approach.
5
Introduction (cont.)
Purposes of this paper:
• Develop a fuzzy MCDM model for the evaluation and selection of
the location of a distribution center.
• Apply maximizing set and minimizing set to the proposed model in
order to develop formulae for ranking procedure.
• Conduct a numerical example to demonstrate the feasibility of the
proposed model.
6
Fuzzy set theory
• 2.1 Fuzzy sets
• 2.2 Fuzzy numbers
• 2.3 α-cuts
• 2.4 Arithmetic operations on fuzzy numbers
• 2.5 Linguistic values
7
2.1 Fuzzy set
• The fuzzy set A can be expressed as:
•
A  {( x, f A ( x)) | x U}
(2.1)
where U is the universe of discourse, x is an element in U, A is a
fuzzy set in U, f x is the membership function of A at x. The
larger f x , the stronger the grade of membership for x in A.
A
A
8
2.2 Fuzzy numbers
• A real fuzzy number A is described as any fuzzy subset of the real
line R with membership function f which possesses the following
properties (Dubois and Prade, 1978):
A
– (a)
fA
– (b)
f A ( x)  0, x  (, a] ;
– (c)
fA
– (d)
f A ( x)  1, x  b, c;
– (e)
fA
– (f)
f A ( x)  0, x  [d , ) ;
is a continuous mapping from R to [0,1];
is strictly increasing on [a ,b];
is strictly decreasing on [c ,d];
where, A can be denoted as a , b , c , d  .
9
2.2 Fuzzy numbers (Cont.)
• The membership function
expressed as:
 f AL ( x), a  x  b,

bxc
1,
f A ( x)   R
 f A ( x), c  x  d ,
0,
otherwise,

where
f AL (x)
and
respectively.
f AR (x)
fA
of the fuzzy number A can also be
(2.2)
are left and right membership functions of A,
10
2.3 α-cut
• The α-cuts of fuzzy number A can be defined as:
A  x | f A x    ,   0, 1
(2.3)
where A is a non-empty bounded closed interval contained in R
and can be denoted by A  [ A , A ] , where A and A are its lower
and upper bounds, respectively.





l
u
l
u
11
2.4 Arithmetic operations on fuzzy numbers
• Given fuzzy numbers A and B,
A  [ Al , Au ]



l
u
A, B  R 
, the α-cuts of A and B are
and B  [ B , B ], respectively. By interval arithmetic, some
main operations of A and B can be expressed as follows (Kaufmann
and Gupta, 1991):

–  A  B
 [ Al  Bl , Au  Bu ]
(2.4)



A

B
–
 [ Al  Bu , Au  Bl ]
(2.5)
–  A  B
 [ Al  Bl , Au  Bu ]
(2.6)
–  A () B 

–  A  r 
Al Au
[  ,  ]
Bu Bl

(2.7)

