HFL-MABAC - Hesitant Fuzzy Linguistic Projection-Based MABAC with Bonferroni Mean (Sun et al. 2018)

Hesitant fuzzy linguistic ranking - HFLTS subscript-symmetric LTS + projection-based signed difference + Bonferroni Mean criterion aggregation

Sun, R., Hu, J., Zhou, J., Chen, X.2018doi:10.1007/s40815-017-0345-7 ↗

Overview

HFL-MABAC ranks alternatives based on performance scores. Higher score = better rank.

Data: Hesitant Fuzzy Linguistic
Weights: Needs a weight source

How it works

1
Convert linguistic CLEs to HFLTSs via transformation function E_{G_H}; assemble HFLTS decision matrix H = (h_S_ij)_{m×n}.
$E_{G_{H}} : l l \to H_{S} p er S u n 2018 D e f .3 p r o d u c t i o n r u l es : E_{G_{H}} (a t m os t s_{i}) = s_{j} ∣ s_{j} \leq s_{i}; E_{G_{H}} (a tl e a s t s_{i}) = s_{j} ∣ s_{j} \geq s_{i}; E_{G_{H}} (b e tw ee n s_{i} an d s_{j}) = s_{k} ∣ s_{i} \leq s_{k} \leq s_{j}; E_{G_{H}} (l ess t han s_{i}) = s_{j} ∣ s_{j} < s_{i}; E_{G_{H}} (g r e a t er t han s_{i}) = s_{j} ∣ s_{j} > s_{i} .$
Sun 2018 Def.3 + Transformation Stage Step 1; Rodríguez et al. 2012.
2
Normalize decision matrix. For maximizing criteria: n_S_ij = h_S_ij (identity). For minimizing criteria: n_S_ij = h̄_S_ij (complement per Def.6 #3): h̄_S = V^{-1}(∪_{γ∈V(h_S)} {1-γ}).
$n_S_ij = h_S_ij if c_j ∈ B (max); n_S_ij = V^{-1}(∪_{γ∈V(h_S_ij)} {1-γ}) if c_j ∈ C (min) [Sun 2018 Eq.(16)]. Transformation: v(δ_l) = δ_l/(2g) + 1/2 [Eq.4]; V^{-1}(γ_l) = (2γ_l - 1)g [Def.4 #2].$
Sun 2018 Eq.(16); Def.6 (complement); Def.4 (V, V^{-1}).
3
Construct weighted HFLTSs decision matrix T = (t_ij)_{m×n}. Each cell is the scalar-multiplied HFLE: t_ij = w_j · n_S_ij.
$t_ij = w_j · n_S_ij = V^{-1}(∪_{γ ∈ V(n_S_ij)} {w_j · γ}) [Sun 2018 Eq.(17)]. Equivalent fast form for HFLE n_S = {s_δ_l}: t = {s_{(2·w_j·v(δ_l)-1)·g}} = {s_{w_j·(δ_l+g) - g}}.$
Sun 2018 Eq.(17); Def.4-5 V/V^{-1} transformations.
4
Compute Border Approximation Area (BAA) vector G = (g_j)_{1×n}. For each criterion column, ARITHMETIC mean of subscripts across alternatives (NOT geometric - design departure from classical Pamucar 2015, per Yu et al. 2017 fuzzy-MABAC precedent).
$g_{j} = \overset{g}{ˉ}_{j l} = (1/ m) \cdot Σ_{i = 1}^{m} \tilde{δ}_{l}^{ij} ∣ l = 1, ..., L [S u n 2018 E q . (18)] . H er e \tilde{δ}_{l}^{ij} i s t h e l - t h s u b scr i pt o f t h ee x t e n d e d - t o - co mm o n - L w e i g h t e d H F L E \tilde{t}_{i} j . E q u a l i z a t i o n r u l e : e x t e n d s h or t er H F L E b y a dd in g m e d ian l in g u i s t i c t er m \overset{s}{ˉ} = (s + + s -) /2 u n t i l a l l ce l l s in co l u mnha v e l e n g t h L = ma x_{i} ∣ t_{i} j ∣.$
Sun 2018 Eq.(18) + Section 2.1 cardinality equalization; Yu et al. 2017 (arithmetic-mean precedent for fuzzy MABAC).
5
Compute distance matrix P = (p_ij)_{m×n} via projection-based SIGNED difference. For each cell: p_ij = d_proj(t_ij, g_j) = (Proj_k(t_ij) - Proj_k(g_j)) / |k|, where k is any HFLE dominating both (default k = {s_g}). Belongingness Eq.(20): a_i ∈ G+ (UAA) if p_ij>0; G (BAA) if p_ij=0; G- (LAA) if p_ij<0.
$I nn er p r o d u c tE q . (8) : h_{S} 1 \cdot h_{S} 2 = Σ_{l = 1}^{L} ((δ_{l}^{1} + g + 1) / (2 g + 1)) \cdot ((δ_{l}^{2} + g + 1) / (2 g + 1)) . M o d u l u s E q . (9 - 10) : ∣ h_{S} ∣ = s q r t (Σ_{l} ((δ_{l} + g + 1) / (2 g + 1))^{2}) . P r o j ec t i o n E q . (12) : P r o j_{h_{S} 2} (h_{S} 1) = ∣ h_{S} 1∣ \cdot C os (h_{S} 1, h_{S} 2) = (h_{S} 1 \cdot h_{S} 2) /∣ h_{S} 2∣. P r o j ec t i o n - ba se dd i f f er e n ce E q . (15) : d_{p} r o j (h_{S} 1, h_{S} 2) = (P r o j_{k} (h_{S} 1) - P r o j_{k} (h_{S} 2)) /∣ k ∣, w h er e k i s an y H F L E w i t h h_{S} 1 ≺ k an d h_{S} 2 ≺ k . B e l o n g in g n ess E q . (20) : p_{i} j > 0 \to G +; p_{i} j = 0 \to G; p_{i} j < 0 \to G - .$
Sun 2018 Def.8-10 + Eqs.(7)-(15), (19)-(20).
6
Aggregate criterion distances via Bonferroni Mean. First shift p_ij to non-negative: p̃_ij = p_ij + θ where θ ≥ |min(p_ij)|. Then CC_i = BM^{p,q}(p̃_i1,...,p̃_in) = ((1/[n(n-1)]) · Σ_{j,k=1, j≠k}^n p̃_ij^p · p̃_ik^q)^{1/(p+q)}. Special case p=1,q=0 (Eq.22): CC_i = S_i/n + θ where S_i = Σ p_ij - reduces to classical MABAC ranking. Larger CC_i → higher rank.
$θ \geq ∣ mi n_{i, j} p_{i} j ∣; \tilde{p}_{i} j = p_{i} j + θ \geq 0. C C_{i} = B M^{p, q} (\tilde{p}_{i} 1, ..., \tilde{p}_{i} n) = ((1/ (n (n - 1))) Σ_{j \neq = k, j, k = 1.. n} \tilde{p}_{i} j^{p} \cdot \tilde{p}_{i} k^{q})^{1/ (p + q)} [S u n 2018 E q . (21)] . S p ec ia l : p = 1, q = 0 \Rightarrow C C_{i} = (1/ n) S_{i} + θ w h er e S_{i} = Σ_{j} p_{i} j [E q . (22)] . R ank in g : A_{i} ≻ A_{i^{'}} ⟺ C C_{i} > C C_{i^{'}} .$
Sun 2018 Eqs.(21)-(22); Bonferroni 1950; Yager 2009.

