HF-MARCOS - Kararsız Bulanık MARCOS

Formül adımları

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

Adım 1 — Girdileri doğrula: uzman bazlı HFE karar matrisleri, kriter ağırlıkları, kriter yönleri, uzman ağırlıkları.

$C h e c k E - 1 . . E - 4; a b o r t o n e r r o r .$

LaTeX
Check E-1..E-4; abort on error.
Adım 2 — K uzman HFE matrisini HFWA ile tek HFE matrisi H'ye topla.

$h_{i} j = H F W A_{l} a m b d a (h_{i} j^{(} 1), . . ., h_{i} j^{(} K)) = u n i o n_{g a m m a_{k} i n h_{i} j^{(} k)} 1 - p r o d_{k = 1}^{K} (1 - g a m m a_{k})^{l a m b d a_{k}}$

LaTeX
h_ij = HFWA_lambda(h_ij^(1),...,h_ij^(K)) = union_{gamma_k in h_ij^(k)} { 1 - prod_{k=1}^K (1 - gamma_k)^{lambda_k} }
Adım 3 — Her hücre için HFE skor x_ij = S(h_ij) = (1/|h_ij|) Σ γ.

$x_{i} j = S (h_{i} j) = (1 / | h_{i} j |) * s u m_{g a m m a i n h_{i} j} g a m m a$

LaTeX
x_ij = S(h_ij) = (1/|h_ij|) * sum_{gamma in h_ij} gamma
Adım 4 — Kriter başına AI/AAI: fayda yönlü C_j için AI_j = max x_ij, AAI_j = min x_ij; maliyet yönlü için tersi. Genişletilmiş X_ext ((m+2)×n) oluştur.

$A I_{j} = m a x_{i} x_{i} j (b e n e f i t) | m i n_{i} x_{i} j (c o s t); A A I_{j} = m i n_{i} x_{i} j (b e n e f i t) | m a x_{i} x_{i} j (c o s t)$

LaTeX
AI_j = max_i x_ij (benefit) | min_i x_ij (cost); AAI_j = min_i x_ij (benefit) | max_i x_ij (cost)
Adım 5 — X_ext'i AI'ye karşı normalleştir: fayda için n_ij = x_ij / AI_j; maliyet için n_ij = AI_j / x_ij.

$n_{i} j = x_{i} j / A I_{j} (b e n e f i t) | A I_{j} / x_{i} j (c o s t); r e p l a c e x w i t h m a x (x, e p s i l o n_{z} e r o) f o r c o s t d e n o m i n a t o r s$

LaTeX
n_ij = x_ij / AI_j (benefit) | AI_j / x_ij (cost); replace x with max(x, epsilon_zero) for cost denominators
Adım 6 — Ağırlıklı normalleştirilmiş matris v_ij = w_j · n_ij; satır toplamı S_i.

$v_{i} j = w_{j} * n_{i} j; S_{i} = s u m_{j} v_{i} j$

LaTeX
v_ij = w_j * n_ij; S_i = sum_j v_ij
Adım 7 — Fayda oranları, fayda fonksiyonları, nihai fayda f(K_i); azalan sırala.

$K p_{i} = S_{i} / S_{A} I; K m_{i} = S_{i} / S_{A} A I; f (K p_{i}) = K m_{i} / (K p_{i} + K m_{i}); f (K m_{i}) = K p_{i} / (K p_{i} + K m_{i}); f (K_{i}) = (K p_{i} + K m_{i}) / (1 + (1 - f (K p_{i})) / f (K p_{i}) + (1 - f (K m_{i})) / f (K m_{i}))$

LaTeX
Kp_i = S_i/S_AI; Km_i = S_i/S_AAI; f(Kp_i) = Km_i/(Kp_i+Km_i); f(Km_i) = Kp_i/(Kp_i+Km_i); f(K_i) = (Kp_i+Km_i) / (1 + (1-f(Kp_i))/f(Kp_i) + (1-f(Km_i))/f(Km_i))

Sezgi

Hesitant Fuzzy compromise ranking - MARCOS extended via Hesitant Fuzzy Elements (HFE). Output typically utility (higher value = preferred).

