HF-MOORA - Hesitant Fuzzy Multi-Objective Optimization by Ratio Analysis

Hesitant fuzzy ratio system ranker (benefit-vs-non-beneficial net score on vector-normalised defuzzified matrix)

Li, Z.-H.2014doi:10.1155/2014/527836 ↗

Overview

HF-MOORA 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; entries may be of varying cardinality.
$H = (h_{i} j)_{m \times n}, h_{i} j \subseteq [0, 1], ∣ h_{i} j ∣ \geq 1.$
Li 2014, Algorithm 20 Step 1; Narayanamoorthy 2020, Section 5.2 Step 1.
2
Defuzzify each HFE to a crisp score via the chosen score operator. Canonical default: arithmetic mean S(h_ij) = (1/|h_ij|) · Σ_{γ ∈ h_ij} γ (Narayanamoorthy 2020 Eq.1, Xia-Xu 2011). Variant: HFPA s(h) = PA(h) (Li 2014 Eq.15).
$S (h_{i} j) = (1/∣ h_{i} j ∣) \cdot Σ_{γ \in h_{i} j} γ (a r i t hm e t i c m e an, c an o ni c a l d e f a u l t) . V a r ian t : s (h_{i} j) = P A (h_{i} j) (H F P A, L i 2014 E q .9) .$
Narayanamoorthy 2020, Eq.(1); Li 2014, Eq.(15); Xia-Xu 2011 score function.
3
Vector-normalise the defuzzified score matrix column-wise: r*_ij = S(h_ij) / √( Σ_{k=1}^m S(h_kj)² ).
$r *_{i} j = S (h_{i} j) /\sqrt (Σ_{k = 1}^{m} S (h_{k} j)^{2}); \forall j : Σ_{i} (r *_{i} j)^{2} = 1. (B r a u er s - Z a v a d s k a s 2006 n or ma l i s a t i o n l i f t e d t oH F E - d e f u z z i f i e d scor es p er N a r a y anam oor t h y 2020 E q .5.)$
Narayanamoorthy 2020, Eq.(5); Brauers-Zavadskas 2006 Eq.(2).
4
Apply criterion weights: T_ij = w_j · r*_ij.
$T_{i} j = w_{j} \cdot r *_{i} j, i = 1.. m, j = 1.. n, Σ_{j} w_{j} = 1.$
Narayanamoorthy 2020, Step 4 / Eq.(24); Brauers-Zavadskas 2006 Eq.(3).
5
Ratio system: y_i = Σ_{j ∈ B} T_ij − Σ_{j ∈ NB} T_ij, where B = {benefit criteria}, NB = {non-beneficial criteria}.
$y_{i} = Σ_{j : d i r_{j} = ma x} w_{j} \cdot r *_{i} j - Σ_{j : d i r_{j} = min} w_{j} \cdot r *_{i} j, i = 1.. m . (N a r a y anam oor t h y 2020 E q .6 - n or ma l i se d v a l u es u se df or b o t hb e n e f i t an d n o n - b e n e f i c ia l s u m s; r eso l v es t h e p a p e r^{'} s t y p o g r a p hi c a l X_{i} j v s X *_{i} j a sy mm e t r y p er t h e i r o w nn u m er i c a l e x am pl e .) E q u i v a l e n tt o L i 2014 E q . (23) r_{i} = s (\oplus_{j \in B} r_{i} j) - s (\oplus_{j \in N B} r_{i} j) f or s in g l e t o n H F E s .$
Narayanamoorthy 2020, Eq.(6); Li 2014, Eq.(23); Brauers-Zavadskas 2006 Eq.(4).
6
Rank alternatives in DESCENDING order of y_i (largest is best).
$A_{i} ≻ A_{i^{'}} ⟺ y_{i} > y_{i^{'}}; D * = a r g ma x_{i} y_{i} .$
Narayanamoorthy 2020, Step 6; Li 2014, Algorithm 20 Step 5.

Fits when

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

Edge cases and pitfalls

•HF-MOORA için kritik edge case'ler: (1) Tek outlier HFE - eski Hamming distance outlier'ı yansıtır (örn h_1={0.8, 0.6, 0.3, 0.2} vs h_2={0.9, 0.4, 0.1}: H=0.15); novel d=|PA(h_1)-PA(h_2)|=0.0019 nedeniyle outlier influence minimize olur. (2) T(h_i) hesabı için d(h_i, h_j) gerekir - circular dependency yok çünkü sup=1-d tüm DM-çiftleri için önceden hesaplanır. (3) Paper Yager support function: sup(a,b) = 0.5(1 - |a-b|) crisp için; HF için sup(h_i,h_j) = 1 - d(h_i,h_j) (Def.18 ile consistent). (4) Cost kriter için N_C: r_i'de doğrudan ÇIKARILIR (Eq.23), TOPSIS gibi cost-flip normalization YAPILMAZ - bu MOORA felsefesinin temel ayrımı. (5) MORV cost kriter: φ_j = min_i r_ij (kabul edilebilir 'en kötü-tolere' yerine 'en iyi-cost' = minimum).

