A fuzzy logic approach to preventive conservation of cultural heritage churches in Popayan, Colombia

A Fuzzy Logic Approach to Preventive Conservation of Cultural Heritage Churches in Popayán, Colombia Andres J. Prieto a , Isabel Turbay b , Rocio Ortiz c , M. J. Chávez d , J. M. Macías-Bernal e , and Pilar Ortiz c a Instituto de Arquitectura y Urbanismo, Facultad de Arquitectura y Artes, Universidad Austral de Chile, Valdivia, Chile; b Facultad de Arquitectura, Fundación Universitaria de Popayán, Popayán, Colombia; c Department of Physical, Chemical and Natural Systems, Universidad Pablo de Olavide, Seville, Spain; d Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingeniería de Edificación, Universidad de Sevilla, Sevilla, Spain; e Departamento de Construcciones Arquitectónicas II, Escuela Técnica Superior de Ingeniería de Edificación, Universidad de Sevilla, Sevilla, España ABSTRACT From a decision-maker’s perspective, the management of cultural heritage is a challenging task because of different objectives pursued, the public/private nature of heritage constructions studied, the wide variety of associated values (artistic, historical, cultural, economic) and diverse interests of different stakeholders. Careful consideration of environmental, social and economic factors is crucial in order to predict the functional service life of heritage constructions. This study outlines a new fuzzy system approach based on expert knowledge, which focuses on the serviceability of heritage buildings using a Mamdani fuzzy model. The method input considers 10 intrinsic variables directly related to the vulnerability of buildings and 9 external hazards, classified according to static-structural features and environmental hazards, which are considered useful for defining emergency programs in historical centres in South America (Colombia). The automation of inspection programs similar to the one described here may reduce the consumption of natural resources and benefit a more rational management of future maintenance actions. In general terms, this type of fuzzy logic model approach is designed to serve as an indicator for the future evolution of the functionality of buildings. This research contributes to the study of new service life models applied to heritage buildings in South America. ARTICLE HISTORY Received 17 September 2019 Accepted 29 February 2020 KEYWORDS Functional service life; fuzzy model; heritage buildings; maintenance programs; preventive conservation 1. Introduction 1.1. Preventive conservation of cultural heritage Cecchi and Gasparoli (2012) argued that preventive conservation analysis focused on architectural heritage buildings is a novel approach, implying that innovation should be based on a set of prioritized steps including conservation focused on maintenance and restoration, and a longterm risk assessment (Della Torre 2018). The vulnerability of architectural heritage is rooted in different external phenomena, such as natural disasters (e.g. earthquakes), social conflicts, climate change, anthropic risks, and daily environmental impacts (Van Balen and Vandesande 2018). On the one hand, Prevention itself cannot be reduced to a single and simple perspective (Della Torre 2018); rather, it should be related to risk management, thus evidencing a specific technological background. Impacts from external risks are correlated with the heritage constructions and the anthropic and environmental context, also considering the conditions of use (Lazzari, Danese, and Masini 2009). On the other hand, meteorological events, such as thunderbolts, heavy rainfall and floods may be exacerbated by climate change phenomena, suggesting the need for different preventive actions (Konsta 2018) and assessment methodologies aimed at predicting the service life of buildings (Ortiz et al. 2016). Further, important hazards might originate from human activities in the surrounding territory (Della Torre 2018; Ortiz et al. 2017) such as pollution, traffic vibrations, shock waves due to explosions or works, subsidence due to groundwater pumping, or nearby constructions. In all these scenarios, mitigating strategies are necessary and should be recommended and established by stakeholders. Moreover, Colombia is situated along the Pacific coastline, also known as the Ring of Fire (or ‘Circum Pacifico’) (Díaz 2017), and is characterized by the largest subduction zone in the world related to the extremely high seismic and volcanic activity in this region. Therefore, new recommendations, guidelines and service life methodologies (Ramos et al. 2018; Silva, de Brito, and Gaspar 2016) are necessary to determine successful plans to safeguard cultural heritage sites at risk due to external hazards (Drdacky 2010). Risk also arises from the absence of systematic policy for the [email protected] Universidad Austral de Chile, Valdivia, Chile planned conservation and service life prediction of heritage constructions in South America. 