Fermatean Fuzzy COPRAS

Three things change; the benefit-cost ratio logic does not.

Cells. In crisp COPRAS every cell is a single number. Here every cell is a pair: a degree of support for a judgement (μ) and a degree of rejection of it (ν). In intuitionistic fuzzy data the sum of these two cannot exceed 1; in Fermatean fuzzy data the limit is wider, the sum of their cubes cannot exceed 1 (μ³+ν³≤1). This allows pairs such as μ=0.8 and ν=0.5, whose sum exceeds 1 but whose cube sum stays below 1, to remain valid; a judgement can be both strongly supported and strongly rejected. Criterion weights are supplied from outside as crisp (single) numbers.

Scale equalisation. DecisionMind first reduces every pair to a single score: this score is μ³−ν³ and ranges from −1 to 1. Because it can be negative, 1 is added to shift it into the [0,2] range. The column-sum-based normalisation of crisp COPRAS is then applied to the scores obtained after this shift. This follows the same logic as the Chen-Tan score-difference shift in IF-COPRAS, except that the score function is extended to μ³−ν³.

Distance / score / combination. The normalised, weighted scores are split into a benefit total and a cost total exactly as in crisp COPRAS; the relative significance value (Q) is built with the same ratio and the same correction term that rewards low cost.

Result and defuzzification. Defuzzification happens right at the start: the support-rejection pair collapses to a single score (μ³−ν³) in the very first step, and the remainder of the calculation follows crisp COPRAS's own steps exactly. The output is a single utility degree, in the same form as crisp COPRAS's own output.

DecisionMind holds the μ³−ν³ score function and the +1 shift fixed for this family; no other score definition, nor a route that normalises support and rejection separately, is followed.

The output is a utility degree and ranking in the same form as crisp COPRAS, and it is read the same way. The difference is this: the score rests on the cube difference of support and rejection, and it also covers the "both high support and high rejection" situations that intuitionistic fuzzy data cannot express. An apparently identical μ³−ν³ difference can arise whether support and rejection are both high or both low; the utility degree does not distinguish between these two cases.

Thus instead of writing:

"Fermatean fuzzy COPRAS found this alternative more reliable"

the report should read:

"The utility degree rests only on the cube difference of support and rejection; A2 leads with 100.00, and this does not by itself mean 'more reliable' unless how strongly A2's support-rejection pair carries both endorsement and reservation is separately examined"

This extension is suitable when criterion assessment rests on a judgement, and the support and rejection degree of that judgement need to be expressed together, more strongly than intuitionistic fuzzy data allow. Example: an expert both strongly endorses a software vendor and holds a serious reservation, and these two sentiments do not fit within intuitionistic fuzzy data's μ+ν≤1 limit. If a criterion states only "how suitable" and carries no separate evidence of rejection, this wider limit adds nothing, and intuitionistic fuzzy or ordinary fuzzy data suffice. The exit condition of crisp COPRAS applies here too: where a criterion admits no compromise, this extension is also fully compensatory.

Entering a cell without checking the cube-sum limit (μ³+ν³≤1). A pair outside this limit is invalid; applying intuitionistic fuzzy data's μ+ν≤1 limit here is wrong, since Fermatean fuzzy data's own limit is wider.

Treating the score function (μ³−ν³) as the only correct choice and failing to state it in the report. Different score definitions can give a different ranking; the function used must be stated in the report.

Ignoring the shift constant (+1) and dividing scores directly by the column sum. If negative scores are divided without shifting, the column sum can turn out negative and the utility degrees become meaningless.

Marking criterion direction wrongly. If a cost criterion is marked "higher is better", high support on that criterion is included in the benefit total, and the result can reverse.

The governing principle is this:

Fermatean fuzzy COPRAS exists to carry the strong support-rejection combinations that intuitionistic fuzzy data cannot hold through to an optimal utility degree; any application that skips checking the cube-sum limit or omits the shift step corrupts this information.