PHF-TOPSIS - Probabilistic Hesitant extension of TOPSIS

1. 1. Compute PHF expected score E(h_ij)=Σ γ_k·p_k for each cell; then vector-normalise columns.

   $$E(h_{ij})=\sum _{k=1}^{|h_{ij}|}{\gamma }_k\cdot p_k\in [0,1];r_{ij}=\frac{E(h_{ij})}{\sqrt{{\sum }_{i=1}^{m}E(h_{ij}{)}^2}}$$

2. 2. Weighted normalised matrix: multiply each column by its weight.

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

3. 3. Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction.

   $$A^{+}=\{v_1^{+},\dots ,v_n^{+}\},v_j^{+}=\{\begin{array}{ll}{\max }_i{v}_{ij}& j\in J\\ {\min }_i{v}_{ij}& j\in J^{\prime }\end{array};A^{-}=\{v_1^{-},\dots ,v_n^{-}\},v_j^{-}=\{\begin{array}{ll}{\min }_i{v}_{ij}& j\in J\\ {\max }_i{v}_{ij}& j\in J^{\prime }\end{array}$$

4. 4. Euclidean separation of each alternative from A⁺ and A⁻.

   $$d_i^{+}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{+}{)}^2},d_i^{-}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{-}{)}^2}$$

5. 5. Relative closeness to the ideal solution; rank by C* descending.

   $$C_i^{\ast }=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},0\le C_i^{\ast }\le 1$$