PROBID - İdeal-Ortalama Uzaklığa Dayalı Tercih Sıralaması

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

Adım 1 — Vektör normalleştirme (yön duyarlı).

$F_{i j} = \frac{f_{i j}}{\sqrt{\sum_{k = 1}^{m} f_{k j}^{2}}} \forall j; criterion direction is applied when constructing A^{(k)}$

LaTeX
F_{ij} = \frac{f_{ij}}{\sqrt{\sum_{k=1}^{m} f_{kj}^{2}}} \quad \forall j; \text{ criterion direction is applied when constructing } A^{(k)}
Adım 2 — Ağırlıklı normalleştirilmiş matris.

$v_{i j} = w_{j} \cdot r_{i j}$

LaTeX
v_{ij} = w_j \cdot r_{ij}
Adım 3 — m adet ideal çözüm oluştur; ortalama ideal hesapla.

$A_{j}^{(k)} = {\begin{matrix} k -th largest v_{i j}, & j \in J^{+} \\ k -th smallest v_{i j}, & j \in J^{-} \end{matrix}; {\bar{A}}_{j} = \frac{1}{m} \sum_{k = 1}^{m} A_{j}^{(k)}$

LaTeX
A^{(k)}_j = \begin{cases}k\text{-th largest }v_{ij},&j\in J^{+}\\k\text{-th smallest }v_{ij},&j\in J^{-}\end{cases};\quad \bar{A}_j = \frac{1}{m}\sum_{k=1}^{m} A^{(k)}_j
Adım 4 — Öklid uzaklıkları: her idealden ve ortalama idealden.

$S_{i (k)} = \sqrt{\sum_{j} (v_{i j} - A_{j}^{(k)})^{2}}; S_{i (a v g)} = \sqrt{\sum_{j} (v_{i j} - {\bar{A}}_{j})^{2}}$

LaTeX
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}
Adım 5 — Harmonik ağırlıklı uzaklıklar, R_i oranı, nihai skor p_i. Azalan sıralama.

$S_{i}^{p o s} = \sum_{k = 1}^{⌈ m / 2 ⌉} \frac{S_{i (k)}}{k}; S_{i}^{n e g} = \sum_{k = ⌊ m / 2 ⌋ + 1}^{m} \frac{S_{i (k)}}{m - k + 1}; R_{i} = \frac{S_{i}^{p o s}}{S_{i}^{n e g}}; p_{i} = \frac{1}{1 + R_{i}^{2}} + S_{i (a v g)}, rank descending$

LaTeX
S^{pos}_{i} = \sum_{k=1}^{\lceil m/2 \rceil}\frac{S_{i(k)}}{k};\quad S^{neg}_{i} = \sum_{k=\lfloor m/2\rfloor+1}^{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}

Sezgi

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

Sonucu okuma: Higher p_i = better. PROBID considers all m ideal solutions (ranked alternatives) and weights closer ideals more heavily. Complexity O(m²n) - suitable for moderate-size problems.

Sık yapılan hatalar

O(m²) uzaklık matrisi - büyük m için hesaplama maliyeti artar.

Hesap adımları ve dayanakları

Apply vector normalization F_ij=f_ij/√Σf_kj² to every criterion; criterion direction is handled while constructing the ordered ideal solutions.

Dayanak: Wang et al. 2021 Eq. (2.1)
Weighted normalized matrix v_ij = w_j · r_ij.

Dayanak: Wang et al. 2021 §2
Construct A(k) from the k-th largest benefit value and k-th smallest cost value; compute the average solution.

Dayanak: Wang et al. 2021 Eqs. (2.3)-(2.5)
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.

Dayanak: Wang et al. 2021 §2
Positive ideals use k=1..⌈m/2⌉; negative ideals use k=⌊m/2⌋+1..m, so odd m shares the middle ideal while even m does not. Then R_i=S_pos/S_neg and P_i=1/(1+R_i²)+S_i(avg).

Dayanak: Wang et al. 2021 Eqs. (2.8)-(2.11)