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

Abstract

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.

1. 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].

1.1. 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. Summary of references, authors, year of publication, topics of study, and type of approach.

Reference	Authors	Year	Topic of Study	Type of Approach
[1,2,3,4]	Zavadskas, Antuchevičienė, Kapliński, Konior	2007–2015	Decision making	Models and methods
[5,6,7]	Hellwig, Morrison, Jackson	2001–2019	Randomness	Statistics
[8,9,10,11,12,13,14]	Zadeh, Yager, Kaufmann, Sanchez, Kacprzyk	1965–2007	Uncertainness	Fuzzy sets
[15,16,17,18]	Knight, Menassa, Konior	2011–2020	Uncertainty	Appraisal of buildings
[19,20,21,22,23,24,25,26,27,28,29]	Nowogońska, Konior, Sawicki, Szóstak	2014–2020	Technical assessment	Diagnosis of buildings
[30,31,32,33,34,35,36,37,38]	Plebankiewicz, Zima, Wieczorek, Frangopol, Lin, Estes, Lee, Kim, Zayed, Chang, Fricker, Oduyemi, Okoroh, Fajana	1997–2019	Cost and risk modeling	LCC of buildings
[39,40]	Chan, Kwong, Dillon, Fung, Nasirzadeh, Afshar, Khanzadi, Howick	2008–2011	Nonlinearity and fuzziness	Fuzzy regression
[41,42,43,44,45,46,47,48,49]	Ibadov, Kulejewski, Knight, Robinson, Fayek, Al-Humaidi, Hadipriono, Andrić, Wang, Zou, Zhang, Dikmen, Birgonul, Han, Leśniak	2002–2019	Fuzzy logic	Construction management
[50,51]	Wieczorek, Kamal, Jain	2012–2018	Fuzzy assessment	LCC of buildings
[52,53,54]	Marzouk, Amin, Sharma, Goyal, Ammar, Zayed, Moselhi	2012–2019	Fuzzy assessment	Construction engineering
[55]	Czapliński	1984–1996	Technical assessment	Wroclaw downtown apartment houses

presents the analyzed literature items, which are divided into engineering processes with fuzzy and random approaches.

1.2. 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. Damage and the probability of its occurrence in 10 elements of the examined buildings.

The Probability of the Occurrence of Elementary Damage p(u), Which is Typical for 10 Selected Building Elements		Foundations	Basement Walls	Solid Floors Above Basements	Structural Walls	Inter-Story Wooden Floors	Stairs	Roof Construction	Window Joinery	Inner Plasters	Facades
No.	Damage	p(u)2	p(u)3	p(u)4	p(u)7	p(u)8	p(u)9	p(u)10	p(u)13	p(u)15	p(u)20
u1	Mechanical damage						0.86		0.89	0.74	0.81
u2	Leaks								0.93
u3	Brick losses	0.78	0.66		0.96		0.78
u4	Mortar losses		0.68	0.71	0.91
u5	Brick decay	0.76	0.63	0.73	0.79
u6	Mortar decay		0.49		0.74					0.8	0.86
u7	Peeling off of paint coatings									0.79	0.85
u8	Falling off of paint coatings									0.31	0.32
u9	Cracks in bricks	0.78	0.52	0.56	0.25
u10	Cracks on plaster		0.39		0.69					0.78	0.82
u11	Scratching on walls				0.11
u12	Scratching on plaster				0.55	0.80				0.73	0.75
u13	Loosening of plaster					0.49				0.44	0.56
u14	Falling off of plaster sheets									0.13	0.22
u15	Dampness	0.35	0.48	0.22	0.66	0.81		0.62	0.47	0.14	0.54
u16	Weeping	0.09	0.24	0.18	0.18	0.53	0.59	0.44	0.29	0.10	0.38
u17	Biological corrosion of bricks	0.66	0.17		0.39
u18	Fungus					0.08				0.03	0.19
u19	Mold and rot	0.03	0.05		0.05				0.11	0.03	0.14
u20	Corrosion raid of steel beams			0.52			0.72
u21	Surface corrosion of steel beams			0.68			0.54
u22	Deep corrosion of steel beams			0.24			0.16
u23	Flooding with water			0.04
u24	Dynamic sensitivity of floor beams					0.64
u25	Deformations of wooden beams					0.35
u26	Skewing of window joinery								0.75
u27	Warping of window joinery								0.59
u28	Delamination of wooden elements							0.51
u29	Partial insect infestation of wooden elements						0.07	0.15	0.10
u30	Complete insect infestation of wooden elements					0.25		0.5	0.30

). 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.

