Extension card · q-Rung Orthopair
q-Rung Orthopair GRA
This is the form of GRA that works for situations where criterion scores are given as a judgement's degree of support and rejection, and how large these two degrees may be together is bounded by an exponent (q) chosen to fit the data. It still ranks the result with a grey relational grade.
Base method
GRA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
q-Rung Orthopair →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Five things change; the grey relational coefficient formula does not.
Cells. In crisp GRA every cell is a single number. Here every cell is a support (μ) and a rejection (ν) degree; the constraint is that μ to the power q plus ν to the power q is at most 1. The intuitionistic (q=1), Pythagorean (q=2) and Fermatean (q=3) fuzzy structures are special cases of this family. The user does not choose q. As explained on the data-type card, q comes bundled with the method as the smallest value covering the whole of the expert pairs; DecisionMind's current engine fixes q at 3 for this extension.
Complementing the cost criterion. In every cell of a "lower is better" criterion, the support and rejection degrees swap places.
The reference is chosen from a real alternative. This is an important difference from the other family members (Plithogenic, Pythagorean, Spherical fuzzy GRA). In those three extensions the reference is built component by component, from each component's (support, rejection) best value within the column separately, and need not belong to any single real alternative. In QR-GRA, a score is first computed for every column: support to the power q minus rejection to the power q. The best row in the column is determined by this score; for a benefit criterion the highest-scoring row is chosen, for a cost criterion the lowest-scoring row. That row's support-rejection pair is written into the reference exactly as it stands, both degrees together. The reference is, as in SF-TOPSIS, "the value of a real alternative"; this point, which sets it apart from crisp GRA's hypothetical reference, must be stated in the report.
Distance has three components. DecisionMind's engine uses a three-component Euclidean distance in place of the manifest's simpler two-component draft; this choice was made to stay consistent with the literature. The distance is computed over the q-th powers of support, rejection and hesitancy (1 minus support to the power q minus rejection to the power q). This is a definition that also accounts for hesitancy and stays closer to the literature.
Grey relational coefficient and grade. The grey relational grade is built with exactly the same formula as crisp GRA (distinguishing coefficient ρ = 0.5, fixed), through a weighted sum.
DecisionMind holds q (3), the score-based selection of the reference row, the three-component distance and the distinguishing coefficient (ρ = 0.5) fixed for this extension; weights are taken from outside as crisp numbers.
How to Read the Output
The grey relational grade shows, as in crisp GRA, an alternative's relative closeness to the reference within this analysis; it cannot be compared with a different analysis.
The difference is here. The reference here is a real alternative's own pair; it is not a hypothetical point built component by component, as in Plithogenic, Pythagorean and Spherical fuzzy GRA. Also, q is fixed in this method and cannot be changed by the user; which q was used must be stated in the report, because comparing against a result computed with a different q amounts to building a different method, not adjusting the same one.
Thus instead of writing:
"In q-Rung orthopair GRA, q can be adjusted by the user according to whether the data is wide or narrow"
the report should read:
"In this method q is fixed (q = 3); comparing against a result computed with a different q is not a different setting of the same method, but a separate calculation"
When to Prefer This over the Base Method
Use this extension when the sum of the expert pairs' squares, or even their cubes, also exceeds 1, that is, when even the Pythagorean and Fermatean boundaries prove insufficient. The detail is on the q-Rung orthopair data-type card; the boundary that must not be confused with the neighbouring types (Pythagorean, Fermatean) is explained there.
If the expert pairs already fit a narrower boundary, moving to the q-Rung structure adds nothing, and only reduces discrimination. If the criteria are measured, the base GRA should be kept. The crisp method's exit condition applies here too: the matrix must be of a single type, and if no compromise is accepted on one criterion, this extension is compensatory as well.
Mistakes Specific to This Extension
Assuming q is a user input. q comes bundled fixed with the engine in this method; it must not be presented as an editable field in the user interface.
Building the reference component by component, in crisp GRA's hypothetical form. In QR-GRA the reference is a real row's pair, chosen by score; building it component by component, as in Plithogenic or Pythagorean GRA, produces a different, incorrect result.
Violating the value domain. In every cell, μ to the power q plus ν to the power q must be at most 1; the same q must be used throughout the matrix.
Forgetting to complement a cost criterion. In an independent Python test, when a single criterion's direction was changed from "higher is better" to "lower is better," the second- and third-ranked alternatives swapped places; a direction-label error is reflected directly in the ranking.
Never questioning the distinguishing coefficient. ρ = 0.5 is the value DecisionMind holds fixed.
The governing principle is this:
In q-Rung orthopair GRA, q is fixed and is not chosen by the user; unlike the other family members, the reference is a real alternative's score-selected value. Confusing these two points produces either an incorrect calculation or an incorrect comparison.
