Fuzzy TOPSIS (Chen and Hwang, 1992)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of four numbers: the lowest plausible value, the lower end of the likely band, the upper end of the likely band, and the highest plausible value. Together these four numbers form a trapezoidal fuzzy number. When an expert says not "one single most likely value" but "every value within this band is equally plausible," the trapezoid is the right shape. When a single most-likely point is given instead, the triangular form (Chen 2000) is sufficient, and a fourth number would be added without purpose. Criterion weights take the same four-number trapezoidal shape.

The method is defined for a single decision-maker. Combining several experts' scores is a preparation step DecisionMind carries out before this stage, not a step of the method itself; in Chen (2000) this combination is done by taking the minimum, mean or maximum of the corresponding corners.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares. Chen-Hwang does not do this, nor does it follow Chen (2000)'s rule of dividing all four components by a single column value; instead it applies crosswise division. For a benefit criterion, the trapezoid's lowest end is divided by the highest end in the column, and the highest end is divided by the lowest end in the column; the two middle points are divided by their own diagonal counterparts. For a cost criterion the direction reverses.

This crosswise division is designed to preserve the ascending order of the four numbers (lowest to highest), and is specific to Chen-Hwang's own paper. In Chen (2000), by contrast, every component is divided by a single column value, which fixes the ideal point at (1,1,1) throughout. The crosswise division used here does not fix the ideal point.

Distance. The ideal and anti-ideal point are built here as in crisp TOPSIS, but over trapezoidal numbers. In each column, the highest- and lowest-ranked alternative is identified using a ranking measure, Chen-Hwang's own generalised mean ranking. This measure serves only to decide which alternative is chosen as ideal or anti-ideal; it does not defuzzify the result.

Each alternative's distance to these two reference points is then computed with Zadeh's supremum-minimum, or sup-min, similarity measure. This looks at how much the membership curves of two trapezoidal numbers overlap: the less the overlap, the greater the distance. This is an entirely different notion of distance from Chen (2000)'s vertex distance, which compares the numbers themselves by taking the square root of the differences between the four corners, whereas sup-min similarity compares the overlapping area of the curves.

Result and defuzzification. The definition is the same here too: the result is computed as the ratio of the distance to the anti-ideal over the sum of the two distances. The result is again a single number between 0 and 1. In this extension, DecisionMind fixes Chen-Hwang's crosswise scale equalisation, the choice of ideal and anti-ideal by generalised mean, and the Zadeh sup-min similarity distance. Weights are taken from outside, in trapezoidal form; the method does not generate weights itself.

The output is a closeness score and a ranking, as in crisp TOPSIS and in Chen (2000); it is read the same way: it is not a percentage, and it cannot be compared with a different analysis. The difference is here: the ideal and anti-ideal point, unlike in Chen (2000), come from the alternative set itself. This makes the rank-reversal risk already present in crisp TOPSIS more pronounced here than in Chen (2000); this risk arises from the reference points shifting when a new alternative is added. Under Chen (2000)'s fixed (1,1,1) ideal, this risk comes only from scale equalisation; here it comes from both scale equalisation and the choice of reference point.

Thus instead of writing:

"Fuzzy TOPSIS (1992) is more reliable than Chen (2000) because it accounts for uncertainty"

the report should read:

"Chen-Hwang (1992) and Chen (2000) are different algorithms; both carry uncertainty through to the final step, and neither is more 'advanced' than the other; which one is used, and why, is stated in the report"

Being dated 1992 does not make the method incomplete; being dated 2000 does not make Chen's version superior. Both solve the same problem with different normalisation and different distance decisions.

Use this extension when experts score criteria "approximately" or verbally, and the nature of this uncertainty is a band. When there is a range that is equally plausible throughout rather than a single most likely point, such as "at least 10, normally between 12 and 16, never more than 20," the trapezoid is the right shape. When the expert gives a single most likely point ("most likely 14 days"), the triangular form is sufficient; see the Chen (2000) card. Compressing a band into a triangle loses information; opening a single point into a trapezoid, that is, setting b = c, adds no artificial distinction but carries an unnecessary component.

The condition for staying with the base method is the same: when criteria are measured. Opening a measured value into a four-component trapezoid, even if all four corners are the same number, is not modelling uncertainty; it is an unnecessary representational burden. If DecisionMind requires a single data type, this embedding is honest but adds no information. If no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold. The exit condition is the same as for crisp TOPSIS.

Confusing Chen-Hwang (1992) with Chen (2000). Both are called "fuzzy TOPSIS", but their scale equalisation and distance measure differ. Using Chen (2000)'s fixed (1,1,1) ideal and vertex distance here, or Chen-Hwang's crosswise equalisation and sup-min similarity in Chen (2000), mixes the steps of two different papers and produces different figures. Which method was run should be stated in the report by name and year.

Defuzzifying the trapezoid before the distance calculation. Reducing the trapezoidal numbers to a single number before the fourth step and then running crisp TOPSIS is not Fuzzy TOPSIS (1992); the overlap information that sup-min similarity measures has already been lost at this point, and score differences are exaggerated (shown in the illustrative example below).

Entering a value that breaks the trapezoid's order. The four components must be ordered from lowest to highest (a ≤ b ≤ c ≤ d). An entry that breaks this order (for example, the lower end of the band being greater than the upper end) renders the crosswise scale equalisation and the generalised mean ranking meaningless.

Opening a measured criterion into a trapezoid. Stretching a measured value such as price or time into a four-cornered trapezoid "to look more thorough" violates the core principle of the Fuzzy data-type card: width must have a genuine source.

The governing principle is this:

The trapezoidal fuzzy number is for situations where the "most likely" information is a band, not a single point; both collapsing the band to a single point (compressing to a triangle) and reducing the whole trapezoid to a single number (early defuzzification) erase the information the method carries.