2. Method of Research

2.1. 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]: A = {(µA(x), x)}, ∀ x∈X.

Therefore, two basic fuzzy sets can be distinguished:

• a fuzzy set of the technical wear of building elements A ⊆ Ze ⇔ Z (to simplify the designations): Z = {(µZ(z), z)}, ∀ z ∈ Z;
• a fuzzy set of damage to building elements B ⊆ U: U = {(µU(u), u)}, ∀ u∈ U.

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:

R ⊆ Z × U = {(z, u): z ∈ Z, u ∈ U}

(1)

and, therefore, it is the set of pairs:

R = {(µR(z, u), (z, u))}, ∀ z∈Z, ∀ u∈U

(2)

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:

$$\mathrm{R}={\displaystyle \sum }_{\mathrm{z},\mathrm{u}}\mathsf{\mu }\mathrm{R}(\mathrm{z},\mathrm{u})/(\mathrm{z},\mathrm{u})$$

(3)

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. Matrix of fuzzy relation R = Z × U for 10 selected building elements and their 30 characteristic types of damage.

	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10	u11	u12	u13	u14	u15	u16	u17	u18	u19	u20	u21	u22	u23	u24	u25	u26	u27	u28	u29	u30
Z2	0.00	0.00	0.13	0.00	0.14	0.00	0.00	0.00	0.05	0.00	0.00	0.00	0.00	0.00	0.70	0.64	0.36	0.00	0.49	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Z3	0.00	0.00	0.23	0.28	0.07	0.05	0.00	0.00	0.01	0.03	0.00	0.00	0.00	0.00	0.74	0.52	0.31	0.00	0.43	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Z4	0.00	0.00	0.08	0.00	0.00	0.00	0.00	0.00	0.05	0.00	0.00	0.00	0.00	0.00	0.58	0.67	0.00	0.00	0.00	0.42	0.29	0.55	0.45	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Z7	0.00	0.00	0.19	0.30	0.17	0.09	0.00	0.00	0.11	0.03	0.21	0.12	0.00	0.00	0.56	0.46	0.67	0.00	0.34	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Z8	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.05	0.09	0.00	0.07	0.27	0.00	0.45	0.00	0.00	0.00	0.00	0.00	0.00	0.12	0.00	0.00	0.00	0.00	0.43
Z9	0.05	0.00	0.03	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.59	0.00	0.00	0.00	0.54	0.61	0.53	0.00	0.00	0.00	0.00	0.00	0.00	0.38	0.00
Z10	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.43	0.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.07	0.28	0.57
Z13	0.29	0.26	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.83	0.74	0.00	0.00	0.49	0.00	0.00	0.00	0.00	0.00	0.00	0.42	0.04	0.00	0.45	0.42
Z15	0.09	0.00	0.00	0.00	0.00	0.47	0.15	0.25	0.00	0.30	0.00	0.18	0.67	0.57	0.70	0.61	0.00	0.38	0.41	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Z20	0.28	0.00	0.00	0.00	0.00	0.48	0.55	0.57	0.00	0.63	0.00	0.63	0.81	0.50	0.84	0.79	0.00	0.60	0.56	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

).

