MULTIMOORA - Oran Analizine ve Tam Çarpımsal Forma Dayalı Çok Amaçlı Optimizasyon

Formül adımları

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

1. Adım 1 — Vektör normalleştirme r_ij.

   $$r_{ij}={\displaystyle \frac{x_{ij}}{\sqrt{{\sum }_{k=1}^m{x}_{kj}^2}}}$$
   LaTeX

   r_{ij} = \dfrac{x_{ij}}{\sqrt{\sum_{k=1}^{m} x_{kj}^{2}}}

2. Adım 2 — Oran Sistemi y_i.

   $$y_i=\sum _{j\in J^{+}}{w}_j{r}_{ij}-\sum _{j\in J^{-}}{w}_j{r}_{ij}$$
   LaTeX

   y_{i} = \sum_{j\in J^{+}} w_{j} r_{ij} - \sum_{j\in J^{-}} w_{j} r_{ij}

3. Adım 3 — Referans Nokta z_i.

   $$z_i={max}_j|w_j{r}_j^{*}-w_j{r}_{ij}|,r_j^{*}={max}_i{r}_{ij}(J^{+})\text{or}{min}_i{r}_{ij}(J^{-})$$
   LaTeX

   z_{i} = \max_{j} \Big|w_{j} r^{*}_{j} - w_{j} r_{ij}\Big|,\quad r^{*}_{j}=\max_{i} r_{ij}\ (J^{+})\ \text{or}\ \min_{i} r_{ij}\ (J^{-})

4. Adım 4 — Tam Çarpımsal Biçim U_i.

   $$U_i={\displaystyle \frac{{\prod }_{j\in J^{+}}(x_{ij}{)}^{w_j}}{{\prod }_{j\in J^{-}}(x_{ij}{)}^{w_j}}}$$
   LaTeX

   U_{i} = \dfrac{\prod_{j\in J^{+}} (x_{ij})^{w_{j}}}{\prod_{j\in J^{-}} (x_{ij})^{w_{j}}}

5. Adım 5 — Üç sıralamanın egemenlik teorisi toplaması.

   $$\text{Final rank}=\text{dominance}(\text{rank}(y_i),\text{rank}(z_i),\text{rank}(U_i))$$
   LaTeX

   \text{Final rank} = \text{dominance}(\text{rank}(y_{i}),\,\text{rank}(z_{i}),\,\text{rank}(U_{i}))

Sezgi

Dominance aggregation of three sub-rankings (RS + RP + FMF). Output typically utility (higher value = preferred).

Sonucu okuma: MULTIMOORA produces three sub-rankings (RS: higher y better; RP: lower d better; FMF: higher U better) and aggregates them via dominance theory - the alternative dominating in ≥2 sub-rankings wins its position. When all three sub-rankings agree, the result is robust. When they disagree, inspect each sub-ranking individually to understand the trade-off.

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

• TÇF tüm x_ij > 0 gerektirir - sıfır veya negatif ham değerler tam çarpımsal formu geçersiz kılar.