Page 318 - Materials Chemistry, Second Edition

15.3 Methods 317
Step 2: Determining the normalized initial influence matrix A.
The normalized initial influence matrix A can be determined by normalizing the initial in-
fluence matrix by Eqs. (15.15), (15.16).
x
ij
a ij ¼ 2 3 ð i ¼ 1, 2, ⋯, N; j ¼ 1, 2, ⋯, NÞ (15.15)
N N
X X
max max a ij , max a ij 5
4
1 i N 1 j N
j¼1 i¼1
B 1 B 2 ⋯ B N
B 1 a 11 a 12 ⋯ a 1N
(15.16)
A ¼ B 2 a 21 a 22 ⋯ a 2N
⋮ ⋮ ⋮ ⋱ ⋮
B N a N1 a N2 ⋮ a NN
where A represents the normalized initial influence matrix, a ij (i¼1,2,⋯,N;j¼1,2,⋯,N),
which is the element of the cell (i, j) in the normalized initial influence matrix, A represents
the normalized influence of the i-th factor on the j-th factor.

Step 3: Calculating the total influence matrix T
Thee total influence matrix T can be determined by Eq. (15.17).
∞
X n 1
T ¼ t ij ¼ A ¼ AI Að Þ (15.17)
N N
n¼1
where T represents the total influence matrix, I is the identity matrix, and t ij represents the
element of cell (i, j) in the total influence matrix.
Step 4: Determining the sum of each row and the sum of each column
The sum of the i-th row and the sum of the j-th column can be determined by Eqs. (15.18),
(15.19), respectively.
N
X
R i ¼ t ij (15.18)
j¼1
N
X
C j ¼ t ij (15.19)
i¼1
where R i represents the sum of the i-th row in the total influence matrix, and C j represents the
sum of the j-th column in the total influence matrix.
R i , as the sum of the i-th row, represents the total direct and indirect effects of the i-th factor
on the other factors, and C j , as the sum of the j-th column, shows the total direct and indirect
effects of all the influential factors on the j-th factor. When i¼j, R i +C j represents the total ef-
fects exerted and received by the i-th factor, and it can be used as an index to show the relative
importance of the i-th factor in the system. R i C j shows the new difference that contributed
by the i-th factor to the system and it is the difference of the influences of the i-th factor exerted
on the other factors from that received by the i-th factor from the other factors. If R i C j is