Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses

The research presented in this article was conducted on a representative and purposefully selected sample of 102 residential buildings that were erected in the second half of the nineteenth and early twentieth centuries in the downtown district of Wroclaw (Poland). The degree of the technical wear of an old residential building is determined by the conditions of its maintenance and operation. The diagnosis of the impact of the maintenance of residential buildings on the degree of their technical wear was carried out using quantitative methods in the categories of fuzzy sets and also by using the authors’ own models created in fuzzy conditions. It was proved that the expression of the operational state of a building, considered as the process that plays the greatest role in its accelerated destruction, is mechanical damage to the internal structure of its elements. This damage is determined in the categories of fuzzy sets and has a high frequency and a cumulative effect of occurrence, which are characteristic for buildings in satisfactory and average maintenance conditions. The use of simple operations in fuzzy set calculus enabled the impact of elementary damage that occurs with a specific frequency, as well as the measure of its correlation on the observed technical wear of building elements to be considered. As a result, it was possible to identify the elementary damage that determines the degree of the technical wear of a building element. For each of the selected building elements, the maximal and minimal fuzzy relational equations (damage and technical wear) were determined. Their solutions were given in the form of clear relational matrixes that constitute big data arrays. They define the domain and range of the maximal and minimal fuzzy relations, the height of the fuzzy relations, their differences, and the place of their occurrence between the maximal and minimal dependencies.


Introduction
Commonly used mathematical methods and broadly understood system analysis deal with real tasks, in which the main goal is the possibility of including all types of indeterminacy among modeled quantities, as well as the relationships between them [1][2][3][4]. Every indeterminacy is traditionally identified with an uncertainty of a random type, which enables the use of known probabilistic and statistical tools [5][6][7]. In the technical assessment of building structures, the indeterminacy of the type of inaccuracy, ambiguity, and imprecision of meanings can be found. In the conditions of fuzziness [8][9][10][11][12][13][14], indeterminacy not only concerns the occurrence of a certain event but also its meaning in general, which cannot be captured using probabilistic methods [15][16][17][18].

Literature Survey
The literature survey was limited to source books and publications, which describe the use of random and fuzzy calculus in multiple applications that have uncertainty and indeterminacy of events occurring in them. Table 1 presents the analyzed literature items, which are divided into engineering processes with fuzzy and random approaches. Table 1. Summary of references, authors, year of publication, topics of study, and type of approach.

Technical Damage and Wear
The problem of the damage and destruction of residential buildings is widely described in the literature [19][20][21][22][23][24][25][26][27][28][29]. There is even a classification of damage that is characteristic for different types of buildings [30][31][32][33][34][35][36][37][38], e.g., those erected using industrialized methods or those with traditional construction. Unfortunately, the problem is recognized worse when it is considered in more detail, i.e., when it concerns individual elements of traditional residential buildings. The individual approach to the issue that is presented in this article may be interesting due to the fact that it associates the type and frequency of damage to the elements of downtown tenement houses with the ongoing process of their technical wear. This aspect of the mutual relationship between damage and technical wear, considered as the fuzzy expression of the maintenance conditions and ways of using residential buildings in the downtown district of Wroclaw (Poland), is presented in this article.
The result of an in-depth analysis of technical reports [55] of the examined group of 102 downtown tenement houses was the identification (at the elementary level) of all the damage to building elements. The identified defects constitute the following groups of damages [26,28]: • I-mechanical defects of the structure and surface of elements • II-defects of elements caused by water penetration and humidity migration • III-defects symptomatic of the loss of the original shape of wooden elements • IV-defects of wooden elements attacked by insects-technical pests of wood The synthesis of the type of elementary failures with regards to their common cause of occurrence (II, IV), as well as the similarity of the consequences that are caused by them (I, II, III), enabled the damage to be semantically and generically ordered into the following groups: I, II, III, IV. The scope of research, which concerned 102 residential buildings, allowed the frequency of the occurrence of 30 elemental examples of damage to be determined ( Table 2). The presentation of the type and probability of the occurrence of damage was limited to the 10 elements that had the highest share in the tested tenement houses.

