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Ranking

HF-COPRAS - Hesitant Fuzzy Complex Proportional Assessment

Hesitant fuzzy benefit/cost proportional ranking

Mishra, A.R., Rani, P., Pardasani, K.R.2018doi:10.1007/s41066-018-0103-8 ↗

Overview

HF-COPRAS 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 Z = (z_ij)_{m×n}. Each z_ij is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA aggregation).
$Z = (z_{i} j), z_{i} j \subseteq [0, 1], ∣ z_{i} j ∣ \geq 1.$
Mishra 2018, Step 1; Rani 2020, Step 1 (Z=(z_ij^(k)) per-DE); Peng 2017, Algorithm 3 Step 1.
2
Step 1b (group mode only) - Aggregate per-DE matrices into a single Z via HFWA with DE weights ϖ_k.
$z_{i} j = H F W A (z_{i}^{1} j, ..., z_{i}^{ℓ} j) = \cup_{γ^{k}} 1 - k = 1 \prod ℓ (1 - γ^{k})^{ϖ_{k}} (X ia - X u 2011; R ani 2020 E q .8 ξ_{i} j)$
Xia & Xu 2011 HFWA operator; Rani 2020 Eq.(8) ξ_ij with DE weights λ_k = (1-e(h_k))/Σ(1-e(h_k)) Eq.(7).
3
Step 2 - Compute benefit aggregate α_i for each alternative via weighted HFE addition over benefit criteria Δ = {j : C_j is benefit}.
$α_{i} = \oplus_{j \in Δ} w_{j} \cdot z_{i} j = \cup_{γ_{j} \in z_{i} j} 1 - j \in Δ \prod (1 - γ_{j})^{w_{j}} (M i s h r a E q .12; R ani E q .12 σ_{i}; P e n g E q .19 P_{i})$
Mishra 2018, Eq.(12); Rani 2020, Eq.(12) benefit aggregate σ_i; Peng 2017, Eq.(19) P_i.
4
Step 3 - Compute cost aggregate β_i via weighted HFE addition over cost criteria V = {j : C_j is cost}.
$β_{i} = \oplus_{j \in V} w_{j} \cdot z_{i} j = \cup_{γ_{j} \in z_{i} j} 1 - j \in V \prod (1 - γ_{j})^{w_{j}} (M i s h r a E q .13; R ani E q .13 v_{i}; P e n g E q .20 R_{i})$
Mishra 2018, Eq.(13); Rani 2020, Eq.(13) cost aggregate v_i; Peng 2017, Eq.(20) R_i.
5
Step 4 - Compute HF score values S(α_i) and S(β_i) using Xia-Xu 2011 score function.
$S (h) = (1/∣ h ∣) Σ_{γ \in h} γ . (X ia - X u 2011 D e f .3/ M i s h r a E q .6/ R ani E q .6/ P e n g E q .1)$
Xia & Xu 2011, Def. 3; Mishra 2018 Def. 3; Rani 2020 Def./Eq.(6); Peng 2017 Eq.(1).
6
Step 5 - Compute relative significance λ_i via COPRAS proportional combination.
$λ_{i} = S (α_{i}) + [Σ_{k = 1}^{m} S (β_{k})] / [S (β_{i}) \cdot Σ_{k = 1}^{m} (1/ S (β_{k}))] (M i s h r a E q .15; r e d u ces t oc l a ss i c a l Z a v a d s k a s C O P R A S f or m) . R ani 2020 E q . (15) γ - s t r a t e g y v a r ian t : θ_{i} = γ \cdot S (σ_{i}) + (1 - γ) \cdot [Σ S (v_{k})] / [S (v_{i}) \cdot Σ (1/ S (v_{k}))], γ \in [0, 1]; γ = 0 r eco v er scos t - s i d e - o n l y r ank in g, γ = 0.5 d e f a u l t mi x es b e n e f i t an d cos t co m p o n e n t s, γ = 1 co l l a p ses t o b e n e f i t - o n l y S (σ_{i}) .$
Mishra 2018, Eq.(14)-(15); Rani 2020, Eq.(14)-(15) γ-strategy form; Zavadskas 1994 crisp form.
7
Step 6 - Rank alternatives in DESCENDING order of λ_i (largest is best).
$A_{i} ≻ A_{i^{'}} ⟺ λ_{i} > λ_{i^{'}}; D * = a r g ma x_{i} λ_{i} . (M i s h r a E q .16; R ani E q .16; P e n g A l g or i t hm 3 S t e p 7)$
Mishra 2018, Step 6 Eq.(16); Rani 2020, Step 8 Eq.(16); Peng 2017, Algorithm 3 Step 7.
8
Step 7 - Compute utility degree η_i = (λ_i / λ_max) × 100%, reported as relative attractiveness 0-100%.
$η_{i} = (λ_{i} / ma x_{k} λ_{k}) \times 100$
Mishra 2018, Eq.(17); Rani 2020, Eq.(17); Peng 2017, Eq.(22).

