Probabilistic hesitant EDAS (manifest name: extended hesitant linguistic EDAS)

Four things change; the logic of locating a position relative to the average does not.

Cells. In classical EDAS every cell is a single number. Here every cell consists of more than one possible value, each given with its own probability: {(γ₁,p₁), (γ₂,p₂), ...}, and the probabilities cannot sum to more than 1. An assessment such as "this supplier's delivery performance is 0.6 with even odds and 0.8 with even odds" corresponds to this structure.

Early reduction to an expected value. Before the average is even built, the kernel reduces every cell, at the very first step of the calculation, to a single number: the expected value found by multiplying every possible value by its own probability and summing. The founding paper's possibility-degree ordering and BUM-weighted aggregation are not implemented here; the kernel instead takes the column's arithmetic mean.

The average solution and deviations. The average is the arithmetic mean, across each column, of these expected values. Positive and negative deviation are computed by the same rule as in classical EDAS: above the average is favourable for a benefit criterion, below the average is favourable for a cost criterion.

Result and defuzzification. Defuzzification sits at the start, not at the end. Every cell's more-than-one possible value descends to a single expected value at the calculation's first step, and the spread among those values feeds into none of the subsequent steps.

DecisionMind fixes the expected-value function and this early-reduction order in this extension. Weights are taken from outside as crisp numbers; the method does not generate weights.

The appraisal score is read exactly as in classical EDAS: a position relative to the set's own average, not a percentage or a probability. The difference is this: beneath the score there is no longer a probability distribution, only that distribution's single expected value. Two alternatives can reach the same expected value through very different distributions; one may arrive at it from a narrow range with high probability, the other from a wide range with uncertain probabilities. The expected value does not show this difference.

Thus instead of writing:

"PHF-EDAS carries probabilistic uncertainty all the way through, so the result is more reliable"

the report should read:

"Every cell's possible values have been reduced to a single expected value at the very first step of the calculation; the resulting score is computed from these expected values, and the width of the distribution is not reflected in the score"

This extension is suitable when more than one plausible value exists for a criterion assessment, and the probability of each of these values occurring is known. If the probability is unknown, and only that more than one value is plausible is known, the classical hesitant structure and an EDAS extension suited to it should be sought instead. For a single measured criterion, classical EDAS is sufficient; opening up a single measurement into a probabilistic set adds no information. If experts give the term linguistically and the possibility-degree ordering itself is wanted (the EHFLTS and ELCOWA mechanism in the founding paper), this is a capability DecisionMind's current engine does not offer, and this limitation should be stated in the report.

Confusing the kernel with the founding paper's linguistic-possibility method. The founding paper works over a set of linguistic terms and a possibility degree; DecisionMind's kernel works over numerical (γ,p) pairs and an expected value. Though called by the same name, the two are mathematically different; the report must state clearly which one was used.

Letting the probabilities in a cell sum to more than 1. If the probabilities in a cell exceed 1, the expected value is no longer an expected value; the input must not be fed into the calculation unchecked.

Weighting the distribution equally, as with a hesitant set without probabilities. Giving every possible value an equal share instead of its real probability erases the one contribution the probabilistic hesitant structure makes, namely bringing the likelihood of occurrence into the information.

The "more sophisticated" fallacy. A probabilistic input does not automatically mean a more accurate result than classical EDAS; because of the early reduction, its contribution lies only in how the input is aggregated.

The governing principle is this:

In PHF-EDAS, every cell's possible values are reduced to a single expected value at the very first step, before the average is even built. This is not the founding paper's linguistic-possibility mechanism; the report must state clearly which mechanism was used.