# Multicriteria Decision Making in Tourism Industry Based on Visualization of Aggregation Operators
## 4 Aggregation-Operator Synthesis Using 3D Graphics  
### Criteria Choice and Normalization  
A synthesized aggregation operator will reflect the point of view of the user (decision maker) concerning his preferences in a set of alternatives. This operator assigns to each alternative the result of aggregation on the unit interval [0, 1]. The initial aggregation phase requires a choice of aggregation criteria. Some of these criteria can be binary and reflect the presence or absence of any properties in a specific alternative. The other criteria part has quantitative character that is justified by the quantitative properties of the alternative. For instance, prices, distances, service levels and so on. To be brief, we shall call normalized criteria attributes. Let us assume that for every alternative the decision maker knows the values of \(y_{1},\dots,\ y_{Y}\) quantitative attributes and \(z_{1},\dots,\ z_{Z}\) binary attributes, where \(Y\) is the number of quantitative and \(Z\) is the number of binary (Boolean) attributes. For example, in a hotel-choice task, the quantitative attributes \(y_{1},\dots,y_{Y}\) can cover the different hotel properties: pricing, location (for instance, travel time to destination), comfort, room cleanliness, service quality. Binary attributes \(z_{1},\dots,\ z_{Z}\) can be assigned as: beach availability, safety risks, hotel events that the tourist wants to visit under safe conditions. In order to synthesize an aggregation operator, it is necessary to preliminarily normalize the quantitative and binary attributes by bringing them to a unit interval. As a result of such normalization, the aggregation criteria are achieved. A fully satisfactory alternative under a certain criterion for the decision maker will have a single value for this criterion which equals 1. And otherwise, if this alternative is totally unacceptable under a certain criterion, the criterion value will be zero. A full singular criterion value of 1 will correspond with a maximum input into the aggregation result while the zero value will represent a minimum influence. There is no uniform approach to aggregation criteria normalization. More than 20 methods of such normalization exist, including linear and non-linear [26]. The normalization method choice can be handled under the assumption that it will bring a positive result in practical tasks [27]. It seems natural to utilize the whole unit interval for criteria variation in our 3D balance model. This is why our approach considers the linear method of max-min attribute normalization [27]. In accordance with this method, criteria normalization in our approach is realized through a procedure consisting of the following steps. Figure 4: Top view of 3D balance model. Step 1. Determine the domains of \(y_{1},\dots,\ y_{Y}\) quantitative attributes as intervals limited by their minimal and maximum values \(\min(y_{1}),\max(y_{1}),\dots,\ \min(y_{Y}),\max(y_{Y})\). Step 2. Build equations \(g[h]=\frac{1}{\max(y_{h})}(y_{h}-\min(y_{h}))\), where \(1\leq h\leq Y\) for those aggregation criteria that have to increase together with the growth of their attributes. Step 3. Build equations \(g[h]=1-\frac{1}{\max(y_{h})}(y_{h}-\min(y_{h}))\), where \(1\leq h\leq Y\) for those aggregation criteria that have to decrease in opposition to positive growth of their corresponding attributes. Step 4. Build equations \(g[h]=z_{h}\), where \(Y+1\leq h\leq Y+Z\) for those binary attributes that, if equal to 1, lead to corresponding criteria values equal to 1. Determine which of these criteria equaling 1 lead to an aggregation result equaling 1 (the \"favor\" effect). Step 5. Build equations \(g[h]=1-z_{h}\), where \(Y+1\leq h\leq Y+Z\) for those binary attributes that, if equal to 1, lead to corresponding criteria values equal to 0. Determine which of these criteria equaling 0 lead to an aggregation result equaling 0 (the \"veto\" effect). The first step of this procedure determines the domains of the quantitative attributes such as pricing, location, comfort and so on.