Bulanık TOPSIS (Chen 2000) - Üçgensel bulanık sayılarla bulanık ortamda grup karar verme için TOPSIS uzantısı

Formül adımları

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

Yerel üçgen bulanık karar matrisini ve kriter ağırlıklarını oluşturun

$\tilde{X} = [{\tilde{x}}_{i j}]_{m \times n}; {\tilde{x}}_{i j} = (a_{i j}, b_{i j}, c_{i j}); \tilde{W} = [{\tilde{w}}_{1}, \dots, {\tilde{w}}_{n}]; {\tilde{w}}_{j} = (w_{j 1}, w_{j 2}, w_{j 3})$

LaTeX
\tilde{X} = [\tilde{x}_{ij}]_{m \times n};\quad \tilde{x}_{ij} = (a_{ij}, b_{ij}, c_{ij});\quad \tilde{W} = [\tilde{w}_1, \ldots, \tilde{w}_n];\ \tilde{w}_j = (w_{j1}, w_{j2}, w_{j3})
Adım 2 — Chen normalleştirmesi: fayda kriterleri için her TFN bileşenini c_j* = max_i c_ij ile böl; maliyet kriterleri için a_j^- = min_i a_ij ile resiprokal formu kullan.

$Benefit: {\tilde{r}}_{i j} = (\frac{a_{i j}}{c_{j}^{*}}, \frac{b_{i j}}{c_{j}^{*}}, \frac{c_{i j}}{c_{j}^{*}}), c_{j}^{*} = {max}_{i} c_{i j}; Cost: {\tilde{r}}_{i j} = (\frac{a_{j}^{-}}{c_{i j}}, \frac{a_{j}^{-}}{b_{i j}}, \frac{a_{j}^{-}}{a_{i j}}), a_{j}^{-} = {min}_{i} a_{i j}$

LaTeX
\text{Benefit: } \tilde{r}_{ij} = \left( \frac{a_{ij}}{c_j^{*}},\ \frac{b_{ij}}{c_j^{*}},\ \frac{c_{ij}}{c_j^{*}} \right),\ c_j^{*} = \max_i c_{ij};\quad \text{Cost: } \tilde{r}_{ij} = \left( \frac{a_j^{-}}{c_{ij}},\ \frac{a_j^{-}}{b_{ij}},\ \frac{a_j^{-}}{a_{ij}} \right),\ a_j^{-} = \min_i a_{ij}
Adım 3 — Ağırlıklandırılmış normalleştirilmiş bulanık matris: ṽ_ij = r̃_ij ⊗ w̃_j (bileşen-bileşen TFN çarpımı).

${\tilde{v}}_{i j} = {\tilde{r}}_{i j} \otimes {\tilde{w}}_{j} = (a_{i j}^{r} w_{j 1}, b_{i j}^{r} w_{j 2}, c_{i j}^{r} w_{j 3}) where {\tilde{r}}_{i j} = (a_{i j}^{r}, b_{i j}^{r}, c_{i j}^{r})$

LaTeX
\tilde{v}_{ij} = \tilde{r}_{ij} \otimes \tilde{w}_j = (a_{ij}^{r} w_{j1},\ b_{ij}^{r} w_{j2},\ c_{ij}^{r} w_{j3})\quad \text{where } \tilde{r}_{ij} = (a_{ij}^{r}, b_{ij}^{r}, c_{ij}^{r})
Adım 4 — Bulanık Pozitif-İdeal Çözüm (FPIS) A⁺ = (ṽ_1*, …, ṽ_n*), ṽ_j* = (1, 1, 1); Bulanık Negatif-İdeal Çözüm (FNIS) A⁻, ṽ_j⁻ = (0, 0, 0).

$A^{+} = ({\tilde{v}}_{1}^{*}, \dots, {\tilde{v}}_{n}^{*}), {\tilde{v}}_{j}^{*} = (1, 1, 1) \forall j; A^{-} = ({\tilde{v}}_{1}^{-}, \dots, {\tilde{v}}_{n}^{-}), {\tilde{v}}_{j}^{-} = (0, 0, 0) \forall j$

LaTeX
A^{+} = (\tilde{v}_1^{*}, \ldots, \tilde{v}_n^{*}),\ \tilde{v}_j^{*} = (1, 1, 1)\ \forall j;\quad A^{-} = (\tilde{v}_1^{-}, \ldots, \tilde{v}_n^{-}),\ \tilde{v}_j^{-} = (0, 0, 0)\ \forall j
Adım 5 — Vertex yöntemiyle FPIS ve FNIS'e bulanık uzaklıklar: d(ã, b̃) = √((1/3)·[(a₁−b₁)² + (a₂−b₂)² + (a₃−b₃)²]); tüm kriterler üzerinden topla.