 Al  r , Au  r , r  R 
(2.8)
12
2.5 Linguistic variable
• According to Zadeh (1975), the concept of linguistic variable is
very useful in dealing with situations which are complex to be
reasonably described by conventional quantitative expressions.
A1
A2
A3
A4
A5
A1=(0,0,0.25)=Unimportant
A2=(0,0.25,0.5)=Less important
A3=(0.25,0.50,0.75)=important
A4=(0.50,0.75,1.00)=More important
A5=(0.75,1.00,1.00)=Very important
0
0.25
0.5
0.75
1
Figure 2-1. Linguistic values and triangular
fuzzy numbers
13
Model development
• 3.1 Aggregate ratings of alternatives versus
qualitative criteria
• 3.2 Normalize values of alternatives versus
quantitative criteria
• 3.3 Average importance weights
• 3.4 Develop membership functions
• 3.5 Rank fuzzy numbers
14
Model development
•
Dt
decision makers
Dt , t  1,2,..., l
• A candidate locations of distribution centers
i
•C j selected criteria,
Ai , i  1,2,..., m
C j , j  1,2,..., n
• In model development process, criteria are categorized into
three groups:
– Benefit qualitative criteria: C j , j  1,..., g
– Benefit quantitative criteria: C j , j  g  1,..., h
– Cost quantitative criteria:
C j , j  h  1,..., n
15
3.1 Aggregate ratings of alternatives versus
qualitative criteria
• Assume
•
xijt  (aijt , bijt , cijt ) ,
i  1,..., m, j  1,..., g , t  1,..., l
1
xij   ( xij1  xij 2  ...  xijl )
l
– where
1 l
aij   aijt ,
l t 1
1 l
bij   bijt ,
l t 1
(3.1)
1 l
cij   cijt
l t 1
– xijt : Ratings assigned by each decision maker for each alternative versus
each qualitative criterion.
– xij : Averaged ratings of each alternative versus each qualitative criterion.
16
3.2Normalize values of alternatives versus
quantitative criteria
• y ij  (oij , pij , qij ) is the value of an alternative Ai , i  1,2,..., m, versus a
benefit quantitative criteria j, j  g  1,..., h, or cost quantitative criteria
j , j  h  1,..., n .
•
xij
•
xij  (
oij pij qij
, * , * ) , qij*  max qij , j  B
*
qij qij qij
xij  (
oij oij oij
, * , * ) , qij*  min oij , j  C
*
qij pij oij
denotes the normalized value of
y ij
• For calculation convenience, assume
(3.2)
xij  (aij , bij , cij ) ,
j  g  1,..., n .
17
3.3 Average importance weights
• Assume w
•
jt
 (d jt , e jt , f jt ) ,
w jt  R  ,
j  1,..., n , t  1,..., l ,
1
w j   ( w j1  w j 2  ...  w jl )
l
• where
1 l
d j   d jt ,
l t 1
1 l
e j   e jt ,
l t 1
(3.3)
1 l
f j   f jt .
l t 1
• w jt : represents the weight assigned by each decision maker for
each criterion.
• w j : represents the average importance weight of each criterion.
18
3.4 Develop membership functions
• The membership function of final fuzzy evaluation value,
Ti , i  1,..., n
of each candidate distribution center can be developed as follows:
• The membership functions are developed as:
•
g
Gi   w j  xij 

j 1


h
w
j  g 1

j

 xij 
n
 w  x
j  h 1
j
ij
,
(3.4)
•
wj  [(e j  d j )  d j , (e j  f )  f j ] ,
(3.5)
•
xij  [(bij  aij )  aij , (bij  cij )  cij ] .
(3.6)
19
3.4 Develop membership functions (cont.)
• From Eqs.(3.5) and (3.6),we can develop Eqs.(3.7) and (3.8)as
follows:
wj  xij  [(e j  d j )(bij  aij ) 2  (aij (e j  d j )  d j (bij  aij ))  aij d ij ,
•
(bij  cij )(e j  f j ) 2  (cij (e j  f j )  f j (bij  cij ))  cij f j ] .
(3.7)
 w  x 
j
•
ij
 [ (e j  d j )(bij  aij ) 2   (aij (e j  d j )  d j (bij  aij ))   aij d j ,
 (b
ij
 cij )(e j  f j ) 2   (cij (e j  f j )  f j (bij  cij ))   cij f j ] .
(3.8)
20
3.4 Develop membership functions (cont.)
• When applying Eq.(3.8)to Eq.(3.4), three equations are developed:
•
g
w
j 1

j
g
j 1
 (b
w
j  g 1

j

 xij  [
h
 (e
j  g 1
 (b
w
j  h 1

j
g
g
j 1
j 1
h
 (a
 d j )(bij  aij ) 
2
j
j  g 1
h
 (c
 cij )(e j  f j ) 
2
ij
j g 1
•
j 1
 cij )(e j  f j )   (cij (e j  f j )  f j (bij  cij ))   cij f j ] .
h
n
j 1
2
ij
j1
•
g
2
g
h
g
 xij  [ (e j  d j )(bij  aij )   (aij (e j  d j )  d j (bij  aij ))   aij d ij ,

j  g 1
n
 xij  [  (e j  d j )(bij  a ij ) 

