Page 38 - Decision Making Applications in Modern Power Systems

Multicriteria decision-making methodologies Chapter | 1 15

Step 3: Calculation of weighted matrix
The decision matrix X mentioned in Eq. (1.13) is associated with the
respective weights, which resembles the significance of corresponding
criteria. Let the weight determined by the decision maker be denoted by
n
w 1 , w 2 , ..., w n ,such that P w i 5 1, and W be the weighted matrix.
i51
W can be calculated as
W 5 XY ð1:14Þ

where Y is a diagonal matrix defined as
2 3
w 1 ? 0
^ & ^
4 5 ð1:15Þ
0 .. . w n
Thus the weighted matrix W can be given by
2 3 2 3
y 11 ? y 1n w 1 x 11 ? w n x 1n
^ & ^ 5 ^ & ^
4 5 4 5 ð1:16Þ
? ?
y m1 y mn w 1 x m1 w n x mn
Step 4: Concordance and discordance matrix formulation
This step involves three substeps.
Step 4a: Calculation of concordance and discordance set
Let P k and P l be two alternatives, m $ k and l $ 1; the concor-
dance set C kl of the two alternatives is such that P k desired over P l is
given by

C kl 5 j; y kj $ y lj ; for j 5 1; 2; 3; ... ; n ð1:17Þ
The discordance (D kl ) set is given by

D kl 5 j; y kj $ y lj ; for j 5 1; 2; 3; .. . ; n ð1:18Þ
Step 4b: Calculation of concordance and discordance indexes
The concordance index presents the relative importance of one
alternative w.r.t. other. It is calculated as a sum of weights associated
with a criterion. Let c kl be the concordance index.
X
c kl 5 w j ; for j 5 1; 2; 3; ...; n ð1:19Þ
jAC kl
Let d kl be the discordance index. This measures the triviality of
one alternative w.r.t. other. It is calculated as
jy kj 2 y lj j
d kl 5 ð1:20Þ
max jAD kl
max j jy kj 2 y lj j
Step 4c: Calculation of concordance and discordance matrices