*3.3. Block of Inference*

**Figure 6.** Linguistic terms of the input variable SE for sports fields. In the inference block in the fuzzy inference model of Mamdani of the MISO type, the resulting membership function is calculated for the output variable µ(y). Its calculation is based on the values of the degree of membership of the sharp input variables µ(x1), µ(x2), and µ(x3) for individual fuzzy sets of linguistic values. The resulting function often has a complex shape and its calculation is done by the so-called inference (inference process). The inference block consists of two basic elements, namely the rule base and the inference mechanism, the operation of which is based on the three following consecutive mathematical operations: aggregation of simple premises, implications of fuzzy inference rules, and aggregation of conclusions of all rules.

The designed base of rules in the cost overrun risk prediction model has a conjunctive form due to the logical conjunction "and" used in conditional sentences, which combines all three simple premises. he model proposes five result conclusions that inform about the size of the calculated risk of cost overruns, i.e., "very low" (Vl), "quite low" (Ql), "average" (Av), "quite high" (Qh), and "very high" (Vh).

For the purpose of developing the rule base, the authors assumed that with an increase in the 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), the value of the risk level of exceeding the costs of a given element in the construction project (R) will naturally and smoothly increase. For this purpose, it was decided to examine the quantities of the products of all combinations of input variables in a set of all 27 possible rules, and then to assign the results to five possible result conclusions on the assumption that the minimum quantities correspond to the "very low" conclusion, the maximum—to the "very high" conclusion, and the intermediate—to the "quite low", "average", and "quite high" conclusions, respectively and proportionally. The following weights were assumed for the linguistic input variables SE, WC, and PC: 1 for "low", 2 for "average", and 3 for "high". Table 5 illustrates the rule base of the inference block consisting of 27 rules, for which equal degrees of fuzzy relationship validity are assumed to be 1.0.

In the interference block, the processes of premise aggregation and rule conclusion aggregation are performed. Aggregation of simple premises consists in calculating the degree of belonging (truthfulness) of the fuzzy rule created by these premises. Due to the fact that in the conditional sentences the logical conjunction "and" was used, which in fuzzy logic is represented by the concept of intersection (product) of the fuzzy sets, the operation of premise aggregation was reduced to searching for the value of the degree of membership to the fuzzy relationship (FR). This value was determined by applying the Mamdani fuzzy implication rule (T-norm), calculated according to the following formula:

$$T\_M = \min(\mu(\mathbf{x\_1}), \mu(\mathbf{x\_2}), \mu(\mathbf{x\_3})) \tag{1}$$

The final stage of the inference block is the aggregation of the conclusions of all running fuzzy rules (the so-called output aggregation). This procedure consists of summing up the conclusions of activated rules that are responsible for the shape of the resulting membership function µ(y). According to the calculation algorithm, the first step is to define separately the modified membership functions of the fuzzy sets of the output variable for the rules involved in the inference, and then sum up these fuzzy sets based on one of the formulas for S-norm. In the cost overrun risk prediction model, the basic S-norm is the following formula of Mamdani:

$$S\_M = \max(\mu(\mathbf{x}\_1), \mu(\mathbf{x}\_2), \mu(\mathbf{x}\_3)) \tag{2}$$

Output variable (y) is described in space (universe) Y. The scope of the Y universe was determined as a percentage [0; 100%]. As in the case of all input variables, the record of the argument domain in the decimal interval was adopted [0; 1]. Sets correspond to the resultant conclusions in the rule database ("very low", "quite low", "average", "quite high", and "very high").

Fuzzy sets for the final result conclusions ("very low" and "very high") and the intermediate internal conclusion ("average") were attempted to be parameterized in such a way that the membership function graphs did not interpenetrate, but were continuous in the full scope of the Y universe. For internal relative conclusions ("quite low" and "quite high"), the same procedure was followed, where the fuzzy sets were entered symmetrically between the extreme (final) and internal (intermediate) conclusions. The parameterization was performed in such a way that the adjacent fuzzy sets overlapped with the membership degree for intermediate elements equal to µ(0.2) = µ(0.4) = µ(0.6) = µ(0.8) = 0.5. Table 6 presents sets of linguistic terms L(Y) for the output variable (y). The membership of all fuzzy sets was defined as in the case of the input variables, that is, using four numbers {α*1*, α*2*, α*3*, α*4*}.

Figure 7 presents a graphic interpretation of the consideration space of the output variable (y), which is represented by the fuzzy sets of all five result conclusions, described in Table 6.


**Table 5.** Rule base of the inference block.

where LV—fuzzy set of linguistic values (fuzzy sets in accordance with Tables 1 and 4), Concl—resulting conclusion for the output variable risk of exceeding the costs of a given element of a construction investment project (R). Vl, very low; Ql, quite low; Av, average; Qh, quite high; Vh, very high.


**Table 6.** Fuzzy interpretation of the linguistic output variable R.

**Fuzzy Evaluation of**

**Fuzzy Set of Linguistic Values for R Description of the**

taken into account, according to which the methods of maxima:

*3.4. Block of Defuzzification* 

and smoothly increase,

output variable.

PC and WC.

method.

The defuzzification process is a mathematical operation performed on the resultant membership function shape (the resulting fuzzy set) obtained after aggregating the conclusions of all inference rules. This operation aims to determine one sharp value of the variable (y) that will appropriately represent the output fuzzy set and indicate unambiguously the result conclusion.

Considering the possibility of using sharpening methods in the cost overrun risk prediction model, the following defuzzification methods were investigated: the first of maxima, middle of maxima, and last of maxima method, the center of gravity method, and the bisector area method. The advantages and disadvantages, as well as the conditions for the application of individual methods, were highlighted. The suggestions and observations contained in [42] were especially

 are not able to implement the assumption adopted for the purposes of building the rule base, that with the increase in the 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), the value of the risk level of exceeding the costs of a given element of the construction investment (R) will naturally

 result in sharp values, which will not in every case adequately represent the output fuzzy set, which is caused by the impact on the sharp result of only the most activated fuzzy set of the

Figure 8 confirms the observations described above with regard to the use of the last of maxima defuzzification method. On the left, there is the result surface for the output variable (R) due to the influence of the input variables PC and SE. The result surface is analogous for the set of input variables WC and SE. On the right, the same result area is shown, but in terms of the input variables

Taking into account the above observations, it was assumed that the proper and basic defuzzification method in the cost overrun risk prediction model would be the center of gravity
