respect to a criterion which is based on the requirements. These classes will be rated based on the importance of vulnerability class with respect to a particular
criterion.
3.5.2 Fuzzy AHP
The steps of Fuzzy AHP for vulnerability analysis are shown in Figure 3.3. PCM from AHP is fuzzified to get the Fuzzy PCM. Fuzzy extent analysis is then
applied to calculate performance ratings and criteria weights for the Fuzzy PCM. Performance ratings are then multiplied by criteria weights according to hierarchy.
The result is a range of values over which any value can be considered as performance value. Decision makers are then requested to give their confidence
level regarding their judgment. The confidence level will be taken as α-cut value,
which will have a range of value between 0 – 1. In addition, decision makers attitude play a role in deciding the performance value. Up to this point, the
performance is still in interval values. Pessimistic, optimistic or middle decision maker attitude will determine which value will be chosen, either the lowest value,
highest value or middle value respectively. Pessimistic decision maker attitude is expressed as Optimism Index
λ which value is between 0 - 1. Using the optimism index, the fuzzy range is once again converted into a crisp range.
38
Figu steps fo
erabilit
3.5.3 uzzy AHP Ap
n Vulnerabil nalysis
the fuzzy AHP approach, triangular fuzzy numbers were used for the fuzzification of the crisp PCM. The crisp PCM is fuzzified using the triangular
fuzzy number f = l, m, u. The l lower bound and u upper bound represents the vague range that might exist in the preferences expressed by the decision
maker or experts. Conversion from crisp re 3.3 Fuzzy AHP
r vuln y Analysis
F proach i
ity A
In
to Fuzzy PCM is shown in Table 3.4.
39
Table 3.4 Conversion of crisp PCM to fuzzy PCM Source: Deng, 1999 in Kuswandari, 2004 and Prakash, 2003
Crisp PCM value
Fuzzy PCM value Crisp PCM
value Fuzzy PCM value
1 1, 1, 1 if diagonal
1, 1,3 otherwise 11
11, 11, 11 if diagonal 13, 11, 11 otherwise
2 1, 2, 4
12 14, 12, 11
3 1, 3, 5
13 15, 13, 11
5 3, 5, 7
15 17, 15, 13
7 5, 7, 9
17 19, 17, 15
9 7, 9, 11
19 111, 19, 17
Given a crisp PCM A, having the values ranging from 19 to 9 ⎟
⎞ ⎜
⎜ ⎜
⎛ =
n
a a
a a
A ...
... ...
21 1
12 11
⎟ ⎟
⎜ ⎜
.. ...
... ..
. ⎟
⎟ ⎟
⎠ ⎜
⎝
n
a a
a a
a
... ...
. ..
.. ...
... ...
... ...
2 22
3.1
parison atrix
mn m
m 2
1
where
11
a to
mn
a are the score values obtained from AHP pairwise com
m The fuzzy PCM
.
A
will be as follows,
⎟ ⎟
.. ...
... ...
... ⎟
⎟
⎠ ⎜
⎜ ⎜
⎜ ⎜
⎜
⎝ ⎛
= ...
... ...
...
l
a a
a a
a a
a a
a a
a a
a a
a a
a a
a 3.2
where pairwise comparison matrix which are fuzzified using the rule from Table 3.4.
Using fuzzy PCM, fuzzy extent analysis then is applied to obtain the fuzzy performance matrix of each alternative with respect to all criteria. The fuzzy
performance X and fuz for vulnerability analysis are shown as
follows: ⎟
⎞ ...
...
1 1
1 12
12 12
11 11
11 u
n m
n l
n u
m l
u m
l
a a
a a
a a
a a
⎟ ..
... ...
... ...
2 2
2 22
22 22
21 21
21 u
n m
n l
n u
m l
u m
A
2 2
2 1
1 1
u mn
m mn
l mn
u m
m m
l m
u m
m m
l m
a
11l
a
11m
a
11u
to a
mnl
a
mnm
a
mnu
are the score values obtained from AHP
zy weights W
40
∑ ∑∑
−
⎤ ⎡
n m
n 1
−
⎤ ⎡
k l
k 1
= j
i 1
= =
⎥ ⎦
⎢ ⎣
⊗ =
i j
j i
j i
a a
x
1 1
= =
⎥ ⎦
⎢ ⎣
⊗ =
j i
j j
i j
i j
b b
w
1 1
1
3.3
here f vulnerable class i
uzzy PCM
∑ ∑∑
=
w x
i
= fuzzy performance value o w
j
= fuzzy weight value of criteria i a
i
= vulnerable level of class i to class j in the f
A
b
i
= importance of criteria i to criteria j in the fuzzy PCM
B
i = 1, 2, 3,…, n; number of rowscolumns vulnerable class in the PCM
j = 1, 2, 3,…,k; number of rowscolumns criteria in the PCM
The result in fuzzy performance matrix X and fuzzy weights W will be like shown below,
⎞ ⎜
⎜ ⎜
⎛
2 2
2 1
1 1
u m
l u
m l
x x
x x
x x
3.4
⎟
⎜ ⎜
= ...
