Fuzzy MABAC - MABAC 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.

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

LaTeX
X̃=[x̃_{ij}],\quad x̃_{ij}=(x_{ij}^{\alpha},x_{ij}^{\beta},x_{ij}^{\gamma});\quad\tilde{w}_j=(w_j^{\alpha},w_j^{\beta},w_j^{\gamma})
Doğrusal bulanık normalleştirme. x+j=max_i(xijγ), x-j=min_i(xijα). Fayda kriteri için pay (xijα-x-j), maliyet kriteri için pay (x+j-xijγ) olarak hesaplanır; payda her iki durumda da (x+j-x-j).

${\tilde{r}}_{i j} = {\begin{matrix} (\frac{x_{i j}^{α} - x_{j}^{-}}{x_{j}^{+} - x_{j}^{-}}, \frac{x_{i j}^{β} - x_{j}^{-}}{x_{j}^{+} - x_{j}^{-}}, \frac{x_{i j}^{γ} - x_{j}^{-}}{x_{j}^{+} - x_{j}^{-}}) & j \in Ω_{+} \\ [6pt] (\frac{x_{j}^{+} - x_{i j}^{γ}}{x_{j}^{+} - x_{j}^{-}}, \frac{x_{j}^{+} - x_{i j}^{β}}{x_{j}^{+} - x_{j}^{-}}, \frac{x_{j}^{+} - x_{i j}^{α}}{x_{j}^{+} - x_{j}^{-}}) & j \in Ω_{-} \end{matrix}$

LaTeX
\tilde{r}_{ij}=\begin{cases}\left(\dfrac{x_{ij}^{\alpha}-x_j^{-}}{x_j^{+}-x_j^{-}},\dfrac{x_{ij}^{\beta}-x_j^{-}}{x_j^{+}-x_j^{-}},\dfrac{x_{ij}^{\gamma}-x_j^{-}}{x_j^{+}-x_j^{-}}\right)&j\in\Omega_{+}\\[6pt]\left(\dfrac{x_j^{+}-x_{ij}^{\gamma}}{x_j^{+}-x_j^{-}},\dfrac{x_j^{+}-x_{ij}^{\beta}}{x_j^{+}-x_j^{-}},\dfrac{x_j^{+}-x_{ij}^{\alpha}}{x_j^{+}-x_j^{-}}\right)&j\in\Omega_{-}\end{cases}
Ağırlıklı matris: ṽij = (1̃+r̃ij)⊗w̃j = ((1+rijα)wjα, (1+rijβ)wjβ, (1+rijγ)wjγ).

${\tilde{v}}_{i j} = ((1 + r_{i j}^{α}) w_{j}^{α}, (1 + r_{i j}^{β}) w_{j}^{β}, (1 + r_{i j}^{γ}) w_{j}^{γ})$

LaTeX
\tilde{v}_{ij}=((1+r_{ij}^{\alpha})w_j^{\alpha},\;(1+r_{ij}^{\beta})w_j^{\beta},\;(1+r_{ij}^{\gamma})w_j^{\gamma})
Sınır Yaklaşım Alanı: m alternatif üzerinden bileşen bazlı geometrik ortalama g̃j. gjk=(Π_i vijkᵏ)^(1/m), k∈{α,β,γ}. Mesafe: D̃ij=ṽij⊖g̃j=(vijα−gjγ, vijβ−gjβ, vijγ−gjα).

$g_{j}^{k} = {(\prod_{i = 1}^{m} v_{i j}^{k})}^{1 / m}, k \in {α, β, γ}; {\tilde{D}}_{i j} = (v_{i j}^{α} - g_{j}^{γ}, v_{i j}^{β} - g_{j}^{β}, v_{i j}^{γ} - g_{j}^{α})$

LaTeX
g_j^k=\left(\prod_{i=1}^{m}v_{ij}^k\right)^{1/m},\;k\in\{\alpha,\beta,\gamma\};\quad \tilde{D}_{ij}=(v_{ij}^{\alpha}-g_j^{\gamma},\;v_{ij}^{\beta}-g_j^{\beta},\;v_{ij}^{\gamma}-g_j^{\alpha})
Değerlendirme skoru Qi=Σj COA(D̃ij). Azalan sıraya göre sırala.

$Q_{i} = \sum_{j = 1}^{n} \frac{D_{i j}^{α} + D_{i j}^{β} + D_{i j}^{γ}}{3}; rank descending$

LaTeX
Q_i=\sum_{j=1}^{n}\frac{D_{ij}^{\alpha}+D_{ij}^{\beta}+D_{ij}^{\gamma}}{3};\quad\text{rank descending}

Sezgi

Sonucu okuma: fuzzy-mabac extends MABAC 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 MABAC directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

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.