HF-ARAS - Hesitant Fuzzy Additive Ratio Assessment

Hesitant fuzzy utility-degree ranker (P_i / P_0 against ideal alternative row)

Mishra, A. R., Rani, P., Krishankumar, R., Ravichandran, K. S., Kar, S.2021doi:10.1016/j.asoc.2021.107155 ↗

Overview

HF-ARAS ranks alternatives based on performance scores. Higher score = better rank.

Data: Hesitant
Weights: Needs a weight source

How it works

1
Construct HF decision matrix H = (h_ij)_{m×n}, where each h_ij is an HFE supplied by the decision-maker (or pre-aggregated from multi-DE input via HFWA Eq.(8)). Entries may be of varying cardinality.
$H = (h_{i} j)_{m \times n}, h_{i} j \subseteq [0, 1], ∣ h_{i} j ∣ \geq 1.$
Mishra 2021, Step 1 (Section 4); Eq.(8) HFWA for optional multi-DE aggregation.
2
Construct optimal performance row M_0 = (h_0j)_{j=1..n}, where h_0j is the column-wise best HFE: max over alternatives by score S(·) for benefit criteria (j ∈ S_b), min for cost criteria (j ∈ S_n). M_0 is appended as a virtual zeroth alternative.
$h_{0} j = a r g ma x_{i} S (h_{i} j) i f j \in S_{b} (b e n e f i t); h_{0} j = a r g mi n_{i} S (h_{i} j) i f j \in S_{n} (cos t) . A ug m e n t e d ma t r i x : \tilde{H} = (h_{i} j)_{(m + 1) \times n}, i \in 0, 1, ..., m . (M i s h r a 2021 E q . (18))$
Mishra 2021, Step 5 (numbered Step 5 in paper despite being Step 2 of ARAS core), Eq.(18); Zavadskas-Turskis 2010 ideal-alternative construct.
3
Linear-normalise the augmented matrix column-wise via Eq.(19). For benefit: divide each entry by the column's maximum score. For cost: take 1 minus that ratio (complement form).
$F or j \in S_{b} : \overset{ˉ}{h}_{i} j = h_{i} j / ma x_{i} S (h_{i} j) = ξ / ma x_{i} S (h_{i} j) : ξ \in h_{i} j . F or j \in S_{n} : \overset{ˉ}{h}_{i} j = 1 - h_{i} j / ma x_{i} S (h_{i} j) = 1 - ξ / ma x_{i} S (h_{i} j) : ξ \in h_{i} j . ma x_{i} t ak e n o v er a ug m e n t e d ma t r i x i \in 0, 1, ..., m . (M i s h r a 2021 E q . (19))$
Mishra 2021, Step 6 / Eq.(19); Zavadskas-Turskis 2010 linear normalisation lifted to HFE entry-wise scalar division/complement.
4
Apply criterion weights via entry-wise HFE scalar power (Xia-Xu 2011 scalar multiplication on HFE): h̃_ij = w_j ⊗ h̄_ij = { 1 − (1 − ξ̄)^{w_j} : ξ̄ ∈ h̄_ij }. Each entry of the normalised matrix is raised to the weight as scalar exponent.
$\tilde{h}_{i} j = w_{j} \otimes \overset{ˉ}{h}_{i} j = 1 - (1 - \overset{ˉ}{ξ})^{w_{j}} : \overset{ˉ}{ξ} \in \overset{ˉ}{h}_{i} j, i = 0.. m, j = 1.. n . Σ_{j} w_{j} = 1, w_{j} \geq 0. (M i s h r a 2021 E q . (20), e n t r y - w i se in t er p r e t a t i o n; t h e p r in t e d Π_{j = 1}^{n} in E q .20 i s a t y p o g r a p hi c a l co n f l a t i o n w i t h t h e a g g r e g a t i o n f or m - e n t r y - w i ser e a d in g r eco n c i l es w i t h S t e p 8/ T ab l e 7 w hi c h r e q u i r e m \times n w e i g h t e d ma t r i x; see P . v er i f i c a t i o n_{s} t a t u s .)$
Mishra 2021, Step 7 / Eq.(20); Xia-Xu 2011 scalar power on HFE.
5
Defuzzify each weighted HFE to a crisp score via arithmetic mean: S(h̃_ij) = (1/g_{h̃_ij}) Σ_{ξ̃ ∈ h̃_ij} ξ̃. Produces an (m+1)×n scalar matrix of weighted normalised scores.
$S (\tilde{h}_{i} j) = (1/ g_{\tilde{h}_{i} j}) \cdot Σ_{\tilde{ξ} \in \tilde{h}_{i} j} \tilde{ξ} . (M i s h r a 2021 E q . (21); i d e n t i c a l t o X ia - X u 2011 scor e f u n c t i o nan d E q . (1) .)$
Mishra 2021, Step 8 / Eq.(21); Xia-Xu 2011 score function.
6
Compute overall performance rating P_i = Σ_{j=1}^n S(h̃_ij) for each alternative (including ideal row P_0). Then compute utility degree U_i = P_i / P_0 (relative to ideal optimum).
$P_{i} = Σ_{j = 1}^{n} S (\tilde{h}_{i} j), i = 0, 1, ..., m . U_{i} = P_{i} / P_{0}, i = 1, ..., m; U_{i} \in [0, 1] . (M i s h r a 2021 E q s . (22) - (23))$
Mishra 2021, Step 9 / Eqs.(22)-(23); Zavadskas-Turskis 2010 utility-degree formula K_i = S_i / S_0.
7
Rank alternatives in DESCENDING order of U_i (largest is best). M* = arg max_i U_i.
$A_{i} ≻ A_{i^{'}} ⟺ U_{i} > U_{i^{'}}; M * = a r g ma x_{i} U_{i} . (M i s h r a 2021 E q . (24))$
Mishra 2021, Step 10 / Eq.(24); Zavadskas-Turskis 2010 ranking.