Sonucu okuma: HF-MARCOS extends crisp MARCOS (Stević 2020) to hesitant fuzzy data. Each cell is an HFE - a finite set of membership degrees in [0,1] representing multiple expert assessments or hesitation. Group decisions are aggregated cell-wise via HFWA (Xia-Xu 2011) before the MARCOS pipeline. HFEs are then collapsed to crisp scores via mean (default) prior to AI/AAI construction. The compromise utility f(K_i) balances proximity to ideal and distance from anti-ideal; higher is better.

Varsayımlar

Each cell is a non-empty HFE with values in [0,1]
Compensation across criteria is acceptable (MARCOS is fully compensatory)
All experts share the same scale/interpretation of HFE membership degrees
Criterion weights are externally sourced (HF-MARCOS does not produce weights)

Ne zaman kullanılmaz

Classical data sufficient - use base MARCOS directly
Non-compensatory or outranking-style ranking is required (use ELECTRE/PROMETHEE family)
Probability distributions are available - prefer probabilistic linguistic / interval-valued methods

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: Inherited from MARCOS family; cf. Stević et al. (2020) Sec. 3.3 on AI/AAI sensitivity to alternative set composition.)
Assumes: Each cell is a non-empty HFE with values in [0,1]
Assumes: Compensation across criteria is acceptable (MARCOS is fully compensatory)
Assumes: All experts share the same scale/interpretation of HFE membership degrees
Assumes: Criterion weights are externally sourced (HF-MARCOS does not produce weights)

Sık yapılan hatalar

Boş HFE: her hücrede en az bir γ ∈ [0,1] olmalı.
Stević 2020 (crisp MARCOS) ile HF-MARCOS'u karıştırmak - Li-Geng-Yuan 2023 HF uzantısıdır.
HFWA toplamadan önce HFE defuzzifiye etmek uzmanlar-arası belirsizliği siler - önce topla.

Hesap adımları ve dayanakları

Validate inputs: per-expert HFE decision matrices {H^(k)}_{k=1..K}, criterion weights w, criterion directions, expert weights λ.

Dayanak: Li-Geng-Yuan 2023, §3.2
Aggregate K expert HFE matrices into a single HFE matrix H via HFWA: h_ij = HFWA_λ(h_ij^(1),…,h_ij^(K)) = ∪_{γ_k ∈ h_ij^(k)} { 1 − ∏_{k=1}^K (1 − γ_k)^{λ_k} }. (K=1 → identity)

Dayanak: Xia-Xu 2011 Def. 5; Li-Geng-Yuan 2023 §3.2 Eq.(7)
Compute HFE score x_ij = S(h_ij) = (1/|h_ij|) Σ_{γ ∈ h_ij} γ for every cell.

Dayanak: Xia-Xu 2011 Def. 6; Li-Geng-Yuan 2023 §3.2 Eq.(8)
Construct AI (ideal) and AAI (anti-ideal) per criterion: benefit C_j: AI_j = max_i x_ij, AAI_j = min_i x_ij; cost C_j: AI_j = min_i x_ij, AAI_j = max_i x_ij. Form extended matrix X_ext of shape (m+2)×n with rows [AAI; A_1; …; A_m; AI].

Dayanak: Stević 2020 Eq.(1); Li-Geng-Yuan 2023 §3.3 Eq.(9)
Normalise X_ext against AI: benefit C_j: n_ij = x_ij / AI_j; cost C_j: n_ij = AI_j / x_ij. Guard against division by zero via epsilon_zero.

Dayanak: Stević 2020 Eq.(2); Li-Geng-Yuan 2023 §3.3 Eq.(10)
Weighted normalised matrix v_ij = w_j · n_ij; row-sum to get S_i = Σ_j v_ij for each row of X_ext (including AAI and AI rows).

Dayanak: Stević 2020 Eqs.(3)-(4); Li-Geng-Yuan 2023 §3.3 Eqs.(11)-(12)
Utility ratios: K^+_i = S_i / S_AI; K^-_i = S_i / S_AAI. Utility functions: f(K^+_i) = K^-_i / (K^+_i + K^-_i); f(K^-_i) = K^+_i / (K^+_i + K^-_i). Final utility f(K_i) = (K^+_i + K^-_i) / (1 + (1 − f(K^+_i))/f(K^+_i) + (1 − f(K^-_i))/f(K^-_i)). Rank by f(K_i) descending.

Dayanak: Stević 2020 Eqs.(5)-(9); Li-Geng-Yuan 2023 §3.3 Eqs.(13)-(17)