SF-EDAS - EDAS yönteminin Spherical uzantısı

Formül adımları

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

1. Adım 3 — Ortalama Çözüm: her kriter sütununu eşit ağırlıklarla (w_i = 1/n) SWAM ile toplayarak SFS ortalama çözümü ãv_j elde et.

   $${\widetilde{av}}_j={\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{M}}_{1/n}({\tilde{x}}_{1j},\dots ,{\tilde{x}}_{mj})=(\sqrt{1-{\prod }_{i=1}^m(1-{\mu }_{ij}^2{)}^{1/m}},\prod _{i=1}^m{\upsilon }_{ij}^{1/m},\sqrt{{\prod }_{i=1}^m(1-{\mu }_{ij}^2{)}^{1/m}-{\prod }_{i=1}^m(1-{\mu }_{ij}^2-{\pi }_{ij}^2{)}^{1/m}})$$
   LaTeX

   \widetilde{av}_j = \mathrm{SWAM}_{1/n}(\tilde{x}_{1j},\ldots,\tilde{x}_{mj}) = \Bigl(\sqrt{1-\prod_{i=1}^{m}(1-\mu_{ij}^2)^{1/m}},\; \prod_{i=1}^{m}\upsilon_{ij}^{1/m},\; \sqrt{\prod_{i=1}^{m}(1-\mu_{ij}^2)^{1/m}-\prod_{i=1}^{m}(1-\mu_{ij}^2-\pi_{ij}^2)^{1/m}}\Bigr)

2. Adım 4 — SF Fark Matrisi: her x̃_ij ile ãv_j arasındaki küresel bulanık fark S̃'_ij'yi hesapla. Maliyet kriterleri için önce eşlenik (υ,μ,π) uygula. dH Hamming mesafesi.

   $$\begin{gathered} d_H({\tilde{x}}_{ij},{\widetilde{av}}_j)=|{\mu }_{ij}-{\mu }_j^{av}|+|{\upsilon }_{ij}-{\upsilon }_j^{av}|+|{\pi }_{ij}-{\pi }_j^{av}|; \\ {\mu }^{\prime }=\frac{|{\mu }_{ij}-{\mu }_j^{av}|}{d_H}\mathbb{1}[{\mu }_{ij}>{\mu }_j^{av}]+\frac{|{\upsilon }_{ij}-{\upsilon }_j^{av}|}{d_H}\mathbb{1}[{\upsilon }_{ij}<{\upsilon }_j^{av}]+\frac{|{\pi }_{ij}-{\pi }_j^{av}|}{d_H}\mathbb{1}[{\pi }_{ij}<{\pi }_j^{av}]; \\ {\upsilon }^{\prime }=\frac{|{\mu }_{ij}-{\mu }_j^{av}|}{d_H}\mathbb{1}[{\mu }_{ij}<{\mu }_j^{av}]+\frac{|{\upsilon }_{ij}-{\upsilon }_j^{av}|}{d_H}\mathbb{1}[{\upsilon }_{ij}>{\upsilon }_j^{av}]; \\ {\pi }^{\prime }=\frac{|{\pi }_{ij}-{\pi }_j^{av}|}{d_H}\mathbb{1}[{\pi }_{ij}>{\pi }_j^{av}] \end{gathered}$$
   LaTeX

   d_H(\tilde{x}_{ij},\widetilde{av}_j)=|\mu_{ij}-\mu_j^{av}|+|\upsilon_{ij}-\upsilon_j^{av}|+|\pi_{ij}-\pi_j^{av}|;\\ \mu'=\frac{|\mu_{ij}-\mu_j^{av}|}{d_H}\mathbb{1}[\mu_{ij}>\mu_j^{av}]+\frac{|\upsilon_{ij}-\upsilon_j^{av}|}{d_H}\mathbb{1}[\upsilon_{ij}<\upsilon_j^{av}]+\frac{|\pi_{ij}-\pi_j^{av}|}{d_H}\mathbb{1}[\pi_{ij}<\pi_j^{av}];\\ \upsilon'=\frac{|\mu_{ij}-\mu_j^{av}|}{d_H}\mathbb{1}[\mu_{ij}<\mu_j^{av}]+\frac{|\upsilon_{ij}-\upsilon_j^{av}|}{d_H}\mathbb{1}[\upsilon_{ij}>\upsilon_j^{av}];\\ \pi'=\frac{|\pi_{ij}-\pi_j^{av}|}{d_H}\mathbb{1}[\pi_{ij}>\pi_j^{av}]