1.2. Service life prediction methodologies Predicting the service life of buildings, heritage buildings and building materials is a very complex process, linked with many parameters such as the interior environment conditions, design, execution, maintenance conditions (corrective or preventive), and quality of building materials, among others (ISO 15686: 2012). In this respect, the current service life prediction models available present certain short-comings, due to (i) the lack of resources; (ii) scarcity of methodologies able to quantify the service life of buildings; (iii) the particular performance of the degradation phenomena. Stakeholders must choose the approach they take to the problem they need to solve, considering the advantages and disadvantages of each service life prediction method (de Freitas et al. 2013). In the literature, service life prediction models have been classified into stochastic, deterministic and computational models (neural networks, fuzzy logic) (Silva, de Brito, and Gaspar 2016). Artificial neural networks and fuzzy logic models are able to work as approximators (Kosko 1994), which can emulate situations of uncertainty such as human reasoning. Fuzzy logic systems have been used in the areas of construction engineering, life cycle analysis, and multi-criteria analysis, among others, and they have yielded successful results. These kinds of models are particularly interesting when they are used to model processes that involve uncertainty. The positive application of fuzzy models is due to their tolerance of inaccurate data (qualitative information), flexibility, their ability to model vague systems, their ability to use input data based on human observation; using linguistic data (human communication). Therefore, in this research, a fuzzy logic model is used, which is able to manage the uncertainty associated with the degradation of heritage buildings. 2. Research objectives The case-study presented here focuses on the protection of the cultural value and functional service life of seven heritage churches in the city centre of Popayan, Colombia. The analytical tool described here will be useful for the stakeholders (users, owners, public and private administrations) involved in the restoration and rehabilitation of heritage buildings in Colombia, since it facilitates the implementation of scientific criteria and local-regional policies for the planning and management of maintenance actions, thus minimizing the risks of losing cultural assets. The software was developed using the expertise of professional experts and computational models based on fuzzy logic and the Delphi method. 3. Materials and methods Generally, the fuzzy logic approach described here is more effective when applied to sets of heritage constructions rather than individual buildings (Prieto et al. 2017a). Nevertheless, the large number of buildings, the area assessed, the number of resources and the amount of data required turned out to be significant when conducting this present study, which prompted the need to simplify building typologies in order to reduce complexity. 3.1. Historical, geographic and climatic conditions of Popayan, Colombia The city of Popayan (altitude 1760 m; 2º27’33”N, 76º36’01”W) is located 600 km southwest of Bogotá (Figure 1). Popayan is the 25 th most populous city in Colombia with over 285,000 inhabitants. Its historic centre has a grid-type perimeter comprising 236 blocks and is the second largest in Colombia (after Cartagena de Indias). The city is located on the road to Quito and Lima in an area rich in resources, both of which contributed to its importance during the Spanish Empire, a status the city maintained until the early 20 th century. The city of Popayan is characterized by colonial-style architecture with a municipal layout comprising mainly religious buildings as part of