HF-COPRAS - Hesitant Fuzzy Complex Proportional Assessment

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

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.

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)

3. 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})

4. 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})

5. 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)

6. 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}) .

7. 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)

8. 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