Transcript 下載/瀏覽

Chapter 4
Model Establishment
Professor: Ta Chung Chu
Student: Nguyen Khoa Tu Uyen – M977 Z215
Thinh Thi Phuong – M977 Z258
1. Average Ratings of Alternatives versus
Subjective Criteria
 k decision-makers (Dt , t=1~k)
 m alternatives ( Ai , i=1~m)
 h subjective criteria (C j , j=1~h)
 Let
xijt  oijt , pijt , qijt  , with
and i =1,2,…; m, j =1,2,…, h;
t =1,2,…, k.
1
xij     xij1  ...  xijt  xijk 
k
k
oij  
oijt
k
k
pij  
pijt
k
xijt is the rating assigned to Ai by Dt for Aj ,
xij is the averaged rating of Ai under Cj .
t 1
t 1
(1)
k
qijt
t 1
k
qij  
2. Normalize Performance of Alternatives
versus Objective Criteria
 Objective criteria can be classified to benefit (B) and cost (C)
 Benefit (B): the larger is the better
 Cost (C): the smaller is the better

 Suppose rij  eij , f ij , g ij

is the performance of alternative Ai
versus objective criterion Cj, the normalized value xij can be
denoted as:
 eij  e*j f ij  e*j g ij  e*j 

xij  
,
,
*
*
*
 d
dj
d j 
j

 g *j  g ij g *j  f ij g *j  eij 

xij  
,
,
*
*
 d*
dj
d j 
j

, jB
(2)
, j C
(3)
3. Average Weights
 Let
w jt  a jt , b jt , c jt ,
w jt  R, j =1,2,…, n; t =1,2,…, k,
1
w j     w j1  w j 2  ...  w jk 
k
(4)
w j  a j , b j , c j 
Where
k
a jt
t 1
k
aj  
k
b jt
t 1
k
bj  
k
c jt
t 1
k
cj  
w jt is the weight assigned by Dt to C j .
w j is the averaged weight of C j assessed by k decision-makers
4. Aggregate the Weighted Ratings
 The aggregation of weighted ratings, , for each alternative is
implemented by the concept of simple additive weighting as:

1
Gi     Ri1  Ri 2  ...  Rit  ...  Rij
n


(5)
 The membership function for Gi can be developed as follows:
n
 1  n 
1 n  1 n

 
Gi    Rij    xij  w j      xij   w j 
 n  j 1
j1
 n  j 1
 n  j 1


(6)
Let:
 eij  e*j f ij  e*j g ij  e*j 
 , jB,
,
,

*
*
*
dj
d j 
 d j
xij  oij , pij , qij   
*
*
*
 g j  g ij g j  f ij g j  eij 
 d * , d * , d *  , j  C
j
j
j


(7)
w j  a j , b j , c j 

xij   pij  oij   oij ,  pij  qij   qij

wj  b j  a j   a j , b j  c j   c j


(8)
(9)

xij  wj   pij  oij b j  a j  2  oij b j  a j   a j  pij  oij   oij a j ,
p
ij
 



 qij b j  c j  2  qij b j  c j  c j pij  qij   qij c j

(10)
The final fuzzy evaluation values can be developed via arithmetic
operation of fuzzy numbers as:
(11)
Two equations to solve, namely:
(12)
(13)
Let:
Equations (12) and (13) can then be expressed as:
I i1 2  J i1  Qi  x  0
(14)
I i 2 2  J i 2  Z i  x  0
(15)
Membership functions
f
L
Gi
can thus be produced as:
x   J i1  J


2
i1

x  Z 
 4I i1 x  Qi 
f GRi x    J i 2  J i22  4I i 2
1/2
1/2
i
 2I
 2I
i1
Qi  x  Yi
(16)
i2
Yi  x  Z i
(17)
5. Ranking Obtains
 Gi is the final fuzzy evaluation value alternative Ai.
 Using centroid ranking method (Ying-Ming Wang, Jian-Bo
Yang, Dong-Ling Xu, Kwai-Sang Chin ):
f A x dx a x f A x dx  b x f A x dx


 b L
xo (Gi)  
c
R
 f A xdx a f A x dx  b f A x dx

b
x
L
~
~
w
R
~
~
0
A
A
y( g ( y)  g ( y))dy

y (Gi) 
 ( g ( y)  g ( y))dy
w
o
0
c
R
~
~
~
~
(18)
L
R
L
~
~
A
A
(19)
 Calculate:
2

x y
i
2
i
(20)
(21)
(22)