Dual hesitant fuzzy EDAS (Ning, Lin, Wei and Chen, 2023)

Five things change.

Cells. In crisp EDAS every cell is a single number. Here every cell carries two probabilistic sets: a support set and a rejection set, each with a total probability of at most 1. Criterion weights are taken from outside as crisp numbers, but are built by combining three sources, as described below.

Expert aggregation (group decision). This extension supports several experts' matrices. A combination rule weighted by expert weight (PDHFWA) reduces p experts' cells to a single collective cell; if there is only one expert, this step changes nothing. For a cost criterion, this collective cell is complemented (the support and rejection sets swap places, the probabilities are preserved).

Score and the hesitation split (α). Each cell's score is built from the difference between the probability-weighted mean of the support set and the probability-weighted mean of the rejection set. If the support and rejection probabilities do not sum to 1 (leaving a residual margin of uncertainty), this margin is split between support and rejection by a parameter α; the default is α = 0.5. This card's author verified this by running the kernel directly. In the literature's supplier-selection example, every cell's probabilities already sum to exactly 1; the residual margin is therefore zero, and changing α among 0.25, 0.50 and 0.75 has NO effect whatsoever on the result. α only comes into play when probability labels are left incomplete.

Weight combined from three sources. Crisp EDAS takes its weight as a single piece from outside. This extension combines three separate weights: a CRITIC weight drawn from the data's own variability, an entropy weight from the probabilistic hesitant set, and the user's own subjective weight. The three are pooled into a single combined weight through a Lagrange solution in square-root-product form. This combination form can be changed: it can rest on objective sources alone (CRITIC + entropy) or on the subjective weight alone. This card's author verified by running the kernel directly that the combination form chosen genuinely changes the ranking (see Case 1).

The result. Once the score matrix has been built, everything that follows is crisp EDAS itself: each criterion's average is taken, each cell's positive and negative deviation from that average is measured, the deviations are summed with the combined weights, normalised, and merged into a single appraisal score.

DecisionMind fixes, for this extension, the PDHFWA combination rule, the score function and crisp EDAS's deviation-normalisation steps. α and the weight-combination form are parameters the user may choose.

The appraisal score is read as in crisp EDAS: a position relative to the set's own average. The difference lies here: the weights behind this score have been combined from three sources (data, entropy, expert opinion), and which source carries how much weight does not show in the score itself. Where CRITIC and the subjective weight point in different directions between two criteria, the choice of combination form can change the ranking.

Thus instead of writing:

"Dual hesitant fuzzy EDAS is more objective because its weights are built jointly from data and expert opinion"

the report should read:

"This score rests on a combined weight set built from data-based (CRITIC, entropy) and expert-based weights through a particular combination form; the ranking can change if the combination form changes, so the report must state which form was chosen"

This extension is appropriate when criterion evaluation carries more than one plausible value on both the supporting and the rejecting side. Its suitability increases further when these values' probabilities of occurrence are known or can be estimated, and when several experts' opinions have been collected separately. This differs from the limitation described on the HF-EDAS card (the engine's reduction to a single triangle): here the engine genuinely carries more than one value, with more than one probability, in two separate sets (support and rejection); there is no early defuzzification. If the probabilities are not known, DecisionMind assigns the entered values equal probability; this is equivalent to a "pure" dual hesitant set and must be stated in the report. Converting a measured criterion into a probabilistic dual set does not model uncertainty, it manufactures it. Crisp EDAS's exit condition applies here as well.

Treating α as an adjustable sensitivity lever. α only affects the result when probability labels are left incomplete (summing to below 1). If the probabilities are complete, as in the literature's supplier example, changing α has no effect whatsoever; this card's author has verified this.

Saying "the weights were combined" without stating the combination form. A combination resting on objective sources alone (CRITIC + entropy) and one resting on the subjective weight alone can give different rankings; this card's author has verified this (see Case 1). Which form was used must be stated explicitly in the report.

Inventing probabilities for a set entered without them. The engine completes sets given without probability using an equal distribution; if the user then tries to distribute probabilities on top of this, the probabilities are counted twice.

Confusing the pure hesitant (probability-free) set with the probabilistic hesitant set. Both can be entered into the engine in the same form, but the "pure" set carries the assumption that "every value is equally likely"; this assumption must be stated in the report.

The governing principle is this:

In this extension the result depends both on which uncertainty-split parameter (α) is in effect and on which weight-combination form was chosen; both must be stated explicitly in the report.