Fuzzy SAW (Bonissone 1982) - L-R yamuk Basit Toplamali Agirliklandirma

Formül adımları

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

1. X_tilde karar matrisini ve w_tilde agirlik vektorunu L-R 4-lulu olarak olustur.

   $${\tilde{x}}_{ij}=(a_{ij},b_{ij},{\alpha }_{ij},{\beta }_{ij});{\tilde{w}}_j=(a_j,b_j,{\alpha }_j,{\beta }_j)$$
   LaTeX

   \tilde{x}_{ij} = (a_{ij}, b_{ij}, \alpha_{ij}, \beta_{ij});\quad \tilde{w}_j = (a_j, b_j, \alpha_j, \beta_j)

2. Bonissone L-R carpimi ile w_tilde_j (x) x_tilde_ij agirlikli katkisini hesapla, Eq.(29).

   $$\tilde{M}\otimes \tilde{N}=(ac,bd,a\gamma +c\alpha -\alpha \gamma ,b\delta +d\beta +\beta \delta )$$
   LaTeX

   \tilde{M} \otimes \tilde{N} = (ac,\; bd,\; a\gamma + c\alpha - \alpha\gamma,\; b\delta + d\beta + \beta\delta)

3. Bonissone L-R toplami ile alternatif basina agirlikli katkilari topla, Eq.(27).

   $${\tilde{U}}_i={⨁}_{j=1}^n({\tilde{w}}_j\otimes {\tilde{x}}_{ij});(a+c,b+d,\alpha +\gamma ,\beta +\delta )$$
   LaTeX

   \tilde{U}_i = \bigoplus_{j=1}^{n} (\tilde{w}_j \otimes \tilde{x}_{ij});\quad (a+c,\; b+d,\; \alpha+\gamma,\; \beta+\delta)

4. Her U_tilde_i icin L-R yamuk centroid (sol ucgen + plato + sag ucgen) ile defuzz uygula.

   $$C_i={\displaystyle \frac{({\alpha }_i/2)(a_i-{\alpha }_i/3)+(b_i-a_i)(a_i+b_i)/2+({\beta }_i/2)(b_i+{\beta }_i/3)}{(b_i-a_i)+({\alpha }_i+{\beta }_i)/2}}$$
   LaTeX

   C_i = \dfrac{(\alpha_i/2)(a_i - \alpha_i/3) + (b_i - a_i)\,(a_i+b_i)/2 + (\beta_i/2)(b_i + \beta_i/3)}{(b_i - a_i) + (\alpha_i + \beta_i)/2}

5. Alternatifleri defuzz C_i merkez degerine gore azalan sirala.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(C_i)$$
   LaTeX

   \text{rank}(A_i) = \text{argsort}_{\downarrow}(C_i)

Sezgi

Fuzzy SAW - L-R trapezoidal (a, b, alpha, beta). Output typically utility (higher value = preferred).

Sonucu okuma: Fuzzy SAW (Bonissone 1982 L-R variant) aggregates weighted L-R 4-tuple contributions per alternative, then defuzzifies via trapezoidal centroid. Output is the descending ranking of centroids.

Varsayımlar

• Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• Compensation assumption holds: a strong score on one criterion can offset a weak score on another

Ne zaman kullanılmaz

• Classical data sufficient - use base SAW
• Multiplicative semantics required - use FUZZY-WPM

Sınırlılıklar

• Assumes: Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• Assumes: Compensation assumption holds: a strong score on one criterion can offset a weak score on another

Sık yapılan hatalar

• Bonissone Eq.(29) carpimi alpha/gamma'da degismeli degildir; operand sirasini koru.
• Kitap p.201'de U_1.beta ve U_3.alpha icin minor typolar var; kanonik algoritma 0.60 ve 0.23 verir.