MULTIMOORA - Multi-Objective Optimisation by Ratio Analysis plus Full Multiplicative Form

1. 1. Vector normalisation r_ij.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{k=1}^m{x}_{kj}^2}}$$

2. 2. Ratio System y_i = Σ benefit − Σ cost.

   $$y_i=\sum _{j\in J^{+}}{w}_j{r}_{ij}-\sum _{j\in J^{-}}{w}_j{r}_{ij}$$

3. 3. Reference Point z_i = max(|w_j r_j* − w_j r_ij|).

   $$z_i=\underset{j}{\max }|w_j{r}_j^{\ast }-w_j{r}_{ij}|,r_j^{\ast }=\underset{i}{\max }{r}_{ij}(J^{+})\text{or}\underset{i}{\min }{r}_{ij}(J^{-})$$

4. 4. Full Multiplicative Form U_i = Π benefit / Π cost.

   $$U_i=\frac{{\prod }_{j\in J^{+}}(x_{ij}{)}^{w_j}}{{\prod }_{j\in J^{-}}(x_{ij}{)}^{w_j}}$$

5. 5. Dominance theory aggregation of three rankings.

   $$\text{Final rank}=\text{dominance}(\text{rank}(y_i),\text{rank}(z_i),\text{rank}(U_i))$$