Fuzzy CODAS (Keshavarz Ghorabaee et al., 2017)

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

Cells. In crisp CODAS every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights are triangular in the same way. The method does not directly support group decisions. Every cell must already be entered as a single, already-aggregated triangle.

Scale equalisation. Crisp CODAS divides every cell by the column's largest value (benefit) or by the ratio of its smallest value (cost). Here the same division is applied separately to the triangle's three components. For a benefit criterion, each component is divided by the largest upper value in the column (Mγ). For a cost criterion, the smallest lower value in the column (mα) is divided by each component. The division works on the same logic as crisp CODAS, only repeated across three components.

Distance. In crisp CODAS, two distances to the negative-ideal point are computed: Euclidean and Taxicab. These distances are computed between single numbers. Here, every weighted triangle is first reduced to its own centre weight. The centre weight is the average of the three components. The negative-ideal is then built from the smallest of these reduced values. The Euclidean and Taxicab distances are computed between these reduced numbers. In other words, the uncertainty stays fuzzy through normalisation and weighting. It descends to a single number only immediately before the distance step.

Assessment score and defuzzification. The pairwise comparison and the total score (h_i) work by the same rule as crisp CODAS. If the Euclidean difference exceeds the threshold (τ), the ranking is made directly from this difference; if not, the Taxicab difference takes over. The result is already a single number. There is no separate defuzzification step besides this, because defuzzification has already been done in the previous step, through the centre weight.

DecisionMind fixes, for classical Fuzzy CODAS, the centre-weight defuzzification (Vi = (Viα+Viβ+Viγ)/3) and the threshold value (τ=0.02). Other defuzzification rules are also supported (the most likely value, the bisector, the alpha-weighted rule), and the rule chosen is stated in the report.

The assessment score is read in the same way as in crisp CODAS: it is a relative positional measure, not a percentage, and it is not compared with a different analysis. The difference is this: because normalisation and weighting proceed in triangular form, the size of the score is slightly sensitive to the choice of defuzzification rule. If the most likely value or the bisector is used instead of the centre weight, the score shifts by a few percentage points. Whether this shift changes the ranking, however, must be tested separately.

Thus instead of writing:

"Because Fuzzy CODAS takes uncertainty into account, the result is more reliable than crisp CODAS"

the report should read:

"Because criterion scores are verbal or approximate, the uncertainty has been carried through normalisation and weighting and then reduced to a single number by the centre weight; the ranking and the margin should be reported against this defuzzification rule"

Use this extension when criterion scores are given approximately or verbally through expert judgement and the need to distinguish close rivals through two different distance measures, CODAS's own rationale, still applies. Measured criteria should not be carried into this extension. If the matrix must be of a single type, a measured value is written with all three components equal to the same triangle, and its width stays at zero. Base CODAS's exit condition applies exactly here too. Where no compromise is acceptable on one criterion, a compensatory method is already unsuitable.

Defuzzifying first and then running crisp CODAS. Reducing the triangles to their centre weight from the outset and running crisp CODAS is not Fuzzy CODAS. This approach skips the fuzziness carried in normalisation and weighting. In the illustrative example below, this route gives A1 −0.4115, A2 0.4068 and A3 0.0047. The correct Fuzzy CODAS gives −0.3769, 0.3733 and 0.0036. The ranking is the same, but the margins inflate somewhat.

Not declaring the defuzzification rule. The centre weight, the most likely value, the bisector and the alpha-weighted rule produce different scores from the same triangle. DecisionMind supporting all four rules does not mean the choice can be made arbitrarily. The choice must be stated in the report.

Using crisp CODAS's threshold value (τ) unquestioned. The threshold operates on the centre-weighted values after normalisation. The width of the fuzzy input changes when the threshold comes into play. The assumption τ=0.02 should therefore be tested separately against the scale of the data.

Giving crisp weights alongside fuzzy scores. The method expects the weights to be triangular too. If a crisp weight is to be given, all three components should be written the same, and this should be stated in the report.

The governing principle is this:

Fuzzy CODAS exists to carry the approximation in criterion judgement through normalisation and weighting; any application that defuzzifies the input from the outset or conceals the defuzzification rule damages not the dual-distance comparison itself but only its apparent precision.