Interval-valued intuitionistic fuzzy SAW (a DecisionMind derivation)

Three things change; the weighted-sum logic does not.

Cells. In crisp SAW 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. 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 data is that support and rejection are themselves intervals. Criterion weights are taken from outside as crisp numbers.

Scale equalisation. Crisp SAW ratios every column to its own best value. There is no such step here, because support-rejection intervals already lie between 0 and 1. Instead, on a cost criterion, the support and rejection intervals swap places: ([μL,μU], [νL,νU]) → ([νL,νU], [μL,μU]). This is this family's counterpart to crisp SAW's ratio-to-best step; it reverses direction, not scale.

Weighted sum and defuzzification. In crisp SAW every column is multiplied by its weight and summed directly. Here, every alternative's support-rejection interval pair across the criteria is reduced directly to a single support-rejection interval pair by interval-valued intuitionistic fuzzy weighted averaging's own aggregation operation (IIFWA). This aggregation is an algebraic operator, different from crisp SAW's linear sum; its result is again a support-rejection interval pair, not yet a single number. Only at this last step does the combined interval pair descend to a single number through a score function. The score is found by subtracting the sum of the rejection bounds from the sum of the support bounds and dividing by two.

DecisionMind fixes, in this classical form, the IIFWA aggregation operator and this mean-like score function. Weights are taken from outside; the method does not generate weights.

The score ranks only this alternative set, exactly as crisp SAW's sum does; it is not a percentage or a probability, and it is not compared with another analysis.

The difference is this. Beneath the score now lies a four-number support-rejection interval that has first been combined by weight and only then reduced to a single number through a score function. Two alternatives with the same score may come from one narrow and one wide support-rejection interval; interval width shows how much experts agree on this judgement, and it does not appear in the score.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy SAW, this alternative is definitively the best"

the report should read:

"This score is a value that emerges from the weighted combination of support-rejection intervals; which alternative's interval is wider, that is, less agreed upon, must be reported separately"

This extension suits situations where a criterion assessment rests on a judgement, and that judgement's own degree of support and rejection is itself uncertain. An expert may answer "how much do you support this" with an interval rather than a single number, for instance when the lower and upper bound of several assessors' support-rejection scores are known but no single consensus point exists. If support and rejection already sit at a single point (interval width zero), this extra width contributes nothing, and intuitionistic fuzzy suffices. Crisp SAW's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is likewise 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; if this is not checked, both the support-rejection constraint (μU + νU ≤ 1) and the aggregation operation become meaningless.

Fabricating ν as 1 − μ. This zeroes the interval's width without justification and silently collapses the structure back into single-point intuitionistic fuzzy data.

Reading the score as "what percentage probability." The score is a combination of support and rejection bounds, not a probability.

Reporting without stating the defuzzification method. The score used here is a mid-point definition; other score definitions also 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 SAW, the support-rejection interval passes through a weighted combination and only descends to a single score at the very last step; interval width shows the degree of agreement among experts, and the report must state this separately.