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Ranking

HF-WASPAS - Hesitant Fuzzy Weighted Aggregated Sum Product Assessment

Hesitant fuzzy WSM+WPM hybrid ranking

Mishra, A.R., Rani, P., Pardasani, K.R., Mardani, A.2019doi:10.1016/j.jclepro.2019.117901 ↗

Overview

HF-WASPAS 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 R = (h_ij)_{m×n}. Each h_ij is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA per Mishra Eq.15).
$R = (h_{i} j), h_{i} j \subseteq [0, 1], ∣ h_{i} j ∣ \geq 1.$
Mishra 2019, Section 4 Step 1; Mardani 2020, Step 1 (per-DE matrices Θ_k); Peng 2017, Algorithm 2 Step 1.
2
Step 1b (group mode only) - Aggregate per-DE matrices into a single R via HFWA with DE weights ϖ_k.
$h_{i} j = H F W A (h_{i}^{1} j, ..., h_{i}^{ℓ} j) = \cup_{γ^{k} \in h_{i}^{k} j} 1 - k = 1 \prod ℓ (1 - γ^{k})^{ϖ_{k}} (M i s h r a E q .15; M a r d ani E q .8; X ia - X u 2011)$
Mishra 2019, Eq.(15); Mardani 2020, Eq.(8) AHF-D matrix with DE weights λ_k = (1-e(h_k))/(ℓ-Σ e(h_k)) Eq.(7); Xia & Xu 2011 HFWA operator.
3
Step 2 - Normalize R by criterion type. Benefit: identity (n_ij = h_ij). Cost: pointwise complement (n_ij = ∪_{γ ∈ h_ij} {1-γ}).
$n_{i} j = h_{i} j i f C_{j} \in Ω_{b} (b e n e f i t); n_{i} j = \cup_{γ \in h_{i} j} 1 - γ i f C_{j} \in Ω_{c} (cos t) . (P e n g E q .5; M i s h r a E q .17 r e d u ce df or m; M a r d ani E q .12)$
Peng 2017, Eq.(5); Mishra 2019, Eq.(17); Mardani 2020, Eq.(12) - Sb=benefit (identity), Sc=cost (1-γ complement).
4
Step 3 - Compute WSM aggregate S_i = ⊕_j w_j n_ij via HFWA.
$S_{i} = \oplus_{j = 1}^{n} w_{j} \cdot n_{i} j = \cup_{γ_{j} \in n_{i} j} 1 - j = 1 \prod n (1 - γ_{j})^{w_{j}} (M i s h r a E q .18; M a r d ani E q .13 A_{i}; P e n g E q .14)$
Mishra 2019, Eq.(18); Mardani 2020, Eq.(13) WSM A_i; Peng 2017, Eq.(14).
5
Step 4 - Compute WPM aggregate P_i = ⊗_j (n_ij)^{w_j} via HFWG.
$P_{i} = \otimes_{j = 1}^{n} (n_{i} j)^{w_{j}} = \cup_{γ_{j} \in n_{i} j} j = 1 \prod n γ_{j}^{w_{j}} (M i s h r a E q .19; M a r d ani E q .14 P_{i}; P e n g E q .15)$
Mishra 2019, Eq.(19); Mardani 2020, Eq.(14) WPM P_i; Peng 2017, Eq.(15).
6
Step 5 - WASPAS combination Q_i = ϑ·score(S_i) + (1-ϑ)·score(P_i) using HF score function S(h) = (1/|h|) Σ γ (Mishra/Torra/Xia-Xu Eq.1).
$Q_{i} = ϑ \cdot S (S_{i}) + (1 - ϑ) \cdot S (P_{i}), w h er e S (h) = (1/∣ h ∣) Σ_{γ \in h} γ . D e f a u l tϑ = 0.5 (M i s h r a E q .20; M a r d ani E q .15 w i t h ι = 0.5; P e n g E q .16) . E x t e n d e df or m s (M i s h r a E q .21/ M a r d ani E q .15/ P e n g E q .17) a l l o w ϑ \in [0, 1] .$
Mishra 2019, Eq.(20)-(21); Mardani 2020, Eq.(15) Q_i = ι·A_i + (1-ι)·P_i; Peng 2017, Eq.(16)-(17); Xia & Xu 2011 score Eq.(1).
7
Step 6 - Rank alternatives in DESCENDING order of Q_i (largest Q_i is best).
$A_{i} ≻ A_{i^{'}} ⟺ Q_{i} > Q_{i^{'}} . (M i s h r a S t e p 8; M a r d ani S t e p 8; P e n g A l g or i t hm 2 S t e p 7)$
Mishra 2019, Step 8; Mardani 2020, Step 8; Peng 2017, Algorithm 2 Step 7.

