*Article* **Substitution of Material Solutions in the Operating Phase of a Building**

**Anna Sobotka, Kazimierz Linczowski and Aleksandra Radziejowska \***

Department of Geomechanics, Civil Engineering and Geotechnics, Faculty of Mining and Geoengineering, AGH University of Science and Technology in Cracow, Al. Mickiewicza 30, 30-059 Cracow, Poland; sobotka@agh.edu.pl (A.S.); klinczowski@gmail.com (K.L.)

**\*** Correspondence: aradziej@agh.edu.pl

**Abstract:** During the operation of buildings, repairs, modernizations, adaptations, renovations, and reconstructions of parts of historic objects are performed. There is often the problem of using a different material or construction technology than was originally used, for a variety of reasons. For example, these are materials not currently manufactured, with necessary higher performance values (insulation, strength). The aim of the article was to analyze and evaluate the possibility of material substitution in repair works and to analyze the cause and effect analysis of its application in the context of different conditions. The article analyzes the causes and conditions of the substitution of materials in various stages of the exploitation phase of buildings, including historic buildings. A SWOT (Strengths, Weaknesses, Opportunities, Threats) matrix was developed for the phenomenon of material substitution during the operational phase. With aid from the DEMATEL (Decision Making Trial and Evaluation Laboratory) method, identification of cause–effect relationships regarding the issue of the possibility of applying the substitution of material solutions in building objects was carried out. The analysis carried out by the authors allows us to conclude that the use of substitution in the construction sector is justified and shows great opportunities in its implementation and development.

**Keywords:** substitution; operation and maintenance phase; cause–effect relationships; historical buildings

#### **1. Introduction**

The phenomenon of substitution is common in various fields of social and economic activity [1–4]. In the case of material economic activity, it is the mutual substitutability of goods with similar properties. The subject of the article is the substitution of constructional and material solutions in the implementation of construction projects, understood as a phenomenon consisting of replacing the designed object structure (element) with another one that meets the same or similar technical and functional requirements, as well as aesthetic requirements [5].

In construction, the application of substitution occurs throughout the life cycle of an object and addresses various issues. Both in the preparation phase, e.g., during choosing the location of a construction investment, variants of functions and/or construction, technology, as well as during the implementation of facilities and construction works, especially when the contractor is left with the choice of construction products. The selection and supply of construction sites with resources is related to the phenomenon of the substitution of suppliers and entire supply chains.

The exploitation phase of a building object is the longest period of its life cycle. However, the scope of construction works, at this stage, is not too large compared to the construction of the facility. Decisions related to undertaking repairs, including reconstruction, changes in the functions of rooms and facilities, and the choice of material solutions

**Citation:** Sobotka, A.; Linczowski, K.; Radziejowska, A. Substitution of Material Solutions in the Operating Phase of a Building. *Appl. Sci.* **2021**, *11*, 2812. https://doi.org/10.3390/ app11062812

Academic Editor: Francesco Colangelo

Received: 8 January 2021 Accepted: 17 March 2021 Published: 22 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the structure, their repair, replacement, or renovation of finishing elements, etc., are difficult and require many aspects to be taken into account.

Despite the phenomenon of substitution that has been present in construction projects for years, there is a need to develop theoretical foundations and methods and tools to support decision-making in construction practice. Analysis and selection of substitute materials should consider the full life cycle of the object. They should also refer to current socio-economic concepts such as sustainable development and the circular economy.

Substitution can significantly affect the quality, cost, and time of individual construction projects. It also has a broader multi-faceted impact on the delivery of construction in environmental, economic, and social contexts. For example, the use of material substitution may make it possible to meet a construction completion date in the event of a market collapse or to purchase equally suitable but less expensive products. This may result in improved user comfort or use of products whose manufacture and use do not result in harmful emissions. This last example has a very large contribution to environmental protection—the implementation of sustainable development principles.

In the presented article, the authors focus on the application of the possibility of substituting construction products. It may be caused by the desire to use materials that raise the standard of the facility and cost conditions, as well as limitations due to the unavailability of original materials used during construction. The last aspect concerns, in particular, the refurbishment of buildings entered in the register of monuments. The use of replacement construction products in these types of buildings is a challenge, not only because of the difficulty in selecting an appropriate substitute, but also because of meeting the procedural requirements approved by the restorer. Thus, many factors and conditions of different natures influence the selection of the best substitute under given conditions, taking into account the consequences in terms of durability, strength, etc. during their further use [6], and, therefore, on the life cycle costs of the facility.

The purpose of this paper is to analyze and evaluate the possibility of material substitution in repair works and to analyze the cause and effect analysis of its application in the context of various conditions. A division of the service life of a building was made in the context of the execution of construction works, their contractors, and investors. Conditions and factors occurring in the decision-making process of maintenance of the object in the deteriorated condition, selection of works, and building materials were analyzed. Attention is drawn to the possibility and necessity of material substitution in relation to historical buildings. The developed SWOT matrix and its analysis allowed us to systematize factors (conditions and limitations) of substitution in the exploitation phase and its influence on the life cycle of buildings. The factors covering various substitution determinants, included in the SWOT matrix, were used for identification of cause–effect relationships in the issue of possibility to apply the substitution of material solutions in building objects. For this purpose, the DEMATEL method was used.

#### **2. Substitution of Construction Products in the Exploitation Phase of a Building Object**

One of the activities aimed at caring for the environment is striving to extend the life cycle. The products of the construction industry are one of the elements that allow us to take care of this trend. Existing buildings are designed for many years, and thanks to appropriate maintenance and refurbishment measures, they can survive many times longer. One of the ways of extending the life cycle of building objects is to carry out a refurbishment policy, during which it is necessary to take care of the proper selection of material solutions.

Depending on the stage of exploitation under consideration, the participant of the investment process, which may be the user, owner/investor, or property manager, will make decisions in which sooner or later will meet the need, or even the necessity, to use the substitution of construction products. Considering the wide market offer of construction products, the decision-maker will have to consider many criteria before deciding to use a product other than the originally built-in product.

Due to many different conditions, it is proposed in the research that the substitution of construction products in the operation phase should be considered by distinguishing its three stages/periods:

Substitution of construction products during the life of a building object is strictly connected with the division presented in Figure 1.

**Figure 1.** Division of substitution during the lifetime of a building structure.

During the warranty and guarantee period in a newly commissioned building, all necessary repairs should be carried out by the contractor who carried out this investment. Consequently, all costs associated with the construction work under consideration are not financially chargeable to the property owner. In the situation described above, due to the short period of time from putting the facility into use, construction products used for repairs and troubleshooting should still be available on the market.

The substitution of construction products during the warranty and guarantee period should result from a possible lack of availability of the original product at the moment of repairing the defect resulting from e.g., the necessity to wait too long for the construction product originally used in the facility, change of the manufacturer's brand, completion of production of a specific construction product, a clear wish of the facility owner, or a change of e.g., fire safety regulations.

However, during the warranty and guarantee period, construction work may already occur that does not merely involve the removal and repair of faults. The owner of the property may decide to reconstruct, expand, or even change the use of a building that has just been put into use. In such a situation, the guarantee and warranty for the current scope of construction works is lost, and as a result, substitutes for the construction products originally used may be introduced.

The next stage of the operation of a building object, after the warranty period, which will usually last for several or even several dozen years, is a natural period during which substitution of construction products is a common phenomenon. It results from the natural wear and tear of a given element and the desire to replace it with other products that raise the standard of use, e.g., safety, convenience, aesthetics, comfort, and even fashion. After the expiry of the warranty and guarantee period, the construction products used for repairs are the responsibility of the property owners and to a large extent their choice is also dependent on the purchase price. In this phase of building operation, all factors that affect the price of the construction service (refurbishment, reconstruction, etc.) are crucial. It can be stated that the investor, when determining the scope of planned works, in most cases initiates a tender procedure, which differs from the one used during the construction of a new facility only in the scope of planned works. The very stage of collecting offers, their consideration, and selection of a potential contractor is analogous to that of any new construction project under construction.

It is important to note that the selection of construction products during this phase is critical in terms of the life cycle of the facility [7]. The proper selection of these for refurbishment and/or modernization works will have a significant impact on the extension or shortening of this phase of the life cycle as well as on costs [8–10]. Saving at the refurbishment stage may result in the necessity to perform another refurbishment quickly.

The use of substitutes for construction products of better quality and technical values may postpone the need for further refurbishment as well as reduce maintenance costs and also raise the standard of the facility.

Figure 2 presents the change of utility values of a building object in its life cycle, which is connected with two main processes, i.e., constant decrease of utility properties from the moment of putting the object into use (curve b) and simultaneous increase of the object users' requirements while taking into account changes in regulations and standards (curve a\*). The drop in the value of curve b is caused by the wear and tear of individual building elements during the operation phase. The continuous line Z shows the performance assessment at the moment the building is put into operation. It was assumed that the building was designed and constructed in accordance with the relevant standards (Eurocodes) with the application of the required supervision procedures and control throughout the construction process. The dashed straight line Z' specifies the minimum level of utility requirements that a building should meet. If the assessment of performance is below the Z' level, further use of the object is unacceptable.

**Figure 2.** Schematic diagram of the increase in building performance requirements (a-curve) and changes in technical condition due to aging and renovation during the building's service life (b-curve) [11,12].

The decrease in the value of curve b is caused by the wear and tear of individual building elements during the exploitation phase. We can observe "jumps" on it, i.e., an increase in the usable value of the object as a result of repairs and renovations—points B1 and B2—and modernization—point D [12]. Modernization is caused not only by the increase of users' requirements but also by the increase of requirements regarding the object's features as a result of stricter legal regulations (e.g., regarding fire protection).

Construction objects are characterized by a long service life when properly operated. Very often they perform a completely different function than those for which they were designed. The durability of their construction exceeds the often assumed periods [13,14]. We have many examples in the world of such age-old buildings and structures. In Europe, in particular, for many years there has been a desire to take care of the historical substance, objects of historical, cultural, and religious significance that bear witness to past eras. Many

of the objects among those existing in the building stock, that due to their exceptional value, are entered in the register of monuments kept by the relevant governmental administration bodies. There is no specific time after which the building is considered a monument. The Act on the Protection and Care of Historical Monuments states that any building which is important for history and science can become a monument, and thus should be preserved [15]. It can also be a building built in the 1950s or 1960s if it presents features characteristic for the architecture of a given period and can be important for its history. Buildings entered in the register of monuments are subject to the Act on the Protection and Care of Historical Monuments [16] and all activities related to the use and in particular their maintenance in a proper technical condition and standard are subject to the supervision of the conservator. Thus, in the phase of exploitation of buildings, the period of their functioning as a monument should be distinguished for a group of exceptional objects.

Substitution starts to appear much more often in the case of buildings already in use for a longer period of time [17–21]. Among the exploited properties, we can observe a certain phenomenon, in which the trend is manifested by the growing deficiencies in the documentation of the exploitation of the building with its age. For example, the documentation of mass-produced buildings built in the 1970s and 1980s in large panel technology is often incomplete and inconsistent. Therefore, owners often look for construction products similar to the original ones while carrying out renovation works, usually guided by the criterion of aesthetics and price. In this case, someone else is also responsible for financing the work on the facility. In cooperative buildings or those owned by housing communities, the costs of all repairs and renovations is borne by the property owners. Most often this is done through the so-called "renovation fund". Such works are very often performed in the order of "from the most urgent", unfortunately in many cases without taking into account the durability of the construction products used for this purpose.

