Interval-valued intuitionistic fuzzy EDAS (DecisionMind derivation)

Four things change; the logic of locating position relative to the average does not.

Cells. In crisp EDAS every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection. The sum of the upper bounds cannot exceed 1. A judgement such as "my confidence in this supplier is between 0.65 and 0.75, my reservation between 0.15 and 0.20" corresponds to these four numbers.

Early reduction to a score. DecisionMind's engine here follows a different route from IV-ARAS. In IV-ARAS the support-rejection interval passes through a weighted aggregation and descends to a score only in the final step. In IV-EDAS, every cell is reduced to a single score in the very first step of the calculation, before the average is even built. The score is found by subtracting the sum of the lower and upper rejection bounds from the sum of the lower and upper support bounds, then dividing by two. On a cost criterion, the support and rejection intervals are swapped before this reduction.

The average solution and the deviations. The average is the arithmetic mean of every column's scores and is now a single exact number. The positive and negative deviations are computed with the same rule as in crisp EDAS: above the average is favourable on a benefit criterion, below the average is favourable on a cost criterion. Since cost direction has already been reversed once, on the way to the score, direction is not reversed again here.

Result and defuzzification. Defuzzification happens at the start, not at the end. The assessment score is a number between 0 and 1, exactly as in crisp EDAS. The width of the support-rejection interval is not reflected in any later step of the calculation; from the moment the score is taken, the interval itself is forgotten.

DecisionMind fixes, in this extension, the score function and the order of early reduction to a score. Weights are taken from outside as crisp numbers; the method does not generate weights.

The assessment score is read in the same way as in crisp EDAS. It is a position relative to the set's own average, not a percentage or a probability, and it cannot be compared with a different analysis. The difference is here: there is no longer any interval uncertainty behind the score. Although the support-rejection interval was gathered as four numbers, the calculation runs on exact numbers from the first step onward.

Thus instead of writing:

"Because IV-EDAS carries the interval uncertainty through to the end, the result is more reliable"

the report should read:

"The support-rejection intervals were reduced to a single score in the very first step of the calculation; the resulting score is computed from these scores, and the interval's width is not reflected in it"

This extension is suitable when a criterion assessment rests on a judgement, and the degree of support and rejection of that judgement is itself uncertain. An expert should be able to answer "how much do I support this" with an interval rather than a single number. If support and rejection are already single points, that is, if the interval width is zero, this extra width adds nothing and intuitionistic fuzzy EDAS is sufficient. Crisp EDAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory and does not eliminate anything below a threshold.

Allowing the lower bound to exceed the upper bound. In every interval the lower bound must be less than or equal to the upper bound. Unless this is checked, both the upper-bound constraint and the score calculation become meaningless.

Mistaking IV-EDAS for IV-ARAS. The two extensions share the same cell shape, but the order of defuzzification is exactly reversed. In IV-ARAS the interval is carried through to the final step; in IV-EDAS it descends to a score in the first step. Reporting one as if it were the other produces a false claim about uncertainty.

Reading the score as a probability. The score is a quantity that can range between minus one and one. It can also come out negative; this is as ordinary as falling below the average.

Trusting a criterion whose score average is very close to zero. In that case the ratio of positive to negative deviation grows large, and the result for that criterion should not be used without checking the underlying data.

The governing principle is this:

In IV-EDAS the support-rejection interval descends to a single score at the very first step, before the average is even built. The interval's width does not return at any later step; the report must not confuse this early defuzzification with IV-ARAS's.