Interval-valued intuitionistic fuzzy TODIM (Krohling & Pacheco, 2014)

Four things change; the amplification of the loss side does not.

Cells. In crisp TODIM 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. Weights are supplied from outside as crisp numbers. DecisionMind does not support group decision-making in this extension.

Scale equalisation. In crisp TODIM, cost criteria are converted to the benefit direction by a column-wise ratio. Here the support–rejection interval on cost criteria is first complemented: the lower and upper support swap with the lower and upper rejection. Every column is then scaled by Xu–Yager vector normalisation: each bound is divided by the square root of the sum of the squares of that same bound across all rows in its column. This is the interval-valued intuitionistic fuzzy counterpart of crisp TODIM's division-by-column-sum normalisation.

Distance and score. In crisp TODIM the difference between two values is a direct subtraction. Here the two aspects separate. The score is found by summing the lower and upper support bounds, subtracting the sum of the lower and upper rejection bounds, and halving the result; where scores are equal, an accuracy term (the sum of all four bounds) decides. Distance, meanwhile, is computed as a Hamming distance: the sum of the absolute differences of the four normalised bounds, divided by four and then rooted. In crisp TODIM one subtraction supplies both magnitude and direction; here magnitude comes from this Hamming distance and direction from the score (accuracy, in a tie) comparison.

Result. The global value is again a single number normalised between 0 and 1; the uncertainty in the support–rejection interval feeds into the normalisation and distance calculations but resolves by the end, and the final number is not itself interval-valued.

DecisionMind fixes, in this extension, Xu–Yager vector normalisation, the score-then-accuracy comparison rule, and the Hamming distance. The loss-aversion coefficient θ defaults to 1 and can be changed by the user. The reference criterion is automatic, the highest-weighted criterion, and cannot be chosen by the user; weights are not rescaled to sum to 1 before this ratio is computed.

The global value is read as in crisp TODIM: the lowest total dominance gets 0, the highest gets 1, and it is not an absolute "good/bad" measure.

The difference is this. The winning-losing direction comes from the score (accuracy, in a tie); the magnitude comes from the normalised Hamming distance. Because the normalisation step scales cells relative to their column's magnitude, the same raw support–rejection interval can make a different contribution in a different column. The report should therefore state only the final global value and its sensitivity to θ, not how each column was affected by normalisation.

Thus instead of writing:

"Interval-valued intuitionistic fuzzy TODIM finds A2 to be the best alternative"

the report should read:

"The support–rejection intervals have been scaled by Xu–Yager normalisation and compared by Hamming distance; with θ = 1, A2 has the highest global value, though this ranking may change as θ increases"

This extension suits situations where only an interval is known about a judgement's degree of support and rejection, that is, where an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." It equally suits situations where the decision-maker's greater sensitivity to losses than to gains matches the nature of the decision.

If support and rejection are given as a single number each, that is, if the interval width is zero, this extra width adds nothing, and intuitionistic fuzzy TODIM (IF-TODIM) suffices. Where criteria are measured, crisp TODIM should be used. TODIM's exit condition applies unchanged: if no compromise is acceptable on one criterion, elimination should be applied first; if the loss-aversion assumption does not fit, a symmetrically compensatory method such as interval-valued intuitionistic fuzzy TOPSIS is preferable.

Constraint violation. In every interval 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.

Converting the cost criterion incorrectly. In this extension the benefit–cost distinction is made by complementing the support–rejection interval (swapping lower–upper support with lower–upper rejection). Applying only a min–max reversal instead breaks the interval's internal structure.

Rescaling the weights to sum to 1. The ratio to the reference criterion (w_c/w_r) is computed with the raw weights; if the weights are first normalised to sum to 1, the reference ratio changes and so does the result.

Confusing this card with IV-TODIM. Both use the same four-number cell format (the support–rejection interval), but IVIF-TODIM first applies Xu–Yager vector normalisation and compares using a normalised Hamming distance, whereas IV-TODIM applies no normalisation and uses Mishra's (2020) divergence measure directly. The two engines follow different steps and produce different numbers.

The governing principle is this:

In interval-valued intuitionistic fuzzy TODIM, the support–rejection interval is first scaled by Xu–Yager normalisation; the winning-losing direction comes from the score (accuracy, in a tie), and the magnitude from the Hamming distance.