IV-VIKOR - VIKOR yönteminin Interval uzantısı

Formül adımları

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

1. normalization

   $${\tilde{a}}_{ij}^c=([{\nu }_{ij}^L,{\nu }_{ij}^U],[{\mu }_{ij}^L,{\mu }_{ij}^U])\text{for cost criteria}$$
   LaTeX

   \tilde{a}_{ij}^c=([\nu^L_{ij},\nu^U_{ij}],[\mu^L_{ij},\mu^U_{ij}])\quad\text{for cost criteria}

2. ranking_computation

   $$f_j^{*}=([{max}_i{\mu }_{ij}^L,{max}_i{\mu }_{ij}^U],[{min}_i{\nu }_{ij}^L,{min}_i{\nu }_{ij}^U]),f_j^{-}=([{min}_i{\mu }_{ij}^L,{min}_i{\mu }_{ij}^U],[{max}_i{\nu }_{ij}^L,{max}_i{\nu }_{ij}^U])$$
   LaTeX

   f_j^*=\bigl([\max_i\mu^L_{ij},\max_i\mu^U_{ij}],[\min_i\nu^L_{ij},\min_i\nu^U_{ij}]\bigr),\quad f_j^-=\bigl([\min_i\mu^L_{ij},\min_i\mu^U_{ij}],[\max_i\nu^L_{ij},\max_i\nu^U_{ij}]\bigr)

3. ranking_computation

   $$f_{ij}=\frac{d({\tilde{f}}_j^{*},{\widehat{a}}_{ij})}{d({\tilde{f}}_j^{*},{\tilde{f}}_j^{-})},d({\alpha }_1,{\alpha }_2)=\sqrt{\frac{1}{4}[({\mu }_1^L-{\mu }_2^L{)}^2+({\mu }_1^U-{\mu }_2^U{)}^2+({\nu }_1^L-{\nu }_2^L{)}^2+({\nu }_1^U-{\nu }_2^U{)}^2]}$$
   LaTeX

   f_{ij}=\frac{d(\tilde{f}_j^*,\hat{a}_{ij})}{d(\tilde{f}_j^*,\tilde{f}_j^-)},\quad d(\alpha_1,\alpha_2)=\sqrt{\tfrac{1}{4}[(\mu^L_1-\mu^L_2)^2+(\mu^U_1-\mu^U_2)^2+(\nu^L_1-\nu^L_2)^2+(\nu^U_1-\nu^U_2)^2]}

4. ranking_computation

   $$S_i=\sum _{j=1}^n{w}_j{f}_{ij},R_i={max}_j(w_j{f}_{ij})$$
   LaTeX

   S_i=\sum_{j=1}^n w_j f_{ij},\quad R_i=\max_j(w_j f_{ij})

5. ranking_computation

   $$Q_i=v\frac{S_i-S^{*}}{S^{-}-S^{*}}+(1-v)\frac{R_i-R^{*}}{R^{-}-R^{*}},v=0.5;DQ=\frac{1}{m-1};C1:Q(A^{(2)})-Q(A^{(1)})\ge DQ;\text{rank ascending};𝒞=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }{A}^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ$$
   LaTeX

   Q_i=v\frac{S_i-S^*}{S^--S^*}+(1-v)\frac{R_i-R^*}{R^--R^*},\quad v=0.5;\quad DQ=\frac{1}{m-1};\quad C1:Q(A^{(2)})-Q(A^{(1)})\geq DQ;\quad\text{rank ascending};\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: iv-vikor extends VIKOR to handle Interval uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Interval Number (IN: [a, b]) algebra. The final scores are defuzzified via midpoint (a+b)/2 before ranking.

Varsayımlar

• Decision matrix entries are valid Interval 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 IN: a ≤ b koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: midpoint (a+b)/2 kanonik seçimdir.