FF-TODIM

FF-TODIM - FF-TODIM yönteminin Fermatean uzantısı

Fermatean üstünlük/sıralama - Fermat Bulanık Küme (FBK: μ, ν; μ³+ν³ ≤ 1)

Formül adımları

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

Cost complement.

$j \in Ω_{c} : {\hat{a}}_{i j} = (ν_{i j}, μ_{i j}); s (α) = μ^{3} - ν^{3}$

LaTeX
j\in\Omega_c:\;\hat{a}_{ij}=(\nu_{ij},\mu_{ij});\;s(\alpha)=\mu^3-\nu^3
Relative weights; reference criterion.

${\bar{w}}_{j r} = w_{j} / w_{r}, w_{r} = {max}_{j} w_{j}$

LaTeX
\bar{w}_{jr}=w_j/w_r,\;w_r=\max_j w_j
Dominance contribution with FF Euclidean distance.

$d_{E} (α_{1}, α_{2}) = \sqrt{[(μ_{1}^{3} - μ_{2}^{3})^{2} + (ν_{1}^{3} - ν_{2}^{3})^{2} + (π_{1}^{3} - π_{2}^{3})^{2}] / 2}; d_{H} = [(| μ_{1}^{3} - μ_{2}^{3} | + | ν_{1}^{3} - ν_{2}^{3} | + | π_{1}^{3} - π_{2}^{3} |)] / 2; ϕ_{j} (A_{i}, A_{k}) = {\begin{matrix} + \sqrt{{\bar{w}}_{j r} d_{j} / \sum {\bar{w}}_{j r}} & s_{i j} > s_{k j} \\ 0 & = \\ - \frac{1}{θ} \sqrt{(\sum {\bar{w}}_{j r}) d_{j} / {\bar{w}}_{j r}} & s_{i j} < s_{k j} \end{matrix}$

LaTeX
d_E(\alpha_1,\alpha_2)=\sqrt{[(\mu_1^3-\mu_2^3)^2+(\nu_1^3-\nu_2^3)^2+(\pi_1^3-\pi_2^3)^2]/2};\;d_H=[(|\mu_1^3-\mu_2^3|+|\nu_1^3-\nu_2^3|+|\pi_1^3-\pi_2^3|)]/2;\;\phi_j(A_i,A_k)=\begin{cases}+\sqrt{\bar{w}_{jr}d_j/\sum\bar{w}_{jr}}&s_{ij}>s_{kj}\\0&=\\-\tfrac{1}{\theta}\sqrt{(\sum\bar{w}_{jr})d_j/\bar{w}_{jr}}&s_{ij}<s_{kj}\end{cases}
Overall dominance δ(A_i,A_k).

$δ (A_{i}, A_{k}) = \sum_{j} ϕ_{j} (A_{i}, A_{k})$

LaTeX
\delta(A_i,A_k)=\sum_j\phi_j(A_i,A_k)
Global normalised value ξ_i; rank descending.

$ξ_{i} = (\sum_{k} δ (A_{i}, A_{k}) - min) / (max - min); rank descending$

LaTeX
\xi_i=(\sum_k\delta(A_i,A_k)-\min)/(\max-\min);\quad\text{rank descending}

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: ff-todim extends FF-TODIM to handle Fermatean uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Fermatean Fuzzy Set (FFS: μ, ν; μ³+ν³ ≤ 1) algebra. The final scores are defuzzified via score function S = μ³ − ν³ before ranking.

Varsayımlar

Decision matrix entries are valid Fermatean 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 TODIM directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin FFS: μ³+ν³ ≤ 1 (q=3 special case of q-ROF) koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ³ − ν³ kanonik seçimdir.