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Outranking

N-PROMETHEE - Neutrosophic extension of PROMETHEE

Neutrosophic outranking/ranking - Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3)

Xu, D., Wei, X., Ding, H., Bin, H.2020doi:10.3390/math8101816 ↗

Overview

n-promethee extends PROMETHEE to handle Neutrosophic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3) algebra. The final scores are defuzzified via score function S = (T − F + 1)/2 before ranking.

Output: preference flow, higher is better
Data: Single-Valued Neutrosophic, uncertainty tuples complete
Weights: Needs a weight source
Size: 2+ alternatives, 3-10 criteria works best
Used for: Single-Valued Neutrosophic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1
SVN matrix; cost complement.
$D = (⟨ T_{ij}, I_{ij}, F_{ij} ⟩); cost: apply complement; S (A) = \frac{2 + T - I - F}{3}$
Xu, Wei, Ding, Bin 2020 (doi:10.3390/math8101816), Sec.3
2
Signed neutrosophic difference for each criterion pair.
$d_{j} (i, k) = sign (s (\tilde{a}_{ij}) - s (\tilde{a}_{k j})) \cdot d (\tilde{a}_{ij}, \tilde{a}_{k j}); d (α_{1}, α_{2}) = \frac{( T _{1} - T _{2} ) ^{2} + ( I _{1} - I _{2} ) ^{2} + ( F _{1} - F _{2} ) ^{2}}{3}$
Brans & Vincke 1985, Sec.2; Abdel-Basset et al. 2018, Sec.3 Step2
3
Apply Brans-Vincke preference function (Type I usual or Type V linear).
$P_{j}^{(I)} (i, k) = {01 d_{j} \leq 0 d_{j} > 0; P_{j}^{(V)} (i, k) = ⎩ ⎨ ⎧ 0 d_{j} / p_{j} 1 d_{j} \leq 0 0 < d_{j} \leq p_{j} d_{j} > p_{j}; π (A_{i}, A_{k}) = j \sum w_{j} P_{j} (i, k)$
Brans & Vincke 1985, Sec.3
4
Positive φ⁺ and negative φ⁻ outranking flows.
$ϕ^{+} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{i}, A_{k}); ϕ^{-} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{k}, A_{i})$
Brans & Vincke 1985, Sec.4
5
Net flow PROMETHEE II; rank descending.
$ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i}); rank descending by ϕ (A_{i}) (PROMETHEE II)$
Brans & Vincke 1985, Sec.4 PROMETHEE II

Fits when / Look elsewhere when

Fits when

•Preserves single_valued_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Small dataset (m<3) - outranking machinery underutilised

Assumptions to verify

Thresholds (q, p, v) can be expressed in Single-Valued Neutrosophic scale
Veto + concordance semantics adapted to fuzzy arithmetic

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy SVNS: T,I,F ∈ [0,1]; 0 ≤ T+I+F ≤ 3 before computation.
Defuzzification method affects ranking: score function S = (T − F + 1)/2 is the canonical choice but alternatives exist.

Works with

Commonly takes its weights from

n_a

How to cite

Xu, D.; Wei, X.; Ding, H.; Bin, H. (2020). A New Method Based on PROMETHEE and TODIM for Multi-Attribute Decision-Making with Single-Valued Neutrosophic Sets. Mathematics. https://doi.org/10.3390/math8101816

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

N-PROMETHEE

Overview

Output

preference flow, higher is better

Data

Single-Valued Neutrosophic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Single-Valued Neutrosophic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

SVN matrix; cost complement.

D = (⟨ T_{ij}, I_{ij}, F_{ij} ⟩); cost: apply complement; S (A) = \frac{2 + T - I - F}{3}

Xu, Wei, Ding, Bin 2020 (doi:10.3390/math8101816), Sec.3

Signed neutrosophic difference for each criterion pair.

d_{j} (i, k) = sign (s (\tilde{a}_{ij}) - s (\tilde{a}_{k j})) \cdot d (\tilde{a}_{ij}, \tilde{a}_{k j}); d (α_{1}, α_{2}) = \frac{( T _{1} - T _{2} ) ^{2} + ( I _{1} - I _{2} ) ^{2} + ( F _{1} - F _{2} ) ^{2}}{3}

Brans & Vincke 1985, Sec.2; Abdel-Basset et al. 2018, Sec.3 Step2

Apply Brans-Vincke preference function (Type I usual or Type V linear).

P_{j}^{(I)} (i, k) = {01 d_{j} \leq 0 d_{j} > 0; P_{j}^{(V)} (i, k) = ⎩ ⎨ ⎧ 0 d_{j} / p_{j} 1 d_{j} \leq 0 0 < d_{j} \leq p_{j} d_{j} > p_{j}; π (A_{i}, A_{k}) = j \sum w_{j} P_{j} (i, k)

Brans & Vincke 1985, Sec.3

Positive φ⁺ and negative φ⁻ outranking flows.

ϕ^{+} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{i}, A_{k}); ϕ^{-} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{k}, A_{i})

Brans & Vincke 1985, Sec.4

Net flow PROMETHEE II; rank descending.

ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i}); rank descending by ϕ (A_{i}) (PROMETHEE II)

Brans & Vincke 1985, Sec.4 PROMETHEE II