TODIM - Etkileşimli ve Çok Kriterli Karar Verme (Portekizce: Tomada de Decisão Interativa e Multicritério)

Formül adımları

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

1. Adım 1 — Pozitif kriterler için p_ij = x_ij / Σ_k x_kj; negatif kriterler için 1/x_ij'nin normalleştirilmesi.

   $$p_{ij}={\displaystyle \frac{x_{ij}}{{\sum }_{k=1}^m{x}_{kj}}}$$
   LaTeX

   p_{ij} = \dfrac{x_{ij}}{\sum_{k=1}^{m} x_{kj}}

2. Adım 2 — Referans kriter (max ağırlık) ve göreli ağırlıklar w_jr = w_j / w_r.

   $$r=\mathrm{\backslash arg}{max}_j{w}_j,w_{jr}={\displaystyle \frac{w_j}{w_r}}$$
   LaTeX

   r = \arg\max_{j} w_{j},\quad w_{jr} = \dfrac{w_{j}}{w_{r}}

3. Adım 3 — Olası kayıp aversiyonu θ ile ikili egemenlik Φ_j(A_i, A_k) (Prospect Theory tabanlı).

   $${\Phi }_j(A_i,A_k)=\{\begin{array}{cc}\sqrt{{\displaystyle \frac{w_{jr}(p_{ij}-p_{kj})}{\sum w_{kr}}}}\hfill & p_{ij}>p_{kj}\hfill \\ 0\hfill & p_{ij}=p_{kj}\hfill \\ -{\displaystyle \frac{1}{\theta }}\sqrt{{\displaystyle \frac{(\sum w_{kr})(p_{kj}-p_{ij})}{w_{jr}}}}\hfill & p_{ij}<p_{kj}\hfill \end{array}$$
   LaTeX

   \Phi_{j}(A_{i},A_{k}) = \begin{cases} \sqrt{\dfrac{w_{jr}(p_{ij}-p_{kj})}{\sum w_{kr}}} & p_{ij}>p_{kj}\\ 0 & p_{ij}=p_{kj}\\ -\dfrac{1}{\theta}\sqrt{\dfrac{(\sum w_{kr})(p_{kj}-p_{ij})}{w_{jr}}} & p_{ij}<p_{kj}\end{cases}

4. Adım 4 — Toplulan 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})

5. Adım 5 — Global değer ξ_i = (Σ δ − min)/(max − min) ∈ [0,1]; azalan sıralama.

   $${\xi }_i={\displaystyle \frac{{\sum }_k\delta (A_i,A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}{{max}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}}$$
   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})}

Sezgi

Prospect theory (loss aversion pairwise dominance). Output typically utility (higher value = preferred).

Sonucu okuma: ξ_i ∈ [0,1]. Higher ξ means greater overall dominance. TODIM is grounded in Prospect Theory - gains and losses are not symmetric: a loss of magnitude x is felt more intensely than a gain of the same magnitude (controlled by θ). θ=1 means standard (symmetric) case. θ>1 amplifies loss aversion; θ→0 reduces it. The reference criterion r is the one with the highest weight.

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

• θ duyarlılığı: θ değiştikçe sıralamalar tersine dönebilir; her zaman θ üzerinde duyarlılık analizi yapın.
• Maliyet kriteri yönü: kazanç/kayıp formülü maliyet kriterleri için tersine çevrilmelidir (yüksek x_ic daha kötüdür).