Pythagorean fuzzy VIKOR

Three things change; the decision logic does not.

Cells. In crisp VIKOR every cell is a single number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree. These two values must satisfy μ² + ν² ≤ 1. The hesitancy margin, π = √(1 − μ² − ν²), is derived from these two degrees; it is not asked of the expert separately. Criterion weights remain crisp numbers and come from outside. The compromise coefficient β also remains a crisp number; its default value is 0.5.

Scale equalisation. Crisp VIKOR finds the best and worst value for every column, then ratios every cell to the range between the two. PF-VIKOR uses the same logic, but the range now lies between two support-rejection pairs rather than two numbers. On a benefit criterion the best pair is built from the highest μ and the lowest ν, and the worst pair is the reverse; on a cost criterion these two roles swap. Every cell's normalised gap from these two references is measured with the Pythagorean Hamming distance; this distance is a third of the sum of the differences between the μ², ν² and π² components.

Group utility, individual regret and the compromise index. Crisp VIKOR's S and R calculation runs here with the same logic. S is the sum of an alternative's weighted normalised gaps across all criteria. R is the largest of these gaps. The difference is that every gap now comes from a Pythagorean distance, rather than from a single number's range ratio. S and R are each drawn onto their own 0-1 scale and combined into the Q index with the coefficient β. This step is identical to crisp VIKOR; β again remains a default preference of 0.5 here.

The compromise test and defuzzification. The output is three columns: S, R and Q. Ranking is done by Q, and a lower Q is better. Crisp VIKOR's two conditions, acceptable advantage and acceptable stability, are applied here exactly as they are there. Uncertainty is carried within the Pythagorean distance inside S and R; once Q enters the calculation it is a single number, but the two-condition test beneath this number does not hide the uncertainty, it only reduces it to a single rank.

DecisionMind fixes, in PF-VIKOR, the Pythagorean Hamming distance, the summation and maximum rules for S and R, the β-and-Q combination, and the two-condition compromise test. Criterion weights and β are taken from outside as crisp numbers; the method does not generate weights.

A low Q is good, but the real result is not the Q ranking but the verdict the two conditions produce. This is the same as in crisp VIKOR. What differs is that this verdict comes from Pythagorean fuzzy inputs. Because S and R are already sums of Pythagorean distances, the S or R gap between two alternatives is sensitive to how wide a region of acceptance the input pairs are spread across. The same Q gap should be judged more robust when it comes from pairs drawn from a narrow region of acceptance, and more fragile when it comes from a wide one.

Thus instead of writing:

"According to PF-VIKOR, this alternative is the single compromise solution"

the report should read:

"With these weights and this β, this alternative is ahead in both the S and the R ranking, and its Q gap to the second alternative exceeds the acceptance threshold; if either condition is not met, the compromise set is stated as it stands"

Use this extension where experts give a judgement both strong support and a marked reservation, and where conflicting stakeholder interests are present. If the two degrees sum to more than 1, the intuitionistic fuzzy constraint forces these pairs to shrink, because that constraint keeps the sum at no more than 1. Pythagorean fuzzy solves this problem. A measured value is not turned directly into a support-rejection pair. It is first converted into a judgement, and only then are μ and ν derived from separate sources; the detail is on the data-type card. The table must hold a single data type throughout; part crisp and part Pythagorean is not allowed. The base method's exit condition applies here too. Where no compromise whatsoever is acceptable on one criterion, PF-VIKOR limits regret but does not eliminate it; where strict elimination is required, outranking methods should be considered instead.

Domain violation. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1. Entering the calculation without this check invalidates the method.

Writing ν as 1 − μ. In that case the sum always comes to exactly 1. The hesitancy margin is zeroed. The extra region the Pythagorean structure offers over the intuitionistic one is never used.

Skipping the two conditions and reporting only the alternative with the smallest Q as the winner. This mistake exists in crisp VIKOR too, and is more easily overlooked with Pythagorean input. Where the input pairs come from a wide region of acceptance, the Q gap between two alternatives comes out small, and the chance of a compromise set emerging increases.

Reversing the best and worst pair's role on a cost criterion. On a benefit criterion the best pair is built from the highest support and the lowest rejection degree. On a cost criterion this is reversed. Skipping this reversal turns the ranking on its head.

Choosing β without justification and not stating it in the report. As β moves away from 0.5, either group utility or individual regret comes to the fore. This is a choice, and it must be written into the report with its reasoning.

The governing principle is this:

PF-VIKOR's result is not a ranking but a conditional compromise proposal. PF-VIKOR has not been properly applied unless the coefficient β, the S and R columns, and the outcome of the two conditions are given together in the report.