$d (\tilde{a}, \tilde{b}) = \sqrt{\frac{1}{3} [(a_{1} - b_{1})^{2} + (a_{2} - b_{2})^{2} + (a_{3} - b_{3})^{2}]}; [4pt] d_{i}^{+} = \sum_{j = 1}^{n} d ({\tilde{v}}_{i j}, {\tilde{v}}_{j}^{*}), d_{i}^{-} = \sum_{j = 1}^{n} d ({\tilde{v}}_{i j}, {\tilde{v}}_{j}^{-})$

LaTeX
d(\tilde{a}, \tilde{b}) = \sqrt{\frac{1}{3} \left[ (a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2 \right]};\\[4pt] d_i^{+} = \sum_{j=1}^{n} d(\tilde{v}_{ij}, \tilde{v}_j^{*}),\quad d_i^{-} = \sum_{j=1}^{n} d(\tilde{v}_{ij}, \tilde{v}_j^{-})
Adım 6 — Yakınlık katsayısı CC_i = d_i⁻ / (d_i⁺ + d_i⁻); alternatifleri CC'ye göre azalan sırada sırala.

$C C_{i} = \frac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}}; rank by descending C C_{i}$

LaTeX
CC_i = \frac{d_i^{-}}{d_i^{+} + d_i^{-}};\quad \text{rank by descending } CC_i

Sezgi

Distance-based (fuzzy, vertex metric). Output typically utility (higher value = preferred).

Sonucu okuma: CC ∈ [0, 1]. Higher CC means the alternative is closer to the FPIS and farther from the FNIS under the fuzzy vertex distance. Rank by descending CC.

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 TOPSIS-CHEN2000 directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: inherits from TOPSIS (Belton-Gear-1983))
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

Bu varyant, beyan edilen bileşen bazlı bulanık normalizasyonu kullanır. Keskin vektör normalizasyonu ikamesi desteklenmez. Ağırlıklı üst uç noktaların bire eşit olması gerekmez.
TFN'leri Adım 5'ten önce durulamak bulanık belirsizliği yıkar ve Bulanık TOPSIS'i klasik TOPSIS'e indirir - durulamayı (varsa) yalnız raporlamaya ertele.

Hesap adımları ve dayanakları

Construct the fuzzy decision matrix X̃ from linguistic ratings (or numeric TFNs) and the fuzzy weight vector W̃.

Dayanak: Chen 2000, p.3 Eqs.(2)-(3)
Chen's normalisation: for benefit criteria divide each TFN component by c_j* = max_i c_ij; for cost criteria use the reciprocal form with a_j^- = min_i a_ij.

Dayanak: Chen 2000, p.3 Eq.(4)
Weighted normalised fuzzy matrix: ṽ_ij = r̃_ij ⊗ w̃_j (component-wise TFN multiplication).

Dayanak: Chen 2000, p.4 Eq.(7)
Define the Fuzzy Positive-Ideal Solution (FPIS) A⁺ = (ṽ_1*, …, ṽ_n*) with ṽ_j* = (1, 1, 1) and the Fuzzy Negative-Ideal Solution (FNIS) A⁻ with ṽ_j⁻ = (0, 0, 0).

Dayanak: Chen 2000, p.4 Eqs.(8)-(9)
Compute fuzzy distances from FPIS and FNIS using the vertex method: d(ã, b̃) = √((1/3)·[(a₁−b₁)² + (a₂−b₂)² + (a₃−b₃)²]); sum over all criteria.

Dayanak: Chen 2000, p.4 Eqs.(10)-(11)
Closeness coefficient CC_i = d_i⁻ / (d_i⁺ + d_i⁻); rank alternatives by CC descending.

Dayanak: Chen 2000, p.4 Eq.(12)