Bulanık BWM - Üçgen Bulanık En İyi-En Kötü Yöntemi

Formül adımları

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

1. Adım 1 — Uzmandan en iyi (B) ve en kötü (W) kriterleri belirle.

   $$B\in \{c_j\},W\in \{c_j\},B\ne W$$
   LaTeX

   B \in \{c_{j}\},\ W \in \{c_{j}\},\ B\neq W

2. Adım 2 — Dilsel skaladan bulanık BO ve OW vektörleri Ã_B = (ã_B1,…,ã_Bn), Ã_W = (ã_1W,…,ã_nW) oluşturulur.

   $${\tilde{A}}_B=({\tilde{a}}_{B1},\dots ,{\tilde{a}}_{Bn}),{\tilde{A}}_W=({\tilde{a}}_{1W},\dots ,{\tilde{a}}_{nW})$$
   LaTeX

   \tilde{A}_{B} = (\tilde{a}_{B1},\ldots,\tilde{a}_{Bn}),\quad \tilde{A}_{W} = (\tilde{a}_{1W},\ldots,\tilde{a}_{nW})

3. Adım 3 — Bulanık BWM NLP'sini çöz: bulanık ağırlıklar w̃_j ve bulanık tutarlılık ξ̃; min R(ξ̃) kısıtlar altında.

   $$minR(\tilde{\xi })\text{s.t.}|{\displaystyle \frac{{\tilde{w}}_B}{{\tilde{w}}_j}}-{\tilde{a}}_{Bj}|\le \tilde{\xi },|{\displaystyle \frac{{\tilde{w}}_j}{{\tilde{w}}_W}}-{\tilde{a}}_{jW}|\le \tilde{\xi },\sum _{j}R({\tilde{w}}_j)=1,l_j^w\le m_j^w\le u_j^w,l_j^w\ge 0$$
   LaTeX

   \min\ R(\tilde{\xi})\quad \text{s.t.}\quad \left|\dfrac{\tilde{w}_{B}}{\tilde{w}_{j}} - \tilde{a}_{Bj}\right| \le \tilde{\xi},\ \left|\dfrac{\tilde{w}_{j}}{\tilde{w}_{W}} - \tilde{a}_{jW}\right| \le \tilde{\xi},\ \sum_{j} R(\tilde{w}_{j}) = 1,\ l_{j}^{w} \le m_{j}^{w} \le u_{j}^{w},\ l_{j}^{w} \ge 0

4. Adım 4 — w̃_j → w_j: w_j = (l_j^w + 4 m_j^w + u_j^w)/6.

   $$w_j={\displaystyle \frac{l_j^w+4m_j^w+u_j^w}{6}}$$
   LaTeX

   w_{j} = \dfrac{l_{j}^{w} + 4 m_{j}^{w} + u_{j}^{w}}{6}

Sezgi

Triangular-fuzzy Best-to-Others and Others-to-Worst pairwise comparison with nonlinearly-constrained programming. Output typically weight (higher value = preferred).

Sonucu okuma: Fuzzy BWM returns crisp criterion weights (Σ w_j = 1, w_j ≥ 0) and a defuzzified consistency R(ξ̃). It needs only 2n−3 fuzzy pairwise comparisons (the BO and OW vectors). Compared with crisp BWM, Fuzzy BWM lets experts express uncertainty in each ratio using linguistic terms (EI/WI/FI/VI/AI). R(ξ̃) = 0 means perfectly consistent comparisons; large R(ξ̃) indicates inconsistency.

Varsayımlar

• Experts can identify Best and Worst criteria
• Each ã_Bj and ã_jW expressible on the linguistic scale
• TFN shape (l ≤ m ≤ u) preserved

Ne zaman kullanılmaz

• Only crisp ratio available → use crisp BWM
• Probabilistic group disagreement modelling needed → use Bayesian BWM

Sınırlılıklar

• Assumes: Experts can identify Best and Worst criteria
• Assumes: Each ã_Bj and ã_jW expressible on the linguistic scale
• Assumes: TFN shape (l ≤ m ≤ u) preserved

Sık yapılan hatalar

• En iyi ≠ en kötü zorunludur; aynı seçim NLP'yi dejenere yapar.
• n=2: dört oran kısıtı tek anlamlı kısıta indirgenir; NLP kapalı-formla çözülür, ξ̃ = 0.