L2T-TODIM - L2T-TODIM yönteminin Linguistic uzantısı

Formül adımları

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

1. Adım 1 — 2-demet dilsel karar matrisini oluştur; kriter ağırlıklarını tanımla.

   $$\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n};{\Delta }^{-1}(s_i,\alpha )=i+\alpha$$
   LaTeX

   \tilde{D} = [(s_{ij}, \alpha_{ij})]_{m \times n};\ \Delta^{-1}(s_{i}, \alpha) = i + \alpha

2. Adım 2 — 2-demet uzaklık d² ve olası kayıp aversiyonu θ (tipik 2.25) ile egemenlik derecesi Φ_j.

   $${\Phi }_j(A_i,A_k)=\{\begin{array}{cc}\sqrt{w_j·d^2((s_{ij},{\alpha }_{ij}),(s_{kj},{\alpha }_{kj}))/{\sum }_j{w}_j}\hfill & {\Delta }^{-1}(s_{ij},{\alpha }_{ij})>{\Delta }^{-1}(s_{kj},{\alpha }_{kj})\hfill \\ 0\hfill & \text{equal}\hfill \\ -{\displaystyle \frac{1}{\theta }}\sqrt{({\sum }_j{w}_j)·d^2((s_{ij},{\alpha }_{ij}),(s_{kj},{\alpha }_{kj}))/w_j}\hfill & {\Delta }^{-1}(s_{ij},{\alpha }_{ij})<{\Delta }^{-1}(s_{kj},{\alpha }_{kj})\hfill \end{array}$$
   LaTeX

   \Phi_{j}(A_{i},A_{k}) = \begin{cases} \sqrt{w_{j}\cdot d^{2}((s_{ij},\alpha_{ij}),(s_{kj},\alpha_{kj}))/\sum_{j} w_{j}} & \Delta^{-1}(s_{ij},\alpha_{ij}) > \Delta^{-1}(s_{kj},\alpha_{kj})\\ 0 & \text{equal}\\ -\dfrac{1}{\theta}\sqrt{(\sum_{j} w_{j})\cdot d^{2}((s_{ij},\alpha_{ij}),(s_{kj},\alpha_{kj}))/w_{j}} & \Delta^{-1}(s_{ij},\alpha_{ij}) < \Delta^{-1}(s_{kj},\alpha_{kj})\end{cases}

3. Adım 3 — Toplam egemenlik δ(A_i,A_k) = Σ_j Φ_j(A_i,A_k).

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

   \delta(A_{i},A_{k}) = \sum_{j=1}^{n} \Phi_{j}(A_{i},A_{k})

4. Adım 4 — Σ_k δ(A_i,A_k) üzerinden [0,1]'e normalleştirilmiş genel beklenti değeri ξ_i; azalan sıralama.

   $${\xi }_i={\displaystyle \frac{{\sum }_k\delta (A_i,A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}{{max}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}};\text{rank}={\text{argsort}}_{\text{desc}}({\xi }_i)$$
   LaTeX

   \xi_{i} = \dfrac{\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} = \text{argsort}_{\text{desc}}(\xi_{i})

Sezgi

Linguistic outranking/ranking - 2-Tuple Linguistic Variable (2TL: (s_i, α)). Output typically utility (higher value = preferred).

Sonucu okuma: l2t-todim extends L2T-TODIM to handle Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using 2-Tuple Linguistic Variable (2TL: (s_i, α)) algebra. The final scores are defuzzified via Δ^{-1}(s_i, α) = i + α before ranking.

Varsayımlar

• Decision matrix entries are valid Linguistic 2-Tuple 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 Linguistic 2-Tuple 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 2TL: s_i ∈ S (linguistic term set), α ∈ [-0.5,0.5) koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: Δ^{-1}(s_i, α) = i + α kanonik seçimdir.