Fits when

•Preserves hesitant_fuzzy_linguistic uncertainty through the pipeline rather than premature crispification at elicitation

Edge cases and pitfalls

•BAA G_j hesabında bazı ν_ij sıfır-üye HFLE ise geometrik ortalama 0'a düşer - paper §3.2'de açıkça "HFLE'ler boş olmayacak" varsayımı yapar; engine boş HFLE girişini reddeder.
•Projection paydası ‖G_j‖² → 0 dejenere: bu yalnız BAA'nın tüm bileşenleri sıfırsa olur (yani tüm alternatifler o kriterde s_0); engine kriteri non-discriminating işaretler ve skora katmaz.
•BM parametreleri p,q ≠ 1 alternatif: paper p=q=1 default kullanır ama p=q=2 (quadratic) veya p=1,q=2 (asymmetric) seçimleri kriter-içi etkileşimi farklı modellenir; bunu değiştirmek hospital case ranking'inde 2.-3. sıralarda swap üretebilir (paper §5.4).
•Cost normalizasyon Def.5'in neg(h_S) = {s_τ−α : s_α ∈ h_S} tanımı LTS'in 2τ+1 sembollü (simetrik 0-merkezli değil, 0..τ aralıklı) bir varyantı varsayar - kullanıcı asimetrik LTS girerse normalizasyon doğru çalışmaz.