the iconic image of modern society in the province of Paia. In addition to their pivotal role in the urban-historical structure of the city, the churches form the axis for the Holy Week processions, one of the most important traditions in the city, classified as Intangible World Heritage by UNESCO. The climate of Popayan is marked by an annual average temperature of around 18–19ºC, with highs in excess of 29ºC during July, August and September. Annual rainfall in the region is around 2127 mm but concentrated mainly between October and December. At a macroclimatic level, the case-study sites correspond to a tropical rainforest climate, classified as “Af”in the Köppen–Geiger system (Kottek et al. 2006). 3.2. Constructive characterization of heritage constructions The case studies presented here focus on seven Baroquestyle churches, or belonging to a later period, but with similar constructive features and built between the 16 th and 18 th centuries in the historical city centre of Popayan (Colombia) (Figure 1). Most of the heritage constructions 1911 analyzed are characterized by a basilica-type layout comprising three naves, where the central nave is separated by semi-circular arches. Four of the seven case studies have three naves and the rest three buildings have only one. All churches were built with brick and stone, with some preserved adobe or mud walls or exposed masonry. The structures are made from coated brick (brick wall with mortar coating), with brick or stone arches and columns and wooden roofs covered with tiles. The churches of La Encarnación (ID-4/PO-ENC), San José (ID-7/PO-SJS), San Francisco (ID-2/PO-FCO), Santo Domingo (ID-1/PO-DMG) and San Agustín (ID-3/POAGT) have bell towers, whereas the El Carmen (ID-5/ PO-CMN) and Jesús Nazareno (ID-6/PO-NAZ) churches are noteworthy for their belfries. Table 1 summarizes the construction periods, architectural styles and certain relevant information related to construction work on these churches from recent centuries. 3.3. The fuzzy logic expert approach In real-life situations, human judgments are vague and cannot be translated into numerical values, because the process of human decision-making is permanently linked to some degree of subjectivity or uncertainty (Kutut, Zavadskas, and Lazauskas 2014). The fuzzy logic approach, introduced by Zadeh (1965), represents a powerful tool for addressing these types of uncertain situations such as built heritage degradation. Fuzzy logic systems perform better and offer greater accuracy than classical linear models (Vieira, Sousa, and Durão 2005). While Boolean logic understands reality in strict terms of ones or zeroes (black or white), fuzzy logic is able to manage subjective hypotheses and the uncertainty associated with real-life processes in a greyscale (Silva, de Brito, and Gaspar 2016) The methodology developed (Art-Risk 3.0) was implemented in the Xfuzzy open-access software program (latest version Xfuzzy 3.5), which can be executed on platforms using Java Runtime Environment—JRE. The tool has recently been updated (at the end of 2017) to include new functionalities and updated documentation on all tools. The Xfuzzy system integrates a set of user-friendly tools to cover the different stages involved in the design process based on fuzzy logic, from the initial description to the final implementation. A total of 18 international experts from Spain, Portugal and Chile specialized in architecture (4), building engineering (10), civil engineering (3) and Figure 1. Location maps and pictures of the case studies analysed in the city of Popayán (Colombia). 1912 A. J. PRIETO ET AL. chemistry (1) were consulted during the design of this new model (Art-Risk 3.0), its development and the implementation process. All advisors were renowned experts in the construction, diagnosis or conservation of cultural heritage. Professional experts presented an average of 10 years’experience in the area. The 21 input variables considered were identified after reviewing the following documents: Spanish Technical Building Code; National Cathedral Plan; Law on Building Planning (Ley de Ordenación de la Edificación —Spain); Heritage Conservation Network; UNE 41805:2009 IN; ISO 15686. The Art-Risk 3.0 method is based on extensive prior research at the University of Seville (Macías-Bernal, Calama, and Chávez 2014; Prieto et al. 2016,2017a,2017b;Prietoetal.2018a)andPablo de Olavide