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TOPSIS family

BN-TOPSIS - Bipolar Neutrosophic TOPSIS

Bipolar neutrosophic extension - revised closeness degree via inferior ratio

Akram, M., Shumaiza, Smarandache, F.2018doi:10.3390/axioms7020033 ↗

Overview

BN-TOPSIS ranks alternatives in ascending order of the inferior ratio IR(i). The alternative with IR=0 (or minimum IR) is the best. The method uses a revised closeness degree instead of the classical TOPSIS coefficient to avoid division-by-zero when an alternative coincides with an ideal solution. BN-TOPSIS can handle bipolar information - each criterion is evaluated both positively (degree of satisfaction) and negatively (degree of dissatisfaction with respect to an implicit counter-property).

Output: rank, lower is better
Data: Bipolar Neutrosophic, bipolar neutrosophic ratings
Size: 2+ alternatives, 3-8 criteria works best
Used for: Supplier selection, HR / employee selection, e-commerce platform selection, medical decision making

How it works

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.
$K = [k_{ij}]_{m \times n}, k_{ij} = ⟨ T_{ij}^{+}, I_{ij}^{+}, F_{ij}^{+}, T_{ij}^{-}, I_{ij}^{-}, F_{ij}^{-} ⟩$
Akram et al. 2018, §2, Step (i)
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.
$w_{j}^{*} = \frac{\sum _{i = 1}^{m} \sum _{l = 1}^{m} ∣ k _{ij} - k _{l j} ∣}{\sum _{j = 1}^{n} \sum _{i = 1}^{m} \sum _{l = 1}^{m} ∣ k _{ij} - k _{l j} ∣}$
Akram et al. 2018, §2, Step (ii); Yang 1998
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})>.
$k_{ij}^{w_{j}} = ⟨ 1 - (1 - T_{ij}^{+})^{w_{j}}, (I_{ij}^{+})^{w_{j}}, (F_{ij}^{+})^{w_{j}}, - ((- T_{ij}^{-})^{w_{j}}), - ((- I_{ij}^{-})^{w_{j}}), - (1 - (1 - (- F_{ij}^{-}))^{w_{j}})⟩$
Akram et al. 2018, §2, Step (iii)
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).
$Benefit:^{+} T_{j}^{w_{j} +} = i max T_{ij}^{w_{j} +},^{+} I_{j}^{w_{j} +} = i min I_{ij}^{w_{j} +}, \dots; Cost: directions reversed$
Akram et al. 2018, §2, Step (iv)
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]).
$d_{N} (S_{i}, BNRPIS) = \frac{1}{6 n} j = 1 \sum n [(T_{ij}^{w_{j} +} -^{+} T_{j}^{w_{j} +})^{2} + (I_{ij}^{w_{j} +} -^{+} I_{j}^{w_{j} +})^{2} + (F_{ij}^{w_{j} +} -^{+} F_{j}^{w_{j} +})^{2} + (T_{ij}^{w_{j} -} -^{+} T_{j}^{w_{j} -})^{2} + (I_{ij}^{w_{j} -} -^{+} I_{j}^{w_{j} -})^{2} + (F_{ij}^{w_{j} -} -^{+} F_{j}^{w_{j} -})^{2}]$
Akram et al. 2018, §2, Step (v)
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.
$ρ (S_{i}) = \frac{d _{N} ( S _{i} , BNRNIS )}{max _{i} { d _{N} ( S _{i} , BNRNIS )}} - \frac{d _{N} ( S _{i} , BNRPIS )}{min _{i} { d _{N} ( S _{i} , BNRPIS )}}$
Akram et al. 2018, §2, Step (vi)
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.
$I R (i) = \frac{ρ ( S _{i} )}{min _{1 \leq i \leq m} ρ ( S _{i} )}, I R (i) \in [0, 1]; rank ascending by I R (i)$
Akram et al. 2018, §2, Steps (vii)-(viii)

Fits when / Look elsewhere when

Fits when

•Preserves bipolar_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation

Look elsewhere when

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

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

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).)