How to cite

Sun, R.; Hu, J.; Zhou, J.; Chen, X. (2018). A Hesitant Fuzzy Linguistic Projection-Based MABAC Method for Patients' Prioritization. International Journal of Fuzzy Systems. https://doi.org/10.1007/s40815-017-0345-7

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HFL-MABAC

How it works

Convert linguistic CLEs to HFLTSs via transformation function E_{G_H}; assemble HFLTS decision matrix H = (h_S_ij)_{m×n}.

E_{G_{H}} : l l \to H_{S} p er S u n 2018 D e f .3 p r o d u c t i o n r u l es : E_{G_{H}} (a t m os t s_{i}) = s_{j} ∣ s_{j} \leq s_{i}; E_{G_{H}} (a tl e a s t s_{i}) = s_{j} ∣ s_{j} \geq s_{i}; E_{G_{H}} (b e tw ee n s_{i} an d s_{j}) = s_{k} ∣ s_{i} \leq s_{k} \leq s_{j}; E_{G_{H}} (l ess t han s_{i}) = s_{j} ∣ s_{j} < s_{i}; E_{G_{H}} (g r e a t er t han s_{i}) = s_{j} ∣ s_{j} > s_{i} .

Sun 2018 Def.3 + Transformation Stage Step 1; Rodríguez et al. 2012.

Normalize decision matrix. For maximizing criteria: n_S_ij = h_S_ij (identity). For minimizing criteria: n_S_ij = h̄_S_ij (complement per Def.6 #3): h̄_S = V^{-1}(∪_{γ∈V(h_S)} {1-γ}).

n_S_ij = h_S_ij if c_j ∈ B (max); n_S_ij = V^{-1}(∪_{γ∈V(h_S_ij)} {1-γ}) if c_j ∈ C (min) [Sun 2018 Eq.(16)]. Transformation: v(δ_l) = δ_l/(2g) + 1/2 [Eq.4]; V^{-1}(γ_l) = (2γ_l - 1)g [Def.4 #2].

Sun 2018 Eq.(16); Def.6 (complement); Def.4 (V, V^{-1}).

Construct weighted HFLTSs decision matrix T = (t_ij)_{m×n}. Each cell is the scalar-multiplied HFLE: t_ij = w_j · n_S_ij.

t_ij = w_j · n_S_ij = V^{-1}(∪_{γ ∈ V(n_S_ij)} {w_j · γ}) [Sun 2018 Eq.(17)]. Equivalent fast form for HFLE n_S = {s_δ_l}: t = {s_{(2·w_j·v(δ_l)-1)·g}} = {s_{w_j·(δ_l+g) - g}}.

Sun 2018 Eq.(17); Def.4-5 V/V^{-1} transformations.

Compute Border Approximation Area (BAA) vector G = (g_j)_{1×n}. For each criterion column, ARITHMETIC mean of subscripts across alternatives (NOT geometric - design departure from classical Pamucar 2015, per Yu et al. 2017 fuzzy-MABAC precedent).

g_{j} = \overset{g}{ˉ}_{j l} = (1/ m) \cdot Σ_{i = 1}^{m} \tilde{δ}_{l}^{ij} ∣ l = 1, ..., L [S u n 2018 E q . (18)] . H er e \tilde{δ}_{l}^{ij} i s t h e l - t h s u b scr i pt o f t h ee x t e n d e d - t o - co mm o n - L w e i g h t e d H F L E \tilde{t}_{i} j . E q u a l i z a t i o n r u l e : e x t e n d s h or t er H F L E b y a dd in g m e d ian l in g u i s t i c t er m \overset{s}{ˉ} = (s + + s -) /2 u n t i l a l l ce l l s in co l u mnha v e l e n g t h L = ma x_{i} ∣ t_{i} j ∣.

Sun 2018 Eq.(18) + Section 2.1 cardinality equalization; Yu et al. 2017 (arithmetic-mean precedent for fuzzy MABAC).

