N-GRA - GRA yönteminin Neutrosophic uzantısı

Formül adımları

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

1. SVN matrix; cost complement for cost criteria.

   $$𝐃=(⟨T_{ij},I_{ij},F_{ij}⟩);\text{cost: }{\tilde{a}}_{ij}^c=⟨F_{ij},1-I_{ij},T_{ij}⟩$$
   LaTeX

   \mathbf{D}=(\langle T_{ij},I_{ij},F_{ij}\rangle);\quad\text{cost: }\tilde{a}_{ij}^c=\langle F_{ij},1-I_{ij},T_{ij}\rangle

2. Neutrosophic ideal reference sequence.

   $${\alpha }_j^0=⟨{max}_i{T}_{ij},{min}_i{I}_{ij},{min}_i{F}_{ij}⟩,j=1,\dots ,n$$
   LaTeX

   \alpha_j^0=\langle\max_i T_{ij},\;\min_i I_{ij},\;\min_i F_{ij}\rangle,\quad j=1,\ldots,n

3. Neutrosophic Euclidean distances to ideal; global extremes.

   $${\Delta }_{ij}=\sqrt{\frac{({max}_k{T}_{kj}-T_{ij}{)}^2+({min}_k{I}_{kj}-I_{ij}{)}^2+({min}_k{F}_{kj}-F_{ij}{)}^2}{3}};{\Delta }_{min}={min}_i{min}_j{\Delta }_{ij};{\Delta }_{max}={max}_i{max}_j{\Delta }_{ij}$$
   LaTeX

   \Delta_{ij}=\sqrt{\tfrac{(\max_k T_{kj}-T_{ij})^2+(\min_k I_{kj}-I_{ij})^2+(\min_k F_{kj}-F_{ij})^2}{3}};\quad\Delta_{\min}=\min_i\min_j\Delta_{ij};\quad\Delta_{\max}=\max_i\max_j\Delta_{ij}

4. Grey Relational Coefficient with ρ=0.5.

   $${\xi }_{ij}=\frac{{\Delta }_{min}+\rho {\Delta }_{max}}{{\Delta }_{ij}+\rho {\Delta }_{max}},\rho =0.5$$
   LaTeX

   \xi_{ij}=\frac{\Delta_{\min}+\rho\Delta_{\max}}{\Delta_{ij}+\rho\Delta_{\max}},\quad\rho=0.5

5. Grey Relational Grade; rank descending.

   $${\Gamma }_i=\sum _{j=1}^n{w}_j{\xi }_{ij};\text{rank descending}$$
   LaTeX

   \Gamma_i=\sum_{j=1}^n w_j\xi_{ij};\quad\text{rank descending}

Sezgi

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

Sonucu okuma: n-gra extends GRA 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

• Decision matrix entries are valid Single-Valued Neutrosophic numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

• Classical data sufficient - use base GRA directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• Assumes: Decision matrix entries are valid Single-Valued Neutrosophic numbers/tuples
• Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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.