The main parameters of the matrix of fuzzy relation R = Z × U are as follows:

• domain of fuzzy relation: dom R(Z) = 0.70/Z2 + 0.74/Z3 + 0.67/Z4 + 0.67/Z7 + 0.45/Z8 + 0.61/Z9 + 0.57/Z10 + 0.83/Z13 + 0.70/Z15 + 0.84/Z20
• scope of fuzzy relation: ran R(u) = 0.29/u1 + 0.26/u2 + 0.23/u3 + 0.30/u4 + 0.17/u5 + 0.48/u6 + 0.55/u7 + 0.57/u8 + 0.11/u9 + 0.63/u10 + 0.21/u11 + 0.63/u12 + 0.81/u13 + 0.57/u14 + 0.84/u15 + 0.79/u16 + 0.67/u17 + 0.60/u18 + 0.56/u19 + 0.54/u20 + 0.61/u21 + 0.55/u22 + 0.45/u23 + 0.00/u24 + 0.12/u25 + 0.42/u26 + 0.04/u27 + 0.07/u28 + 0.45/u29 + 0.57/u30
• height of fuzzy relation: h{R(Z,u)} = 0.84

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:

$${\mathsf{\mu }}_{\mathrm{domR}(\mathrm{Z})}=\underset{\mathrm{z}\in \mathrm{Z}}{{\displaystyle \vee }}{\mathsf{\mu }}_{\mathrm{R}(\mathrm{z},\mathrm{u})}\stackrel{}{\iff }\underset{\mathrm{i}=1}{\overset{\mathrm{n}}{{\displaystyle \vee }}}{\mathrm{z}}_{\mathrm{i}}=\max \left\{{\mathrm{z}}_1,{\mathrm{z}}_2,\dots ,{\mathrm{z}}_{\mathrm{n}}\right\}$$

(4)

the scope of the fuzzy relation R ⊆ Z × U, which is called the second projection of the fuzzy relation, and denoted by ranR:

$${\mathsf{\mu }}_{\mathrm{ranR}(\mathrm{u})}=\underset{\mathrm{u}\in \mathrm{U}}{{\displaystyle \vee }}{\mathsf{\mu }}_{\mathrm{R}(\mathrm{z},\mathrm{u})}\stackrel{}{\iff }\underset{\mathrm{j}=1}{\overset{\mathrm{m}}{{\displaystyle \vee }}}{\mathrm{u}}_{\mathrm{j}}=\max \left\{{\mathrm{u}}_1,{\mathrm{u}}_2,\dots ,{\mathrm{u}}_{\mathrm{m}}\right\}$$

(5)

the height of the fuzzy relation R ⊆ Z × U, which is called the global projection of the fuzzy relation, and marked as h(R)

$$\mathrm{h}(\mathrm{R})=\underset{\mathrm{z}\in \mathrm{Z}}{{\displaystyle \vee }}{\mathsf{\mu }}_{\mathrm{domR}(\mathrm{z})}=\underset{\mathrm{u}\in \mathrm{U}}{{\displaystyle \vee }}{\mathsf{\mu }}_{\mathrm{ranR}(\mathrm{u})}=\underset{\mathrm{z}\in \mathrm{Z}}{{\displaystyle \vee }}\underset{\mathrm{u}\in \mathrm{U}}{{\displaystyle \vee }}{\mathsf{\mu }}_{\mathrm{R}(\mathrm{z},\mathrm{u})}$$

(6)

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 domain, scope, and height of fuzzy relations for 10 selected elements in the II, III, and IV maintenance class of a residential building.

