The task of the inference block in models based on the assumptions of the Mamdani fuzzy model of inference (Mamdani, 1974; Mamdani, 1977) is the calculation of the resultant membership function for the output variable *μ*_{res}(*y*) based on membership levels for the values of sharp input variables *x*_{1}^{*} i *x*_{2}^{*}. This function often has a complex shape and its calculation is performed by means of the process of inference.

The inference block forms the basic two elements, namely the rule base and the inferential mechanism.

The rule base developed within the fuzzy risk assessment module has a form of a conjunction due to the use of the operator „*And*”. It consists of 25 rules presented in . The main assumption when designing the rule base was that with the increase of the probability and the effect of an adverse event, the impact of the identified risk factor on the size of the corresponding component of the life cycle cost of the building is to grow in a gentle (non-stop) manner.

Table 1 Rule base for the inferential block

The operation of the inferential mechanism is, in turn, based on three consecutive mathematical operations: aggregation of simple premises, implications of fuzzy inference rules and aggregation of the conclusions of all rules.

Aggregation of simple premises concerns the calculation of the degree of truth of a fuzzy rule created by premises. The degree of truth is understood here as the degree of membership to a given relation *R*_{N} fuzzy values of sharp input variables *x*_{1}^{*} and *x*_{2}^{*}. Due to the fact that in conditional sentences a conditional link „*And*” was used, which in fuzzy logic is represented by the concept of the intersection (product) of fuzzy sets, the operation of aggregation of premises is reduced to seeking the value of the degree of membership to the relation *R*_{N}. This value is determined by using one selected rule of the fuzzy implication (a formula for the *T-norm*), which can be recorded as:

$$\begin{array}{r}\begin{array}{c}{\mu}_{{R}_{N}}\left({x}_{1}^{\ast},{x}_{2}^{\ast}\right)={\mu}_{{A}_{i}{\cap B}_{j}}\left({x}_{1}^{\ast},{x}_{2}^{\ast}\right)=T\left[{\mu}_{{A}_{i}}({x}_{1}^{\ast}),{{\mu}_{{B}_{j}}(x}_{2}^{\ast})\right]\end{array}\end{array}$$(3)where: *R*_{N} – fuzzy rule; *A*_{i}, *B*_{j} – fuzzy sets for input variables; *T* – selected formula on *T-norm*; *x*_{1}^{*}, *x*_{2}^{*} – sharp inputs values.

Where the degree of membership to a relation *R*_{N} equals 0, the given rule is not started and does not take part in the process of inference.

The next stage is the application of the fuzzy implication method for each activated inference rule to determine its resultant conclusion. This process leads to a change in the shape of the membership function of fuzzy sets describing the resulting conclusions according to the degree of truthfulness of meeting the corresponding rules.

The concluding stage of the inference block is the aggregation of the conclusions of all the rules (aggregation of outputs). This action consists in combining the conclusions of the rules that are responsible for the shape of the resultant membership function *μ*_{res}(*y*). The first step determines, one by one, the modified membership functions of fuzzy sets for the output conclusions of the rules participating in the inference, and then the resulting fuzzy sets are summed up based on one of the formulas for the *S-norm*. The general form of the equation for aggregation of outputs can be presented as follows:

$$\begin{array}{r}\begin{array}{c}{\mu}_{res}\left(y\right)=S\left[{\mu}_{RN}\left({x}_{1}^{\ast},{x}_{2}^{\ast}\right)\right]\end{array}\end{array}$$(4)where: *μ*_{RN} – degree of truthfulness of activated rules; *S* – selected formula on *S-norm*.

compiles the selected fuzzy implications (*T-norms* and *S-norms*), which are most frequently used for models built according to the assumptions of the Mamdani fuzzy model of inference (Mamdani, 1974; Mamdani, 1977). The fuzzy risk assessment module involves the basic applications of Mamdani *T-norm* and *S-norm*, that is minimum and maximum of Mamdani, respectively. The authors wrote about this assumption in (Plebankiewicz, & Wieczorek, 2016).

Table 2 Selected rules for the intersection and sum operation of fuzzy sets

The figures from Figure 9 to Figure 12 present a graphical interpretation of all sets of membership functions for the output variable *y*, namely the impact of an identified risk factor on a given component of the life cycle cost of a building object – *IRF(Uph)*.

Figure 9 Membership function for the output variable *y* – *IRF(Uph)*, polygonal functions (piecewise linear)

Figure 10 Membership function for the output variable *y* – *IRF(Uph)*, complex functions (linear quadratic)

Figure 11 Membership function for the output variable *y* – *IRF(Uph)*, harmonic functions

Figure 12 Membership function for the output variable *y* – *IRF(Uph)*, Gaussian functions

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