# Multicriteria Decision Making in Tourism Industry Based on Visualization of Aggregation Operators
## 4 Aggregation-Operator Synthesis Using 3D Graphics
### Aggregation-Operator Synthesis Based on the 3D Balance Model  
The suggested 3D balance model can be applied for different aggregation-operator synthesis including the Choquet integral. As mentioned earlier, the Choquet integral is well suited for multicriteria decision making because it allows one to model different interdependencies between criteria. This is the justification of our choice of it as the basis of aggregation-operator synthesis through visualization involving the 3D balance model. As well as that, in order to use the Choquet integral, it is necessary to identify a fuzzy measure according to the decision maker's preferences. Such identification is troubled by the exponential difficulty growth of a fuzzy measure definition for each subset of criteria. Definition of all \(2^{H}\) fuzzy measure coefficients is a rather hard or even an impossible task for the decision maker. Notice that even using three criteria, we have to define a fuzzy measure with \(2^{3}=8\) coefficients. This is why [PERSON] introduced the concept of k-additivity that helps to simplify the fuzzy measure by excluding criteria dependencies of more than \(k\) criteria. In accordance with the abovementioned, we have chosen the 2-order Choquet integral that suits most practical cases [29]. In order to define a fuzzy measure, we have chosen the dispersion minimization method [30] among other methods [31] as, in contrast to other methods, it provides a uniqueness of the resulting fuzzy measure (or its absence, in the case of contradictions between the decision maker's preferences or their incompatibility with the Choquet integral properties) and a lack of subjectivism in the aggregation result apart from the subjectivism of the decision maker themself. As a result, the 3D balance model will help us build the aggregation operator based on the Choquet integral with respect to 2-order fuzzy measure. The dispersion minimization method will be used for this fuzzy measure identification. The initial balance model only has to reflect the aggregation criteria set without any additional information. Consequently, at this initial stage, we need to build a balance model that would include the chosen quantitative and binary criteria with equal weights, the criteria values should be neutral. For binary criteria, such a value would be zero, for quantitative 0.5. The aggregation result will correspond to the arithmetic mean and will also be equal to 0.5. Such a structure would be the initial point for further reasoning by the decision maker. At the next phase, the binary criteria that possess the \"veto\" and \"favor\" effect need to be singled out, in order to examine them separately with the balance model and build the dependencies for the corresponding auxiliary weights. After that, based on the input of the decision maker's preferences, the fuzzy measure with the minimal dispersion method needs to be identified, and the acquired fuzzy measure used to compile the Choquet integral and, finally, build the 3D balance model corresponding to this Choquet integral, considering the influence of the binary criteria. The decision maker could then check the resulting balance model for compliance with personal preferences by watching its output during criteria changes in virtual reality. In the case of the model being compliant with the expert's preferences, the aggregation-operator synthesis is finished. Otherwise the decision maker has to return to the previous steps and change the structure. Given the above considerations, we suggest the aggregation-operator synthesis procedure based on expert preferences using the 3D balance model. The input information for this procedure is the list of aggregated criteria formed by the normalization procedure described in the previous section and also the decision maker's preferences. The mentioned procedure consists of the following steps. Step 1. Every quantitative criterion \(g[h]\), \(1\leq h\leq Y\) receives a corresponding weight on the balance model.