Moreover, one of the major problems of substitution is the choice of substitute material. During the design phase, the architect is almost free to choose a replacement material. In contrast, there are many more factors to consider during the renovation phase. Thanks to advances in material engineering, manufacturers offer a large selection of substitution products with different properties. There is a need to select criteria to evaluate possible alternate materials and make a decision. This is done by multi-criteria analysis using different methods [22]. Among the adopted criteria, the important ones are those that take into account the principles of sustainable development. Therefore, ecological materials, modern technologies or modernization of traditional ones with addition of raw materials from different branches of economy (tea to brick) are being sought.

Two types of approach to substitution can be observed. The first, in a more general sense, is an attempt to:

− produce new materials and building elements capable of performing the appropriate function in the construction of a building. Improve their physical, chemical, etc. properties and usability (durability, aesthetics, usability, operation, etc.), thanks to the development of materials engineering, using the achievements of science, nanotechnology, etc. [23–26]. They can be used interchangeably with traditional materials (instead of clay bricks, e.g., cellular concrete).

The second, however, related to the idea of sustainable development through:


The second approach to substitution is a partial restriction on the choice of a substitute by, for example, an architect, developer, or user by placing a condition (of an aesthetic, logistical, etc. nature). This situation relates to a specific building or material solution [34,35]. Here also the selection can be made in terms of one or more optimization criteria [36]. The

criteria are based on the individual requirements of the user, the investor or on current social and economic concepts: sustainable development, circular economy [37]. Applying material substitution, it is useful to have knowledge about the determinants of its use, the cause–effect relationships of the factors that have an impact on its use. Such research and results are presented in Section 4 of the article.

#### **3. Substitution in Historical Buildings**

It should be noted that in most European countries it is obligatory to replace the materials used in historic buildings with the same ones that were used originally. In Poland, however, the law allows the use of substitutes [16] depending on various conditions.

Factors that affect the possibility of using construction product substitutes are defined in the so-called conservation program, which is developed for each renovation of an object entered in the register of monuments. Each proposed substitute for a construction product must be prepared in the form of a sample and accepted by the Conservator.

Positive aspects of the application of construction product substitution in the renovation of buildings entered in the register of monuments are the factors that primarily enable the refurbishment. Historic sites were built in different construction realities, at a time when available building products were based on natural resources (e.g., stone, rock, clay) and the technology to produce them was simpler. It was common practice to import construction products from other areas of Europe. Even today it is costly and environmentally unfriendly and, due to the environmental protection of certain areas, exploitation is prohibited. However, it is worth considering the use of a substitute material and conducting an analysis of the impact of using such a solution on social, environmental, and economic factors [38–41].

Ownership of the most valuable objects entered in the register of monuments is mostly in the hands of the State or various institutions such as the churches. It should be remembered that the number of facilities under consideration is large and the possibilities of financing renovations are limited, hence the price will always be an important component of planning a refurbishment. The use of original construction products in one object may lead to abandonment or postponement of the renovation in other objects. Such a situation may lead to degradation of the remaining buildings and, consequently, increase the costs of renovations that are planned in them. Therefore, the introduction of substitutes for construction products in historic buildings, which give positive aesthetic and visual values and are less of a financial burden, gives the opportunity to conduct a more effective and larger-scale renovation policy.

In the case of the described refurbishments, a significant price-creating factor is also the time of completion. It is obvious that a longer period of renovation of one object can postpone the start of renovation in another object, which also requires this renovation. Substitution of construction products may increase the pace of renovation works in connection with, among others, less complicated technology of conducting works, faster pace of assembly of built-in elements, and the possibility of conducting works in less favorable weather conditions. Reducing the duration of the renovation gives further savings, thanks to which it is possible to predict that the renovation of a monumental object (the process of renovation of an object entered in the register of monuments takes a very long time because not all the necessary construction works can be predicted at the stage of designing the renovation) will be completed within the assumed time.

Construction products used in historic buildings have often survived years or even centuries. Thus, these are durable products that have been subject to gradual degradation over the years due to lack of refurbishment or minor damage, which has increased the impact zone from year to year [42,43]. The weakness of the construction product substitutes may be their durability and resistance to weather conditions in comparison with primary products and other influences e.g., related to the intensity of car exhaust or air pollution [44]. Renovations of buildings included in the register of monuments should be carried out by companies specializing in this type of construction works. Due to the specific nature of

renovation work in historic buildings, the contractor may encounter problems at each stage of the work that are unusual for newly erected buildings.

Substitutes of construction products used in the renovation of historic buildings give a wide range of possibilities. Substitutes can be manufactured from recycled, environmentallyfriendly materials and produced by local entrepreneurs [7,45]. The current technology of conducting construction works and the variety of construction products makes it possible to carry out a renovation of basically any building, including historic buildings.

In the current market situation, the cost and time of implementation are critical in any type of construction project. In the case of renovations of objects entered in the register of monuments, the specificity of the conducted construction works and a certain unpredictability of additional construction works, which may appear at each stage of the renovation, are still imposed.

The authors met with an opinion that a historic object that has undergone renovation with the use of construction product substitutes loses its historical value and should no longer be treated as a monument. The basic issue to consider in such a situation is the possibility of renovation.

In old, historic buildings, especially those protected by law (in Poland the register of historic monuments), the use of substitution of materials has a long history. There is an extensive literature in this area, including concepts of substitution principles, developed and proven methodologies for design, testing, analysis, and selection of substitutes [46–48].

#### **4. Evaluation of the Possibility of Using Material Substitution in the Maintenance of Buildings**

Preceding the decision to use substitution, the authors suggest performing an assessment and identifying key factors that influence the effectiveness of its use. On the basis of the presented conditions and factors influencing the application of substitution in construction objects in the exploitation phase, a SWOT (Strengths, Weaknesses, Opportunities, Threats) matrix was developed (see Table 1). It contains factors that constitute strengths and weaknesses of the substitution phenomenon and opportunities and threats in its application.



The analysis of the matrix, in particular the comparison of factors from different fields of the matrix gives an opportunity to determine the type of a possible general strategy in substitution activity, as well as detailed strategies in organizations dealing with the management of building real estate, including historic buildings.

If a strategy is established, reference should be made to a specific object.

On the basis of the analysis of information from the presented SWOT matrix, conclusions can be drawn with regard to the possibilities for developing material substitution in the construction industry. Undoubtedly, in the situation of emerging new and modern technologies, more and more diversified offers of the manufacturers' market allows for flexible and quick adaptation of investors to the dynamics of social and economic changes, especially for such long-lasting products as building structures. Undoubtedly, there is an advantage to the benefits of substitution in various aspects of the investment and construction process, both in terms of execution and ancillary activities, including logistics. Out of the four presented threats, two factors concern historic buildings, and one needs to be supplemented in legal regulations. The fourth one related to the requirements of relevant competences requires the support of the educational system.

The information contained in the presented SWOT matrix can be used in two ways. It can be used to analyze and generally evaluate the development of a certain phenomenon. It can also be used in the strategic analysis of an individual specific enterprise, company, or system.

This paper will use the data from the SWOT matrix to assess the overall feasibility of using substitution in building repair work (Section 5). The factors collected in the SWOT matrix can be used to establish cause–effect relationships between them. Identifying the causal chain will allow us to identify those factors that have the greatest impact on the process of substitution.

For an individual facility, on the other hand, this analysis will determine whether the planned substitution will have a more positive or negative impact on the renovated facility. It will also allow the investor to look at all the pros and cons of using substitution and assist him in making a final decision on the renovation policy on the chosen facility.

#### **5. Cause-and-Effect Analysis of the Use of Substitution**

#### *5.1. Research Methodology*

To identify cause–effect relationships in the issue of possible substitution of material solutions of buildings the authors propose to use the DEMATEL method [49–52]. When analyzing a multi-factor problem, a multi-criteria analysis is used to evaluate the problem using different methods that allow ranking of solutions. On the other hand, the DEMATEL method chosen by the authors also enables a cause-and-effect analysis of the phenomenon under study.

The computational flow is as follows:


4. Determination of the normalized direct influence matrix *A <sup>D</sup>*, which contains all parameters that take values that are in the range [0, 1] (Table 2). The normalizing number (*n*) is taken as the largest of the sum of the rows or columns of the matrix *AD*:

$$A'\_D = \frac{A\_D}{n},\tag{1}$$

$$m = \max\left\{ \sum\_{i=1}^{\text{II}} a\_{ij}; \sum\_{j=1}^{\text{II}} a\_{ij}; \right\},\tag{2}$$

5. It is also possible to develop an indirect impact matrix Δ*T*:

$$
\Delta T = A\_D^{\prime 2} \cdot \left( I - A\_D^{\prime} \right)\_{\prime} \tag{3}
$$

6. Determination of the total influence matrix T (Table 3):

$$T = A\_D' \cdot (I - A\_D'),\tag{4}$$

7. On the basis of the above matrices, the determination of the indices of position and relationship, respectively, which express in turn: *s*+—tells about the role of a given factor in the process of determining the structure of links between objects, while *s*−—expresses the total influence of a given factor on the others. These values are determined according to the formulas (Table 4):

$$s^{+} = \sum\_{j=1}^{n} t\_{ij} + \sum\_{j=1}^{n} t\_{ji} = R\_{T\_i} + \mathbb{C}\_{T\_{i'}} \tag{5}$$

$$s^{-} = \sum\_{j=1}^{n} t\_{ij} - \sum\_{j=1}^{n} t\_{ji} = R\_{T\_i} - \mathbb{C}\_{T\_i \prime} \tag{6}$$

When these values are plotted on a graphical representation, it is easy to see which factors have the greatest influence on the others and to determine which are the causes and which are the effects of the actions taken (Figure 5).

8. Finally, the net impact value is also determined, which tells the factor that has the greatest impact on the others considering both the causal and effect nature (Table 4):

$$netto = \ s^{+} + s^{-} \tag{7}$$


**Table 2.** The fragment of normalized direct influence matrix *A D*.


**Table 3.** Total influence matrix T (fragment).

**Table 4.** Summary of DEMATEL analysis results.


**Figure 3.** Direct influence graph—expert evaluation results.

**Figure 4.** The matrix of direct effects of factors on each other.

**Figure 5.** Graphical interpretation of DEMATEL results.

#### *5.2. Study Results and Its Analysis*

In supporting the decision to use substitution to examine the cause and effect relationships, all the factors summarized in the SWOT matrix were considered. These factors, as in the case of the SWOT matrix, were divided into the same four groups. To simplify the recording of the factors in further analysis with the help of the DEMATEL method, only the number from the SWOT table is marked (see Table 1).

The factors identified during the SWOT analysis are subjected to an assessment of the strength of their impact on each other.

For the analyzed issue—application of substitution, e.g., in repair works, the form of direct influence graph is presented in Figure 3. The intensity of relationships was coded using different hatchings of arc lines.

Based on the relationships illustrated above, a direct influence matrix *AD* was created (step 3).

Table 2 shows fragment of element values of the normalized matrix (step 4):

Next, based on Equation (3), the matrix of total relations T was determined:

A summary of the values to build an illustration of the causal nature is shown in Table 4 (step 7).

The analysis was performed by using summative, linear aggregation of the values of the position and relationship indicators (*s*<sup>+</sup> and *s*−). The calculations in general are expressed in the graph shown in Figure 5, which shows the values of the position and relationship indicators. Based on the aggregated values of the item index, it was found that the greatest role in determining the nature of the factors is played by: 3.1 (market-access and development of modern construction products) and 1.5 (replacement of products no longer manufactured).