Fits when

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

Edge cases and pitfalls

•λ=0: saf WPM (multiplicative, herhangi bir kriterde score=0 → toplam Q=0; alternatif elenir). λ=1: saf WSM (additive, sıfırlar etkisiz). λ=0.5 default: paper'da yeşil tedarikçi case'inde A_3 birinci.
•score(h_ij) hesabı için Xia-Xu 2011 score function s(h) = (1/l)·Σγ kullanılır; paper bu temel skor üzerinden λ-kombinasyonu yapar.
•Exponential entropi e_1(M) sıfır = HFE crisp tek bir değer; entropi maksimum (≈1) = HFE tüm γ ∈ [0,1] aralığında dağıtılmış. Entropi düşük kriter = ayırt edici, ağırlığı yüksek olur (paper §3.5).
•Exponential divergence L = 0 iff M = N veya M = N^c (Eq.E3); bu, paper'ın Teorem 3.1 ile kanıtladığı bir özellik - KL-divergence'tan farklı (KL = 0 sadece P=Q için).
•Yeşil tedarikçi case (paper §5.1): 4 tedarikçi × 12 kriter, management commitment (0.3119), environmental management system (0.2259), green product (0.2010) en üst 3 kriter; A_3 (Supplier 3) Q_3 max ile birinci.

How to cite

Mishra, A.R.; Rani, P.; Pardasani, K.R.; Mardani, A. (2019). A novel hesitant fuzzy WASPAS method for assessment of green supplier problem based on exponential information measures. Journal of Cleaner Production. https://doi.org/10.1016/j.jclepro.2019.117901

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-WASPAS

How it works

Construct HF decision matrix R = (h_ij)_{m×n}. Each h_ij is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA per Mishra Eq.15).

R = (h_{i} j), h_{i} j \subseteq [0, 1], ∣ h_{i} j ∣ \geq 1.

Mishra 2019, Section 4 Step 1; Mardani 2020, Step 1 (per-DE matrices Θ_k); Peng 2017, Algorithm 2 Step 1.

Step 1b (group mode only) - Aggregate per-DE matrices into a single R via HFWA with DE weights ϖ_k.

h_{i} j = H F W A (h_{i}^{1} j, ..., h_{i}^{ℓ} j) = \cup_{γ^{k} \in h_{i}^{k} j} 1 - k = 1 \prod ℓ (1 - γ^{k})^{ϖ_{k}} (M i s h r a E q .15; M a r d ani E q .8; X ia - X u 2011)

Mishra 2019, Eq.(15); Mardani 2020, Eq.(8) AHF-D matrix with DE weights λ_k = (1-e(h_k))/(ℓ-Σ e(h_k)) Eq.(7); Xia & Xu 2011 HFWA operator.

Step 2 - Normalize R by criterion type. Benefit: identity (n_ij = h_ij). Cost: pointwise complement (n_ij = ∪_{γ ∈ h_ij} {1-γ}).

n_{i} j = h_{i} j i f C_{j} \in Ω_{b} (b e n e f i t); n_{i} j = \cup_{γ \in h_{i} j} 1 - γ i f C_{j} \in Ω_{c} (cos t) . (P e n g E q .5; M i s h r a E q .17 r e d u ce df or m; M a r d ani E q .12)

Peng 2017, Eq.(5); Mishra 2019, Eq.(17); Mardani 2020, Eq.(12) - Sb=benefit (identity), Sc=cost (1-γ complement).

