L2T-COPRAS - L2T-COPRAS yönteminin Linguistic uzantısı

Formül adımları

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

1. Adım 1 — S={s_0,…,s_q} üzerinde x̃_ij=(s_ij,α_ij) 2-demet karar matrisi ve Δ⁻¹(s_i,α)=i+α; ağırlıklar w_j ile yön kümeleri Ω_max/Ω_min.

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

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

2. Adım 2 — 2-demet ağırlıklı karar matrisi d̃_ij = Δ(w_j · Δ⁻¹(s_ij,α_ij)).

   $${\tilde{d}}_{ij}=\Delta (w_j·{\Delta }^{-1}(s_{ij},{\alpha }_{ij}))$$
   LaTeX

   \tilde{d}_{ij} = \Delta\!\left(w_{j}\cdot \Delta^{-1}(s_{ij},\alpha_{ij})\right)

3. Adım 3 — Fayda toplamı P_i = Σ_{j∈Ω_max} Δ⁻¹(d̃_ij) ve maliyet toplamı R_i = Σ_{j∈Ω_min} Δ⁻¹(d̃_ij).

   $$P_i=\sum _{j\in {\Omega }_{max}}{\Delta }^{-1}({\tilde{d}}_{ij}),R_i=\sum _{j\in {\Omega }_{min}}{\Delta }^{-1}({\tilde{d}}_{ij})$$
   LaTeX

   P_{i} = \sum_{j \in \Omega_{max}} \Delta^{-1}(\tilde{d}_{ij}),\quad R_{i} = \sum_{j \in \Omega_{min}} \Delta^{-1}(\tilde{d}_{ij})

4. Adım 4 — Göreli önem Q_i = P_i + (Σ_k R_k)/(R_i · Σ_k 1/R_k); fayda ve maliyetin birleşimi.

   $$Q_i=P_i+{\displaystyle \frac{{\sum }_{k=1}^m{R}_k}{R_i·{\sum }_{k=1}^m{\displaystyle \frac{1}{R_k}}}}$$
   LaTeX

   Q_{i} = P_{i} + \dfrac{\sum_{k=1}^{m} R_{k}}{R_{i}\cdot \sum_{k=1}^{m} \dfrac{1}{R_{k}}}

5. Adım 5 — Fayda derecesi N_i = (Q_i / max_k Q_k) × 100%; azalan sıralama.

   $$N_i={\displaystyle \frac{Q_i}{{max}_k{Q}_k}}\times 100\%;\text{rank}={\text{argsort}}_{\text{desc}}(N_i)$$
   LaTeX

   N_{i} = \dfrac{Q_{i}}{\max_{k} Q_{k}} \times 100\%;\ \ \text{rank} = \text{argsort}_{\text{desc}}(N_{i})

Sezgi

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

Sonucu okuma: l2t-copras extends L2T-COPRAS 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 COPRAS 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.