University (Ortiz et al. 2014; Ortiz, Macias- Bernal, and Ortiz 2018; Ortiz and Ortiz 2016; Ortiz et al. 2016). Art-Risk 3.0 is a new tool based on the previous studies developed by the research groups. Previous research focused on applying the same inference rules, intending to validate the model by applying it to new sets of buildings with homogenous constructive characteristics in Spain (Prieto et al. 2016), Portugal (Prieto et al. 2018b) and Chile (Prieto et al. 2019). The previous system (Art-Risk2.0 or FBSL) was validated and correlated to another predictive method, which evaluates the degradation of building components. The results demonstrated that when the functionality index decreases, the Table 1. Historical information on the group of heritage buildings studied. Identification Parish church Construction time Architectural style Data recovered and dates of most relevant historical events PO-DMG Santo Domingo 1736 Baroque 1572—Construction of the first church. 1736—Earthquake damage. The building collapses. 1885—Earthquake, which destroys the church tower. 1741—Reconstruction of the buildings. 1983—After an earthquake, the building, bell tower, walls and main entrance are repaired. 1996—Declaration as a National Monument of Colombia and building of cultural interest. PO-FCO San Francisco 1775–1795 Baroque 1500—Construction of the first church. 1736—Earthquake damage. The building collapses. 1775–1795—Construction of the modern-day church. 1983—Earthquake damage. The church suffers considerable damage, also affecting the bell tower and apse. 1996—Declaration as a National Monument of Colombia and building of cultural interest. 1998—Completion of restoration work. 2017—Inclusion in the “Religious Buildings”category. PO-AGT San Agustín 1583 Baroque 1583—Construction (during the 16 th century). 1736—Earthquake damage. Reconstruction by the architect Gregorio Causi. 1738—Reconstruction of the façades. 1827 —Different earthquakes damage the building’s construction. 1834 —New restoration and consolidation of the building. 1836—The bell tower is completed. 1846—The central nave is erected. 1858—The main entrance is completed. 1983—The church suffers damage to the columns and the roof of the central nave. PO-ENC Ntra. Sra. Encarnación 1764–1766 Baroque 1591—Completion of the first building. 1736—Destruction of the building by an earthquake 1764–1766—Construction of the new church. 1983—Earthquake damage. Restoration work is performed. 1985—Temple after restorations. 1996—Declaration of the church as a National Monument of Colombia and building of cultural interest. PO-CMN Ntra. Sra. Carmen 1729–1744 Baroque 1729–1744—Construction starts under the supervision of the architect Gregorio Causi. 1736—An earthquake damages the building’s construction. 1983—Earthquake damage. 1996—Declaration of the church as a National Monument of Colombia and building of cultural interest. 1998—Restoration work on the church begins. PO-NAZ Jesús Nazareno 1612–1617 Baroque 1612–1617—Construction of the original structure. 1736 —The church survives an earthquake. 1782–1784—The church is used as a cathedral. 1983—An earthquake damages the belfry and the roof. 1996—Declaration of the church as a National Monument of Colombia and building of cultural interest. PO-SJS San José 1744–1767 Baroque 1642—Construction of the building starts. 1702—Reconstruction of the building. 1736—Earthquake damage. 1744–1767—Construction of the current church on the foundations of the old one. 1885—The buildings suffer deterioration due to the last earthquake. 1983—An earthquake destroys the main vaults of the church and even the lateral vaults. 1913 degradation of the building components increases (Prieto et al. 2018b). Moreover, the previous fuzzy model (Art- Risk 2.0/FBSL)inwhichArt-Risk 3.0 is supported was enriched and reviewed through the analysis of the international standard ISO 31000:2009 (2009). The software designed is coupled with on-site inspections that could be carried out by experts or even end-users (owners, priests, local organizations, etc.). The information obtained in these kinds of inspections can be included in the fuzzification process in order to simultaneously evaluate three variables: the intrinsic vulnerability of the constructions, the external risk levels or affections, and the integral functional service life of the buildings studied. The comparison between these parameters will allow professional experts to make more efficient decisions related to the preventive conservation of a set of buildings. The Art-Risk 3.0 model can be complemented by user inputs and could be upgraded in a sequence of continuous improvements. 