Fuzzy TOPSIS (Chen 2000) (Chen, 2000)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Experts give their score in words ("high," "medium"); every word is converted into a triangle from a pre-declared scale. Chen's scale has seven terms and sits between 0 and 1: "very low" (0; 0; 0.1), "low" (0; 0.1; 0.3), "medium-low" (0.1; 0.3; 0.5), "medium" (0.3; 0.5; 0.7), "medium-high" (0.5; 0.7; 0.9), "high" (0.7; 0.9; 1), "very high" (0.9; 1; 1). Criterion weights are triangular in the same way; the method can combine several experts' scores and weights, and its founding article was itself written for a group decision.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares. Chen does not use this method. For a benefit criterion, every triangle's three components are divided by the column's largest upper value; for a cost criterion, the column's smallest lower value is instead ratioed against the components. This division ensures the ideal alternative comes out exactly (1; 1; 1) on every criterion, and the anti-ideal exactly (0; 0; 0); the ideal point is fixed, unlike in crisp TOPSIS, where it is built from the alternatives. If the fuzzy numbers already come from verbal scores on the 0-1 scale, the column's upper value is already 1, and this step changes nothing.

Distance. The distance between two triangular numbers is the square root of one-third of the sum of the squared differences of the three components; this is called vertex distance. Every alternative's distance to the ideal and to the anti-ideal is the sum of these vertex distances across the criteria. Crisp TOPSIS used the Euclidean distance of single numbers; here it is the vertex distance of triangles.

Closeness score. The definition here is the same as before; the result is calculated 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, because the distances have already collapsed to single numbers. The method does not defuzzify the uncertainty; it consumes it in the distance calculation.

DecisionMind fixes Chen's scale equalisation and vertex distance for classical Fuzzy TOPSIS. Weights are taken from outside as triangles; the method does not generate weights.

The output is a closeness score and a ranking, as in crisp TOPSIS, and is read the same way: it is not a percentage, it cannot be compared with a different analysis, and even though the ideal point is fixed, changing the alternative set changes the scale equalisation on cost criteria and can shift the ranking.

The difference is here: beneath the score now lies a three-layered uncertainty that the score conceals. Whether the gap between two alternatives is meaningful depends on the width of the inputs. In crisp TOPSIS, a gap between 0.62 and 0.53 reads as "sensitive to the weights"; here it should also read as "may close if experts give one score a term lower." The report should therefore show whether the ranking changes if any score or weight shifts by one term.

Thus instead of writing:

"Because Fuzzy TOPSIS accounts for uncertainty, the result is more reliable"

the report should read:

"Because expert scores were verbal, uncertainty has been carried through the calculation; A1 leads with 0.62, A2 follows with 0.53, and the gap is sensitive to a single term's change in score"

Carrying uncertainty does not make the result more accurate; it prevents a false precision.

Use this when experts score criteria verbally or approximately, and reducing these scores to a single number would create an artificial precision. Typical situations: unmeasurable criteria such as "service quality" in supplier evaluation, "communication skills" in personnel selection, or "technical risk" in project selection; and board decisions where several experts' differing scores must be combined without collapsing them into an average.

The base method should be kept when criteria are measured. Converting a measured price or time into a triangle is not modelling uncertainty but manufacturing it. If the table is mixed, meaning some criteria are measured and some verbal, DecisionMind requires a single data type; a measured criterion is also written as a triangle, and this triangle's three components are the same number: no uncertainty. This embedding is honest but adds no information.

The exit condition is the same as for crisp TOPSIS: if no compromise is acceptable on one criterion, this extension too is compensatory and will not screen out anything below a threshold.

Defuzzifying first, then running crisp TOPSIS. Reducing the triangles to a single number at the outset and applying the crisp method is not Fuzzy TOPSIS; uncertainty is erased at the first step and score gaps are exaggerated. In the illustrative example below, this route gives A1 0.75 and A2 0.25; the fuzzy method gives 0.62 and 0.53. The ranking is the same, but the first route shows a gap between the two alternatives that does not exist.

Not declaring the scale, or changing it from expert to expert. Five-term and seven-term scales convert the same "good" score into different triangles. The scale is chosen before the analysis, stated in the report, and applied identically to every expert.

Using crisp TOPSIS's scale equalisation. Dividing by the square root of the sum of squares moves the ideal point away from (1; 1; 1) and makes Chen's fourth step meaningless. This is the first place to check when software packages give different results.

Giving weights as crisp numbers while scores stay fuzzy. The method expects weights as triangles too; if a crisp weight is to be used, its three components are written as identical and this is stated in the report.

The governing principle is this:

Fuzzy TOPSIS exists to carry the approximation in expert judgement through to the very last step; any application that renders the input precise from the outset, or leaves the scale undeclared, destroys the extension's one contribution.