Cubic Pythagorean fuzzy EDAS (Paul, Jana & Pal, 2023)

Four things change.

Cells. In crisp EDAS every cell is a single number. Here every cell consists of six numbers: a Pythagorean interval (support [Y⁻,Y⁺], rejection [F⁻,F⁺]) AND a Pythagorean point (support Y, rejection F). The interval part carries the information "this assessment is plausible within this band"; the point part carries "the single most likely value"; neither substitutes for the other, and both are kept together. The constraint is Pythagorean: at the interval's upper end, (Y⁺)²+(F⁺)² ≤ 1, and for the point part, Y²+F² ≤ 1. Criterion weights are taken from outside as crisp numbers.

Combining experts (group decision). This extension supports more than one expert's matrix. A geometric combination rule weighted by expert weights (CuPyFWG) reduces K experts' cells to a single collective cell; with a single expert this step changes nothing. Crisp EDAS does not directly support group decisions and already treats the input as combined; here, combination is a step of the method itself.

Weight source: optimisation on paper, direct input in the engine. The founding article (Paul, Jana & Pal, 2023) derives criterion weights through a non-linear optimisation. That optimisation runs under partial weight constraints and requires a solver. DecisionMind's engine does NOT run this optimisation; weights are supplied directly as input. This card's author has checked the β and q parameters the manifest reports (belonging to the article's optimisation objective) by reading the kernel code, and found that the engine uses them in no step at all. Changing these two parameters has no effect whatsoever on the result; changing the weight is what is required.

Score, deviation and combination. Every cell's score is the average of its interval part's Pythagorean score and its point part's Pythagorean score. The column average is built on this same score, as the alternatives' Pythagorean weighted geometric mean. The rest is crisp EDAS itself: every cell's positive and negative deviation from this average is measured, weighted and summed, normalised, and combined into a single appraisal score. Because the score can come out negative (the Pythagorean score lies between −1 and 1), the column average can also be negative; in that case the denominator of the deviation calculation becomes a negative number, and unless built carefully, the positive/negative signs can flip. This is a risk the manifest itself reports (see Mistakes Specific to This Extension).

DecisionMind fixes, for this extension, the score function, the geometric-mean rule and crisp EDAS's own deviation-normalisation steps. Weights come from outside; the method does not generate weights itself.

The appraisal score is read as in crisp EDAS: it is a position relative to the set's own average. The difference lies here: this average now comes from the combination of an interval AND a point assessment. Two alternatives can share the same point value while differing in interval width; that width merges into the single number from the moment the score is computed, and does not appear separately in the score itself.

Thus instead of writing:

"Cubic Pythagorean fuzzy EDAS gives a more reliable result because it uses interval and point information together"

the report should read:

"This score comes from the average of the interval assessment and the point assessment; which alternative has a wide interval and which has a narrow one does not show up in the score itself and must be reported separately"

This extension is worth considering when experts give a criterion both as a band (at least this, at most that) and as a single most-likely value together. The aim is to carry this pair of pieces of information without reducing one to the other. If more than one expert's matrix exists, the method combines them internally; where weights are wanted derived by optimisation as in the article, this must be done as a separate step outside DecisionMind and the result entered directly as a weight. Converting a measured criterion into an interval-point pair models uncertainty rather than manufacturing it. Crisp EDAS's exit condition applies here too: where one criterion cannot be traded away, this extension remains compensatory as well.

Expecting to change the result by adjusting β and q. These two parameters belong to the article's weight-optimisation step; DecisionMind's engine does not run that step, so these parameters are read at no computational step at all. The only way to change the result is to change the weight directly.

Not checking deviation signs when the column average comes out negative. Because the score lies between −1 and 1, a criterion's average can come out negative; in that case the denominator of the positive/negative deviation calculation becomes negative, and unless built carefully, the signs can flip. This is a risk the manifest itself reports; the sign of every criterion's average score must be checked before the result is accepted.

Value-domain violation. At the interval's upper end, (Y⁺)²+(F⁺)² ≤ 1 must hold, and for the point part, Y²+F² ≤ 1; this is checked from the sum of squares, not from the intuitive sum (Y+F≤1).

Assuming equal expert weights and not reporting it. Where more than one expert is involved, the combination is sensitive to expert weights; saying "the experts were combined" without stating the weights is an incomplete report.

The governing principle is this:

In this engine, weights always arrive as direct input; the article's optimisation parameters (β, q) are never read by the engine and have no effect whatsoever on the result. The report must state this limit plainly.