BN-ELECTRE-I - Bipolar Neutrosophic ELECTRE-I

Overview

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.

Output

partial order

Data

Bipolar Neutrosophic, bipolar neutrosophic ratings

Size

2+ alternatives, 3-8 criteria works best

Used for

Supplier selection, site selection, HR selection

How it works

1. 1

   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.

   Akram et al. 2018, §4, Step (i-iii) referencing §2

2. 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}.

   $${\rho }_{ij}=T_{ij}^{+}+I_{ij}^{+}+F_{ij}^{+}+T_{ij}^{-}+I_{ij}^{-}+F_{ij}^{-};E_{xy}=\{j\mid {\rho }_{xj}\ge {\rho }_{yj}\},F_{xy}=\{j\mid {\rho }_{xj}\le {\rho }_{yj}\}$$

   Akram et al. 2018, §4, Step (iv)

3. 3

   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 (-).

   $$e_{xy}=\sum _{j\in E_{xy}}{w}_j$$

   Akram et al. 2018, §4, Step (v)

4. 4

   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).

   $$f_{xy}=\frac{{\max }_{j\in F_{xy}}{d}_N(k_{xj}^{w_j},k_{yj}^{w_j})}{{\max }_j{d}_N(k_{xj}^{w_j},k_{yj}^{w_j})}$$

   Akram et al. 2018, §4, Step (vi)

5. 5

   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).

   $$\hat{e}=\frac{1}{m(m-1)}\sum _{x\ne y}{e}_{xy},\hat{f}=\frac{1}{m(m-1)}\sum _{x\ne y}{f}_{xy}$$

   Akram et al. 2018, §4, Step (vii)

6. 6

   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.

   $${\varphi }_{xy}=\{\begin{array}{ll}1& e_{xy}\ge \hat{e}\\ 0& \text{otherwise}\end{array},{\psi }_{xy}=\{\begin{array}{ll}1& f_{xy}\le \hat{f}\\ 0& \text{otherwise}\end{array},{\pi }_{xy}={\varphi }_{xy}\cdot {\psi }_{xy}$$

   Akram et al. 2018, §4, Steps (viii)-(x)

7. 7

   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.

   $${\pi }_{xy}=1\Rightarrow S_x\succ S_y;\text{kernel}=\{S_x\mid \nexists S_y:{\pi }_{yx}=1\}$$

   Akram et al. 2018, §4, Step (xi)

Fits when / Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

• •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.

• The BNS score ρ_ij = T+ + I+ + F+ + T- + I- + F- can be zero or negative (since T-,I-,F- ∈ [-1,0]). This is the defining score for concordance set membership - do not confuse with other BNS score functions.

• The discordance formula normalises by max_j d_N(x,y,j) over ALL criteria pairs (not just F_xy). If all distances are zero (identical alternatives), discordance is undefined.

• BN-ELECTRE-I may not produce a complete ranking. If needed, apply successive ELECTRE runs removing the kernel to get a full ranking.

How to cite