Fuzzy CODAS - CODAS yönteminin Fuzzy uzantısı

Formül adımları

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

TFN matrisini x̃ij=(xijα,xijβ,xijγ) ve ağırlıkları w̃j=(wjα,wjβ,wjγ) olarak al.

$X ̃ = [x ̃_{i j}]; {\tilde{w}}_{j} = (w_{j}^{α}, w_{j}^{β}, w_{j}^{γ})$

LaTeX
X̃=[x̃_{ij}];\quad\tilde{w}_j=(w_j^{\alpha},w_j^{\beta},w_j^{\gamma})
Normalize et: fayda r̃ij=(xijα/Mγj, xijβ/Mγj, xijγ/Mγj), Mγj=max_i(xijγ). Maliyet r̃ij=(mαj/xijγ, mαj/xijβ, mαj/xijα), mαj=min_i(xijα).

${\tilde{r}}_{i j} = {\begin{matrix} (x_{i j}^{α} / M_{j}^{γ}, x_{i j}^{β} / M_{j}^{γ}, x_{i j}^{γ} / M_{j}^{γ}) & j \in Ω_{+} \\ [4pt] (m_{j}^{α} / x_{i j}^{γ}, m_{j}^{α} / x_{i j}^{β}, m_{j}^{α} / x_{i j}^{α}) & j \in Ω_{-} \end{matrix}$

LaTeX
\tilde{r}_{ij}=\begin{cases}\bigl(x_{ij}^{\alpha}/M_j^{\gamma},\;x_{ij}^{\beta}/M_j^{\gamma},\;x_{ij}^{\gamma}/M_j^{\gamma}\bigr)&j\in\Omega_{+}\\[4pt]\bigl(m_j^{\alpha}/x_{ij}^{\gamma},\;m_j^{\alpha}/x_{ij}^{\beta},\;m_j^{\alpha}/x_{ij}^{\alpha}\bigr)&j\in\Omega_{-}\end{cases}
Ağırlıklı: ṽij=r̃ij⊗w̃j=(rα·wα, rβ·wβ, rγ·wγ).

${\tilde{v}}_{i j} = (r_{i j}^{α} w_{j}^{α}, r_{i j}^{β} w_{j}^{β}, r_{i j}^{γ} w_{j}^{γ})$

LaTeX
\tilde{v}_{ij}=(r_{ij}^{\alpha}w_j^{\alpha},\;r_{ij}^{\beta}w_j^{\beta},\;r_{ij}^{\gamma}w_j^{\gamma})
NIS (negatif ideal): her kriter için n̊j=min_i COA(ṽij). Öklid Êi=√(Σj(COA(ṽij)-n̊j)²). Taksi T̂i=Σj|COA(ṽij)-n̊j|.

${\tilde{n}}_{j} = \arg {min}_{{\tilde{v}}_{i j}} C O A ({\tilde{v}}_{i j}); E_{i} = \sqrt{\sum_{j} \frac{(v_{i j}^{l} - n_{j}^{l})^{2} + 4 (v_{i j}^{m} - n_{j}^{m})^{2} + (v_{i j}^{u} - n_{j}^{u})^{2}}{6}}; T_{i} = \sum_{j} \frac{| v_{i j}^{l} - n_{j}^{l} | + 4 | v_{i j}^{m} - n_{j}^{m} | + | v_{i j}^{u} - n_{j}^{u} |}{6}$

LaTeX
\tilde n_j=\arg\min_{\tilde v_{ij}} COA(\tilde v_{ij});\quad E_i=\sqrt{\sum_j\frac{(v^l_{ij}-n^l_j)^2+4(v^m_{ij}-n^m_j)^2+(v^u_{ij}-n^u_j)^2}{6}};\quad T_i=\sum_j\frac{|v^l_{ij}-n^l_j|+4|v^m_{ij}-n^m_j|+|v^u_{ij}-n^u_j|}{6}
İkili H̃ik=(Êi-Êk)+ψ(|Êi-Êk|≥τ)·(T̂i-T̂k), τ=0.02. Değerlendirme skoru hi=Σk H̃ik.

$H_{i k} = ({\hat{E}}_{i} - {\hat{E}}_{k}) + ψ (| {\hat{E}}_{i} - {\hat{E}}_{k} | \geq τ) \cdot ({\hat{T}}_{i} - {\hat{T}}_{k}); h_{i} = \sum_{k = 1}^{m} H_{i k}$

LaTeX
H_{ik}=(\hat{E}_i-\hat{E}_k)+\psi(|\hat{E}_i-\hat{E}_k|\geq\tau)\cdot(\hat{T}_i-\hat{T}_k);\quad h_i=\sum_{k=1}^{m}H_{ik}
Alternatifleri hi'ye göre azalan sıraya göre sırala.

$rank descending by h_{i}$

LaTeX
\text{rank descending by }h_i

Sezgi

Fuzzy outranking/ranking - Triangular Fuzzy Number (TFN: l, m, u). Output typically utility (higher value = preferred).

Sonucu okuma: fuzzy-codas extends CODAS to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Varsayımlar

Decision matrix entries are valid Fuzzy (Triangular) 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 CODAS directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

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

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin TFN: l ≤ m ≤ u ve tümü ≥ 0 koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: centroid (l+m+u)/3 kanonik seçimdir.

Hesap adımları ve dayanakları

Extract TFN matrix x̃ij=(xijα,xijβ,xijγ), weights w̃j=(wjα,wjβ,wjγ).
Normalize: benefit r̃ij=(xijα/Mγj, xijβ/Mγj, xijγ/Mγj), Mγj=max_i(xijγ). Cost r̃ij=(mαj/xijγ, mαj/xijβ, mαj/xijα), mαj=min_i(xijα).
Weighted: ṽij=r̃ij⊗w̃j=(rα·wα, rβ·wβ, rγ·wγ).
NIS (negative ideal): n̊j=min_i COA(ṽij) per criterion. Euclidean Êi=√(Σj(COA(ṽij)-n̊j)²). Taxicab T̂i=Σj|COA(ṽij)-n̊j|.
Pairwise H̃ik=(Êi-Êk)+ψ(|Êi-Êk|≥τ)·(T̂i-T̂k) where τ=0.02. Appraisal score hi=Σk H̃ik.
Rank alternatives descending by hi.