BN-TOPSIS: Bipolar Neutrosophic TOPSIS

Overview

Bipolar neutrosophic extension: revised closeness degree via inferior ratio. Output typically rank (lower value = preferred).

Strengths

• •Method-specific: Bipolar neutrosophic extension: revised closeness degree via inferior ratio
• •Preserves bipolar_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from TOPSIS family; increasing alternatives may cause rank reversals (noted in Akram et al. 2018 §5).)
• •Assumes: Decision maker can provide bipolar neutrosophic assessments (6 components per cell)
• •Assumes: All T+,I+,F+ ∈ [0,1] and T-,I-,F- ∈ [-1,0]
• •Assumes: Criteria are mutually preferentially independent

Assumptions to verify

• •Decision maker can provide bipolar neutrosophic assessments (6 components per cell)
• •All T+,I+,F+ ∈ [0,1] and T-,I-,F- ∈ [-1,0]
• •Criteria are mutually preferentially independent

When not to use

• •Decision maker cannot provide bipolar (positive + negative) membership degrees
• •Crisp or simple fuzzy data: use TOPSIS or N-TOPSIS instead

Edge cases

• •If weights are unknown, derive them via the maximizing deviation method. The normalised weight w*_j of criterion T_j is: w*_j = Σ_i Σ_l |k_ij - k_lj| / Σ_j (Σ_i Σ_l |k_ij - k_lj|). Here |k_ij - k_lj|
• •zero denominators.

Common pitfalls

• •Hatalı: 'BN-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision maker can provide bipolar neutrosophic assessments (6 components per cell)
• •Hatalı: 'BN-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All T+,I+,F+ ∈ [0,1] and T-,I-,F- ∈ [-1,0]
• •Hatalı: 'BN-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria are mutually preferentially independent
• •Hatalı: BN-TOPSIS'yi 'Decision maker cannot provide bipolar (positive + negative) membership degrees' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BN-TOPSIS'yi 'Crisp or simple fuzzy data' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct the bipolar neutrosophic decision matrix K = [k_ij]_{m×n}. Each entry k_ij = <T+_ij, I+_ij, F+_ij, T-_ij, I-_ij, F-_ij> is a BNS rating of alternative S_i on criterion T_j. Formül: K = [k_{ij}]_{m \times n},\quad k_{ij} = \langle T^+_{ij}, I^+_{ij}, F^+_{ij}, T^-_{ij}, I^-_{ij}, F^-_{ij} \rangle Anchor: Akram et al. 2018, §2, Step (i)
2. 2.Adım 2 (F2): Step 2: If weights are unknown, derive them via the maximizing deviation method. The normalised weight w*_j of criterion T_j is: w*_j = Σ_i Σ_l |k_ij - k_lj| / Σ_j (Σ_i Σ_l |k_ij - k_lj|). Here |k_ij - k_lj| is the Euclidean-type BNS distance. If weights are given, skip this step. Formül: w^*_j = \dfrac{\sum_{i=1}^{m}\sum_{l=1}^{m}|k_{ij}-k_{lj}|}{\sum_{j=1}^{n}\sum_{i=1}^{m}\sum_{l=1}^{m}|k_{ij}-k_{lj}|} Anchor: Akram et al. 2018, §2, Step (ii); Yang 1998
3. 3.Adım 3 (F3): Step 3: Build the weighted BN decision matrix K⊗W = [k^{w_j}_{ij}] by raising each BNS entry to the power w_j using BNS scalar multiplication. The weighted BNN is: k^{w_j}_{ij} = <1-(1-T+)^{w_j}, (I+)^{w_j}, (F+)^{w_j}, -((-T-)^{w_j}), -((-I-)^{w_j}), -(1-(1-(-F-))^{w_j})>. Formül: 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 Anchor: Akram et al. 2018, §2, Step (iii)
4. 4.Adım 4 (F4): Step 4: Determine BNRPIS (+) and BNRNIS (-) for each criterion. For benefit criteria (max): BNRPIS_j = <max T+, min I+, min F+, min T-, max I-, max F->; BNRNIS_j = <min T+, max I+, max F+, max T-, min I-, min F->. Directions reversed for cost criteria (min). Formül: \text{Benefit: } {^+}T^{w_j+}_j = \max_i T^{w_j+}_{ij},\; {^+}I^{w_j+}_j = \min_i I^{w_j+}_{ij},\; \ldots;\quad \text{Cost: directions reversed} Anchor: Akram et al. 2018, §2, Step (iv)
5. 5.Adım 5 (F5): Step 5: Compute the normalised Euclidean distance of each alternative from BNRPIS and BNRNIS. Distance is averaged over 6 components × n criteria: d_N(S_i, BNRPIS) = sqrt(1/(6n) × Σ_j [(T+−+T+)² + (I+−+I+)² + (F+−+F+)² + (T-−+T-)² + (I-−+I-)² + (F-−+F-)²]). Formül: d_N(S_i, \text{BNRPIS}) = \sqrt{\dfrac{1}{6n}\sum_{j=1}^{n}\left[(T^{w_j+}_{ij}-{^+}T^{w_j+}_j)^2 + (I^{w_j+}_{ij}-{^+}I^{w_j+}_j)^2 + (F^{w_j+}_{ij}-{^+}F^{w_j+}_j)^2 + (T^{w_j-}_{ij}-{^+}T^{w_j-}_j)^2 + (I^{w_j-}_{ij}-{^+}I^{w_j-}_j)^2 + (F^{w_j-}_{ij}-{^+}F^{w_j-}_j)^2\right]} Anchor: Akram et al. 2018, §2, Step (v)
6. 6.Adım 6 (F6): Step 6: Compute the revised closeness degree ρ(S_i) = d_N(S_i, BNRNIS)/max{d_N(S_i, BNRNIS)} − d_N(S_i, BNRPIS)/min{d_N(S_i, BNRPIS)}. This is a revision of the classical TOPSIS closeness coefficient; it avoids the issue of zero denominators. Formül: \rho(S_i) = \dfrac{d_N(S_i, \text{BNRNIS})}{\max_i\{d_N(S_i, \text{BNRNIS})\}} - \dfrac{d_N(S_i, \text{BNRPIS})}{\min_i\{d_N(S_i, \text{BNRPIS})\}} Anchor: Akram et al. 2018, §2, Step (vi)
7. 7.Adım 7 (F7): Step 7: Compute the inferior ratio IR(i) = ρ(S_i) / min_{1≤i≤m} ρ(S_i). Each IR(i) ∈ [0,1]. Step 8: Rank alternatives in ascending order of IR; the alternative with minimum IR (= 0) is the best. Formül: IR(i) = \dfrac{\rho(S_i)}{\min_{1 \le i \le m}\rho(S_i)},\quad IR(i) \in [0,1];\quad \text{rank ascending by } IR(i) Anchor: Akram et al. 2018, §2, Steps (vii)-(viii)

Commonly paired with

• • (high)

How to cite