*3.4. Block of Defuzzification*

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


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 PC and WC. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 11 of 17

**Figure 8.** Result area in terms of input variables PC and SE (left) and PC and WC (right)—last of maxima defuzzification method. **Figure 8.** Result area in terms of input variables PC and SE (**left**) and PC and WC (**right**)—last of maxima defuzzification method.

**4. Discussion**  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 method.

#### A cost overrun risk prediction model was developed for each type of construction site separately using the "Fuzzy Logic Designer" application that is available in the MATLAB R2013a **4. Discussion**

0.5,

buildings.

software package (The MathWorks, Inc., Natick, MA, USA) for scientific and engineering calculations. In order to investigate the correctness of the assumption made at the design stage of the rule A cost overrun risk prediction model was developed for each type of construction site separately using the "Fuzzy Logic Designer" application that is available in the MATLAB R2013a software package (The MathWorks, Inc., Natick, MA, USA) for scientific and engineering calculations.

base (i.e., that as 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) increase, the value of the risk In order to investigate the correctness of the assumption made at the design stage of the rule base (i.e., that as the share of element costs in the building costs (SE), predicted changes in the number of

objects on the value of the results obtained for the output variable (R)), the following result diagrams

 diagrams of the result area for the output variable (R) due to the influence of the input variables PC and SE in the cross-section, when WC = 0.5, and WC and SE in the cross-section, when PC =

diagrams of the result area for the output variable (R) taking into account the set of input

 flat diagrams of the resultant curves for the output variable (R) due to the influence of PC input variables in the cross-section, when WC = SE = 0.5, WC in the cross-section, when PC = SE = 0.5,

The following figures show flat and spatial diagrams for the relationships between the output variable (R) and the input variables (SE, WC, and PC) for all types of buildings under analysis (single- and multi-family residential buildings, office buildings, highways and expressways, and sports fields). Figure 9 shows the result area for the output variable (R) in terms of PC and WC variables (left diagram) and the relationship between the output variable (R) and the PC input variable (right diagram). It should be noted that both the result areas as well as dependencies on the output variable (R) are analogous for each type of building object because, in the cost overrun risk prediction model, it was assumed that PC and WC input variables would remain the same for all

were generated for the relationships between the variable R and the input variables:

variables PC and WC in the cross-section, when SE = 0.5,

and SE in the cross-section, when PC = WC = 0.5.

of construction objects.

residential buildings.

works (WC) and expected changes in the unit price (PC) increase, the value of the risk level of exceeding the costs of a given element of the construction project (R) will increase naturally and smoothly) and also to examine the impact of the change of the membership function for the input variable (i.e., share of element costs in the building costs (SE) for individual types of building objects on the value of the results obtained for the output variable (R)), the following result diagrams were generated for the relationships between the variable R and the input variables:


The following figures show flat and spatial diagrams for the relationships between the output variable (R) and the input variables (SE, WC, and PC) for all types of buildings under analysis (singleand multi-family residential buildings, office buildings, highways and expressways, and sports fields). Figure 9 shows the result area for the output variable (R) in terms of PC and WC variables (left diagram) and the relationship between the output variable (R) and the PC input variable (right diagram). It should be noted that both the result areas as well as dependencies on the output variable (R) are analogous for each type of building object because, in the cost overrun risk prediction model, it was assumed that PC and WC input variables would remain the same for all buildings. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 12 of 17

**Figure 9.** The result area for the output variable (R) in terms of the PC and WC variables in the cross-section, when SE = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable PC in the cross-section, when WC = SE = 0.5 (right diagram). **Figure 9.** The result area for the output variable (R) in terms of the PC and WC variables in the cross-section, when SE = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable PC in the cross-section, when WC = SE = 0.5 (**right** diagram).

Figures 10–14 show the result area for the output variable (R) in terms of the variables PC and SE (diagrams on the left, respectively) and the relationships between the output variable (R) and the input variable SE (diagrams on the right, respectively). It should be noted that both the result area and the dependencies with respect to the output variable (R) are analogous for the set of input Figures 10–14 show the result area for the output variable (R) in terms of the variables PC and SE (diagrams on the left, respectively) and the relationships between the output variable (R) and the input variable SE (diagrams on the right, respectively). It should be noted that both the result area and the dependencies with respect to the output variable (R) are analogous for the set of input variables WC and SE.

variables WC and SE. Diagrams of the result areas and of the relationship between the output variable (R) and the input variables confirm the correctness of the assumptions made when designing the rule base of the cost overrun risk prediction model. Figures 9–14 indicate unequivocally 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 of a construction investment (R) increases naturally and smoothly. Diagrams of the result areas and of the relationship between the output variable (R) and the input variables confirm the correctness of the assumptions made when designing the rule base of the cost overrun risk prediction model. Figures 9–14 indicate unequivocally 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 of a construction investment (R) increases naturally and smoothly.

