N-PROMETHEE: Neutrosophic extension of PROMETHEE

Overview

Neutrosophic outranking/ranking: Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3). Output typically preference_flow (higher value = preferred).

Strengths

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

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Thresholds (q, p, v) can be expressed in Single-Valued Neutrosophic scale
• •Assumes: Veto + concordance semantics adapted to fuzzy arithmetic

Assumptions to verify

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

When not to use

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

Edge cases

• •See F.steps and D.parameters for N-PROMETHEE-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'N-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Thresholds (q, p, v) can be expressed in Single-Valued Neutrosophic scale
• •Hatalı: 'N-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Veto + concordance semantics adapted to fuzzy arithmetic
• •Hatalı: N-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): SVN matrix; cost complement. Formül: \mathbf{D}=(\langle T_{ij},I_{ij},F_{ij}\rangle);\quad\text{cost: apply complement };\quad S(A)=\tfrac{2+T-I-F}{3} Anchor: Xu, Wei, Ding, Bin 2020 (doi:10.3390/math8101816), Sec.3
2. 2.Adım 2 (F2): Signed neutrosophic difference for each criterion pair. Formül: d_j(i,k)=\mathrm{sign}(s(\tilde{a}_{ij})-s(\tilde{a}_{kj}))\cdot d(\tilde{a}_{ij},\tilde{a}_{kj});\quad d(\alpha_1,\alpha_2)=\sqrt{\tfrac{(T_1-T_2)^2+(I_1-I_2)^2+(F_1-F_2)^2}{3}} Anchor: Brans & Vincke 1985, Sec.2; Abdel-Basset et al. 2018, Sec.3 Step2
3. 3.Adım 3 (F3): Apply Brans-Vincke preference function (Type I usual or Type V linear). Formül: P_j^{(I)}(i,k)=\begin{cases}0&d_j\leq 0\\1&d_j>0\end{cases};\quad P_j^{(V)}(i,k)=\begin{cases}0&d_j\leq 0\\d_j/p_j&0<d_j\leq p_j\\1&d_j>p_j\end{cases};\quad\pi(A_i,A_k)=\sum_j w_j P_j(i,k) Anchor: Brans & Vincke 1985, Sec.3
4. 4.Adım 4 (F4): Positive φ⁺ and negative φ⁻ outranking flows. Formül: \phi^+(A_i)=\frac{1}{m-1}\sum_{k\neq i}\pi(A_i,A_k);\quad\phi^-(A_i)=\frac{1}{m-1}\sum_{k\neq i}\pi(A_k,A_i) Anchor: Brans & Vincke 1985, Sec.4
5. 5.Adım 5 (F5): Net flow PROMETHEE II; rank descending. Formül: \phi(A_i)=\phi^+(A_i)-\phi^-(A_i);\quad\text{rank descending by }\phi(A_i)\;\text{(PROMETHEE II)} Anchor: Brans & Vincke 1985, Sec.4 PROMETHEE II

Commonly paired with

• •n_a + N-PROMETHEE (common)

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