2
j  h 1
n
 (b
j h 1
ij
 cij )(e j  f j ) 
2
ij
ij
n
 (a
j  h 1
n
 (c
j  h 1
ij
h
a
(e j  d j )  d j (bij  aij )) 
(e j  f j )  f j (bij  cij )) 
ij
(3.9)
j  g 1
ij
d ij ,
h
c
j  g 1
(e j  d j )  d j (bij  a ij )) 
(e j  f j )  f j (bij  cij )) 
ij
f j ].
(3.10)
n
a
j  h 1
ij
d ij ,
n
c
j  h 1
ij
f j ].
(3.11)
21
Assume :
Ai1 
Ai 3 
Bi 2 
C i1 
Ci 3 
Di 2 
Oi1 
Oi 3 
Pi 2 
Qi1 
Qi 3 
g
 (e
 d j )(bij  a ij )
j
j 1
k
 (e
 a e
h
ij
j  g 1
g
 (b
k
 (b
ij
j  h 1
 c
h
j  g 1
ij (e j  f j )  f j (bij  c ij )
 aij d j
j 1
k
a
ij
dj
h
b
ij
j  g 1
ej
g
c
j 1
ij

 d j   d j bij  a ij 
fj
j  g 1
 a e


 d j   d j bij  a ij 
j
 a e
k
ij
j  h 1
h
 (b
j
 c e
g
ij
j 1
 c
k
j  h 1

 d j   d j bij  a ij 
 cij )( e j  f j )
ij
j  g 1
Di 3 
j

 f j   f j bij  cij 
ij
(e j  f j )  f j (bij  cij )
ij
dj

h
a
Oi 2 
Pi1 
ij
j 1
Bi 3 
Di1 
 d j )(bij  a ij )
j
g
Ci 2 
 cij )( e j  f j )
g
j  h 1
Bi1 
 cij )( e j  f j )
ij
j 1
j
 (e
Ai 2 
 d j )(bij  a ij )
j
j  h 1
h
j  g 1
g
b
Pi 3 
Qi 2 
ij
j 1
ej
k
b
j  h 1
ij
ej
h
c
j  g 1
ij
fj
k
c
j  h 1
ij
fj
22
3.4 Develop membership functions (cont.)
• By applying the above equations, Eqs.(3.9)-(3.11) can be arranged
as Eqs.(3.12)-(3.14)as follows:
g
wj  xij
• 
j 1
h
•  wj  xij
j  g 1
n
wj  xij
• j
 h 1
 [ Ai1 2  Bi1  Oi1 , Ci1 2  Di1  Qi1 ] .
(3.12)
 [ Ai 2 2  Bi 2  Oi 2 , Ci 2 2  Di 2  Qi 2 ] .
(3.13)
 [ Ai 3 2  Bi 3  Oi 3 , Ci 3 2  Di 3  Qi 3 ] .
(3.14)
23
3.4 Develop membership functions (cont.)
a
• Applying Eqs.(3.12)-(3.14) to Eq.(3.4) to produce Eq.(3.15):
Gi  [( Ai1  Ai 2  Ci 3 ) 2  ( Bi1  Bi 2  Di 3 )  (Oi1  Oi 2  Qi 3 ) ,
(3.15)
• The left and right membership function of G can be obtained as
shown in Eq. (3.16) and Eq. (3.17) as follows:
(Ci1  Ci 2  Ai 3 ) 2  ( Di1  Di 2  Bi 3 )  (Qi1  Qi 2  Oi 3 )] .
i

If

If
fGLi ( x)
( Bi1  Bi 2  Di 3 )  [( Bi1  Bi 2  Di3 )2  4( Ai1  Ai 2  Ci3 )( x  (Oi1  Oi 2  Qi 3 ))]

2( Ai1  Ai 2  Ci 3 )
1
2
(3.16)
Oi1  Oi 2  Qi 3  x  Pi1  Pi 2  Pi 3 ;
fGRi ( x)
( Di1  Di 2  Bi 3 )  [( Di1  Di 2  Bi 3 )2  4(Ci1  Ci 2  Ai 3 )( x  (Qi1  Qi 2  Oi 3 ))]
2(Ci1  Ci 2  Ai 3 )
1
2
(3.17)
Pi1  Pi 2  Pi 3  x  Qi1  Qi 2  Oi 3 .
24
3.5 Rank fuzzy numbers
• In this research, Chen’s maximizing set and minimizing set (1985)
is applied to rank all the final fuzzy evaluation values.
• Definition 1.
The maximizing set M is defined as:
 x Ri  xmin k
) , xmin  x Ri  xmax ,
(
f M x    xmax  xmin
0, otherwise .

