Fuzzy SIWEC - Bulanık Basit Ağırlık Hesaplama (F-SIWEC)

Formül adımları

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

1. Each of the k decision makers evaluates all n criteria using linguistic variables (e.g. VB, B, MB, M, MG, G, VG). Convert linguistic labels to TFNs using the 7-level scale. The result is a k×n matrix of TFNs (rows=DMs, cols=criteria). No pairwise comparisons or criterion ranking required.

   $$\tilde{X}=[{\tilde{x}}_{ij}{]}_{k\times n};{\tilde{x}}_{ij}=(x_{ij}^l,x_{ij}^m,x_{ij}^u)$$
   LaTeX

   \tilde{X} = [\tilde{x}_{ij}]_{k×n}; \tilde{x}_{ij} = (x^l_{ij}, x^m_{ij}, x^u_{ij})

2. For each decision maker i, normalize all TFN scores by dividing by the maximum upper bound across all their criteria scores. This preserves the TFN shape while scaling to [0,1]. The denominator is a crisp scalar (max of u-values for that DM).

   $${\tilde{n}}_{ij}=(\frac{x_{ij}^l}{{max}_j{x}_{ij}^u},\frac{x_{ij}^m}{{max}_j{x}_{ij}^u},\frac{x_{ij}^u}{{max}_j{x}_{ij}^u})$$
   LaTeX

   \tilde{n}_{ij} = \left(\frac{x^l_{ij}}{\max_j x^u_{ij}},\ \frac{x^m_{ij}}{\max_j x^u_{ij}},\ \frac{x^u_{ij}}{\max_j x^u_{ij}}\right)

3. Compute the standard deviation of the midpoint (modal) values of each DM's normalized TFNs across all criteria. This scalar σ_i measures the DM's discriminating power: higher σ_i means the DM more clearly differentiates between criteria.

   $${\sigma }_i=\sqrt{\frac{1}{n}{\sum }_{j=1}^n(n_{ij}^m-{\overline{n}}_i^m{)}^2};{\overline{n}}_i^m=\frac{1}{n}\sum _{j=1}^n{n}_{ij}^m$$
   LaTeX

   \sigma_i = \sqrt{\frac{1}{n}\sum_{j=1}^{n}(n^m_{ij} - \bar{n}^m_i)^2}; \quad \bar{n}^m_i = \frac{1}{n}\sum_{j=1}^{n} n^m_{ij}

4. Multiply each normalized TFN by the scalar standard deviation of the corresponding DM. TFN × scalar multiplication is component-wise. This downweights DMs who assign uniform scores across criteria.

   $${\tilde{v}}_{ij}={\tilde{n}}_{ij}\times {\sigma }_i=(n_{ij}^l·{\sigma }_i,n_{ij}^m·{\sigma }_i,n_{ij}^u·{\sigma }_i)$$
   LaTeX

   \tilde{v}_{ij} = \tilde{n}_{ij} \times \sigma_i = (n^l_{ij}\cdot\sigma_i,\ n^m_{ij}\cdot\sigma_i,\ n^u_{ij}\cdot\sigma_i)

5. Sum the dispersion-weighted TFNs over all DMs for each criterion j using TFN addition (component-wise sum). Result is a single TFN s̃_j per criterion representing aggregate importance.

   $${\tilde{s}}_j=\sum _{i=1}^k{\tilde{v}}_{ij}=({\sum }_i{v}_{ij}^l,{\sum }_i{v}_{ij}^m,{\sum }_i{v}_{ij}^u)$$
   LaTeX

   \tilde{s}_j = \sum_{i=1}^{k} \tilde{v}_{ij} = \left(\sum_i v^l_{ij},\ \sum_i v^m_{ij},\ \sum_i v^u_{ij}\right)

6. Apply TFN division normalization: the lower bound divides by the sum of upper bounds, the modal divides by sum of modals, the upper bound divides by sum of lower bounds. This cross-division follows standard TFN division (a/b where a is TFN and b is TFN → l/u_sum, m/m_sum, u/l_sum). Result is a weight TFN w̃_j ∈ [0,1]^3 for each criterion.

   $${\tilde{w}}_j=(\frac{s_j^l}{{\sum }_k{s}_k^u},\frac{s_j^m}{{\sum }_k{s}_k^m},\frac{s_j^u}{{\sum }_k{s}_k^l})$$
   LaTeX

   \tilde{w}_j = \left(\frac{s^l_j}{\sum_k s^u_k},\ \frac{s^m_j}{\sum_k s^m_k},\ \frac{s^u_j}{\sum_k s^l_k}\right)

Sezgi

Sonucu okuma: Higher w̃_j modal = more important criterion. Fuzzy weights can be passed directly to fuzzy ranking methods (F-RAWEC, F-TOPSIS, F-VIKOR, etc.) without defuzzification. For crisp downstream methods, defuzzify using centroid or (l+4m+u)/6.

Varsayımlar

• Domain experts available and willing to score criteria
• Experts can meaningfully differentiate between criteria (non-zero σ)
• Triangular fuzzy numbers adequately capture uncertainty

Ne zaman kullanılmaz

• No domain experts available - use objective weighting (FUZZY-CRITIC, FUZZY-ENTROPY)
• Experts tend to rate all criteria equally (zero variance problem)
• Higher-order uncertainty (IFS, hesitant) needed - use IF-/HF- weight methods

Sık yapılan hatalar

• Normalizasyon paydası üst sınırların maksimumunu kullanır (modal maksimumunu değil) - tüm bileşenler için TFN ≤ 1 sağlar.
• Ağırlık normalizasyonu çapraz bölme kullanır (l/Σu, m/Σm, u/Σl) - basit bileşen bölmesi kullanmayın.