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Ranking
PHFS-EHVaR - Expected Hesitant Value-at-Risk for Probabilistic Hesitant Fuzzy Sets (Zhou-Xu 2017)
Extended tail decision-making method for probabilistic hesitant fuzzy environments. EHVaR improves upon HVaR by computing the expected (weighted sum) value over the entire left tail, not just the boundary point. EHVaR(h, X) = Σ_{i=1}^{k-1} c_i·p_i + c_k·(X - Σ_{i=1}^{k-1} p_i) where k satisfies P_{k-1} < X ≤ P_k. Always strictly separates PHFEs that HVaR cannot distinguish. Supports group decision-making via dynamic weight programming model.
Zhou, W., Xu, Z.2017doi:10.1016/j.asoc.2017.06.057 ↗
Overview
EHVaR is the recommended tail risk measure for PHFE-based decision making. Unlike HVaR (boundary-only), EHVaR integrates the full expected value over the tail - analogous to CVaR/Expected Shortfall in classical finance. A higher EHVaR at certainty degree X means the alternative has better expected performance in its worst-X% scenarios. Results may differ from overall-score ranking - always report both.
- Data
- Probabilistic Hesitant Fuzzy
Edge cases and pitfalls
EHVaR is sensitive to probability assignments p_l - even small changes in p_l can shift the ranking. Verify probability elicitation quality.
EHVaR at X=1 equals the overall score s(h) = Σ c_l p_l - so for X→1, EHVaR ranking approaches score ranking.
Group mode dynamic weights are alternative-specific (w_ij varies per i) - this means different investors get different weights for different alternatives, which may seem counterintuitive.
Do not confuse EHVaR (tail expected value) with EVaR (entropic VaR) from classical finance - they are different concepts.
How to cite
Zhou, W.; Xu, Z. (2017). Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment. Applied Soft Computing. https://doi.org/10.1016/j.asoc.2017.06.057
System ID, as it appears in reports and the API
PHFS-EHVAR