HFL-PROMETHEE - Kararsız Bulanık Dilsel PROMETHEE (Liang-Wang-Zhang 2018)

Formül adımları

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

1. Collect HFLTS evaluations from DMs. Apply negation operator to convert cost criteria to benefit type: neg({s_φ_1,...,s_φ_l}) = {s_{2t-φ_l},...,s_{2t-φ_1}} on linguistic term set S = {s_0,...,s_{2t}}.

   $$D=[H_S^{ij}{]}_{m\times n};H{̃}_S^{ij}=H_S^{ij}(benefit)orneg(H_S^{ij})(cost)(Eq\mathrm{.14})$$
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   D = [H^{ij}_S]_{m×n}; H̃^{ij}_S = H^{ij}_S (benefit) or neg(H^{ij}_S) (cost) (Eq.14)

2. Positive ideal for criterion C_j = HFLTS with maximum linguistic value across all alternatives (upper-bound of the normalized HFLTSs).

   $$H_S^{j+}=ma{x}_{i=1,...,m}(H_S^{ij+})—upperboundof{H}_S^{ij}foreachcriterion{C}_j$$
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   H^{j+}_S = max_{i=1,...,m}(H^{ij+}_S) — upper bound of H^{ij}_S for each criterion C_j

3. Normalized projection = cosine similarity of H^{ij}_S onto H^{j+}_S divided by norm. RC_{ij} ∈ [0,1] blends projection closeness with risk preference ν: ν < 0.5 = risk-seeking; ν > 0.5 = risk-averse; ν = 0.5 = neutral.

   $$NProj{H}_S^{j+}(H_S^{ij})=Proj{H}_S^{j+}(H_S^{ij})/\Vert H_S^{j+}\Vert (Def\mathrm{.7},Eqs\mathrm{.3},10);R{C}_{ij}=(1-v)·NProj{H}_S^{j+}(H_S^{ij})/[(1-v)·NProj{H}_S^{j+}(H_S^{ij})+v·(1-NProj{H}_S^{j+}(H_S^{ij}))](Def\mathrm{.8},Eq\mathrm{.12})$$
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   NProjH^{j+}_S(H^{ij}_S) = ProjH^{j+}_S(H^{ij}_S) / ‖H^{j+}_S‖ (Def.7, Eqs.3,10); RC_{ij} = (1−v)·NProjH^{j+}_S(H^{ij}_S) / [(1−v)·NProjH^{j+}_S(H^{ij}_S) + v·(1−NProjH^{j+}_S(H^{ij}_S))] (Def.8, Eq.12)

4. Preference function H(Diff) applies one of six PROMETHEE generalized criteria (usual, U-shape, V-shape, level, V-shape-with-indifference, Gaussian). Paper uses usual (Eq.16) and V-shaped-with-indifference (Eq.17).

   $$Dif{f}_j(A_i,A_k)=R{C}_{ij}-R{C}_{kj}(Def\mathrm{.9},Eq\mathrm{.13});H(Diff)=p_j(A_i,A_k)ifDiff\ge 0,else{p}_j(A_k,A_i)(Eq\mathrm{.15});p(A_i,A_k)={\Sigma }_j{\omega }_j·p_j(A_i,A_k)/{\Sigma }_j{\omega }_j(Eq\mathrm{.18})$$
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   Diff_j(A_i, A_k) = RC_{ij} − RC_{kj} (Def.9, Eq.13); H(Diff) = p_j(A_i,A_k) if Diff ≥ 0, else p_j(A_k,A_i) (Eq.15); p(A_i,A_k) = Σ_j ω_j·p_j(A_i,A_k) / Σ_j ω_j (Eq.18)

5. Sum of all pairwise comprehensive preference indices where A_i outranks others. Larger φ⁺ → stronger overall preference for A_i.

   $$\phi ⁺(A_i)={\Sigma }_{k=1,k\ne i}^{m}p(A_i,A_k)(Eq\mathrm{.19})$$
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   φ⁺(A_i) = Σ_{k=1,k≠i}^{m} p(A_i, A_k) (Eq.19)

6. Sum of pairwise indices where other alternatives outrank A_i. Smaller φ⁻ → A_i is less dominated.

   $$\phi ⁻(A_i)={\Sigma }_{k=1,k\ne i}^{m}p(A_k,A_i)(Eq\mathrm{.20})$$
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   φ⁻(A_i) = Σ_{k=1,k≠i}^{m} p(A_k, A_i) (Eq.20)

7. Net flow balances outgoing and incoming flows. Represents overall competitiveness of A_i.

   $$\phi (A_i)=\phi ⁺(A_i)-\phi ⁻(A_i)(Eq\mathrm{.21})$$
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   φ(A_i) = φ⁺(A_i) − φ⁻(A_i) (Eq.21)

8. PROMETHEE I produces a partial ranking; some alternatives may remain incomparable.

   $$A_i\succ ¹A_k(prefer)iff[\phi ⁺(A_i)>\phi ⁺(A_k)AND\phi ⁻(A_i)<\phi ⁻(A_k)]OR[\phi ⁺(A_i)>\phi ⁺(A_k)AND\phi ⁻(A_i)=\phi ⁻(A_k)]OR[\phi ⁺(A_i)=\phi ⁺(A_k)AND\phi ⁻(A_i)<\phi ⁻(A_k)];A_i\sim ¹A_{k}iff\phi ⁺(A_i)=\phi ⁺(A_k)AND\phi ⁻(A_i)=\phi ⁻(A_k);elseincomparable$$
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   A_i ≻¹ A_k (prefer) iff [φ⁺(A_i) > φ⁺(A_k) AND φ⁻(A_i) < φ⁻(A_k)] OR [φ⁺(A_i) > φ⁺(A_k) AND φ⁻(A_i) = φ⁻(A_k)] OR [φ⁺(A_i) = φ⁺(A_k) AND φ⁻(A_i) < φ⁻(A_k)]; A_i ∼¹ A_k iff φ⁺(A_i) = φ⁺(A_k) AND φ⁻(A_i) = φ⁻(A_k); else incomparable

9. PROMETHEE II produces a complete linear ranking using net flows. No incomparability.

   $$A_i\succ ²A_{k}iff\phi (A_i)>\phi (A_k);A_i\sim ²A_{k}iff\phi (A_i)=\phi (A_k);rankdescendingby\phi (A_i)$$
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   A_i ≻² A_k iff φ(A_i) > φ(A_k); A_i ∼² A_k iff φ(A_i) = φ(A_k); rank descending by φ(A_i)

Sezgi

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