Group Number	Building Element	Parameters of Fuzzy Relations	Characteristics of Maximal Fuzzy Relations	Characteristics of Minimal Fuzzy Relations
Z2	Foundations	dom R(Z)	1.00/ZII + 1.00/ZIII + 1.00/ZIV	0.50/ZIII + 0.61/ZIV
		ran R(U)	1.00/UII + 0.61/UIII + 0.61/UIV	0.61/UIII + 0.61/UIV
		h{R(Z,U)}	1.00	0.61
Z3	Basement walls	dom R(Z)	1.00/ZII + 1.00/ZIII + 1.00/ZIV	0.48/ZIII + 0.61/ZIV
		ran R(U)	1.00/UII + /0.61UIII + 1.00/UIV	0.61/UIII
		h{R(Z,U)}	1.00	0.61
Z4	Solid floors above basements	dom R(Z)	0.28/ZII + 0.46/ZIII + 1.00/ZIV	0.28/ZII + 0.46/ZIII
		ran R(U)	1.00/UII + 1.00/UIII + 1.00/UIV	0.41/UII + 0.46/UIII
		h{R(Z,U)}	1.00	0.46
Z7	Structural walls	dom R(Z)	1.00/ZII + 1.00/ZIII + 1.00/ZIV	0.48/ZIII + 0.62/ZIV
		ran R(U)	1.00/UII + 0.62/UIII + 1.00/UIV	0.62/UIII + 0.43/UIV
		h{R(Z,U)}	1.00	0.62
Z8	Inter-story wooden floors	dom R(Z)	1.00/ZII + 1.00/ZIII + 1.00/ZIV	0.36/ZIII + 0.52/ZIV
		ran R(U)	1.00/UII + 1.00/UIII + 1.00/UIV	0.45/UIII + 0.48/UIV
		h{R(Z,U)}	1.00	0.54
Z9	Stairs	dom R(Z)	0.28/ZII + 1.00/ZIII + 1.00/ZIV	0.28/ZII + 0.45/ZIII
		ran R(U)	1.00/UII + 1.00/UIII + 1.00/UIV	0.43/UII + 0.45/UIII
		h{R(Z,U)}	1.00	0.45
Z10	Roof construction	dom R(Z)	1.00/ZII + 1.00/ZIII + 1.00/ZIV	0.30/ZII + 0.44/ZIII
		ran R(U)	1.00/UII + 1.00/UIII + 1.00/UIV	0.41/UIII + 0.44/UIV
		h{R(Z,U)}	1.00	0.44
Z13	Window joinery	dom R(Z)	0.29/ZII + 0.50/ZIII + 1.00/ZIV	0.29/ZII + 0.50/ZIII + 0.65/ZIV
		ran R(U)	1.00/UII + 0.65/UIII + 1.00/UIV	0.40/UII + 0.65/UIII + 0.50/UIV
		h{R(Z,U)}	1.00	0.65
Z15	Inner plasters	dom R(Z)	0.30/ZII + 1.00/ZIII + 1.00/ZIV	0.30/ZII + 0.44/ZIII + 0.65/ZIV
		ran R(U)	1.00/UII + 0.65/UIII + 1.00/UIV	0.30/UII + 0.65/UIII
		h{R(Z,U)}	1.00	0.65
Z20	Facades	dom R(Z)	0.28/ZII + 0.45/ZIII + 1.00/ZIV	0.28/ZII + 0.45/ZIII + 0.68/ZIV
		ran R(U)	1.00/UII + 1.00/UIII + 0.68/UIV	0.45/UII + 0.44/UIII + 0.68/UIV
		h{R(Z,U)}	1.00	0.68

. The table presents quantitative and measurable values that can be easily quantified.

2.2. 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:

A ¤ R = B

(7)

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.,

$${\mathsf{\mu }}_{\mathrm{B}(\mathrm{z},\mathrm{w})}={{\displaystyle \vee }}_{\mathrm{u}\in \mathrm{U}}({\mathsf{\mu }}_{\mathrm{R}(\mathrm{z},\mathrm{u})}\wedge {\mathsf{\mu }}_{\mathrm{A}(\mathrm{u},\mathrm{w})}),\forall \mathrm{z}\in \mathrm{Z},\forall \mathrm{w}\in \mathrm{W}$$

(8)

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:

$$\begin{array}{c}{\mathrm{R}}^{-1}=\mathrm{A}·\mathrm{B}=\begin{array}{c} \\ u1\\ u2\\ \cdots \\ um\end{array}\begin{array}{c}\begin{array}{ccc}wII & wIII& wIV\end{array}\\ \left[\begin{array}{ccc}\mathrm{u}1 \mathrm{wII} & \mathrm{u}1 \mathrm{wIII} & \mathrm{u}1 \mathrm{wIV}\\ \mathrm{u}2 \mathrm{wII} & \mathrm{u}2 \mathrm{wIII} & \mathrm{u}2 \mathrm{wIV}\\ \cdots & \cdots & \cdots \\ \mathrm{um} \mathrm{wII} & \mathrm{um} \mathrm{wIII} & \mathrm{um} \mathrm{wIV}\end{array}\right]\end{array}·\begin{array}{c} \\ wIV\\ wIV\\ wIV\end{array}\begin{array}{c}\begin{array}{cccc}z1 & z2& & zn\end{array}\\ \left[\begin{array}{cccc}\mathrm{wII}\mathrm{z}1& \mathrm{wII}\mathrm{z}2& \cdots & \mathrm{u}1\mathrm{wIV}\\ \mathrm{wIII}\mathrm{z}1& \mathrm{wIII}\mathrm{z}2& \cdots & \mathrm{u}2\mathrm{wIV}\\ \mathrm{wIV}\mathrm{z}1& \mathrm{wIV}\mathrm{z}2& \cdots & \mathrm{um}\mathrm{wIV}\end{array}\right]\end{array}=\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\begin{array}{c} \\ U II\\ U III\\ U IV\end{array}\begin{array}{c}\begin{array}{cccc}Z II & Z III& & Z IV\end{array}\\ \left[\begin{array}{ccc}\mathrm{UII}\times \mathrm{ZII}& \mathrm{UII}\times \mathrm{ZIII} & \mathrm{UII}\times \mathrm{ZIV}\\ \mathrm{UIII}\times \mathrm{ZII}& \mathrm{UIII}\times \mathrm{ZIII}& \mathrm{UIII}\times \mathrm{ZIV}\\ \mathrm{UIV}\times \mathrm{ZII}& \mathrm{UIV}\times \mathrm{ZIII}& \mathrm{UIV}\times \mathrm{ZIV}\end{array}\right]\end{array};\end{array}$$

(9)

$$\mathrm{and} \mathrm{therefore} \mathrm{R}={(\mathrm{A}·{\mathrm{B}}^{-1})}^{-1}=\begin{array}{c} \\ Z II\\ Z III\\ Z IV\end{array}\begin{array}{c}\begin{array}{cccc}U II & U III& & U IV\end{array}\\ \left[\begin{array}{ccc}\mathrm{ZII}\times \mathrm{UII}& \mathrm{ZII}\times \mathrm{UIII} & \mathrm{ZII}\times \mathrm{UIV}\\ \mathrm{ZIII}\times \mathrm{UII}& \mathrm{ZIII}\times \mathrm{UIII}& \mathrm{ZIII}\times \mathrm{UIV}\\ \mathrm{ZIV}\times \mathrm{UII}& \mathrm{ZIV}\times \mathrm{UIII}& \mathrm{ZIV}\times \mathrm{UIV}\end{array}\right]\end{array}$$

(10)

if the fuzzy relation R − 1 ⊆ U × Z (inverse to the searched relation R ⊆ Z × U) is the relation defined as:

µ_R⁻¹(u, z) = µ_R(z, u), ∀ z∈Z, ∀ u∈U.

(11)

2.2.1. The Maximal Relational Equation

The maximal cause-effect solution that satisfies Equation (7) is determined by relationship:

$$\stackrel{\nabla }{\mathrm{R}} = \mathrm{A}-1 \mathsf{\alpha } \mathrm{B} = \mathrm{A} \mathsf{\alpha } \mathrm{B}$$

(12)

and equivalently in the language of the membership function as the α-product of the two fuzzy sets A ⊆ U and B ⊆ Z:

µ A α B (u, z) = µA(u) α µ_B(z), ∀ u∈U, ∀ z∈Z

(13)

where parameter α, called the Sanchez operator [12], is defined as:

$$\mathrm{a} \mathsf{\alpha } \mathrm{b} = \{\begin{array}{c}1, \mathrm{if}:\mathrm{a}\le \mathrm{b}\\ \mathrm{b}, \mathrm{if}:\mathrm{a}>\mathrm{b}\end{array}\mathrm{for} \mathrm{the} \mathrm{two} \mathrm{numbers} (\mathrm{non}-\mathrm{fuzzy}) \mathrm{a}, \mathrm{b} \in [0, 1]$$

(14)

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:

$$\mu \mathrm{R}\nabla (\mathrm{z},\mathrm{u}=\mu \mathrm{R}\nabla - 1 (\mathrm{u}, \mathrm{z}) =\underset{\mathrm{w}\in \mathrm{W}}{{\displaystyle \wedge }}(\mu \mathrm{A}(\mathrm{u}, \mathrm{w}) \mathsf{\alpha } \mu \mathrm{B} - 1(\mathrm{w}, \mathrm{z})), \forall \mathrm{u}\in \mathrm{U}, \forall \mathrm{z}\in \mathrm{Z}$$

(15)

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. Maximal and minimal equations of relational damage to elements in the II, III, and IV maintenance class of a residential building.