Factors 3.2 (recycled products) and 3.3 (more environmentally friendly products) have slightly less influence. The clearly positive values of the relationship index for these factors indicate their causal character.

Almost half of the analyzed factors show a negative value of the relationship index, hence they should be treated as possible effects of the causes.

Of the factors with a negative relationship index value, a significantly outstanding negative value was obtained by 4.1 (loss of authenticity and historical value and object), which represents the largest negative possible effect of using substitution.

Factors with a positive sign but close to the zero value can be treated as elements of a mixed nature, partly causal, partly effectual, but both as causes and effects of far less importance.

The situation is different if we look at the values of the factors they obtain in the position axis (Figure 5). Factors with above average values of the item index testify to their leading role in determining the nature of individual factors. Among the prominent factors of the position indicator are 1.1 (possibility of renovation), 4.1 (loss of authenticity and historical value of the object), 1.2 (lower price of renovation), 1.5 (replacement of products no longer manufactured), and 3.1 (market-access and development of modern construction products). Again, as far as the others are concerned, they have far less active participation in the process of identifying the role of factors.

The aggregated values of the relation index allowed for distinguishing three groups of factors: key, average and insignificant for shaping the renovation policy. In particular, the key factors as reasons for decision-making turned out to be: 3.1, 1.5. Key factors as reasons for taking the group of average significant factors form: 3.2 *i* 3.3. The other factors can be considered by far the least important.

The possible impacts of the decision are definitely influenced by factor 4.1, which reflects the fear of losing the authenticity of the historic substance, as well as 1.1, which represents the opportunity for renovation. The fear of loss of authenticity should be the starting point in selecting the right, in this case the closest substitution to the original. The effect of being able to renovate is a decisive advantage of substitution and can often be the only solution to improve the technical condition of an object and extend its life cycle.

#### **6. Summary**

Substitution of construction products is a common phenomenon in the construction industry at every stage of a building's life cycle. Moreover, sometimes the use of substitution may be the only feasible solution to save a facility. In a wider context, it can have a great impact on the implementation of the principles of sustainability and circular economy in the maintenance of building stock.

The conducted observations show that during the warranty and guarantee period in newly constructed buildings, the substitution of construction products is much lower than in the next period of the facility's operation. The phenomenon of substitution, however, is often encountered in the long-term perspective of facility operation, especially during all repair and overhaul works.

One of the ways of extending the life cycle of buildings is a proper renovation policy, which through proper selection of material solutions will ensure longer durability of components and the entire facility.

A special case is the substitution in the renovation works of objects entered in the register of monuments, which gives the possibility to protect the historic substance while maintaining the structural and aesthetic values.

Product substitution, which is often cheaper than the original, may also allow for a wider range of renovations, which directly contributes to improving the technical and functional condition of the object and thus allows for extending its life cycle.

The SWOT analysis conducted by the authors allows us to conclude that substitution in the construction industry is justified and that there are great opportunities for its implementation and development. A detailed analysis of the SWOT matrix factors, using the DEMATEL method, allowed for an overall assessment of substitution possibilities with the determination of the cause–effect relationship of factors from particular groups characterizing the strengths and weaknesses of substitution as well as the opportunities and threats. Undoubtedly, the use of substitute materials, especially in historic buildings, will result in a decrease in their authenticity, but will ultimately restore them to safe operation and in other buildings allow for an extended life cycle.

Despite the possibility of product substitution thanks to materials engineering and technology development, the use of substitution should not be approached uncritically. The authors recommend a case-by-case approach, conducting a comprehensive analysis and making decisions based on, among other things, the tools proposed in the article and evaluating the cause-and-effect relationships that will occur when substitution is applied.

The above comment also applies to using a different approach to material substitution in historic buildings. The decision whether or not to use substitution and the freedom to choose substitution solutions are influenced by the conservation concepts and legal regulations of the respective country or type of object.

**Author Contributions:** Conceptualization, A.S., K.L. and A.R.; methodology, A.S., K.L. and A.R.; validation, A.S., K.L. and A.R.; formal analysis, A.S., K.L. and A.R.; investigation, A.S., K.L. and A.R.; resources, K.L. and A.R.; writing—original draft preparation, A.S., K.L. and A.R.; writing—review and editing, A.S. and A.R.; visualization, K.L. and A.R.; supervision, A.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

**Acknowledgments:** The work was carried out as part of statutory research no. 11.11.100.197 in the Department of Geomechanics, Civil Engineering and Geotechnics of Faculty of Mining and Geoengineering, AGH University of Science and Technology in Cracow.

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

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### *Article* **Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings**

**Jarosław Konior \* and Tomasz Stacho ´n**

Department of Building Engineering, Faculty of Civil Engineering, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland; tomasz.stachon@pwr.edu.pl

**\*** Correspondence: jaroslaw.konior@pwr.edu.pl; Tel.: +48-71-320-23-69

**Abstract:** The purpose of the research presented in the article is to identify the impact of the processes associated with the broadly understood maintenance of old residential buildings with a traditional construction on the size and intensity of the wear of their elements. The goal was achieved by analyzing the symptoms of the technical wear process, which involved the understanding of the mechanism of the occurrence of the phenomenon of damage, and the identification of the size and intensity of the damage to the elements of the evaluated buildings. The consequence of systematizing the most important processes that influence the loss of functional properties of residential buildings was the creation of the authors' own qualitative model and its transformation into a quantitative model. This, in turn, enabled a multi-criteria quantitative analysis of the cause and effect phenomena—"damagetechnical wear"—of the most important elements of downtown tenement buildings to be carried out in fuzzy conditions, i.e., uncertainty concerning the occurrence of damage and the wear process. The following key question was answered in the subjective expert assessment of the technical condition of an evaluated residential building: what is the probability of the wear of an element, which may be more or less correlated with its average maintenance conditions, or more simply, what is the probability that the element is more or less (approximately) worn? It has been proven that the conditional probability of the technical wear of an element in relation to its damage increases with the deterioration of the maintenance conditions of the building, and this increase is very regular, even in the case of different building elements. This probability is characterized by a low standard deviation and a narrow range of the dispersion of results in the case of various elements with regards to each of the considered building maintenance conditions.

**Keywords:** residential buildings; technical wear; damage; Bayes conditional probability; fuzzy sets

#### **1. Introduction**

*1.1. Literature Survey*

The literature survey was based on the theory of decision-making in the conditions of uncertainty and fuzziness, which is given by Kacprzyk [1], and which defines the following decision situations [2]:


**Citation:** Konior, J.; Stacho ´n, T. Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings. *Appl. Sci.* **2021**, *11*, 2518. https://doi.org/10.3390/ app11062518

Academic Editor: Marco Vona

Received: 17 February 2021 Accepted: 8 March 2021 Published: 11 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

• Fuzziness: indeterminacy not only relates to the occurrence of an event, but also to its meaning in general, which can no longer be described using probabilistic methods. Of course, further extensions are possible here, such as adding risk to fuzziness.

When assessing the degree of the technical wear of building elements, apart from measurable (quantitative) criteria, non-measurable (qualitative) criteria are also used. They are expressed in the analysis of symptoms by, i.e., damage that reduces the technical condition and utility value of building elements. Only some of these criteria can be roughly quantified. However, most of these criteria are qualitative. Their value is determined verbally by using terms such as "significant", "poor", "strong", "almost not at all", "partial", or "complete", and always appears in the description of damage phenomena. The interpretation of the effects of these phenomena, which is performed according to subjective and qualitative premises, leads to the indiscriminate categorization of the technical maintenance conditions for buildings and their elements, i.e., good, satisfactory, average, poor, or bad. Striving for a quantification of criteria that are inherently qualitative and immeasurable, and trying to determine the relations between them, led to the use of the category of fuzzy sets (the basis of which were formulated by Zadeh [3,4] and Yager [5,6]) with regards to this issue. Their properties enable damage to building elements, as well as the conditions of their technical maintenance, to be described within an unambiguous measurable quantitative aspect.

In the methodical approach to the technical assessment of residential buildings, research by Nowogonska [7–11] was used, which provides methods and models for the estimation of the degree of the technical wear of buildings. However, it should be remembered that the presented methodical approach of Nowogonska is exclusively deterministic, and therefore simplified and also practical. This approach is confirmed by the research of Lee and Kim [12], who indicated the degree of risk that is associated with damage to a building element. The assessment of the entire service life of a building structure includes a fuzzy calculation, which was presented in the publications of Plebankiewicz, Wieczorek, and Zima [13–16] in order to determine the impact and significance of the risk of the emergency operation of a building. The works by Ibadov [17–20] concerning the building investment process with a fuzzy phase allowed for the practical application of uncertain and subjective events when determining the degree of damage to the tested tenement houses. The assessment of the risk and costs of maintaining construction facilities, and also the conducting of the construction process in fuzzy conditions, were also presented by Kamal and Jain [21], Andri´c, Wang, Zou and Zhang [22], J. Marzouk and Amin [23], Knight, Robinson, and Fayek [24], Sharma and Goyal [25], Al-Humaidi and Hadipriono [26], Ammar, Zayed and Moselhi [27], Chan, Kwong, Dillon and Fung [28], and Naszrzadeh, Afshar, Khanzadi, and Howick [29].

Methods, models, and methodological tools for the assessment of the technical condition of buildings, which are considered in the article with regards to the research sample, were described and summarized by Konior in papers [30–34] with co-authors [35–37] and in a collective study under the supervision of Kapli ´nski, which is entitled "Methods and Models in the Engineering of Building Processes" [38].

#### *1.2. Research Sample*

The research sample, which included 102 technically assessed residential buildings from the "Srodmiescie" district of Wroclaw, was selected from a group of 160 examined objects [39]. The overriding criterion for sampling involved the obtaining of a comparable group of objects. Mutual comparability of the downtown tenement houses meant:


• identical functional solutions, understood as the standard of apartment amenities and furnishings (for that time), and a defined standard of living for residents.

The method of selecting the research sample at the level of greater detail was based on the mutual similarity of all the technical solutions of the downtown tenement houses. The selected research sample, according to the criteria presented above, is a representative sample with regards to the concept of representativeness that is specific to the adopted purpose of the study [40,41]. It contains all the values of the variables that could be recreated from the research carried out earlier using a different objective function than the one adopted in the study.

These values and variables were then compiled and processed in such a way that it was possible to make conclusions about the cause–effect relationships between them in the general population.

Therefore, it can be considered as a typologically representative sample that includes the desired types of homogeneous variables. Due to the fact that the structure of the population and its properties were previously well recognized, such a selection of the research sample can also be seen as a deliberate selection. It should be noted that the sample may not be representative in terms of the distributions of the studied variables, which may—for the adopted level of significance—not correspond to analogous distributions in the general population. It is also not known—at this stage of the research—whether the selected sample is representative due to the relationship between its variables and the identically defined variables in the entire set of downtown residential buildings. Therefore, at the very beginning of the research, it was assumed that a specific research sample occurs in the existing population with the fuzzy phase.

Tested buildings have been classified into classes, determined by the degree of the technical wear. The technical wear 0–15% has been classified to the class I, 16–30% to the class II, 31–50% to the class III, 51–70% to the class IV, 71–100% to the class V. Owing to the fact that all considering apartment houses belong to the same group of their age it is possible to assume that the class of the technical wear corresponds to the conditions of building maintenance. Therefore, the equivalence has been defined: a poor maintenance—the class IV, V, an average maintenance—the class III, an above than an average maintenance—the class II, a very well cared maintenance—the class I.

