Fuzzy ELECTRE III - ELECTRE-III yönteminin Fuzzy uzantısı

Formül adımları

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

Adım 1 — Fuzzy karar matrisi X̃ oluştur.

$\tilde{X} = [{\tilde{x}}_{i j}]$

LaTeX
\tilde{X} = [\tilde{x}_{ij}]
Adım 2 — Performans ve eşikleri centroid ile defuzzify et.

$g_{j} (A_{i}) = x_{i j}^{*} = (l_{i j} + m_{i j} + u_{i j}) / 3; q_{j}, p_{j}, v_{j} analogously if fuzzy$

LaTeX
g_j(A_i) = x_{ij}^{*} = (l_{ij}+m_{ij}+u_{ij})/3; \quad q_j, p_j, v_j \text{ analogously if fuzzy}
Adım 3 — Her kriter için eşik sırası doğrulaması.

$0 \leq q_{j} \leq p_{j} \leq v_{j} \forall j$

LaTeX
0 \leq q_j \leq p_j \leq v_j \quad \forall j
Adım 4 — Kriter bazlı concordance c_j(a,b).

$Δ_{j} (a, b) = g_{j} (A_{b}) - g_{j} (A_{a}); c_{j} (a, b) = {\begin{matrix} 1 & Δ_{j} \leq q_{j} \\ 0 & Δ_{j} \geq p_{j} \\ (p_{j} - Δ_{j}) / (p_{j} - q_{j}) & q_{j} < Δ_{j} < p_{j} \end{matrix}$

LaTeX
\Delta_j(a,b)=g_j(A_b)-g_j(A_a); \quad c_j(a,b)=\begin{cases}1 & \Delta_j\leq q_j\\0 & \Delta_j\geq p_j\\(p_j-\Delta_j)/(p_j-q_j) & q_j<\Delta_j<p_j\end{cases}
Adım 5 — Global concordance C(a,b).

$C (a, b) = \sum_{j = 1}^{n} w_{j} c_{j} (a, b) (\sum_{j} w_{j} = 1)$

LaTeX
C(a,b) = \sum_{j=1}^{n} w_j c_j(a,b) \quad (\sum_j w_j = 1)
Adım 6 — Kriter bazlı discordance d_j(a,b).

$d_{j} (a, b) = {\begin{matrix} 0 & Δ_{j} \leq p_{j} \\ 1 & Δ_{j} \geq v_{j} \\ (Δ_{j} - p_{j}) / (v_{j} - p_{j}) & p_{j} < Δ_{j} < v_{j} \end{matrix}$

LaTeX
d_j(a,b)=\begin{cases}0 & \Delta_j\leq p_j\\1 & \Delta_j\geq v_j\\(\Delta_j-p_j)/(v_j-p_j) & p_j<\Delta_j<v_j\end{cases}
Adım 7 — Credibility σ(a,b).

$σ (a, b) = C (a, b) \prod_{j \in J (a, b)} \frac{1 - d_{j} (a, b)}{1 - C (a, b)}; J (a, b) = {j ∣ d_{j} (a, b) > C (a, b)}$

LaTeX
\sigma(a,b) = C(a,b) \prod_{j \in J(a,b)} \dfrac{1-d_j(a,b)}{1-C(a,b)}; \quad J(a,b) = \{j \mid d_j(a,b) > C(a,b)\}
Adım 8 — Credibility matrisi Σ.

$Σ = [σ (a, b)]_{m \times m}, σ (a, a) = -$

LaTeX
\Sigma = [\sigma(a,b)]_{m \times m}, \quad \sigma(a,a) = -
Adım 9 — Kanonik Roy 1978 ayırma eşiği (adaptif, sabit λ değil). λ_max = en büyük credibility; s(λ_max) = 0.30 − 0.15·λ_max; λ_L = (λ_max − s) altındaki en yüksek credibility; outranking ⇔ σ(i,j) > λ_L ∧ σ(i,j) > σ(j,i) + s.

$λ_{max} = {max}_{i \neq j} σ (i, j); s (λ_{max}) = α - β λ_{max} (α = 0.30, β = 0.15); λ_{L} = max {σ (i, j) : σ (i, j) < λ_{max} - s}; r (i, j) = 1 [σ (i, j) > λ_{L} \land σ (i, j) > σ (j, i) + s]$

LaTeX
\lambda_{\max} = \max_{i\neq j} \sigma(i,j); \quad s(\lambda_{\max}) = \alpha - \beta \lambda_{\max} \;\; (\alpha=0.30, \beta=0.15); \quad \lambda_L = \max\{\sigma(i,j): \sigma(i,j) < \lambda_{\max} - s\}; \quad r(i,j) = \mathbb{1}\big[\sigma(i,j) > \lambda_L \;\wedge\; \sigma(i,j) > \sigma(j,i) + s\big]
Adım 10 — Descending distillation: tüm Σ üzerinde qualification q(i) = giden−gelen outranking sayısı; max-q sınıfını çek (eşitlikte alt-matrikste yeniden hesapla), tepeden sıraya ekle, sil; tekrarla.

