PL-TODIM - TODIM yönteminin Probabilistic Linguistic uzantısı

Formül adımları

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

1. Adım 1 — PLTS karar matrisi L_ij(p_ij); Σ_k p_ij^(k)=1; ağırlıklar w_j ve referans kriter r = arg max_j w_j; göreli ağırlıklar w_jr = w_j/w_r.

   $$L_{ij}(p_{ij})=\{L_{ij}^{(k)}(p_{ij}^{(k)})\mid k=1,\dots ,\#L_{ij}\},\sum _k{p}_{ij}^{(k)}=1;r=\mathrm{\backslash arg}{max}_j{w}_j,w_{jr}=w_j/w_r$$
   LaTeX

   L_{ij}(p_{ij}) = \{L_{ij}^{(k)}(p_{ij}^{(k)}) \mid k=1,\dots,\#L_{ij}\},\ \sum_{k} p_{ij}^{(k)} = 1;\ \ r = \arg\max_{j} w_{j},\ w_{jr} = w_{j}/w_{r}

2. Adım 2 — PLTS uzaklığı d ve θ ile ikili egemenlik Φ_j; E(L_ij) ile E(L_kj) karşılaştırmasına göre 3-durumlu.

   $${\Phi }_j(A_i,A_k)=\{\begin{array}{cc}\sqrt{{\displaystyle \frac{w_{jr}d(L_{ij}(p_{ij}),L_{kj}(p_{kj}))}{{\sum }_{j^{\prime }}{w}_{j^{\prime }r}}}}\hfill & E(L_{ij})>E(L_{kj})\hfill \\ 0\hfill & E(L_{ij})=E(L_{kj})\hfill \\ -{\displaystyle \frac{1}{\theta }}\sqrt{{\displaystyle \frac{({\sum }_{j^{\prime }}{w}_{j^{\prime }r})d(L_{ij}(p_{ij}),L_{kj}(p_{kj}))}{w_{jr}}}}\hfill & E(L_{ij})<E(L_{kj})\hfill \end{array}$$
   LaTeX

   \Phi_{j}(A_{i},A_{k}) = \begin{cases} \sqrt{\dfrac{w_{jr}\, d(L_{ij}(p_{ij}), L_{kj}(p_{kj}))}{\sum_{j'} w_{j'r}}} & E(L_{ij}) > E(L_{kj})\\ 0 & E(L_{ij}) = E(L_{kj})\\ -\dfrac{1}{\theta}\sqrt{\dfrac{(\sum_{j'} w_{j'r})\, d(L_{ij}(p_{ij}), L_{kj}(p_{kj}))}{w_{jr}}} & E(L_{ij}) < E(L_{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

Probabilistic Linguistic outranking/ranking - Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}). Output typically utility (higher value = preferred).

Sonucu okuma: pl-todim extends TODIM to handle Probabilistic Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}) algebra. The final scores are defuzzified via expected linguistic value E = Σ p_k · index(L_k) before ranking.

Varsayımlar

• Decision matrix entries are valid Probabilistic Linguistic 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 Probabilistic Linguistic 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 PLTS: {L_k|p_k} where L_k is linguistic label, Σ p_k ≤ 1 koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: expected linguistic value E = Σ p_k · index(L_k) kanonik seçimdir.