(3.18)
The minimizing set N is defined as:
 x Li  xmax k
) , xmin  x Li  xmax ,
(
f N x    x min  x max
0, otherwise ,

where
x min  inf S ,
x
xmax  sup S ,
x
S   in1 S i ,
(3.19)
S i  {x f Ai ( x)  0} ,
usually k is set to 1.
25
3.5 Rank fuzzy numbers (cont.)
Definition 2.
The right utility of A is defined as:
i
U M ( Ai )  sup ( f M ( x)  f Ai ( x)) , i  1 ~ n .
(3.20)
x
The left utility of
Ai
is defined as:
U N ( Ai )  sup ( f M ( x)  f Ai ( x)) , i  1 ~ n .
x
The total utility of
U T ( Ai ) 
Ai
(3.21)
is defined as:
1
(U M ( Ai )  1  U N ( Ai )) , i  1 ~ n .
2
(3.22)
26
3.5 Rank fuzzy numbers (cont.)
Figure 3-1. Maximizing and minimizing set
27
3.5 Rank fuzzy numbers (cont.)
• Applying Eqs.(3.16)~ (3.22), the total utility of fuzzy number
can be obtained as:
U T (Gi ) 
Gi
1
(U M (Gi )  1  U N (Gi )) , i  1 ~ n ,
2
1  ( Di1  Di 2  Bi 3 )  [( Di1  Di 2  Bi 3 )  4(C i1  C i 2  Ai 3 )( x RI  (Qi1  Qi 2  Oi 3 ))]
 [
2
2(C i1  C i 2  Ai 3 )
2
 ( Bi1  Bi 2  Di 3 )  [( Bi1  Bi 2  Di 3 )  4( Ai1  Ai 2  C i 3 )( x Li  (Oi1  Oi 2  Qi 3 ))]
2
1
2( Ai1  Ai 2  C i 3 )
1
2
1
2
].
(3.23)
28
3.5 Rank fuzzy numbers (cont.)
• x R is developed as follows:
Assume:
i
xRi  xmin
xmax  xmin

 ( Di1  Di 2  Bi 3 )
2(Ci1  Ci 2  Ai 3 )
[( Di1  Di 2  Bi 3 )  4(Ci1  Ci 2  Ai 3 )( xRi  (Qi1  Qi 2  Oi 3 ))]
2

2(Ci1  Ci 2  Ai 3 )
1
2
.
(3.24)
x Ri  (2(Ci1  C i 2  Ai 3 ) x min  ( x min  x max )( Di1  Di 2  Bi 3  x min  x max ))
 ( x max  x min )[( Di1  Di 2  Bi 3  x min  x max ) 2
1
2
 4(Ci1  Ci 2  Ai 3 )( x min  Qi1  Qi 2  Oi 3 )] / 2(C i1  C i 2  Ai 3 ).
(3.25)
29
3.5 Rank fuzzy numbers (cont.)
•
is developed as follows:
Assume:
xLi
x Li  x max
x min  x max

 ( Bi1  Bi 2  Di 3 )
2( Ai1  Ai 2  C i 3 )
[( Bi1  Bi 2  Di 3 )  4( Ai1  Ai 2  Ci 3 )( xLi  (Oi1  Oi 2  Qi 3 ))]
2

