Brief specification of FLMICMAC

The output of traditional MICMAC consists of four rankings (direct and indirect influence/dependence rankings) and two influence/dependence 2D charts to categorize the variables. Both the rankings and the charts only show relative information, that is, they give information about how is a variable respect to the others, but there is no measure that states whether the influence/dependence of a variable is strong or weak in absolute terms. Furthermore, although the input data are based on a very basic scale, the values from which we build the rankings are just numbers without a meaning. Both aspects are improved in our novel method, FLMICMAC.

As done in previous related works, we will employ a linguistic variable to measure the degree of relation between the variables. The possible values of such variable are linguistic labels over which a strict ordering must exist so all the values are comparable. The entries of our input MDI matrix will now be linguistic labels as in the former works. However, differently from them, we associate to every linguistic label a fuzzy number, which is the mathematical structure that enables calculations with the linguistic labels and represents a number with uncertainty. To be precise, we will make use of Triangular Fuzzy Numbers (TFNs).

Steps of the method

Step 1: definition of a set of linguistic labels for the MDI. We will refer to them as L = {l1,..., lN}, assuming the label None is not included in L because it represents no influence at all, in a crisp sense. Therefore there are N+1 terms. Following the common practice in Computing with Words, N must be an odd natural number (usually 3 or 5) so there is a central label dividing the set in two parts of the same number of labels each.

Step 2: definition of the TFNs for each input label. Ideally, the user indicates only the number of input labels he/she wants to use, and they are given a default representation as TFNs, without requiring further information. However, although the focus of the method is on the linguistic input and output, it is possible to customize the TFNs that will be used to represent each input label to match the user's needs.

Step 3: computation of direct influence/dependence. The row and column sum relies on the component-by-component sum of TFNs of the corresponding labels of the MDI.

Step 4: computation of the TFNs supporting reference output labels. When computing the (fuzzy) direct and indirect relations, the system is aimed at providing a linguistic value as a result, not a TFN, even when all the underlying computations involve TFNs. Therefore it is necessary to first calculate a set of output TFNs that will support the linguistic labels employed for the output of the direct and indirect relations. In other words, we need to determine what is considered Weak, what is Strong and what Very strong when giving the output of the fuzzy direct influences/dependences, and the same (slightly more complicated) for the output of the fuzzy indirect influences/dependences. The values defining such output TFNs depend on the entries of the linguistic MDI. This step is explained with more detail in our paper A new fuzzy linguistic approach to qualitative cross impact analysis, currently under review in Applied Soft Computing.

Step 5: fuzzy direct method. Once the output TFNs for the output reference labels of direct influence/dependence have been calculated, the fuzzy sums of the the values of each row and values of each column are computed. As a result, two lists of TFNs are obtained. Each TFN is assigned the label of the closest output reference TFN for the direct method, which is an absolute linguistic measure of the influence or dependence of the variable. Apart from this, the influence and dependence TFNs are sorted according to their defuzzified value to build the direct influence/dependence rankings. The pairs of defuzzified values of each variable are depicted in a 2D chart to uncover the role of every variable.

Step 6: fuzzy indirect method. Again we assume the output reference labels of the indirect method, which are of course different from those of the direct method, have been calculated. Then, sucessive fuzzy matrix multiplication of the MDI by itself is carried out based on TFN sums and products, until the rankings of two consecutive iterations match. Then the same procedure of step 5 is applied to the final matrix to compute the indirect rankings and 2D chart.