Pythagorean fuzzy TODIM (Ren, Xu and Gou, 2016)

Three things change; the reference-criterion mechanism and loss aversion's magnifying of losses do not.

Cells. In crisp TODIM every cell is a single number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree; it must satisfy μ² + ν² ≤ 1. Criterion weights come from outside as crisp numbers; uncertainty is carried only in the decision matrix's cells.

Scale equalisation and the reference criterion. Crisp TODIM divides every column by its own sum. PF-TODIM does not do this; in its place comes the swapping of μ and ν on a cost criterion. Crisp TODIM's reference-criterion step then runs exactly as before: the heaviest criterion is chosen as the reference, and the other criteria's weights are ratioed against it.

Comparison and distance. To determine which of two pairs "wins," every pair's score (μ² − ν²) is first calculated; where scores are equal, the pair with the higher accuracy degree (μ² + ν²) is taken as superior. The magnitude of gain or loss is a distance based on the difference between the μ², ν² and squared-hesitancy (π²) components. On the winning side a positive contribution is summed; on the losing side a negative contribution, magnified by the loss-aversion coefficient θ, is summed. This is the same logic as crisp TODIM's third step.

Result. The global value is again a number normalised between 0 and 1; the lowest total dominance takes 0, the highest takes 1. The hesitancy margin, π = √(1 − μ² − ν²), enters the calculation only indirectly, through the distance; there is no separate defuzzification step.

DecisionMind fixes, in this extension, the score-based comparison (μ² − ν², with μ² + ν² used at ties) and the distance definition. θ is fixed internally at the value used in the source paper, θ=2.5; in this extension of DecisionMind, θ is not exposed as a user-facing parameter.

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

The difference is this: this value is a combination of both θ and a difference calculated through the square of every cell's support-rejection pair. When reading the score gap between two alternatives, attention should be paid to which alternative comes out ahead on which criterion, in the support or the rejection direction; two pairs with the same score can carry a different hesitancy margin, and this shows up in the distance calculation.

Thus instead of writing:

"The PF-TODIM result, with θ=2.5, is A2 first"

the report should read:

"Expert scores were entered as support-rejection pairs, ranked with relative weights ratioed to the heaviest criterion and a loss-aversion coefficient of θ=2.5; A2 has the highest global value"

Use this extension where experts give a judgement both strong support and a marked reservation, where these pairs' sum exceeds the intuitionistic fuzzy constraint (that the sum be at most 1), and where the assumption that the decision-maker is more sensitive to losses than to gains fits the nature of the decision.

Where the sum does not already exceed 1, intuitionistic fuzzy TODIM (if-todim) is sufficient; moving to Pythagorean adds no information. Where the sum of squares also exceeds 1, Pythagorean falls short, and q-Rung orthopair TODIM is required. A measured value is not turned directly into a support-rejection pair; it is first converted into a judgement, and only then are μ and ν derived from separate sources. Crisp TODIM's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold. Where the loss-aversion assumption does not fit the nature of the decision, a symmetrically compensatory method such as Pythagorean fuzzy TOPSIS is simpler.

Domain violation. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1.

Writing ν as 1 − μ. In that case the sum always comes to exactly 1 and the hesitancy margin is zeroed; the calculation may look like PF-TODIM, but it actually reverts to crisp TODIM.

Assuming θ is adjustable. In this extension θ is fixed internally at the source paper's value of 2.5, and there is no input field for it in the interface.

Skipping the accuracy degree at a score tie. If two pairs' μ² − ν² scores are equal but their μ² + ν² differ, the one with the higher accuracy degree must be taken as superior; skipping this step leaves which alternative "wins" undetermined.

Skipping the μ-ν swap on a cost criterion. If the calculation is done without this step, the alternative with the highest cost appears to have moved closer to the ideal.

The governing principle is this:

PF-TODIM exists to carry the expert's support and rejection degrees, without clipping them, into loss-aversion logic; writing ν as 1 − μ, skipping the accuracy degree at a score tie, or assuming θ is adjustable, damages this contribution or its correctness.