PL-MARCOS - MARCOS 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 yön kümeleri J^+/J^-.

   $$L_{ij}(p_{ij})=\{L_{ij}^{(k)}(p_{ij}^{(k)})\},\sum _k{p}_{ij}^{(k)}=1;J^{+}\cup J^{-}=\{1,\dots ,n\}$$
   LaTeX

   L_{ij}(p_{ij}) = \{L_{ij}^{(k)}(p_{ij}^{(k)})\},\ \sum_{k} p_{ij}^{(k)} = 1;\ \ J^{+}\cup J^{-}=\{1,\dots,n\}

2. Adım 2 — E(L(p))=Σ_k r(L^(k))·p^(k) ile PL anti-ideal L_AI(p) ve ideal L_I(p); fayda için L_I=arg max_i, L_AI=arg min_i; matrise satır olarak eklenir.

   $$E(L(p))=\sum _{k}r(L^{(k)})·p^{(k)};L_{AI,j}=\mathrm{\backslash arg}min/{max}_{i}E(L_{ij}),L_{I,j}=\mathrm{\backslash arg}max/{min}_{i}E(L_{ij})$$
   LaTeX

   E(L(p)) = \sum_{k} r(L^{(k)})\cdot p^{(k)};\ \ L_{AI,j} = \arg\min/\max_{i} E(L_{ij}),\ L_{I,j} = \arg\max/\min_{i} E(L_{ij})

3. Adım 3 — Olasılıksal dilsel ağırlıklı toplam ile ağırlıklı PLTS matrisi v_ij(p).

   $$v_{ij}(p)=w_j\otimes L_{ij}(p_{ij})$$
   LaTeX

   v_{ij}(p) = w_{j}\otimes L_{ij}(p_{ij})

4. Adım 4 — Alternatif başına ağırlıklı toplam S_i = Σ_j E(v_ij(p)); S_AI ve S_I.

   $$S_i=\sum _{j=1}^{n}E(v_{ij}(p));S_{AI}=\sum _{j}E(v_{AI,j}(p)),S_I=\sum _{j}E(v_{I,j}(p))$$
   LaTeX

   S_{i} = \sum_{j=1}^{n} E(v_{ij}(p));\ \ S_{AI} = \sum_{j} E(v_{AI,j}(p)),\ S_{I} = \sum_{j} E(v_{I,j}(p))

5. Adım 5 — Fayda dereceleri K_i^- = S_i/S_AI ve K_i^+ = S_i/S_I.

   $$K_i^{-}={\displaystyle \frac{S_i}{S_{AI}}},K_i^{+}={\displaystyle \frac{S_i}{S_I}}$$
   LaTeX

   K_{i}^{-} = \dfrac{S_{i}}{S_{AI}},\quad K_{i}^{+} = \dfrac{S_{i}}{S_{I}}

6. Adım 6 — Fayda fonksiyonları f(K_i^+) ve f(K_i^-).

   $$f(K_i^{+})={\displaystyle \frac{K_i^{-}}{K_i^{+}+K_i^{-}}},f(K_i^{-})={\displaystyle \frac{K_i^{+}}{K_i^{+}+K_i^{-}}}$$
   LaTeX

   f(K_{i}^{+}) = \dfrac{K_{i}^{-}}{K_{i}^{+}+K_{i}^{-}},\quad f(K_{i}^{-}) = \dfrac{K_{i}^{+}}{K_{i}^{+}+K_{i}^{-}}

7. Adım 7 — Nihai fayda fonksiyonu f(K_i) ve azalan sıralama.

   $$f(K_i)={\displaystyle \frac{K_i^{+}+K_i^{-}}{1+{\displaystyle \frac{1-f(K_i^{+})}{f(K_i^{+})}}+{\displaystyle \frac{1-f(K_i^{-})}{f(K_i^{-})}}}};\text{rank}={\text{argsort}}_{\text{desc}}(f(K_i))$$
   LaTeX

   f(K_{i}) = \dfrac{K_{i}^{+}+K_{i}^{-}}{1+\dfrac{1-f(K_{i}^{+})}{f(K_{i}^{+})}+\dfrac{1-f(K_{i}^{-})}{f(K_{i}^{-})}};\ \ \text{rank} = \text{argsort}_{\text{desc}}(f(K_{i}))

Sezgi

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

Sonucu okuma: pl-marcos extends MARCOS 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 MARCOS 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 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.