Fits when

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

Edge cases and pitfalls

•Cost kriteri yok (S_− = ∅): COPRAS Q_i = S_+i'ye düşer; bu durumda yöntem etkin olarak saf benefit toplamı sıralamasıdır, COPRAS'ın orantısal-fayda zenginliği kaybolur. Paper implicit varsayım: en az 1 cost kriteri var.
•Shapley measure additive (μ(A∪B) = μ(A)+μ(B)) ise w_j^S klasik weighted-sum ağırlıklarına indirgenir; non-additive supermodular μ ile kriter sinerjisi modellenir (paper §3.4 örneği).
•Entropi e_1(M) = 0 dejenere durumu: HFE crisp tek bir değer içeriyorsa entropi sıfır → kriter "tam belirli" sayılır → nesnel ağırlığı (1−e_1)/Σ paydasından yüksek pay alır. Yorumlama: belirsizliği düşük kriter daha çok ayırt eder (paper'ın motivasyonu).
•Vehicle insurance case (paper §4): 4 şirket × 4 kriter, sonuç A_3 N_3=100% birinci. Bu reproduce edilebilir; Block J fixture'a anchored.
•S_− = 0 dejenere (bir alternatif tüm cost kriterlerde 0): Q_i hesabında bölme tanımsız → engine bu alternatifi 'ideal-no-cost' işaretler ve kriterler arası dengeyi raporlar.

How to cite

Mishra, A.R.; Rani, P.; Pardasani, K.R. (2018). Multiple-criteria decision-making for service quality selection based on Shapley COPRAS method under hesitant fuzzy sets. Granular Computing. https://doi.org/10.1007/s41066-018-0103-8

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-COPRAS

How it works

Construct HF decision matrix Z = (z_ij)_{m×n}. Each z_ij is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA aggregation).

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

Mishra 2018, Step 1; Rani 2020, Step 1 (Z=(z_ij^(k)) per-DE); Peng 2017, Algorithm 3 Step 1.

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

z_{i} j = H F W A (z_{i}^{1} j, ..., z_{i}^{ℓ} j) = \cup_{γ^{k}} 1 - k = 1 \prod ℓ (1 - γ^{k})^{ϖ_{k}} (X ia - X u 2011; R ani 2020 E q .8 ξ_{i} j)

Xia & Xu 2011 HFWA operator; Rani 2020 Eq.(8) ξ_ij with DE weights λ_k = (1-e(h_k))/Σ(1-e(h_k)) Eq.(7).

Step 2 - Compute benefit aggregate α_i for each alternative via weighted HFE addition over benefit criteria Δ = {j : C_j is benefit}.

α_{i} = \oplus_{j \in Δ} w_{j} \cdot z_{i} j = \cup_{γ_{j} \in z_{i} j} 1 - j \in Δ \prod (1 - γ_{j})^{w_{j}} (M i s h r a E q .12; R ani E q .12 σ_{i}; P e n g E q .19 P_{i})

Mishra 2018, Eq.(12); Rani 2020, Eq.(12) benefit aggregate σ_i; Peng 2017, Eq.(19) P_i.

Step 3 - Compute cost aggregate β_i via weighted HFE addition over cost criteria V = {j : C_j is cost}.