#### **2. Research Method**

#### *2.1. Problem Identification*

The research methodology at a level of greater detail was prepared in such a way that allowed the previously prepared qualitative model to be transformed into a quantitative model. Therefore, the diagnosis of the impact of the maintenance of the residential buildings on the amount of their technical wear was carried out using quantitative methods in fuzzy set categories, and also by using the authors' own model that was created in the conditions of fuzziness. The model allowed for the determination of the conditional probabilities of the process of technical wear, and also the set of damage according to both Bayes formulas [40–42] and the combined approach of Zadeh [3,4] and Yager [5,6];

As mentioned in the introduction, when visually assessing the technical wear of building elements, the symptoms of their destruction are taken into account, i.e., individual damage that can be categorized into the following groups (groups) of damage:


The purpose of such a conceptual and technical systematization of damage is a comprehensive diagnosis of the extent to which a building element is worn. This assessment, in turn, leads to the implication of stating under what technical conditions—good, satisfactory, average, poor, or bad—the building element was (is) maintained. It is difficult to define a fuzzy set with such a broad meaning as "average technical condition of maintenance"

using one membership function. In this case, a semantic analysis of the term "technical wear of a building element" was used, which was denoted with the symbol of a fuzzy set "Z". Let the technical wear of building element Z consist of: mechanical wear of its structure and texture (fuzzy set ZM), its technical wear caused by water penetration and moisture penetration (fuzzy set ZW), technical wear resulting from the loss of its original shape (fuzzy set ZD), and technical wear caused by the attack of biological pests (fuzzy ZP harvest). This sum can then be expressed as follows:

$$\mathbf{Z} = \mathbf{Z}\mathbf{M} \cup \mathbf{Z}\mathbf{W} \cup \mathbf{Z}\mathbf{D} \cup \mathbf{Z}\mathbf{P} \tag{1}$$

and when assuming the identity of the degree of technical wear and its visual symptom (Z ⇔ U)—damage to a building element that is integrated into the above-described damage sets—this expression takes the following form:

$$\mathbf{U} = \mathbf{U}\mathbf{M} \cup \mathbf{U}\mathbf{W} \cup \mathbf{U}\mathbf{D} \cup \mathbf{U}\mathbf{P} \tag{2}$$

If technical wear was assumed in the observed states with its measure—the degree of wear as a fuzzy set with no crisp membership boundary of {z} = Z—then the visual image of this wear—global damage to a building element—should be treated as a fuzzy set, the fuzzy events of which are arguments—distinguished types of damage {u} = U. Therefore, fuzzy random events are fuzzy sets that express the degree of technical wear, for which there is no complete (measurable) certainty of membership to the II, III or IV class of the technical maintenance of an element. The question then arises: what is the probability of an element being worn, which will more or less represent its average maintenance conditions. To put it simply, what is the probability that an element is more or less (approximately) worn?

The approach of Zadeh [3,4], who defined the probabilities of fuzzy events in the form of real numbers from interval [0, 1], was used in the research. Therefore, the probability of a fuzzy event, which is the technical wear of a building element, which corresponds to satisfactory, average, and poor maintenance conditions, was defined as:

$$\mathbf{P}(\mathbf{Z})\mathbf{I}\mathbf{I}\text{, III, IV = }\sum\_{i=1}^{n}\mathbf{p}(\mathbf{z}\_{i})\boldsymbol{\mu}\_{\text{zi}}(\mathbf{z}\_{i})\text{, if }\mathbf{Z} = \{\mathbf{z}\_{i}\} = \{\mathbf{z}\_{1},\mathbf{z}\_{2},\dots,\mathbf{z}\_{n}\} \tag{3}$$

For the global damage of a structural element, which is assumed equivalently to the event of technical wear, the probability of its occurrence is expressed by the following analogous relationship:

$$\text{P(U)II, III, IV} = \sum\_{\mathbf{j}=1}^{\text{m}} \mathbf{p}(\mathbf{u}\_{\mathbf{j}}) \,\upmu\_{\mathbf{u}\mathbf{j}}(\mathbf{u}\_{\mathbf{j}}) , \,\text{gdy} \,\mathbf{U} = \,\left\{\mathbf{u}\_{\mathbf{j}}\right\} \,\, = \left\{\mathbf{u}\_{\mathbf{1}\prime} \,\upmu\_{\mathbf{2}\prime} \,\dots , \,\text{u}\_{\mathbf{m}\prime}\right\} \tag{4}$$

The probabilities p(ui) of the occurrence of elementary damage ui in sets II, III, IV were calculated and then presented in Table 1.



**Table 1.** *Cont.*



It should be noted that a slightly simplified approach, in which a fuzzy number is assigned to the probability of fuzzy events, was used here. This is opposed to the *Yager* approach [5,6], in which the probabilities are fuzzy events. It is important that the study did not consider the differences between the concepts of fuzziness and randomness. It was only assumed, although these phenomena are different and described differently, that they may nevertheless occur together as two types of uncertainty.

#### *2.2. Model of Determining the Conditional Probabilities of the Process of Technical Wear in Relation to the Occurrence of Damage*

The preliminary assumption: the process of technical wear of building elements occurs when there is identifiable damage: {ZII, ZIII, ZIV} = Z ⇔ U.

The technical wear of building elements, determined by a group of experts in the II, III, and IV state of their technical maintenance, takes the following argument values:


Technical inspections of the residential buildings were executed by a team of experts consisted of:


In order to simplify the calculations, it was assumed that the domain of sets defined as fuzzy (ZII, ZIII, and ZIV) is interval [0.2, 0.7], and each of the sets contains a sum of N arguments of z1, z2, z3, z4. Each of these arguments occurs n times in the set. Without complicating the method with operations performed on fuzzy sets, it can be assumed that the degree to which arguments z1, z2, z3, z4 belong to fuzzy sets ZII, ZIII, ZIV is equal to the frequency of their occurrence in the sets:

$$
\mu\_{\rm xi} = \mathbf{n}\_{\rm i} / \mathbf{N}\_{\rm k}, \text{ where } \mathbf{i} = 1, 2, 3, 4, \text{ and } \mathbf{k} = 1, 2, 3 \Leftrightarrow \mathbf{II}, \text{III}, \text{IV} \tag{5}
$$

Each of the fuzzy sets ZII, ZIII, ZIV can be written with the use of the membership function as follows:

$$\text{ZII} = (\mu\_{\text{x1}}/\text{z1} + \mu\_{\text{x2}}/\text{z2} + \mu\_{\text{x3}}/\text{z3})\text{II} \tag{6}$$

$$\text{ZIII} = (\mu\_{\text{z1}}/\text{z1} + \mu\_{\text{z2}}/\text{z2} + \mu\_{\text{z3}}/\text{z3} + \mu\_{\text{z4}}/\text{z4})\text{III} \tag{7}$$

$$\text{ZIV} = (\mu\_{\text{x1}}/\text{z1} + \mu\_{\text{x2}}/\text{z2} + \mu\_{\text{x3}}/\text{z3} + \mu\_{\text{x4}}/\text{z4})\text{IV} \tag{8}$$

and when supplementing the output data with the values of their intersections:

$$\text{ZII} \bullet \text{ZIII} = (\mu\_{\text{x1}} \mu\_{\text{x1}} / \text{z1} + \mu\_{\text{x2}} \mu\_{\text{x2}} / \text{z2} + \mu\_{\text{x3}} \mu\_{\text{x3}} / \text{z3}) \text{II}, \text{III} \tag{9}$$

$$\text{ZII} \bullet \text{ZIV} = (\mu\_{\text{x1}} \mu\_{\text{x1}} / \text{z1} + \mu\_{\text{x2}} \mu\_{\text{x2}} / \text{z2} + \mu\_{\text{x3}} \mu\_{\text{x3}} / \text{z3}) \text{II}, \text{IV} \tag{10}$$

$$\text{ZZII}\bullet\text{ZIV} = (\mu\_{\text{x1}}\mu\_{\text{x1}}/\text{z1} + \mu\_{\text{x2}}\mu\_{\text{x2}}/\text{z2} + \mu\_{\text{x3}}\mu\_{\text{x3}}/\text{z3} + \mu\_{\text{x4}}\mu\_{\text{x4}}/\text{z4})\text{III}, \text{IV} \tag{11}$$

$$\text{ZII} \bullet \text{ZII} \bullet \text{ZIV} = (\mu\_{\text{z1}} \mu\_{\text{z1}} \mu\_{\text{z1}} / \text{z1} + \mu\_{\text{z2}} \mu\_{\text{z2}} \mu\_{\text{z2}} / \text{z2} + \mu\_{\text{z3}} \mu\_{\text{z3}} \mu\_{\text{z3}} / \text{z3}) \text{II}, \text{III}, \text{IV} \tag{12}$$

The probabilities of the occurrence of individual arguments in sets ZII, ZIII, ZIV are as follows:

$$\mathbf{p(z1)I}\mathbf{I} = 1/3; \mathbf{p(z2)I}\mathbf{I} = 1/3; \mathbf{p(z3)I} = 1/3; \mathbf{p(z4)I} = 0\tag{13}$$

$$\mathbf{p(z1)}\text{III}=1/4;\;\mathbf{p(z2)}\text{III}=1/4;\;\mathbf{p(z3)}\text{III}=1/4;\;\mathbf{p(z4)}\text{III}=1/4\tag{14}$$

$$\mathbf{p(z1)IV} = 1/4; \; \mathbf{p(z2)IV} = 1/4; \; \mathbf{p(z3)IV} = 1/4; \; \mathbf{p(z4)IV} = 1/4 \tag{15}$$

When using dependence (3), the degrees of membership of arguments z1, z2, z3, and z4 (6–12), and the probabilities of the occurrence of particular arguments in sets ZII, ZIII, ZIV (13–15), the partial probabilities of the occurrence of technical wear processes were calculated as fuzzy events in the satisfactory, average, and poor technical maintenance conditions of the analyzed residential buildings:

$$\mathbf{P(Z\Pi)} = (\sum\_{i=1}^{3} \mathbf{p(zi)} \,\mu\_{\text{zi}}(\text{zi})) \Pi \tag{16}$$

$$\text{P(ZIII)} = \sum\_{\text{i=1}}^{4} \mathbf{p}(\text{zi}) \,\updownarrow\!\!\mathbf{p}\_{\text{zi}}(\text{zi}) \,\text{III} \tag{17}$$

$$\text{P(ZIV)} = \sum\_{\text{i=1}}^{4} \mathbf{p}(\text{zi}) \,\updownarrow\text{p}\_{\text{zi}}(\text{zi})) \text{IV} \tag{18}$$

and their products:

$$\Pr(\text{ZII} \bullet \text{ZIII}) \stackrel{\text{3}}{=} \sum\_{\text{i}=1}^{3} [(\text{p}(\text{zi}) \,\mu\_{\text{zi}}(\text{zi})) \text{II} \bullet (\text{p}(\text{zi}) \,\mu\_{\text{zi}}(\text{zi})) \text{III}] \tag{19}$$

$$\mathbf{P}(\mathbf{ZII}\bullet\mathbf{ZIV}) = \sum\_{i=1}^{3} [(\mathbf{p}(\text{zi})\ \mu\_{\text{zi}}(\text{z}\_{i}))\mathbf{II}\bullet(\mathbf{p}(\text{zi})\ \mu\_{\text{zi}}(\text{zi}))\mathbf{IV}] \tag{20}$$

$$\Pr(\text{ZIII}\bullet\text{ZIV}) = \sum\_{i=1}^{4} [(\text{p}(\text{zi})\,\mu\_{\text{zi}}(\text{zi})) \text{III}\bullet (\text{p}(\text{zi})\,\mu\_{\text{zi}}(\text{zi})) \text{IV}] \tag{21}$$

$$\mathbf{P}(\mathbf{ZII}\bullet\mathbf{ZIII}\bullet\mathbf{ZIV}) \stackrel{\text{\tiny{\tiny{\tiny{3}}}}}{=} \sum\_{\mathbf{i}=1}^{3} [(\mathbf{p}(\text{zi})\,\upmu\_{\text{zi}}(\text{zi})) \mathbf{II}\bullet(\mathbf{p}(\text{zi})\,\upmu\_{\text{zi}}(\text{zi})) \mathbf{III}\bullet(\mathbf{p}(\text{zi})\,\upmu\_{\text{zi}}(\text{zi})) \text{IV}] \tag{22}$$

abilities of the occurrence of a set of damage to residential building elements in relation to the processes of their wear. It was assumed that the conditional probabilities, defined in such a way, correspond to the frequency of the occurrence of all elementary damage related to a single element in the following building maintenance conditions:


The above calculations of conditional and partial probabilities (16—22) allowed the probability of the occurrence of a group of damage to be determined in the middle, nonacute technical maintenance states of the analyzed residential buildings:

$$\begin{aligned} \mathrm{P(U)} &= \mathrm{P(U/ZII)} \bullet \mathrm{P(ZII)} + \mathrm{P(U/ZIII)} \bullet \mathrm{P(ZIII)} + \mathrm{P(U/ZIV)} \bullet \mathrm{P(ZIV)} - \mathrm{P(U/ZII \bullet ZIII)} \bullet \mathrm{P(ZII \bullet ZIII)} \\ &- \mathrm{P(U/ZII \bullet ZIV)} \bullet \mathrm{P(ZII \bullet ZIV)} - \mathrm{P(U/ZII \bullet ZIV)} \bullet \mathrm{P(ZIII \bullet ZIV)} \\ &+ \mathrm{P(U/ZII \bullet ZII \bullet ZIV)} \bullet \mathrm{P(ZII \bullet ZIII \bullet ZIV)} \end{aligned} \tag{23}$$

In the last stage of the developed model, the Bayes formula [37–39] for a posteriori probabilities was used. It determines the conditional probabilities of fuzzy events (i.e., the processes of the technical wear of building elements) in relation to another fuzzy event, i.e., the occurrence of their damage. The Bayes formula under satisfactory, average, and poor fuzziness conditions is as follows:

$$\Pr(\text{ZII}/\text{U}) = \frac{\Pr(\text{U}/\text{ZII}) \bullet \Pr(\text{ZII})}{\Pr(\text{U})} \tag{24}$$

$$\Pr(\text{ZIII}/\text{U}) = \frac{\Pr(\text{U}/\text{ZIII}) \bullet \Pr(\text{ZIII})}{\Pr(\text{U})} \tag{25}$$

$$\text{P(ZIV/U)} = \frac{\text{P(U/ZIV)} \bullet \text{P(ZIV)}}{\text{P(U)}} \tag{26}$$

The defined conditional probabilities of fuzzy event Z = {z1, z2, z3, z4} were supplemented, using the relationships (3) and (13)–(15), with the calculations of its mean value in relation to the probabilistic measure P(Z) in classes II, III, and IV of the technical maintenance of building elements:

$$\text{Im}\_{\mathbb{P}}(\mathbf{Z})\Pi\_{\prime}\Pi\_{\prime}\Pi\_{\prime} = \mathbf{1}/\mathbf{P}(\mathbf{Z})\Pi\_{\prime}\Pi\_{\prime}\Pi\_{\prime}\mathbf{V}\bullet\sum\_{i=1}^{4}\mathbf{p}(\text{zi})\,\mu\_{\text{zi}}(\text{zi})\mathbf{zi}\tag{27}$$

The values of the conditional probabilities of the technical wear processes Z, which correspond to the II, III, and IV maintenance conditions of 10 selected elements of the analyzed buildings, in relation to the occurrence of their damage U, and with their mean values in relation to the probabilistic measure P(Z), are given in Table 2.


**Table 2.** Fuzzy conditional

probabilities

 of the technical wear process in relation to the

probabilistic

 measure and the occurrence of damage, as well as the fuzzy


98

**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*

#### *2.3. The Model for Determining the Conditional Probabilities of a Set of Damage in Relation to the Process of Their Technical Wear*

The preliminary assumption: damage to building elements occurs when there is a process of their technical wear, which can be estimated within the range of 0–100%: {UII, UIII, UIV} = U ⇔ Z.

Damage to building elements, which is identified by experts in classes II, III, and IV of their technical maintenance, is defined as being dichotomous variables that assume values "0" (damage does not occur) or "1" (damage occurs). The domain of the set of damage, defined as fuzzy UII, UIII, UIV, is binary {0}, {1}.

It was assumed that the measure of the degree of membership of a single damage μuj to the set of a group of damage U, which is the symptom of the ongoing wear processes Z, is the feature that most fully expresses the correlation between these variables. This can be a point two-series correlation coefficient r(Z) ⇔ r(U), which is determined in each of the states II, III, and IV of the technical maintenance.

Each of the fuzzy sets UII, UIII, UIV can therefore be written using the membership function as follows:

$$\text{UIII, III, IV} = \left(\sum\_{\mathbf{j}=1}^{\mathbf{m}} \mathbf{r}(\mathbf{u}\_{\mathbf{j}})/\mathbf{u}\_{\mathbf{j}}\right) \text{II, III, IV, gdzie j} \rightarrow \text{ m} \in [5, 12] \tag{28}$$

and when supplementing the output data with the values of their products:

$$\text{UIII}\bullet\text{UIII}=\left(\sum\_{\mathbf{j}=1}^{\text{m}}\mathbf{r}(\mathbf{u}\_{\mathbf{j}})\bullet\mathbf{r}(\mathbf{u}\_{\mathbf{j}})/\mathbf{u}\_{\mathbf{j}}\right)\text{II}\,,\text{III}\tag{29}$$

$$\text{UIII}\bullet\text{UIV}=\left(\sum\_{\mathbf{j}=1}^{\text{m}}\mathbf{r}(\mathbf{u}\_{\mathbf{j}})\bullet\mathbf{r}(\mathbf{u}\_{\mathbf{j}})/\mathbf{u}\_{\mathbf{j}}\right)\text{II}\text{. IV}\tag{30}$$

$$\text{UIII}\bullet\text{UIV}=(\sum\_{\mathbf{j=1}}^{\text{m}}\mathbf{r}(\mathbf{u}\_{\mathbf{j}})\bullet\mathbf{r}(\mathbf{u}\_{\mathbf{j}})/\mathbf{u}\_{\mathbf{j}})\text{III},\text{IV}\tag{31}$$

$$\mathbf{U}\mathbf{III}\bullet\mathbf{U}\mathbf{III}\bullet\mathbf{U}\mathbf{IV} = \left(\sum\_{\mathbf{j}=1}^{m} \mathbf{r}(\mathbf{u}\_{\mathbf{j}}) \bullet \mathbf{r}(\mathbf{u}\_{\mathbf{j}}) \bullet \mathbf{r}(\mathbf{u}\_{\mathbf{j}}) / \mathbf{u}\_{\mathbf{j}}\right) \mathbf{II}\prime\prime\prime\tag{32}$$

When using relationship (4), the degrees of memberships of individual damage μuj = r(uj) to sets of groups of damage U (28)–(32), and by having data concerning the probabilities of individual damage in sets ZII, ZIII, ZIV (13)–(15), the partial probabilities of the damage were calculated as fuzzy events in the satisfactory, average, and poor technical maintenance conditions of the analyzed residential buildings:

$$P(\text{UII}) = (\sum\_{\mathbf{j=1}}^{\text{m}} \mathbf{p}(\mathbf{u}\_{\mathbf{j}}) \mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{II} \tag{33}$$

$$\text{P(UIII)} \;=\underset{\mathbf{j}=1}{\text{m}}\sum\_{\mathbf{j}=1}^{\text{m}}\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})\,\text{III}\tag{34}$$

$$\mathbf{P(UIV)} = \sum\_{\mathbf{j=1}}^{\mathbf{m}} \mathbf{p(u\_{\mathbf{j}})} \mathbf{r(u\_{\mathbf{j}})}) \text{IV} \tag{35}$$

and their products:

$$\Pr(\text{UIII}\bullet\text{UIII}) = \sum\_{\mathbf{j}=1}^{\text{m}} [\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{II}\bullet\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{III}] \tag{36}$$

$$\mathbf{P}(\mathbf{U}\mathbf{II}\bullet\mathbf{U}\mathbf{IV}) \stackrel{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\Gamma}\_{\Gamma}}}}}}}}}{\sum\_{j=1}^{\infty}} [\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})] \mathbf{II}\bullet\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})] \mathbf{IV}]} \tag{37}$$

$$\Pr(\text{UIII}\bullet\text{UIV}) = \sum\_{\mathbf{j}=1}^{\text{m}} [\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{III}\bullet\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{IV} \tag{38}$$

$$\Pr(\text{ZII}\bullet\text{ZII}\bullet\text{ZIV}) \stackrel{\text{m}}{=} \sum\_{\mathbf{j}=1}^{\text{m}} [\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{II}\bullet\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\mathbf{r}(\mathbf{u}\_{\mathbf{j}})) \text{III}\bullet\mathbf{p}(\mathbf{u}\_{\mathbf{j}})\text{r}(\mathbf{u}\_{\mathbf{j}})) \text{IV}] \tag{39}$$

The next stage of the created model involved the calculation of the conditional probabilities of the wear processes of the residential buildings' elements in relation to the occurrence of their damage. Due to the assumption that damage is an expression of technical wear, it was assumed, as in the case of defining the wear processes, that the conditional probabilities of the technical wear correspond to the frequency of the occurrence of all the elementary damage ({uj} = 1) of a selected building element in the II, III, IV conditions of its maintenance: P(Z/UII), P(Z/UIII), P(Z/UIV), P(Z/UII•UIII), P(Z/UII•UIV), P(Z/UIII•UIV), P(Z/UII•UIII•UIV).

The above calculations of the conditional and partial probabilities (33–39) enabled the probability of the occurrence of technical wear processes to be determined in the middle, non-acute technical maintenance states of the analyzed residential buildings:

#### P(Z) = P(Z/UII)•P(UII) + P(Z/UIII)•P(UIII) + P(Z/UIV)•P(UIV) − P(Z/UII•UIII)•P(UII•UIII) − P(Z/UII•UIV)•P(UII•UIV) − P(Z/UIII•UIV)•P(UIII•UIV) + P(Z/UII•UIII•UIV)•P(UII•UIII•UIV) (40)

In the last stage of the developed model, the Bayes formula for a posteriori probabilities was used again, which determines the conditional probabilities of fuzzy events (i.e., the occurrence of damage to building elements) in relation to another fuzzy event (i.e., the processes of their technical wear) [38]. The Bayes formula under satisfactory, moderate, and poor fuzziness conditions is as follows:

$$\Pr(\text{UII}/Z) = \frac{\Pr(Z/\text{UII}) \cdot \Pr(\text{UII})}{\Pr(Z)} \tag{41}$$

$$\Pr(\text{UIII}/\text{Z}) = \frac{\Pr(\text{Z}/\text{UIII}) \cdot \Pr(\text{UIII})}{\Pr(\text{Z})} \tag{42}$$

$$\Pr(\text{UIV}/\text{Z}) = \frac{\Pr(\text{Z}/\text{UIV}) \cdot \Pr(\text{UIV})}{\Pr(\text{Z})} \tag{43}$$

In this case, the mean value mp(U) of fuzzy event U = {u} in relation to the probabilistic measure P(U) is a constant value equal to one, because only the cases in which the dichotomous variable occurred were taken into account.

