CODAS - Combinative Distance-Based Assessment

1. 1. Linear normalisation (max for benefit, min/x for cost).

   $$n_{ij}=\{\begin{array}{ll}\frac{x_{ij}}{{\max }_i{x}_{ij}}& j\in J^{+}\\ \frac{{\min }_i{x}_{ij}}{x_{ij}}& j\in J^{-}\end{array}$$

2. 2. Weighted normalised matrix r_ij = w_j · n_ij.

   $$r_{ij}=w_j\cdot n_{ij}$$

3. 3. Negative-Ideal solution NI per criterion.

   $$N{I}_j=\underset{i}{\min }{r}_{ij}$$

4. 4. Euclidean E_i and Taxicab T_i distances from NI.

   $$E_i=\sqrt{\sum _{j=1}^n(r_{ij}-N{I}_j{)}^2},T_i=\sum _{j=1}^n|r_{ij}-N{I}_j|$$

5. 5. Relative Assessment matrix h_ik with threshold function ψ.

   $$h_{ik}=(E_i-E_k)+\psi (E_i-E_k)\cdot (T_i-T_k),\psi (z)=\{\begin{array}{ll}1& |z|\ge \tau \\ 0& |z|<\tau \end{array}$$

6. 6. Assessment score H_i and descending ranking.

   $$H_i=\sum _{k=1}^m{h}_{ik}$$