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

PHFS-HVaR is a risk-focused ranking method used when you hold, for an alternative's performance, not a single definite figure but several possible values, each with its own probability of occurring. Its output is the boundary value that the alternative is "guaranteed to stay below with this probability" at a chosen confidence level (X); alternatives are ranked by this boundary. Zhou and Xu proposed it in 2017, from the observation that classical average-based comparisons can conceal poor scenarios.

PHFS-HVaR's underlying idea is to carry the Value-at-Risk concept from finance into decision analysis. Even where an alternative's average performance looks good, this average can conceal a low-probability but very poor scenario. Rather than asking "what is the average," PHFS-HVaR asks "what is the worst case guaranteed under a given probability"; it finds the point at which probabilities, accumulated starting from the lowest values, reach that threshold, and takes the value at that point as the boundary.

This idea carries a philosophical consequence: the method reflects a risk-averse decision stance. PHFS-HVaR does not look at an optimistic average but at the boundary of the poor tail; two alternatives with the same average return can therefore produce different PHFS-HVaR results if their poor-scenario distributions differ.

The method proceeds through four steps.

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; this answers the question "what percentage of probability's worst case do we want to see."

Second, cumulative probability. Starting from the smallest value, probabilities are accumulated over the sorted values.

Third, finding the boundary. The value at which cumulative probability first crosses, or exactly matches, the X threshold is found. This value is that alternative's PHFS-HVaR.

Fourth, ranking. Alternatives are ranked by this boundary value; for a benefit-direction measure (such as return), a higher boundary is safer or better, while for a loss-direction measure a lower boundary is better. Where alternatives' boundary values come out equal, PHFS-HVaR is not discriminating, and EHVaR should then be used.

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-HVaR is a point estimate; it states only the boundary at the chosen confidence level and gives no information about how poor the scenarios below that boundary could be. Two alternatives can have an equal PHFS-HVaR without their entire poor-scenario distributions being the same; one may have few scenarios just below the boundary while the other has far poorer ones.

The confidence level (X) must be stated in the report; as X changes the boundary changes too, and different X values can give different rankings.

Thus instead of writing:

"PHFS-HVaR proved with certainty that this alternative is safer"

the report should read:

"At the twenty per cent probability threshold X, this alternative's guaranteed boundary is this figure; this boundary does not show the severity of the scenarios below it"

PHFS-HVaR works with a probabilistic form of hesitant data: for every alternative, several possible values are given not merely as "possible" but 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 (the probabilities must sum to 1); a confidence level (X); and whether the values represent a gain or a loss. PHFS-HVaR does not combine several criteria in the classical sense; it processes the possible scenarios of a single measurement dimension (such as rate of return). Where several criteria exist, a separate PHFS-HVaR is computed for each, or the criteria are combined beforehand.

PHFS-HVaR is a sound choice if you hold several scenarios for an alternative's performance and the probability of each (expert estimates, historical data distributions, scenario analysis), and you particularly want a cautious decision against poor scenarios.

It should not be used where you hold only "possible" values whose probability of occurring is unknown; PHFS-HVaR requires probabilities and cannot be applied to hesitant data without them. Where the boundary values PHFS-HVaR produces come out equal, the method is also not discriminating.

Several probabilistic scenarios, particular interest in the poor tail → PHFS-HVaR

PHFS-HVaR values come out equal, a more discriminating risk measure is needed → PHFS-EHVaR

Only possible values exist, probability unknown → ordinary hesitant methods

Average performance is enough, no particular interest in the poor tail → expected value or classical weighted methods

PHFS-HVaR's greatest strength is its direct interpretability. The statement "this is the worst case guaranteed under this probability" is easy to convey to a risk-averse decision-maker, and is familiar to anyone acquainted with the Value-at-Risk concept in finance. Its calculation is simple; it only requires finding the point at which cumulative probability crosses a threshold.

Its limitations stem from its point-estimate structure. PHFS-HVaR disregards all information below the boundary entirely; two very different scenario distributions can give the same boundary value, and the method cannot distinguish between them (Zhou and Xu, 2020, Example 6.1). The choice of confidence level (X) strongly affects the result, and this choice is left largely to the decision-maker. Because the method was proposed in 2017, its literature is limited and it does not yet carry a broad body of comparison.

The most common mistake is sharing a PHFS-HVaR result without stating the confidence level (X); different X values can produce different rankings on the same data.

A second mistake is treating alternatives with equal PHFS-HVaR as "equally good" and never checking EHVaR; equality usually shows a situation the method cannot distinguish, not that the alternatives are genuinely equal. A third mistake is inventing probabilities for data where only possible values are known but probabilities are not, and applying PHFS-HVaR anyway; probability data must come from a genuine source, such as historical frequency or expert estimation.

The governing principle is this:

PHFS-HVaR shows only a boundary at the chosen confidence level; a PHFS-HVaR result presented without stating this level in the report, and without discussing the severity of the scenarios below the boundary, is incomplete.