How to cite

Li, Z.-H. (2014). An Extension of the MULTIMOORA Method for Multiple Criteria Group Decision Making Based upon Hesitant Fuzzy Sets. Journal of Applied Mathematics. https://doi.org/10.1155/2014/527836

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System ID, as it appears in reports and the API

HF-MOORA

How it works

Construct HF decision matrix H = (h_ij)_{m×n}, where each h_ij is an HFE supplied by the decision-maker; entries may be of varying cardinality.

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

Li 2014, Algorithm 20 Step 1; Narayanamoorthy 2020, Section 5.2 Step 1.

Defuzzify each HFE to a crisp score via the chosen score operator. Canonical default: arithmetic mean S(h_ij) = (1/|h_ij|) · Σ_{γ ∈ h_ij} γ (Narayanamoorthy 2020 Eq.1, Xia-Xu 2011). Variant: HFPA s(h) = PA(h) (Li 2014 Eq.15).

S (h_{i} j) = (1/∣ h_{i} j ∣) \cdot Σ_{γ \in h_{i} j} γ (a r i t hm e t i c m e an, c an o ni c a l d e f a u l t) . V a r ian t : s (h_{i} j) = P A (h_{i} j) (H F P A, L i 2014 E q .9) .

Narayanamoorthy 2020, Eq.(1); Li 2014, Eq.(15); Xia-Xu 2011 score function.

Vector-normalise the defuzzified score matrix column-wise: r*_ij = S(h_ij) / √( Σ_{k=1}^m S(h_kj)² ).

r *_{i} j = S (h_{i} j) /\sqrt (Σ_{k = 1}^{m} S (h_{k} j)^{2}); \forall j : Σ_{i} (r *_{i} j)^{2} = 1. (B r a u er s - Z a v a d s k a s 2006 n or ma l i s a t i o n l i f t e d t oH F E - d e f u z z i f i e d scor es p er N a r a y anam oor t h y 2020 E q .5.)

Narayanamoorthy 2020, Eq.(5); Brauers-Zavadskas 2006 Eq.(2).

Apply criterion weights: T_ij = w_j · r*_ij.

T_{i} j = w_{j} \cdot r *_{i} j, i = 1.. m, j = 1.. n, Σ_{j} w_{j} = 1.

Narayanamoorthy 2020, Step 4 / Eq.(24); Brauers-Zavadskas 2006 Eq.(3).

Ratio system: y_i = Σ_{j ∈ B} T_ij − Σ_{j ∈ NB} T_ij, where B = {benefit criteria}, NB = {non-beneficial criteria}.

y_{i} = Σ_{j : d i r_{j} = ma x} w_{j} \cdot r *_{i} j - Σ_{j : d i r_{j} = min} w_{j} \cdot r *_{i} j, i = 1.. m . (N a r a y anam oor t h y 2020 E q .6 - n or ma l i se d v a l u es u se df or b o t hb e n e f i t an d n o n - b e n e f i c ia l s u m s; r eso l v es t h e p a p e r^{'} s t y p o g r a p hi c a l X_{i} j v s X *_{i} j a sy mm e t r y p er t h e i r o w nn u m er i c a l e x am pl e .) E q u i v a l e n tt o L i 2014 E q . (23) r_{i} = s (\oplus_{j \in B} r_{i} j) - s (\oplus_{j \in N B} r_{i} j) f or s in g l e t o n H F E s .

Narayanamoorthy 2020, Eq.(6); Li 2014, Eq.(23); Brauers-Zavadskas 2006 Eq.(4).

Rank alternatives in DESCENDING order of y_i (largest is best).

A_{i} ≻ A_{i^{'}} ⟺ y_{i} > y_{i^{'}}; D * = a r g ma x_{i} y_{i} .

Narayanamoorthy 2020, Step 6; Li 2014, Algorithm 20 Step 5.

Edge cases and pitfalls

•HF-MOORA için kritik edge case'ler: (1) Tek outlier HFE - eski Hamming distance outlier'ı yansıtır (örn h_1={0.8, 0.6, 0.3, 0.2} vs h_2={0.9, 0.4, 0.1}: H=0.15); novel d=|PA(h_1)-PA(h_2)|=0.0019 nedeniyle outlier influence minimize olur. (2) T(h_i) hesabı için d(h_i, h_j) gerekir - circular dependency yok çünkü sup=1-d tüm DM-çiftleri için önceden hesaplanır. (3) Paper Yager support function: sup(a,b) = 0.5(1 - |a-b|) crisp için; HF için sup(h_i,h_j) = 1 - d(h_i,h_j) (Def.18 ile consistent). (4) Cost kriter için N_C: r_i'de doğrudan ÇIKARILIR (Eq.23), TOPSIS gibi cost-flip normalization YAPILMAZ - bu MOORA felsefesinin temel ayrımı. (5) MORV cost kriter: φ_j = min_i r_ij (kabul edilebilir 'en kötü-tolere' yerine 'en iyi-cost' = minimum).