*3.2. Mathematical Model of Data Analysis*

To assess the impact of risk factors on the construction project's life cycle, it is required not only to allocate the risk to the desired stage and conduct an expert assessment of the risk factor, but the data of the expert survey should be calculated mathematically to determine not only the degree of impact, but also the likelihood of the factor impacting such parameters as cost and duration.

The mathematical model for the analysis of expert evaluation is based on two theories:


The fuzzy set theory is a method of experiment planning that is widely used in quantitative analysis of a machine process, especially for quality and risk assessment in engineering [29]. The main limitation of the method is related to the use of statistical mathematics and probability theory in the analysis. A probabilistic attempt is insufficient when the data are scant, as knowledge of their values becomes inaccurate or incomplete [30].

One of the possible solutions for cases where the data is scant is a non-parametric maximum likelihood estimate [31,32]. At the end of the twentieth century, a method based on the idea of fuzzy logic associated with L.A. Zadeh [33] was developed taking into account the possibility of describing the so-called linguistic variable. An example of applying the idea to the perceived risk assessment in the project is presented in the articles [34,35].

Fuzzy logic was first introduced by Professor L.A. Zadeh in 1965 and began to be applied in the 1970s [33,34]. Fuzzy logic is a successful application in the context of fuzzy sets in which the variables are linguistic rather than numeric. Since its development in 1965, it has become the optimal choice for handling data-related inaccuracies and uncertainties in risk assessment tasks [36].

Fuzzy logic is different from binary or Aristotelian logic, which sees everything as binary: yes or no, black or white, zero or one. The values in this logic vary from zero to one [37]. Figure 2 shows the architecture of the fuzzy inference system.

**Figure 2.** Fuzzy inference system [17].

A fuzzy inference system [17] usually consists of the following components:


The components of the fuzzy inference system for risk assessment are described below [17,37]. The process of converting explicit variables into linguistic variables is called fuzzification.

Fuzzification is the establishment of a correspondence between the numerical value of the input variable of the fuzzy inference system and the value of the membership function of the corresponding term of the linguistic variable [38].

The input and output data of the fuzzy inference system must first be fuzzy in a fuzzy inference system. The probability of occurrence and the severity of the impact of the risk are considered as two inputs, and the level of risk is considered as the output of the system of fuzzy inferences.

The linguistic expressions and fuzzy sets used for defining the input and output data of a fuzzy inference system are presented in Table 1 [17,38,39].


**Table 1.** Linguistic terms.

For the functioning of the fuzzy logic system, referring to the standard risk matrix is required.

The risk matrix is a tool of the threat management process designed to increase the objectivity of its interpretation [17]. To place an item in the matrix, you must assign it a probability and damage rating.

The degree of risk is determined on the basis of the risk matrix [13] and, accordingly, this component of the developed fuzzy inference system for risk assessment is a knowledge base and fuzzy rules, including 25 fuzzy "if" rules, which are presented in Table 2.

**Table 2.** Mathematical model rule table.


Tables 3 and 4 shows the indicators of the standard risk matrix.


**Table 3.** Risk Matrix.

**Table 4.** Risk matrix with ranks.


The next component of the developed fuzzy inference system for risk assessment is the fuzzy inference mechanism. The inference engine evaluates and makes logical inference to the rules using inference algorithms, and after the inference rules are aggregated by the defuzzifier block they are converted to an explicit or numeric value. The fuzzy inference mechanism is the Mamdani algorithm [17]. The optimum method is used to aggregate the output data, and the center of gravity method is used for defuzzification.

The fuzzy risk assessment index is considered as an output parameter, and varies from 0 to 5. In this article, the risk is divided into five equal parts, as shown in Figures 3–5. Risks are represented by fuzzy sets, the ranges of which coincide with the linguistic terms given in Table 1. Using the appropriate transformation scale, the linguistic terms are converted into fuzzy ratings. One of the key points in fuzzy modeling is the definition of fuzzy numbers, which are vague concepts and expressed in inaccurate terms in natural language [36].

**Figure 3.** Membership function for the probability level.

**Figure 4.** Membership function for the influence level.

**Figure 5.** Membership function for the risk level.

In this work, fuzzification distributes system variables, including probability (P), impact level (I) and risk levels (R) with clear numbers. The structure of the fuzzy model is shown in Figure 6.