The values of the conditional probabilities of the occurrence of a group of damage, which correspond to the II, III, and IV maintenance conditions of 10 selected elements of the analyzed buildings, in relation to the processes of their technical wear, are presented in Table 2.

#### **3. Results**

The results of research concerning the impact of damage to building elements on their technical wear in the Bayes conditional probability domain (for damage and technical wear as fuzzy events) led to the following conclusions (within two aspects A and B—Table 2):

A. the probability of the conditional process of the technical wear, which corresponds to the three middle states of maintenance of the building elements, with regards to damage—P (Z/U) II, III, IV—is as follows:

	- for foundations: dampness of foundations 0.40
	- for basement walls: crack in bricks 0.39
	- for solid floors above basements: dampness of floors 0.38
	- for structural walls: cracks of plaster 0.40
	- for wooden inter-storey floors: weeping on floors 0.44
	- for internal stairs: weeping on stairs 0.46
	- for roof constructions: delamination of beams 0.35
	- for window joinery: mold and rot on windows 0.37
	- for inner plasters: scratches on plaster 0.36
	- for facades: scratches on plaster 0.37

The above values are therefore a fuzzy value of the probability of the degree of the technical wear, which was determined as an average degree, i.e., within the range of 35–50%—in the case of the occurrence of a fuzzy damage to the building element;

	- the conditional probability of damage to the element in relation to its technical wear increases with the deterioration of the building maintenance conditions;
	- the probability of such a conditionally defined fuzzy event is indicated by the damage that most intensely affects the technical wear of the following elements of the tested residential buildings, and it amounts in their average maintenance condition P(UIII/Z) to:
		- for foundations: dampness of foundations 0.27
		- for basement walls: crack in bricks 0.55
		- for solid floors above basements: dampness of floors 0.46
		- for structural walls: cracks of plaster 0.45
		- for wooden inter-storey floors: weeping on floors 0.31
		- for internal stairs: weeping on stairs 0.58
		- for roof constructions: delamination of beams 0.66
		- for window joinery: mold and rot on windows 0.50
		- for inner plasters: scratches on plaster 0.46
		- for facades: scratches on plaster 0.36

The above values are therefore a fuzzy value of the probability of damage to a building element, but only in the case that its fuzzy technical wear is determined to be an average degree, i.e., within the range of 35–50%;

• the irregularity of this increase and the too-high coefficients of variation indicate only a partial identity of the fuzzy event defined within aspect B with the reverse event; the fuzzy event determined within aspect A is characterized by a much greater consistency of the obtained results.

#### **4. Discussion and Conclusions**

Quantitative damage analysis, which was carried out using empirical methods of assessing the technical condition of a building, indicates the type and size of damage to the building's elements, which are characteristic of the appropriate maintenance conditions. Research concerning the cause–effect relationships ("damage-technical wear") in fuzzy calculus allowed for a numerical approach to the impact of building maintenance conditions on the degree of technical wear of its elements. The analysis of fuzzy cause–effect relationships ("damage-technical wear") created the possibility of determining conditional probabilities of these dependencies that are treated as fuzzy events. The fuzzy conditional probabilities of the technical wear process in relation to the occurrence of damage (with a probabilistic measure), as well as conditional probabilities of the occurrence of a group of damage in relation to the process of technical wear, were determined.

The research methodology has been prepared in such a way that allowed the previously prepared qualitative model to be transformed into a quantitative model. Therefore, the diagnosis of the impact of the maintenance of the residential buildings on the amount of their technical wear was executed using quantitative methods in fuzzy set categories, and also by using the authors' own model that was created in the conditions of fuzziness. The model allowed for the determination of the conditional probabilities of the process of technical wear, and also the set of damage according to both Bayes formulas applied to fuzzy sets operations.

The research procedure was developed in a way that allowed for the transition of a previously prepared qualitative model into a quantitative model. The diagnosis of the impact of the maintenance of residential buildings on the amount of their technical wear was carried out using quantitative methods in the categories of fuzzy sets, and also by using the authors' own model of determining the mutually dependent probabilities created in the conditions of fuzziness. The model enabled the conditional probabilities of the process of the technical wear, as well as the set of damage, to be determined according to probabilistic Bayes formulas. Moreover, it also allowed the fuzzy approach of Zadeh to be combined with the Yager approach. In such a multi-criteria fuzzy technical assessment of residential buildings, a simplified approach was used. In this approach, the probability of fuzzy events was assigned to a fuzzy measure, as opposed to the Yager approach, in which the probabilities are fuzzy events. The differences between the concepts of fuzziness and randomness were not considered in the study. It was assumed that these phenomena are different and described differently, however, they may—as two types of uncertainty occur together.

The methods and results of the research presented in the article indicated a way that allows for the transition of the previously prepared qualitative model into a quantitative model. The diagnosis of the impact of the maintenance of residential buildings on the amount of their technical wear was carried out using quantitative methods in the categories of fuzzy sets, and also using the authors' own models created in fuzzy conditions. The key question from the subjective expert assessment of the technical condition of the evaluated residential buildings was answered: what is the probability of the wear of an element that may be more or less represented by its average maintenance conditions? Therefore, the probability that the element is more or less worn was determined. It was proven that the conditional probability of the technical wear of an element in relation to its failure increases with the deterioration of the maintenance conditions of the building, and this increase is extremely regular, even in the case of different building elements. This probability is characterized by a low standard deviation and a narrow range of the dispersion of the results in the case of various elements within each of the considered building maintenance conditions.

**Author Contributions:** Conceptualization, J.K., T.S.; methodology, J.K.; software, T.S.; validation, J.K.; formal analysis, J.K., T.S.; investigation, J.K., T.S.; resources, J.K., T.S.; writing—original draft preparation, J.K.; writing—review and editing, J.K., T.S.; supervision, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

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

#### **References**


### *Article* **Damage and Technical Wear of Tenement Houses in Fuzzy Set Categories**

**Jarosław Konior \*, Marek Sawicki and Mariusz Szóstak**

Department of Building Engineering, Faculty of Civil Engineering, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland; marek.sawicki@pwr.edu.pl (M.S.); mariusz.szostak@pwr.edu.pl (M.S.) **\*** Correspondence: jaroslaw.konior@pwr.edu.pl; Tel.: +48-71-320-23-69

**Abstract:** The results and conclusions of the research presented in the article concern the topic of the technical maintenance and wear of traditionally erected residential buildings. The cause and effect relations between the occurrence of damage to the elements of tenement houses, which are treated as an expression of their maintenance conditions, and the size of the technical wear of these elements were determined in a representative and purposefully selected sample of 102 apartment houses built in the second half of the 19th and early 20th centuries in the Wroclaw, Poland downtown district "Srodmiescie". Recognition of the impact of the maintenance of residential buildings on the level of their technical wear was carried out using quantitative methods from fuzzy set categories, and also with the use of the authors' own model. The created model, based on the Zadeh function, was created in fuzzy conditions for the purpose of assessing the degree of damage to selected building elements. The treatment of the problem with regard to fuzzy criteria allowed for the synthesis of elementary criteria, which give the greatest approximations at the technical research stage of a residential building, into a global assessment of the degree of the wear of its elements. Moreover, it also significantly reduced the subjective factor of this assessment, which had a significant impact on the results of the research obtained in the case of good, medium and poor conditions of tenement houses. It was proven that the conditions of maintenance and use of buildings determine the amount of technical wear of their elements. The state of exploitation of the examined tenement houses is reflected in the mechanical damage to the internal structure of the elements (determined in fuzzy categories). This damage has a significant frequency and cumulative effects, and is characteristic for buildings with satisfactory and average maintenance.

**Keywords:** tenement houses; technical wear; damage; maintenance; fuzzy sets

#### **1. Introduction**

*1.1. Source Literature*

The aim of the research was to identify the impact of the processes associated with the broadly understood maintenance of old residential buildings with a traditional construction on the size and intensity of the wear of their elements. The degree of technical wear of residential building elements is a parameter of fundamental importance in the comprehensive assessment of their technical condition, regardless of the approach that was used in the test method. The aim of the research was achieved through the analysis of the symptoms of the technical wear process—understanding the mechanism of the phenomenon of damage and identifying the size and intensity of damage to the elements of the evaluated buildings.

Essential research of tenement houses aims to undertake a qualitative analysis of detected defects and identify all particular defects of their elements. Therefore, the "reason– effect" model is applied as follows:

> [ REASONS] <sup>→</sup> [observed SYMPTOMS ←−−−−−→ measured] <sup>→</sup> [EFFECTS ]

> > 107

**Citation:** Konior, J.; Sawicki, M.; Szóstak, M. Damage and Technical Wear of Tenement Houses in Fuzzy Set Categories. *Appl. Sci.* **2021**, *11*, 1484. https://doi.org/10.3390/ app11041484

Academic Editor: Asterios Bakolas Received: 18 January 2021 Accepted: 4 February 2021 Published: 6 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Commonly used mathematical methods and the broadly understood system analysis deal with real tasks in which the basic goal is the possibility of including all the types of indeterminacy among modeled quantities and the relationships between them. Every indeterminacy has traditionally been equated with the uncertainty of a random type, which has enabled known probabilistic and statistical tools to be used. In practice, however, there are many cases in which the indeterminacy of the type of inaccuracy, ambiguity and imprecision of meanings can be found. However, these situations are not of a random nature, and therefore traditional probabilistic models may not be adequate [1–7]. When assessing the possibility of random and/or fuzzy events occurring in construction investment projects, apart from immeasurable (qualitative) criteria, measurable (quantitative) criteria are also used. These quantitative criteria are expressed in a mathematical model that describes multiple phenomena of construction engineering processes. Only some of these criteria are strictly defined concepts—boundary, extreme. Most of these criteria are approximate. Their value is determined using descriptive methods, e.g., "good quality", "short term", "low budget". Therefore, concepts of this type cannot be adequately represented as a conventional set. To overcome this difficulty, in 1965, *Lotfi A. Zadeh* of the University of California in Berkeley introduced the concept of a fuzzy set with its membership function [8–10].

*Zadeh* [8,9], when developing the foundations of fuzzy set theory, formulated the following principle: "in general, complexity and precision are inversely related to each other in the sense that if the complexity of the problem under consideration increases, the possibility of its precise analysis decreases". *Yager* [11,12] independently came to a similar conclusion when examining the uncertainty in probability. However, people can cope with situations in which all attempts at the mathematical formalization of a task and its solutions are unsuccessful due to the fact that, e.g., it is impossible to build an exact mathematical model, or it would take too long to solve it. *Zadeh* saw the reasons for this in the ability of the human mind to think in approximate categories, which microprocessors do not have. Due to this, a person can process approximate and ambiguous data, create models of the most complex processes, determine approximate solutions, etc. Fuzzy set theory, according to *Zadeh* and *Yager*, is therefore a tool used to formalize this approximate reasoning in vague and ambiguous terms.

