Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings

Round 1

Reviewer 1 Report

The paper uses the probability of the dommage on residential buildings in order to organize the maintenance. The probability equations used are known, but the values applied on the type of each domage are not verified and not compared with other values presented in literature on good journals.  

1/I don’t understand how authors obtain the values of probabilities presented in table 1. For this reason, I want that authors give some references in literature where other authors used the same method.

2/lines 195-196 : give some references of this group of experts.

3/Eqs (13)-(15) : give more explanations in order to justify the values obtained.

4/the conclusion must be re-written in order to refere to probability obtained in this paper. Also, it is capital to give some practical conclusions obtained in order to improve the maintenance of the residential buildings.

Author Response

Wrocław, Poland, 5^th March 2021

Dear Reviewer of Applied Sciences,

Thank you for the review of our paper applsci-1132609 entitled “Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings” to be published in the journal Applied Sciences, special issue “Probabilistic and Fuzzy Approaches for Estimating the Life Cycle Costs of Buildings”.

We appreciate your thoughtful and  accurate comments as well as appreciation of our research works. We have carefully considered all comments and have now completed the revisions incorporating  your suggestions in the revised uploaded manuscript.

We hope that the revised paper meets your expectations.

Kind regards,

Jarosław Konior and Tomasz Stachoń

Department of Building Engineering, Faculty of Civil Engineering, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland

Here are answers to reviewer’s comments:

REVIEWER 1

General Comments. The paper uses the probability of the damage on residential buildings in order to organize the maintenance. The probability equations used are known, but the values applied on the type of each damage are not verified and not compared with other values presented in literature on good journals?

Answers to General Comments. We appreciate the reviewer’s valuable view on the presented topic but cannot agree to full extant. As a matter of fact the approach to estimate conditional probability of the technical wear of building elements in relation to their damages increases (or the other way round) is absolutely novel and do not exist in accessible literature, though Bayes conditional probability formulas are known. However, the point is to apply them sensibly to determine probabilities of defects and wears which are related to each other as fuzzy events (we do not know if some damage appears so technical wear degree lessens). 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], 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].

Comment 1. I don’t understand how authors obtain the values of probabilities presented in table 1. For this reason, I want that authors give some references in literature where other authors used the same method.

Answer 1. The probabilities p(u_i) of elementary damages u_i for 10 selected building elements were determined by a group of experts inspecting residential buildings. The approach was purely probabilistic with the commonly met assumption that defects stand for random values with interval [0,1]. Probabilistic approach to the research has been previously developed by the authors’ in published papers:

2. Konior, J. Decision assumptions on building maintenance management. Probabilistic methods, Civ. Eng. 2007, 53, 403–423.
3. Konior, J. Technical Assessment of old buildings by probabilistic approach., Archives of Civil Engineering, 2020, 66(3), pp. 443–466, https://0-doi-org.brum.beds.ac.uk/https://doi.org/10.24425/ace.2020.134407.
4. Konior, J. Maintenance of apartment buildings and their measurable deterioration, Trans. Czas. Tech. 2017, 6, 101–107, https://0-doi-org.brum.beds.ac.uk/10.4467/2353737xct.17.090.6566.
5. Konior, K. Bi-serial correlation of civil engineering building elements under constant technical deterioration, Sci. Gen. Tadeusz Kosiuszko Mil. Acad. L. Forces. 2016, 179, 142–150.
6. Konior, J. Intensity of defects in residential buildings and their technical wear, Tech. Trans. Civ. Eng. 2014, 111(2-B), 137–146.
7. Konior, J.; Sawicki, M.; Szóstak, M. Intensity of the Formation of Defects in Residential Buildings with Regards to Changes in Their Reliability.  Sci.2020, 10, 6651. doi:10.3390/app10196651
8. Konior, J.; Sawicki, M.; Szóstak, M. Influence of Age on the Technical Wear of Tenement Houses. Sci. 2020, 10, 6651. doi:10.3390/app11010297

which represent the current state of the art on buildings’ deterioration. However, randomness has fundamental limits while assessing technical state during technical inspections of tenement houses by experts with their natural subjective / descriptive way of measurement. Experts may not know if a hidden damage = event exists. Therefore fuzziness makes possible to convert qualitative appraisal of damages into quantitative / measurable ones.

