HFEA (Hesitant Fuzzy Envelopment Analysis)

HFEA is an efficiency-analysis method for when you have several options (companies, branches, projects) and want to compare their performance across multiple criteria. Unlike TOPSIS, it does not rank by a single distance measure; it asks each option "how efficient can you appear at best, if weighted entirely in your own favour?" and turns the answer into an efficiency score between 0 and 1. Options scoring 1 are considered to be "on the efficient frontier"; those below 1 show how far behind that frontier they fall. Zhou, Chen, Xu and Meng proposed it in 2018 as the form classical data envelopment analysis (DEA) takes when expert opinion is given not as a single number but as several possible values (hesitant).

The core idea of classical data envelopment analysis (Charnes, Cooper and Rhodes, 1978) is to evaluate every unit with the weights most favourable to it: whichever criterion a unit is strong on, more weight is given to that criterion, and the condition under which the unit could look its best is sought. HFEA carries the same idea but uses two things in place of inputs and outputs: the mean expert score on each criterion (score, the higher the better) and the inconsistency among these scores (deviation, the lower the better, since agreement among experts increases confidence). For every option, the method searches for the weights that maximise the score relative to the deviation; if this ratio reaches 1 the option is "on the efficient frontier," and if it cannot, the option lies behind that frontier.

This idea carries a philosophical consequence: fair comparison. HFEA imposes no predetermined, externally set weight on any option; each option is evaluated on the dimension where it is strong. If an option turns out inefficient, this means that "even at its best" it falls behind the others; no weighting could rescue it.

The method proceeds through six steps; the last is optional.

First, gathering the hesitant assessments. Where more than one expert has scored the same cell separately, these scores are combined into a single hesitant set (a set containing more than one possible value).

Second, computing score and deviation. For every option, the mean (score) and internal inconsistency (deviation) of the hesitant set on each criterion are computed separately.

Third, forming the efficiency ratio. For every option, the ratio of the weighted sum of the criteria's scores to the weighted sum of the criteria's deviations is defined. This ratio measures how efficient that option can appear at best, under weights chosen entirely in its own favour.

Fourth, finding the best weights. For every option separately, the weights that maximise this ratio are sought; the only constraint is that no option's ratio, computed with the same weights, may exceed 1. This is solved by converting it into a mathematically solvable linear programming problem.

Fifth, classification and ranking. Options whose efficiency ratio comes out at 1 are considered to be on the efficient frontier, those below 1 are deemed inefficient, and all are ranked from highest to lowest ratio.

Sixth (optional), an improvement target. For an option found inefficient, the amount by which the score on a given criterion must rise, or the deviation must fall, to reach the efficient frontier can be computed.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The closer the efficiency score is to 1, the closer the option is to the efficient frontier; a score of 1 does not mean "perfect" but "within this option set, even under the best possible weighting in its own favour, no other option's ratio can exceed 1." A score of 0.875 shows that even under the best weighting, that option falls 12.5 per cent short of the efficient frontier. More than one option can score 1 at the same time; this does not mean all of them are equally good, but that this method cannot distinguish between them. The score cannot be compared with another option set or another analysis, because the efficient frontier is built afresh in every analysis from that analysis's own options.

Thus instead of writing:

"HFEA found the most efficient company"

the report should read:

"With this option set and these criteria, the following companies are on the efficient frontier; if a distinction between them is needed, cross-efficiency analysis (HFPE) is required"

HFEA works with hesitant data: each option's value on each criterion is not a single number but a set of several possible values (such as scores given separately by more than one expert). DecisionMind holds no extension of this method; the base method works on its own. The method neither requests nor produces weights from outside; each option's efficiency is determined by its own optimisation. All criteria must be benefit-oriented (higher value is better); a cost-oriented criterion must be converted to benefit-oriented by taking its complement (1 minus the value). The number of options must clearly exceed the number of criteria; if the number of options is close to or fewer than the number of criteria, nearly all options come out efficient and the analysis loses its power to discriminate.

HFEA is appropriate if you want to compare several options fairly, on their own strengths, without setting weights from outside, and your data has a hesitant structure such as expert opinion. If your data consists of exact numbers and the classical input/output distinction is clear (production volume, cost, and so on), classical data envelopment analysis (CCR, BCC) is sufficient; using this method when there is no hesitation adds unnecessary complexity. If your goal is not merely a ranking but a full distinction among the options on the efficient frontier as well, HFEA alone is not enough.

Efficiency comparison weighted in each option's own favour, data hesitant → HFEA

Data an exact number, classical input/output efficiency → classical DEA (CCR, BCC)

A full distinction is needed among options on the efficient frontier → HFPE (cross-efficiency)

A priority order among criteria is also to be specified → HFPEA

HFEA's greatest strength is that it evaluates each option on the criterion where it is strong, without imposing an external weight on any of them; this makes the comparison unbiased. It does not collapse the hesitation in expert opinion (the several possible values) into a single number at an early stage and lose it, but accounts for it as deviation. For options found inefficient, it can show not just a score but also how much improvement is needed, and on which criterion, to reach the efficient frontier; this makes the result actionable.

HFEA's fundamental limitation is that it inherits a problem from classical data envelopment analysis: if the number of options is low relative to the number of criteria (roughly, if there are not about three options for every criterion), nearly all options come out efficient and the analysis loses its meaning (Banker, Charnes and Cooper, 1984). A second limitation is that more than one option can appear on the efficient frontier at the same time, and the method cannot distinguish between them; this requires a cross-evaluation step such as HFPE. Third, the score-and-deviation calculation collapses all the information in a hesitant set into two numbers; two sets of different structure can give the same score and deviation, in which case the method cannot tell them apart (Xia and Xu, 2011). Fourth, since all criteria must be treated as benefit-oriented, failing to take the correct complement of a cost-oriented criterion leads to a serious error.

The most common mistake is treating a cost-oriented criterion directly as benefit-oriented without taking its complement; this can reverse the efficiency ranking. A second mistake is interpreting the fact that nearly everyone comes out efficient, in a small data set where the number of options is close to the number of criteria, as "everyone is genuinely excellent"; this is generally a sign of a small sample. A third mistake is treating more than one option scoring 1 on efficiency as equal in quality; if a distinction is needed among these options, a cross-efficiency step such as HFPE should be applied. The governing principle is this:

An HFEA score is the highest efficiency an option can attain even under the best weights that could favour it; every score below 1 shows that no weighting could rescue that option.