Hesitant TODIM (Zhang and Xu, 2016)

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 is a set: it holds several plausible values for the same criterion together, for instance {0.4; 0.7}. Sets belonging to different alternatives in the same column can be of different length. DecisionMind completes every column to the length of the longest set in that column; this is called length equalisation. The missing place is filled in one of two ways: the cautious mode (the default) repeats the set's smallest value, the assertive mode repeats its largest value. Weights are crisp. This extension supports a group decision: several decision-makers' sets are merged and reduced to a single table in DecisionMind's input layer.

Scale equalisation. In crisp TODIM, cost criteria are turned to the benefit direction by ratioing on a column basis. Here the same purpose is served by the set's complement: every set in a cost criterion is complemented so that every element becomes "1 minus the element." This is not a division but a reversal of direction.

Measurement and distance. In crisp TODIM the difference between two values is a direct subtraction. Here the process splits into two steps. First, every set is reduced to a single number (Z_δ) with a δ measurement function; this number determines which alternative wins on that criterion. At δ = 1 this is the set's arithmetic mean; as δ shrinks it weights the set's lower end more heavily, and as it grows it weights the upper end more heavily. Then the distance between two sets is computed with the Euclidean distance between the length-equalised sets' sorted elements. In crisp TODIM the difference gives both magnitude and direction in a single step; here direction is determined by the Z_δ comparison, and magnitude by the distance.

Result. The global value is once again a single number, normalised between 0 and 1; the hesitation in the set enters the distance calculation, but it settles into a single figure in the end, and the number itself does not remain hesitant.

DecisionMind takes the cautious completion (the default) and the δ = 1 measurement function as defaults in this extension; both can be changed by the user. The loss-aversion coefficient θ also defaults to 1 and can be changed. The reference criterion, unlike in crisp TODIM, is automatic here: it is always the most heavily weighted criterion and cannot be chosen by the user.

The global value is read exactly as in crisp TODIM: the lowest total superiority 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 Z_δ measurement function, its magnitude from the distance. The choice of length-equalisation mode (cautious or assertive) and of δ can affect the result; the same table can give a different distance, and hence a different global value, in a different mode. The report must therefore state which mode and which δ were used.

Thus instead of writing:

"According to HF-TODIM, R2 is the best alternative"

the report should read:

"With the cautious completion and the δ = 1 measurement function, at a loss-aversion coefficient of θ = 1, R2 has the highest global value; this ranking is preserved between θ = 0.5 and θ = 2"

This extension is suitable when there is more than one plausible and defensible value for a criterion, for instance when several experts have each given an independent score and these are to be kept together rather than reduced to an average. It is equally suitable when the intuition that the decision-maker is more sensitive to losses than to gains, TODIM's basic assumption, fits the nature of the decision.

If there is a single measured value, a single number is written into the cell and no set is built. If the criteria are measured, stay with crisp TODIM. TODIM's exit condition applies unchanged: if no compromise is accepted on a criterion, screening is applied first; if the loss-aversion assumption does not fit, a symmetric compensatory method such as hesitant TOPSIS should be preferred instead.

Failing to declare the length-equalisation mode. The cautious mode and the assertive mode can give the same table a different distance, and hence a different global value. The mode used must be stated in the report.

Choosing δ without justification. It must be stated that the Z_δ measurement function has shifted, with a δ different from the arithmetic mean (δ=1), to another member of Zhang and Xu's (2014) family of precursor measures; a different δ can produce a different ranking.

Reporting the reference criterion as if the user could choose it. While the reference criterion can be changed in crisp TODIM, in this extension it is automatic and is always the most heavily weighted criterion.

Ignoring rank reversal. When the alternative set changes (an alternative is added or removed), the global values are rescaled; this is a known limitation of the TODIM family and must be stated in the report.

The governing principle is this:

In Hesitant TODIM, the winner-loser direction comes from the Z_δ measurement function, and the magnitude from the distance between the length-equalised sets; if the length-equalisation mode and δ are not stated in the report, the same data can produce a different ranking.