MARCOS - Measurement of Alternatives and Ranking according to Compromise Solution

1. 1. Form extended decision matrix with AAI (anti-ideal) and AI (ideal) rows.

   $$X^{\ast }=[X^T;AAI;AI{]}^T,AA{I}_j=\min /\max ,A{I}_j=\max /\min$$

2. 2. Normalise vs ideal AI per criterion direction.

   $$n_{ij}=\{\begin{array}{ll}{x}_{ij}/x_{AI,j}& j\in J^{+}\\ x_{AI,j}/x_{ij}& j\in J^{-}\end{array}$$

3. 3. Weighted normalised matrix v_ij = w_j · n_ij.

   $$v_{ij}=w_j{n}_{ij}$$

4. 4. Sum each alternative's weighted matrix row: S_i = Σ v_ij.

   $$S_i=\sum _{j=1}^n{v}_{ij}$$

5. 5. Utility ratios vs ideal K+ and anti-ideal K− rows.

   $$K_i^{+}=\frac{S_i}{S_{AI}},K_i^{-}=\frac{S_i}{S_{AAI}}$$

6. 6. Utility functions f(K+_i) and f(K−_i) of the utility ratios.

   $$f(K_i^{+})=\frac{K_i^{-}}{K_i^{+}+K_i^{-}},f(K_i^{-})=\frac{K_i^{+}}{K_i^{+}+K_i^{-}}$$

7. 7. Final utility f(K_i) and descending ranking.

   $$f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+\frac{1-f(K_i^{+})}{f(K_i^{+})}+\frac{1-f(K_i^{-})}{f(K_i^{-})}}$$