... ...
W 3.5
⎟ ⎠
⎜ ⎝
u i
m i
l i
x x
x
⎟ ⎟
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎜ ⎜
⎝ ⎛
... ...
...
2 2
2 1
1 1
ju jm
jl u
m l
u m
l
w w
w w
w w
w w
w
where X
⎟ ⎟
⎜ ⎜
= ...
... ...
... ...
...
j
X ⎟
⎟ ⎟
⎟
j
= performance matrix of vulnerable class for criteria j W
= weight matrix of criteria i = 1, 2, 3, …, n; number of vulnerable class
j = 1, 2, 3, …, k; number of criteria
41
A fuzzy weighted performance matrix P can thus be obtained by multiplying the weight from the weight vector with the decision matrix over the
hierarchy.
⎜ ⎜
= ⎟
⎟ ⎜
⎜ =
= ..
... ...
... ...
...
2 2
2 2
2 2
m l
u m
l
w w
w w
X P
⎟ ⎟
⎠ ⎜
⎜ ⎝
⎟ ⎟
⎠ ⎜
⎜ ⎝
... ...
... ...
... ...
u n
m n
l n
u i
m i
l i
where P ⎟
⎟ ⎟
⎟ ⎞
⎜ ⎛
⎟ ⎞
⎜ ⎛
1 1
1 1
1 1
u m
l u
m l
p p
p x
x x
⎜ ⎟
⎜ .
u ju
jm jl
j j
j
p p
p p
p p
x x
x x
x x
3.6
α
preference or judgment and the result will be a single value having the membership 1 in the fuzzy performance set. It means no further step is required.
If α-cut is less than 1 then there is degree of vagueness from the expert or decision
maker related to the preference or judgment. On the other hand if the α-cut = 0, it
expresses the highest vagueness of the expert or decision maker regarding the preference and judgment; therefore the possible performance will be whole
n 1 will need further evaluat
j
= fuzzy weighted performance matrix of criteria j The next step is to calculate the interval performance matrix using the -cut
over the result above. Alpha-cut is known for enabling to include the decision maker or expert confidence about the preference or judgment that has been made.
Applying the α-cut will result to the interval performances. The α-cut value
ranges from 0 to 1. If the α-cut = 1 then the expert is very sure about the
support of the fuzzy performance. Any value of α less tha
ion to get the crisp performance.
42
⎟ ⎟
⎟ ⎟
⎜ ⎝
] ,
[ ...
] ,
[
2 2
ir il
r l
p p
p p
p p
⎟ ⎠
⎜ ⎜
⎜ ⎜
⎜ ⎛
= ]
, [
3 3
r l
j
P
α
3.7
where l and r represent the left and right value of the interval set. Th
ptimum index. Lambda function enables to include the decision maker or expert attitude
about judgment that has been made. The λ value ranges from 0 to 1. If the λ= 1
then the expert is very sure about the judgment. On the other hand if the λ = 0, it
expresses the lowest attitude of the expert or decision maker regarding the judgment. In the current studies of hazard vulnerability, this function will serve to
represent boundaries of vulnerability classes. Optimism index λ is applied over
the interval performance set that will result in the performance matrix C.
α-cut analysis P
α
il
= the left most value obtained from α-cut analysis.
The equation will give the crisp performance matrix,
3.9
where = crisp performance matrix of criteria i
c
1
…c
i
= crisp performance value of vulnerable class i ⎟
⎞ ]
, [
1 1
r l
p p
e crisp performance matrix is obtained by applying λ the o
1
l j
r j
j
P P
C
α
λ λ
∗ −
+ ∗
=
, with λ = [0, 1]
3.8 where P
α α
λ
α
ir
= the right most value obtained from
⎟⎟ ⎟
⎟ ⎟
⎠ ⎞
⎜⎜ ⎜
⎜ ⎜
⎝ ⎛
=
i j
c c
c C
...
2 1
λ α
λ α
Cj
43
3.6 Generating Vulnerability Map
