Intuitionistic fuzzy TODIM (Krohling, Pacheco and Siviero, 2013)

Four things change; the reference-criterion mechanism and the magnification of the loss side by θ do not.

Cells. In crisp TODIM every cell is a single number. Here every cell is a pair: a degree of support μ and a degree of rejection ν, whose sum does not exceed 1. The hesitancy margin is not written into the cell separately; it is derived as what remains after support and rejection. Weights are crisp numbers, not intuitionistic. This extension supports group decisions: several decision-makers' pairs are combined into a single matrix with the intuitionistic fuzzy weighted average (IFWA), weighted by each decision-maker's own weight. With a single decision-maker this step is an identity transformation.

Scale equalisation. In crisp TODIM, cost criteria are turned to the benefit direction by column-wise scaling. Here the same aim is achieved through the intuitionistic fuzzy complement: every (μ, ν) pair in a cost criterion swaps places with (ν, μ). This is a reversal of direction, not a numerical division; as a result, every criterion becomes comparable in the same, benefit, direction.

Distance and score. In crisp TODIM the difference between two values is a direct subtraction. Here this splits into two steps. First, the Szmidt-Kacprzyk Euclidean distance is calculated from the difference between two pairs' support, rejection and hesitancy components; this distance is symmetric and does not say which alternative comes out ahead. The winner-loser direction is determined by the Chen-Tan score function: whichever alternative retains the higher value once its rejection degree is subtracted from its support degree is deemed to "win" on that criterion. In crisp TODIM the difference gives both magnitude and direction in one step; here the magnitude comes from the distance and the direction from the score comparison, separately.

Result. The global value is again a single number normalised between 0 and 1; the uncertainty in the support-rejection pair enters the distance calculation but settles by the end, so the number no longer remains intuitionistic fuzzy.

DecisionMind takes the Szmidt-Kacprzyk Euclidean distance as the default distance function and Chen-Tan's support-minus-rejection difference as the default score function in this extension; both can be changed by the user. θ defaults to 1.

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

The difference is this. The winner-loser direction comes from the score difference, not the distance. So even where two alternatives' support and rejection degrees are very close, if their support-minus-rejection differences point in opposite orders, a small distance can still carry a major directional decision. The report should therefore show not only the global value, but also which alternative comes out ahead by score on which criterion. Two pairs looking "close to each other" does not mean the gain-loss direction is uncertain too.

Thus instead of writing:

"According to IF-TODIM, A1 is the best alternative"

the report should read:

"Support and rejection degrees have been compared with the Szmidt-Kacprzyk distance, and the gain-loss direction has been determined with the Chen-Tan score; A1 holds the highest global value, and this order is preserved between θ=0.5 and θ=5"

Use this extension when a criterion comes not from a measurement but from a judgement, and the expert's opposing view, that is, rejection as much as support, carries its own information. It is equally suitable when the intuition that the decision-maker is more sensitive to losses than to gains fits the nature of the decision.

If the expert only states "how suitable" and there is no separate source for the opposing view, there is no need to write ν as 1 − μ and move to this extension; that carries the same information as crisp or fuzzy TODIM. Stay with crisp TODIM where criteria are measured. DecisionMind requires a single data type if the table is mixed; a measured criterion is written as the pair (t, 0). Here support is t, rejection 0, and hesitancy is (1−t); this embedding is honest but adds no information. TODIM's exit condition applies unchanged: if no compromise is acceptable on one criterion, elimination is applied first; if the loss-aversion assumption does not fit, a symmetrically compensatory method such as intuitionistic fuzzy TOPSIS should be preferred instead.

Forgetting the complement on a cost criterion. If the (μ, ν) pair is not replaced with (ν, μ) on a cost criterion, the gain-loss direction reverses and the ranking becomes meaningless; this is the manifest's most critical warning.

Mistaking the distance for the gain-loss direction. The Szmidt-Kacprzyk distance is symmetric; which alternative wins is determined by the Chen-Tan score. Using the distance directly as if it were a signed difference discards the source of loss aversion, the score comparison.

Choosing θ too small. A value such as θ < 0.1 magnifies the loss side disproportionately; a small ζ difference turns into an extreme ξ difference. Krohling and colleagues (2013) used θ=1 and noted that θ ∈ [1, 2.5] is typical in the TODIM literature.

Choosing the reference criterion as anything other than the highest-weighted criterion. The reference criterion is the one with the highest weight; choosing a different criterion as reference can distort the relative weights, and with them the ranking.

The governing principle is this:

In IF-TODIM, distance and direction come from two separate calculations; the distance gives magnitude, the Chen-Tan score gives direction, and the complement on a cost criterion is required for this direction to be marked correctly.