Interval-valued intuitionistic fuzzy TODIM (Mishra et al., 2020)

Four things change; the logic of magnifying 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 given from outside as crisp numbers and need not sum to 1 in this extension; the method uses only their ratio to one another. DecisionMind does not support group decisions in this extension.

Scale equalisation. In crisp TODIM, cost criteria are turned towards the benefit direction by ratioing them column-wise. There is no separate normalisation step in this extension; cells enter the comparison as raw support-rejection intervals. The cost-benefit distinction is instead achieved during the score comparison below, by reversing the sign of the score.

Divergence and score. In crisp TODIM, the difference between two values is a direct subtraction. Here the two steps split apart. The score is found by subtracting the sum of the rejection bounds from the sum of the support bounds and dividing by two; this determines which alternative wins on that criterion. The magnitude of the divergence is instead computed with a divergence measure proposed by Mishra and colleagues (2020). This measure sums, in a symmetric way, the information gap between two alternatives across each of the four components (lower support, upper support, lower rejection, upper rejection). In crisp TODIM, the difference gives both magnitude and direction in a single step; here, magnitude comes from this divergence measure and direction comes from the score comparison.

Result. The global value is again a single number normalised between 0 and 1; the uncertainty in the support-rejection interval enters the divergence calculation but is resolved into a crisp result, and the number does not remain interval-valued.

DecisionMind fixes, in this extension, the score function and Mishra's (2020) divergence measure. The loss-aversion coefficient θ defaults to 1 and can be changed by the user. The reference criterion is automatic, always the most heavily weighted criterion, and cannot be chosen by the user.

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

The difference is this. The winner-loser direction comes from the score difference, and its magnitude comes from the divergence measure. Because this divergence measure evaluates all four components together, two cells with the same score but a different interval width can produce a different divergence, and therefore a different contribution. The report should therefore state not only the global value but also which criterion has the wider intervals.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy TODIM, P4 is the best alternative"

the report should read:

"The support-rejection intervals were compared with Mishra's (2020) divergence measure, and the gain-loss direction was set by the score difference; P4 has the highest global value, and this ranking is not affected by the choice of θ"

This extension 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." It is equally appropriate when the decision-maker's intuition that losses weigh more heavily than gains fits the nature of the decision.

If support and rejection are each given as a single number (interval width zero), this extra width contributes nothing, and intuitionistic fuzzy TODIM (IF-TODIM) suffices. If the criteria are measured, stay with crisp TODIM. TODIM's exit condition applies exactly as it does there: if no compromise is acceptable on one criterion, screening should be applied first; if the loss-aversion assumption does not fit, a symmetrically compensatory method such as interval-valued intuitionistic fuzzy TOPSIS should be preferred instead.

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.

Reporting the reference criterion as user-selectable. In crisp TODIM the reference criterion can be changed; in this extension it is automatic and is always the most heavily weighted criterion.

Assuming the weights must sum to 1. This extension uses only the weights' ratio to one another; a set of weights that does not sum to 1 is also a valid input.

Confusing this card with IVIF-TODIM. Both use the same four-number cell format (the support-rejection interval), but IV-TODIM compares cells directly with a divergence measure without normalising them first, whereas IVIF-TODIM first applies Xu-Yager vector normalisation and makes the comparison with a normalised Hamming distance. The two engines follow different steps and produce different numbers; using one in place of the other changes the result.

The governing principle is this:

In interval-valued intuitionistic fuzzy TODIM, the winner-loser direction comes from the support-rejection score difference, and the magnitude comes from Mishra's (2020) divergence measure. This measure does not normalise the cells beforehand.