Fuzzy DEMATEL

Cells. In crisp DEMATEL every cell, how much one criterion influences another, is a single number. Here every cell is a triangular fuzzy number; the expert gives the influence with verbal terms such as "no influence", "low", "medium", "high", and each term is converted into a triangle using a pre-declared scale. Where there is more than one expert, the matrices are combined with an arithmetic mean; DecisionMind takes this average at the first step.

Scale equalisation / defuzzification. Crisp DEMATEL normalises directly. Fuzzy DEMATEL first converts the fuzzy matrix into a single crisp matrix: this is done with the CFCS method (a weighted average that uses the left and right normalised values, drawing on both the triangle's ends and its middle together). This defuzzification happens right at the start of the calculation, before normalisation. Unlike Fuzzy MOORA's defuzzification, which averages at the last step, here the uncertainty descends to a single number in the very first step, and the remaining four steps (normalisation, total-relation matrix, D/R, weight) proceed exactly as in crisp DEMATEL.

Total relation and weight. From the crisp matrix produced by CFCS, the total-relation matrix is built, and the influence given (D) and influence received (R) are computed; total interaction (D+R) and net role (D−R) are derived from these. DecisionMind computes this weight as the normalised form of D+R (prominence). This is a different weighting rule from the ‖(D+R, D−R)‖ form on the crisp DEMATEL card, and the two methods can give a different weight ranking even applied to the same data.

Result and defuzzification. The final output is not a ranking but two numbers per criterion (D+R, D−R) and a weight vector derived from them, exactly as in crisp DEMATEL. The only difference is that the input (the influence matrix) is fuzzy and is defuzzified at the very start with CFCS.

DecisionMind fixes, in classical Fuzzy DEMATEL, the CFCS defuzzification, the use of normalised prominence (D+R's share of the total) as the weight, and keeps this weight deliberately different from crisp DEMATEL's ‖(D+R,D−R)‖ form.

The output is, as in crisp DEMATEL, the total interaction (D+R) and net role (D−R) per criterion; it is not a ranking but a structural map, and is read the same way. D+R shows centrality, D−R shows the cause/effect direction.

The difference is this. These two numbers now rest on a fuzzy input that has passed through the expert's verbal judgement and been reduced to a single crisp matrix (CFCS). The weight ranking, that is, which criterion is most central, is generally about as robust to small changes in the influence scores as crisp DEMATEL is. But whether a criterion is a cause or an effect (the sign of D−R) can change with a single expert scoring one notch differently.

Thus instead of writing:

"Fuzzy DEMATEL found criterion C2 to be both the most important and definitively in the cause group"

the report should read:

"C2 has the highest weight (total interaction); this ranking is robust to small changes in the influence scores. But whether some criteria are causes or effects can change direction if a single expert changes their score by one notch, and this should be checked separately"

This extension is used when cross-criterion influence is not given by experts as a measured quantity but as a verbal or approximate judgement. For example, it is suitable when the answer to "how much does this factor affect that one" comes back as "high", "medium" and the like, or when several experts' differing scores are wanted to be recorded separately rather than averaged beforehand. If the influence scores are already numerical and crisp (for instance, correlation coefficients computed from past data), fuzzifying them produces needless uncertainty; crisp DEMATEL is sufficient. The exit condition is the same as for crisp DEMATEL: the method does not rank alternatives, it compares criteria; if a final ranking is needed, the weights it produces should be fed as input into a ranking method (such as Fuzzy TOPSIS or Fuzzy VIKOR).

Not declaring the linguistic scale, or letting it vary from expert to expert. What the term "high" corresponds to is fixed before the analysis, written in the report, and applied identically to all experts; otherwise CFCS produces a different crisp matrix.

Entering a fuzzy value other than zero on the diagonal cells. A factor influencing itself is undefined; the diagonal should stay fixed at the "no influence" triangle (with a width close to zero).

Skipping CFCS and running crisp DEMATEL directly by averaging the triangles (taking the centre value). This skips CFCS's left–right normalisation steps (the weighted transformation that uses not just the triangle's middle but also its ends); this is not the defuzzification rule DecisionMind fixes, and the result can lead to a different crisp matrix and hence different D/R values.

Reading the robustness of the weight ranking as meaning the cause/effect distinction is equally robust. In the illustrative example below, the weight ranking is unaffected by a one-notch score change, while whether a criterion is a cause or an effect can reverse with a single-notch change; these are two separate robustness questions.

The governing principle is this:

In Fuzzy DEMATEL, defuzzification (CFCS) happens at the very start, and the rest of the calculation is identical to crisp DEMATEL; consistency in the linguistic scale and the diagonal is therefore a more critical source of accuracy than the weight itself.