HFPEA (Hesitant Fuzzy Preference-Constrained Envelopment Analysis)

HFPEA is an extension of HFEA (Hesitant Fuzzy Envelopment Analysis). HFEA allows every option to choose entirely free weights in its own favour; this can sometimes lead to unrealistic results, because an option can appear artificially efficient simply by loading all its weight onto a criterion the decision-maker considers unimportant. HFPEA adds, as a constraint on the weights, a priority order the decision-maker has stated in the form "this criterion is more important than that one"; the efficiency calculation thereby both preserves the idea of self-favouring evaluation and stays tied to the decision-maker's actual priorities. Zhou, Chen, Xu and Meng proposed it in 2018 as HFEA strengthened with preference information.

HFEA's principle of "evaluate everyone in their own favour" is sometimes too generous: if an option is strong on a criterion the decision-maker regards as unimportant, it can load all its weight onto that criterion and still reach the efficient frontier. HFPEA bounds this latitude. A ranking is obtained from the decision-maker among the criteria (for instance, "growth prospects matter more than market share; market share matters more than team quality"), and this ranking is turned into constraints fixing the relative order of magnitude among the weights. An option can no longer assign more weight to an unimportant criterion than to the others.

This idea carries a philosophical consequence: it carries fair evaluation and the decision-maker's value judgement together. HFEA looks only at the data; HFPEA also factors in the answer to "which criterion is more important." This means an option that looks efficient under HFEA can turn out inefficient once the priority order is added, because the criterion on which that option is strong may rank low in the decision-maker's priorities.

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

First, gathering the hesitant assessments. As in HFEA, every option's value on each criterion is combined into a single hesitant set.

Second, computing score and deviation. As in HFEA, the mean (score) and inconsistency (deviation) of the set on each criterion are computed for every option.

Third, converting the priority order into a constraint. The decision-maker's stated ranking, of the form "this criterion is more important than that one," is turned into inequalities fixing the relative order of magnitude among the weights of successive criterion pairs.

Fourth, forming the constrained efficiency ratio. The same efficiency ratio as in HFEA (the ratio of the weighted sum of scores to the weighted sum of deviations) is formed, but this time the priority-order constraints are added as well.

Fifth, solving and classifying. For every option, the weights that maximise the efficiency ratio under the priority constraints are found; options scoring 1 are considered efficient under this priority order, those below 1 inefficient, and all are ranked.

Sixth (optional), a priority-sensitive improvement target. For an option found inefficient, the change in score and deviation needed to reach the efficient frontier while preserving the priority order 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 efficiency score lies between 0 and 1, as in HFEA, but now shows the highest efficiency attainable "under weights most favourable to the option itself, while also respecting the decision-maker's priority order." An option that appeared on the efficient frontier under HFEA can come out inefficient once priority constraints are added; this is not an error, it shows that the criterion on which the option was strong ranks low in the decision-maker's priorities. The score is valid only for this priority order and this option set; if the priority order changes, the set of efficient options changes too.

Thus instead of writing:

"HFPEA found the most efficient option objectively"

the report should read:

"Under this priority order, the following options are efficient; if the priority order changes, the set of efficient options changes too"

HFPEA uses the same hesitant data structure as HFEA: every option's value on each criterion is a set of several possible values. DecisionMind holds no extension of this method; the base method works on its own. The method does not ask for numerical weights from outside, but it does require an ordinal priority among the criteria (which is more important than which); it works with information no more detailed than "this matters more than that," without producing full numerical weights of the kind AHP or SWARA would. Requiring all criteria to be included in the priority order can render the problem infeasible in small data sets; in that case, only part of the ordering (a partial ranking) can be used.

HFPEA is appropriate if, having applied HFEA, you feel the result does not reflect the decision-maker's real priority order among the criteria, or if you already have such a priority order from the outset. If you have no priority judgement among the criteria, or wish to treat them all as equally important, HFEA is sufficient; moving to HFPEA adds an unnecessary constraint. If your priority order is already clear in the form of numerical weights (for instance, derived from AHP), it may be more direct to move to classical weighted methods (such as TOPSIS) that use those weights directly.

A priority order exists among the criteria, and efficiency should reflect it → HFPEA

No priority among the criteria, all can be treated equally → HFEA

Priority already known as full numerical weights → weighted ranking methods (TOPSIS, VIKOR)

Distinction among options on the efficient frontier needed through peer evaluation → HFPE

HFPEA's greatest strength is that it preserves HFEA's objective flexibility while also taking the decision-maker's value judgement into account; this offers a middle path between a purely data-driven result and a purely subjective weighting. It prevents an option from appearing artificially efficient by over-weighting an unimportant criterion. By showing which options' efficiency position changes once the priority order is added, it makes clear, in which direction, the decision-maker's priorities affect the result.

HFPEA's fundamental limitation is that it requires a full ranking over all the criteria; in small data sets this can render the problem infeasible, in which case a partial ranking must be used instead. A second limitation is the assumption that the priority order is stable; if the ranking changes, so does the set of efficient options, so the result is valid only for that particular ranking. Third, HFPEA's improvement targets differ from HFEA's, because the feasible region under priority constraints is smaller; conflating the two produces the wrong improvement advice. Fourth, the priority order itself usually rests on the view of a limited number of decision-makers; a different decision-maker might propose a different order, and this is a general limitation of preference-constrained methods (Zhu, 1996).

The most common mistake is interpreting an option that was efficient under HFEA but comes out inefficient under HFPEA as "the method made an error"; this is a correct result showing that the priority constraints have pushed back the criterion on which the option was strong. A second mistake is assuming that changing the priority order will not change the result; the ordering is an input to the analysis, and the result changes when it does. A third mistake is assuming HFPEA's improvement targets are the same as HFEA's and copying them directly; the targets under priority constraints are calculated differently. The governing principle is this:

An HFPEA result reflects the decision-maker's priority order in addition to the data; if the priority order is contested, the efficiency result is contested too.