Comment 2. Lines 195-196 : give some references of this group of experts.

Answer 2. Technical inspections of the residential buildings were executed by a team of experts consisted of:

• 1 architect;
• 1 structural engineer;
• 1 mechanical / sanitary engineer;
• 1 electrical engineer;
• 2 quantity surveyors;
• 1 technician / administrator.

The average workload needed for the technical assessment of each tenement houses has been calculated as follows:

• desk top study of multidiscipline design and archive documents - 2 days for 5 people;
• technical investigations and surveys – 3 days for 7 people;
• generating calculations and reports - 2 days for 5 people.

The above extra information was inserted between the lines 195-196.

Comment 3. Eqs (13)-(15) : give more explanations in order to justify the values obtained.

Answer 3. Equations (13)-(15) are self-explanatory as arguments z1, z2, z3, and z4 are assumed as being equally probable if the arguments appear in the sets ZII, ZIII, ZIV, so their values are either 1/4 or 1/3, as follows:

p(z1)II = 1/3; p(z2)II = 1/3; p(z3)II = 1/3; p(z4)II = 0                         (13)

p(z1)III = 1/4; p(z2)III = 1/4; p(z3)III = 1/4; p(z4)III = 1/4                     (14)

p(z1)IV = 1/4; p(z2)IV = 1/4; p(z3)IV = 1/4; p(z4)IV = 1/4                     (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 analysed residential buildings.

Comment 4. The conclusion must be re-written in order to refer to probability obtained in this paper. Also, it is capital to give some practical conclusions obtained in order to improve the maintenance of the residential buildings.

Answer 4. We are sorry but are not getting the point made in the comment. The conclusion does refer to probability obtained in this paper to full extent:

• the probability of such a conditionally defined fuzzy event indicates the state of the technical wear for which the fuzzy damage occurs with the highest intensity, and it amounts, for the following elements of the tested residential buildings in their average maintenance condition  P(ZIII/U), to:

• 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 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

These are accurate conditional probabilities determined on and reverse ways and presented in the table 2. 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 (this is an exceptionally steady increase, even in the case of different building elements). Also, the conditional probability of damage to the element in relation to its technical wear increases with the deterioration of the building maintenance conditions. Practical conclusions obtained in order to improve the maintenance of the residential buildings were given as well: “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 Response File: Author Response.doc

Reviewer 2 Report

This study could serve to determine the probability that a construction element is more or less worn or damaged, which will allow progress in the evaluation of the degree of technical wear of the construction elements and an improvement in the monitoring of damage and its relationship with the maintenance of certain elements of the buildings.

However, the document presents some deficiencies with respect to the clarity in the wording to facilitate its reading, understanding and reproduction of the model, as well as regarding some data, its obtaining and its presentation. It is also suggested that English, writing, citations and literature be checked. Perhaps a deeper analysis of the results is missing, and points 3 and 4 should also be reordered into 3 Results, 4 Discussion and 5. Conclusions.

Here are some observations:

-Summary: Improve the summary in terms of methodology and results. On the other hand, although the objectives are identified, they could be written more succinctly without losing clarity.

1. Introduction

1.1. Literary review: they seem correct references, although some of them could be discussed a little more.

1.2. Research sample: Some of the constructive characteristics of the chosen buildings could be specified a little more, and if they are frequent or not in other areas (autonomous, national or international) for extrapolation purposes.

2. Research method

2.1 Identification problem: it is necessary to better clarify the concept of  II, III and IV of the maintenance conditions of an element. It is understood that clarity is not reached as to how the probabilistic results of table I are produced, referring to the data entered into them to produce the results.

