Extension card · q-Rung Orthopair
q-Rung Orthopair CODAS
This is the form of CODAS for situations where an expert gives a judgement both strong support and a strong reservation. It is used when the sum of these two exceeds the intuitionistic or Pythagorean boundary; it still ranks the result with an assessment score.
Base method
CODAS →
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?
Four things change. The decision logic stays the same.
Cells. In crisp CODAS every cell is a single number. Here every cell is a pair: a support degree μ and a rejection degree ν. The sum of the two degrees raised to the q-th power cannot exceed 1. The user does not choose q; the method works with q = 3, and DecisionMind holds this value fixed. Weights are supplied from outside as single numbers; the method does not generate weights.
Scale equalisation. Crisp CODAS divides columns by their largest or smallest value. No such division happens here. For a cost criterion the method inverts the pair instead: the support and rejection degrees swap places. Every pair is then scaled by its criterion's weight through a q-ROF weighted-average operation.
Negative-ideal and distance. In crisp CODAS the negative-ideal is built from each criterion's single worst number. Here a negative-ideal pair is built from the lowest support and highest rejection degree on each criterion; this pair need not be a real alternative's own pair, it is the two extremes brought together. Every alternative's Euclidean distance (E) and city-block distance (T) to this point are then computed; both use all three components at once, support, rejection and hesitancy. In crisp CODAS, single numbers carried a straight-line distance and a horizontal-and-vertical distance; here the distance has three components.
The threshold value is closed off. In crisp CODAS and in the picture-fuzzy extension, the threshold value (τ) is an input the user can set. In the q-Rung extension τ is not open to the user; DecisionMind holds it fixed at 0.02. This difference matters: if the data's scale is unusual (distances very small or very large), the user cannot change the threshold.
DecisionMind holds the standard q-ROF Euclidean and Hamming distance (Du, 2018) and the Liu-Wang (2018) weighting operation fixed for this family. A different distance measure has also been proposed in the literature, and it can give a different magnitude, and in some tables even a different ranking.
How to Read the Output
The assessment score is read as in crisp CODAS: a relative score, coming out negative is not a failure, and the negative-ideal shifts when the alternative set changes. The difference is here: beneath the score now lies a tension between support and rejection. When an expert gives the same judgement both high support and high reservation, this duality enters the calculation but does not show up in the score.
Thus instead of writing:
"Because q-Rung CODAS models the uncertainty, the result is more accurate"
the report should read:
"The criteria on which the expert gave both strong support and strong reservation have been marked separately; the assessment score has come out sensitive to the weight on these criteria"
When to Prefer This over the Base Method
Use this extension when an expert gives a judgement both strong support and a strong reservation, and this pair exceeds both the intuitionistic and the Pythagorean boundary. Which pair exceeds which boundary is settled by the short decision rule on the q-Rung data-type card. Converting a measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. The base CODAS exit condition applies here too: if no compromise is accepted on one criterion, this compensatory family is not the right choice.
Mistakes Specific to This Extension
Entering data without checking the constraint. In every cell, the sum of the support and rejection degrees' third powers cannot exceed 1. This check is performed before the calculation.
Inverting the cost direction twice. For a cost criterion the method swaps the support and rejection degrees once. Repeating this swap when choosing the negative-ideal breaks the direction from end to end.
Forgetting that the threshold value is fixed. τ = 0.02 is closed off to the user. If the data's scale is unusually small (distances well below 0.02), the city-block distance is brought into every comparison, and the user cannot change this; this must be stated in the report.
Changing the distance measure and expecting the same result. DecisionMind uses the standard Du (2018) q-ROF distance. An alternative distance formula from the literature gives a different score, and in some tables a different ranking too.
The governing principle is this:
q-Rung orthopair CODAS exists to carry the tension between support and rejection through to the final step. Any application that ignores the rejection degree or fails to check the constraint erases this contribution.
Cases
No verified literature application has been found for this method; the only q-Rung CODAS paper on record in the manifest (Naz et al., 2022) is a two-tuple linguistic variant, not pure q-ROF CODAS. The first case is therefore DecisionMind's own validation example; its figures were obtained by independently rewriting the engine. The second case is an illustrative construction.