Step 3 - Compute WSM aggregate S_i = ⊕_j w_j n_ij via HFWA.

S_{i} = \oplus_{j = 1}^{n} w_{j} \cdot n_{i} j = \cup_{γ_{j} \in n_{i} j} 1 - j = 1 \prod n (1 - γ_{j})^{w_{j}} (M i s h r a E q .18; M a r d ani E q .13 A_{i}; P e n g E q .14)

Mishra 2019, Eq.(18); Mardani 2020, Eq.(13) WSM A_i; Peng 2017, Eq.(14).

Step 4 - Compute WPM aggregate P_i = ⊗_j (n_ij)^{w_j} via HFWG.

P_{i} = \otimes_{j = 1}^{n} (n_{i} j)^{w_{j}} = \cup_{γ_{j} \in n_{i} j} j = 1 \prod n γ_{j}^{w_{j}} (M i s h r a E q .19; M a r d ani E q .14 P_{i}; P e n g E q .15)

Mishra 2019, Eq.(19); Mardani 2020, Eq.(14) WPM P_i; Peng 2017, Eq.(15).

Step 5 - WASPAS combination Q_i = ϑ·score(S_i) + (1-ϑ)·score(P_i) using HF score function S(h) = (1/|h|) Σ γ (Mishra/Torra/Xia-Xu Eq.1).

Q_{i} = ϑ \cdot S (S_{i}) + (1 - ϑ) \cdot S (P_{i}), w h er e S (h) = (1/∣ h ∣) Σ_{γ \in h} γ . D e f a u l tϑ = 0.5 (M i s h r a E q .20; M a r d ani E q .15 w i t h ι = 0.5; P e n g E q .16) . E x t e n d e df or m s (M i s h r a E q .21/ M a r d ani E q .15/ P e n g E q .17) a l l o w ϑ \in [0, 1] .

Mishra 2019, Eq.(20)-(21); Mardani 2020, Eq.(15) Q_i = ι·A_i + (1-ι)·P_i; Peng 2017, Eq.(16)-(17); Xia & Xu 2011 score Eq.(1).

Step 6 - Rank alternatives in DESCENDING order of Q_i (largest Q_i is best).

A_{i} ≻ A_{i^{'}} ⟺ Q_{i} > Q_{i^{'}} . (M i s h r a S t e p 8; M a r d ani S t e p 8; P e n g A l g or i t hm 2 S t e p 7)

Mishra 2019, Step 8; Mardani 2020, Step 8; Peng 2017, Algorithm 2 Step 7.

Edge cases and pitfalls

•λ=0: saf WPM (multiplicative, herhangi bir kriterde score=0 → toplam Q=0; alternatif elenir). λ=1: saf WSM (additive, sıfırlar etkisiz). λ=0.5 default: paper'da yeşil tedarikçi case'inde A_3 birinci.

•score(h_ij) hesabı için Xia-Xu 2011 score function s(h) = (1/l)·Σγ kullanılır; paper bu temel skor üzerinden λ-kombinasyonu yapar.

•Exponential entropi e_1(M) sıfır = HFE crisp tek bir değer; entropi maksimum (≈1) = HFE tüm γ ∈ [0,1] aralığında dağıtılmış. Entropi düşük kriter = ayırt edici, ağırlığı yüksek olur (paper §3.5).

•Exponential divergence L = 0 iff M = N veya M = N^c (Eq.E3); bu, paper'ın Teorem 3.1 ile kanıtladığı bir özellik - KL-divergence'tan farklı (KL = 0 sadece P=Q için).

•Yeşil tedarikçi case (paper §5.1): 4 tedarikçi × 12 kriter, management commitment (0.3119), environmental management system (0.2259), green product (0.2010) en üst 3 kriter; A_3 (Supplier 3) Q_3 max ile birinci.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-WASPAS