2.2 Model to determine the conditional probabilities of the technical wear process in relation to the occurrence of damage: More data from the group of experts, their characteristics and how the values ​​of the aforementioned argument were determined should be provided. Perhaps it would be possible to better clarify the proposed model to facilitate its reproduction. As already mentioned for table I, it is understood that clarity is not reached as to how the results of table II are produced based on the same argument indicated in the previous point. On the other hand, it is not very well understood because this data has not been placed in the results section.

3 Results and conclusions: the conclusions should go to the end of the document after the discussion and before the references. Since some of the statements about the probabilistic results presented are evident, in the sense of being able to be deduced without the need for the study carried out, these statements, in addition to being verified by the different probabilities presented, must be supported by a presentation of the data in the form of graphs, especially the effect of a deeper analysis on whether it is possible to represent and even predict, with other types of mathematical functions, how the different elements behave in a generic way to apply the results to other situations with the proposed model or with others, even some of them simpler or simplified.

4 Discussion and summary: It does not seem like a discussion but rather a summary or conclusions as already mentioned. Although the discussion results from commenting on the results, perhaps it is a section that should be carried out together with the presentation of these in a more precise way for a better understanding.

Author Response

Wrocław, Poland, 5^th March 2021

Dear Reviewer of Applied Sciences,

Thank you for the review of our paper applsci-1132609 entitled “Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings” to be published in the journal Applied Sciences, special issue “Probabilistic and Fuzzy Approaches for Estimating the Life Cycle Costs of Buildings”.

We appreciate your thoughtful and  accurate comments as well as appreciation of our research works. We have carefully considered all comments and have now completed the revisions incorporating  your suggestions in the revised uploaded manuscript.

We hope that the revised paper meets your expectations.

Kind regards,

Jarosław Konior and Tomasz Stachoń

Department of Building Engineering, Faculty of Civil Engineering, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland

Here are answers to reviewer’s comments:

REVIEWER 2

General Comments. This study could serve to determine the probability that a construction element is more or less worn or damaged, which will allow progress in the evaluation of the degree of technical wear of the construction elements and an improvement in the monitoring of damage and its relationship with the maintenance of certain elements of the buildings.

However, the document presents some deficiencies with respect to the clarity in the wording to facilitate its reading, understanding and reproduction of the model, as well as regarding some data, its obtaining and its presentation. It is also suggested that English, writing, citations and literature be checked. Perhaps a deeper analysis of the results is missing, and points 3 and 4 should also be reordered into 3 Results, 4 Discussion and 5. Conclusions.

Answers to General Comments. We appreciate the reviewer’s valuable view on the presented topic and addressed the his / her suggestions in the revised paper. However, we do not fully agree with the observation that the document presents some deficiencies with respect to the clarity in the wording to facilitate its reading, understanding and reproduction of the model. The paper is a natural continuity of the research works exposed in the following set of references:

30. Konior, J. Technical assessment of old buildings by fuzzy approach, Archives of Civil Engineering, 2019, 65(1), pp.129–142, https://0-doi-org.brum.beds.ac.uk/10.2478/ace-2019-0009.
31. Konior, J.; Sawicki, M.; Szóstak, M. Intensity of the Formation of Defects in Residential Buildings with Regards to Changes in Their Reliability.  Sci.2020, 10, 6651. doi:10.3390/app10196651
32. Konior, J.; Sawicki, M.; Szóstak, M. Influence of Age on the Technical Wear of Tenement Houses. Sci. 2020, 10, 6651. doi:10.3390/app11010297
33. Konior, J.; Sawicki, M.; Szóstak, M. Damage and Technical Wear of Tenement Houses in Fuzzy Set Categories. Sci. 2021, 11, 1484. doi: 10.3390/app11041484.

but does not need to be read in conjunction with one another. Identification of the problem, method and model, results and conclusions seem to be comprehensive, understandable and linked to each other. English language has been rechecked and improved to the higher level of technical language and its comprehensiveness for the reader. 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 modelled 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. 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. 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. In turn, in fuzzy sets, the observed states of cause-effect phenomena in the fuzzy conditions (i.e. uncertainty as to the very fact of their occurrence) were analysed.

