IF-TOPSIS - Sezgisel Bulanık TOPSIS

Formül adımları

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

1. Yerel sezgisel bulanık karar girdilerini oluşturun

   $$R=(r_{ij}{)}_{m\times n},r_{ij}=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\nu }_{ij}\le 1$$
   LaTeX

   R = (r_{ij})_{m \times n},\ r_{ij} = (\mu_{ij}, \nu_{ij}),\ \mu_{ij}+\nu_{ij} \le 1

2. Uzman değerlendirmelerini IFWA ile birleştirin

   $$r_{ij}={\text{IFWA}}_{\lambda }(r_{ij}^{(1)},\dots ,r_{ij}^{(K)})=(1-{\prod }_{k=1}^K(1-{\mu }_{ij}^{(k)}{)}^{{\lambda }_k},{\prod }_{k=1}^K({\nu }_{ij}^{(k)}{)}^{{\lambda }_k})$$
   LaTeX

   r_{ij} = \text{IFWA}_{\lambda}(r_{ij}^{(1)}, \ldots, r_{ij}^{(K)}) = \left(1 - \prod_{k=1}^{K}(1-\mu_{ij}^{(k)})^{\lambda_k},\ \prod_{k=1}^{K}(\nu_{ij}^{(k)})^{\lambda_k}\right)

3. Yerel kriter ağırlıklarını IFWA ile toplayın

   $$w_j={\text{IFWA}}_{\lambda }(w_j^{(1)},\dots ,w_j^{(K)})=(1-{\prod }_{k=1}^K(1-{\mu }_{w_j}^{(k)}{)}^{{\lambda }_k},{\prod }_{k=1}^K({\nu }_{w_j}^{(k)}{)}^{{\lambda }_k})$$
   LaTeX

   w_j = \text{IFWA}_{\lambda}(w_j^{(1)},\ldots,w_j^{(K)}) = \left(1 - \prod_{k=1}^{K}(1-\mu_{w_j}^{(k)})^{\lambda_k},\ \prod_{k=1}^{K}(\nu_{w_j}^{(k)})^{\lambda_k}\right)

4. Sezgisel bulanık kriter ağırlıklarını uygula

   $$r_{ij}^{\prime }=r_{ij}\otimes w_j=({\mu }_{ij}·{\mu }_{w_j},{\nu }_{ij}+{\nu }_{w_j}-{\nu }_{ij}·{\nu }_{w_j});{\pi }_{ij}^{\prime }=1-{\mu }_{ij}^{\prime }-{\nu }_{ij}^{\prime }$$
   LaTeX

   r'_{ij} = r_{ij} \otimes w_j = \left(\mu_{ij} \cdot \mu_{w_j},\ \nu_{ij} + \nu_{w_j} - \nu_{ij} \cdot \nu_{w_j}\right);\quad \pi'_{ij} = 1 - \mu'_{ij} - \nu'_{ij}

5. Pozitif ve negatif ideal çözümleri belirleyin

   $$A^{*}=\{({\mu }_j^{*},{\nu }_j^{*}){\}}_{j=1}^n,({\mu }_j^{*},{\nu }_j^{*})=\{\begin{array}{cc}({max}_i{\mu }_{ij}^{\prime },{min}_i{\nu }_{ij}^{\prime })\hfill & j\in J_b\hfill \\ ({min}_i{\mu }_{ij}^{\prime },{max}_i{\nu }_{ij}^{\prime })\hfill & j\in J_c\hfill \end{array};A^{-}\text{ swaps max}\leftrightarrow min.$$
   LaTeX

   A^{*} = \{(\mu^{*}_j, \nu^{*}_j)\}_{j=1}^{n},\ (\mu^{*}_j, \nu^{*}_j) = \begin{cases} (\max_i \mu'_{ij},\ \min_i \nu'_{ij}) & j \in J_b \\ (\min_i \mu'_{ij},\ \max_i \nu'_{ij}) & j \in J_c \end{cases};\ A^{-} \text{ swaps max} \leftrightarrow \min.

6. Normalize edilmiş Öklid ayrımlarını hesaplayın

   $$S_i^{*}=\sqrt{\frac{1}{2n}{\sum }_{j=1}^n[({\mu }_{ij}^{\prime }-{\mu }_j^{*}{)}^2+({\nu }_{ij}^{\prime }-{\nu }_j^{*}{)}^2+({\pi }_{ij}^{\prime }-{\pi }_j^{*}{)}^2]};S_i^{-}\text{ analogous with }{A}^{-}.$$
   LaTeX

   S_i^{*} = \sqrt{\frac{1}{2n}\sum_{j=1}^{n}\left[(\mu'_{ij}-\mu^{*}_j)^2 + (\nu'_{ij}-\nu^{*}_j)^2 + (\pi'_{ij}-\pi^{*}_j)^2\right]};\quad S_i^{-} \text{ analogous with } A^{-}.

7. Göreli yakınlığı hesapla

   $$C_i^{*}=\frac{S_i^{-}}{S_i^{*}+S_i^{-}},0\le C_i^{*}\le 1$$
   LaTeX

   C_i^{*} = \frac{S_i^{-}}{S_i^{*} + S_i^{-}},\quad 0 \le C_i^{*} \le 1

8. Alternatifleri göreli yakınlığa göre sıralayın

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(C_i^{*})$$
   LaTeX

   \text{rank}(A_i) = \text{argsort}_{\downarrow}(C_i^{*})

Sezgi

Distance-based ranking under Intuitionistic Fuzzy uncertainty. Output typically utility (higher value = preferred).

Sonucu okuma: IF-TOPSIS extends crisp TOPSIS to settings where each rating is an Intuitionistic Fuzzy Number (μ, ν) representing membership/non-membership. The algorithm aggregates expert ratings and weights via IFWA (Steps 2-3), weights the matrix with Atanassov's ⊗ operator (Step 4), extracts IF positive/negative ideal solutions per criterion direction (Step 5), then ranks alternatives by their crisp closeness coefficient C* ∈ [0,1] (Steps 6-7). Higher C* = better. NO defuzzification step is involved; the ranking emerges directly from IF-distance separations.

Varsayımlar

• Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• All decision-makers use the same linguistic-to-IFN scale (e.g. Boran 2009 Table 1)
• Criterion directions (benefit/cost) are explicitly labelled

Ne zaman kullanılmaz

• Classical data sufficient - use base TOPSIS directly (avoid unnecessary uncertainty layer)
• Hesitation degree π must be modelled explicitly rather than implicitly - consider Pythagorean Fuzzy TOPSIS or q-ROF TOPSIS when μ+ν > 1 is observed in raw judgements

Sınırlılıklar

• Alternative-set dependence has not been independently validated in this review
• Assumes: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• Assumes: All decision-makers use the same linguistic-to-IFN scale (e.g. Boran 2009 Tables 2-3)
• Assumes: Criterion directions (benefit/cost) are explicitly labelled

Sık yapılan hatalar

• IFN (μ, ν ikili) ile TFN (a, b, c üçlü) karıştırmak: IF-TOPSIS değerleri μ+ν ≤ 1 koşulunu sağlayan IFN ikilileri OLMALIDIR, üçgensel bulanık sayılar DEĞİL.
• Skaler bileşen ölçeklendirmesi Boran Denklem (8)’deki Atanassov çarpımı değildir; yerel IFN kriter ağırlıkları kullanın.
• Boran Eqs. (15)-(16)’da tüm üç bileşen mu, nu ve pi ile 1/(2n) faktörünü kullanınız.