Z-number TOPSIS (Gardashova, 2019)

Three things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is a Z-number: Z = (A, B). A is a triangular fuzzy number stating the criterion value. B is a separate triangular fuzzy number stating how far that value can be trusted. Criterion weights stay crisp numbers; the method does not generate weights, it takes them from outside.

Scale equalisation. DecisionMind reduces the Z-number to a single crisp number before normalisation. This reduction follows the route proposed by Kang and colleagues (2012). A's membership curve is scaled by a coefficient drawn from B's centroid. The centroid of this scaled curve is then taken. Classical TOPSIS's vector normalisation is applied to the crisp number obtained this way: every column is divided by the square root of the sum of the squares of the values in that column. This is a fundamental difference from the Fuzzy TOPSIS (Chen 2000) extension, which carries the uncertainty all the way to the last step. There the uncertainty stays triangular and is consumed at the distance calculation. Here the uncertainty collapses into a single number as early as the first step.

This reduction has a verified and important consequence. In the Kang transformation, B's contribution only scales the height of A's membership curve, not the curve's horizontal position, its centroid. Because this constant multiplier appears in both the numerator and denominator of the centroid ratio, it cancels out. The crisp value obtained is therefore equal only to A's centroid, and B's numerical value does not affect the result at all. This has been verified by running DecisionMind's Z-TOPSIS engine: lowering B from (0.70; 0.80; 0.90) to (0.01; 0.02; 0.03) in the example below leaves all three closeness scores exactly the same.

Distance and closeness score. Nothing changes at these steps. On the crisp numbers obtained, classical TOPSIS's ideal alternative, anti-ideal alternative, Euclidean distance and closeness score all run exactly as before.

DecisionMind holds the Kang transformation and the centroid defuzzification, followed by classical TOPSIS's vector normalisation and Euclidean distance, fixed in this extension. Weights are taken from outside as crisp numbers.

The output is a closeness score and a rank, as in crisp TOPSIS, and reads the same way. It is not a percentage, it is not compared with a different analysis, and the ideal and anti-ideal points shift when the alternative set changes.

The difference is here. The purpose of the Z-number data structure is to carry trust in the source into the decision. But in the current DecisionMind engine, this trust is eliminated at the defuzzification step and does not reach the result. For this reason, the report must not state that "reliability has been taken into account."

Thus instead of writing:

"Because a Z-number was used, the reliability of the source is reflected in the result"

the report should read:

"Criterion values were entered as Z-numbers, and the calculation has run on the value's centroid; the degree of reliability is not reflected in the result after defuzzification in the current engine"

If your data is genuinely Z-number in structure, that is, if every criterion value is recorded together with the source's reliability, this extension is the formally correct choice. But if the decision-maker expects that "a reliability difference will affect the ranking," this expectation is not met by the current engine. This limit should be stated before data collection. Converting a measured value into a Z-number generates uncertainty; the base method should be kept for that case.

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

Assuming reliability (B) will change the ranking. In the current engine, B drops out of the calculation entirely after defuzzification. If two experts with sources of differing reliability give the same A value, the current engine treats the two as identical. This is not a software fault; it is a mathematical consequence of the Kang transformation, and it should be stated plainly in the report.

Writing the same B into every cell and using a Z-number. If B is the same everywhere, it already carries no discriminating information. Even where the current engine carries none of it through, this wastes the effort of collecting the data.

Writing a low B out of doubt for a measured value. Writing "medium reliability" for a value that has actually been measured adds an uncertainty to the measurement that does not belong there.

The governing principle is this:

Z-number TOPSIS exists to record the criterion value and the trust placed in that value separately. But in the current engine, trust makes no contribution to the result after defuzzification; this limit must be told to the decision-maker before data collection.