CF-VIKOR - VIKOR yönteminin Complex uzantısı

Formül adımları

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

1. normalization

   $${\tilde{a}}_{ij}^c=(r_{\nu ij}{e}^{i{\omega }_{\nu ij}},r_{\mu ij}{e}^{i{\omega }_{\mu ij}})\text{for cost criteria}$$
   LaTeX

   \tilde{a}_{ij}^c=(r_{\nu ij}e^{i\omega_{\nu ij}},r_{\mu ij}e^{i\omega_{\mu ij}})\quad\text{for cost criteria}

2. ranking_computation

   $$f_j^{*}=\mathrm{\backslash arg}{max}_{i}s({\widehat{\tilde{a}}}_{ij}),f_j^{-}=\mathrm{\backslash arg}{min}_{i}s({\widehat{\tilde{a}}}_{ij}),s(\tilde{\alpha })=\frac{(r_{\mu }-r_{\nu })+({\omega }_{\mu }-{\omega }_{\nu })/(2\pi )}{2}$$
   LaTeX

   f_j^*=\arg\max_i s(\hat{\tilde{a}}_{ij}),\quad f_j^-=\arg\min_i s(\hat{\tilde{a}}_{ij}),\quad s(\tilde{\alpha})=\frac{(r_\mu-r_\nu)+(\omega_\mu-\omega_\nu)/(2\pi)}{2}

3. ranking_computation

   $$f_{ij}=\frac{d_E(f_j^{*},{\widehat{\tilde{a}}}_{ij})}{d_E(f_j^{*},f_j^{-})},d_E=\frac{1}{2}\sqrt{(r_{\mu 1}-r_{\mu 2}{)}^2+(r_{\nu 1}-r_{\nu 2}{)}^2+\frac{({\omega }_{\mu 1}-{\omega }_{\mu 2}{)}^2+({\omega }_{\nu 1}-{\omega }_{\nu 2}{)}^2}{4{\pi }^2}}$$
   LaTeX

   f_{ij}=\frac{d_E(f_j^*,\hat{\tilde{a}}_{ij})}{d_E(f_j^*,f_j^-)},\quad d_E=\frac{1}{2}\sqrt{(r_{\mu1}-r_{\mu2})^2+(r_{\nu1}-r_{\nu2})^2+\frac{(\omega_{\mu1}-\omega_{\mu2})^2+(\omega_{\nu1}-\omega_{\nu2})^2}{4\pi^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: cf-vikor extends VIKOR to handle Complex uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Complex Fuzzy Set (CFS: amplitude r ∈ [0,1], phase ω ∈ [0,2π]) algebra. The final scores are defuzzified via real part: r·cos(ω) before ranking.

Varsayımlar

• Decision matrix entries are valid Complex Fuzzy 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 CFS: r(x)·e^{iω(x)} where r ∈ [0,1], ω ∈ [0,2π] koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: real part: r·cos(ω) kanonik seçimdir.