**Figure 6.** Fuzzy model inference structure.

Twenty-five rules were introduced to the mathematical model in Table 2, performing the defuzzification process [38,39]. Defuzzification in fuzzy inference systems is the process of transition from the membership function of an output linguistic variable to its clear (numerical) value. The purpose of defuzzification is to use the results of the accumulation of all output linguistic variables to obtain quantitative values for each output variable used by devices external to the fuzzy inference system [17,39].

The last step in the approximation is defuzzification. This step contains the process of replacing a fuzzy value with a clear inference, consisting of a procedure for weighing and averaging the outputs of all individual fuzzy rules. In total, there are six defuzzification methods [40]:

Centroid Average (CA) Center of Gravity (COG) Maximum Center Average (MCA) Medium of the Maximum (MOM) Smallest of the Maximum (SOM) Largest of the Maximum (LOM)

Center of gravity (COG) is one of the most popular defuzzification methods, chosen because of its simple calculations and intuitive plausibility [41].

COG is defined by the following equation:

$$Z = \frac{\int \mu\_i(\mathbf{x}) \mathbf{x} d\mathbf{x}}{\int \mu\_i(\mathbf{x}) d\mathbf{x}} \tag{1}$$

where:

Z—defuzzified result.

x—output variable.

μ<sup>i</sup> (x)—aggregated membership function.

The defuzzification process creates a clear value from fuzzy sets that reflect the risk of the project, as in Figure 7.

The data of mathematical calculation of expert assessments are presented in Table 5.

The DS method is a more general form of the Bayesian approach that retains all its advantages. For example, in the DS method, as in the Bayesian method, available a priori information can be included in the inaccurate output of uncertain indicators and inferred results. Nevertheless, the use of a priori information in the DS method is not mandatory. This is one of the advantages of the DS theory [42,43].

**Figure 7.** Output from Polyspace software package based on 25 rules.

$$\text{DS} = \text{m}(\text{A}) = \frac{\text{n} - \text{minF}}{\text{maxF} - \text{minF}} \tag{2}$$

where:

m(A)—degree of reliability. maxF = max{*fj*| *j* ∈ [1, *n*]}; minF = min{*fj*| *j* ∈ [1, *n*]}; n—number of factors

Compared with other probabilistic methods, such as the Bayesian method, the DS method does not require the calculation of a priori probability; it has a flexible and understandable mass function, and the formation of a mass function is convenient and simple. The computational complexity of this method is much less than that of the Bayesian method [41,44].

For the processing of expert data, a risk matrix, the Dempster-Shaffer theory and a mathematical model of fuzzy logic were used. The processed results of the study are presented in Section 3.

Dempster-Shafer cell data were obtained by mathematical calculation according to formula 2. Fuzzy logic output data, defuzzification results, were obtained by mathematical modeling through the Polyspace software package. The inference algorithm was used. After aggregating the output rules by the defuzzifier block, we obtained an explicit numerical value as the result of fuzzy inference.

The data of mathematical calculations are presented in Section 3. The values of FLRC and FLRT are ranked in ascending order, according to the fuzzy inference group.

#### **4. Result and Discussion**

The following values of probability and impact on project parameters were obtained during the expert survey of Section 3.1. The results of the expert survey are presented in Table 5.


#### **Table 5.** Analysis of the expert survey results.

**Table 5.** *Cont.*


To understand the operation of the mathematical model in the life cycle of a multistorey residential building, each factor at different stages of the project was considered and the rank of the factor was determined by fuzzy logic, as this was the main tool in our study, with 25 preprogrammed rules.

The data of the expert survey are the input data for the mathematical model presented in Section 3.2. The results of the mathematical model are presented in Table 6.


**Table 6.** Comparative analysis of the obtained data.


**Table 6.** *Cont.*

P—probability; IoC—impact on cost; IoT—impact on timeline; RC—risk cost; RT—risk of timeline; DCRS— Dempster Schafferis risk cost; DCRT—Dempster Schafferis risk of timeline; FLRC—fuzzy logic risk cost; FLRT fuzzy logic risk of timeline.

After analyzing the results of mathematical calculations, a diagram with factors and their ranks can be constructed as shown in Figures 8 and 9. The data are presented without ranking by the magnitude of the influence.

Z ^Z &>Z

Zd ^Zd &>Zd

**Figure 9.** Diagram of the distribution of the impact of risk factor on the duration by ranks.

The diagram shows 64 factors, with each ranked in relation to another; due to this we see a clearer picture of the distribution of risk factors by measurement value, both in cost and in time.

The study identified the most dangerous risk factors that affect the key parameters of the life cycle of a multi-storey residential building.

