## **Integrated-mixed normalization techniques**

| Notation/Technique              | Formula                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| :------------------------------ | :  |
| Ideal Linear Normalization      | $n_{ij} = \begin{cases} \min_i x_{ij} & \text{for cost criteria} \\ x_{ij} & \\ \frac{x_{ij}}{\text{maks}_i x_{ij}} & \text{for benefit criteria} \\ \frac{\min\{x_{ij}, R_j\}}{\text{maks}\{x_{ij}, R_j\}} & \text{for reference-based criteria} \end{cases}$                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| Enhanced Accuracy Normalization | $$n_{ij} = \begin{cases} 1 - \frac{\text{maks}_i x_{ij} - x_{ij}}{\sum_{i=1}^m (\text{maks}_i x_{ij} - x_{ij})} & \text{for benefit criteria} \\ 1 - \frac{x_{ij} - \min_i x_{ij}}{\sum_{i=1}^m (x_{ij} - \min_i x_{ij})} & \text{for cost criteria} \\ \frac{x_{ij} - (\mu + 1,96\sigma)}{\sum_{i=1}^m (x_{ij} - (\mu + 1,96\sigma))} & \text{for criteria having normal distribution, if } x_{ij} > (\mu + 1,96\sigma) \\ \frac{(\mu - 1,96\sigma) - x_{ij}}{\sum_{i=1}^m ((\mu - 1,96\sigma) - x_{ij})} & \text{for criteria having normal distribution, if } x_{ij} < (\mu - 1,96\sigma) \\ 1 & \text{for criteria having normal distribution, if } x_{ij} \in (\mu \pm 1,96\sigma) \end{cases}$$ |

## **Normalization techniques independent of the optimization orientation**

| Technique                                        | All criteria                                                                                                            |
| :----------------------------------------------- | :---------------------------------------------------------------------------------------------------------------------- |
| Z-Score (Non-Monotonic) Normalization            | $n_{ij} = e^{-\frac{(x_{ij}-R_j)^2}{2\sigma_j^2}}$                                                                      |
| Comprehensive Normalization                      | $n_{ij} = 1 - e^{-\frac{\|x_{ij}-R_j\|}{(\max\{\max_{i} x_{ij}, R_j\} - \min\{\min_{i} x_{ij}, R_j\})}}$                |
| Normalization Equalizing the Average to 1        | $n_{ij} = \frac{x_{ij}}{\mu_j}$                                                                                         |
| Normalization Equalizing Standard Deviation to 1 | $n_{ij} = \frac{x_{ij}}{\sigma_j}$                                                                                      |
| Decimal Normalization                            | $n_{ij} = \frac{x_{ij}}{10^\rho}, \quad \rho \ge 0$                                                                     |
| Reference Based Normalization                    | $n_{ij} = 1 - \frac{\|x_{ij} - R_j\|}{\max\{\max_{i} x_{ij}, R_j\} - \min\{\min_{i} x_{ij}, R_j\}}$                     |
| Wu's Reference Based Normalization               | $n_{ij} = \frac{\|x_{ij} - R_j\|}{\max_{i} x_{ij} - R_j}$                                                               |
| Aytekin's Reference Based Normalization          | $n_{ij} = 1 - \frac{\|x_{ij} - R_j\|}{\|R_j\| + 10^\rho}, \rho \ge 0$                                                   |
| Range Normalization Between -1 and +1            | $n_{ij} = \frac{x_{ij} - \frac{\text{max}_i x_{ij} + \min_i x_{ij}}{2}}{\frac{\text{max}_i x_{ij} - \min_i x_{ij}}{2}}$ |
| Range Normalization Between 0 and +1             | $n_{ij} = \frac{x_{ij} - \min_i x_{ij}}{\text{max}_i x_{ij} - \min_i x_{ij}}$                                           |
| (Zero-) Centering - Mean Normalization           | $\left(\mu\right)_{j} = \frac{1}{m} \sum_{i = 1}^{m} x_{j}^{\left(\right. i \left.\right)}$                             |
| (Scaling to) Unit Length Normalization           | $x^{'} = \frac{x}{\parallel x \parallel}$                                                                               |