Setting the Problem-Fuzzy Relations
The basic concept of the theory that was used in this chapter is the concept of a fuzzy set [8][9][10][11][12][13][14]. The definition of a fuzzy set can be formulated as follows: a fuzzy set is set A, the elements x of which are characterized by a 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 µ A (x), which is called the membership function. The µ A (x) function takes values from the interval of [0,1], where: µ A (x) = 0 means that x is not a member of A; µ A (x) = 1 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 [8][9][10][11]14]: Therefore, two basic fuzzy sets can be distinguished: With the use of conventional (non-fuzzy) sets, certain strictly defined properties of theoretical and observed states were adequately expressed, and with the use of the (non-fuzzy) relation, interrelationships between the variables of these states, if they existed, were defined [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54].
The problem arose when there was a need to express the interdependencies that are not very strictly defined, e.g., what is the impact (cause) of the ways of maintaining residential buildings on their technical wear (effect) or damage (to the elements), which is its symptom. The concept of a fuzzy relation was used to define the problem and to record the parameters of this phenomenon. Each pair of arguments (x,y)⇔(z,u) had a membership degree (measure) assigned to it, which expressed the intensity of the relationship between Z and U; that is, how Z depends on U, i.e., what the interrelationships (correlations) between them are. It was assumed that Z and U, as non-crisp sets, which are defined in fuzzy conditions, may have a certain relationship with each other. This led to the determination of the concept of a fuzzy relation in the following form [8][9][10][11]14]: A fuzzy binary relation R between two sets Z = {z} and U = {u} is a relation defined as a fuzzy set that is determined by the Cartesian product Z × U: and, therefore, it is the set of pairs: where µ R : Z × U→[0, 1] is the membership function of fuzzy relation R, which assigns to each pair (z,u): z∈Z u∈U its degree of membership µ R (z,u)∈[0, 1], which in turn is a measure of the intensity of the fuzzy relation R between Z and U. Thus, the fuzzy relation can be written as: The basic fuzzy relation, which is defined in the research at a general level, is the relation concerning the technical wear Z and damage U of the entire residential building that consists of the 23 elements subjected to technical tests. Therefore, Z = {Z1, Z2, ..., Z23} and U = {u1, u2, ..., u30}. The measure of the relationship between Z (for 10 selected building elements) and U are the values of the point two-series correlation coefficient r(Z), which are identical to the degree of the membership of µ R ⇔µ Zu [9]. Fuzzy relation R = Z × U, which was defined in this way, is presented in the form of a fuzzy relation matrix ( Table 3).
The main parameters of the matrix of fuzzy relation R = Z × U are as follows: The matrix's components, which represent the fuzzy relations between Z and U, provide the extent to which the maintenance of residential buildings (expressed as the intensity of the occurrence of damage to their elements) affects the amount (degree) of their technical wear. For all the fuzzy relations R ⊆ Z × U, which are determined later in the paper for the fuzzy sets that represent satisfactory, average, and poor maintenance conditions of residential buildings, the following were determined [14]: the domain of the fuzzy relation R ⊆ Z × U, which is called the first projection of the fuzzy relation, and denoted by domR: the scope of the fuzzy relation R ⊆ Z × U, which is called the second projection of the fuzzy relation, and denoted by ranR: the height of the fuzzy relation R ⊆ Z × U, which is called the global projection of the fuzzy relation, and marked as h(R) where if h(R) = 1, the fuzzy relation is normal, and if h(R) < 1, it is subnormal. The domain, scope, and height of the maximal and minimal fuzzy relations R ⊆ Z × U for 10 selected elements in the II, III, and IV maintenance class of a residential building are given in Table 4. The table presents quantitative and measurable values that can be easily quantified.