Mathematical calculations showed that the most effective mathematical apparatus is fuzzy logic based on 25 given rules. Dempster Schaffer's theory has small deviations from fuzzy logic, but this spread is within the acceptable limit. The standard risk matrix has large deviations in data reliability, as it excludes the presence of the said risk factor definition rules.

The final step was to determine the magnitude of the impact of the main factors identified in Table 6, on the parameters of cost and duration of the construction project.

Table 7 shows the results of the analysis of the obtained data. The cost and duration values were determined by the experts in Supplementary, Section 3.1.


**Table 7.** Critical risk factor analysis.

The factors in Table 7 are in the Significant category of Table 1. Mitigation measures must be taken to reduce the risks. These are critical risks that have serious consequences and a high probability of occurrence. High priority means immediate action is required to eliminate or mitigate possible consequences.

Factors not included in the table are not excluded; they are part of the whole project system and are subject to the rules of Table 1.

The following data were obtained as a result of the analysis of key risk factors:


The difference in the rank values of the risk factors presented in Figures 8 and 9 shows that the choice of mathematical tool plays an important role in determining the rank of risk factors.

The results obtained during the study will help to predict project risks and allow taking the right steps in due time to manage them and to adjust the budget and resources.

#### **5. Discussion**

A key component of the experiment was focused on the analysis of the influence of various risk factors that affect the stages of important parameters of the building life cycle. The experiment showed the performance of the mathematical model and identified critical factors. This technique allows work on one structure, that is, the life cycle of an object with all its parameters and the mathematical apparatus for taking into account the influence of factors, which allows a response to their impact in a timely manner. In general, the study of the influence of factors on the life cycle of an object will allow creating a common interconnection environment focused on successful implementation and improvement of informed decisions that can bring maximum benefit to stakeholders.

The use of two mathematical models for assessing the risk factor is not comprehensive today, but it copes well with the tasks set; namely, it takes into account the requirements and rules laid down by the operator for each object. However, for future buildings, actual data on behavior is not available, there are no public registries, no record of the maximum influencing factors is kept, which is a hot topic these days, and often the data are confidential. Hence, co-modeling by integrating BIM with a robust risk analysis model is one of the most appropriate methods to solve this problem. It is convenient when each factor has its own individual number, tracked in real time at each stage of work in the BIM system.

This study has the following unsolved problems. The scope of the simulation experiment was limited both in terms of the simulation time period and the space coverage of the object data. Over time, more participation from experts from the construction industry is required. More designs, materials and design approaches need to be evaluated as the pace of construction continues to be high and every year we see new technologies emerging in the construction industry. Simulation results will be more coherent and informative if it is possible to expand the range of data collection on the objects under study; the functions of joint modeling can be improved as research progresses.

Since the proposed model is relatively new, it should be considered a starting point for a new assessment of the impact of risk factors on the project. The methodology is subject to improvement, and many aspects remain to be studied. Of course, this model will allow managers of organizations to significantly reduce costs, correctly form the tasks set, identify and eliminate risk factors in a timely manner, and identify weaknesses in the company that will lead to financial losses.

Future research in this area should focus on identifying risk factors and managing them during the project cycle. It is worth introducing an electronic database of risk factors, so the percentage of risks can be reduced and projects implemented more efficiently.

#### **6. Conclusions**

This article proposes a scientifically justified mathematical model of the life cycle of a multi-storey residential building. The model allows competent determination and ranking of the influence of risk factors at each stage of the project. The presented methodology was developed to assess the impact of risk factors on the main parameters of the project. The stages of the life cycle for a residential building were analyzed, the risk factors arising at each stage identified, and their impact assessed by an expert survey. The expert survey involved 60 experts who are professionals in the construction industry, with more than 10 years of experience. The experts were requested to assess the impact of factors on both the cost and the duration. As a result, the following conclusions can be made.


fication number to track them. This data will help to predict the consequences in a timely manner and take measures to eliminate them.


**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/buildings12040484/s1, Table S1: Expert Questionnaire.

**Author Contributions:** Conceptualization, A.L.; methodology, A.L. and D.T.; software, O.C.; data analysis, D.T.; investigation, O.C. and T.K.; data curation, D.T. and T.K.; writing—original draft preparation, O.C. and T.K.; writing—review and editing, O.C. and T.K.; final conclusions, O.C. and T.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by the Ministry of Science and Higher Education of Russian Federation (grant NO. 075-15-2021-686).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