Compute distance matrix P = (p_ij)_{m×n} via projection-based SIGNED difference. For each cell: p_ij = d_proj(t_ij, g_j) = (Proj_k(t_ij) - Proj_k(g_j)) / |k|, where k is any HFLE dominating both (default k = {s_g}). Belongingness Eq.(20): a_i ∈ G+ (UAA) if p_ij>0; G (BAA) if p_ij=0; G- (LAA) if p_ij<0.

I nn er p r o d u c tE q . (8) : h_{S} 1 \cdot h_{S} 2 = Σ_{l = 1}^{L} ((δ_{l}^{1} + g + 1) / (2 g + 1)) \cdot ((δ_{l}^{2} + g + 1) / (2 g + 1)) . M o d u l u s E q . (9 - 10) : ∣ h_{S} ∣ = s q r t (Σ_{l} ((δ_{l} + g + 1) / (2 g + 1))^{2}) . P r o j ec t i o n E q . (12) : P r o j_{h_{S} 2} (h_{S} 1) = ∣ h_{S} 1∣ \cdot C os (h_{S} 1, h_{S} 2) = (h_{S} 1 \cdot h_{S} 2) /∣ h_{S} 2∣. P r o j ec t i o n - ba se dd i f f er e n ce E q . (15) : d_{p} r o j (h_{S} 1, h_{S} 2) = (P r o j_{k} (h_{S} 1) - P r o j_{k} (h_{S} 2)) /∣ k ∣, w h er e k i s an y H F L E w i t h h_{S} 1 ≺ k an d h_{S} 2 ≺ k . B e l o n g in g n ess E q . (20) : p_{i} j > 0 \to G +; p_{i} j = 0 \to G; p_{i} j < 0 \to G - .

Sun 2018 Def.8-10 + Eqs.(7)-(15), (19)-(20).

Aggregate criterion distances via Bonferroni Mean. First shift p_ij to non-negative: p̃_ij = p_ij + θ where θ ≥ |min(p_ij)|. Then CC_i = BM^{p,q}(p̃_i1,...,p̃_in) = ((1/[n(n-1)]) · Σ_{j,k=1, j≠k}^n p̃_ij^p · p̃_ik^q)^{1/(p+q)}. Special case p=1,q=0 (Eq.22): CC_i = S_i/n + θ where S_i = Σ p_ij - reduces to classical MABAC ranking. Larger CC_i → higher rank.

θ \geq ∣ mi n_{i, j} p_{i} j ∣; \tilde{p}_{i} j = p_{i} j + θ \geq 0. C C_{i} = B M^{p, q} (\tilde{p}_{i} 1, ..., \tilde{p}_{i} n) = ((1/ (n (n - 1))) Σ_{j \neq = k, j, k = 1.. n} \tilde{p}_{i} j^{p} \cdot \tilde{p}_{i} k^{q})^{1/ (p + q)} [S u n 2018 E q . (21)] . S p ec ia l : p = 1, q = 0 \Rightarrow C C_{i} = (1/ n) S_{i} + θ w h er e S_{i} = Σ_{j} p_{i} j [E q . (22)] . R ank in g : A_{i} ≻ A_{i^{'}} ⟺ C C_{i} > C C_{i^{'}} .

Sun 2018 Eqs.(21)-(22); Bonferroni 1950; Yager 2009.

Edge cases and pitfalls

•BAA G_j hesabında bazı ν_ij sıfır-üye HFLE ise geometrik ortalama 0'a düşer - paper §3.2'de açıkça "HFLE'ler boş olmayacak" varsayımı yapar; engine boş HFLE girişini reddeder.

•Projection paydası ‖G_j‖² → 0 dejenere: bu yalnız BAA'nın tüm bileşenleri sıfırsa olur (yani tüm alternatifler o kriterde s_0); engine kriteri non-discriminating işaretler ve skora katmaz.

•BM parametreleri p,q ≠ 1 alternatif: paper p=q=1 default kullanır ama p=q=2 (quadratic) veya p=1,q=2 (asymmetric) seçimleri kriter-içi etkileşimi farklı modellenir; bunu değiştirmek hospital case ranking'inde 2.-3. sıralarda swap üretebilir (paper §5.4).

•Cost normalizasyon Def.5'in neg(h_S) = {s_τ−α : s_α ∈ h_S} tanımı LTS'in 2τ+1 sembollü (simetrik 0-merkezli değil, 0..τ aralıklı) bir varyantı varsayar - kullanıcı asimetrik LTS girerse normalizasyon doğru çalışmaz.