BN-ELECTRE-I - Bipolar Nötrosofik ELECTRE-I

Formül adımları

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

Adım 1–3 BN-TOPSIS ile aynı: karar matrisi, ağırlıklar, ağırlıklı matris.

$K = [k_{i j}]_{m \times n}, k_{i j} = ⟨ T_{i j}^{+}, I_{i j}^{+}, F_{i j}^{+}, T_{i j}^{-}, I_{i j}^{-}, F_{i j}^{-} ⟩; w_{j}^{*} = \frac{\sum_{i} \sum_{l} | k_{i j} - k_{l j} |}{\sum_{j} \sum_{i} \sum_{l} | k_{i j} - k_{l j} |}; k_{i j}^{w_{j}} = ⟨ 1 - (1 - T_{i j}^{+})^{w_{j}}, (I_{i j}^{+})^{w_{j}}, (F_{i j}^{+})^{w_{j}}, - ((- T_{i j}^{-})^{w_{j}}), - ((- I_{i j}^{-})^{w_{j}}), - (1 - (1 - (- F_{i j}^{-}))^{w_{j}}) ⟩$

LaTeX
K = [k_{ij}]_{m \times n},\; k_{ij} = \langle T^+_{ij}, I^+_{ij}, F^+_{ij}, T^-_{ij}, I^-_{ij}, F^-_{ij} \rangle;\quad w^*_j = \dfrac{\sum_i \sum_l |k_{ij}-k_{lj}|}{\sum_j \sum_i \sum_l |k_{ij}-k_{lj}|};\quad k^{w_j}_{ij} = \langle 1-(1-T^+_{ij})^{w_j},\, (I^+_{ij})^{w_j},\, (F^+_{ij})^{w_j},\, -((-T^-_{ij})^{w_j}),\, -((-I^-_{ij})^{w_j}),\, -(1-(1-(-F^-_{ij}))^{w_j}) \rangle
Adım 4 — Her giriş için BNS skoru ρ_ij hesapla. Her (x,y) çifti için uyuşum seti E_xy ve uyuşmazlık seti F_xy belirle.

$ρ_{i j} = T_{i j}^{+} + I_{i j}^{+} + F_{i j}^{+} + T_{i j}^{-} + I_{i j}^{-} + F_{i j}^{-}; E_{x y} = {j ∣ ρ_{x j} \geq ρ_{y j}}, F_{x y} = {j ∣ ρ_{x j} \leq ρ_{y j}}$

LaTeX
\rho_{ij} = T^+_{ij} + I^+_{ij} + F^+_{ij} + T^-_{ij} + I^-_{ij} + F^-_{ij};\quad E_{xy} = \{j \mid \rho_{xj} \ge \rho_{yj}\},\quad F_{xy} = \{j \mid \rho_{xj} \le \rho_{yj}\}
Adım 5 — m×m uyuşum matrisi E kur. e_xy = x'nin y'yi geçtiği kriterlerin ağırlık toplamı.

$e_{x y} = \sum_{j \in E_{x y}} w_{j}$

LaTeX
e_{xy} = \sum_{j \in E_{xy}} w_j
Adım 6 — m×m uyuşmazlık matrisi F kur. f_xy = F_xy'deki maksimum normalise uzaklık bölü tüm kriterler arası maksimum uzaklık.

$f_{x y} = \frac{{max}_{j \in F_{x y}} d_{N} (k_{x j}^{w_{j}}, k_{y j}^{w_{j}})}{{max}_{j} d_{N} (k_{x j}^{w_{j}}, k_{y j}^{w_{j}})}$

LaTeX
f_{xy} = \dfrac{\max_{j \in F_{xy}} d_N(k^{w_j}_{xj}, k^{w_j}_{yj})}{\max_{j} d_N(k^{w_j}_{xj}, k^{w_j}_{yj})}
Adım 7 — Uyuşum eşiği ê ve uyuşmazlık eşiği f̂ hesapla (tüm çiftlerin ortalaması).

$\hat{e} = \frac{1}{m (m - 1)} \sum_{x \neq y} e_{x y}, \hat{f} = \frac{1}{m (m - 1)} \sum_{x \neq y} f_{x y}$

LaTeX
\hat{e} = \dfrac{1}{m(m-1)} \sum_{x \ne y} e_{xy},\qquad \hat{f} = \dfrac{1}{m(m-1)} \sum_{x \ne y} f_{xy}
Adım 8 — Uyuşum baskınlık matrisi φ, uyuşmazlık baskınlık matrisi ψ ve birleşik baskınlık matrisi π kur.

$ϕ_{x y} = {\begin{matrix} 1 & e_{x y} \geq \hat{e} \\ 0 & otherwise \end{matrix}, ψ_{x y} = {\begin{matrix} 1 & f_{x y} \leq \hat{f} \\ 0 & otherwise \end{matrix}, π_{x y} = ϕ_{x y} \cdot ψ_{x y}$

LaTeX
\phi_{xy} = \begin{cases}1 & e_{xy} \ge \hat{e}\\ 0 & \text{otherwise}\end{cases},\quad \psi_{xy} = \begin{cases}1 & f_{xy} \le \hat{f}\\ 0 & \text{otherwise}\end{cases},\quad \pi_{xy} = \phi_{xy} \cdot \psi_{xy}
Adım 9 — π matrisini yönlü üstünlük grafiği olarak yorumla. π_xy=1 → S_x, S_y'yi üstünler. Baskın olmayan alternatifler nihai seçim setini oluşturur.

