AHSPR (Asymmetric Hesitant Fuzzy Sigmoid Preference Relations)

AHSPR is a method for when you hold pairwise-comparison data between alternatives that admits more than one plausible value at once, hesitant fuzzy data ("which stock," "which supplier," "which project"). It produces a priority ranking for each criterion, and, once the criteria are combined, an overall priority ranking. It builds on the pairwise-comparison idea of classical AHP, but instead of Saaty's (1977) 1-9 scale applied identically to everyone, it uses an asymmetric scale that bends according to the decision-maker's attitude to risk. Zhou and Xu proposed it in 2016; the method's detailed derivation and application are given in the second chapter of the authors' 2020 book.

The idea behind AHSPR is the observation that the same pairwise-comparison data can yield a different priority ranking for two decision-makers with different attitudes to risk. Classical AHP assumes everyone reasons on the same 1-9 scale. AHSPR instead shows the decision-maker a handful of example differences and asks which they clearly prefer and which they clearly reject. From these answers it derives two curve parameters: one shows how quickly preference strengthens, the other how quickly rejection strengthens. For a risk-neutral decision-maker these two curves are symmetric; for one who avoids or seeks risk, the curve becomes markedly asymmetric.

This idea carries a philosophical consequence. In AHSPR there is no single "correct priority ranking": the same raw data can produce one ranking for a risk-averse investor and another for a risk-seeking one. This is not an error but the method's core claim, that preference arises not from the data alone but also from the risk attitude of whoever is interpreting it.

The method proceeds through six steps.

First, learning the risk attitude. The decision-maker is shown a handful of examples and asked which differences they clearly prefer and which they clearly reject. From these answers, two curve parameters are derived, showing how quickly preference and rejection each strengthen. If the decision-maker gives just one preference example and one rejection example, these two parameters are computed exactly.

Second, assembling the pairwise-comparison table. For each criterion, the pairwise preferences between alternatives are collected not as a single number but as a set of more than one plausible value. This carries forward the disagreement between several experts, or a single expert's own indecision, without collapsing it into one number too early. The table must satisfy indifference on the diagonal and complementarity in reciprocal cells.

Third, the criterion-level priority vector. From this raw pairwise-comparison table, a priority vector giving the importance order of the alternatives on that criterion is computed by a closed formula, without requiring optimisation.

Fourth, the personalised transformation. The same raw table is passed through the asymmetric curve shaped by the risk parameters derived in the first step, and turned into a preference table specific to that decision-maker. For a risk-neutral decision-maker this transformation barely changes the table; the more pronounced the risk attitude, the more pronounced the transformation.

Fifth, combining across criteria. The personalised tables obtained for each criterion are averaged, weighted by the criterion weights, into a single combined preference table.

Sixth, overall priority and ranking. The criterion-level priority vectors are combined with the criterion weights to compute each alternative's overall priority; alternatives are ranked by this overall priority.

The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The overall priority value shows how far an alternative stands out relative to the others for this decision-maker's attitude to risk; it is not an objective "true value." It is normal, not a computational error, for two different risk profiles to produce different priority rankings from the same raw data.

Thus instead of writing:

"AHSPR found the objectively best alternative"

the report should read:

"For this decision-maker's risk attitude, the alternative with the highest priority is this one; a different risk attitude could produce a different priority ranking from the same raw data"

AHSPR works with hesitant fuzzy data: pairwise-comparison cells hold not a single number but a set of more than one plausible value. DecisionMind carries no separate data-type extension for this method; it stands alone in its base form.

You need: for each criterion, an n×n pairwise-comparison table between alternatives (diagonal indifference, reciprocal cells complementary); and criterion weights. You also need at least one "I clearly prefer" and one "I clearly do not prefer" example, to derive the decision-maker's risk parameters. The manifest records no general limit on the number of alternatives or criteria for this method; as the pairwise-comparison table grows, however, collecting examples from the decision-maker also becomes harder.

AHSPR is a suitable choice if your data is collected through pairwise comparisons, if there is hesitancy in the comparisons (more than one plausible value), and if a marked difference between decision-makers, or in a single decision-maker's own attitude to risk, would affect the outcome.

It should not be used when the decision-maker is neutral about risk and this personalisation is not needed; classical AHP is enough there. If the pairwise comparisons are given as a single exact number, with no hesitancy, a simpler weighting method should be preferred. If you cannot derive the risk parameters from a genuine example and can only guess at them, the claim to personalisation in the result has no foundation.

Pairwise comparison, hesitancy present, risk attitude affects the outcome → AHSPR

Pairwise comparison, no hesitancy, no need for risk personalisation → AHP

Not pairwise comparison but a direct decision matrix → TOPSIS, SAW and weighting methods

The relationship between criteria also needs modelling → ANP, DEMATEL

AHSPR's most important strength is that it folds the decision-maker's attitude to risk into the result directly and measurably, a dimension most weighting methods ignore. Because the priority vector is computed by a closed formula, its computational burden is lighter than AHP's consistency-ratio check or BWM's optimisation solution. Working with hesitant fuzzy data lets it carry forward the disagreement between experts, or within a single expert's own judgement, without collapsing it into one number too early.

Its limitations arise from the fragility of the risk parameters and from the structure of the data. First, the risk parameters are derived exactly from a single preference example and a single rejection example; a small change in that example can noticeably change the curve, and hence the result. Second, the priority calculation in the closed formula can in some cases produce a negative value; this invalidates the priority vector and must be checked before it is reported. Third, the pairwise-comparison tables collected for several criteria are in practice often filled with identical or very similar numbers; this can give the impression that there is no real distinction between the criteria, and calls for the data-collection process to be reviewed. Fourth, that the same raw data yields different results under different risk profiles can confuse a reader who assumes the result has a single right answer; the report must state this personalisation explicitly.

The most common mistake is deriving the risk parameters just once and presenting the result as though it were certain, without testing how much a small change in the parameters affects it.

A second mistake is reporting the personalised result as the "objectively best alternative"; the result holds only for that decision-maker's attitude to risk. A third is using the pairwise-comparison table without verifying its diagonal and complementarity conditions. A fourth is ignoring a negative value that appears in the priority vector and reporting the ranking as though nothing were wrong; a negative priority is a sign that the comparisons are inconsistent.

The governing principle is this:

An AHSPR result must be read together with the decision-maker's attitude to risk; the same raw data can give a different priority ranking under a different risk profile, and this is not a flaw in the method but a designed feature.