BN-ELECTRE-I: Bipolar Neutrosophic ELECTRE-I

Overview

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

Strengths

• •Method-specific: Bipolar neutrosophic outranking: concordance/discordance dominance
• •Preserves bipolar_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

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

Assumptions to verify

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

When not to use

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

Edge cases

• •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.

Common pitfalls

• •Hatalı: 'BN-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision maker can provide bipolar neutrosophic assessments
• •Hatalı: 'BN-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: A partial order output (shortlist) is acceptable
• •Hatalı: BN-ELECTRE-I'yi 'A complete total ranking is required' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BN-ELECTRE-I'yi 'Number of alternatives > 15 (pairwise comparison becomes unwieldy)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): 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. Anchor: Akram et al. 2018, §4, Step (i-iii) referencing §2
2. 2.Adım 2 (F2): 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}. Formül: \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}\} Anchor: Akram et al. 2018, §4, Step (iv)
3. 3.Adım 3 (F3): 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 (-). Formül: e_{xy} = \sum_{j \in E_{xy}} w_j Anchor: Akram et al. 2018, §4, Step (v)
4. 4.Adım 4 (F4): 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). Formül: 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})} Anchor: Akram et al. 2018, §4, Step (vi)
5. 5.Adım 5 (F5): 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). Formül: \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} Anchor: Akram et al. 2018, §4, Step (vii)
6. 6.Adım 6 (F6): 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. Formül: \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} Anchor: Akram et al. 2018, §4, Steps (viii)-(x)
7. 7.Adım 7 (F7): 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. Formül: \pi_{xy} = 1 \Rightarrow S_x \succ S_y;\quad \text{kernel} = \{S_x \mid \nexists S_y : \pi_{yx} = 1\} Anchor: Akram et al. 2018, §4, Step (xi)

Commonly paired with

• • (high)

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