FF-ARAS - FF-ARAS yönteminin Fermatean uzantısı

Formül adımları

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

1. Cost complement; augment with optimal row A0.

   $$j\in {\Omega }_c:{\widehat{a}}_{ij}=({\nu }_{ij},{\mu }_{ij});{\widehat{a}}_{0j}=({max}_i{\widehat{\mu }}_{ij},{min}_i{\widehat{\nu }}_{ij})$$
   LaTeX

   j\in\Omega_c:\;\hat{a}_{ij}=(\nu_{ij},\mu_{ij});\;\hat{a}_{0j}=(\max_i\hat{\mu}_{ij},\min_i\hat{\nu}_{ij})

2. Column-sum normalisation on scores; include A0.

   $${\widehat{s}}_{ij}=s({\widehat{a}}_{ij})/\sum _{i=0}^{m}s({\widehat{a}}_{ij})$$
   LaTeX

   \hat{s}_{ij}=s(\hat{a}_{ij})/\sum_{i=0}^m s(\hat{a}_{ij})

3. FFWA aggregate score S_i.

   $${\mathrm{F}\mathrm{F}\mathrm{W}\mathrm{A}}_w({\alpha }_1,\dots ,{\alpha }_n)=((1-\prod _j(1-{\mu }_j^3{)}^{w_j}{)}^{1/3},\prod _j{\nu }_j^{w_j});{\tilde{S}}_i={\mathrm{F}\mathrm{F}\mathrm{W}\mathrm{A}}_w({\widehat{a}}_{i1}^n,\dots ,{\widehat{a}}_{in}^n);S_i=s({\tilde{S}}_i)$$
   LaTeX

   \mathrm{FFWA}_w(\alpha_1,\ldots,\alpha_n)=\bigl((1-\prod_j(1-\mu_j^3)^{w_j})^{1/3},\;\prod_j\nu_j^{w_j}\bigr);\;\tilde{S}_i=\mathrm{FFWA}_w(\hat{a}_{i1}^n,\ldots,\hat{a}_{in}^n);\;S_i=s(\tilde{S}_i)

4. Degree of utility K_i = S_i / S_0.

   $$K_i=S_i/S_0$$
   LaTeX

   K_i=S_i/S_0

5. Rank descending by K_i.

   $$\text{rank descending by }{K}_i$$
   LaTeX

   \text{rank descending by }K_i

Sezgi

Sonucu okuma: ff-aras extends FF-ARAS 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 ARAS 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.