Methods · Outranking
HF-QUALIFLEX (Hesitant Fuzzy QUALIFLEX)
When experts cannot agree on a single value for a criterion and report several plausible values instead, HF-QUALIFLEX tries out every possible ranking of the alternatives one by one and picks the one most consistent with the criteria.
Base method's data type: Fuzzy
What Is the Method?
HF-QUALIFLEX is a ranking method designed for cases where the cells of your decision table hold not a single number but several plausible values (a hesitant fuzzy set). This is not measurement uncertainty; it represents cases where several experts propose different scores for the same cell and none of these differing opinions is to be discarded. Its output is a complete ranking of the alternatives. It is the hesitant fuzzy adaptation of the classical QUALIFLEX method (Paelinck, 1978), proposed by Zhang and Xu in 2015.
The Philosophy Behind It
The underlying idea of the QUALIFLEX family is, rather than scoring and ranking alternatives one by one, to try out every possible ranking one by one and measure how consistent each ranking is with the criteria. The most consistent ranking is chosen as the final result. This asks, before "which alternative is best," the question "which ordering conflicts least with the criteria."
HF-QUALIFLEX carries this idea into hesitant fuzzy data. Every alternative's performance on every criterion is measured by a signed distance from that criterion's best-possible or worst-possible end, and the hesitation (several plausible values) is not reduced to a single number until this distance is computed. This sets the method apart from fully compensatory additive methods: HF-QUALIFLEX does not search for a "best alternative," it searches for the ordering that conflicts least with the criteria.
How It Works
The method proceeds through four steps.
First, equalising length. Experts may report different numbers of plausible values for a cell (one proposing two values, another three). For comparison to be possible, every cell must hold the same number of values. HF-QUALIFLEX extends the shorter set, generally by repeating its most pessimistic (lowest) value; this is a modelling choice, and an optimistic completion rule (repeating the highest value) can equally be defined. Values are sorted from the smallest to the largest.
Second, computing the signed distance. Every alternative's hesitant value on every criterion is measured against a reference point set by that criterion's direction. For a benefit criterion, the reference is "the best possible" (a set whose every element is 1); for a cost criterion, it is "the worst possible" (a set whose every element is 0). This distance is signed: the closer the alternative sits to the reference, the more positive the value it takes.
Third, computing consistency for every possible ranking. For every possible ordering (permutation) of the alternatives, how far ahead or behind each alternative sits relative to the one below it is multiplied by the criterion weights and summed. This gives a single number showing how consistent that ordering is with the criteria.
Fourth, choosing the most consistent ordering. Among all possible orderings, the one with the highest consistency number is chosen as the final ranking.
The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
The chosen ranking is the ordering that, given these weights, these reference points and this length-equalisation rule, conflicts least with the criteria. The consistency number itself is not a "success percentage"; it is only a search measure used to compare orderings against one another, and it cannot be compared with the consistency number of another analysis.
If the gap between the most consistent ordering and the second most consistent one is small, the ranking is sensitive: it can change if the reference point or the length-equalisation rule changes. The hesitation information (several plausible values) has been carried all the way to the result here; the report must show where this hesitation ends up, that is, which alternatives' performance is most contested.
Thus instead of writing:
"HF-QUALIFLEX found the best ordering"
the report should read:
"With these weights, these reference points and this length-equalisation rule, the ordering most consistent with the criteria is this one; if the gap to the second most consistent ordering is small, the ranking is sensitive"
Data Type and Inputs
HF-QUALIFLEX works with hesitant fuzzy data: every cell is a set of values proposed by several experts, none of them discarded. This differs from a single "approximate" number (a fuzzy number); a hesitant set holds several distinct possibilities together on which no agreement has been reached. DecisionMind holds only the base HF-QUALIFLEX; it has no extension.
You need: one or more membership degrees (between 0 and 1) in every cell; a direction for every criterion; and weights that sum to 1. HF-QUALIFLEX does not produce weights, it asks for them. The number of alternatives should be kept to six at most; at seven alternatives the number of possible orderings reaches five thousand, and at eight it reaches forty thousand, so the computational burden grows quickly. There is no strict upper bound on the number of criteria.
When to Use It, When Not To
HF-QUALIFLEX is a sound choice if your experts cannot agree on a single value for a criterion, several plausible values carry genuine information (real disagreement, not random error), and your number of alternatives does not exceed six. Its typical territory is small-scale decisions, based on an expert panel, where hesitation must not be discarded.
