PHFS-EHVaR - Expected Hesitant Value-at-Risk for Probabilistic Hesitant Fuzzy Sets (Zhou-Xu 2017)

1. Collect PHFE evaluations h_i(c_l|p_l) for K alternatives. Sort each PHFE ascending by c_l. Set certainty degree X.
$h_{i} (c_{l}^{i} ∣ p_{l}^{i}), c_{1}^{i} \leq \dots \leq c_{# h_{i}}^{i}, l \sum p_{l}^{i} = 1$
2. Construct cumulative distribution P_k = Σ_{l=1}^k p_l for each alternative. Find index k such that P_{k-1} < X ≤ P_k (same as HVaR step). Set P_0 = 0.
$P_{k}^{i} = l = 1 \sum k p_{l}^{i}, P_{0}^{i} = 0, find k_{i} : P_{k_{i} - 1}^{i} < X \leq P_{k_{i}}^{i}$
3. Compute EHVaR for each alternative using Eq.(6.4): EHVaR(h_i, X) = Σ_{l=1}^{k-1} g(c^i_l) + c^i_k · (X - Σ_{l=1}^{k-1} p^i_l), where g(c_l) = c_l · p_l is the expected contribution of element c_l. If k=1, EHVaR = c^i_1 · X.
$EHVaR (h_{i}, X) = l = 1 \sum k - 1 g (c_{l}^{i}) + c_{k}^{i} \cdot (X - l = 1 \sum k - 1 p_{l}^{i}), g (c_{l}^{i}) = c_{l}^{i} \cdot p_{l}^{i}, p_{0} = 0, 1 \leq k \leq # h_{i}$
4. Rank alternatives by EHVaR (higher = better for benefit/gain criteria; lower = better for loss criteria). If mode='group', first aggregate PHFE evaluations from multiple investors using dynamic weights (§6.3), then compute EHVaR on aggregated PHFEs.
$h_{i} ≻ h_{j} ⟺ EHVaR (h_{i}, X) > EHVaR (h_{j}, X)$
5. Compute dynamic investor weights w_ij for alternative i and investor j via the dynamic programming model (§6.3): principle 'larger weight to investor with smaller deviation'. Solve: min e_{g,t} = |w_ig·d(h_ig) - w_it·d(h_it)| s.t. Σ_j w_ij=1, w_ij≥0 per alternative i. This gives w_ij ∝ 1/d(h_ij) after normalization.
$w_{i g}, w_{i t} min e_{g, t} = w_{i g} \cdot d (h_{i g}) - w_{i t} \cdot d (h_{i t}), j = 1 \sum k w_{ij} = 1, w_{ij} \geq 0 \Rightarrow w_{ij} \propto \frac{1}{d ( h _{ij} )}$

1. Collect PHFE evaluations h_i(c_l|p_l) for K alternatives. Sort each PHFE ascending by c_l. Set certainty degree X.

h_{i} (c_{l}^{i} ∣ p_{l}^{i}), c_{1}^{i} \leq \dots \leq c_{# h_{i}}^{i}, l \sum p_{l}^{i} = 1

2. Construct cumulative distribution P_k = Σ_{l=1}^k p_l for each alternative. Find index k such that P_{k-1} < X ≤ P_k (same as HVaR step). Set P_0 = 0.

P_{k}^{i} = l = 1 \sum k p_{l}^{i}, P_{0}^{i} = 0, find k_{i} : P_{k_{i} - 1}^{i} < X \leq P_{k_{i}}^{i}

3. Compute EHVaR for each alternative using Eq.(6.4): EHVaR(h_i, X) = Σ_{l=1}^{k-1} g(c^i_l) + c^i_k · (X - Σ_{l=1}^{k-1} p^i_l), where g(c_l) = c_l · p_l is the expected contribution of element c_l. If k=1, EHVaR = c^i_1 · X.

EHVaR (h_{i}, X) = l = 1 \sum k - 1 g (c_{l}^{i}) + c_{k}^{i} \cdot (X - l = 1 \sum k - 1 p_{l}^{i}), g (c_{l}^{i}) = c_{l}^{i} \cdot p_{l}^{i}, p_{0} = 0, 1 \leq k \leq # h_{i}

4. Rank alternatives by EHVaR (higher = better for benefit/gain criteria; lower = better for loss criteria). If mode='group', first aggregate PHFE evaluations from multiple investors using dynamic weights (§6.3), then compute EHVaR on aggregated PHFEs.

h_{i} ≻ h_{j} ⟺ EHVaR (h_{i}, X) > EHVaR (h_{j}, X)

5. Compute dynamic investor weights w_ij for alternative i and investor j via the dynamic programming model (§6.3): principle 'larger weight to investor with smaller deviation'. Solve: min e_{g,t} = |w_ig·d(h_ig) - w_it·d(h_it)| s.t. Σ_j w_ij=1, w_ij≥0 per alternative i. This gives w_ij ∝ 1/d(h_ij) after normalization.

w_{i g}, w_{i t} min e_{g, t} = w_{i g} \cdot d (h_{i g}) - w_{i t} \cdot d (h_{i t}), j = 1 \sum k w_{ij} = 1, w_{ij} \geq 0 \Rightarrow w_{ij} \propto \frac{1}{d ( h _{ij} )}