PHFS-EHVaR (Expected Hesitant Value-at-Risk under Probabilistic Hesitant Fuzzy Sets)

PHFS-EHVaR is a refined form of PHFS-HVaR, used when you hold several possible values for an alternative's performance, each with its own probability of occurring. Its output is the alternative's expected (probability-weighted average) value in the poor-scenario region at a chosen confidence level (X); alternatives are ranked by this value. Zhou and Xu proposed it in 2017 to solve the indistinguishability problem created by PHFS-HVaR, which looks only at a single boundary point and disregards the scenarios below that boundary.

PHFS-EHVaR's underlying idea is that a single boundary point does not, on its own, carry enough information. Two alternatives can share the same poor-scenario boundary, yet the scenarios falling below that boundary may be mildly poor for one and severely poor for the other. Rather than looking only at the value at the boundary, PHFS-EHVaR looks at the probability-weighted average of every scenario below it; it thereby turns the question "how bad could it get" into "how bad is it on average below the boundary."

This idea carries a philosophical consequence: PHFS-EHVaR is more discriminating than PHFS-HVaR, but slightly more complex to compute. As the confidence level X approaches 1, PHFS-EHVaR approaches the probability-weighted average of all of the alternative's possible values, its overall expected value; as X shrinks, it turns into a risk measure focused only on the worst scenarios.

The method proceeds through four steps (a fifth is added for a group decision).

First, gathering the possible values. For every alternative, the possible values and their probabilities of occurring are sorted from smallest to largest; a confidence level (X) is set.

Second, cumulative probability. Probabilities are accumulated over the sorted values; this is the same step as in PHFS-HVaR.

Third, computing the expected value. All values falling below the X threshold are multiplied by their own probabilities and summed; the final value that crosses the threshold is counted only in proportion to the probability share remaining. PHFS-EHVaR thus gives a probability-weighted summary of every scenario below the boundary, not a single point.

Fourth, ranking. Alternatives are ranked by this expected value. In group-decision mode, several assessors' PHFEs are first combined under the principle "a smaller deviation earns a larger weight," and PHFS-EHVaR is then computed on this combined PHFE.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

PHFS-EHVaR is a probability-weighted average; unlike PHFS-HVaR, it also reflects the severity of the scenarios below the boundary. Two alternatives sharing the same PHFS-HVaR boundary can come out different under PHFS-EHVaR; this shows their poor-scenario distributions differ.

The confidence level (X) must be stated in the report here too. When X = 1, PHFS-EHVaR equals the alternative's overall expected value; this corresponds to a decision-maker concerned with general performance rather than tail risk. As X shrinks, the method moves towards a more cautious stance.

Thus instead of writing:

"PHFS-EHVaR showed this alternative is better overall"

the report should read:

"At the twenty per cent probability threshold X, this alternative's expected value in the poor-scenario region is this figure; this measures something different from the alternative's overall average performance"

PHFS-EHVaR works with the same probabilistic hesitant data as PHFS-HVaR: several possible values for every alternative, each with its own probability of occurring, summing to 1. DecisionMind holds no extension of this base method.

You need: a set of possible values for every alternative and each value's probability of occurring; a confidence level (X); and whether the values represent a gain or a loss. If a group decision is wanted, several assessors' separate PHFEs can also be entered; the method automatically weights and combines them. PHFS-EHVaR, like PHFS-HVaR, does not combine several criteria in the classical sense; it processes the possible scenarios of a single measurement dimension.

PHFS-EHVaR is a sound choice if you hold several scenarios for an alternative's performance and the probability of each scenario occurring, you want to take a cautious approach to poor scenarios, and the boundary values PHFS-HVaR produces come out close to or equal to one another.

It should not be used where probabilities are unknown or unreliable; PHFS-EHVaR is sensitive to the quality of the probability estimates, and small changes in probability can change the ranking. Where only overall average performance matters and there is no particular interest in tail risk, a simpler expected-value comparison is sufficient.

PHFS-HVaR gives equal or close results, a more discriminating risk measure is needed → PHFS-EHVaR

Only the worst boundary matters, no interest in the distribution below it → PHFS-HVaR

Probabilities unreliable or unknown → ordinary hesitant methods

Overall average performance is enough → expected value or classical weighted methods

PHFS-EHVaR's greatest strength is its ability to resolve situations PHFS-HVaR cannot distinguish; it can separate two alternatives sharing the same boundary by looking at the probability-weighted average of the scenarios below it (Zhou and Xu, 2020, Example 6.2). Its smooth reduction to the overall expected value as the confidence level X approaches 1 makes it a tool adjustable between tail risk and general performance. Its group-decision extension combines several assessors' views automatically and on a principled basis (a smaller deviation earns a larger weight).

Its limitations stem from its dependence on probability information. PHFS-EHVaR is sensitive to small changes in probability values; if the quality of how the probabilities were obtained is doubtful, so is the result. In group-decision mode, each assessor's weight varies by alternative (the same assessor can receive a different weight for different alternatives), which can seem counter-intuitive at first and needs explaining. Because the method was proposed in 2017, its literature is limited.

The most common mistake is confusing PHFS-EHVaR with PHFS-HVaR; PHFS-HVaR is a boundary point, PHFS-EHVaR is an average below that boundary, and the two produce different figures.

A second mistake is sharing a PHFS-EHVaR result without stating the confidence level (X); a value of X close to 1 corresponds to general performance, a small value to tail risk, and without this distinction stated the result is misunderstood. A third mistake is confusing PHFS-EHVaR with the classical finance concept of entropic value-at-risk (EVaR); the two have different mathematical definitions.

The governing principle is this:

PHFS-EHVaR is a probability-weighted summary of the poor-scenario region at the chosen confidence level; a PHFS-EHVaR result presented without stating this level and without distinguishing it from overall average performance is incomplete.