PSI - Tercih Seçim İndeksi

Formül adımları

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

1. Adım 1 — Kriter yönüne göre doğrusal normalleştirme.

   $$r_{ij}=\{\begin{array}{cc}{x}_{ij}/{max}_i{x}_{ij}\hfill & j\in J^{+}\hfill \\ {min}_i{x}_{ij}/x_{ij}\hfill & j\in J^{-}\hfill \end{array}$$
   LaTeX

   r_{ij} = \begin{cases} x_{ij}/\max_{i} x_{ij} & j\in J^{+} \\ \min_{i} x_{ij}/x_{ij} & j\in J^{-} \end{cases}

2. Adım 2 — Sütun ortalamaları ve karesel sapmalar.

   $${\overline{r}}_j=\frac{1}{m}\sum _i{r}_{ij},V_j=\sum _{i=1}^m(r_{ij}-{\overline{r}}_j{)}^2$$
   LaTeX

   \bar{r}_{j} = \tfrac{1}{m}\sum_{i} r_{ij},\quad V_{j} = \sum_{i=1}^{m}(r_{ij}-\bar{r}_{j})^{2}

3. Adım 3 — Tercih varyasyonu φ_j ve self-weighted ağırlıklar.

   $${\varphi }_j=1-V_j,w_j={\displaystyle \frac{{\varphi }_j}{{\sum }_{k=1}^n{\varphi }_k}}$$
   LaTeX

   \phi_{j} = 1 - V_{j},\quad w_{j} = \dfrac{\phi_{j}}{\sum_{k=1}^{n} \phi_{k}}

4. Adım 4 — Tercih Seçim İndeksi I_i ve azalan sıralama.

   $$I_i=\sum _{j=1}^n{w}_j{r}_{ij}$$
   LaTeX

   I_{i} = \sum_{j=1}^{n} w_{j}\,r_{ij}

Sezgi

Preference variation index (weight-free statistical). Output typically utility (higher value = preferred).

Sonucu okuma: Higher I means better. PSI derives criterion weights from Preference Variation Values rather than consuming user-specified weights. Because φ_j=1−V_j, larger squared variation V_j produces a smaller weight basis φ_j.

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

• Sabit kriter sütunu: Φ_j = 0 yani Ω_j = 1/n (eşit ağırlık) - bu kabul edilebilir ancak kriterin hiçbir ayrım sağlamadığını gösterir.