2( Ai1  Ai 2  Ci 3 )
1
2
.
(3.26)
x Li  (2( Ai1  Ai 2  Ci 3 ) x max  ( x max  x min )( Bi1  Bi 2  Di 3  x max  x min ))
 ( x min  x max )[( Bi1  Bi 2  Di 3  x max  x min ) 2
1
2
 4( Ai1  Ai 2  Ci 3 )( x max  Oi1  Oi 2  Qi 3 )] / 2( Ai1  Ai 2  C i 3 ) .
(3.27)
30
NUMERICAL EXAMPLE
• 4.1 Ratings of alternatives versus qualitative
criteria
• 4.2 Normalization of quantitative criteria
• 4.3 Averaged weights of criteria
• 4.4 Development of membership function
• 4.5 Defuzzification
31
NUMERICAL EXAMPLE
• Assume that a logistics company is looking for a suitable city to set
up a new distribution center.
• Suppose three decision makers, D1, D2 and D3 of this company is
responsible for the evaluation of three distribution center
candidates, A1, A2 and A3.
32
Criteria
Qualitative criteria
Quantitative criteria
Benefit
Benefit
Cost
Expansion
possibility
Square measure
of area
Investment cost
Availability of
acquirement
material
Closeness to
demand market
Human resource
Figure 4-1. Selected criteria
33
Table 4-1 Linguistic values and fuzzy numbers for importance weights
Unimportant
(0.00,0.00,0.25)
Less important
(0.00,0.25,0.50)
Important
(0.25,0.50,0.75)
More important
(0.50,0.75,1.00)
Very important
(0.75,1.00,1.00)
Table 4-2 Linguistic values and fuzzy numbers for ratings
Very low (VL) /Very difficult (VD) /Very far (VF)
(0.00,015,0.30)
Low (L)/Difficult (D)/Far (F)
(0.15,0.30,0.50)
Medium (M)
(0.30,0.50,0.70)
High (H)/Easy (E)/Close (C)
(0.50,0.70,0.85)
Very high (VH)/Very easy (VE)/Very close (VC)
(0.70,0.85,1.00)
34
4.1 Ratings of alternatives versus qualitative criteria
• Ratings of distribution center candidates versus qualitative criteria
are given by decision makers as shown in Table 4-3. Through Eq.
(3.1), averaged ratings of distribution center candidates versus
qualitative criteria can be obtained as also displayed in Table 4-3.
Candidates
A1
A2
A3
Criteria
D1
D2
D3
C1
C2
C3
C4
C1
C2
C3
C4
C1
C2
C3
C4
VH
VE
C
M
VH
M
C
VH
L
VE
M
L
H
E
VC
H
VH
M
C
VH
L
E
M
M
VH
M
VC
H
H
E
VC
VH
H
VE
C
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)
35
4.2 Normalization of quantitative criteria
• Evaluation values under quantitative criteria are objective. Suppose
values of distribution center candidates versus quantitative criteria
are present as in Table 4-4.
Table 4-4 Values of distribution center candidates versus quantitative
criteria
Distribution Center Candidates
Criteria
Units
A1
A2
A3
C5
100
80
90
hectare
C6
2
5
10
million
36
4.2 Normalization of quantitative criteria (Cont.)
• According to Eq. (3.2), values of alternatives under benefit and
cost quantitative criteria can be normalized as shown in Table 4-5.
Table 4-5 Normalization of quantitative criteria
Distribution Center Candidates
Criteria
A1
A2
A3
C5
1
0.8
0.9
C6
1
0.4
0.2
37
4.3 Averaged weights of criteria
• The linguistic values and its corresponding fuzzy numbers, shown in
Table 4-1, are used by decision makers to evaluate the importance of