## **Normalization techniques depending on the optimization orientation**

| Techniques                                          | Benefit Criteria                                                                                                                                 | Cost Criteria                                                                    |
| :-------------------------------------------------- | :----------------------------------------------------------------------------------------------------------------------------------------------- | :------------------------------------------------------------------------------- |
| Sum-Based Linear Normalization                      | $n_{ij} = \frac{x_{ij}}{\sum_{i=1}^{m} x_{ij}}$                                                                                                  | $n_{ij} = \frac{1/x_{ij}}{\sum_{i=1}^{m} 1/x_{ij}}$                              |
| Vector Normalization                                | $n_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$                                                                                         | $n_{ij} = 1 - \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$                     |
| Logarithmic Normalization                           | $n_{ij} = \frac{\ln x_{ij}}{\ln(\prod_{i=1}^{m} x_{ij})}$                                                                                        | $n_{ij} = \frac{1 - \frac{\ln x_{ij}}{\ln(\prod_{i=1}^{m} x_{ij})}}{m-1}$        |
| Maximum-Linear Normalization                        | $n_{ij} = \frac{x_{ij}}{\max_{i} x_{ij}}$                                                                                                        | $n_{ij} = 1 - \frac{x_{ij}}{\max_{i} x_{ij}}$                                    |
| Sum-Based Linear Normalization                      | $n_{ij} = \frac{x_{ij}}{\sum_{i=1}^{m} x_{ij}}$                                                                                                  | $n_{ij} = \frac{1/x_{ij}}{\sum_{i=1}^{m} 1/x_{ij}}$                              |
| Vector Normalization                                | $n_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$                                                                                         | $n_{ij} = 1 - \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$                     |
| Logarithmic Normalization                           | $n_{ij} = \frac{\ln x_{ij}}{\ln(\prod_{i=1}^{m} x_{ij})}$                                                                                        | $n_{ij} = \frac{1 - \frac{\ln x_{ij}}{\ln(\prod_{i=1}^{m} x_{ij})}}{m-1}$        |
| Maximum-Linear Normalization                        | $n_{ij} = \frac{x_{ij}}{\max_{i} x_{ij}}$                                                                                                        | $n_{ij} = 1 - \frac{x_{ij}}{\max_{i} x_{ij}}$                                    |
| Minimum-Linear Normalization                        | $n_{ij} = 1 - \frac{\min_{i} x_{ij}}{x_{ij}}$                                                                                                    | $n_{ij} = \frac{\min_{i} x_{ij}}{x_{ij}}$                                        |
| 0-1 Interval Normalization Using Max-Min            | $n_{ij} = \frac{x_{ij}}{\max_{i} x_{ij}}$                                                                                                        | $n_{ij} = \frac{\min_{i} x_{ij}}{x_{ij}}$                                        |
| Jüttler-Körth Normalization                         | $n_{ij} = 1 - \left\| \frac{\max_{i} x_{ij} - x_{ij}}{\max_{i} x_{ij}} \right\|$                                                                 | $n_{ij} = 1 - \left\| \frac{\min_{i} x_{ij} - x_{ij}}{\max_{i} x_{ij}} \right\|$ |
| Stopp Normalization                                 | $n_{ij} = \frac{100 x_{ij}}{\max_{i} x_{ij}}$                                                                                                    | $n_{ij} = \frac{100 \min_{i} x_{ij}}{x_{ij}}$                                    |
| Nonlinear (Peldschus) Normalization                 | $n_{ij} = \left( \frac{x_{ij}}{\max_{i} x_{ij}} \right)^2$                                                                                       | $n_{ij} = \left( \frac{x_{ij}}{\max_{i} x_{ij}} \right)^3$                       |
| Weitendorf's Linear Normalization                   | $n_{ij} = \frac{x_{ij} - \min_{i} x_{ij}}{\max_{i} x_{ij} - \min_{i} x_{ij}}$                                                                    | $n_{ij} = \frac{\max_{i} x_{ij} - x_{ij}}{\max_{i} x_{ij} - \min_{i} x_{ij}}$    |
| Z Score Normalization Depending Optimization Aspect | $n_{ij} = \frac{x_{ij} - \mu_j}{\sigma_j} = \frac{x_{ij} - \frac{\sum_{i=1}^{m} x_{ij}}{m}}{\sqrt{\frac{\sum_{i=1}^{m} (x_{ij} - \mu_j)^2}{m}}}$ | $n_{ij} = -\frac{x_{ij} - \mu_j}{\sigma_j}$                                      |

Sources: 
[Ahmet Aytekin](https://doi.org/10.31181/dmame210402001a)
[Ahmad Badary](https://ahmedbadary.github.io/work_files/research/dl/concepts/norm_methods)