1. Illustrative example: Assessment of three candidates on three criteria (DecisionMind's validation example)
All three criteria are in the benefit direction, with weights of 0.40, 0.35 and 0.25 respectively. A1's support degree on every criterion is higher than A2's, and its rejection degree lower than A2's. The same relationship holds between A2 and A3.
| Candidate | C1 (μ, ν) | C2 (μ, ν) | C3 (μ, ν) |
|---|---|---|---|
| A1 | 0.90 · 0.20 | 0.85 · 0.30 | 0.80 · 0.40 |
| A2 | 0.70 · 0.40 | 0.65 · 0.50 | 0.60 · 0.55 |
| A3 | 0.50 · 0.55 | 0.45 · 0.60 | 0.40 · 0.65 |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method scales every column by its weight, builds the negative-ideal pair, and computes every candidate's Euclidean and city-block distance to it.
| Candidate | Assessment score | Rank |
|---|---|---|
| A1 | 0.9642 | 1 |
| A2 | -0.1412 | 2 |
| A3 | -0.8230 | 3 |
The result reads as follows. A1 has both the higher support degree and the lower rejection degree on every criterion, and is the candidate furthest from the negative-ideal. A3 sits in exactly the opposite position and finishes last.
The decision's hesitation: even if the weights are reversed (C1 = 0.25, C2 = 0.35, C3 = 0.40), A1 still comes first, its score dropping to 0.9261, but the ranking does not change. In this example the ranking withstands the weight swap, because every candidate outranks the next on every criterion; this robustness should not be expected in every table.
In the report: "Among the three candidates, A1 sits furthest from the negative-ideal, with a higher support degree and a lower rejection degree on every criterion (score 0.9642). The ranking does not change under a weight swap."
Source: DecisionMind's validation example for the QR-CODAS engine. The figures were computed by independently rewriting the manifest's steps in Python and matched against the engine's own output.
2. Defence industry: Choosing a munitions supplier
A defence institution will choose among three munitions suppliers. Three criteria are set: technical capability, production capacity and safety risk (lower is better). The expert team gave both a support and a rejection degree to each supplier on these three criteria; particularly on the safety-risk criterion, suppliers carry both strong capability and a serious reservation at once, and these pairs exceed the Pythagorean boundary.
| Supplier | Technical capability | Production capacity | Safety risk (lower is better) |
|---|---|---|---|
| S1 | 0.85 · 0.30 | 0.75 · 0.30 | 0.80 · 0.30 |
| S2 | 0.65 · 0.45 | 0.60 · 0.45 | 0.35 · 0.70 |
| S3 | 0.50 · 0.55 | 0.55 · 0.50 | 0.55 · 0.50 |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method first swaps the support and rejection degrees in the safety-risk column, then computes the weights, the negative-ideal pair and the two distances.
| Supplier | Assessment score | Rank |
|---|---|---|
| S1 | 0.2252 | 1 |
| S2 | 0.1902 | 2 |
| S3 | -0.4154 | 3 |
The result reads as follows. S1 is the strongest supplier on technical capability and capacity, and with these weights beats S2 by a small margin (0.0350). But S1's support degree on the safety-risk criterion (0.80) is high; this is a serious reservation for the committee.
The committee's hesitation: if the weights shift towards safety risk (safety 0.60, technical capability 0.20, capacity 0.20), S2 moves into first place, its score rising to 0.7557. S1's score drops to -0.7460 and it falls to last place. S1's high capability score cannot sufficiently offset its high risk score once the weight shifts towards safety.
In the report: "With the weight given to technical capability and capacity, S1 leads, but the gap between S1 and S2 is small (0.0350). Once the weight shifts to safety risk, S2 moves into first place and S1 drops to last; the decision depends directly on this weight choice."
3. What Not to Do
If the rejection degree is ignored in the illustrative example and crisp CODAS is run on the support degrees alone, the gap between A1 and A2 comes out different from its true value; the hesitancy margin (π) is left out of the calculation. The second error, in the defence example, is forgetting to swap the support and rejection degrees for a cost criterion such as safety risk; in that case the riskiest supplier looks like the ideal alternative. The third error is writing in the report "the threshold was set to 0.05" without knowing that the threshold value (τ) is fixed at 0.02 in DecisionMind; this parameter is not open to the user in this family.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/qr-codas
Yager, R. R. (2017). Generalized orthopair fuzzy sets. IEEE Transactions on Fuzzy Systems, 25(5), 1222–1230. DOI: 10.1109/TFUZZ.2016.2604005
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (No DOI.)
Naz, S., Akram, M., Sattar, A., & Al-Shamiri, M. M. A. (2022). 2-tuple linguistic q-rung orthopair fuzzy CODAS approach and its application in arc welding robot selection. AIMS Mathematics, 7(9), 17529–17569. DOI: 10.3934/math.2022966
Liu, P., & Wang, P. (2018). Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. International Journal of Intelligent Systems, 33(2), 259–280. DOI: 10.1002/int.21927