Fuzzy Relational Equations
All the relations on the fuzzy sets that were analyzed in this article are anti-transitive and weakly antisymmetric [14]. Checking transitivity with regards to the min-max and max-min is cumbersome, and due to its ambiguity (there may be many fuzzy relations), it does not lead to interesting observations. Therefore, it was decided (with the assumption of treating the U and Z sets as fuzzy) to attempt a practical solution: the fuzzy relational equations, which can be particularly useful in the diagnosis of conditional phenomena, i.e., cause-effect relationships. The scope of considerations was limited to the inclusion of the maximal and minimal solutions in the fuzzy relations.
The following preliminary assumption was made: in the process of the practical evaluation of the utility value of building elements, when making a decision on the further future of a residential building, the analysis of the set of damage that occurs in them is cause (A), and the estimation of their technical wear is result (B).
Using the simplification of the general cause-effect model (changed from three to two stages), it was decided to consider the case of a relational equation of the following type: where A ⊆ U is a fuzzy set that corresponds to the cause, R ⊆ Z × U is a fuzzy cause-effect relation, and B ⊆ Z is a fuzzy set that corresponds to the effect. Moreover, the symbol ¤ denotes a composition of the max-min type of three fuzzy relations: A ⊆ U × W, R ⊆ Z × U, and B ⊆ Z × W, Z = {z}, U = {u}, W = {w}, which have membership functions of µA(u,w), µ R (z,u), and µ B (z,w), respectively, i.e., µ B(z,w) = u∈U (µ R(z,u) ∧ µ A(u,w) ), ∀z ∈ Z, ∀w ∈ W where the fuzzy set W = {w} corresponds to class II, III, and IV of the residential buildings' maintenance class in the conditions of fuzziness when determining U and Z. By assuming the identity of cause A ⊆ U with its inverse (A = A − 1), and by transforming Equation (7) according to the rules of matrix calculus in order to determine the cause-effect relationship, the following was obtained: u1 wIII u1 wIV u2 wII u2 wIII u2 wIV · · · · · · · · · um wII um wIII um wIV     · wIV wIV wIV z1 z2 zn   wII z1 wII z2 · · · u1 wIV wIII z1 wIII z2 · · · u2 wIV wIV z1 wIV z2 · · · um wIV and therefore R = (A·B −1 ) if the fuzzy relation R − 1 ⊆ U × Z (inverse to the searched relation R ⊆ Z × U) is the relation defined as:

The Maximal Relational Equation
The maximal cause-effect solution that satisfies Equation (7) is determined by relationship: and equivalently in the language of the membership function as the α-product of the two fuzzy sets A ⊆ U and B ⊆ Z: where parameter α, called the Sanchez operator [12], is defined as: Due to the fact that fuzzy relations A ⊆ U × W and B ⊆ Z × W were determined two-dimensionally using the matrixes given in transformation (8), the definition of the α (α-composition) type composition with the following membership function was used in order to determine the maximum cause-effect solution R ⊆ Z × U: The last stage of determining the maximum cause-and-effect relationship is the adjustment of empirical data (research results U and Z in three common maintenance states of residential buildings {wII, wIII, wIV} = W) to the membership functions that correspond to: A-cause, where µ A (u, w) ⇔ r(z i ) = r(u j ); B-effect, where µ B (z, w) ⇔ m(Z) ⇐ u j = 1; where A and B have the advantageous feature of being defined in the same domain of [0,1].
In the fuzzy cause relation A ⊆ U × W, the logical justification of the proposed identity is the statistical interpretation of the correlation as a measure of the interdependence (association) between U and Z. In turn, the fuzzy effect relation B ⊆ Z × W is an implication of such an estimation of the technical wear of z i building elements (by experts), which is determined by the occurrence of damage u j . Single values of technical wear z i were averaged while giving the average value m(Z) of only wears z i for which u j = 1. After receiving pairs of fuzzy relations Rj = {(m(Z)¤r(u))j}II, III, IV and α-composition according to (15), the sought maximum cause-effect relationship R∇ ⊆ (Z × U) II, III, IV was obtained. The maximum relational solutions for 10 selected elements of the analyzed buildings are given in Table 5. Table 5. Maximal and minimal equations of relational damage to elements in the II, III, and IV maintenance class of a residential building.

The Minimal Relational Equation
The last empirical stage of the model for determining the minimal cause-effect relationship was based on the same adjustments as in the case of determining the maximal solution (see Section 2.2.1). However, the assumptions and the defined composition of the membership function are different.
The minimal cause-effect solution, which satisfies Equation (7), is determined by relationship: and equivalently in the language of the membership function as the σ-product of the two fuzzy sets A ⊆ U i B ⊆ Z: µ A σ B (u, z) = µA(u) σ µB(z), ∀ u∈U, ∀ z∈Z (17) where the parameter called the Kaufman operator [13] is defined as: In this case, fuzzy relations A ⊆ U × W and B ⊆ Z × W were also determined two-dimensionally using the matrixes given in transformation (8). Therefore, in order to determine the minimal cause-effect solution R ⊆ Z × U, the definition of the σ (σcomposition) type composition with the following membership function was used: As a result of these transformations, the required minimal cause-effect relationship R∆ ⊆ (Z × U)II, III, IV was obtained. The minimal relational solutions for 10 selected elements of the analyzed buildings are given in Table 5.

