PROBID - Preference Ranking on the Basis of Ideal-Average Distance

1. Vector-normalize every criterion column. Criterion direction is not applied here; benefit/cost ordering is applied when the ideal solutions are constructed in F3.
$F_{ij} = \frac{f _{ij}}{\sum _{k = 1}^{m} f _{k j}^{2}} \forall j; criterion direction is applied when constructing A^{(k)}$
2. Weighted normalized matrix v_ij = w_j · r_ij.
$v_{ij} = w_{j} \cdot r_{ij}$
3. Construct m ideal solutions A(k): each column of wnmatrix sorted in descending order. Average ideal solution Ā_j = (1/m) Σ_k A(k)_j.
$A_{j}^{(k)} = {k -th largest v_{ij}, k -th smallest v_{ij}, j \in J^{+} j \in J^{-}; \overset{ˉ}{A}_{j} = \frac{1}{m} k = 1 \sum m A_{j}^{(k)}$
4. Euclidean distances: S_i(k) = distance from alt i to k-th ideal A(k); S_i(avg) = distance from alt i to average ideal Ā.
$S_{i (k)} = j \sum (v_{ij} - A_{j}^{(k)})^{2}; S_{i (a v g)} = j \sum (v_{ij} - \overset{ˉ}{A}_{j})^{2}$
5. Harmonic-weighted positive and negative ideal distances. S_pos uses k=1,…,⌈m/2⌉; S_neg uses k=⌊m/2⌋+1,…,m, so the middle ideal is shared only when m is odd. Then R_i=S_pos_i/S_neg_i and p_i=1/(1+R_i²)+S_i(avg), ranked descending.
$S_{i}^{p os} = k = 1 \sum ⌈ m /2 ⌉ \frac{S _{i (k)}}{k}; S_{i}^{n e g} = k = ⌊ m /2 ⌋ + 1 \sum m \frac{S _{i (k)}}{m - k + 1}; R_{i} = \frac{S _{i}^{p os}}{S _{i}^{n e g}}; p_{i} = \frac{1}{1 + R _{i}^{2}} + S_{i (a v g)}, rank descending$

1. Vector-normalize every criterion column. Criterion direction is not applied here; benefit/cost ordering is applied when the ideal solutions are constructed in F3.