For a long time, "uncertainty" and "ambiguity" have been used as synonyms for a lack of knowledge, which is decreasing as research progresses. Relatively recently, starting from the 1970s, these terms began to be treated as a reflection of reality, without the previous clearly negative meaning. It was then that the first major works in the field of multiple applications of fuzzy sets occurred, including *Zadeh* [8,9], *Yager* [11,12] and *Sanchez* [13]. Summing up, among the formal apparatuses that led to the development of fuzzy set theory, the first place is occupied by multi-valued logics. Previously, since ancient times, almost the entire development of logic could have been equated with two-valued logic, in which a statement can only be either true or false. The fact of such a polarization of truth and falsehood was considered as an essential feature of any "logical" reasoning. Many logicians, represented especially by *Lukaszewicz* (a co-founder of the Polish School of Logic), were already aware of the mismatch between such "rigid" logic and reality. The explosion of interest in multi-valued logics also aroused a significant increase in the interest of fuzziness and its origins, which was widely described in the later works of *Zadeh* [9], *Yager* [12], *Sanchez* [13] and *Kasprzyk* [14].

The stage preceding the main scope of the work was the conducting of a qualitative analysis of damage to the elements of the tested residential buildings [1,15]. The technical characteristics and typological ordering of this damage, understood as an expression of the quality of maintenance of residential buildings, enabled the exploitation conditions of the considered objects to be identified.

A number of works by *Nowogo ´nska* [16–20] were used in the methodical approach to the technical assessment of tenement houses, and the fuzzy calculus presented in the publications of *Plebankiewicz, Wieczorek* and *Zima* [21–26] was used in the assessment of the whole service life of a building object. The works of *Ibadov* [27–30] and other authors [31–39], which concerned the construction investment process with the fuzzy phase, allowed for the practical application of uncertain and subjective events when determining the degree of damage to the tested tenement houses.

#### *1.2. Subject of Study*

A group of old tenement houses (that is, those erected before the First World War) takes an important place in Polish building resources. This group includes about 10.1% of the whole number of urban flats. What is more, the importance of this type of building relies on the fact that it takes part in creating an urban environment. At present, an action needs to be directed to the repair of the old land development. Doubtless, cultural aspects motivate all this action. To estimate its technical and economic justification, the degree of the technical wear of the old land development must be recognized and calculated.

This paper is a result of technical research and analyses on the old apartment houses in Wrocław, Poland [40]. The aim of the analysis is to provide information, which should help to direct an action, connected with this group of residential buildings. They are the apartment houses which were built at the turn of the nineteenth and twentieth centuries. The buildings are situated in the part of the city which (as a district from very few ones) was not completely destroyed by the war activities. The apartment houses are three- or four-storey buildings, made of bricks, erected in longitudinal, usually three-row, structural systems. Apart from the floors over the basement, which are solid ones, all the inter-storey floors represent typical wooden floors. All the buildings are covered with wooden rafter framing, usually a purlin–collar one. The staircases are composed of wooden or steel structural elements with wooden flights of steps.

#### *1.3. Research Problem*

While appraising building elements' technical wear—apart from applying the measurable (qualitative) criteria—the immeasurable (quantitative) criteria representing symptoms (pinpointed defects) of their deterioration have been taken into account. Only very few of these criteria can be classified at a high level of probability. There are symptoms of extreme characters, described by extreme dichotomic divisions. It is, however, agreed that between, e.g., a total pest attack to wooden elements and a lack of pests, the mid-states appear. Their value is often appreciated in a verbal way, e.g., "substantially", considerably", "significantly", "partially", "hardly" and it is always used in a description of detected defects as a result of a building object's technical inspections.

When assessing the degree of technical wear of building elements, apart from measurable (quantitative) criteria, immeasurable (qualitative) criteria are also used. They are expressed in the analysis of symptoms, i.e., damage, which lowers the technical condition and utility value of building elements. Only some of these criteria can be quantified with a big approximation. These are the symptoms with an extreme character, e.g., inter-story ceilings that are replaced with new elements that are not damp. It can then be assumed that the damage, and the technical wear it causes, take a value of zero.

In turn, flooding of the floors above basements does not raise doubts regarding the occurrence of the total dampness, and therefore the degrees of damage and technical wear caused by moisture take values equal to one within the variability interval of [0, 1]. Most of these criteria, however, are qualitative. Their value is determined verbally, e.g., as "significant", "poor", "strong", "almost not at all", "partial" or "complete", and it always appears in the description of damage phenomena. The interpretation of the effects of these phenomena, which is performed according to qualitative (i.e., subjective) premises, leads to the indiscriminate categorization of the technical maintenance conditions for buildings and their elements, i.e., good, satisfactory, average, poor or bad. Therefore, can a building element with a degree of technical wear of, e.g., 15%, be considered good or satisfactory from the point of view of the technical maintenance quality? Does significant biological contamination of wooden floor beams determine their 100% wear?

Striving for a quantification of criteria that are inherently qualitative (and therefore immeasurable), and trying to determine the relations between them, led to the use of the category of fuzzy sets with regard to this issue. Their properties enable damage to building elements, as well as the conditions of their technical maintenance, to be described within an unambiguous quantitative (measurable) aspect.

Therefore, the research led towards looking at the problem from this angle, which allowed the description of naturally qualitative (immeasurable) variables and the determination of existing relations between them in fuzzy set categories [15,25–39]. The advantages of fuzzy theory made it possible to describe the defects, representing three middle states (II, III, IV) of conditions of the building elements' maintenance, in a clear quantitative (measurable) aspect. Doubtless, fuzzy conditions are fully represented in these mid-states.

#### **2. Research Methodology**

#### *2.1. Fuzzy Set Theory*

The basic concept of the theory that was used in this paper is the concept of a fuzzy set [8,9,11–14]. The definition of a fuzzy set can be formulated as follows: a fuzzy set is set A, the x elements of which are characterized by the lack of a clear boundary between the membership and non-membership of x to A. The degree of the membership of element x to fuzzy set A is described by function μA(x), which is called the membership function. The μA(x) function takes values from the interval of [0, 1], where:

μA(x) = 0, which means that x is not a member of A;

μA(x) = 1, which means that x is a full member of A.

Fuzzy set A in a certain space (in this paper, it is the area of considerations concerning the observed states) X = {x}, which is written as A ⊆ X, is called the set of pairs:

$$\mathcal{A} = \langle (\mu \mathcal{A}(\mathbf{x}), \mathbf{x}) \rangle\_{\prime} \,\forall \, \mathbf{x} \in \mathcal{X}\_{\prime}$$

Therefore, two basic fuzzy sets can be distinguished in a problem (each one is described in the three following observed states—II, III, IV):


The basic operations performed on the fuzzy sets defined in the article are presented below:

• the absolute complement of the fuzzy set A ⊆ X, denoted as −A:

$$
\mu\_{-\mathcal{A}}(\mathbf{x}) = 1 - \mu\_{\mathcal{A}}(\mathbf{x}), \forall \mathbf{x} \in \mathcal{X} \tag{1}
$$

• the multiple sum of fuzzy sets A,B ⊆ X, denoted as A ∪ B:

$$
\mu\_{\text{A}\cup\text{B}}(\mathbf{x}) = \mu\_{\text{A}}(\mathbf{x}) \lor \mu\_{\text{B}}(\mathbf{x}),
\forall \mathbf{x} \in \mathcal{X} \text{ (symbol} \lor \text{ denotes } \text{\textquotedblleft} \text{\textquotedblright})\tag{2}
$$

• the intersection of fuzzy sets A,B ⊆ X, denoted as A∩B:

$$
\mu\_{\mathbf{A}\cap\mathbf{B}}(\mathbf{x}) = \mu\_{\mathbf{A}}(\mathbf{x}) \land \mu\_{\mathbf{B}}(\mathbf{x}), \forall \mathbf{x} \in \mathcal{X} \text{ (symbol}\land \text{ denotes } \text{\textquotedblleft}\text{\textquotedblright)}\tag{3}
$$

• the k-th power (k > 0) of fuzzy set A ⊆ X, denoted as Ak:

$$
\mu\_{\mathcal{A}}{}^{\mathbf{k}}(\mathbf{x}) = (\mu(\mathbf{x}))^{\mathbf{k}}, \forall \, \mathbf{x} \in \mathcal{X} \tag{4}
$$

Special cases of exponentiation include:

• the concentration of fuzzy set A ⊆ X, denoted as CON (A):

$$
\mu\_{\text{CON}(\Lambda)}(\mathbf{x}) = \left(\mu\_{\Lambda}(\mathbf{x})\right)^2, \forall \mathbf{x} \in \mathcal{X} \tag{5}
$$

• the dilution of fuzzy set A ⊆ X, denoted as DIL (A):

$$
\mu\_{\rm DIL(A)}(\mathbf{x}) = (\mu\_A(\mathbf{x}))^{0.5}, \forall \, \mathbf{x} \in \mathcal{X} \tag{6}
$$

All these operations, which are of great importance in linguistic semantics, are interpreted as:


When visually assessing the technical wear of building elements that inspected tenement houses consist of, the symptoms of their damage are taken into account, i.e., individual damage that can be categorized into the following groups of damage:


The purpose of such a conceptual and technical systematization of damage is a comprehensive diagnosis of the extent to which a building element is worn. This assessment, in turn, leads to the implication of stating under what technical conditions—good, satisfactory, average, poor or bad—the building element was (is) maintained. The terms "good technical condition of maintenance", "satisfactory technical condition of maintenance", etc., can be considered as fuzzy sets with regard to semantic (qualitative) and technical (quantitative) aspects.

It is difficult to define a fuzzy set with such a broad meaning as "average technical condition of maintenance" using one membership function. In this case, a semantic analysis of the term "technical wear of a building element" was used, which was denoted with the symbol of a fuzzy set "Z". Let the technical wear of building element Z consist of: mechanical wear of its structure and texture (fuzzy set ZM), its technical wear caused by water penetration and moisture penetration (fuzzy set ZW), technical wear resulting from the loss of its original shape (fuzzy set ZD) and technical wear caused by the attack of biological pests (fuzzy ZP harvest). This sum can then be expressed as follows:

$$\mathbf{Z} = \mathbf{Z}\mathbf{M} \cup \mathbf{Z}\mathbf{W} \cup \mathbf{Z}\mathbf{D} \cup \mathbf{Z}\mathbf{P} \tag{7}$$

and when assuming the identity of the degree of technical wear and its visual symptom (Z ⇔ U)—damage to a building element that is integrated into the above-described damage sets (Expression (7))—it takes the following form:

$$\mathbf{U} = \mathbf{U}\mathbf{M} \cup \mathbf{U}\mathbf{W} \cup \mathbf{U}\mathbf{D} \cup \mathbf{U}\mathbf{P} \tag{8}$$

#### *2.2. Research Model*

The aim of the proposed model is to assess the technical wear of a building element with regard to the overriding criterion, i.e., "slightly worn, worn, significantly worn". The concepts defined in this way at the basic level best describe the behavior of a building element in its three middle maintenance states. It is in them, after rejecting the extreme states (i.e., good and bad) that have the most reliable evaluation principles from the technical point of view, that the fuzzy conditions are most fully represented. The basic principles of fuzzy logic and approximate reasoning were applied [8,9,11–15], and the fuzzy state was described as follows: its damage means that it can be classified as being in a satisfactory (II), average (III) and poor (IV) technical condition of maintenance. Therefore, in each maintenance state, the total damage to a building element is a multiplicity sum of the sets of damage, and it is expressed by Formula (9):