Results and Conclusions
The analysis of the results of research concerning the impact of damage to building elements on their technical wear, which was conducted in fuzzy set categories, leads to the following conclusions regarding the cause-effect dependencies of the fuzzy events (technical wear and damage) that are considered as fuzzy relations-R ⊆ Z × U/II, III, IV (Tables 3-5): • for each of the selected building elements, the maximal and minimal fuzzy relational equations (damage and technical wear) were determined and given in the form of clear relational matrixes that constitute big data arrays; they define the domain and range of the maximal and minimal fuzzy relations, the height of the fuzzy relations, their differences, and the place of their occurrence between the maximal and minimal dependencies; • the calculations in the fuzzy sets enabled the cause-effect relationships (technical wear and damage) of the fuzzy events, which are defined in the categories of fuzzy relations for the three middle maintenance states of buildings, to be found; • all the analyzed relations on the fuzzy sets are anti-transitive and weakly antisymmetric; the domain and scope were numerically determined in the complete matrix of the fuzzy relations, which are as follows: • the search for fuzzy relations in wooden inter-story floors did not bring the expected results. This can be explained by significant disproportions in their state of preservation in different apartments and also by the averaging (blurring) of the results of the technical assessment of these floors.

Discussion and Summary
The degree of the technical wear of an old residential building is determined by the conditions of its maintenance and operation. An expression of the state of operation of such a building, seen as the process that plays the greatest role in its accelerated destruction, is the mechanical damage to the internal structure of its elements. This damage is determined in the categories of fuzzy sets and has a significant frequency and cumulative effect of occurrence, which are characteristic for buildings that have a satisfactory and average maintenance condition.
The use of simple operations in fuzzy set calculus enabled the influence of elementary damage that occurs with a certain frequency (probability), as well as the measure of its interdependence (correlation) on the observed technical wear of building elements to be considered. As a result, it was possible to identify the elementary damage that determines the degree of destruction of a building element (Item 1.2, Table 2).
The research procedure in this paper (at the level of greater detail) was prepared in a way that allows the previously prepared qualitative model to be transformed into a quantitative model. The diagnosis of the impact of the maintenance of residential buildings on the degree of their technical wear was carried out using quantitative methods in the categories of fuzzy sets, as well as by using the authors' own models created in fuzzy conditions.
It was established that the expression of the operational state of a building, considered as the process that plays the greatest role in its accelerated destruction, is mechanical damage to the internal structure of its elements. This damage is determined in the categories of fuzzy sets and has a high frequency and a cumulative effect of occurrence, which are characteristic for buildings in satisfactory and average maintenance conditions. The fuzzy set operations made it possible to determine the impact of elementary damage that occurs with a specific frequency in building elements. As a result, it was possible to identify the elementary damage that determines the degree of the technical wear of a building element. For each of the selected building elements, the maximal and minimal fuzzy relational equations (damage and technical wear) were determined. Their solutions were given in the form of clear relational matrixes that constitute big data arrays. They define the domain and range of the maximal and minimal fuzzy relations, the height of the fuzzy relations, their differences, and the place of their occurrence between the maximal and minimal dependencies.
The relationship of technical wear and damage was defined as fuzzy relational equations [8][9][10][11]14,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. In these equations, maximal cause-effect solutions based on Sanchez α-compositions [12] and minimal solutions using Kaufman σ-compositions [13] were determined. It was proved that the sought relationship has a significant strength: its minimum height and value in average compositions exceeds 0.50 within the range [0,1], which indicates strong fuzzy relations of damage and technical wear. The study and the determination of the fuzzy relations between the technical wear of residential buildings and their maintenance conditions, which are manifested by damage to elements, is justified and leads to numerical conclusions regarding the adopted research sample.