PF-PROMETHEE

PF-PROMETHEE - PROMETHEE yönteminin Pythagorean uzantısı

Pythagorean üstünlük/sıralama - Pisagoryan Bulanık Sayı (PFS: μ, ν; μ²+ν² ≤ 1)

Formül adımları

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

Cost complement.

$j \in Ω_{c} : {\hat{α}}_{i j} = (ν_{i j}, μ_{i j}); s (α) = μ^{2} - ν^{2}$

LaTeX
j\in\Omega_c:\;\hat{\alpha}_{ij}=(\nu_{ij},\mu_{ij});\;s(\alpha)=\mu^2-\nu^2
Signed PF differences.

$d_{E} (α_{1}, α_{2}) = \sqrt{[(μ_{1}^{2} - μ_{2}^{2})^{2} + (ν_{1}^{2} - ν_{2}^{2})^{2} + (π_{1}^{2} - π_{2}^{2})^{2}] / 2}; π = \sqrt{1 - μ^{2} - ν^{2}}; d_{j} (i, k) = s i g n (s ({\hat{α}}_{i j}) - s ({\hat{α}}_{k j})) \cdot d_{E} ({\hat{α}}_{i j}, {\hat{α}}_{k j})$

LaTeX
d_E(\alpha_1,\alpha_2)=\sqrt{[(\mu_1^2-\mu_2^2)^2+(\nu_1^2-\nu_2^2)^2+(\pi_1^2-\pi_2^2)^2]/2};\;\pi=\sqrt{1-\mu^2-\nu^2};\;d_j(i,k)=\mathrm{sign}(s(\hat{\alpha}_{ij})-s(\hat{\alpha}_{kj}))\cdot d_E(\hat{\alpha}_{ij},\hat{\alpha}_{kj})
Type V preference function; aggregated π.

$P_{j} (i, k) = 0 (d_{j} \leq 0); d_{j} / p_{j} (0 < d_{j} \leq p_{j}); 1 (d_{j} > p_{j}); π (A_{i}, A_{k}) = \sum_{j} w_{j} P_{j} (i, k)$

LaTeX
P_j(i,k)=0\;(d_j\leq 0);\;d_j/p_j\;(0<d_j\leq p_j);\;1\;(d_j>p_j);\;\pi(A_i,A_k)=\sum_j w_j P_j(i,k)
Positive and negative flows.

$ϕ^{+} (A_{i}) = \frac{1}{m - 1} \sum_{k \neq i} π (A_{i}, A_{k}); ϕ^{-} (A_{i}) = \frac{1}{m - 1} \sum_{k \neq i} π (A_{k}, A_{i})$

LaTeX
\phi^+(A_i)=\tfrac{1}{m-1}\sum_{k\neq i}\pi(A_i,A_k);\;\phi^-(A_i)=\tfrac{1}{m-1}\sum_{k\neq i}\pi(A_k,A_i)
Net flow PROMETHEE II; rank descending.

$ϕ (A_{i}) = ϕ^{+} - ϕ^{-}; rank descending$

LaTeX
\phi(A_i)=\phi^+-\phi^-;\quad\text{rank descending}
Final ranking output.

$A_{σ (1)} ≻ \dots ≻ A_{σ (m)}$

LaTeX
A_{\sigma(1)}\succ\cdots\succ A_{\sigma(m)}

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: pf-promethee extends PROMETHEE to handle Pythagorean uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Varsayımlar

Thresholds (q, p, v) can be expressed in Pythagorean Fuzzy scale
Veto + concordance semantics adapted to fuzzy arithmetic

Ne zaman kullanılmaz

Small dataset (m<3) - outranking machinery underutilised

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ² − ν² kanonik seçimdir.