$π_{x y} = 1 \Rightarrow S_{x} ≻ S_{y}; kernel = {S_{x} ∣ ∄ S_{y} : π_{y x} = 1}$

LaTeX
\pi_{xy} = 1 \Rightarrow S_x \succ S_y;\quad \text{kernel} = \{S_x \mid \nexists S_y : \pi_{yx} = 1\}

Sezgi

Bipolar neutrosophic outranking - concordance/discordance dominance. Output typically partial_order.

Sonucu okuma: BN-ELECTRE-I produces a partial order (outranking graph), not a total ranking. The output is a set of non-dominated alternatives (kernel). When comparing two alternatives, three outcomes are possible: (a) outranking (π_xy=1), (b) indifference (π_xy=π_yx=1), or (c) incomparability (π_xy=π_yx=0). This is more conservative than BN-TOPSIS which forces a total order. Use BN-ELECTRE-I when you want to identify a shortlist rather than a single winner.

Varsayımlar

Decision maker can provide bipolar neutrosophic assessments
A partial order output (shortlist) is acceptable

Ne zaman kullanılmaz

A complete total ranking is required - use BN-TOPSIS instead
Number of alternatives > 15 (pairwise comparison becomes unwieldy)

Sınırlılıklar

Assumes: Decision maker can provide bipolar neutrosophic assessments
Assumes: A partial order output (shortlist) is acceptable

Sık yapılan hatalar

BNS skoru ρ = T+ + I+ + F+ + T- + I- + F- sıfır veya negatif olabilir. Bu uyuşum seti üyeliğini belirler.
Uyuşmazlık formülü TÜM kriter çiftleri üzerinden normalise eder. Tüm uzaklıklar sıfırsa uyuşmazlık tanımsız.
BN-ELECTRE-I tam sıralama üretmeyebilir. Tam sıralama için çekirdeği çıkararak tekrarlayan ELECTRE çalıştır.

Hesap adımları ve dayanakları

Steps 1-3 are identical to BN-TOPSIS: (i) construct BN decision matrix K; (ii) compute weights by maximizing deviation or use given weights; (iii) build weighted BN decision matrix K⊗W. See BN-TOPSIS.json F1-F3 for formulas.

Dayanak: Akram et al. 2018, §4, Step (i-iii) referencing §2
Step 4 - Compute the BNS score ρ_ij = T+_ij + I+_ij + F+_ij + T-_ij + I-_ij + F-_ij for each entry of the weighted matrix. For each ordered pair (x,y) with x≠y, partition criteria into bipolar neutrosophic concordance set E_xy = {j | ρ_xj ≥ ρ_yj} and discordance set F_xy = {j | ρ_xj ≤ ρ_yj}.

Dayanak: Akram et al. 2018, §4, Step (iv)
Step 5 - Build the m×m bipolar neutrosophic concordance matrix E. The concordance index e_xy = Σ_{j∈E_xy} w_j (sum of weights of criteria on which x outscores y). Diagonal entries are undefined (-).

Dayanak: Akram et al. 2018, §4, Step (v)
Step 6 - Build the m×m bipolar neutrosophic discordance matrix F. The discordance index f_xy = max_{j∈F_xy} d_N(x,y,j) / max_j d_N(x,y,j), where d_N(x,y,j) is the normalised Euclidean distance between the weighted BNS entries of x and y on criterion j (same formula as in BN-TOPSIS: sqrt of 6 squared differences / 6n, here with n=1 per criterion).

Dayanak: Akram et al. 2018, §4, Step (vi)
Step 7 - Compute the concordance level ê = (1 / m(m-1)) × Σ_{x≠y} e_xy (mean of all concordance indices) and the discordance level f̂ = (1 / m(m-1)) × Σ_{x≠y} f_xy (mean of all discordance indices).

Dayanak: Akram et al. 2018, §4, Step (vii)
Step 8 - Build the concordance dominance matrix φ: φ_xy = 1 if e_xy ≥ ê, else 0. Build the discordance dominance matrix ψ: ψ_xy = 1 if f_xy ≤ f̂, else 0. Build the aggregated dominance matrix π: π_xy = φ_xy × ψ_xy.

Dayanak: Akram et al. 2018, §4, Steps (viii)-(x)
Step 9 - Interpret the aggregated dominance matrix π as a directed outranking graph: an arrow from S_x to S_y exists iff π_xy = 1. Three cases: (a) unique arrow S_x→S_y: S_x preferred to S_y; (b) arrows both directions: S_x and S_y indifferent; (c) no arrow: S_x and S_y incomparable. Identify the kernel (non-dominated alternatives) as the final choice set.

Dayanak: Akram et al. 2018, §4, Step (xi)