3.3.1. Fuzzification process Thefuzzificationstageinvolvestheconversionofcrisp values into membership grades for linguistic terms of fuzzy sets. Şen (1998) specified that partial assignment to a particular set can be described numerically by a membership function, which can take an infinite number of values in the range [0, 1]. In this kind of model, the input vector may be translated into linguistic terms, such as Very Low (VL), Low (L), Medium (M), High (H) or Very High (VH). The membership function, which has diverse types of non-linear and linear shapes, is used to associate a grade with each linguistic term. These applications assign a degree of membership to each component in the universe of discourse U, on which the fuzzy set is stated. The membership function μ A (u) of Fuzzy Set Acantakeanyvalueinthe range [0, 1] [Equation (1)]: μAuðÞ:U!I0;1½ (1) The software model is set up with 19 inputs—10 vulnerability variables and 9 hazards (Table 2). The total variables of the system were fuzzified by means of the Gaussian membership functions shown in Figure 2,whichareconsidered to be the most appropriate for modelling building degradation (Prieto et al. 2017b;Ross2010). In this study, visual inspections as a diagnosis assessment and an in situ survey were used to collect data on vulnerability, risks and functional service life of heritage constructions. Specifically, the input variables (AR 2 ,AR 3 , AR 4 ,AR 5 ,AR 9 ,AR 10 ,AR 11 ,AR 12 ,AR 13 ,AR 14 ,AR 15 )were studied during each in-situ visual inspection of the buildings. The other variables were obtained as follows: variable AR 1 from the Colombian Geological Service (“SGC”—“Servicio Geológico Colombiano”in Spanish, 2015); variable AR 6 from the National Administrative Statistics Department (“DANE”—“Departamento Administrativo Nacional de Estadística”in Spanish), (“Proyecciones censo”, 2005); variable AR 7 from the Columbian Ministry of Culture, Cultural Interest Assets (updated, 2018); variables AR 16 and AR 17 from the Institute of Hydrology, Meteorology and Environment Studies of Columbia (“Instituto de Hidrología, Meteorología y Estudios Ambientales de Colombia”—“IDEAM”—in Spanish) (“Atlas Climatológico Colombiano”, 2018); variable AR 20 is obtained from the Colombian Regulation on Seismic-Resistant Constructions, NSR-10, of the Ministry of Environment, Housing and Territorial Development; variable AR 21 from the Special Management Plan (“Plan Especial de Manejo”—“PEMP”—in Spanish), Popayán (“Amenazas por inundación”, 2007). The input, intermediate and output variables were correlated with a linguistic label from minimum values (Very Low (VL), Low (L)), which reveal minimal vulnerability-risk impact, but high functional service life, to (High (H), Very High (VH)), which indicate maximum vulnerability-risk impact, but low functional service life. 3.3.2. Knowledge base and inference rules This stage is the main part of the fuzzy expert system Art- Risk 3.0 and combines the facts derived from the fuzzification step with the base rule previously carried out in the modelling process. Babuška (1998) refers to the fuzzy expert system as grey boxes since they describe relationships using a set of IF-THEN rules. The methodology described here is based on the Mamdani fuzzy model, which is one of the most accepted and applied algorithms (Mamdani and Assilian 1975). The Mamdani model involves fuzzy inference rules, where each rule describes a local input–output relationship. The knowledge base is a set of fuzzy control rules, including linguistic descriptions that represent the expert knowledge of the system (Prieto et al. 2017b). The fuzzy model works as a generalized modus ponens Equation (2). Rule jðÞ:IF AR1is X1jAND AR2is X2j...ARnisXnj THEN yisZj(2) where AR 1 ,AR 2 ,AR n are the input variables, X 1j ,X 2j ,X nj the output variables, Z j the linguistic terms, nthe number of inputs, ythe outputs and jthe number of rules. The Mamdani inference mechanism is used in the composition of fuzzy propositions. In contrast to a conventional fuzzy expert system, various rules can be activated simultaneously. These types of methodologies can work with the