Edge cases and pitfalls

•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.

The minimum of ρ(S_i) is always ≤ 0 for non-trivial cases (by construction). If min ρ = 0, IR(i) is undefined - apply a small perturbation or use ρ directly for ranking.
Maximizing deviation weights assume all criteria are informative. Constant criterion → w_j = 0 → criterion excluded from analysis.
The factor 1/(6n) in the distance formula normalises over 6 BNS components × n criteria. Do not confuse with 1/(2n) used in interval-valued or single-valued NS distances.

How to cite

Akram, M.; Shumaiza; Smarandache, F. (2018). Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I. Axioms. https://doi.org/10.3390/axioms7020033

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

BN-TOPSIS

Overview

Output

rank, lower is better

Data

Bipolar Neutrosophic, bipolar neutrosophic ratings

Size

2+ alternatives, 3-8 criteria works best

Used for

Supplier selection, HR / employee selection, e-commerce platform selection, medical decision making

How it works

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.

K = [k_{ij}]_{m \times n}, k_{ij} = ⟨ T_{ij}^{+}, I_{ij}^{+}, F_{ij}^{+}, T_{ij}^{-}, I_{ij}^{-}, F_{ij}^{-} ⟩

Akram et al. 2018, §2, Step (i)

w_{j}^{*} = \frac{\sum _{i = 1}^{m} \sum _{l = 1}^{m} ∣ k _{ij} - k _{l j} ∣}{\sum _{j = 1}^{n} \sum _{i = 1}^{m} \sum _{l = 1}^{m} ∣ k _{ij} - k _{l j} ∣}

Akram et al. 2018, §2, Step (ii); Yang 1998

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})>.

k_{ij}^{w_{j}} = ⟨ 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}})⟩

Akram et al. 2018, §2, Step (iii)

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).

Benefit:^{+} T_{j}^{w_{j} +} = i max T_{ij}^{w_{j} +},^{+} I_{j}^{w_{j} +} = i min I_{ij}^{w_{j} +}, \dots; Cost: directions reversed

Akram et al. 2018, §2, Step (iv)

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]).

d_{N} (S_{i}, BNRPIS) = \frac{1}{6 n} j = 1 \sum n [(T_{ij}^{w_{j} +} -^{+} T_{j}^{w_{j} +})^{2} + (I_{ij}^{w_{j} +} -^{+} I_{j}^{w_{j} +})^{2} + (F_{ij}^{w_{j} +} -^{+} F_{j}^{w_{j} +})^{2} + (T_{ij}^{w_{j} -} -^{+} T_{j}^{w_{j} -})^{2} + (I_{ij}^{w_{j} -} -^{+} I_{j}^{w_{j} -})^{2} + (F_{ij}^{w_{j} -} -^{+} F_{j}^{w_{j} -})^{2}]

Akram et al. 2018, §2, Step (v)

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.

ρ (S_{i}) = \frac{d _{N} ( S _{i} , BNRNIS )}{max _{i} { d _{N} ( S _{i} , BNRNIS )}} - \frac{d _{N} ( S _{i} , BNRPIS )}{min _{i} { d _{N} ( S _{i} , BNRPIS )}}

Akram et al. 2018, §2, Step (vi)

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.

I R (i) = \frac{ρ ( S _{i} )}{min _{1 \leq i \leq m} ρ ( S _{i} )}, I R (i) \in [0, 1]; rank ascending by I R (i)

Akram et al. 2018, §2, Steps (vii)-(viii)

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

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).)

Edge cases and pitfalls

•zero denominators.

The minimum of ρ(S_i) is always ≤ 0 for non-trivial cases (by construction). If min ρ = 0, IR(i) is undefined - apply a small perturbation or use ρ directly for ranking.

Maximizing deviation weights assume all criteria are informative. Constant criterion → w_j = 0 → criterion excluded from analysis.

The factor 1/(6n) in the distance formula normalises over 6 BNS components × n criteria. Do not confuse with 1/(2n) used in interval-valued or single-valued NS distances.