Picture fuzzy TODIM (Wei, 2018)

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

Cells. In crisp TODIM every cell holds a single number. Here every cell holds three degrees: support (yes), abstention and rejection (no); their sum cannot exceed 1. This triple comes from a committee's or a survey's vote distribution on the same judgement. Criterion weights are taken from outside, as crisp numbers.

Scale equalisation and the reference criterion. Crisp TODIM divides every column by its own sum. This extension does not: for a cost criterion, the support and rejection degrees swap places, and the abstention degree stays as it is. From there, crisp TODIM's reference-criterion step runs unchanged: the heaviest criterion is chosen as the reference, and the other criteria's weights are expressed relative to it.

Comparison and distance. Which of two triples "wins" is decided by a score based on the difference between support and rejection (the score lies between 0 and 1; a higher score means support-weighted). Magnitude is a distance based on the average of the differences across the three degrees (support, abstention, rejection). The winning side accumulates a positive contribution, the losing side a negative contribution amplified by the loss-aversion coefficient θ; this is the same logic as crisp TODIM's third step.

Unlike other members of this family, θ is exposed to the user here. In the 2-tuple linguistic, plithogenic and Pythagorean fuzzy TODIM extensions, θ is fixed internally by DecisionMind and carries no input field in the interface. In this extension, by contrast, θ is a mandatory user input: the user must enter a value between 0.01 and 100, and there is no default.

Result. The global value is again a number normalised between 0 and 1: the lowest total dominance takes 0, the highest takes 1. How much the abstention degree leaves the judgement "open" is carried only indirectly, through the fact that all three components of the triple enter the distance calculation together.

DecisionMind fixes, for this extension, the score definition and the distance formula. Weights come from outside; the method does not generate weights.

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. Because θ is a mandatory input here, which value was chosen is an integral part of the report.

The difference is this: this value is computed via a score based on the support-minus-rejection difference of an opinion distribution; the abstention share does not enter the score directly, it leaves only an indirect trace in the distance calculation. How robust the score gap between two alternatives is should also be read against the question of which direction the abstention share might shift towards, for alternatives where that share is large, once further information becomes available.

Thus instead of writing:

"With picture fuzzy TODIM, and θ = 2.5, A1 comes first"

the report should read:

"The committee's votes were entered as support-abstention-rejection triples; the ranking uses relative weights expressed against the heaviest criterion and a loss-aversion coefficient of θ = 2.5; A1 has the highest global value, and this depends on the choice of θ"

Use this extension when criterion evaluations come from a committee's, survey's or panel's yes, abstain, no vote distribution on the same judgement, and the assumption that the decision-maker is more sensitive to losses than to equivalent gains fits the nature of the decision. For this, the three shares must be separately countable; deriving the other two degrees from a single support percentage erases this extension's sole contribution.

There is no need to expand a measured criterion into a picture fuzzy triple; DecisionMind requires a single data type. Crisp TODIM's exit condition also applies here: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold. If the loss-aversion assumption does not fit the nature of the decision, a symmetrically compensatory method such as picture fuzzy MOORA is simpler.

Violating the value space. In every cell, the sum of support, abstention and rejection must not exceed 1; this check must be made before the calculation begins.

An unresolved tie in choosing the reference criterion. The reference criterion is the one carrying the heaviest weight. If more than one criterion carries the same highest weight, which one counts as the reference must be stated clearly and consistently in the report; otherwise the relative weights, and hence the result, become indeterminate.

Forgetting to enter θ, or choosing it arbitrarily. θ is mandatory in this extension and has no default; if θ is chosen small (below 1), losses are shown as smaller than gains, which amounts to a behavioural assumption at odds with prospect theory's own idea of loss aversion. Which value of θ was chosen, and why, must be stated in the report.

Skipping the complement for a cost criterion. If the support and rejection degrees are not swapped before the calculation, the highest-cost alternative appears to have been pulled towards the ideal.

The governing principle is this:

Picture fuzzy TODIM carries the abstention share of a committee's or survey's vote distribution while also running the loss-aversion logic; θ is a mandatory input in this extension, and which value it was used with is an integral part of the report.