HFPE (Hesitant Fuzzy Cross-Efficiency Evaluation)

HFPE picks up where HFEA (Hesitant Fuzzy Envelopment Analysis) leaves off. Because HFEA evaluates each option under the weights most favourable to itself, more than one option can come out "on the efficient frontier" (a score of 1) at the same time, leaving unanswered the question of which is better than the other. HFPE evaluates each option not only under its own weights but also under the best weights every other option has found for itself, and then averages these cross-scores. The result is a full ranking that distinguishes even the options on the efficient frontier from one another. Zhou, Chen, Xu and Meng proposed it in 2018 as the cross-efficiency idea from classical data envelopment analysis (Sexton, Silkman and Hogan, 1986) carried into a hesitant fuzzy setting.

HFEA's principle of "evaluate everyone in their own favour" is fair but not discriminating; two options can each look excellent through their own eyes. HFPE introduces a peer-evaluation idea: to know whether an option is genuinely good, you must look not only at its own view but also at how its peers see it. Each option first finds the weights most favourable to itself; these weights are then applied to every other option as well, giving each option a score "through the others' eyes." An option's final score is the average of all these cross-views.

This idea carries a philosophical consequence: self-praise alone is not sufficient. HFPE offers two extreme strategies: the benevolent strategy searches for weights that preserve an option's own efficiency while also scoring the others as highly as possible; the aggressive strategy does the opposite, searching for weights that score the others as low as possible. Both answer a different question of "how do you look through others' eyes" from the same data.

The method proceeds through five steps.

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

Second, score, deviation and the self-evaluation ratio. For every option, the score (mean value) and deviation (inconsistency) are computed; their weighted ratio forms that option's initial self-assessment.

Third, finding the weights most favourable to oneself. For every option, as in HFEA, the weights that maximise its own efficiency ratio are found by a linear programming problem.

Fourth, cross-evaluation. The weights each option has found are also applied to every other option, forming a cross-score table of the kind "how efficient does option B look under option A's weights." This step can be set up in two different ways, according to the benevolent or aggressive strategy.

Fifth, averaging and ranking. For every option, the average of the scores it received under all the other options' weights is taken; this average is the final cross-efficiency score, and options are ranked by it.

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 cross-efficiency score shows how good an option looks, on average, not only through its own eyes but through every other option's eyes as well. Options that scored 1.0 and could not be distinguished under HFEA now receive different scores under HFPE, and a genuine ranking emerges. A high score means not only "good in its own favour" but "good by others' standards too"; this is a stronger claim than HFEA's. The benevolent and aggressive strategies can produce different rankings; which strategy was used must always be stated in the report, because the two answer different questions from the same data.

Thus instead of writing:

"HFPE measured true efficiency"

the report should read:

"Under this strategy (benevolent/aggressive), viewed through the other options' eyes, the ranking is as follows; the ranking can change if the strategy changes"

HFPE 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 request weights from outside; each option's weights are found through its own optimisation and then applied crosswise. Which strategy (benevolent or aggressive) to use must be chosen at the start of the analysis; this is an analytical decision, not a data input. The computational load grows with the square of the number of options, because every option requires a separate linear programming solution for every other option; this cost should be kept in mind when working with more than fifty options.

HFPE is appropriate once HFEA has been applied and more than one option comes out on the efficient frontier at the same time, requiring a choice among them. It is more economical to run HFEA first and check whether a genuine need for discrimination exists, rather than starting directly with HFPE; if HFEA already gives a single clear efficient option, HFPE is not needed. If the number of options is very large (more than fifty), the sample should first be reduced, or approximate methods considered, because of the computational cost.

More than one option on the efficient frontier under HFEA, discrimination needed → HFPE

HFEA already gives a single, clear, efficient option → HFEA is sufficient, HFPE unnecessary

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

Data an exact number, classical cross-efficiency → classical DEA cross-efficiency (Sexton et al., 1986)

HFPE's greatest strength is turning the "everyone is first" ambiguity that HFEA leaves behind into a genuine ranking. Because it offers an evaluation through peers' eyes and not only through an option's own, it balances out results that one-sided self-assessment might exaggerate. By offering both a benevolent and an aggressive strategy, it presents the decision-maker with the "most optimistic" and "most pessimistic" extreme scenarios together, giving a sensitivity range rather than relying on a single number.

HFPE's fundamental limitation is that the linear programming's optimal weights may not be unique; different solvers or different tolerance settings can give different cross-efficiency matrices for the same data (Doyle and Green, 1994). A second limitation is that the benevolent and aggressive strategies can produce very different rankings; if which strategy was used is not clearly stated, the result becomes misleading. Third, the computational load grows with the square of the number of options; this may be impractical for large data sets. Fourth, HFEA's inherited limitations (loss of discrimination in small samples, information loss from collapsing to score and deviation) remain valid in HFPE too.

The most common mistake is presenting the result simply as "a cross-efficiency score" without stating which strategy (benevolent or aggressive) was used; the two answer different questions and must not be confused. A second mistake is directly comparing results from different linear programming solvers; since optimal weights are not unique, small differences should be expected. A third mistake is starting directly with HFPE without ever running HFEA; whether HFEA on its own already gives a clear result should be checked first. The governing principle is this:

An HFPE score is a composite of the chosen strategy (benevolent/aggressive) and the peer options; if the strategy or the option set changes, the ranking can change too.