DHF-TOPSIS - Wang2020 özgün ortalama/oynaklık mesafe profili

Formül adımları

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

1. Yerel çift tereddütlü girdileri doğrulayın

   $$D=(a_{ij}{)}_{m\times n}=(⟨h_{ij},g_{ij}⟩{)}_{m\times n}$$
   LaTeX

   D = \big( a_{ij} \big)_{m \times n} = \big( \langle h_{ij}, g_{ij} \rangle \big)_{m \times n}

2. Kriter katsayılarını çözümleyin

   $$w_j=\{\begin{array}{cc}{w}_j^{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d}},\hfill & \text{if provided}\hfill \\ Y_j/\sum _{k=1}^n{Y}_k,\hfill & \text{otherwise}\hfill \end{array},Y_j=\sum _i\sum _k{d}_h(a_{ij},a_{kj}),d_h(a,b)=\frac{|\Delta G_m|+|\Delta G_n|+|\Delta V_m|+|\Delta V_n|}{4}$$
   LaTeX

   w_j=\begin{cases}w_j^{\mathrm{supplied}},&\text{if provided}\\Y_j/\sum_{k=1}^{n}Y_k,&\text{otherwise}\end{cases},\quad Y_j=\sum_i\sum_k d_h(a_{ij},a_{kj}),\quad d_h(a,b)=\frac{|\Delta G_m|+|\Delta G_n|+|\Delta V_m|+|\Delta V_n|}{4}

3. Pozitif ve negatif idealleri oluşturun

   $$a_j^{+}=⟨\{{max}_{i}max{h}_{ij}\},\{{min}_{i}min{g}_{ij}\}⟩,a_j^{-}=⟨\{{min}_{i}min{h}_{ij}\},\{{max}_{i}max{g}_{ij}\}⟩$$
   LaTeX

   a_j^+=\langle\{\max_i\max h_{ij}\},\{\min_i\min g_{ij}\}\rangle,\quad a_j^-=\langle\{\min_i\min h_{ij}\},\{\max_i\max g_{ij}\}\rangle

4. Ortalama/oynaklık mesafelerini hesapla

   $$G_m(e)=\frac{1}{l(h)}\sum _{\gamma \in h}\gamma ,G_n(e)=\frac{1}{k(g)}\sum _{\eta \in g}\eta ;V_m(e)=\frac{1}{2}[\frac{1}{l(h)}\sum _{\gamma \in h}(\gamma -G_m{)}^2+\sqrt{1-\frac{1}{1+\ln l(h)}}],V_n(e)=\frac{1}{2}[\frac{1}{k(g)}\sum _{\eta \in g}(\eta -G_n{)}^2+\sqrt{1-\frac{1}{1+\ln k(g)}}];d_{wpg}(A_i,A^{\pm })=\left(\frac{1}{2}\sum _{j=1}^n{w}_j\right[\alpha |G_m(a_{ij})-G_m(a_j^{\pm }){|}^{\lambda }+\alpha |G_n(a_{ij})-G_n(a_j^{\pm }){|}^{\lambda }+\beta |V_m(a_{ij})-V_m(a_j^{\pm }){|}^{\lambda }+\beta |V_n(a_{ij})-V_n(a_j^{\pm }){|}^{\lambda }]{)}^{1/\lambda },\alpha +\beta =1,\lambda \ge 1$$
   LaTeX

   G_{m}(e) = \tfrac{1}{l(h)}\sum_{\gamma \in h}\gamma, \quad G_{n}(e) = \tfrac{1}{k(g)}\sum_{\eta \in g}\eta; \quad V_{m}(e) = \tfrac{1}{2}\Big[\tfrac{1}{l(h)}\sum_{\gamma \in h}(\gamma - G_{m})^{2} + \sqrt{1 - \tfrac{1}{1 + \ln l(h)}}\Big], \quad V_{n}(e) = \tfrac{1}{2}\Big[\tfrac{1}{k(g)}\sum_{\eta \in g}(\eta - G_{n})^{2} + \sqrt{1 - \tfrac{1}{1 + \ln k(g)}}\Big]; \quad d_{wpg}(A_{i}, A^{\pm}) = \Big(\tfrac{1}{2}\sum_{j=1}^{n} w_{j}\big[\alpha\,|G_{m}(a_{ij}) - G_{m}(a_{j}^{\pm})|^{\lambda} + \alpha\,|G_{n}(a_{ij}) - G_{n}(a_{j}^{\pm})|^{\lambda} + \beta\,|V_{m}(a_{ij}) - V_{m}(a_{j}^{\pm})|^{\lambda} + \beta\,|V_{n}(a_{ij}) - V_{n}(a_{j}^{\pm})|^{\lambda}\big]\Big)^{1/\lambda}, \quad \alpha + \beta = 1,\ \lambda \ge 1

5. Yakınlığı hesaplayın ve alternatifleri sıralayın

   $$C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},d_i^{+}+d_i^{-}>0,rank(i)=1+\#\{k:C{C}_k>C{C}_i\}$$
   LaTeX

   CC_i=\frac{d_i^-}{d_i^++d_i^-},\quad d_i^++d_i^->0,\qquad\operatorname{rank}(i)=1+\#\{k:CC_k>CC_i\}

Sezgi

Sonucu okuma: Benefit-only native Wang2020 feature distance; use supplied coefficients or derive Equation27 coefficients. Printed example inconsistencies and raw-set feature collisions are disclosed.

Varsayımlar

• Native dual hesitant sets and benefit-only criteria.
• Accept source mean/variance/cardinality feature distance and disclosed example inconsistencies.

Ne zaman kullanılmaz

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

Sık yapılan hatalar

• Kaynak lambda1’i belirtir; eski lambda2 düzeni ve 0.15 tolerans arşivlenmiş olup kaynak doğrulaması değildir.
• Farklı yerel kümeler aynı ortalama, varyans ve kardinalite özelliklerine sahip olabilir; sıfır uzaklık ham küme eşitliğini kanıtlamaz.
• Maliyet dönüşümü ve uzman toplulaştırması bu kaynak profilinde desteklenmemektedir.