N-TODIM - TODIM yönteminin Neutrosophic uzantısı

Formül adımları

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

1. SVN matrix; cost complement.

   $$𝐃=(⟨T_{ij},I_{ij},F_{ij}⟩);\text{cost: }{\tilde{a}}_{ij}^c=⟨F_{ij},1-I_{ij},T_{ij}⟩$$
   LaTeX

   \mathbf{D}=(\langle T_{ij},I_{ij},F_{ij}\rangle);\quad\text{cost: }\tilde{a}_{ij}^c=\langle F_{ij},1-I_{ij},T_{ij}\rangle

2. Relative criterion weights; reference criterion w_r = max w_j.

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

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

3. Dominance contribution: gain (√) and loss (−1/θ·√) with neutrosophic distance.

   $$d_j(i,k)=d({\tilde{a}}_{ij},{\tilde{a}}_{kj})=\sqrt{\frac{(T_{ij}-T_{kj}{)}^2+(I_{ij}-I_{kj}{)}^2+(F_{ij}-F_{kj}{)}^2}{3}};{\varphi }_j(A_i,A_k)=\{\begin{array}{cc}+\sqrt{{\overline{w}}_{jr}{d}_j/{\sum }_j{\overline{w}}_{jr}}\hfill & s_{ij}>s_{kj}\hfill \\ 0\hfill & s_{ij}=s_{kj}\hfill \\ -\frac{1}{\theta }\sqrt{({\sum }_j{\overline{w}}_{jr})·d_j/{\overline{w}}_{jr}}\hfill & s_{ij}<s_{kj}\hfill \end{array}$$
   LaTeX

   d_j(i,k)=d(\tilde{a}_{ij},\tilde{a}_{kj})=\sqrt{\tfrac{(T_{ij}-T_{kj})^2+(I_{ij}-I_{kj})^2+(F_{ij}-F_{kj})^2}{3}};\quad\phi_j(A_i,A_k)=\begin{cases}+\sqrt{\bar{w}_{jr}d_j/\sum_j\bar{w}_{jr}}&s_{ij}>s_{kj}\\0&s_{ij}=s_{kj}\\-\tfrac{1}{\theta}\sqrt{(\sum_j\bar{w}_{jr})\cdot d_j/\bar{w}_{jr}}&s_{ij}<s_{kj}\end{cases}

4. Overall dominance δ(A_i,A_k) summed over criteria.

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

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

5. Global normalised value ξ_i; rank descending.

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

   \xi_i=\frac{\sum_k\delta(A_i,A_k)-\min_k\sum_t\delta(A_k,A_t)}{\max_k\sum_t\delta(A_k,A_t)-\min_k\sum_t\delta(A_k,A_t)};\quad\text{rank descending}

Sezgi

Neutrosophic outranking/ranking - Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3). Output typically utility (higher value = preferred).

Sonucu okuma: n-todim extends TODIM to handle Neutrosophic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3) algebra. The final scores are defuzzified via score function S = (T − F + 1)/2 before ranking.

Varsayımlar

• Decision matrix entries are valid Single-Valued Neutrosophic 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

Sınırlılıklar

• Assumes: Decision matrix entries are valid Single-Valued Neutrosophic 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 SVNS: T,I,F ∈ [0,1]; 0 ≤ T+I+F ≤ 3 koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = (T − F + 1)/2 kanonik seçimdir.