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

Overview

Multi-ideal distance ranking with harmonic weighting. Output typically ranking.

Strengths

• •Method-specific: Multi-ideal distance ranking with harmonic weighting

Edge cases

• •See F.steps and D.parameters for PROBID-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Bkz. PROBID F.steps citation_anchor'lar ve P.verification_status.

Worked example

1. 1.Adım 1 (F1): Step 1: Vector normalization (direction-aware): benefit r_ij = x_ij/√Σx²; cost r_ij = (1/x_ij)/√Σ(1/x)². (pymcdm uses vector_normalization with types.) Formül: r_{ij} = \frac{x_{ij}}{\sqrt{\sum_k x_{kj}^2}} \text{ (benefit)};\quad r_{ij} = \frac{1/x_{ij}}{\sqrt{\sum_k (1/x_{kj})^2}} \text{ (cost)} Anchor: Wang et al. 2021 §2
2. 2.Adım 2 (F2): Step 2: Weighted normalized matrix v_ij = w_j · r_ij. Formül: v_{ij} = w_j \cdot r_{ij} Anchor: Wang et al. 2021 §2
3. 3.Adım 3 (F3): Step 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. Formül: A^{(k)}_j = k\text{-th largest } v_{ij} \text{ in col } j;\quad \bar{A}_j = \frac{1}{m}\sum_k A^{(k)}_j Anchor: Wang et al. 2021 §2
4. 4.Adım 4 (F4): Step 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. Formül: S_{i(k)} = \sqrt{\sum_j (v_{ij} - A^{(k)}_j)^2};\quad S_{i(avg)} = \sqrt{\sum_j (v_{ij} - \bar{A}_j)^2} Anchor: Wang et al. 2021 §2
5. 5.Adım 5 (F5): Step 5: Harmonic-weighted positive and negative ideal distances. lim = ⌈m/2⌉. S_pos_i = Σ_{k=1}^{lim} S_i(k)/k; S_neg_i = Σ_{k=lim}^{m} S_i(k)/(m-k+1). R_i = S_pos_i / S_neg_i. Final score p_i = 1/(1+R_i²) + S_i(avg). Rank descending. Formül: S^{pos}_{i} = \sum_{k=1}^{\lceil m/2 \rceil}\frac{S_{i(k)}}{k};\quad S^{neg}_{i} = \sum_{k=\lceil m/2\rceil}^{m}\frac{S_{i(k)}}{m-k+1};\quad R_i = \frac{S^{pos}_i}{S^{neg}_i};\quad p_i = \frac{1}{1+R_i^2} + S_{i(avg)},\text{ rank descending} Anchor: Wang et al. 2021 §2 Eq.(main)

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