Selection of Water Quality Monitoring Stations - Water Quality Management: Assessment and Interpretation - page 8

Table 2.1 Matrix of MADM

X 1 X 2 X 3 … . X n

A 1 r 11 r 12 r 13 … . r 1n

A 2 r 21 r 22 r 23 … . r 2n

A m r m1 r m2 r m3 … . r mn

A 1 ; A 2 ; ... ; A m , in decision making matrix D, indicate m predetermined alterna-
tives (Such as sampling stations in this work), X 1 ; X 2 ; ... ; X n show n attributes (such
as population, area of basin, water qualitative parameters,

…

) to assess desirability
of each attribute. The members of matrix describe the special values of jth attribute
for jth alternative. The optimal solution for a MADM consists of the most suitable
assumed alternative A .

A ¼ f A 1 ; A 2 ; ... :; A n g

X j ¼ max iuj ð r ij Þ

ð 2 : 5 Þ

i ¼ 1

; ; ... :; m

2

A consists of the most preferable desirability of every existing alternative in
decision (Asgharpoor 2004 ). The various steps of this method will be presented as
follows.

2.3.1 Making Dimensionless

To compare various measurement scales (for various attributes), it is necessary to
use a dimensionless method (Asgharpoor 2004 ). There are several techniques to
make dimensionless (none missing), but the normal method is explained. First
normality of data is checked using the Shapiro-Wilk test. If data is not normal,
Box - Cox technique can be used for normality. Finally, Uniform Function is applied
to unify and perform dimensions of data.
In Box Cox method, it is necessary to estimate a value for , and then the

- k

following equation can be applied for normality.

y i ¼ x i
k

for k 6 ¼ 0

ð 2 : 6 Þ

where y i = normalized data, xi i = original data and λ = a value which we substitute

in Eq. ( 2.6 ).