Fits when

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

Edge cases and pitfalls

•Optimal satır x_0j extraction (Eq.15): max_i h_ij komponent-wise alındığında, l(h_ij) farklı kolonlarda farklı olabilir → length-equalization Adım 2'de HFWA içinde otomatik yapılır (Xia-Xu 2011 §3.1 konvansiyonu); paper bunu örtük varsayar.
•α=0 (saf nesnel) veya α=1 (saf subjektif) uç durumlarda K_i hâlâ iyi tanımlıdır ama paper §5.3 her iki uçta da farklı kazanan gösterebileceğini kanıtlar (Remdesivir vs Favipiravir swap'ı).
•S_0 = 0 durumu: hipotetik optimum satırı tüm-sıfır olamaz çünkü x_0j = max_i h_ij ve en az bir HFE pozitif değer içerir (boş HFE Torra 2010'da tanımlı değil). Pratik kullanımda engine boş-HFE girişi reddeder.
•Cost normalizasyonu Eq.(1)'in HFE kompleman (1-γ) yorumu Xia-Xu 2011 §2.2'ye dayanır; alternatif konvansiyon (max/min flip) ARAS'ta kullanılmaz.

How to cite

Mishra, A. R.; Rani, P.; Krishankumar, R.; Ravichandran, K. S.; Kar, S. (2021). A multi-criteria framework for evaluating the sustainable drug selection for COVID-19 patients using hesitant fuzzy information and ARAS method. Applied Soft Computing Journal. https://doi.org/10.1016/j.asoc.2021.107155

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-ARAS

How it works

Construct HF decision matrix H = (h_ij)_{m×n}, where each h_ij is an HFE supplied by the decision-maker (or pre-aggregated from multi-DE input via HFWA Eq.(8)). Entries may be of varying cardinality.

H = (h_{i} j)_{m \times n}, h_{i} j \subseteq [0, 1], ∣ h_{i} j ∣ \geq 1.

Mishra 2021, Step 1 (Section 4); Eq.(8) HFWA for optional multi-DE aggregation.

Construct optimal performance row M_0 = (h_0j)_{j=1..n}, where h_0j is the column-wise best HFE: max over alternatives by score S(·) for benefit criteria (j ∈ S_b), min for cost criteria (j ∈ S_n). M_0 is appended as a virtual zeroth alternative.

h_{0} j = a r g ma x_{i} S (h_{i} j) i f j \in S_{b} (b e n e f i t); h_{0} j = a r g mi n_{i} S (h_{i} j) i f j \in S_{n} (cos t) . A ug m e n t e d ma t r i x : \tilde{H} = (h_{i} j)_{(m + 1) \times n}, i \in 0, 1, ..., m . (M i s h r a 2021 E q . (18))

Mishra 2021, Step 5 (numbered Step 5 in paper despite being Step 2 of ARAS core), Eq.(18); Zavadskas-Turskis 2010 ideal-alternative construct.

Linear-normalise the augmented matrix column-wise via Eq.(19). For benefit: divide each entry by the column's maximum score. For cost: take 1 minus that ratio (complement form).