Here are some observations:

Summary: Improve the summary in terms of methodology and results. On the other hand, although the objectives are identified, they could be written more succinctly without losing clarity.

Answer: Good point. The following improvement to better understanding of summary has been inserted: “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.”

1. Introduction

1.1. Literary review: they seem correct references, although some of them could be discussed a little more.

Answer: Valuable remark. It is planned to allow for the following digest of considered literature:

1.2. Research sample: Some of the constructive characteristics of the chosen buildings could be specified a little more, and if they are frequent or not in other areas (autonomous, national or international) for extrapolation purposes.

Answer: It is being considered to include the following supplementary characteristic of selected buildings: “The subject of the research concerning the accelerated wear of residential buildings [7-8] involves tenement houses in a separate part of Wroclaw's "Srodmiescie" district. The buildings are situated along downtown streets of secondary importance in an urban layout that has remained unchanged for years. They are front buildings, and also outbuildings with a modest architectural design and economical functional standard. They were built of brick in longitudinal, usually three-bay, structural systems. The 102 tenement houses were mainly built in the second half of the twentieth century, until the outbreak of World War I, although three of them are 170 years old. Due to the enormous scale of war damage that took place in this area in 1945, it is difficult to determine with certainty the type of building development. It can be assumed that at the time of the examination, almost 2/3 of the buildings were built in compact developments, 1/5 in semi-compact developments, with 1/6 being erected as free-standing buildings. The number of storeys varies from 2 to 5, where 9% are two-story buildings, 10% are three-story buildings, 39% are four-story buildings, and 42% are five-story buildings. The vast majority of the tenement houses (84%) have a basement under their entire ground floor, 9% have a basement under a part of their ground floor, and 7% have no basement at all. With the exception of three buildings, all of them have usable attics: 83% of them are used as a drying room, and 17% have been converted into apartments. The apartments were designed without sanitary installations. Water intake points, as well as sinks and toilets (c.c.) were later installed on the staircase landings and even in the kitchens of the apartments. Most of the apartments are heated by furnaces, and only a few have central heating made by the residents themselves. Electrical installations, originally designed as surface-mounted, after the unprofessional modifications of tenants, are placed under the plaster. Gas installations were introduced gradually (with the development of the city network) to almost all apartments. The above-described downtown residential buildings with construction and material solutions typical for the turn of the 19th and 20th centuries, with similar functions and standards, and with a specific form of ownership (the so-called pre-war "tenement houses") are defined in all parts of this article by the term "tenement houses". Extrapolation of results is out the scope of that specific paper and its correctness has been already developed in the papers:

2. Konior, J. Decision assumptions on building maintenance management. Probabilistic methods, Civ. Eng. 2007, 53, 403–423.
3. Konior, J. Technical Assessment of old buildings by probabilistic approach., Archives of Civil Engineering, 2020, 66(3), pp. 443–466, https://0-doi-org.brum.beds.ac.uk/https://doi.org/10.24425/ace.2020.134407
4. Research method

2.1 Identification problem: it is necessary to better clarify the concept of  II, III and IV of the maintenance conditions of an element. It is understood that clarity is not reached as to how the probabilistic results of table I are produced, referring to the data entered into them to produce the results.

Answer: OK, fine. The following explanation has been inserted: “Tested buildings have been classified to the classes, which determine 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”. The probabilities p(ui) presented in Table 1 of elementary damages ui in sets II, III, IV were calculated as simple occurrences / frequencies of random events and afterwards transferred to conditional fuzzy probabilities of Bayes.

