LMAW - Katkılı Ağırlıkların Logaritma Metodolojisi

Formül adımları

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

1. Adım 1 — Yön farkındalıklı standartlaştırma ϑ_ij. Fayda: ϑ_ij = (x_ij + max_i x_ij)/max_i x_ij. Maliyet: ϑ_ij = (x_ij + min_i x_ij)/x_ij.

   $${\vartheta }_{ij}=\{\begin{array}{cc}{\displaystyle \frac{x_{ij}+{max}_i{x}_{ij}}{{max}_i{x}_{ij}}},\hfill & j\in J^{+}(\text{benefit})\hfill \\ [\mathrm{6pt}]{\displaystyle \frac{x_{ij}+{min}_i{x}_{ij}}{x_{ij}}},\hfill & j\in J^{-}(\text{cost})\hfill \end{array}$$
   LaTeX

   \vartheta_{ij} = \begin{cases} \dfrac{x_{ij} + \max_{i} x_{ij}}{\max_{i} x_{ij}}, & j \in J^{+}\ (\text{benefit}) \\[6pt] \dfrac{x_{ij} + \min_{i} x_{ij}}{x_{ij}}, & j \in J^{-}\ (\text{cost}) \end{cases}

2. Adım 2 — Logaritmik dönüşüm ϕ_ij = ln(ϑ_ij)/ln(Π_i ϑ_ij).

   $${\varphi }_{ij}={\displaystyle \frac{\ln ({\vartheta }_{ij})}{\ln \left({\prod }_{i=1}^m{\vartheta }_{ij}\right)}}$$
   LaTeX

   \phi_{ij} = \dfrac{\ln(\vartheta_{ij})}{\ln\big(\prod_{i=1}^{m} \vartheta_{ij}\big)}

3. Adım 3 — Sigmoid biçimli ağırlıklı toplulaştırma ξ_ij = 2·ϕ_ij^{w_j} / [(2−ϕ_ij)^{w_j} + ϕ_ij^{w_j}].

   $${\xi }_{ij}={\displaystyle \frac{2{\varphi }_{ij}^{w_j}}{(2-{\varphi }_{ij}{)}^{w_j}+{\varphi }_{ij}^{w_j}}}$$
   LaTeX

   \xi_{ij} = \dfrac{2\,\phi_{ij}^{w_{j}}}{(2-\phi_{ij})^{w_{j}} + \phi_{ij}^{w_{j}}}

4. Adım 4 — Toplulaştırma Q_i = Σ_j ξ_ij; azalan sıralama (yüksek Q = iyi).

   $$Q_i=\sum _{j=1}^n{\xi }_{ij}$$
   LaTeX

   Q_{i} = \sum_{j=1}^{n} \xi_{ij}

Sezgi

Logarithm-based additive weighting. Output typically utility (higher value = preferred).

Sonucu okuma: Q_i ∈ [0, n] where n = number of criteria. Higher Q means better alternative. Q_i is the sum over criteria of the sigmoid-weighted ξ_ij ∈ (0, 2). Direction is handled in F1 via Pamucar 2021 Eq.(2): benefit standardisation (x+max)/max, cost standardisation (x+min)/x - both produce ϑ_ij > 1 with best alternative receiving largest ϑ. F1 → F2 (log-norm) → F3 (sigmoid weighting) → F4 (sum) is the canonical pipeline.

Varsayımlar

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Ne zaman kullanılmaz

• Criteria strongly correlated → consider DEMATEL/ANP for interdependence
• Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Sınırlılıklar

• Assumes: Criteria preferences are independent (no synergistic interactions)
• Assumes: Compensation is acceptable: high score on one criterion can offset low on another
• Assumes: Decision matrix is complete (no missing values)

Sık yapılan hatalar

• Sıfır veya negatif değerler: F2 logaritması ϑ_ij > 0 gerektirir; E-2, x_ij > 0 zorunlu kılar.
• F1 yön işleme zorunludur - F1 olmadan algoritma fayda ve maliyet kriterlerini ayırt edemez.