U(II, III, IV) = UM(II, III, IV) ∪ UW(II, III, IV) ∪ UD(II, III, IV) ∪ UP(II, III, IV) (9)

where each set of damage, in each of the three maintenance states (II, III, IV), is a set of basic damage uj, which represents elementary lower order criteria:


Multiplication sum (9) can be written in each of the three maintenance states (II, III, IV) using the membership function:

$$
\mu\text{U} = \mu\text{U}\mathfrak{M} \lor \mu\text{U}\mathfrak{W} \lor \mu\text{U}\mathfrak{D} \lor \mu\text{U}\mathfrak{P} \tag{10}
$$

There is an intermediate stage between identifying damage at the elementary level, which occurs in everyday construction practice, and merging it into sets of damage in terms of their similarity regarding the wear processes. This stage involves the selection of damage of the same type but of different intensity (e.g., pitting corrosion, surface corrosion, deep corrosion of steel beams), or damage occurring to complex elements (e.g., structural walls—decay of brick or mortar). This method of combining elementary damage was used in the research, which led to the obtaining of greater possibilities of using operations of system analysis in fuzzy sets. In the considered sample of downtown tenement houses, this division is as follows:


In each of the damage types distinguished in this way, there is an intersection of two or three elementary fuzzy sets. Between them, as is the case between sets of damage, there is a multiple sum of the fuzzy sets that are defined above. All these dependencies can be described by the general formula for assessing the degree of damage to the elements of the analyzed residential buildings in their middle maintenance states which, when using the membership function, is as follows:

```
μU = (μu1 ∧ μu2) ∨ (μu3 ∧ μu4) ∨ (μu5 ∧ μu6) ∨ (μu7 ∧ μu8) ∨ (μu9 ∧ μu10) ∨
∨ (μu11 ∧ μu12) ∨ (μu13 ∧ μu14) ∨ (μu15 ∧ μu16∧ μu23) ∨ (μu17 ∧ μu18 ∧μu19) ∨
∨ (μu20 ∧ μu21 ∧ μu22) ∨ (μu24 ∧ μu25) ∨ (μu26 ∧ μu27∧ μu28) ∨ (μu29 ∧ μu30)
                                                                                  (11)
```
Due to the fact that the greatest approximations of the observed states can be obtained at the level of elementary criteria, the degrees of membership of damage u1 ÷ u30 to fuzzy sets UM, UW, UD, UP were calculated at the stage of the basic comparative analysis, in which the fundamental probabilistic measure is the probability of the occurrence of a single damage p(uj) in the II, III and IV maintenance states. The probability of p(uj) is therefore a feature that determines the membership to elementary sets u1 ÷ u30. It would not be a mistake to simply identify the probabilities p(uj) with the degrees of memberships μuj, which are described linearly by the membership function operating on the domain [0, 1]. However, in order to present the properties of fuzzy sets more closely, the function used by *Zadeh* [8–10] was chosen for intensifying the contrast of the fuzzy set A ⊆ X:

$$\mu\_{INT(A)}(\mathbf{x}) = \begin{cases} \quad 2\left(\mu\_A(\mathbf{x})\right)^2, \forall \mathbf{x} : \mu\_A(\mathbf{x}) < 0.5\\ \quad 1 - 2\left(1 - \mu\_A(\mathbf{x})\right)^2, \forall \mathbf{x} : \mu\_A(\mathbf{x}) \ge 0.5 \end{cases} \tag{12}$$

Therefore, the intensification of contrast increases the membership degrees that are greater than or equal to 0.5, while reducing the membership degrees that are lower than 0.5 (Figure 1).

**Figure 1.** The effect of the contrast intensification of the degrees of damage memberships.

The final stage of the created model for assessing the technical wear (damage degree) of selected building elements in the three middle states of their technical maintenance is to estimate the size of the impact of elementary damage on the total damage. The study of the observed states and the conclusions from the proposed method of associating the occurring damage with the occurrence of the process of technical wear indicate a significant range of the strength of this relation within one building element in the maintenance states II, III and IV [1–7]. None of the values of the bi-serial correlation coefficient r(Z), which is a measure of this relationship, reaches a value of 1 in domain [0, 1]. Therefore, when taking the extreme value from this range as a reference point, it can be assumed that none of the values of the correlation coefficient r(Z) concentrates the fuzzy set U, while each of them—to a different degree—dilutes it. Considerations regarding the relationship between these dependencies and the analysis of the effects of the dilution process of fuzzy sets have led to the determination of the weights of the degrees of membership of elementary damage uj as a function of the correlation coefficient r(Z):

$$
\mu\_{\rm trj} = [\mathbf{f}(\mu\_{\rm trj})]^{1/\mathbf{r}(\mathbf{Z})} \tag{13}
$$

The result of the proposed Formula (13) is the following change in the membership function:


The application of the original procedures of the intensification and dilution of membership functions, according to Formulas (12) and (13), to the general Formula (11) of the model for assessing the degree of damage to the elements of the analyzed tenement houses in terms of fuzzy sets allowed for the transition from the data recorded using non-measurable variables to results defined by measurable values. The proposed model gives a numerical answer to the question of to what extent is a building element damaged. The total degrees of damage to the ten selected elements of the analyzed buildings S(U) in the maintenance states II, III and IV are presented in Table 1.

**Table 1.** The degree of fuzzy damage to building elements in their middle maintenance states. (grey backgroud is necessary to distinguish extreme values)



**Table 1.** *Cont.*


**Table 1.** *Cont.*

#### **3. Results**

The analysis of the results of the research concerning the impact of damage to building elements on their technical wear with regard to fuzzy sets leads to the following conclusions (Table 1):

	- the development of the model presented in the article allowed the fundamental question of to what extent a building element is worn (damaged), when knowing that it is (more or less) satisfactorily, moderately or poorly maintained, to be answered;
	- the use of simple operations in the fuzzy set calculus enabled the influence of both elementary damage that occurs with a specific frequency (probability) and the measure of its interdependence (correlation) on the observed technical wear of building elements to be considered;
	- as a result of the proposed model, which is based on fuzzy set theory, it was possible to identify the elementary damage that determines the degree of destruction of the building's elements;
	- the degree of damage to the element increases with the deterioration of its maintenance conditions (although not proportionally to the maintenance conditions and not equally for different types of elements). For instance, degrees of fuzzy damage set S(U) corresponding to the maintenance states II, III and IV grow in the following way: Z3—basement walls—u3—brick losses: 0.05; 0.25; 0.67. It most often differs from the observed values of the degree of the technical wear that was determined using the probabilistic approach [1]—in particular, in poor conditions of building maintenance, the degree of damage exceeds 70% of its technical wear threshold;
	- elementary damage that determines the degree of destruction of an element comes much more often from group I (mechanical damage to the structure and texture of elements) than was the case in the analysis of the observed states. Only under poor conditions of building maintenance does the analysis of the observed random [1] and fuzzy [15] phenomena show a great similarity—the decisive damage is the destruction of the element caused by water penetration and moisture penetration (group II);
	- at the level of the greatest detail, the type of damage and the degrees of fuzzy damage to the elements of the downtown tenement houses were determined. In the most representative, i.e., average/satisfactory condition of maintenance—S (U) III—the degrees were as follows:
		- for foundations: brick decay 0.59
		- for basement walls: brick decrements 0.25
		- for solid floors above basements: brick decrements 0.22
		- for structural walls: mortar decrements 0.93
		- for wooden inter-storey floors: weeping 0.64
		- for internal stairs: mechanical damage 0.56
		- for roof constructions: weeping on wooden elements 0.43
		- for window joinery: mechanical damage 0.85
		- for inner plasters: plaster decay 0.85
		- for facades: cracks on plaster 0.94

#### **4. Summary and Discussion**

At the beginning, general methodological conclusions were formulated. They resulted from the modeling of the impact of the maintenance of tenement houses on the technical wear of their elements in fuzzy conditions. Such an approach gives much greater possibilities of studying cause and effect relationships than the probabilistic analysis [1]:


The consequence of systematizing the most important processes that influence the loss of functional properties of residential buildings was the creation of the authors; own qualitative model and its transformation into a quantitative model. This, in turn, enabled a multi-criteria quantitative analysis of the cause–effect phenomena—"damage–technical wear"—of the most important elements of downtown residential buildings to be conducted in the so-called conventional and fuzzy sets. In conventional sets, in which attempts were made to describe the observed (empirical) states with the use of theoretical formulas, the probabilistic side of the problem and its random nature were considered [1]. In turn, in fuzzy sets, the observed states of cause–effect phenomena in the fuzzy conditions [15] (i.e., uncertainty as to the very fact of their occurrence) were analyzed.

The fact that the membership function of a fuzzy set assumes values from interval [0, 1] leads to the hasty conclusion that fuzziness is a hidden form of randomness, and therefore fuzzy set theory is basically nothing new in relation to probability. The differences between fuzziness and randomness, however, concern both their nature and the formal differences between probabilistic calculus and fuzzy sets. The nature of these phenomena lies in the problem of the uncertainty of the type of randomness and fuzziness. In the case of randomness, the event is strictly defined, while its occurrence is uncertain. Therefore, randomness can be equated with the uncertainty regarding an element's membership or non-membership. This is not the case with fuzziness, which concerns the very degree of membership of an element to a set, and therefore an event is no longer strictly defined. Such events are the ones analyzed in the paper—the occurring damage of building elements and the processes of their technical wear. Their nature, in the authors' opinion, is more fuzzy than random.

The differences between randomness and fuzziness can be presented with regard to the following three points of view:


Regarding "the degree of uncertainty", the following decision-making situations, with an increasing degree of uncertainty, can be distinguished:


The sense of a fuzzy set can therefore be used to formally determine and quantitatively express ambiguous concepts that are always present in the programming and analysis of a construction process. Thus, fuzzy set theory is a theory of classes in which the transition from membership to non-membership does not have a jumping character, as is the case in a conventional set, but instead it is gradual. Striving for a quantification of criteria that are inherently qualitative (and therefore immeasurable), and trying to determine the relations between them, led to the use of the category of fuzzy sets with regard to this issue. Their properties enable elementary construction processes to be mathematically described as fuzzy events within an unambiguous quantitative (measurable) aspect.

To sum up, the approach of the creator of fuzzy set theory [8–10] (*Lofti Zadeh*, who, unlike *Yager* and *Kaufmann* [11,12], assumed the fuzzy set as a random event) was consciously used by the authors. This enabled the question of what is the probability that a building element is more or less (approximately) worn to be answered. Therefore, the differences between the concepts of fuzziness and randomness were not considered. It was only assumed that although these phenomena are different and described differently, they may nevertheless occur together as two types of uncertainty.

**Author Contributions:** Conceptualization, J.K., M.S. (Marek Sawicki) and M.S. (Mariusz Szóstak); methodology, J.K.; software, J.K. and M.S. (Mariusz Szóstak); validation, J.K., M.S. (Marek Sawicki) and M.S. (Mariusz Szóstak); formal analysis, J.K., M.S. (Marek Sawicki) and M.S. (Mariusz Szóstak); investigation, J.K. and M.S. (Marek Sawicki); resources, J.K. and M.S. (Marek Sawicki); writing—original draft preparation, J.K.; writing—review and editing, J.K. and M.S. (Mariusz Szóstak); supervision, M.S. (Marek Sawicki). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

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

#### **References**