**Figure 10.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—single-family

residential buildings.

expressways.

expressways.

variables WC and SE.

**Figure 9.** The result area for the output variable (R) in terms of the PC and WC variables in the cross-section, when SE = 0.5 (left diagram), and the relationship between the output variable (R) and

Figures 10–14 show the result area for the output variable (R) in terms of the variables PC and SE (diagrams on the left, respectively) and the relationships between the output variable (R) and the input variable SE (diagrams on the right, respectively). It should be noted that both the result area and the dependencies with respect to the output variable (R) are analogous for the set of input

Diagrams of the result areas and of the relationship between the output variable (R) and the input variables confirm the correctness of the assumptions made when designing the rule base of the cost overrun risk prediction model. Figures 9–14 indicate unequivocally 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

In contrast, the diagrams of the dependence between the output variable (R) and the input

the input variable PC in the cross-section, when WC = SE = 0.5 (right diagram).

given element of a construction investment (R) increases naturally and smoothly.

**Figure 10.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—single-family residential buildings. **Figure 10.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (**right** diagram)—single-family residential buildings. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 17 *Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 17

**Figure 11.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—multi-family **Figure 11.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (**right** diagram)—multi-family residential buildings. **Figure 11.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—multi-family residential buildings.

**Figure 12.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and **Figure 12.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—office buildings. **Figure 12.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (**right** diagram)—office buildings.

**Figure 13.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—highways and

**Figure 13.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—highways and

the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—office buildings.

residential buildings.

**Figure 12.** The result area for the output variable (R) in terms of the PC and SE variables in the

the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—office buildings.

**Figure 11.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—multi-family

**Figure 13.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—highways and expressways. **Figure 13.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (**right** diagram)—highways and expressways. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 14 of 17

**Figure 14.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—sports. **Figure 14.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (**left** diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (**right** diagram)—sports. **.** 

In contrast, the diagrams of the dependence between the output variable (R) and the input variable SE in the cross-section were superimposed on Figure 15 when WC = PC = 0.5 for all five types of construction objects. **Figure 14.** The result area for the output variable (R) in terms of the PC and SE variables in the cross-section, when WC = 0.5 (left diagram), and the relationship between the output variable (R) and the input variable SE in the cross-section, when WC = PC = 0.5 (right diagram)—sports.

From the comparison of flat dependence diagrams (Figure 15), the input variable share of element costs in the building costs (SE), adjusted individually to the model for each building type, should be considered crucial in the context of the impact on the result value of the output variable **Figure 15.** The diagrams of the dependence between the output variable (R) and the input variable SE for all five types of construction objects. **Figure 15.** The diagrams of the dependence between the output variable (R) and the input variable SE for all five types of construction objects.

(R). The lower the membership for the values of the arguments of the X1 universe domain for the

From the comparison of flat dependence diagrams (Figure 15), the input variable share of

sports fields (purple line).

sports fields (purple line).

**5. Conclusions** 

**5. Conclusions** 

particular by the comparison of the course of the result curves for office buildings (blue line) and

linguistic terms "average" and "high" of the SE variable, the more the resulting value of the risk of construction investment cost overrun (R) increases for the arguments of the X1 variable universe with smaller values—the SE interval approximately [0.1; 0.3]. This conclusion is confirmed in particular by the comparison of the course of the result curves for office buildings (blue line) and

The phenomenon of exceeding planned investment costs is often encountered in the construction industry, and the determination of the risk associated with it may be of key importance for achieving the objectives of the project. This paper discusses a cost overrun risk prediction model, the development of which was based on the fuzzy inference model of Mamdani. The model input variables include the following: 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 basic problem is to

The phenomenon of exceeding planned investment costs is often encountered in the construction industry, and the determination of the risk associated with it may be of key importance for achieving the objectives of the project. This paper discusses a cost overrun risk prediction model, the development of which was based on the fuzzy inference model of Mamdani. The model input variables include the following: 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 basic problem is to

From the comparison of flat dependence diagrams (Figure 15), the input variable share of element costs in the building costs (SE), adjusted individually to the model for each building type, should be considered crucial in the context of the impact on the result value of the output variable (R). The lower the membership for the values of the arguments of the X1 universe domain for the linguistic terms "average" and "high" of the SE variable, the more the resulting value of the risk of construction investment cost overrun (R) increases for the arguments of the X1 variable universe with smaller values—the SE interval approximately [0.1; 0.3]. This conclusion is confirmed in particular by the comparison of the course of the result curves for office buildings (blue line) and sports fields (purple line).
