Page 54 - Decision Making Applications in Modern Power Systems
P. 54
Multicriteria decision-making methodologies Chapter | 1 29
where w j is the weights of the fuzzy values, and n ij is calculated as
0 1
2
a ij b ij c ij d ij
6 y ij 1ð Þy j 5 @ ; ; ; A
d j c j b j a j
6
n ij 5 6 0 1 ð1:45Þ
6
a b c d
6 2 2 2 2
2
6 j j j j
y ð @ A
4 1 Þy ij 5 ; ; ;
j
d jj c jj b jj a jj
The calculation of WN ij becomes simple for crisp values, but when
the values of n ij are fuzzy, the formulae of WN ij are updated as follows:
0 1
2
a ij b ij c ij d ij
y ij :ðÞy j 5 α j ; β ; γ ; for the benefit attribute
6 @ j j δ j A
6
d j c j b j a j
6
n ij 5 6 0 1
a b c d
6 2 2 2 2
6 2 j j j j
y :ðÞy ij 5 α j ; β ; γ ; for the cost attribute
4 @ δ j A
j j j
d jj c jj b jj a jj
ð1:46Þ
The final weighted normalized decision matrix is given in the follow-
ing form:
2 ? 3
WN 11 WN 1n
WN 5 4 ^ WN ij ^ 5 ð1:47Þ
?
WN m1 WN mn
Step 4: Calculation of positive and negative ideal solution
As calculated in the conventional TOPSIS, and hybrid technique,
these values are calculated. The values are easily calculated for systems
with the crisp values as given in the following equations:
½
P 5 WN 1 ; ... ; WN n ð1:48Þ
2 2
N 5 ½WN ; ...; WN ð1:49Þ
1 n
2
where WN n is equivalent to max(WN ij ) and WN 5 minðWN ij Þ. In case the
n
values are fuzzy, the calculation becomes complex and is calculated as a
generalized mean for the fuzzy number. This is calculated in numerous
ways. The mathematical equation of the Lee and Li’s ranking method is
given as follows:
2 2 2 2
2a 2 b 1c 1d 2 a ij b ij 1 c ij d ij
ij
ij
ij
ij
Mean WN ij 5 ð1:50Þ
½3ð2a ij 2 b ij 1 c ij 1 d ij Þ
is equivalent to max(MeanðWN ij Þ) and
The value of WN n
2
WN 5 minðMean WN ij Þ.
n