It should not be used where the number of alternatives exceeds six, or where the data is already crisp. Where the number of alternatives is large, the computational burden explodes and the method stops being practical; where the data is crisp, there is no hesitation-bearing information to carry, and a crisp-data method such as classical QUALIFLEX or TOPSIS is sufficient.
Experts cannot agree on a single value, the number of alternatives is small → HF-QUALIFLEX
Same hesitation situation, but the number of alternatives exceeds six → narrow the alternatives first, or consider an additive hesitant method
Data is crisp, no hesitation → TOPSIS and similar crisp-data methods
Data is an "approximate" fuzzy number, not a hesitant set → the relevant fuzzy extension
Strengths
HF-QUALIFLEX's greatest strength is that it does not reduce disagreement among experts to a single number at an early stage and lose it; the hesitation is preserved until the signed distance is computed. Because it tries out every possible ordering one by one, it does not get stuck at a local optimum; it searches the whole of the search space. Because the reference point and the length-equalisation rule are stated explicitly, which assumptions the result rests on can be traced.
Weaknesses
Its limitations stem largely from the cost of exhaustive search and the nature of the input. As the number of alternatives grows, the number of possible orderings grows very fast; this confines the method to small-scale decisions only. The rule used to equalise the length of the hesitant sets (a pessimistic or an optimistic completion) can change the result; the general finding that the choice of operation affects the result in hesitant fuzzy sets is well known in the literature (Xia and Xu, 2011). Fixing the reference point as "best possible / worst possible" is a stricter assumption than an ideal point derived from the actual dataset; this choice, too, affects the result.
Common Mistakes
The most common mistake is presenting the result as the single correct ranking without stating the length-equalisation rule (pessimistic or optimistic) used in the report. A second mistake is forcing the method on a decision with more than six alternatives and ignoring that the computation time grows exponentially. A third is averaging the values in a hesitant set down to a single number before running the method; this destroys the hesitation information, HF-QUALIFLEX's central contribution, before the data is even entered. A fourth is reporting the ranking as a definitive result when the gap between the most consistent ordering and the second most consistent one is small.
The governing principle is this:
The ranking HF-QUALIFLEX gives is a consequence of the reference point and the length-equalisation rule you set; if these choices change, the ranking drawn from the hesitation information can change too.
Cases
Each case opens with a decision table, describes in words what the method does to it, and shows how to read the result. The first case is drawn from the method's founding source; its figures are the article's own. The remaining cases are illustrative constructions.
1. Fuzzy Decision Analysis: A three-alternative, three-criterion hesitant example (Zhang and Xu, 2015)
Zhang and Xu's founding article uses three alternatives (A1, A2, A3) and three criteria (C1, C2 benefit; C3 cost) to show HF-QUALIFLEX's mechanics. Every cell consists of two plausible values on which the experts could not agree. The weights are 0.40 for C1, 0.35 for C2 and 0.25 for C3.
| Alternative | C1 (benefit) | C2 (benefit) | C3 (cost) |
|---|---|---|---|
| A1 | {0.4, 0.6} | {0.5, 0.7} | {0.3, 0.4} |
| A2 | {0.3, 0.5} | {0.6, 0.8} | {0.2, 0.3} |
| A3 | {0.5, 0.7} | {0.3, 0.5} | {0.4, 0.6} |
| Weight | 0.40 | 0.35 | 0.25 |
The method first computes every cell's signed distance: for the benefit criteria, "best possible" (every element 1) is taken as the reference; for the cost criterion, "worst possible" (every element 0). It then computes the consistency number for each of the six possible orderings of the alternatives.
| Ordering | Consistency Number |
|---|---|
| A2 ≻ A1 ≻ A3 | 0.175 |
| A1 ≻ A2 ≻ A3 | 0.135 |
| A2 ≻ A3 ≻ A1 | 0.040 |
| A1 ≻ A3 ≻ A2 | −0.040 |
| A3 ≻ A2 ≻ A1 | −0.135 |
| A3 ≻ A1 ≻ A2 | −0.175 |
The ordering with the highest consistency number is A2 ≻ A1 ≻ A3; this is chosen as the final ranking.
The result reads as follows. Although A2 holds the lowest values on C1, it carries the highest values on C2 (the second-heaviest criterion) and the lowest, that is best, values on C3 (cost). These three together put A2 ahead. A3, despite holding the best values on C1, sits weakest on C2 and most expensive on C3, and so finishes last.