minimum operator as the implication function and the maximum as the aggregation operator (Ross 2010). The methodology was established with a set of 1914 A. J. PRIETO ET AL. 581 fuzzy inference rules. An example of a total of 33 fuzzy inference rules extracted from the new model (Art-Risk 3.0) is presented in Table 3. The hierarchical structure of the new fuzzy model can be analyzed based on the level of interrelation between variables (Figure 3): Table 2. Input variables (vulnerability and external hazard impacts) of the new method (Art-Risk 3.0). Categories Inputs variables Ids Quantitative and Qualitative valuation Descriptive valuation of the input parameters Vulnerability Geological location AR 1 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Very favourable ground conditions Acceptable ground conditions Very unfavourable ground conditions Built context AR 2 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Buildings without complex surrounding constructions. Buildings between surrounding constructions. Buildings with complex surrounding constructions. Construction system AR 3 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Homogeneous characteristics of constructive system. Heterogeneous features of constructive system. Very heterogeneous features of constructive system. Roof design AR 4 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Fast evacuation of water. Normal evacuation of water. complex and slow evacuation of water. Conservation AR 5 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Optimal state of conservation. Need conservation. Neglected state of conservation. Anthropic considerations Population growth AR 6 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Population growth greater than 15% (very favourable). Population growth equal to 0% (acceptable). Population growth less than 5% (very unfavourable). Heritage value AR 7 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Great historical value. Medium historical value. Low historical value. Furniture value and Occupancy Value of movable assets AR 8 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Very high furniture value. Medium furniture value. Very low furniture value. Occupancy AR 9 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Very high activity in the building. Medium activity in the building. Very low activity in the building. Maintainability Maintenance AR 10 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Good maintenance of the building. Average maintenance of the building. Bad maintenance of the building. Static-Structural hazards Ventilation AR 11 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Natural cross-ventilation in all areas. Natural cross-ventilation only in some area. Not natural cross-ventilation inside the construction. Overloads AR 12 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Overloads below than original state. Overloads equal to original state. Overloads higher than the original state. Fire AR 13 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Low fire load —combustible structure. Medium fire load —combustible structure. High fire load in relation to the combustible structure. Facilities AR 14 1—Very low (VL) 3—Medium (M) 5—Very high (VH) All facilities are in use and under standard. Some facilities are in use. Facilities are not in use. Structural modification AR 15 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Symmetric and balanced structural modifications. Apparently balanced structural modifications. Disorderly structural modifications. Environmental hazards Average Rainfall AR 16 1—Very low (VL) 3—Medium (M) 5—Very high (VH) <600 mmm (favourable). 600 mm < 1000 mm (acceptable). >1000 mm (unfavourable). Raindrop impact AR 17 1—Very low (VL) 3—Medium (M) 5—Very high (VH) <7 (favourable). 8 < 9 (acceptable). >10 (unfavourable). (Torrential rain index) Thermal stress AR 18 1—Very low (VL) 3—Medium (M) 5—Very high (VH) <6 (favourable). 6 < 10 (acceptable). <10 (unfavourable). (Thermal amplitude) Frozen damage AR 19 1—Very low (VL) 3—Medium (M) 5—Very high (VH) <1 day (favourable). 5 days < 20 days (acceptable). >60 days (unfavourable). (days below 0) Natural hazards Seismic hazard AR 20 1—Very low (VL) 3—Medium (M) 5—Very high (VH) <0.04 g (favourable). 0.08 g < 0.12 g (acceptable). >0.16 g (unfavourable). (acceleration) Floods AR 21 1—Very low (VL) 3—Medium (M) 5—Very high (VH) Never (favourable). 