Page 390 - Biofuels for a More Sustainable Future

346 Biofuels for a More Sustainable Future

Assuming that there are a total of n decision attributes C 1 , C 2 ,…, C n to
assess the n alternatives, namely, A 1 , A 2 ,…, A m , then, the interval decision-
making matrix can be determined, as presented in Eq. (12.17).

C 1 C 2 ⋯ C n

A 1 x x ⋯ x

11 12
1n
¼ A 2 x x ⋮ x (12.17)

ij 21 22
X ¼ x
m n 2n
⋮ ⋮ ⋯ ⋱ ⋮

A m x x ⋯ x
m1 m2 mn
h i
where x ¼ x x + represents the value of the ith alternative with respect

ij ij ij
to the jth attribute.
Step 2: Normalizing the decision-making matrix. In order to avoid the
effects caused by the unit gaps existing in the data of the interval decision-
making matrix, all the data determined by step 1 can be normalized accord-
ing to Eqs. (12.18),(12.19).
As for the benefit-type criteria,
x
8
ij
>
>
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> m
1 X
>
> +
x 2
>
>
> ij
n
>
<
h i j¼1
+
r ¼ r r + (12.18)

ij ij ij ¼ x
> ij
>
>
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 X
> m
>
> +
x 2
>
> ij
n
>
:
j¼1
As for the cost-type criteria,
8 +
1=x
> ij
>
m
> s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
> 2
X
>
>
1=x
>
> ij
n
>
h i < j¼1
+
r ¼ r r (12.19)

ij ij ij ¼ 1=x
> ij
>
>
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> m
> 2
X
> 1
1=x
>
>
n
> ij
:
j¼1
Step 3: Determining the weighted normalized decision-making matrix.
After determining the normalized decision-making matrix, the weighted
normalized decision-making matrix, Eq. (12.20), can be obtained by incor-
porating the criterion weights.