HF-VIKOR - Kararsız Bulanık VIKOR (Liao-Xu 2013)

Formül adımları

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

1. Each h_ij is a HFE containing all possible membership degrees assigned by DMs to alternative A_i on criterion C_j. Criteria weights ω_j are given by the DMs.

   $$H=[h_{i}j{]}_{I\times J};\omega =({\omega }_1,...,{\omega }_J{)}^{T}with\Sigma {\omega }_j=1$$
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   H = [h_ij]_{I×J}; ω = (ω_1,...,ω_J)^T with Σω_j=1

2. Score = mean membership; variance = mean squared pairwise differences. Higher score and lower variance is better. Ties on score are broken by variance (lower variance → preferred). HFEs of different lengths are extended by repeating minimum values.

   $$s(h)=(1/l_h)·{\Sigma }_{\gamma \in h}\gamma (Def\mathrm{.5});v(h)=(1/l_h)·{\Sigma }_{{\gamma }_i,{\gamma }_j\in h}({\gamma }_i-{\gamma }_j)²(Def\mathrm{.6});h{*}_j=ma{x}_i{h}_{i}j(benefit)ormi{n}_i{h}_{i}j(cost);h{⁻}_j=mi{n}_i{h}_{i}j(benefit)orma{x}_i{h}_{i}j(cost)$$
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   s(h) = (1/l_h)·Σ_{γ∈h} γ (Def.5); v(h) = (1/l_h)·Σ_{γ_i,γ_j∈h}(γ_i−γ_j)² (Def.6); h*_j = max_i h_ij (benefit) or min_i h_ij (cost); h⁻_j = min_i h_ij (benefit) or max_i h_ij (cost)

3. Distance d(h_M, h_N) = (1/l)·Σ|h_M^(σj) − h_N^(σj)| (Manhattan, Eq.4) after extending shorter HFE. S̃* = min_i S̃_i, S̃⁻ = max_i S̃_i; same for R̃. υ=0.5 by default (balanced majority/regret).

   $$S{̃}_i={\Sigma }_j{\omega }_j·d(h{*}_j,h_{i}j)/d(h{*}_j,h{⁻}_j)(Eq\mathrm{.13});R{̃}_i=ma{x}_j[{\omega }_j·d(h{*}_j,h_{i}j)/d(h{*}_j,h{⁻}_j)](Eq\mathrm{.14});Q{̃}_i=\upsilon ·(S{̃}_i-S̃*)/(S̃⁻-S̃*)+(1-\upsilon )·(R{̃}_i-R̃*)/(R̃⁻-R̃*)(Eq\mathrm{.15})$$
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   S̃_i = Σ_j ω_j · d(h*_j, h_ij) / d(h*_j, h⁻_j) (Eq.13); R̃_i = max_j [ω_j · d(h*_j, h_ij) / d(h*_j, h⁻_j)] (Eq.14); Q̃_i = υ·(S̃_i−S̃*)/(S̃⁻−S̃*) + (1−υ)·(R̃_i−R̃*)/(R̃⁻−R̃*) (Eq.15)

4. If both C1 (acceptable advantage) and C2 (acceptable stability) are satisfied, A_(1) is the unique compromise solution. Otherwise proceed to Step 5.

   $$RankascendingbyQ{̃}_i;C1:Q̃(A_{(}2))-Q̃(A_{(}1))\ge 1/(I-1);C2:A_{(}1)isalsobestrankedbyS̃andR̃$$
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   Rank ascending by Q̃_i; C1: Q̃(A_(2)) − Q̃(A_(1)) ≥ 1/(I−1); C2: A_(1) is also best ranked by S̃ and R̃

5. When strict conditions are not met, multiple alternatives form the compromise set. The final ranking uses Q̃ as the primary ordering.

   $$IfC1fails:findmaxNs.t.Q̃(A_{(}N))-Q̃(A_{(}1))<1/(I-1);all{A}_{(}1),...,A_{(}N)arecompromisesolutions.IfonlyC2fails:A_{(}1)and{A}_{(}2)arecompromisesolutions.$$
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   If C1 fails: find max N s.t. Q̃(A_(N)) − Q̃(A_(1)) < 1/(I−1); all A_(1),...,A_(N) are compromise solutions. If only C2 fails: A_(1) and A_(2) are compromise solutions.

Sezgi

Compromise ranking on Hesitant Fuzzy Elements; Manhattan L_p-metric defuzzification; S/R/Q measures with compromise-solution conditions. Output typically ranking.

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

Sık yapılan hatalar

• Bkz. HF-VIKOR F.steps citation_anchor'lar ve P.verification_status.