HF-ELECTRE-I

HF-ELECTRE I - Kararsız Bulanık ELECTRE I (Chen-Xu-Xia 2015)

Kararsız Bulanık Elemanlar için uyum/uyumsuzluk dışlama sıralaması; skor fonksiyonu ve sapma derecesine dayalı HF uyum ve uyumsuzluk indeksleri; seçim için güvenilirlik matrisi.

Formül adımları

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

Collect HFE evaluations h_ij from a group of DMs. Specify criterion importance weights w_k and the four relative weights for strong/weak concordance and discordance sets.

$H = [h_{i} j]_{m \times n}; w = (w_{1}, . . ., w_{n})^{T} w i t h Σ w_{k} = 1; ω = (ω_{C}, ω_{C}^{'}, ω_{D}, ω_{D}^{'})^{T} f o r c o n c o r d a n c e / d i s c o r d a n c e s e t w e i g h t s$

LaTeX
H = [h_ij]_{m×n}; w = (w_1,...,w_n)^T with Σw_k=1; ω = (ω_C, ω'_C, ω_D, ω'_D)^T for concordance/discordance set weights
Score = mean membership value; deviation = standard deviation of membership values. If two HFEs have different lengths, extend the shorter by repeating its minimum value (pessimistic rule).

$s (h_{i} j) = (1 / | h_{i} j |) \cdot Σ_{γ \in h_{i} j} γ; σ (h_{i} j) = s q r t ((1 / | h_{i} j |) \cdot Σ_{γ \in h_{i} j} (γ - s (h_{i} j)) ²)$

LaTeX
s(h_ij) = (1/|h_ij|)·Σ_{γ∈h_ij} γ; σ(h_ij) = sqrt((1/|h_ij|)·Σ_{γ∈h_ij}(γ − s(h_ij))²)
For each ordered pair (i,j) with i≠j: J^c_ij contains criteria where A_i outperforms A_j with higher consistency (lower deviation); J'^c_ij contains criteria where A_i scores at least as well but with lower or equal consistency.

$J_{i}^{c} j = k | s (h_{i} k) \geq s (h_{j} k) A N D σ (h_{i} k) < σ (h_{j} k) (E q .4 .19); J_{i}^{' c} j = k | s (h_{i} k) \geq s (h_{j} k) A N D σ (h_{i} k) \geq σ (h_{j} k) (E q .4 .20)$

LaTeX
J^c_ij = {k | s(h_ik) ≥ s(h_jk) AND σ(h_ik) < σ(h_jk)} (Eq.4.19); J'^c_ij = {k | s(h_ik) ≥ s(h_jk) AND σ(h_ik) ≥ σ(h_jk)} (Eq.4.20)
J^d_ij collects criteria where A_i is inferior to A_j with lower consistency (more discordant); J'^d_ij collects criteria where A_i is inferior but has higher consistency (weak discordance).

$J_{i}^{d} j = k | s (h_{i} k) < s (h_{j} k) A N D σ (h_{i} k) \geq σ (h_{j} k) (E q .4 .21); J_{i}^{' d} j = k | s (h_{i} k) < s (h_{j} k) A N D σ (h_{i} k) < σ (h_{j} k) (E q .4 .22)$

LaTeX
J^d_ij = {k | s(h_ik) < s(h_jk) AND σ(h_ik) ≥ σ(h_jk)} (Eq.4.21); J'^d_ij = {k | s(h_ik) < s(h_jk) AND σ(h_ik) < σ(h_jk)} (Eq.4.22)
The concordance index c_ij ∈ [0,1] measures how much A_i outranks A_j. Weighted sum of criteria in the concordance sets divided by total weights. c_ii is undefined (diagonal excluded).

$c_{i} j = (ω_{C} \cdot Σ_{k \in J_{i}^{c} j} w_{k} + ω_{C}^{'} \cdot Σ_{k \in J_{i}^{' c} j} w_{k}) / Σ_{k = 1}^{n} w_{k}; C = [c_{i} j]_{n \times n} (E q s .4 .23 - 4.24)$

LaTeX
c_ij = (ω_C·Σ_{k∈J^c_ij} w_k + ω'_C·Σ_{k∈J'^c_ij} w_k) / Σ_{k=1}^{n} w_k; C = [c_ij]_{n×n} (Eqs.4.23-4.24)
d_ij measures worst-case weighted distance on discordant criteria relative to maximum distance across all criteria. Distance d(·,·) is the HFS distance measure (Def.2.2 in book).

$d_{i} j = m a x ω_{D} \cdot d (w_{k} h_{i} k, w_{k} h_{j} k) : k \in J_{i}^{d} j, ω_{D}^{'} \cdot d (w_{k} h_{i} k, w_{k} h_{j} k) : k \in J_{i}^{' d} j / m a x_{k = 1 . . n} d (w_{k} h_{i} k, w_{k} h_{j} k); D = [d_{i} j]_{n \times n} (E q s .4 .25 - 4.26)$

LaTeX
d_ij = max{ω_D·d(w_k h_ik, w_k h_jk): k∈J^d_ij, ω'_D·d(w_k h_ik, w_k h_jk): k∈J'^d_ij} / max_{k=1..n} d(w_k h_ik, w_k h_jk); D = [d_ij]_{n×n} (Eqs.4.25-4.26)
Threshold c̄ is the average concordance index over all ordered pairs. f_ij=1 indicates A_i concordance-dominates A_j.

$c ̄ = Σ_{i = 1, j \neq i}^{n} c_{i} j / (n (n - 1)); f_{i} j = 1 i f c_{i} j \geq c ̄ e l s e 0; F = [f_{i} j] (E q s .4 .27 - 4.29)$

LaTeX
c̄ = Σ_{i=1,j≠i}^{n} c_ij / (n(n−1)); f_ij = 1 if c_ij ≥ c̄ else 0; F = [f_ij] (Eqs.4.27-4.29)
Threshold d̄ is the average discordance index. q_ij=1 indicates A_i is NOT strongly discordant relative to A_j (i.e., discordance is within acceptable range).

$d ̄ = Σ_{i = 1, j \neq i}^{n} d_{i} j / (n (n - 1)); q_{i} j = 1 i f d_{i} j \leq d ̄ e l s e 0; Q = [q_{i} j] (E q s .4 .30 - 4.32)$

LaTeX
d̄ = Σ_{i=1,j≠i}^{n} d_ij / (n(n−1)); q_ij = 1 if d_ij ≤ d̄ else 0; Q = [q_ij] (Eqs.4.30-4.32)
p_ij=1 iff A_i concordance-dominates A_j AND is not excessively discordant w.r.t. A_j. Interpretation: p_ij=1 → A_i preferred to A_j; p_ij=p_ji=1 → indifferent; p_ij=p_ji=0 → incomparable.

$P = F \otimes Q; p_{i} j = f_{i} j \cdot q_{i} j (E q s .4 .33 - 4.34)$

LaTeX
P = F ⊗ Q; p_ij = f_ij · q_ij (Eqs.4.33-4.34)
Identify the kernel (undominated set) from the aggregation dominance graph. Alternatives in the kernel are the recommended choices.

$D r a w d i r e c t e d g r a p h f r o m P : e d g e A_{i} \to A_{j} i f f p_{i} j = 1 . K e r n e l = s e t o f a l t e r n a t i v e s n o t d o m i n a t e d b y a n y o t h e r s e l e c t e d a l t e r n a t i v e .$

LaTeX
Draw directed graph from P: edge A_i→A_j iff p_ij=1. Kernel = set of alternatives not dominated by any other selected alternative.

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

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