Page 53 - Decision Making Applications in Modern Power Systems

28 Decision Making Applications in Modern Power Systems

1.3.2 Fuzzy technique for order preference by similarity to ideal
solutions

This method is based on the idea that the geometric distance between the
1
best alternative should be the shortest from the PIA . Also, it should have
2
the longest geometric distance from the NIA . A relative closeness or simi-
larity index is determined considering the distance. This method was devel-
oped by Hwang and Yoon [93]. This method was later modified by Chen
and Hwang [86]. Several versions of fuzzy TOPSIS can be found in the lit-
erature [86,93]. In this chapter the method developed by Chen and Hwang
[86] is described in detail considering the shape of the membership function
trapezoidal. Most of the methods developed later have been possible due to
this method with a minor modification in the ranking method.
Step 1: Formulation of decision matrix
Let D be the decision matrix formed by the decision makers. Let the
elements of the matrix be represented by y ij that is either a fuzzy or crisp
value for a given value of i and j.

2 3
y 11 ? y 1n
D 5 4 ^ y ij ^ 5 ð1:42Þ
?
y m1 y mn
Step 2: Normalization of the decision matrix
Each element of the decision matrix is normalized based on the fol-
lowing formula:

2
y ij
for a ideal positive value of jth attribute
y j
6
6 2
y ð1:43Þ
j
n ij 5 6
for a ideal negative value of jth attribute
4
y jj
The normalized value calculated in the previous equation is also
called benefit attribute for the positive value and cost attribute for the
negative value.
Step 3: Calculation of weighted normalized decision matrix
For the case when the elements of decision matrix are a crisp value, it
is easy to proceed, but if the values are fuzzy, each element is to be
represented in terms of fuzzy values. Let y ij 5 ða ij ; b ij ; c ij ; d ij Þ and
y j 5 ðy j ; y j ; y j ; y j Þ. The weighted normalized decision matrix for the matrix
is obtained as given by
WN ij 5 n ij w j ð1:44Þ