Interval-valued intuitionistic fuzzy ARAS (Büyüközkan and Göçer, 2018)

Five things change; the decision logic does not.

Cells. In crisp ARAS 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 support and the upper rejection cannot exceed 1. This structure is the interval-extended form of the single support-single rejection pair in the intuitionistic fuzzy extension: an expert now says "support is at least 0.5, at most 0.7" rather than "support is 0.6." This card presents this four-number cell on DM3's site under the fuzzy data-type family; conceptually it is the interval form of the intuitionistic fuzzy structure, distinct from a three-cornered triangular fuzzy number.

Scale equalisation. In crisp ARAS every column is divided by its own sum. Here the division is done with Xu's vector normalisation: every bound is divided by the square root of the sum of the squares of that same bound across all rows in that column (including the optimal alternative). This differs from crisp ARAS's sum-division logic; the column's magnitude (its norm), not its sum, is the basis.

Optimal alternative. In crisp ARAS the optimal alternative is built from every criterion's best crisp value. Here, on a benefit criterion, the lower and upper support are taken from the highest values and the lower and upper rejection from the lowest; on a cost criterion, the reverse is taken. Cost direction is reversed not by swapping places, as in the intuitionistic fuzzy extension, but by separately choosing the best and worst of these four bounds.

Weighting and summation. The equalised four-number cells are weighted by the criterion weight using intuitionistic fuzzy algebra's own rule (multiplicative combination, not exponential). Every row's cells (including the optimal alternative's) are then summed across the criteria with this same algebra. The result is a single four-number "optimality" cell for every row.

Result and defuzzification. The four-number optimality cell descends to a single number through a score function that adds the complement of the mean rejection to the mean support. The degree of utility Q carries the same meaning as K in crisp ARAS: the ratio of the real alternative's defuzzified score to the optimal alternative's defuzzified score. DecisionMind fixes this score function and Xu's normalisation; the four-number cells are never ranked directly at any stage, ranking is done only on the defuzzified Q.

The degree of utility Q carries the same meaning here. The best alternative is taken as 100, and the others receive a percentage relative to it; this percentage holds only for this alternative set and these weights. The difference is here: beneath Q lie a two-bounded support and a two-bounded rejection. Because these four numbers are reduced to a single number at the last step, they do not appear in the report. An alternative with a wide support or rejection interval may have a Q value more sensitive to the weight distribution than the others.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy ARAS, A2 is the best alternative"

the report should read:

"With the given weights (0.35; 0.25; 0.40), A2 has the highest degree of utility (Q=0.945); when the weight is shifted to the third criterion (0.15; 0.15; 0.70), A3 takes first place, so the weight distribution must be separately justified in the report"

This method is appropriate when only an interval is known for a judgement's degree of support and degree of rejection, that is, when an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." Examples include supplier audits where several assessors' support-rejection scores are summarised as an interval, and committee evaluations that are deliberately left unreduced to a single crisp ratio.

If support and rejection are given as a single number, opening them into an interval produces uncertainty rather than modelling it; in that case intuitionistic fuzzy ARAS suffices. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is likewise fully compensatory and will not eliminate anything below a threshold.

Constraint violation. In every cell the lower bound must be less than or equal to the upper bound, and the sum of the upper support and the upper rejection must not exceed 1. If this condition is broken, the optimal alternative and the normalisation become meaningless.

Reversing cost direction by a swap. Swapping support and rejection, as in the intuitionistic fuzzy extension, is not enough here; the best and worst of the four bounds (lower support, upper support, lower rejection, upper rejection) must each be chosen separately.

Leaving the optimal alternative out of Xu's normalisation. The column norm must be computed over all (m+1) rows, including the optimal-alternative row; computing it with only the real alternatives makes the degrees of utility deviate from the source paper's own definition.

Confusing weight elicitation (IVIF-AHP) with this card. DecisionMind's engine takes weights from outside as crisp numbers; the source paper's own full method also generates the weights themselves from an interval-valued intuitionistic fuzzy AHP process. These are two separate stages, and this card covers only the second one, the ARAS calculation once weights have been supplied.

The governing principle is this:

Interval-valued intuitionistic fuzzy ARAS exists to carry a judgement's two-layered uncertainty in its degree of support and rejection (interval width and rejection share) all the way through to the ratio against the optimal alternative; any application that breaks the constraint or leaves the optimal alternative out of the normalisation corrupts the information the four-number cell carries.