HF-TradeOff-Portfolio - Hesitant Fuzzy Score-Deviation Trade-Off Portfolio Selection (Zhou-Xu 2018)

1. Collect HFS decision matrix H = [h_ij]_{n×m}. Aggregate per-criterion HFEs into per-stock HFEs h_i = ⊗_j h_ij (identical to HF-MaxScore-Portfolio F1-F2).
$h_{i} = j = 1 ⨁ m h_{ij}$
2. Compute deviation and score range bounds by solving auxiliary models: max/min d(⊗_i w_i h_i) (Model 3.17) to get [min_D, max_D]; and max/min s(⊗_i w_i h_i) (Model 3.19) to get [min_S, max_S]. These define the feasible risk/return envelope.
$min D = W \in Δ min d (i ⨁ w_{i} h_{i}), max D = W \in Δ max d (i ⨁ w_{i} h_{i})$
3. Set D (or S) based on investor risk type via trisection: risk-seeker: D1=max_D; neutral: D2=min_D+(2/3)(max_D-min_D); risk-averse: D3=min_D+(1/3)(max_D-min_D). Score trisection (Def.3.2) is symmetric.
$D_{1} = max D, D_{2} = min D + \frac{2}{3} (max D - min D), D_{3} = min D + \frac{1}{3} (max D - min D)$
4. Solve Model 3.16 (max_score formulation): max s(⊗_i w_i h_i) s.t. d(⊗_i w_i h_i) ≤ D, Σw_i=1, w_i≥0. OR solve Model 3.18 (min_deviation formulation): min d(⊗_i w_i h_i) s.t. s(⊗_i w_i h_i) ≥ S, Σw_i=1, w_i≥0.
$Model 3.16: W \in Δ max s (i ⨁ w_{i} h_{i}) s.t. d (i ⨁ w_{i} h_{i}) \leq D Model 3.18: W \in Δ min d (i ⨁ w_{i} h_{i}) s.t. s (i ⨁ w_{i} h_{i}) \geq S$

1. Collect HFS decision matrix H = [h_ij]_{n×m}. Aggregate per-criterion HFEs into per-stock HFEs h_i = ⊗_j h_ij (identical to HF-MaxScore-Portfolio F1-F2).