β_{i} = \oplus_{j \in V} w_{j} \cdot z_{i} j = \cup_{γ_{j} \in z_{i} j} 1 - j \in V \prod (1 - γ_{j})^{w_{j}} (M i s h r a E q .13; R ani E q .13 v_{i}; P e n g E q .20 R_{i})

Mishra 2018, Eq.(13); Rani 2020, Eq.(13) cost aggregate v_i; Peng 2017, Eq.(20) R_i.

Step 4 - Compute HF score values S(α_i) and S(β_i) using Xia-Xu 2011 score function.

S (h) = (1/∣ h ∣) Σ_{γ \in h} γ . (X ia - X u 2011 D e f .3/ M i s h r a E q .6/ R ani E q .6/ P e n g E q .1)

Xia & Xu 2011, Def. 3; Mishra 2018 Def. 3; Rani 2020 Def./Eq.(6); Peng 2017 Eq.(1).

Step 5 - Compute relative significance λ_i via COPRAS proportional combination.

λ_{i} = S (α_{i}) + [Σ_{k = 1}^{m} S (β_{k})] / [S (β_{i}) \cdot Σ_{k = 1}^{m} (1/ S (β_{k}))] (M i s h r a E q .15; r e d u ces t oc l a ss i c a l Z a v a d s k a s C O P R A S f or m) . R ani 2020 E q . (15) γ - s t r a t e g y v a r ian t : θ_{i} = γ \cdot S (σ_{i}) + (1 - γ) \cdot [Σ S (v_{k})] / [S (v_{i}) \cdot Σ (1/ S (v_{k}))], γ \in [0, 1]; γ = 0 r eco v er scos t - s i d e - o n l y r ank in g, γ = 0.5 d e f a u l t mi x es b e n e f i t an d cos t co m p o n e n t s, γ = 1 co l l a p ses t o b e n e f i t - o n l y S (σ_{i}) .

Mishra 2018, Eq.(14)-(15); Rani 2020, Eq.(14)-(15) γ-strategy form; Zavadskas 1994 crisp form.

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

A_{i} ≻ A_{i^{'}} ⟺ λ_{i} > λ_{i^{'}}; D * = a r g ma x_{i} λ_{i} . (M i s h r a E q .16; R ani E q .16; P e n g A l g or i t hm 3 S t e p 7)

Mishra 2018, Step 6 Eq.(16); Rani 2020, Step 8 Eq.(16); Peng 2017, Algorithm 3 Step 7.

Step 7 - Compute utility degree η_i = (λ_i / λ_max) × 100%, reported as relative attractiveness 0-100%.

η_{i} = (λ_{i} / ma x_{k} λ_{k}) \times 100

Mishra 2018, Eq.(17); Rani 2020, Eq.(17); Peng 2017, Eq.(22).

Edge cases and pitfalls

•Cost kriteri yok (S_− = ∅): COPRAS Q_i = S_+i'ye düşer; bu durumda yöntem etkin olarak saf benefit toplamı sıralamasıdır, COPRAS'ın orantısal-fayda zenginliği kaybolur. Paper implicit varsayım: en az 1 cost kriteri var.

•Shapley measure additive (μ(A∪B) = μ(A)+μ(B)) ise w_j^S klasik weighted-sum ağırlıklarına indirgenir; non-additive supermodular μ ile kriter sinerjisi modellenir (paper §3.4 örneği).

•Entropi e_1(M) = 0 dejenere durumu: HFE crisp tek bir değer içeriyorsa entropi sıfır → kriter "tam belirli" sayılır → nesnel ağırlığı (1−e_1)/Σ paydasından yüksek pay alır. Yorumlama: belirsizliği düşük kriter daha çok ayırt eder (paper'ın motivasyonu).

•Vehicle insurance case (paper §4): 4 şirket × 4 kriter, sonuç A_3 N_3=100% birinci. Bu reproduce edilebilir; Block J fixture'a anchored.

•S_− = 0 dejenere (bir alternatif tüm cost kriterlerde 0): Q_i hesabında bölme tanımsız → engine bu alternatifi 'ideal-no-cost' işaretler ve kriterler arası dengeyi raporlar.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HF-COPRAS