100 years (acceptable). 10 years (unfavourable). (return period) 1915 ●Hierarchical level 1: input variables (AR 1 -AR 19 ) ●Hierarchical level 2: Vulnerability 1 and Vulnerability 2, Anthropic vulnerability, Maintainability, Static-Structural Risk 1 and Static-Structural Risk 2; ●Hierarchical level 3: Static-Structural Risks and Environmental Risks; ●Hierarchical level 4: Vulnerability and Risks; ●Hierarchical level 5: Durability. Finally, the model outputs (intrinsic vulnerability, external risk impacts and functional service life of the constructions) are produced in the final level of the set of rules. Impacts from natural hazards, such as floods (AR 20 ) and seismic hazard (AR 21 ), are not included in the fuzzy methodology, because both variables involve several grades of uncertainty. When flood and/or seismic risks are detected (value ≥3), an emergency plan is recommended to the end-users. An example is shown Figure 2. Membership functions of the fuzzy system: i) input variables; ii) intermediate variables; iii) output variables. 1916 A. J. PRIETO ET AL. in Figure 4 demonstrating the association between membership functions (static-structural risks 1, staticstructural risks 2 and environmental risks). 3.3.3. Defuzzification process The final defuzzification phase was used to obtain a (crisp) value representing the fuzzy information produced by the inference rules. The Art-Risk 3.0 model uses the Centre of the Area, which is one of the most successful defuzzification methods (Moreno-Velo et al. 2007; Prieto et al. 2017a). One of the most important properties of this process is continuity: small changes in input parameters do not involve an unexpected change in the modelling outputs. It can be interpreted as a Riemann sum [Equation (3)]. Art Risk3:0¼PiARiμBARi ðÞ PiμBARi ðÞ (3) The fuzzy model outputs are interpreted in terms of Vulnerability (V), Risks (R) and Functional Service Life (FSL) allowing for owners, building users, public administrations and companies to manage the deterioration of constructions. This model supports an interdisciplinary approach for cultural heritage conservation managers (Lobovikov-Katz et al. 2018). 4. Results and discussion 4.1. Theoretical application of the fuzzy model considering vulnerability, risks and functional service life First, the sensitivity of the model was analyzed to obtain information about the influence of each parameter on the modelling results. This section shows the functionality evolution of seven theoretical buildings based on various vulnerabilities and external hazard situations, showing which contexts influence the output of the model to a greater or lesser extent: vulnerability, risks and functional service life of buildings. On the one hand, contexts with a lower functional service life level are those that should present higher vulnerabilities and risks. Thus, these kinds of situations must be prioritized in order to create preventive maintenance actions. Dann and Wood (2004) commented in this regard that maintenance priorities should include preserving the social and cultural importance of each construction and its components. On the other hand, contexts with a higher functional service life present lower levels of risk and vulnerability. In these situations, preventive maintenance actions or programs should be considered without priority. The new model is a fuzzy inference system, which performs an integrative analysis of heritage constructions, considering 10 vulnerabilities (AR 1-10 ) and nine hazard factors (AR 11-19 )(Table 2). Different theoretical states of heritage constructions were considered. The set of variables ranged from 1 (most favourable scenarios) to 5 (least favourable scenarios) (Table 4). The sensitivity approach aims to demonstrate the relevance of each input variable in the model output by considering seven scenarios: ●Scenario I: all input variables present the best possible valuation; ●Scenario II: major impact of the intrinsic vulnerabilities (geological location, built context, construction system, roof design and conservation); ●Scenario III: major impact of anthropic vulnerabilities (population growth, heritage value, furniture value, occupancy); ●Scenario IV: maintenance presents the worst possible valuation; ●Scenario V: major impact of static-structural hazards (ventilation, facilities, overloads, fire, structural modification); ●Scenario