Fuzzy Relational Equations of Damage (U), Which Correspond to the II, III, and IV Conditions of the Maintenance of the Element and Its Technical Wear (Z)
			Maximal Fuzzy Relational Equation			Minimal Fuzzy Relational Equation
Z2	Foundations		UII	UIII	UIV	UII	UIII	UIV
		ZII	1.00	0.00	0.00	0.00	0.00	0.00
		ZIII	1.00	0.50	0.00	0.00	0.50	0.00
		ZIV	1.00	0.61	0.61	0.00	0.61	0.61
Z3	Basement walls		UII	UIII	UIV	UII	UIII	UIV
		ZII	1.00	0.00	0.00	0.00	0.00	0.00
		ZIII	1.00	0.48	0.00	0.00	0.48	0.00
		ZIV	1.00	0.61	1.00	0.00	0.61	0.00
Z4	Solid floors above basements		UII	UIII	UIV	UII	UIII	UIV
		ZII	0.28	0.00	0.00	0.28	0.00	0.00
		ZIII	0.41	0.46	0.00	0.41	0.46	0.00
		ZIV	1.00	1.00	1.00	0.00	0.00	0.00
Z7	Structural walls		UII	UIII	UIV	UII	UIII	UIV
		ZII	1.00	0.00	0.00	0.00	0.00	0.00
		ZIII	1.00	0.48	0.43	0.00	0.48	0.43
		ZIV	1.00	0.62	1000	0.00	0.62	0.00
Z8	Inter-story wooden floors		UII	UIII	UIV	UII	UIII	UIV
		ZII	1.00	0.00	0.00	0.00	0.00	0.00
		ZIII	1.00	1.00	0.00	0.00	0.00	0.00
		ZIV	1.00	1.00	1.00	0.00	0.00	0.00
Z9	Stairs		UII	UIII	UIV	UII	UIII	UIV
		ZII	0.28	0.00	0.00	0.28	0.00	0.00
		ZIII	0.43	0.45	1.00	0.43	0.45	0.00
		ZIV	1.00	1.00	1.00	0.00	0.00	0.00
Z10	Roof construction		UII	UIII	UIV	UII	UIII	UIV
		ZII	1.00	0.00	0.30	0.00	0.00	0.30
		ZIII	1.00	0.41	0.44	0.00	0.41	0.44
		ZIV	1.00	1.00	1.00	0.00	0.00	0.00
Z13	Window joinery		UII	UIII	UIV	UII	UIII	UIV
		ZII	0.29	0.00	0.00	0.29	0.00	0.00
		ZIII	0.4	0.50	0.50	0.40	0.50	0.50
		ZIV	1.00	0.65	1.00	0.00	0.65	0.00
Z15	Inner plasters		UII	UIII	UIV	UII	UIII	UIV
		ZII	0.30	0.00	0.00	0.30	0,00	0.00
		ZIII	1.00	0.44	0.00	0.00	0.44	0.00
		ZIV	1.00	0.65	1.00	0.00	0.65	0.00
Z20	Facades		UII	UIII	UIV	UII	UIII	UIV
		ZII	0.28	0.00	0.00	0.28	0.00	0.00
		ZIII	0.45	0.44	0.00	0.45	0.44	0.00
		ZIV	1.00	1.00	0.68	0.00	0.00	0.68

.

