HF-ELECTRE-II

HF-ELECTRE II - Kararsız Bulanık ELECTRE II (Chen-Xu 2015)

Kararsız Bulanık Elemanlar için iki geçişli dışlama sıralaması; HF uyum indeksleri üzerinden güçlü/zayıf dışlama ilişkileri; iki ön-sıralamadan kesişim ile nihai sıralama.

Formül adımları

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

Collect HFE evaluations from DMs. Specify criterion weights and seven attitude weights determining sensitivity to concordance/discordance strength levels. HFEs are extended to equal length using the optimistic rule (add max value) when comparing pairs.

$H = [h_{i} j]_{m \times n} (E q .4); W = (w_{1}, . . ., w_{n})^{T} w i t h Σ w_{j} = 1; χ = (χ_{C}, χ_{C^{'}}, χ_{C^{″}}, χ_{D}, χ_{D^{'}}, χ_{D^{″}}, χ_{J} =)^{T} — a t t i t u d e w e i g h t s f o r s t r o n g / m e d i u m / w e a k c o n c o r d a n c e, d i s c o r d a n c e, a n d i n d i f f e r e n t s e t s$

LaTeX
H = [h_ij]_{m×n} (Eq.4); W = (w_1,...,w_n)^T with Σw_j=1; χ = (χ_C, χ_{C'}, χ_{C''}, χ_D, χ_{D'}, χ_{D''}, χ_J=)^T — attitude weights for strong/medium/weak concordance, discordance, and indifferent sets
Score = mean membership; deviation = standard deviation. Used to classify concordance/discordance. For HFE length extension: add max element of shorter HFE until lengths match (optimistic principle).

$s (h) = (1 / l_{h}) \cdot Σ_{c \in h} c (D e f .2); r (h) = s q r t ((1 / l_{h}) \cdot Σ_{c \in h} (c - s (h)) ²) (E q .5)$

LaTeX
s(h) = (1/l_h)·Σ_{c∈h} c (Def.2); r(h) = sqrt((1/l_h)·Σ_{c∈h}(c − s(h))²) (Eq.5)
Strong concordance: A_k beats A_l in both score and consistency. Medium: beats in score, lower or equal consistency. Weak: same score but A_k is more consistent. Strength order: J^C > J^{C'} > J^{C''}.

$J_{k l}^{C} = j | s (h_{k} j) > s (h_{l} j) A N D r (h_{k} j) < r (h_{l} j) (E q .6); J_{k l}^{C^{'}} = j | s (h_{k} j) > s (h_{l} j) A N D r (h_{k} j) \geq r (h_{l} j) (E q .7); J_{k l}^{C^{″}} = j | s (h_{k} j) = s (h_{l} j) A N D r (h_{k} j) < r (h_{l} j) (E q .8)$

LaTeX
J^C_{kl} = {j | s(h_kj) > s(h_lj) AND r(h_kj) < r(h_lj)} (Eq.6); J^{C'}_{kl} = {j | s(h_kj) > s(h_lj) AND r(h_kj) ≥ r(h_lj)} (Eq.7); J^{C''}_{kl} = {j | s(h_kj) = s(h_lj) AND r(h_kj) < r(h_lj)} (Eq.8)
Strong discordance: A_k is worse in both score and consistency. Medium: worse in score only. Weak: same score but less consistent. Strength order: J^D > J^{D'} > J^{D''}.

$J_{k l}^{D} = j | s (h_{k} j) < s (h_{l} j) A N D r (h_{k} j) > r (h_{l} j) (E q .9); J_{k l}^{D^{'}} = j | s (h_{k} j) < s (h_{l} j) A N D r (h_{k} j) \leq r (h_{l} j) (E q .10); J_{k l}^{D^{″}} = j | s (h_{k} j) = s (h_{l} j) A N D r (h_{k} j) > r (h_{l} j) (E q .11)$

LaTeX
J^D_{kl} = {j | s(h_kj) < s(h_lj) AND r(h_kj) > r(h_lj)} (Eq.9); J^{D'}_{kl} = {j | s(h_kj) < s(h_lj) AND r(h_kj) ≤ r(h_lj)} (Eq.10); J^{D''}_{kl} = {j | s(h_kj) = s(h_lj) AND r(h_kj) > r(h_lj)} (Eq.11)
Criteria where A_k and A_l are perfectly equivalent in both score and deviation.

