Fuzzy SPOTIS (Shekhovtsov, Paradowski, Więckowski, Kizielewicz & Sałabun, 2022)

Three things change; the philosophy of "distance to an ideal built from fixed bounds" does not.

Cells. In crisp SPOTIS every cell is a single number. Here every cell is a triangular fuzzy number (l, m, u): the lowest plausible, most likely, and highest plausible value. Criterion weights, too, may be given as triangles. The fixed bounds, each criterion's "best/worst possible" value, are now also triangles; in crisp SPOTIS these bounds were single numbers.

Defuzzification (COA) and scaled distance. Crisp SPOTIS takes the absolute difference between each cell and the fixed ideal directly and divides it by the width of the fixed-bound interval. Here, every triangle, cell, ideal and bounds alike, is first reduced to a single number by the centroid (COA: (l+m+u)/3); distance and scaling are then computed on these reduced numbers using the same formula as crisp SPOTIS. Uncertainty is thus consumed before the distance calculation, right at the start. This is the reverse of Fuzzy TOPSIS, where defuzzification is not carried out even at the end, because the distances are already calculated corner by corner and the closeness score emerges that way.

Weighted sum and ranking. The scaled distances are multiplied by the centroid of the weights and summed; as in crisp SPOTIS, a smaller distance is better, and the ranking runs by increasing distance.

DecisionMind fixes the centroid defuzzification ((l+m+u)/3) for Fuzzy SPOTIS. A critical point: if the fixed bounds (s̃ⱼ^min, s̃ⱼ^max) are not explicitly given in the analysis input (extras), DecisionMind, contrary to SPOTIS's core principle, automatically builds these bounds from that analysis's own alternatives' observed lowest and highest triangle. As Case 1 below shows, this effectively pulls the method towards fuzzy TOPSIS and invalidates SPOTIS's promise of "rank reversal free" for that analysis; giving the fixed bounds explicitly in the analysis input is the user's own responsibility.

The distance value shows, as in crisp SPOTIS, how far an alternative sits from the fixed ideal; a small value is good, a large value is bad.

The difference is here. This distance is now the distance of a triangle that has been reduced to its centroid, and the triangle's width, that is, the size of the uncertainty, does not enter the distance visibly; only the centroid does. If two alternatives' centroids sit close together, but one comes from a narrow triangle (0.58; 0.60; 0.62) and the other from a wide triangle (0.40; 0.60; 0.80), the distance values come out nearly identical, even though the second alternative's input is far more uncertain. Furthermore, whether the fixed bounds truly came from outside or were built from the dataset determines whether the distance should be read "against a fixed ruler" or "against this particular set."

Thus instead of writing:

"According to Fuzzy SPOTIS, A2's score of 0.175 is a fixed and reliable result"

the report should read:

"A2's distance, at 0.175, is the lowest; this result is stable if fixed bounds (s̃ⱼ^min, s̃ⱼ^max) were defined in the analysis input, and if they were not, it has been derived from this set's own observed bounds and may change once a new alternative is added"

This extension is especially suitable when criterion values come from expert judgement or an approximate estimate, and the alternative set is expected to change over time, that is, when a new candidate or a new supplier will be added later. If experts are giving "approximate" scores and the comparison will be repeated over the years, these two conditions together make the strongest case.

If criteria are measured, meaning crisp SPOTIS is needed rather than fuzzy SPOTIS, or if fixed bounds cannot be set beforehand in a defensible way, the base method should be kept; in that case a data-driven method such as fuzzy TOPSIS is more honest. If the table is mixed, meaning some criteria are crisp and some fuzzy, DecisionMind requires a single data type; a measured value is written as (v, v, v), with zero width. The base SPOTIS's exit condition applies here in exactly the same way: if the fixed bounds are unknown or contested, or if no compromise is acceptable on one criterion, the method should not be used.

Violating the value space. Every triangle must satisfy l ≤ m ≤ u with all components ≥ 0; if this order is broken, the cell is invalid.

Not supplying the fixed bounds as input. As described above, this is the most critical mistake specific to this extension. If s̃ⱼ^min / s̃ⱼ^max are not stated in the analysis input, DecisionMind silently uses the dataset's own lowest/highest triangle, and SPOTIS no longer behaves like SPOTIS. This is the fuzzy counterpart of crisp SPOTIS's "most common mistake" on the base card, and here it happens as a silent default rather than a warning.

Changing the defuzzification method. DecisionMind uses the centroid ((l+m+u)/3); a different defuzzification (weighted average, most-likely value) can produce a different ranking, and which one was used should be stated in the report.

Reading the distance value the wrong way round. In the SPOTIS family, a smaller distance is better; reading it the way fuzzy TOPSIS's closeness score is read, "the larger one wins," reverses the order.

Giving weights as crisp numbers while cells stay fuzzy. If a crisp weight is to be used, its three components are written as identical (w, w, w) and this is stated in the report; otherwise the weight's centroid, equal to the crisp number, gives the impression of coming from a different source.

The governing principle is this:

Fuzzy SPOTIS's promise of "rank reversal free" holds only when the fixed bounds have been explicitly and justifiably defined in the analysis input; if the bounds are built automatically from the dataset, the method has silently turned into fuzzy TOPSIS.

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