2.2.2. 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:

$$\stackrel{\mathrm{\Delta }}{\mathrm{R}} = \mathrm{A} - 1 \mathsf{\sigma } \mathrm{B} = \mathrm{A} \mathsf{\sigma } \mathrm{B}$$

(16)

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:

$$\mathrm{a} \mathsf{\sigma } \mathrm{b} =\{\begin{array}{c}1, \mathrm{if}:\mathrm{a}<\mathrm{b}\\ \mathrm{b}, \mathrm{if}:\mathrm{a}\ge \mathrm{b}\end{array} \mathrm{for} \mathrm{the} \mathrm{two} \mathrm{numbers} (\mathrm{non}-\mathrm{fuzzy}) \mathrm{a}, \mathrm{b} \in [0, 1]$$

(18)

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 (σ-composition) type composition with the following membership function was used:

$$\mu \mathrm{R}\mathrm{\Delta }(\mathrm{z},\mathrm{u}=\mu \mathrm{R}\mathrm{\Delta }- 1 (\mathrm{u}, \mathrm{z}) =\underset{\mathrm{w}\in \mathrm{W}}{{\displaystyle \vee }}(\mu \mathrm{A}(\mathrm{u}, \mathrm{w}) \mathsf{\sigma } \mu \mathrm{B} - 1(\mathrm{w}, \mathrm{z})), \forall \mathrm{u}\in \mathrm{U}, \forall \mathrm{z}\in \mathrm{Z}$$

(19)

As a result of these transformations, the required minimal cause-effect relationship RΔ ⊆ (Z × U)II, III, IV was obtained. The minimal relational solutions for 10 selected elements of the analyzed buildings are given in Table 5.

3. 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 (Table 3, Table 4 and Table 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:

  o

  dom R(Z) = 0.70/Z2 + 0.74/Z3 + 0.67/Z4 + 0.67/Z7 + 0.45/Z8 + 0.61/Z9 + 0.57/Z10 + 0.83/Z13 + 0.70/Z15 + 0.84/Z20

  o

  ran R(u) = 0.29/u1 + 0.26/u2 + 0.23/u3 + 0.30/u4 + 0.17/u5 + 0.48/u6 + 0.55/u7 + 0.57/u8 + 0.11/u9 + 0.63/u10 + 0.21/u11 + 0.63/u12 + 0.81/u13 + 0.57/u14 + 0.84/u15 + 0.79/u16 + 0.67/u17 + 0.60/u18 + 0.56/u19 + 0.54/u20 + 0.61/u21 + 0.55/u22 + 0.45/u23 + 0.00/u24 + 0.12/u25 + 0.42/u26 + 0.04/u27 + 0.07/u28 + 0.45/u29 + 0.57/u30

and prove the complete range of the domain and the scope of the fuzzy relation R ⊆ Z × U with its height h(R) = 0.84, which in turn confirms the close (tending to 1) normality of this relation between Z and U;

• for each of the 10 selected building elements, the maximal and minimal fuzzy relational equations were determined, and their solutions were given in the form of clear relational matrixes (3 × 3), which define:

  o

  the height of the fuzzy relations, which in the case of the maximal solutions in the paper is equal to 1 for all the analyzed elements, and in the case of the minimal solutions range from 0.44 to 0.68; the span of values is relatively narrow, which indicates the minimal heights of fuzzy relations may be expected above 0.5 (average 0.56) in the interval [0; 1];

  o

  differences and their place of occurrence between the maximal and minimal dependencies, or, if these differences do not exist, the exact value of the fuzzy relation that occurs in different compositions that are characteristic for different building elements. However, there is almost always the middle composition of (ZIII-UIII), in which the fuzzy relations take the values from 0.41 to 0.50, and additionally also 2–4 compositions in other places;

• it was observed that the middle compositions (ZIII-UIII) are identical and that they unambiguously define the value of fuzzy relation (technical wear and damage) R ⊆ Z × U in interval [0, 1], which is equal to:

  o

  for foundations: 0.50

  o

  for basement walls: 0.48

  o

  for solid floors above basements: 0.46

  o

  for structural walls: 0.48

  o

  for internal stairs: 0.45

  o

  for roof constructions: 0.41

  o

  for window joinery: 0.50

  o

  for inner plasters: 0.44

  o

  for facades: 0.44

  All fuzzy relations R ⊆ Z × U are in the middle of the membership function, which means close to 0.5 (strong relations)

• 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.

4. 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.