DNMA - Çift Normalleştirme Tabanlı Çoklu Toplama

Formül adımları

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

Adım 1 — Üç normalleştirme.

$r_{i j}^{(1)} = x_{i j} / {max}_{i} x_{i j}; r_{i j}^{(2)} = x_{i j} / \sqrt{\sum_{i} x_{i j}^{2}}; r_{i j}^{(3)} = (x_{i j} - {min}_{i} x_{i j}) / ({max}_{i} x_{i j} - {min}_{i} x_{i j}), direction-adjusted$

LaTeX
r^{(1)}_{ij}=x_{ij}/\max_i x_{ij};\quad r^{(2)}_{ij}=x_{ij}/\sqrt{\sum_i x_{ij}^{2}};\quad r^{(3)}_{ij}=(x_{ij}-\min_i x_{ij})/(\max_i x_{ij}-\min_i x_{ij})\text{, direction-adjusted}
Adım 2 — Üç alt-fayda hesaplama.

$u_{i}^{(1)} = \sum_{j} w_{j} r_{i j}^{(1)}, u_{i}^{(2)} = \prod_{j} (r_{i j}^{(2)})^{w_{j}}, u_{i}^{(3)} = d_{i}^{-} / (d_{i}^{+} + d_{i}^{-})$

LaTeX
u^{(1)}_{i}=\sum_j w_jr^{(1)}_{ij},\quad u^{(2)}_{i}=\prod_j(r^{(2)}_{ij})^{w_j},\quad u^{(3)}_{i}=d_i^-/(d_i^++d_i^-)
Adım 3 — Üç sıralama ve sapma ölçüsü.

$R_{i}^{(k)} = rank (u_{i}^{(k)}); S_{i}^{(k)} = u_{i}^{(k)} - {min}_{l} u_{l}^{(k)}$

LaTeX
R^{(k)}_{i}=\text{rank}(u^{(k)}_{i});\ S^{(k)}_{i}=u^{(k)}_{i}-\min_{l} u^{(k)}_{l}
Adım 4 — Birleştirilmiş DNMA skoru.

$D_{i} = \sum_{k = 1}^{3} α_{k} (u_{i}^{(k)})^{2} - \sum_{k = 1}^{3} β_{k} (R_{i}^{(k)})^{2}$

LaTeX
D_{i} = \sum_{k=1}^{3} \alpha_{k}\,(u^{(k)}_{i})^{2} - \sum_{k=1}^{3} \beta_{k}\,(R^{(k)}_{i})^{2}

Sezgi

Dual-normalisation aggregation (linear + vector). Output typically utility (higher value = preferred).

Sonucu okuma: U_i ∈ [0,1] approximately. Higher U means better. DNMA combines two normalisation paradigms - linear (min-max) and vector (Euclidean) - to reduce the influence of any single normalisation method on the ranking. λ=0.5 gives equal weight to both. λ=1 reduces to a min-max SAW; λ=0 gives a vector-normalised SAW.

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: doğrusal normalleştirme paydası sıfırdır.

Hesap adımları ve dayanakları

Three normalisations: linear-max, vector, and target-based.

Dayanak: Liao-Wu 2020, p.6 Eqs.(1)-(3)
Three sub-utilities: complete (WSM), incomplete (WPM), and TOPSIS-like.

Dayanak: Liao-Wu 2020, p.7 Eqs.(4)-(6)
Compute three ranks R^{(k)}_i; deviation measure considers rank spread.

Dayanak: Liao-Wu 2020, p.8 Eq.(7)
Combined DNMA score D_i = Σ α_k (u^{(k)}_i)^2 − Σ β_k (R^{(k)}_i)^2; descending ranking.

Dayanak: Liao-Wu 2020, p.8 Eq.(8)