HF-DFT - Kararsız Bulanık Karar Alan Teorisi (Song-Xu 2021)

Formül adımları

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

1. Each expert provides HFE evaluations. Aggregate using HFWA with expert weights ζ = (ζ_1,...,ζ_k)^T, Σζ_l=1. Result is the collective hesitant fuzzy decision matrix Ḣ.

   $$H^l=[h_{ij}^{(l)}{]}_{m\times n}forexpert{p}_l;{ḣ}_{ij}=HFWA(h_{ij}^{(1)},h_{ij}^{(2)},...,h_{ij}^{(k)})(Eq\mathrm{.3});Ḣ=[{ḣ}_{ij}{]}_{m\times n}$$
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   H^l = [h^{(l)}_{ij}]_{m×n} for expert p_l; ḣ_{ij} = HFWA(h^{(1)}_{ij}, h^{(2)}_{ij},...,h^{(k)}_{ij}) (Eq.3); Ḣ = [ḣ_{ij}]_{m×n}

2. Transform cost/negative criteria to benefit type via complement. All subsequent steps use the normalized matrix.

   $$H(x{)}^N=H(x)forpositivecriteria;H(x{)}^N=neg(H(x))=1-\gamma |\gamma \in H(x)fornegativecriteria(Def\mathrm{.9},Eq\mathrm{.13})$$
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   H(x)^N = H(x) for positive criteria; H(x)^N = neg(H(x)) = {1−γ | γ∈H(x)} for negative criteria (Def.9, Eq.13)

3. If weights are known, use directly. If partially known (linear inequality constraints Ω, Eq.12), solve the linear programming model per alternative then average. If completely unknown, use entropy or maximize total score.

   $$Model1:max{\Sigma }_i{s}_i(\omega )={\Sigma }_i{\Sigma }_j{\omega }_j·s_{ij}s.t.\omega \in \Omega ,\Sigma {\omega }_j=1(Eq\mathrm{.11});s_{ij}=s({ḣ}_{ij})=(1/\#h_{ij})·\Sigma \gamma (Eq\mathrm{.1})$$
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   Model 1: max Σ_i s_i(ω) = Σ_i Σ_j ω_j·s_{ij} s.t. ω∈Ω, Σω_j=1 (Eq.11); s_{ij} = s(ḣ_{ij}) = (1/#h_{ij})·Σγ (Eq.1)

4. Generalized hesitant weighted Euclidean distance (λ=2) using attribute weights ω. HFEs of unequal length are extended with minimum value (pessimistic principle). D* is an m×m symmetric matrix.

   $$d_1(H,M)=[{\Sigma }_i{\omega }_i·(1/\#h_{xi})·{\Sigma }_j|h^{\rho (j)}(x_i)-m^{\rho (j)}(x_i){|}^{\lambda }{]}^{1/\lambda }(Def\mathrm{.5},Eq\mathrm{.5},\lambda =2);D{*}_{ik}=d(S_i,S_k)$$
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   d_1(H,M) = [Σ_i ω_i·(1/#h_{xi})·Σ_j |h^{ρ(j)}(x_i) − m^{ρ(j)}(x_i)|^λ]^{1/λ} (Def.5, Eq.5, λ=2); D*_{ik} = d(S_i, S_k)

5. Gaussian-based feedback matrix. φ controls competitive influence strength between alternatives (smaller φ → weaker competition). δ controls discriminability (larger δ for more similar alternatives). Eigenvalues of S* must be < 1 for stability.

   $$S*=I-\phi ·e^{-\delta ·D*²}(Def\mathrm{.7},Eq\mathrm{.9});0\le \phi \le 1;\delta \in [0.01,1000]$$
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   S* = I − φ·e^{−δ·D*²} (Def.7, Eq.9); 0 ≤ φ ≤ 1; δ ∈ [0.01, 1000]

6. Valence captures the momentary psychological advantage of each alternative over others. Contrast matrix C: self-comparison=1, cross-comparison=−1/(n−1). M*⊗W uses HFS operational laws (Def.3-4) to weight the hesitant fuzzy decision matrix.

   $$V*(t)=C\otimes M*\otimes W(t)(Def\mathrm{.6},Eq\mathrm{.8});C:C_{ii}=1,C_{ij}=-1/(n-1)fori\ne j;M*\otimes W=HFWA/HFWGweightedaggregationofHFEs$$
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   V*(t) = C ⊗ M* ⊗ W(t) (Def.6, Eq.8); C: C_{ii}=1, C_{ij}=−1/(n−1) for i≠j; M*⊗W = HFWA/HFWG weighted aggregation of HFEs

7. s = small time step. Preference accumulates dynamically. Positive P*_i indicates tendency toward alternative S_i. Decision stops at deliberation time T or when any preference exceeds threshold θ.

   $$P*(t+s)=S*·P*(t)+V*(t+s)(Def\mathrm{.8},Eq\mathrm{.10});P*(0)=0;iterateuntilt=T(timerule)orma{x}_i|P{*}_i(t)|\ge \theta (thresholdrule)$$
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   P*(t+s) = S*·P*(t) + V*(t+s) (Def.8, Eq.10); P*(0) = 0; iterate until t = T (time rule) or max_i|P*_i(t)| ≥ θ (threshold rule)

8. Alternative with the highest accumulated preference value at deliberation end T is optimal. If preference values tie, compare by score (higher = better) then by standard deviation (lower = better).

   $$best=argma{x}_{i}P{*}_i(T);comparebyscores(h)thenstddev\sigma (h)iftied(Defs\mathrm{.1}-2,Eqs\mathrm{.1}-2)$$
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   best = argmax_i P*_i(T); compare by score s(h) then stddev σ(h) if tied (Defs.1-2, Eqs.1-2)

Sezgi

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