each criterion as displayed in Table 4-6. The average weight of each
criterion can be obtained using Eq. (3.3) and can also be shown in
Table 4-6.
Table 4-6 Averaged weight of each criterion
D1
D2
D3
Averaged weights
C1
MI
VI
IM
(0.50,0.75,0.92)
C2
IM
MI
LI
(0.25,0.50,0.75)
C3
LI
LI
VI
(0.25,0.53,0.67)
C4
UI
IM
VI
(0.33,0.50,0.67)
C5
MI
VI
IM
(0.50,0.75,0.92)
C6
VI
VI
VI
(0.75,1.00,1.00)
38
• Apply Eqs. (3.4)-(3.15) and g=4, h=5, n=6 to the numerical
example to produce the following values Ai1, Ai 2, Ai3, Bi1, Bi 2, Bi3, Ci1,Ci 2,Ci3,
Di1, Di 2, Di 3, Oi1,Oi 2,Oi 3, Pi1, Pi 2, Pi 3,Qi1,Qi 2,Qi 3, For
each candidate as displayed in
Table 4-7
Table 4-7 Values for Ai1, Ai 2, Ai 3, Bi1, Bi 2, Bi3, Ci1,Ci 2,Ci3, Di1, Di 2, Di 3, Oi1,Oi 2,Oi3, Pi1, Pi 2, Pi 3,Qi1,Qi 2,Qi3,
A1
A2
A3
A1
A2
A3
A1
A2
A3
Ai1
0.17
0.17
0.17
Di1
-1.11
-1.11
-1.08
Qi1
2.68
2.70
2.23
Ai2
0.00
0.00
0.00
Di2
-0.17
-0.13
-0.15
Qi2
0.92
0.73
0.83
Ai3
0.00
0.00
0.00
Di3
0.00
0.00
0.00
Qi3
1.00
0.40
0.20
Bi1
0.77
0.76
0.62
Oi1
0.74
0.78
0.49
Bi2
0.25
0.16
0.23
Oi2
0.50
0.40
0.45
Bi3
0.25
0.10
0.05
Oi3
0.75
0.30
0.15
Ci1
0.12
0.12
0.12
Pi1
1.68
1.71
1.28
Ci2
0.00
0.00
0.00
Pi2
0.75
0.60
0.68
Ci3
0.00
0.00
0.00
Pi3
1.00
0.40
0.20
39
• the calculation values for Ai1  Ai 2  Ci3, Bi1  Bi 2  Di3, Oi1  Oi 2  Oi3,
Ci1  Ci 2  Ai 3, Di1  Di 2  Bi 3, Pi1  Pi 2  Pi 3, Qi1  Qi 2  Oi 3, are shown in table 48.
Table 4-8 Values for
Ai1  Ai 2  Ci 3, Bi1  Bi 2  Di 3, Oi1  Oi 2  Oi 3,Ci1  Ci 2  Ai 3, Di1  Di 2  Bi 3,
Pi1  Pi 2  Pi 3,Qi1  Qi 2  Oi 3,
A1
A2
A3
Ai1+Ai2-Ci3
0.17
0.17
0.17
Bi1+Bi2-Di3
1.02
0.92
0.84
Oi1+Oi2-Qi3
0.24
0.78
0.74
Ci1+Ci2-Ai3
0.12
0.12
0.12
Di1+Di2-Bi3
-1.53
-1.34
-1.28
Pi1+Pi2-Pi3
1.43
1.91
1.75
Qi1+Qi2-Oi3
2.84
3.13
2.91
40
4.4 Development of membership function
• Through Eqs. (3.17) - (3.18) , the left, f ( x) , and right, f ( x) ,
membership functions of the final fuzzy evaluation value, G , i  1,..., n ,
of each distribution center candidate can be obtained and displayed
in Table 4-9.
L
Gi
R
Gi
i
Table 4-9 Left and right membership functions of Gi
41
4.5 Defuzzification
• By Eqs. (3.23)-(3.28) in Model development for defuzzification,
the total utilities, U T (Gi ), x R and x L can be obtained and shown in
Table 4-10.
i
Table 4-10 Total utilities
Alternatives
U T (Gi ),
i
x Ri
and
xLi
T1
T2
T3
x Ri
1.97
2.26
2.12
x Li
1.39
1.40
1.33
0.315
0.551
0.517
U T (Gi )
Then according to values in Table 4-10, candidate A2 has the largest
total utility, 0.551. Therefore A becomes the most suitable
distribution center candidate.
42
2
CONCLUSIONS
• A fuzzy MCDM model is proposed for the evaluation and selection
of the locations of distribution centers
• Chen’s maximizing set and minimizing set is applied to the model
in order to develop ranking formulae.
• Ranking formulae are clearly developed for better executing the
decision making
• A numerical example is conducted to demonstrate the
computational procedure and the feasibility of the proposed model.
43