PF-MARCOS

PF-MARCOS - MARCOS 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; AAI and AI.

$j \in Ω_{c} : α_{i j} = (ν_{i j}, μ_{i j}); {A A I}_{j} = ({min}_{i} μ_{i j}, {max}_{i} ν_{i j}) (j \in Ω_{b}); {A I}_{j} = ({max}_{i} μ_{i j}, {min}_{i} ν_{i j}) (j \in Ω_{b})$

LaTeX
j\in\Omega_c:\;\alpha_{ij}=(\nu_{ij},\mu_{ij});\;\mathrm{AAI}_j=(\min_i\mu_{ij},\max_i\nu_{ij})\;(j\in\Omega_b);\;\mathrm{AI}_j=(\max_i\mu_{ij},\min_i\nu_{ij})\;(j\in\Omega_b)
Normalise using AI.

${\hat{μ}}_{i j} = (μ_{i j}^{2} / μ_{A I, j}^{2})^{1 / 2}; {\hat{ν}}_{i j} = (ν_{A I, j}^{2} / ν_{i j}^{2})^{1 / 2} clipped to [0, 1]$

LaTeX
\hat{\mu}_{ij}=(\mu_{ij}^2/\mu_{\mathrm{AI},j}^2)^{1/2};\;\hat{\nu}_{ij}=(\nu_{\mathrm{AI},j}^2/\nu_{ij}^2)^{1/2}\;\text{clipped to }[0,1]
PFWA row aggregate.

${P F W A}_{w} (α_{1}, \dots, α_{n}) = ((1 - \prod_{j} (1 - μ_{j}^{2})^{w_{j}})^{1 / 2}, \prod_{j} ν_{j}^{w_{j}}); {\tilde{S}}_{i} = {P F W A}_{w} ({\hat{α}}_{i 1}, \dots, {\hat{α}}_{i n}); S_{i} = s ({\tilde{S}}_{i})$

LaTeX
\mathrm{PFWA}_w(\alpha_1,\ldots,\alpha_n)=\bigl((1-\prod_j(1-\mu_j^2)^{w_j})^{1/2},\;\prod_j\nu_j^{w_j}\bigr);\;\tilde{S}_i=\mathrm{PFWA}_w(\hat{\alpha}_{i1},\ldots,\hat{\alpha}_{in});\;S_i=s(\tilde{S}_i)
K± and f(K±).

$K_{i}^{-} = S_{i} / S_{A A I}; K_{i}^{+} = S_{i} / S_{A I}; f (K_{i}^{+}) = K_{i}^{-} / (K_{i}^{-} + K_{i}^{+}); f (K_{i}^{-}) = K_{i}^{+} / (K_{i}^{-} + K_{i}^{+})$

LaTeX
K_i^-=S_i/S_{\mathrm{AAI}};\;K_i^+=S_i/S_{\mathrm{AI}};\;f(K_i^+)=K_i^-/(K_i^-+K_i^+);\;f(K_i^-)=K_i^+/(K_i^-+K_i^+)
Composite f(K_i).

$f (K_{i}) = (K_{i}^{+} + K_{i}^{-}) / (1 + (1 - f (K_{i}^{+})) / f (K_{i}^{+}) + (1 - f (K_{i}^{-})) / f (K_{i}^{-}))$

LaTeX
f(K_i)=(K_i^++K_i^-)/(1+(1-f(K_i^+))/f(K_i^+)+(1-f(K_i^-))/f(K_i^-))
Rank descending by f(K_i).

$rank descending$

LaTeX
\text{rank descending}
Final ranking.

$A_{σ (1)} ≻ \dots ≻ A_{σ (m)}$

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

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: pf-marcos extends MARCOS 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

Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
Underlying crisp method's compensation assumption holds in uncertain space
All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

Classical data sufficient - use base MARCOS directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

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.