Hesitant Fuzzy CODAS

Four things change. The two-distance comparison logic does not.

Cells. In crisp CODAS every cell is a single number. Here every cell is a set that holds more than one plausible value on the same criterion together. In the same column, different alternatives' sets can have different lengths. DecisionMind completes every column to the length of the longest set in that column. The gap is filled using a coefficient η. The default η is zero, filling the gap with the set's smallest value; this is called cautious completion. Criterion weights remain crisp numbers. The method does not directly support group decisions.

Scale equalisation. Crisp CODAS divides every cell by the ratio of the column's largest or smallest value. There is no such division here, because the set's values already lie between 0 and 1. For a cost criterion, every member of the set is converted to its complement (one minus the member). Every member of the set is then shrunk by an exponential operation using the criterion's weight (Xia and Xu, 2011). This operation is a shrinking, not the division used in crisp CODAS, and the result remains a set between 0 and 1.

Distance. There is a point of difference to note here. In crisp CODAS the negative-ideal is built artificially from the column's worst value. In Hesitant CODAS, by contrast, the negative-ideal is the observed set itself of whichever alternative has the lowest average on that criterion; it is not artificially assembled from the ends of the column. The Euclidean and Hamming distances are computed from the difference between the average values of the weighted sets. That is, every set is reduced to its own average immediately before entering the distance calculation. This makes the set's internal spread, the gap between its lowest and highest member, invisible once the distance step has passed.

Result and defuzzification. The assessment score is built with the same pairwise comparison rule as crisp CODAS (threshold τ, Euclidean first, Hamming if that is not enough). There is no separate defuzzification step, because the sets are already reduced to single numbers, by averaging, within the distance calculation itself.

DecisionMind holds fixed, in classical Hesitant CODAS, cautious completion (η=0) and the threshold value (τ=0.02).

The assessment score is read exactly as in crisp CODAS. It is a relative measure of position, not a percentage, and it cannot be compared with a different analysis. The difference is this: because the score comes from the average of the set, the set's width, that is, the disagreement among experts, is not reflected directly in the score. Two alternatives can reach the same average with sets of different widths and end up with the same score.

Thus instead of writing:

"The Hesitant CODAS score also shows how uncertainly the alternatives were assessed"

the report should read:

"The score has been computed from the average of the sets; two alternatives can share the same average while their sets differ in width, and this width should be reported separately, outside the score"

Use this extension when more than one plausible and defensible value exists for a criterion, for instance when different experts or different measurement rounds give separate scores for the same criterion. Measured criteria should not be carried into this extension. If the internal width of a set matters to the decision itself, that is, if which alternative was assessed more consistently needs to show up in the result, Hesitant TOPSIS may be more suitable. The reason is this: that method carries the distance between sets in ordered form all the way to the end, without reducing it to an average early on. If the matrix must be of a single type, a measured value is written as a single-element set. The base CODAS's exit condition applies here in exactly the same way.

Not declaring the length-equalisation coefficient (η). η=0 and η=1 can give the same table a different negative-ideal and a different score. The value used must be stated in the report.

Assuming the set's width is not lost. Because the score comes from the average of the set, width enters the calculation invisibly. On a criterion where width matters, this information should be reported separately.

Marking criterion direction wrongly. Skipping the complement step on a cost criterion attaches the negative-ideal to the wrong alternative, and the ranking breaks. In the illustrative example below, this mistake swaps the first- and second-ranked alternatives.

The "more advanced" fallacy. Where no genuine source supports more than one value, opening a single crisp number arbitrarily into several nearby values adds no information.

The governing principle is this:

A set must genuinely come from more than one defensible source for the same criterion-alternative pair. Hesitant CODAS compares this set by reducing it to its average; the set's internal width is lost as separate information, and the report must show this.