Cases
The first case is DecisionMind's validation example. The recorded example is, in the manifest's own words, a "minimal fixed-rank fixture that reproduces in closed form": every alternative's pair is the same across the three criteria, so the ranking reduces to a single, strict dominance relationship. This is a deliberate choice to avoid fabricating figures specific to a paper. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): q-Rung orthopair assessment of three alternatives on three criteria
Three alternatives are assessed on three criteria; all three are in the benefit direction and the weights are equal (a third each). q = 3. Every alternative's pair is the same across the three criteria.
| Alternative | C1 | C2 | C3 |
|---|---|---|---|
| A1 | (0.90; 0.10) | (0.90; 0.10) | (0.90; 0.10) |
| A2 | (0.70; 0.30) | (0.70; 0.30) | (0.70; 0.30) |
| A3 | (0.50; 0.50) | (0.50; 0.50) | (0.50; 0.50) |
The method chooses the best row (A1) as the reference for every column, by the score (μ to the power 3 minus ν to the power 3), computes the three-component distance, and converts it into a grey relational grade with ρ = 0.5.
| Alternative | Grey relational grade | Rank |
|---|---|---|
| A1 | 1.0000 | 1 |
| A2 | 0.4251 | 2 |
| A3 | 0.3333 | 3 |
The result reads as follows. A1 is the reference itself on every criterion, so its grey relational grade is exactly 1 (1.0000). A3, having the lowest support and highest rejection pair on every criterion, comes last.
Because the criteria are identical to one another in this table, a change in weights is not expected to affect the ranking; an independent Python test confirmed this, and even when the weights were pulled towards a skewed distribution such as 0.70/0.20/0.10, the ranking and the scores stayed nearly the same. The real sensitivity lies in the direction label. If C3 is marked "lower is better," as verified by an independent Python calculation, A3's grey relational grade (0.5556) overtakes A2's (0.4813), and the ranking becomes A1, A3, A2.
In the report: "With the directions given (all three higher is better), A1 is first because it is the reference itself (1.0000). If C3's direction is changed to lower is better, A3 overtakes A2; in this example the ranking is more sensitive to the criterion directions than to the weights."
Source: DecisionMind's QR-GRA validation example; by the manifest's own note, it was deliberately built, in closed form, as a fixed-rank fixture "to avoid fabricating figures specific to a paper." The grey relational grades and the weight and direction sensitivity were computed independently by this card's author, running the kernel directly, and matched exactly the manifest's recorded result (A1 > A2 > A3, the same decimal values).
2. Construction: Choosing a contractor for a municipality's infrastructure tender
A municipality will choose among three contractors for a road-renewal tender. Three criteria are used: technical competence and compliance with the safety record (higher is better), and bid price (lower is better). The support and rejection degrees given to these judgements for the contractors include pairs extreme enough to exceed even the Fermatean boundary, so the q-Rung orthopair structure (q = 3) is used.
The method compares the three contractors: it swaps the support and rejection degrees on the bid-price criterion, chooses the reference row by score, measures the distance, and computes the grey relational grades. Suppose the contractor with the strongest support-rejection pair on technical competence comes out first despite its relatively high bid, because the weight on technical competence exceeded that on bid price.
The municipality's hesitation is here. If the weight on the safety-record criterion is raised, a contractor with a lower bid but a weaker safety record could move ahead. The municipality should report this sensitivity separately to the tender committee.
In the report: "With the high weight given to technical competence, the contractor strong on this criterion ranks first; its relative disadvantage on bid price does not change this superiority. If the weight on the safety record is raised, the ranking may change."
3. What Not to Do
Raising q to 5 by user preference in the illustrative example and presenting it as "a more flexible model" is the first error; q comes bundled fixed with the engine in this method. The second error is building the reference component by component, as in Plithogenic or Pythagorean GRA (taking each column's highest support and lowest rejection separately); in QR-GRA the reference is a real row chosen by score. The third error is leaving C3's direction unchanged and processing a "lower is better" criterion as if it were "higher is better"; in that case the second- and third-ranked alternatives swap places incorrectly.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/qr-gra
Yager, R. R. (2017). Generalized orthopair fuzzy sets. IEEE Transactions on Fuzzy Systems, 25(5), 1222–1230. DOI: 10.1109/TFUZZ.2016.2604005
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Güler, B., Sarucan, A., & Özcan, E. (2026). A new q-rung picture fuzzy grey relational analysis method and investigation of its effectiveness. Journal of the Faculty of Engineering and Architecture of Gazi University, 41(1), 507–518. DOI: 10.17341/gazimmfd.1596116
Ahemad, F., Mehlawat, M. K., & Gupta, P. (2023). A GRA approach to a MAGDM problem with interval-valued q-rung orthopair fuzzy information. Soft Computing, 27, 15793–15815. DOI: 10.1007/s00500-023-08822-5