N-PROMETHEE - PROMETHEE yönteminin Neutrosophic uzantısı

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. SVN matrix; cost complement.

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

   \mathbf{D}=(\langle T_{ij},I_{ij},F_{ij}\rangle);\quad\text{cost: apply complement };\quad S(A)=\tfrac{2+T-I-F}{3}

2. Signed neutrosophic difference for each criterion pair.

   $$d_j(i,k)=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s({\tilde{a}}_{ij})-s({\tilde{a}}_{kj}))·d({\tilde{a}}_{ij},{\tilde{a}}_{kj});d({\alpha }_1,{\alpha }_2)=\sqrt{\frac{(T_1-T_2{)}^2+(I_1-I_2{)}^2+(F_1-F_2{)}^2}{3}}$$
   LaTeX

   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}}

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

   $$P_j^{(I)}(i,k)=\{\begin{array}{cc}0\hfill & d_j\le 0\hfill \\ 1\hfill & d_j>0\hfill \end{array};P_j^{(V)}(i,k)=\{\begin{array}{cc}0\hfill & d_j\le 0\hfill \\ d_j/p_j\hfill & 0<d_j\le p_j\hfill \\ 1\hfill & d_j>p_j\hfill \end{array};\pi (A_i,A_k)=\sum _j{w}_j{P}_j(i,k)$$
   LaTeX

   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)

4. Positive φ⁺ and negative φ⁻ outranking flows.

   $${\varphi }^{+}(A_i)=\frac{1}{m-1}\sum _{k\ne i}\pi (A_i,A_k);{\varphi }^{-}(A_i)=\frac{1}{m-1}\sum _{k\ne i}\pi (A_k,A_i)$$
   LaTeX

   \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)

5. Net flow PROMETHEE II; rank descending.

   $$\varphi (A_i)={\varphi }^{+}(A_i)-{\varphi }^{-}(A_i);\text{rank descending by }\varphi (A_i)\text{(PROMETHEE II)}$$
   LaTeX

   \phi(A_i)=\phi^+(A_i)-\phi^-(A_i);\quad\text{rank descending by }\phi(A_i)\;\text{(PROMETHEE II)}

Sezgi

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).

Sonucu okuma: 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.

Varsayımlar

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

Ne zaman kullanılmaz

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

Sınırlılıklar

• 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

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin SVNS: T,I,F ∈ [0,1]; 0 ≤ T+I+F ≤ 3 koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = (T − F + 1)/2 kanonik seçimdir.