Hesitant Fuzzy ARAS (Mishra, Rani, Krishankumar, Ravichandran and Kar, 2021)

Five things change; the logic of taking a ratio to the optimal alternative does not.

Cells. In crisp ARAS every cell is a single, positive number. Here every cell is a set that holds, for the same criterion-alternative pair, more than one plausible degree; for example {0.3; 0.3; 0.4; 0.5}. In the same column, different alternatives' sets can have different lengths; unlike HF-TOPSIS, no length-equalisation step is needed here, because the set is processed by averaging rather than by distance.

Optimal alternative. The optimal-alternative row is built, for each criterion, from whichever set is best according to the average of the observed sets. For a benefit criterion the set with the highest average is chosen, for a cost criterion the one with the lowest average, and it is carried into this row unchanged; this row is then added among the real alternatives.

Scale equalisation. For a benefit criterion, every element of every set is divided by the highest average in the column (including the optimal row). For a cost criterion, this ratio is subtracted from 1. This operation is applied to each element of the set separately; once normalised, the set remains a set, and has not yet been reduced to a single number.

Weighting. Every element of the normalised set goes through an operation in which the criterion weight becomes an exponent: 1 minus (1 minus the element) raised to the power of the weight. This magnifies or shrinks every element of the set separately; the set itself still carries more than one value. Unlike HF-TOPSIS, weighting here is accepted only fully and from outside; there is no option to derive it from the data or to supply it only partially.

Result and defuzzification. Only after this step is the weighted set reduced to a single number, by taking the arithmetic mean of its own elements. These single numbers are summed across the row (over the criteria), divided by the optimal row's sum, and give the utility degree (U). The order matters here: averaging first and weighting afterwards gives a different number from weighting first and averaging afterwards, because the weighting operation is not linear.

DecisionMind holds fixed, in this classical form, the selection of the optimal row by average score, the normalise-then-weight-then-average order, and the requirement of a fully external weight.

The utility degree U carries the same meaning as K in crisp ARAS: a percentage-like benefit relative to the optimal alternative, meaningful only for this alternative set and these weights. The difference is this: beneath U now lies a set that once held more than one plausible value, first weighted and then reduced to a single number by averaging. The width of the set, that is, the agreement or disagreement among sources, becomes invisible in that single number.

Thus instead of writing:

"According to Hesitant ARAS, A1 is the best alternative"

the report should read:

"Relative to the optimal alternative derived from this alternative set, A1 has the highest utility degree (U=0.75); this ratio rests on the average of the set of plausible values on each criterion, and the width of these sets should be stated separately in the report"

This extension is suitable when more than one plausible and defensible value exists for the same criterion-alternative pair, for instance when several experts or several scenarios have produced this value separately. The principle on the Hesitant data-type card applies here too: the set must rest on a genuine source, and width should not be added, without justification, to a single measured value. Crisp ARAS's exit condition applies here too: this extension is also fully compensatory and performs no elimination if no compromise is acceptable on one criterion.

Applying crisp ARAS's 1/x operation to a cost criterion. Here the cost direction is reversed by subtracting the ratio from 1; the 1/x operation does not hold here and can push the set outside [0,1].

Averaging the set first and only then weighting. The correct order is to normalise, weight, and only then average. If the order is reversed, a different utility degree results, because the weighting operation is not linear.

Assuming the weight can be partial or derived from the data. Unlike HF-TOPSIS, this classical form accepts the weight only fully and from outside; an incomplete or interval-form weight is not an option here.

Never reporting the width of the set. When two alternatives' utility degrees come out close, whether that closeness depends on narrow or wide sets should be stated separately; U alone does not show this.

The governing principle is this:

In Hesitant ARAS, a set is normalised and weighted before it is reduced to a single number by averaging; if this order is reversed, or if the width of the set is never reported, part of the information it carries is lost.