Cubic-VIKOR - VIKOR yönteminin Cubic uzantısı

Formül adımları

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

1. normalization

   $${\widehat{c}}_{ij}=([1-{\mu }_{ij}^U,1-{\mu }_{ij}^L],1-{\lambda }_{ij})\text{for cost criteria}$$
   LaTeX

   \hat{c}_{ij}=([1-\mu_{ij}^U,1-\mu_{ij}^L],\,1-\lambda_{ij})\quad\text{for cost criteria}

2. ranking_computation

   $$f_j^{*}=\mathrm{\backslash arg}{max}_{i}s({\widehat{c}}_{ij}),f_j^{-}=\mathrm{\backslash arg}{min}_{i}s({\widehat{c}}_{ij}),s(\tilde{c})=\frac{{\mu }^L+{\mu }^U}{2}+\frac{{\mu }^U-{\mu }^L}{4}-\frac{1}{2}(1-\lambda -\frac{{\mu }^L+{\mu }^U}{2})$$
   LaTeX

   f_j^*=\arg\max_i s(\hat{c}_{ij}),\quad f_j^-=\arg\min_i s(\hat{c}_{ij}),\quad s(\tilde{c})=\frac{\mu^L+\mu^U}{2}+\frac{\mu^U-\mu^L}{4}-\frac{1}{2}\!\left(1-\lambda-\frac{\mu^L+\mu^U}{2}\right)

3. ranking_computation

   $$f_{ij}=\frac{d_E(f_j^{*},{\widehat{c}}_{ij})}{d_E(f_j^{*},f_j^{-})},d_E({\tilde{c}}_1,{\tilde{c}}_2)=\sqrt{\frac{({\mu }_1^L-{\mu }_2^L{)}^2+({\mu }_1^U-{\mu }_2^U{)}^2}{4}+\frac{({\lambda }_1-{\lambda }_2{)}^2}{2}}$$
   LaTeX

   f_{ij}=\frac{d_E(f_j^*,\hat{c}_{ij})}{d_E(f_j^*,f_j^-)},\quad d_E(\tilde{c}_1,\tilde{c}_2)=\sqrt{\frac{(\mu_1^L-\mu_2^L)^2+(\mu_1^U-\mu_2^U)^2}{4}+\frac{(\lambda_1-\lambda_2)^2}{2}}

4. ranking_computation

   $$S_i=\sum _j{w}_j{f}_{ij},R_i={max}_j(w_j{f}_{ij}),Q_i=v\frac{S_i-S^{*}}{S^{-}-S^{*}}+(1-v)\frac{R_i-R^{*}}{R^{-}-R^{*}},v=0.5$$
   LaTeX

   S_i=\sum_j w_j f_{ij},\quad R_i=\max_j(w_j f_{ij}),\quad Q_i=v\frac{S_i-S^*}{S^--S^*}+(1-v)\frac{R_i-R^*}{R^--R^*},\quad v=0.5

5. ranking_computation

   $$DQ=\frac{1}{m-1};C1:Q(A^{(2)})-Q(A^{(1)})\ge DQ;C2:A^{(1)}\text{ is also best in }S\text{ or }R;\text{rank ascending by }Q;𝒞=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }{A}^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ$$
   LaTeX

   DQ=\frac{1}{m-1};\quad C1:Q(A^{(2)})-Q(A^{(1)})\geq DQ;\quad C2:A^{(1)}\text{ is also best in }S\text{ or }R;\quad\text{rank ascending by }Q;\quad \mathcal{C}=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }A^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ

Sezgi

Sonucu okuma: cubic-vikor extends VIKOR to handle Cubic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Cubic Fuzzy Set (CuFS: interval [a,b] for internal membership, scalar λ for external) algebra. The final scores are defuzzified via P-order: (a⁻+a⁺)/2 and λ combined before ranking.

Varsayımlar

• Decision matrix entries are valid Cubic Set numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

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

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin CuFS: A = ⟨[a⁻,a⁺], λ⟩ where [a⁻,a⁺] ⊆ [0,1], λ ∈ [0,1] koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: P-order: (a⁻+a⁺)/2 and λ combined kanonik seçimdir.