The analyst hesitates here: the consistency gap between A2 ≻ A1 ≻ A3 and the second most consistent ordering, A1 ≻ A2 ≻ A3, is only 0.04 (0.175 against 0.135), whereas A3's third-place position does not change across any ordering. This shows that the relative order of A2 and A1 is less certain than A3's third place. If the weight on C2 is lowered slightly, A1 and A2 could swap places.
In the report: "With these weights and reference points, the most consistent ordering is A2 ≻ A1 ≻ A3; the relative position of A2 and A1 is sensitive to the weight on C2, while A3's third place does not change across any ordering."
Source: Zhang and Xu (2015), §4, Example. The distance and consistency figures are the article's own values; this example serves as the validation case for DecisionMind's HF-QUALIFLEX engine, and the engine reproduces the same result.
2. Examination Centre: Choosing a new exam-security protocol at an assessment body
An assessment and evaluation body must choose one of three protocols to strengthen exam security: tightened identity verification, in-room electronic-device scanning, and question-booklet rotation. Members of the security committee could not agree on the deterrence and practicality scores for each protocol; some members gave high scores, others gave middling ones, and this disagreement was recorded as a hesitant set rather than reduced to a single number. A similar disagreement exists for the third criterion, implementation cost (lower is better).
The method converts the three protocols' hesitant sets on the three criteria into signed distances and computes the consistency number for each of the six possible orderings. Suppose in-room electronic-device scanning takes the most consistent first place, thanks to the high upper end of its hesitant deterrence set; tightened identity verification, despite its advantage on cost, finishes second because it trails on deterrence.
The committee hesitates here: if the consistency gap between the two protocols is small, this gap may depend on how the committee members' disagreement was recorded, whether it was completed pessimistically or optimistically. The committee should change the completion rule and check whether the result changes.
In the report: "In-room electronic-device scanning ranks most consistently first; this ranking is sensitive to the completion rule applied to the disagreement among committee members."
3. Care Home: Choosing a new care-home care model in a municipality
A municipality must choose one of three care models to apply in its care homes: an intensive medically supported model, a socially active model, and a mixed model. The care-quality assessment board could not reach agreement among its members on the quality-of-life perception and family-satisfaction criteria, each member proposing a different score range; these views were recorded as a hesitant set. Figures for the third criterion, cost per person (lower is better), are clearer but still carry a small range.
The method converts the three models' hesitant sets into signed distances and searches for the most consistent ordering. Suppose the mixed model, despite holding a wide hesitant range on quality-of-life perception, takes the most consistent first place because the upper end of this range is high and its cost stays at a middling level.
The board hesitates here: if the mixed model's lead rests on the optimistic end of the hesitant range rather than the pessimistic one, this may sideline the more cautious view held by some board members. The board should recheck the result with the pessimistic completion rule.
In the report: "The mixed model ranks most consistently first; this result is sensitive to whether the disagreement on quality-of-life perception is completed optimistically or pessimistically, and should be separately verified with the pessimistic rule."
4. What Not to Do
In the Zhang and Xu example, had the hesitant sets, such as A1's {0.4, 0.6} on C1, been averaged down to a single number (0.5), the information about disagreement among experts would have been lost before the calculation even began, and HF-QUALIFLEX's central contribution would have vanished. A second error is ignoring the 0.04 consistency gap between A2 ≻ A1 ≻ A3 and A1 ≻ A2 ≻ A3 and presenting the result as a definitive ordering. A third error is forcing this method on seven or eight alternatives without noticing that the computation time grows exponentially; at this scale, the number of alternatives should be narrowed first.
Sources
For the formulas behind each step, the intermediate tables and citation formats (BibTeX, RIS, APA), see the DecisionMind method page: decisionmind.app/library/hf-qualiflex
Zhang, Z. X., & Xu, Z. S. (2015). Hesitant fuzzy QUALIFLEX approach with a signed distance-based comparison method for multiple criteria decision analysis. Expert Systems with Applications, 42(3), 873–884. DOI: 10.1016/j.eswa.2014.08.056
Paelinck, J. H. P. (1978). Qualiflex: A flexible multiple-criteria method. Economics Letters, 1(3), 193–197. DOI: 10.1016/0165-1765(78)90023-x
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395–407. DOI: 10.1016/j.ijar.2010.09.002