IV-TODIM - TODIM yönteminin Aralık Değerli Sezgisel Bulanık uzantısı

Formül adımları

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

1. normalization

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

   \tilde{a}_{ij}^c=([\nu^L_{ij},\nu^U_{ij}],[\mu^L_{ij},\mu^U_{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

   $${\varphi }_j(A_i,A_k)=\{\begin{array}{cc}{\overline{w}}_{jr}\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\overline{w}}_{jr}}{{\overline{w}}_{jr}}}\hfill & \text{gain}\hfill \\ [\mathrm{4pt}]-\frac{1}{\theta }\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\overline{w}}_{jr}}{{\overline{w}}_{jr}}}\hfill & \text{loss}\hfill \\ 0\hfill & \text{tie}\hfill \end{array}$$
   LaTeX

   \phi_j(A_i,A_k)=\begin{cases}\bar{w}_{jr}\sqrt{\frac{d(\tilde{a}_{ij},\tilde{a}_{kj})\sum_j\bar{w}_{jr}}{\bar{w}_{jr}}} & \text{gain}\\[4pt]-\frac{1}{\theta}\sqrt{\frac{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

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\varphi }_j(A_i,A_k),d({\alpha }_1,{\alpha }_2)=\sqrt{\frac{1}{4}[({\mu }_1^L-{\mu }_2^L{)}^2+({\mu }_1^U-{\mu }_2^U{)}^2+({\nu }_1^L-{\nu }_2^L{)}^2+({\nu }_1^U-{\nu }_2^U{)}^2]}$$
   LaTeX

   \delta(A_i,A_k)=\sum_{j=1}^n\phi_j(A_i,A_k),\quad d(\alpha_1,\alpha_2)=\sqrt{\tfrac{1}{4}[(\mu^L_1-\mu^L_2)^2+(\mu^U_1-\mu^U_2)^2+(\nu^L_1-\nu^L_2)^2+(\nu^U_1-\nu^U_2)^2]}

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

Interval outranking/ranking - Interval Number (IN: [a, b]). Output typically utility (higher value = preferred).

Sonucu okuma: iv-todim extends TODIM to handle Interval uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Interval Number (IN: [a, b]) algebra. The final scores are defuzzified via midpoint (a+b)/2 before ranking.

Varsayımlar

• Decision matrix entries are valid Interval 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 Interval 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 IN: a ≤ b koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: midpoint (a+b)/2 kanonik seçimdir.