2.2 Model to determine the conditional probabilities of the technical wear process in relation to the occurrence of damage: More data from the group of experts, their characteristics and how the values ​​of the aforementioned argument were determined should be provided. Perhaps it would be possible to better clarify the proposed model to facilitate its reproduction. As already mentioned for table I, it is understood that clarity is not reached as to how the results of table II are produced based on the same argument indicated in the previous point. On the other hand, it is not very well understood because this data has not been placed in the results section.

Answer: Well, we cannot agree with losing clarity how the how the results of table II are produced. It has been clearly, step by step, formula by formula, from (13) to (43) laid out how conditional probabilities were determined and presented in the table 2. Please note that the table 2 consists of two type of Bayes probabilities:

• 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 analysed buildings, in relation to the occurrence of their damage U, and with their mean values in relation to the probabilistic measure P(Z);
• 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 analysed buildings, in relation to the processes of their technical wear.

3. Results and conclusions: the conclusions should go to the end of the document after the discussion and before the references. Since some of the statements about the probabilistic results presented are evident, in the sense of being able to be deduced without the need for the study carried out, these statements, in addition to being verified by the different probabilities presented, must be supported by a presentation of the data in the form of graphs, especially the effect of a deeper analysis on whether it is possible to represent and even predict, with other types of mathematical functions, how the different elements behave in a generic way to apply the results to other situations with the proposed model or with others, even some of them simpler or simplified.

Answer: Items 3 and 4 have been re-named for the better conjunction the paper structure with its content, therefore item 3 is Results and item 4 is Discussion and Conclusions. As regard suggestions of better presentation of findings and transformations (already 43 ! mathematical functions presented and operated) please note that they have already been enclosed in our previous papers, especially in:

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

It was already driven and proved that as a result of the proposed model, which is based on fuzzy set theory, it is possible to identify the elementary damage that determines the degree of destruction of the building element. The result of the cumulative effects of frequently occurring mechanical damage to the structure and texture of elements indicates that this group of damage is no less important in the process of the technical wear of elements of downtown tenement houses. When determining the degree of damage of 10 selected building elements according to fuzzy criteria, it was indicated that there is a need for an individual approach to each of the elements (especially structural) during the process of their technical assessment. However, several regularities can be identified:

• 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). It most often differs from the observed values of the degree of the technical wear that was deter-mined using the probabilistic approach - especially in poor conditions of maintaining a building, 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 maintaining the building does the analysis of the observed random and fuzzy 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. Discussion and summary: It does not seem like a discussion but rather a summary or conclusions as already mentioned. Although the discussion results from commenting on the results, perhaps it is a section that should be carried out together with the presentation of these in a more precise way for a better understanding.

Answer: Items 3 and 4 have been re-named for the better conjunction the paper structure with its content, therefore item 3 is Results and item 4 is Discussion and Conclusions.

All comments and suggestions are accurate, valuable and professional, however the authors’ request is to include some of them in further publications as the topic is very precise and quite specific (Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings) and the manuscript is swelling out of 22 pages already. Also, some of remarks are well spotted but answers and solutions are presented in the past publications of J. Konior and it would be pretty difficult to allow for them without repetitions. We hope that the Reviewer will accept our justification and will respect our own point of view on presented research works in the current form.

Author Response File: Author Response.doc

Reviewer 3 Report

In this paper, authors presented a method to identify the impact of the processes associated with the broadly understood maintenance of old residential buildings.

The research sample was 102. The theory was novelty and the new information was provided. However, it lacked of the independent validation of their model and theory.

Two data sets were required for the development and validation. The first data sets was used to validate the model that have been presented in this manuscript. However, authors should select others samples (buildings) with 102 or more and less samples to perform the model validation. With the evaluation of independent samples, the prediction performance then could be confirmed.

Author Response

Wrocław, Poland, 5^th March 2021

Dear Reviewer of Applied Sciences,

Thank you for the review of our paper applsci-1132609 entitled “Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings” to be published in the journal Applied Sciences, special issue “Probabilistic and Fuzzy Approaches for Estimating the Life Cycle Costs of Buildings”.