3. Adım 5 — Ağırlıklı SF Farkları: her S̃'_ij'yi SFS çarpımı (⊗) kullanarak kriter ağırlığı w̃_j ile çarp.

   $${\tilde{S}}_{ij}={\tilde{w}}_j\otimes {\tilde{S}}_{ij}^{\prime }$$
   LaTeX

   \tilde{S}_{ij} = \tilde{w}_j \otimes \tilde{S}'_{ij}

4. Adım 6 — SFAgg ile Değerlendirme Skoru: her alternatif için ağırlıklı farkları SFAgg ile topla (dengeli toplulaştırma, eşit ağırlıklar w_j = 1/m), SFS değerlendirme demeti ãs_i elde et.

   $${\widetilde{as}}_i={\mathrm{S}\mathrm{F}\mathrm{A}\mathrm{g}\mathrm{g}}_{1/m}({\tilde{S}}_{i1},\dots ,{\tilde{S}}_{im})=(\sqrt{\frac{1}{m}{\sum }_{j=1}^m{\mu }_{{\tilde{S}}_{ij}}^2},\sqrt{\frac{1}{m}{\sum }_{j=1}^m{\upsilon }_{{\tilde{S}}_{ij}}^2},\sqrt{\frac{1}{m}{\sum }_{j=1}^m{\pi }_{{\tilde{S}}_{ij}}^2})$$
   LaTeX

   \widetilde{as}_i = \mathrm{SFAgg}_{1/m}(\tilde{S}_{i1},\ldots,\tilde{S}_{im}) = \Bigl(\sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\mu^2_{\tilde{S}_{ij}}},\; \sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\upsilon^2_{\tilde{S}_{ij}}},\; \sqrt{\tfrac{1}{m}\sum_{j=1}^{m}\pi^2_{\tilde{S}_{ij}}}\Bigr)

5. Adımlar 7–8 — Defuzzify ve Sırala: her ãs_i'ye skor fonksiyonu uygula; beraberlik durumunda doğruluk fonksiyonuna başvur. Azalan sırayla sırala.

   $$\begin{gathered} \mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}({\widetilde{as}}_i)=({\mu }_{{\widetilde{as}}_i}-{\pi }_{{\widetilde{as}}_i}{)}^2-({\upsilon }_{{\widetilde{as}}_i}-{\pi }_{{\widetilde{as}}_i}{)}^2; \\ \mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}({\widetilde{as}}_i)={\mu }_{{\widetilde{as}}_i}^2+{\upsilon }_{{\widetilde{as}}_i}^2+{\pi }_{{\widetilde{as}}_i}^2; \\ \text{rank descending by Score (Accuracy as tie-break)} \end{gathered}$$
   LaTeX

   \mathrm{Score}(\widetilde{as}_i) = (\mu_{\widetilde{as}_i}-\pi_{\widetilde{as}_i})^2-(\upsilon_{\widetilde{as}_i}-\pi_{\widetilde{as}_i})^2;\\ \mathrm{Accuracy}(\widetilde{as}_i) = \mu^2_{\widetilde{as}_i}+\upsilon^2_{\widetilde{as}_i}+\pi^2_{\widetilde{as}_i};\\ \text{rank descending by Score (Accuracy as tie-break)}

Sezgi

Spherical outranking/ranking - Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1). Output typically utility (higher value = preferred).

Sonucu okuma: sf-edas extends EDAS to handle Spherical uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Varsayımlar

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

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• Assumes: Decision matrix entries are valid Spherical Fuzzy numbers/tuples
• Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin SFS: μ,ν,π ∈ [0,1]; μ²+ν²+π² ≤ 1 koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ² − ν² kanonik seçimdir.