F or j \in S_{b} : \overset{ˉ}{h}_{i} j = h_{i} j / ma x_{i} S (h_{i} j) = ξ / ma x_{i} S (h_{i} j) : ξ \in h_{i} j . F or j \in S_{n} : \overset{ˉ}{h}_{i} j = 1 - h_{i} j / ma x_{i} S (h_{i} j) = 1 - ξ / ma x_{i} S (h_{i} j) : ξ \in h_{i} j . ma x_{i} t ak e n o v er a ug m e n t e d ma t r i x i \in 0, 1, ..., m . (M i s h r a 2021 E q . (19))

Mishra 2021, Step 6 / Eq.(19); Zavadskas-Turskis 2010 linear normalisation lifted to HFE entry-wise scalar division/complement.

Apply criterion weights via entry-wise HFE scalar power (Xia-Xu 2011 scalar multiplication on HFE): h̃_ij = w_j ⊗ h̄_ij = { 1 − (1 − ξ̄)^{w_j} : ξ̄ ∈ h̄_ij }. Each entry of the normalised matrix is raised to the weight as scalar exponent.

\tilde{h}_{i} j = w_{j} \otimes \overset{ˉ}{h}_{i} j = 1 - (1 - \overset{ˉ}{ξ})^{w_{j}} : \overset{ˉ}{ξ} \in \overset{ˉ}{h}_{i} j, i = 0.. m, j = 1.. n . Σ_{j} w_{j} = 1, w_{j} \geq 0. (M i s h r a 2021 E q . (20), e n t r y - w i se in t er p r e t a t i o n; t h e p r in t e d Π_{j = 1}^{n} in E q .20 i s a t y p o g r a p hi c a l co n f l a t i o n w i t h t h e a g g r e g a t i o n f or m - e n t r y - w i ser e a d in g r eco n c i l es w i t h S t e p 8/ T ab l e 7 w hi c h r e q u i r e m \times n w e i g h t e d ma t r i x; see P . v er i f i c a t i o n_{s} t a t u s .)

Mishra 2021, Step 7 / Eq.(20); Xia-Xu 2011 scalar power on HFE.

Defuzzify each weighted HFE to a crisp score via arithmetic mean: S(h̃_ij) = (1/g_{h̃_ij}) Σ_{ξ̃ ∈ h̃_ij} ξ̃. Produces an (m+1)×n scalar matrix of weighted normalised scores.

S (\tilde{h}_{i} j) = (1/ g_{\tilde{h}_{i} j}) \cdot Σ_{\tilde{ξ} \in \tilde{h}_{i} j} \tilde{ξ} . (M i s h r a 2021 E q . (21); i d e n t i c a l t o X ia - X u 2011 scor e f u n c t i o nan d E q . (1) .)

Mishra 2021, Step 8 / Eq.(21); Xia-Xu 2011 score function.

Compute overall performance rating P_i = Σ_{j=1}^n S(h̃_ij) for each alternative (including ideal row P_0). Then compute utility degree U_i = P_i / P_0 (relative to ideal optimum).

P_{i} = Σ_{j = 1}^{n} S (\tilde{h}_{i} j), i = 0, 1, ..., m . U_{i} = P_{i} / P_{0}, i = 1, ..., m; U_{i} \in [0, 1] . (M i s h r a 2021 E q s . (22) - (23))

Mishra 2021, Step 9 / Eqs.(22)-(23); Zavadskas-Turskis 2010 utility-degree formula K_i = S_i / S_0.

Rank alternatives in DESCENDING order of U_i (largest is best). M* = arg max_i U_i.

A_{i} ≻ A_{i^{'}} ⟺ U_{i} > U_{i^{'}}; M * = a r g ma x_{i} U_{i} . (M i s h r a 2021 E q . (24))

Mishra 2021, Step 10 / Eq.(24); Zavadskas-Turskis 2010 ranking.

Edge cases and pitfalls

•Optimal satır x_0j extraction (Eq.15): max_i h_ij komponent-wise alındığında, l(h_ij) farklı kolonlarda farklı olabilir → length-equalization Adım 2'de HFWA içinde otomatik yapılır (Xia-Xu 2011 §3.1 konvansiyonu); paper bunu örtük varsayar.

•α=0 (saf nesnel) veya α=1 (saf subjektif) uç durumlarda K_i hâlâ iyi tanımlıdır ama paper §5.3 her iki uçta da farklı kazanan gösterebileceğini kanıtlar (Remdesivir vs Favipiravir swap'ı).

•S_0 = 0 durumu: hipotetik optimum satırı tüm-sıfır olamaz çünkü x_0j = max_i h_ij ve en az bir HFE pozitif değer içerir (boş HFE Torra 2010'da tanımlı değil). Pratik kullanımda engine boş-HFE girişi reddeder.

•Cost normalizasyonu Eq.(1)'in HFE kompleman (1-γ) yorumu Xia-Xu 2011 §2.2'ye dayanır; alternatif konvansiyon (max/min flip) ARAS'ta kullanılmaz.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-ARAS