3.2. Linear ratio based

3.2.1. Max normalization

The maximum method considers the maximum ratings of the given criteria in normalization. This normalization can be defined by the Equation (3.9) for profit-type criteria and by the Equation (3.10) for cost-type criteria.

(3.9)\[\begin{equation} r_{i j}=\frac{x_{i j}}{\max _{j}\left(x_{i j}\right)} \end{equation}\]

(3.10)\[\begin{equation} r_{i j}=1-\frac{x_{i j}}{\max _{j}\left(x_{i j}\right)} \end{equation}\]

where \(x_{ij}\) is the \(i-th\) value of the alternative and the \(j-th\) value of the criterion in the decision matrix.

3.2.2. Linear normalization

The linear normalization method is similar to max normalization, where profit-type criteria are normalized as in max normalization, while cost-type criteria are normalized using the Equation (3.11).

(3.11)\[\begin{equation} \label{equ:maxben} r_{i j}=\frac{\min _{j}\left(x_{i j}\right)}{x_{i j}} \end{equation}\]

where \(x_{ij}\) is the \(i-th\) value of the alternative and the \(j-th\) value of the criterion in the decision matrix.

3.2.3. Nonlinear normalization

The nonlinear method considers the normalization of the maximum and minimum ratings of the criteria in question and exponentiation. This normalization can be defined by the Equation (3.12) for profit-type criteria and by the Equation (3.13) for cost-type criteria.

(3.12)\[\begin{equation} r_{i j}= \left ( \frac{x_{ij} }{ \max_i x_{ij}} \right )^2 \end{equation}\]

(3.13)\[\begin{equation} r_{i j}= \left ( \frac{ \min_i x_{ij}}{ x_{ij}} \right )^3 \end{equation}\]

where \(x_{ij}\) is the \(i-th\) value of the alternative and the \(j-th\) value of the criterion in the decision matrix.