We appreciate your thoughtful and  accurate comments as well as appreciation of our research works. We have carefully considered all comments and have now completed the revisions incorporating  your suggestions in the revised uploaded manuscript.

We hope that the revised paper meets your expectations.

Kind regards,

Jarosław Konior and Tomasz Stachoń

Department of Building Engineering, Faculty of Civil Engineering, Wroclaw University of Science and Technology, 50-370 Wrocław, Poland

Here are answers to reviewer’s comments:

REVIEWER 3

General Comments. In this paper, authors presented a method to identify the impact of the processes associated with the broadly understood maintenance of old residential buildings. The research sample was 102. The theory was novelty and the new information was provided.

Answers to General Comments. Thank you for your openness and appreciation of our research works as well as the fuzzy approach. While assessing 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 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 its lack the mid-states appear. Their value is often appreciate in verbal way, e.g. “substantially”, considerably”, “significantly”, “partially”, “hardly” and it is always met in a description of detected defects as a result of a building objects technical inspections. Therefore, the research led towards looking at the problem from the angle, which gave right to describe naturally qualitative variables (so immeasurable) and determine existing conditional probabilities in fuzzy sets categories [30, 36-37]. This is novel approach and as fuzziness supplements the random methodology developed by author’s in previous papers [2, 31-36].

Comment 1. However, it lacked of the independent validation of their model and theory. Two data sets were required for the development and validation. The first data sets was used to validate the model that have been presented in this manuscript. However, authors should select others samples (buildings) with 102 or more and less samples to perform the model validation. With the evaluation of independent samples, the prediction performance then could be confirmed.

Answer 1. This is a good point in terms of professional rules on validation applied models as to treat results as conclusion absolutely trustworthy, e.g. per [40]. Please note, that the research sample, which included 102 technically assessed residential buildings was selected from a group of 160 examined objects [39]. 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 for the adopted purpose of the study [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. 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. 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. Thus, after excluding 58 non-homogenous building there were no more objects to be selected for required validation of the adopted model. So, when it is not possible to verify the model by similar topologically sample then the equivalent = probabilistic approach to alliance between damages and technical wear may be applied – see papers:

2. Konior, J. Decision assumptions on building maintenance management. Probabilistic methods, Civ. Eng. 2007, 53, 403–423.
3. Konior, J. Technical Assessment of old buildings by probabilistic approach., Archives of Civil Engineering, 2020, 66(3), pp. 443–466, https://0-doi-org.brum.beds.ac.uk/https://doi.org/10.24425/ace.2020.134407.
4. Konior, J. Maintenance of apartment buildings and their measurable deterioration, Trans. Czas. Tech. 2017, 6, 101–107, https://0-doi-org.brum.beds.ac.uk/10.4467/2353737xct.17.090.6566.
5. Konior, K. Bi-serial correlation of civil engineering building elements under constant technical deterioration, Sci. Gen. Tadeusz Kosiuszko Mil. Acad. L. Forces. 2016, 179, 142–150.
6. Konior, J. Intensity of defects in residential buildings and their technical wear, Tech. Trans. Civ. Eng. 2014, 111(2-B), 137–146.
7. Konior, J.; Sawicki, M.; Szóstak, M. Intensity of the Formation of Defects in Residential Buildings with Regards to Changes in Their Reliability.  Sci.2020, 10, 6651. doi:10.3390/app10196651
8. Konior, J.; Sawicki, M.; Szóstak, M. Influence of Age on the Technical Wear of Tenement Houses. Sci. 2020, 10, 6651. doi:10.3390/app11010297

Validity of our research has been already proved by means of modern statistical software tools and was published in the papers applying probabilistic and correlation methodology [2, 31, 40-42]. The fuzzy contributions are only supplementing probabilistic analysis and are bearing out the basic findings concluded in random approach. The correctness of the test results for a representative group of old downtown apartment houses with traditional structure, erected in Wrocław (Poland) at the turn of the 19th and 20th centuries, can be therefore summed up by the following conclusions:

