*3.2. Block of Fuzzification*

The input variables, namely share of element costs in the building costs (SE), predicted changes in the number of works (WC), and expected changes in the unit price (PC), are described with appropriate linguistic terms (fuzzy sets) in the consideration spaces on the so-called universes X1, X2, and X3. The domain (range of arguments) of the universes was determined as a percentage within the interval

[0; 100%] for each input variable, with the model using the decimal notation corresponding to the interval [0; 1]. In defining the X consideration spaces, for all variables described by the linguistic terms "high", "average", and "low", it was assumed that the adjacent fuzzy sets (representing consecutive linguistic terms) would overlap. According to Hovde and Moser [41], only this modelling of the linguistic terms for the input variables gives a favorable effect in the inference process. the interval [0; 100%] for each input variable, with the model using the decimal notation corresponding to the interval [0; 1]. In defining the X consideration spaces, for all variables described by the linguistic terms "high", "average", and "low", it was assumed that the adjacent fuzzy sets (representing consecutive linguistic terms) would overlap. According to Hovde and Moser [41], only this modelling of the linguistic terms for the input variables gives a favorable effect in the inference process.

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and X3. The domain (range of arguments) of the universes was determined as a percentage within

Table 1 represents the fuzzy sets for the linguistic terms L(X2) and L(X3), that is, for the input variables WC and PC. For the description of linguistic terms, membership functions with line graphs were used (triangular functions and classes Γ and L). The qualitative definition of fuzzy sets was based on the selection of appropriate types of membership functions. The quantitative definition was performed on the basis of the selection of the values of parameters characterizing the functional curves, which made it possible to precisely determine the degrees of membership of individual fuzzy sets. Degrees of membership for fuzzy sets are described in Table 1 (in the last column) by means of four numbers {α1, α2, α3, α4}. These parameters indicate, respectively, the intervals of achieving the value of membership degree 1.0 {α2, α3} and the left or right width of the distribution of the membership function to the value of the membership degree 0.0 {α1, α4}. It was assumed that linguistic values for both input variables (WC and PC) would remain unchanged regardless of the type of the building object. Table 1 represents the fuzzy sets for the linguistic terms L(X2) and L(X3), that is, for the input variables WC and PC. For the description of linguistic terms, membership functions with line graphs were used (triangular functions and classes Γ and L). The qualitative definition of fuzzy sets was based on the selection of appropriate types of membership functions. The quantitative definition was performed on the basis of the selection of the values of parameters characterizing the functional curves, which made it possible to precisely determine the degrees of membership of individual fuzzy sets. Degrees of membership for fuzzy sets are described in Table 1 (in the last column) by means of four numbers {*α*1, *α*2, *α*3, *α*4}. These parameters indicate, respectively, the intervals of achieving the value of membership degree 1.0 {*α*2, *α*3} and the left or right width of the distribution of the membership function to the value of the membership degree 0.0 {*α*1, *α*4}. It was assumed that linguistic values for both input variables (WC and PC) would remain unchanged regardless of the type of the building object.

**Table 1.** Fuzzy interpretations of the linguistic input variables "predicted changes in the number of works" (WC) or expected changes in the unit price (PC). **Table 1.** Fuzzy interpretations of the linguistic input variables "predicted changes in the number of works" (WC) or expected changes in the unit price (PC).


The data presented in Table 1 correspond to the graphic interpretation of fuzzy sets of linguistic values for WC and PC, which is illustrated in Figure 1. The data presented in Table 1 correspond to the graphic interpretation of fuzzy sets of linguistic values for WC and PC, which is illustrated in Figure 1.

**Figure 1.** Linguistic terms of the input variables WC and PC. **Figure 1.** Linguistic terms of the input variables WC and PC.

Input variable: share of element costs in the building costs (SE) should be subject to the process of adjusting the shapes of fuzzy sets described by the linguistic terms "high", "average", and "low" individually, depending on the type of the building object. The authors decided to analyze the following types of building objects in the context of determining the parameters denoting the intervals of attaining the value of the membership degree of 1.0 and the left or right width of the distribution of the membership function to the value of the membership degree 0.0. The following Input variable: share of element costs in the building costs (SE) should be subject to the process of adjusting the shapes of fuzzy sets described by the linguistic terms "high", "average", and "low" individually, depending on the type of the building object. The authors decided to analyze the following types of building objects in the context of determining the parameters denoting the intervals of attaining the value of the membership degree of 1.0 and the left or right width of the distribution of the membership function to the value of the membership degree 0.0. The following types of buildings were analyzed:


Each of the buildings was divided according to cost elements following the tables of billing elements for an average of five buildings of each type. Table 2 presents the range of cost elements for cubature facilities, highways and expressways, as well as sports fields.

**Table 2.** Range of cost elements for individual buildings.


For each building object, based on the data from an average of five objects, the average percentage of each cost component was determined. Then, the values of quartiles Q1 and Q3 and the median were calculated using statistical measures. The results are presented in Table 3.


**Table 3.** Values of statistical measures for cost elements of building objects.

It should be noted that the research sample (five objects) is relatively small. However, it can be concluded that for standard material and technological solutions, the deviations from the results obtained for a given type of building are small. In the case of non-standard solutions, the share of component costs should be modified, taking into account the specificity of a given building object.

On the basis of the data presented in Table 3, a fuzzy interpretation of the linguistic input variable SE for each of the buildings was proposed. It was assumed that for fuzzy sets:


Table 4 depicts a fuzzy interpretation of the linguistic input variable SE for all types of buildings.


**Table 4.** Fuzzy interpretation of the linguistic input variable share of element costs in the building costs (SE). costs (SE). **Fuzzy Set of Linguistic Values for SE Description Fuzzy Evaluation** 

**Table 4.** Fuzzy interpretation of the linguistic input variable share of element costs in the building

In Figures 2–6, graphical interpretations of the input variable consideration space are presented for the subsequent types of buildings subjected to analysis. These interpretations accurately reproduce the fuzzy sets for linguistic terms "high", "average", and "low", which are described in Table 4. In Figures 2–6, graphical interpretations of the input variable consideration space are presented for the subsequent types of buildings subjected to analysis. These interpretations accurately reproduce the fuzzy sets for linguistic terms "high", "average", and "low", which are described in Table 4.

**Figure 4.** Linguistic terms of the input variable SE for office buildings.

**Figure 5.** Linguistic terms of the input variable SE for highways and expressways.

**Figure 6.** Linguistic terms of the input variable SE for sports fields.

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**Figure 3.** Linguistic terms of the input variable SE for multi-family residential buildings.

**Figure 3.** Linguistic terms of the input variable SE for multi-family residential buildings.

**Figure 6.** Linguistic terms of the input variable SE for sports fields.
