SF-TODIM - TODIM yönteminin Spherical uzantısı

Formül adımları

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

1. normalization

   $${\tilde{a}}_{ij}^c=({\nu }_{ij},{\mu }_{ij},{\pi }_{ij})\text{for cost criteria}$$
   LaTeX

   \tilde{a}_{ij}^{c}=(\nu_{ij},\mu_{ij},\pi_{ij})\quad\text{for cost criteria}

2. ranking_computation

   $${\overline{w}}_{jr}=\frac{w_j}{w_r},w_r={max}_j{w}_j$$
   LaTeX

   \bar{w}_{jr}=\frac{w_j}{w_r},\quad w_r=\max_j w_j

3. ranking_computation

   $${\varphi }_j(A_i,A_k)=\{\begin{array}{cc}{\overline{w}}_{jr}\sqrt{{\displaystyle \frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\overline{w}}_{jr}}{{\overline{w}}_{jr}}}}\hfill & \text{gain}\hfill \\ [\mathrm{6pt}]-{\displaystyle \frac{1}{\theta }}\sqrt{{\displaystyle \frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\overline{w}}_{jr}}{{\overline{w}}_{jr}}}}\hfill & \text{loss}\hfill \\ 0\hfill & \text{tie}\hfill \end{array}$$
   LaTeX

   \phi_j(A_i,A_k)=\begin{cases}\bar{w}_{jr}\sqrt{\dfrac{d(\tilde{a}_{ij},\tilde{a}_{kj})\sum_j\bar{w}_{jr}}{\bar{w}_{jr}}} & \text{gain}\\[6pt]-\dfrac{1}{\theta}\sqrt{\dfrac{d(\tilde{a}_{ij},\tilde{a}_{kj})\sum_j\bar{w}_{jr}}{\bar{w}_{jr}}} & \text{loss}\\0 & \text{tie}\end{cases}

4. ranking_computation

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\varphi }_j(A_i,A_k)$$
   LaTeX

   \delta(A_i,A_k)=\sum_{j=1}^n\phi_j(A_i,A_k)

5. ranking_computation

   $${\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{min}_i{\sum }_k\delta (A_i,A_k)}{{max}_i{\sum }_k\delta (A_i,A_k)-{min}_i{\sum }_k\delta (A_i,A_k)},\text{rank descending}$$
   LaTeX

   \xi_i=\frac{\sum_k\delta(A_i,A_k)-\min_i\sum_k\delta(A_i,A_k)}{\max_i\sum_k\delta(A_i,A_k)-\min_i\sum_k\delta(A_i,A_k)},\quad\text{rank descending}

Sezgi

Sonucu okuma: sf-todim extends TODIM 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 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 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.