1. Age of elements of old apartment houses with a traditional structure:

• is of secondary importance in the process of the intensity of loss of its useful values;
• is not the essential size determining the course of their technical wear and tear;

1. The technical extent of wear and tear of the components of an old residential building is determined by the conditions for its maintenance and use;
2. The previous theoretical methods for measuring the technical wear and tear of the building and its components do not sufficiently describe the actual states, which is called into question:

• how these methods are assigned to the maintenance conditions of the building;
• not precise selection of too general forms of mathematical functions;

1. Quantitative damage analysis carried out by experiential methods of assessing the technical condition of the building shall indicate the nature and magnitude of the damage to its components which are characteristic of the relevant maintenance conditions;
2. A study analysis of the processes of operation of residential objects and the basic dependencies of reliability theory made in it indicates that for the useful life of an object in which the working time to damage has an exponential distribution (this is in principle the life expectancy corresponding to the length of service of the dwellings concerned), the average remaining time of unsafe operation is constant at all times. Theoretically, therefore, residential objects, after some time of trouble-free operation, perform their functions as new. The age of the elements of an old residential building is then of secondary importance in the process of the intensity of loss of its useful value;
3. If assumed that the measure of matching the nonlinear mathematical models tested in the nonlinear regression method, as a function of the technical consumption of building elements over time, is the determination factor, then no more than 30% of the destruction of the elements is explained by the passage of time. Age is therefore not a determinant of the technological consumption of the elements of the buildings analysed;
4. Previous theoretical methods for measuring the technical wear and tear of the building and its components do not reflect the actual course of the deterioration process over time. Two facts pay attention:

• an assessment of the significance of the differences between the theoretical and observed technical consumption distribution values of building elements by WILCOXON test and the Sign Test in most cases confirmed the conclusions of their comparative analysis and showed the significance of the differences between the distributions of theoretical and observed wear, although the Sign Test indicated their identity in the case of foundation distributions, underground walls and structural walls in the all five conditions of maintenance, while both WILCOXON test and the Sign Test confirmed the identity of the distributions only in individual maintenance groups of the building;
• adopting too general and not always appropriate forms of parabolic and linear functions to describe the theoretical side of the progress of the technical consumption of the building's elements with age; of the four nonlinear regression separable tested, new mathematical models, none of the power (parabolic) models represent the nature of the designated trend of the time-consuming process (very low determination factor and unnatural size of parameterized durability); analysis of variance in the nonlinear regression method also indicates a much better representation of the modelled trend by exposive and hyperbolic dependencies and slightly worse by linear functions;

1. The quantitative analysis of damage, carried out by empirical (visual) methods of assessing the technical condition of the buildings, indicates the type and determines the magnitude of these damage to its components which are characteristic of the appropriate conditions of maintenance. Studies of cause - effect "damage - technical wear" in observed states allow a numerical recognition of the impact of the building's maintenance conditions on the degree of technical wear of its components:

• the direction of the relationship is right-hand (positive) for all test elements of the building, but the correlation force between the defect occurring and their technical wear shows a significant span (from 0.00 to 0.84) depending on the conditions of the buildings maintenance;
• the rule is that correlations of at least moderate strength always show damage caused by water penetration and moisture penetration (on average 0,54); only in the case of internal plasters and façade, individual mechanical damage to their structure and texture can also be considered moderate and quite strong;
• for the accepted confidence level of 95%, the dependence of moderate force can be applied to 34-48% of the general population size, and the correlations quite strong - to 49-71%.

Therefore, we reckon that our research works – presented both in random and fuzzy approaches – are coherent, consistent, comprehensive, solid and reliable.

Author Response File: Author Response.doc

Round 2

Reviewer 1 Report

I'm globally satisfied by all responses given by Authors.

Reviewer 3 Report

The content of revised paper have been improved significantly. It could be accepted as a research paper.