$q (i) = \sum_{j \neq i} r (i, j) - \sum_{j \neq i} r (j, i); {class}_{t} = \arg {max}_{i} q (i) (tie: recompute on sub-credibility); append, remove, iterate$

LaTeX
q(i) = \sum_{j\neq i} r(i,j) - \sum_{j\neq i} r(j,i); \quad \text{class}_t = \arg\max_i q(i) \text{ (tie: recompute on sub-credibility)}; \quad \text{append, remove, iterate}
Adım 11 — Ascending distillation: F10 simetrik, min-q sınıfını çek (kötüden başlayarak), ters çevir.

$Symmetric to F10 with \arg {min}_{i} q (i); reverse final order to obtain ascending preorder$

LaTeX
\text{Symmetric to F10 with } \arg\min_i q(i); \quad \text{reverse final order to obtain ascending preorder}
Adım 12 — İki ön-sıralamanın kesişimi → nihai preorder.

$F i n a l_p r e o r d e r = D e s c e n d i n g \cap A s c e n d i n g; relations: P (preferred), I (indifferent), R (incomparable)$

LaTeX
Final\_preorder = Descending \cap Ascending; \text{ relations: } P \text{ (preferred)}, I \text{ (indifferent)}, R \text{ (incomparable)}

Sezgi

Sonucu okuma: fuzzy-electre-iii extends ELECTRE-III 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

Thresholds (q, p, v) can be expressed in Fuzzy (Triangular) scale
Veto + concordance semantics adapted to fuzzy arithmetic

Ne zaman kullanılmaz

Small dataset (m<3) - outranking machinery underutilised

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

Build/normalize decision matrix m×n (Montazer §6 Table 2 normalized 4 vendors × 6 criteria). Define per-criterion thresholds q_j ≤ p_j ≤ v_j (Montazer used identical 0.2 ≤ 0.5 ≤ 0.9 across all criteria) and weights w_j (Montazer used q_1=7,q_2=4,q_3=5,q_4=3,q_5=2,q_6=5 then normalize).
For each ordered pair (a,b), compute coalitions J^S = {j: g_j(b)-g_j(a) ≤ q_j} (strong support), J^Q = {j: q_j < g_j(b)-g_j(a) ≤ p_j} (weak support). Eqs.5-6.
Partial concordance per criterion c_j(a,b) - 1 if j∈J^S, linear ramp if j∈J^Q, 0 otherwise. Eq.7.
Aggregate to comprehensive concordance c(a,b) = Σ w_j · c_j(a,b). Eq.8. (Montazer §6 Table 3 shows c values mostly 1.0 with some 0.88-0.99.)
Per-criterion discordance d_j(a,b) - 0 if Δ ≤ p_j, linear ramp if p_j<Δ<v_j, 1 if Δ ≥ v_j. Eq.9. (Montazer §6 Table 4: ALL ZEROS for OIEC fixture - no veto triggered, comfortable thresholds.)
Compute V = {j: d_j > c}; if V empty, ρ = c (Eq.11). Else compute D_j = (1-d_j)/(1-c) for j∈V.
Build triangular fuzzy credibility ρ(a,b) = [C⊖D, C•D, C∩D] via three intersection methods: bounded subtraction (Eq.14, lower) gives a; algebraic product (Eq.13, middle) gives m; natural definition min (Eq.12, upper) gives b. Eq.16. Lemma 1: a ≤ m ≤ b. (Montazer §6 Table 5 has full ρ(a,b) for 4×4 = 16 pairs.)
Form outranking strength sets φ_1(a_i) = {ρ(a_i, a_j) : j≠i} (a_i outranking), φ_2(a_i) = {ρ(a_j, a_i) : j≠i} (a_i being-outranked). Eqs.17-18.
Triangular representatives Index1 = [Yager(min ρ over φ_1), Yager(avg ρ over φ_1), Yager(max ρ over φ_1)]; Index2 symmetric over φ_2. Eqs.19-20. (Montazer §6 Tables 6-7 give Min/Avg/Max with Yager defuzz per vendor.)
Net strength Q(a) = Index1 - Index2; Yager defuzzify Q(a) per Eq.21: Yager(triangle [a,m,b]) = ((3·a) - (m-a) + (b-m))/D where D is normalization factor (paper text). Rank by descending Yager(Q). (Montazer Table 8: Q(vendor 1)=0.008, Q(vendor 2)=0.009, Q(vendor 3)=0.026, Q(vendor 4)=-0.021 → ranking 3≻2≻1≻4.)
Optional final weighted score for multi-bid pipeline: A_v = Σq_i·A_i/Σq_i (Eq.23) where q_i = factor weights, A_i = match score per criterion. Used for system-level scoring in Montazer 2009 §6.1 fuzzy expert evaluation module.