3.3. Linear max-min based

3.3.1. Min-max normalization

The minimum-maximum method considers the maximum and minimum ratings of the given criteria in normalization. This normalization can be defined by the Equation (3.14) for profit-type criteria and by the Equation (3.15) for cost-type criteria.

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

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

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

3.3.2. Zavadskas and Turkis normalization

Zavadskas and Turskis in 2008 proposed a new normalization method. This normalization can be defined by the Equation (3.16) for profit-type criteria and by the Equation (3.17) for cost-type criteria.

(3.16)\[\begin{equation} r_{i j}= 1 - \frac{\left | \max_j \left( x_{ij} \right) - x_{ij} \right |}{\max_j \left( x_{ij} \right)} \end{equation}\]

(3.17)\[\begin{equation} r_{i j}= 1 - \frac{\left | \min_j \left( x_{ij} \right) - x_{ij} \right |}{\min_j \left( x_{ij} \right)} \end{equation}\]

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