$J_{k l}^{=} = j | s (h_{k} j) = s (h_{l} j) A N D r (h_{k} j) = r (h_{l} j) (E q .12)$

LaTeX
J^=_{kl} = {j | s(h_kj) = s(h_lj) AND r(h_kj) = r(h_lj)} (Eq.12)
Weighted sum of criterion weights in concordance and indifferent sets, modulated by attitude weights. c_{kl} ∈ [0,1]. Large c_{kl} → A_k is superior to A_l.

$c_{k l} = χ_{C} \cdot Σ_{j \in J^{C}} w_{j} + χ_{C^{'}} \cdot Σ_{j \in J^{C^{'}}} w_{j} + χ_{C^{″}} \cdot Σ_{j \in J^{C^{″}}} w_{j} + χ_{J =} \cdot Σ_{j \in J^{=}} w_{j} (E q .13); C = [c_{k l}] (E q .14)$

LaTeX
c_{kl} = χ_C·Σ_{j∈J^C} w_j + χ_{C'}·Σ_{j∈J^{C'}} w_j + χ_{C''}·Σ_{j∈J^{C''}} w_j + χ_{J=}·Σ_{j∈J^=} w_j (Eq.13); C = [c_{kl}] (Eq.14)
Discordance index is max weighted distance on discordant criteria normalized by max distance across all criteria. d_{kl} ∈ [0,1]. HFE comparison uses optimistic extension (add max element).

$d_{1} (a, b) = (1 / l) \cdot Σ | a^{q (i)} - b^{q (i)} | (E q .2, H a m m i n g) o r d_{2} = s q r t - v e r s i o n (E q .3, E u c l i d e a n); d_{k l} = m a x χ_{D} \cdot d (w_{j} h_{k j}, w_{j} h_{l j}) : j \in J^{D} \cup J^{D^{'}} \cup J^{D^{″}} / m a x_{j} d (w_{j} h_{k j}, w_{j} h_{l j}) (E q .15); D = [d_{k l}] (E q .16)$

LaTeX
d_1(a,b) = (1/l)·Σ|a^{q(i)} − b^{q(i)}| (Eq.2, Hamming) or d_2 = sqrt-version (Eq.3, Euclidean); d_{kl} = max{χ_D·d(w_j h_{kj}, w_j h_{lj}): j∈J^D∪J^{D'}∪J^{D''}} / max_j d(w_j h_{kj}, w_j h_{lj}) (Eq.15); D = [d_{kl}] (Eq.16)
Three concordance thresholds c* > c' > c⁻ and two discordance thresholds d' < d* are user-set (0 < c⁻ < c' < c* < 1; 0 < d' < d* < 1). Strong outranking requires strict conditions; weak is more permissive.

$A_{k} S F A_{l} (s t r o n g) i f f [C (A_{k}, A_{l}) \geq c * A N D D (A_{k}, A_{l}) \leq d * A N D C (A_{k}, A_{l}) \geq C (A_{l}, A_{k})] O R [C (A_{k}, A_{l}) \geq c^{'} A N D D (A_{k}, A_{l}) \leq d^{'} A N D C (A_{k}, A_{l}) \geq C (A_{l}, A_{k})] (E q .17); A_{k} S f A_{l} (w e a k) i f f C (A_{k}, A_{l}) \geq c ⁻ A N D D (A_{k}, A_{l}) \leq d * A N D C (A_{k}, A_{l}) \geq C (A_{l}, A_{k}) (E q .18)$

LaTeX
A_k SF A_l (strong) iff [C(A_k,A_l) ≥ c* AND D(A_k,A_l) ≤ d* AND C(A_k,A_l) ≥ C(A_l,A_k)] OR [C(A_k,A_l) ≥ c' AND D(A_k,A_l) ≤ d' AND C(A_k,A_l) ≥ C(A_l,A_k)] (Eq.17); A_k Sf A_l (weak) iff C(A_k,A_l) ≥ c⁻ AND D(A_k,A_l) ≤ d* AND C(A_k,A_l) ≥ C(A_l,A_k) (Eq.18)
Build strong and weak outranking directed graphs. Run forward distillation (best → worst) and reverse distillation (worst → best) using both graphs. Average the two rankings. The alternative with lowest m(x) is best.

$m^{'} (x) = f o r w a r d r a n k i n g s c o r e (r a n k i n d e s c e n d i n g o u t r a n k i n g p a t h); m^{″} (x) = r e v e r s e r a n k i n g s c o r e; m (x) = (m^{'} (x) + m^{″} (x)) / 2$

LaTeX
m'(x) = forward ranking score (rank in descending outranking path); m''(x) = reverse ranking score; m(x) = (m'(x) + m''(x)) / 2

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: HF-ELECTRE-II ranks alternatives based on performance scores. Higher score = better rank.