L2T-TOPSIS - L2T-TOPSIS yönteminin Linguistic uzantısı

Formül adımları

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

1. Adım 1 — Dilsel terim kümesi S = {s_0,…,s_q}'yi tanımla ve her x̃_ij = (s_i, α_ij) olan 2-demet karar matrisini oluştur.

   $$S=\{s_0,s_1,\dots ,s_q\};\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n},\Delta (\beta )=(s_i,\alpha ),i=\text{round}(\beta ),\alpha =\beta -i;{\Delta }^{-1}(s_i,\alpha )=i+\alpha$$
   LaTeX

   S = \{s_{0}, s_{1}, \ldots, s_{q}\};\ \tilde{D} = [(s_{ij}, \alpha_{ij})]_{m\times n},\ \Delta(\beta) = (s_{i}, \alpha),\ i = \text{round}(\beta),\ \alpha = \beta - i;\ \Delta^{-1}(s_{i}, \alpha) = i + \alpha

2. Adım 2 — Δ⁻¹ ile kriter başına ağırlıklı 2-demet agregasyonu.

   $${\overline{x}}_j=\Delta ({\sum }_{i=1}^m{w}_i·{\Delta }^{-1}(s_{ij},{\alpha }_{ij}))$$
   LaTeX

   \bar{x}_{j} = \Delta\!\left(\sum_{i=1}^{m} w_{i}\cdot \Delta^{-1}(s_{ij}, \alpha_{ij})\right)

3. Adım 3 — 2-Demet Dilsel Pozitif İdeal Çözüm (TLPIS) A⁺ ve Negatif İdeal Çözüm (TLNIS) A⁻.

   $$A^{+}=\{(s_1^{+},{\alpha }_1^{+}),\dots ,(s_n^{+},{\alpha }_n^{+})\};A^{-}=\{(s_1^{-},{\alpha }_1^{-}),\dots ,(s_n^{-},{\alpha }_n^{-})\}$$
   LaTeX

   A^{+} = \{(s_{1}^{+}, \alpha_{1}^{+}), \ldots, (s_{n}^{+}, \alpha_{n}^{+})\};\ \ A^{-} = \{(s_{1}^{-}, \alpha_{1}^{-}), \ldots, (s_{n}^{-}, \alpha_{n}^{-})\}

4. Adım 4 — Δ⁻¹ tabanlı 2-demet uzaklık d((s_i,α_i),(s_j,α_j)) = |Δ⁻¹(s_i,α_i) − Δ⁻¹(s_j,α_j)| ile ayrım ölçüleri.

   $$d((s_i,{\alpha }_i),(s_j,{\alpha }_j))=|{\Delta }^{-1}(s_i,{\alpha }_i)-{\Delta }^{-1}(s_j,{\alpha }_j)|;D_i^{+}=\sqrt{{\sum }_{j=1}^n{d}^2((r_{ij},{\alpha }_{ij}),(s_j^{+},{\alpha }_j^{+}))},D_i^{-}=\sqrt{{\sum }_{j=1}^n{d}^2((r_{ij},{\alpha }_{ij}),(s_j^{-},{\alpha }_j^{-}))}$$
   LaTeX

   d((s_{i},\alpha_{i}),(s_{j},\alpha_{j})) = |\Delta^{-1}(s_{i},\alpha_{i}) - \Delta^{-1}(s_{j},\alpha_{j})|;\ \ D_{i}^{+} = \sqrt{\sum_{j=1}^{n} d^{2}((r_{ij},\alpha_{ij}),(s_{j}^{+},\alpha_{j}^{+}))},\ \ D_{i}^{-} = \sqrt{\sum_{j=1}^{n} d^{2}((r_{ij},\alpha_{ij}),(s_{j}^{-},\alpha_{j}^{-}))}

5. Adım 5 — Göreli yakınlık RC_i = D_i⁻ / (D_i⁺ + D_i⁻); azalan sıralama.

   $$R{C}_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}},0\le R{C}_i\le 1;\text{rank}={\text{argsort}}_{\text{desc}}(R{C}_i)$$
   LaTeX

   RC_{i} = \dfrac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}},\ 0 \le RC_{i} \le 1;\ \ \text{rank} = \text{argsort}_{\text{desc}}(RC_{i})

Sezgi

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

Sonucu okuma: l2t-topsis extends L2T-TOPSIS 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 TOPSIS 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 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.