HFLPR-PRIORITY (Hesitant Fuzzy Linguistic Preference Relation Priority Programming)

HFLPR-PRIORITY is a method for when you have pairwise comparisons between criteria, and between options, expressed in words ("this criterion is somewhat more important than that one," "this station's vibration problem is markedly worse than that one's," and so on). It derives both the criterion weights and the option priorities under each criterion at the same time. What sets it apart from classical pairwise-comparison methods such as AHP is that, instead of first checking whether the comparisons are consistent and asking the decision-maker to redo them if not, it uses a direct optimisation to find the priority vector that best matches the comparisons as given. Ren, Zhu and Xu proposed it in 2018, together with an application assessing the environmental impact of hydropower stations.

The idea behind HFLPR-PRIORITY is to accept that human judgement will always be somewhat inconsistent. Classical AHP computes a consistency ratio; if this ratio exceeds a threshold, it asks the decision-maker to revise the comparisons. In practice this is difficult: an environmental board or a management committee cannot easily "redo" dozens of pairwise judgements it has given. HFLPR-PRIORITY skips this step entirely. Instead, it finds a single priority vector that minimises the worst disagreement across all the comparisons given (every pairwise judgement, including every reasonable term under hesitation).

This idea carries a philosophical consequence. The method does not treat the decision-maker's judgement as "an error to be corrected"; it accepts it as given and produces the single interpretation that best matches that judgement. It also allows judgements to be given as a set of terms (not just a single word, but something like "between somewhat good and very good"), letting the decision-maker carry their own uncertainty forward without collapsing it into a number too early.

The method proceeds through seven steps.

First, gathering the pairwise comparisons. Two tables are requested from the decision-maker: a pairwise comparison table among the criteria, and a separate pairwise comparison table among the options for each criterion. Each cell may contain one or more reasonable terms from a shared linguistic term set (for example, a nine-point scale running from "very poor" to "very good").

Second, structural checking. Each table's diagonal must represent indifference, and reciprocal cells (criterion A's comparison against B and B's comparison against A) must complement one another. Tables failing this condition are flagged.

Third, expressing disagreement. For a given table and a candidate priority vector, how much each pairwise comparison agrees or disagrees with that priority vector is computed. If the priority vector matches the table exactly, this disagreement is zero.

Fourth, minimising the worst disagreement. A linear programming model finds the priority vector that minimises the largest disagreement across all the pairwise comparisons (including every reasonable term under hesitation). The constraint is that priority values must sum to one and be non-negative.

Fifth, criterion weights. The fourth step is applied to the pairwise comparison table among the criteria, yielding the criterion weights. These weights are not assigned by the board; they emerge directly from the pairwise judgements between criteria.

Sixth, option priorities under each criterion. The fourth step is applied separately to the pairwise comparison table among the options for each criterion, yielding the option priorities under that criterion.

Seventh, overall priority and ranking. The criterion weights and the per-criterion option priorities are combined by weighting; a single overall priority value is obtained for every option, and the options are ranked from highest to lowest.

The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The overall priority value shows how far ahead an option stands relative to the others given the pairwise comparisons supplied; it is not a percentage or a probability. Criterion weights are read the same way: if a criterion's weight is high, this criterion has been consistently found more important in the pairwise comparisons between criteria; it was not assigned in advance by the board.

Thus instead of writing:

"HFLPR-PRIORITY found the best option"

the report should read:

"With the pairwise comparisons given, the option with the highest overall priority is this one; the criterion weights, too, are derived from these comparisons and were not assigned in advance"

HFLPR-PRIORITY works with hesitant fuzzy linguistic data: pairwise comparison cells may contain more than one reasonable term from a linguistic term set. DecisionMind holds no separate data-type extension of this method; it stands alone in its base form.

You need the following: a pairwise comparison table among the criteria; a separate pairwise comparison table among the options for each criterion; and a shared linguistic term set on which all the tables are based (for example, a nine-point scale). The method assumes a single decision-maker; for a group decision, individual tables must first be combined into a single group table. HFLPR-PRIORITY does not require a decision matrix, only pairwise comparison tables; weights are not taken from outside, they are produced directly.

HFLPR-PRIORITY is a suitable choice if your data is collected through pairwise comparisons, those comparisons are expressed in words, and it is not practical to ask the decision-maker to redo their judgements because of inconsistency.

The situations in which it should not be used are as follows. If you already have a direct decision matrix (an options-by-criteria table), there is no need to collect pairwise comparisons. If you have more than one decision-maker and want to treat them separately, a group-aggregation step is needed first. If the pairwise comparison tables fail the diagonal and reciprocity condition, the data-collection process should be reviewed first.

Pairwise comparison, expressed in words, inconsistency acceptable → HFLPR-PRIORITY

Pairwise comparison, consistency ratio checked and judgements re-requested if needed → AHP

A direct decision matrix exists → TOPSIS, SAW and similar ranking methods

More than one decision-maker, group aggregation needed → a consensus or aggregation method first

HFLPR-PRIORITY's greatest strength is that it produces a priority vector directly, without putting the decision-maker through a consistency check that forces them to redo their judgements; in practice this reduces meeting time and decision-maker fatigue. Producing criterion weights and option priorities within the same framework and the same pairwise-comparison logic removes the need for a separate weighting method. Working with hesitant linguistic data lets the decision-maker remain undecided between more than one reasonable word.

Its limitations arise from its data structure and its reliance on a single decision-maker. First, the method assumes a single decision-maker; group decisions require individual tables to be combined first, which is a separate source of uncertainty. Second, criterion weights being derived directly from pairwise comparisons between criteria means a criterion's weight can be overly sensitive to a handful of judgements given during comparison. Third, a table that fails the diagonal and reciprocity condition undermines the method's foundation; this condition must be reverified on every use. Fourth, the approach of minimising the worst disagreement targets only the worst case, not the average disagreement; this can let a single extreme judgement affect the entire priority vector.

The most common mistake is feeding a pairwise comparison table straight into the method without verifying the diagonal and reciprocity condition; if this condition is not met, the result is unreliable.

A second mistake is running the tables of more than one decision-maker separately, as if there were a single decision-maker, without combining them, and then comparing the results. A third mistake is forgetting that criterion weights come directly from pairwise comparisons and presenting them as though they were "predetermined" by the board. A fourth mistake is failing to look back at which pairwise judgements a criterion's disproportionately high weight stems from.

The governing principle is this:

An HFLPR-PRIORITY result is the best-matching interpretation of the pairwise comparisons given; if one comparison changes, both the criterion weights and the option priorities can change.