IF-DFT - Sezgisel Bulanık Karar Alan Teorisi (Hao-Xu-Zhao-Zhang 2017)

Formül adımları

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

1. Each element α_{ij} is an IFN representing the evaluation of alternative Y_i on attribute G_j. If multiple DMs, aggregate individual matrices via IFHA or IFHG operators. Weights may be given or optimized via model M-3.1. Criteria the user marks as cost are mapped to the Atanassov IFN complement α^c = (ν, μ) before any further step, so every column reads higher is better from F2 on. The complement leaves the hesitancy π = 1 − μ − ν unchanged and therefore leaves the F2 distance matrix and the F3 feedback matrix bit-identical; it negates only the F4 attribute score μ − ν of that column, which is where less is better belongs in a valence.

   $$M=[{\alpha }_{ij}{]}_{m\times n}where{\alpha }_{ij}=({\mu }_{ij},{\nu }_{ij});costcolumnsarecomplementedfirst:{\alpha }_{ij}^c=({\nu }_{ij},{\mu }_{ij});W=(w_1,...,w_n{)}^{T}with\Sigma w_j=1$$
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   M = [α_{ij}]_{m×n} where α_{ij} = (μ_{ij}, ν_{ij}); cost columns are complemented first: α^c_{ij} = (ν_{ij}, μ_{ij}); W = (w_1,...,w_n)^T with Σw_j=1

2. Normalized Euclidean distance between IFNs including hesitancy π=1−μ−ν. Used to build the feedback matrix.

   $$d({\alpha }_1,{\alpha }_2)=sqrt(0.5·[({\mu }_1-{\mu }_2)²+({\nu }_1-{\nu }_2)²+({\pi }_1-{\pi }_2)²])(Eq\mathrm{.5}with\lambda =2);D=[d_{ik}{]}_{m\times m}where{d}_{ik}={\Sigma }_j{w}_j·d({\alpha }_{ij},{\alpha }_{kj})$$
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   d(α_1, α_2) = sqrt(0.5·[(μ_1−μ_2)² + (ν_1−ν_2)² + (π_1−π_2)²]) (Eq.5 with λ=2); D = [d_{ik}]_{m×m} where d_{ik} = Σ_j w_j·d(α_{ij}, α_{kj})

3. Contrast matrix C extracts pairwise comparisons (self-contrast=1, cross-contrast=−1/(m−1)). Feedback matrix S encodes memory (diagonal: self-influence, α) and inter-alternative competition (off-diagonal: β). Parameters α ∈ [0.01, 1000] and β ∈ (0,1).

   $$C:C_{ii}=1,C_{ij}=-1/(m-1)fori\ne j;S=e^{-\alpha ·D²}-\beta ·I(Eq\mathrm{.14},Gaussianfeedback)$$
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   C: C_{ii} = 1, C_{ij} = −1/(m−1) for i≠j; S = e^{−α·D²} − β·I (Eq.14, Gaussian feedback)

4. Standard IFN subtraction loses information; instead use directed distance as contrast. Valence vector V = C·(weighted contrast matrix) represents instantaneous preference pressure on each alternative.

   $$contrast({\alpha }_1,{\alpha }_2)=(d({\alpha }_1,{\alpha }_2),0)if{\alpha }_1\ge {\alpha }_{2}else(-d({\alpha }_1,{\alpha }_2),0)(Def\mathrm{.3}\mathrm{.1},Eq\mathrm{.13});V(t)=C·M^{\prime }·W(Eq\mathrm{.12})$$
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   contrast(α_1, α_2) = (d(α_1,α_2), 0) if α_1 ≥ α_2 else (−d(α_1,α_2), 0) (Def.3.1, Eq.13); V(t) = C·M'·W (Eq.12)

5. h = small time step. Iterate until deliberation time T is reached (time rule) or any |P_i(t)| exceeds threshold θ (threshold rule). At each step, attention weight W is updated (stochastic process; mean remains constant).

   $$P(t+h)=S·P(t)+h·V(Eq\mathrm{.10});P(0)=0$$
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   P(t+h) = S·P(t) + h·V (Eq.10); P(0) = 0

6. The alternative with the highest accumulated preference P_i(T) at deliberation end is the optimal choice. Choice probabilities may be computed from preference values for probabilistic interpretation.

   $$best=argma{x}_i{P}_i(T);Choiceprobability\propto P_i(T)(normalized,seeAppendix)$$
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   best = argmax_i P_i(T); Choice probability ∝ P_i(T) (normalized, see Appendix)

Sezgi

Sonucu okuma: IF-DFT ranks alternatives based on performance scores. Higher score = better rank.