Interval-valued intuitionistic fuzzy ARAS (aggregative, DecisionMind derivation)

Five things change; the logic of taking a ratio to the optimal alternative does not.

Cells. In crisp ARAS every cell is a single positive 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. 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. The difference from intuitionistic fuzzy is that support and rejection are themselves each an interval; the expert states not only "how much I support" but also how certain this support is.

Cost direction. On a cost criterion, direction is reversed by swapping the support and rejection intervals; this must not be confused with crisp ARAS's 1/x operation.

Optimal alternative. The optimal-alternative row is not copied from a single real alternative, as it is in crisp ARAS. For every criterion, the largest of all observed alternatives' lower support bounds, the largest of their upper support bounds, the smallest of their lower rejection bounds and the smallest of their upper rejection bounds are taken separately. This is an "envelope" value that assembles the best of each criterion and does not match any real alternative exactly.

Weighted aggregation instead of scale equalisation. Crisp ARAS first divides columns by their sum and then multiplies by the weight. Here there is no separate column normalisation; every alternative's (and the optimal row's) support-rejection intervals across criteria are reduced directly to a single support-rejection interval pair by interval-valued intuitionistic fuzzy's own weighted aggregation operation (IIFWA). This aggregation is an algebraic sum that treats criterion weights as exponents; its result is again a support-rejection interval pair, not yet a single number.

Result and defuzzification. Only in this last step does the aggregated support-rejection interval pair descend to a single number through a score function. 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. The utility degree K is the ratio of a real alternative's score to the optimal row's score.

DecisionMind fixes, in this classical form, the construction of the optimal row as a criterion-by-criterion envelope, the weighted aggregation operator (IIFWA), and this average-like score function.

The utility degree K is read as in crisp ARAS: a percentage-like benefit relative to the optimal alternative, meaningful only for this particular alternative set and these weights. The difference is here: beneath K now lies a four-number support-rejection interval that has first been aggregated with weights and then reduced to a single number by a score function. When a criterion is given excessive weight this score can come out negative; in that case K's interpretation as "what percentage of benefit" breaks down and the weight distribution should be reviewed.

Thus instead of writing:

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

the report should read:

"Relative to the optimal alternative derived from this alternative set, A2 has the highest utility degree (K=0.91); this ratio is meaningful only with these weights, and if weight is shifted excessively onto one criterion the score can turn negative, invalidating the interpretation"

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 may answer "how much do I support this" not with a single number but with an interval. Example: when the lower and upper bounds of support-rejection scores given by several assessors are known but there is no single point of consensus. If support and rejection are already single points (zero interval width), this extra width adds nothing and intuitionistic fuzzy is sufficient. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory.

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 support-rejection bound (μᵁ+νᵁ≤1) and the aggregation operation become meaningless.

Copying the optimal row from a single real alternative. The optimal row is an envelope built criterion by criterion; "making the best alternative's row the optimal row as it stands" is correct in crisp ARAS but wrong here, because no real alternative may be best on every criterion at once.

Reading the score as "what percentage probability." The score is a quantity that can range between −1 and 1, not a probability; it can come out negative when weight is shifted excessively onto one criterion.

Reporting without stating the defuzzification method. The score used here is a midpoint-form definition; other score definitions exist in the literature, and which one was used must be stated in the report.

The governing principle is this:

In interval-valued intuitionistic fuzzy ARAS, the support-rejection interval passes through weighted aggregation and descends to a single score only in the final step. The optimal row is not a real alternative but a criterion-by-criterion envelope; it must also be remembered that the score can turn negative under extreme weights.