Fuzzy MARCOS - MARCOS 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})
İdeal AI*j'yi tanımla: max kriteri için max_i(xijγ); min kriteri için min_i(xijα). Anti-ideal AIj: max kriteri için min_i(xijα); min kriteri için max_i(xijγ).

$A I_{j}^{*} = {\begin{matrix} {max}_{i} (x_{i j}^{γ}) & j \in Ω_{+} \\ {min}_{i} (x_{i j}^{α}) & j \in Ω_{-} \end{matrix}, A I_{j} = {\begin{matrix} {min}_{i} (x_{i j}^{α}) & j \in Ω_{+} \\ {max}_{i} (x_{i j}^{γ}) & j \in Ω_{-} \end{matrix}$

LaTeX
AI^*_j=\begin{cases}\max_i(x_{ij}^{\gamma})&j\in\Omega_{+}\\\min_i(x_{ij}^{\alpha})&j\in\Omega_{-}\end{cases},\quad AI_j=\begin{cases}\min_i(x_{ij}^{\alpha})&j\in\Omega_{+}\\\max_i(x_{ij}^{\gamma})&j\in\Omega_{-}\end{cases}
İdeale göre normalize et. Max kriteri: r̃ij=(xijα/AI*j, xijβ/AI*j, xijγ/AI*j). Min kriteri: r̃ij=(AI*j/xijγ, AI*j/xijβ, AI*j/xijα). AI* satırı (→tümü 1) ve AI satırına aynı normalizasyonu uygula.

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

LaTeX
\tilde{r}_{ij}=\begin{cases}\left(\dfrac{x_{ij}^{\alpha}}{AI^*_j},\dfrac{x_{ij}^{\beta}}{AI^*_j},\dfrac{x_{ij}^{\gamma}}{AI^*_j}\right)&j\in\Omega_{+}\\[6pt]\left(\dfrac{AI^*_j}{x_{ij}^{\gamma}},\dfrac{AI^*_j}{x_{ij}^{\beta}},\dfrac{AI^*_j}{x_{ij}^{\alpha}}\right)&j\in\Omega_{-}\end{cases}
Ağırlıklı matris ṽij=r̃ij⊗w̃j. Toplam fayda T̃i=⊕j ṽij=(Σwjαrjα, Σwjβrjβ, Σwjγrjγ).

${\tilde{v}}_{i j} = (r_{i j}^{α} w_{j}^{α}, r_{i j}^{β} w_{j}^{β}, r_{i j}^{γ} w_{j}^{γ}); {\tilde{T}}_{i} = ⨁_{j} {\tilde{v}}_{i j} = (\sum_{j} v_{i j}^{α}, \sum_{j} v_{i j}^{β}, \sum_{j} v_{i j}^{γ})$

LaTeX
\tilde{v}_{ij}=(r_{ij}^{\alpha}w_j^{\alpha},r_{ij}^{\beta}w_j^{\beta},r_{ij}^{\gamma}w_j^{\gamma});\quad\tilde{T}_i=\bigoplus_j\tilde{v}_{ij}=\left(\sum_j v_{ij}^{\alpha},\sum_j v_{ij}^{\beta},\sum_j v_{ij}^{\gamma}\right)
Fayda dereceleri: K̃i+ = T̃i⊘T̃AI* = (Tiα/TAI*γ, Tiβ/TAI*β, Tiγ/TAI*α). K̃i- = T̃i⊘T̃AI = (Tiα/TAIγ, Tiβ/TAIβ, Tiγ/TAIα). Bulanıklıktan arındır: ki+=COA(K̃i+), ki-=COA(K̃i-).

${\tilde{K}}_{i}^{+} = {\tilde{T}}_{i} ⊘ {\tilde{T}}_{A I^{*}}; {\tilde{K}}_{i}^{-} = {\tilde{T}}_{i} ⊘ {\tilde{T}}_{A I}; k_{i}^{\pm} = \frac{K_{i}^{α \pm} + K_{i}^{β \pm} + K_{i}^{γ \pm}}{3}$

LaTeX
\tilde{K}_i^+=\tilde{T}_i\oslash\tilde{T}_{AI^*};\quad\tilde{K}_i^-=\tilde{T}_i\oslash\tilde{T}_{AI};\quad k_i^{\pm}=\frac{K_i^{\alpha\pm}+K_i^{\beta\pm}+K_i^{\gamma\pm}}{3}
Fayda fonksiyonu: f(K̃i+)=ki+/(ki++ki-); f(K̃i-)=ki-/(ki++ki-).

$f ({\tilde{K}}_{i}^{+}) = \frac{k_{i}^{+}}{k_{i}^{+} + k_{i}^{-}}; f ({\tilde{K}}_{i}^{-}) = \frac{k_{i}^{-}}{k_{i}^{+} + k_{i}^{-}}$

LaTeX
f(\tilde{K}_i^+)=\frac{k_i^+}{k_i^++k_i^-};\qquad f(\tilde{K}_i^-)=\frac{k_i^-}{k_i^++k_i^-}
MARCOS skoru: Di = (ki++ki-)/(1+(1-f+)/f++(1-f-)/f-). Azalan sıraya göre sırala.

$D_{i} = \frac{k_{i}^{+} + k_{i}^{-}}{1 + \frac{1 - f ({\tilde{K}}_{i}^{+})}{f ({\tilde{K}}_{i}^{+})} + \frac{1 - f ({\tilde{K}}_{i}^{-})}{f ({\tilde{K}}_{i}^{-})}}; rank descending$

LaTeX
D_i=\frac{k_i^++k_i^-}{1+\dfrac{1-f(\tilde{K}_i^+)}{f(\tilde{K}_i^+)}+\dfrac{1-f(\tilde{K}_i^-)}{f(\tilde{K}_i^-)}};\quad\text{rank descending}

Sezgi

Sonucu okuma: fuzzy-marcos extends MARCOS 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 MARCOS 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.