VI: major impact of environmental hazards (average rainfall, raindrop impact, thermal stress, frozen damage); Table 3. If-then fuzzy inference rules established by international professional experts. Inference rules IF AND AND THEN Rule 1 AR 1 is VL AR 2 is VL AR 3 is VL Vulnerability_1 is VVL Rule 2 AR 1 is VH AR 2 is VH AR 3 is VH Vulnerability_1 is VVH Rule 3 AR 1 is VL AR 2 is VL AR 3 is L Vulnerability_1 is VL Rule 4 AR 1 is VL AR 2 is VL AR 3 is M Vulnerability_1 is L Rule 5 AR 1 is VL AR 2 is VL AR 3 is VH Vulnerability_1 is VH Rule 6 AR 1 is VL AR 2 is L AR 3 is VL Vulnerability_1 is VL Rule 7 AR 1 is VL AR 2 is M AR 3 is VL Vulnerability_1 is L Rule 8 AR 1 is VL AR 2 is VH AR 3 is VL Vulnerability_1 is M Rule 9 AR 1 is L AR 2 is VL AR 3 is VL Vulnerability_1 is M Rule 10 AR 1 is M AR 2 is VL AR 3 is VL Vulnerability_1 is MH Rule 11 AR 1 is M AR 2 is VL AR 3 is VL Vulnerability_1 is VH Rule 12 AR 1 is VL AR 2 is L AR 3 is L Vulnerability_1 is LM Rule 13 AR 1 is VL AR 2 is L AR 3 is L Vulnerability_1 is L Rule 14 AR 1 is VL AR 2 is M AR 3 is L Vulnerability_1 is LM Rule 15 AR 1 is VL AR 2 is VH AR 3 is L Vulnerability_1 is M Rule 16 AR 1 is VL AR 2 is L AR 3 is M Vulnerability_1 is MH Rule 17 AR 1 is VL AR 2 is L AR 3 is VH Vulnerability_1 is VH Rule 18 AR 1 is VL AR 2 is H AR 3 is H Vulnerability_1 is H Rule 19 AR 1 is VL AR 2 is M AR 3 is M Vulnerability_1 is MH Rule 20 AR 1 is VL AR 2 is VH AR 3 is VH Vulnerability_1 is VVH Rule 21 AR 1 is L AR 2 is L AR 3 is L Vulnerability_1 is M Rule 22 AR 1 is M AR 2 is L AR 3 is L Vulnerability_1 is MH Rule 23 AR 1 is H AR 2 is L AR 3 is L Vulnerability_1 is VH Rule 24 AR 1 is VH AR 2 is L AR 3 is L Vulnerability_1 is VH Rule 25 AR 1 is M AR 2 is M AR 3 is M Vulnerability_1 is MH Rule 26 AR 1 is M AR 2 is M AR 3 is M Vulnerability_1 is M Rule 27 AR 1 is M AR 2 is VH AR 3 is VH Vulnerability_1 is VVH Rule 28 AR 1 is M AR 2 is H AR 3 is H Vulnerability_1 is VH Rule 29 AR 1 is VH AR 2 is VH AR 3 is H Vulnerability_1 is H Rule 30 AR 1 is VH AR 2 is H AR 3 is VH Vulnerability_1 is H Rule 31 AR 1 is L AR 2 is M AR 3 is M Vulnerability_1 is MH Rule 32 AR 1 is L AR 2 is H AR 3 is M Vulnerability_1 is MH Rule 33 AR 1 is L AR 2 is M AR 3 is VH Vulnerability_1 is VVH 1917 Figure 3. Hierarchical structure of the fuzzy inference system. Figure 4. Inference level of the hierarchical structure of the fuzzy system. 1918 A. J. PRIETO ET AL. To answer these needs, the Art-Risk 3.0 method for assessing vulnerability, hazard impacts and functionality of cultural heritage was defined, tested and applied to historical buildings in Colombia. The system incorporates the following innovative characteristics, making it a versatile and useful tool for managing architectural heritage in South America: (i) it is based on a survey of international professional experts; (ii) it Figure 9. Exterior images of the current state of San Francisco parish church (Popayán, Colombia). 1925 can be adapted to different environmental, economic, social and cultural contexts; (iii) it integrates fuzzy logic approaches and a survey of professional experts; and (iv) it is able to classify sets of heritage buildings (with homogeneous constructive features), prioritizing interventions and considering the serviceability of buildings as a whole. In the context of Colombia, the main cause for building degradation is the lack of investment by stakeholders. The analysis performed here provides a firstorder approach to define criteria for the development of preventive maintenance plans for groups of buildings with similar characteristics and to design effective and efficient strategies for the conservation of heritage buildings in regional contexts across South America. Semi-automated maintenance activities can minimize the consumption of natural resources, enabling more rational management of future maintenance processes. This information is crucial since it can be applied by different stakeholders (local end-users, communities, managers, engineers, architects) from the construction sector, promoting an effective method for safeguarding heritage constructions. In future research, the methodology may be adapted to new scenarios and environments, including local contexts, and could take into account the prevailing circumstances and identify new potential input variables that could be important in explaining functional service life, vulnerability and risks affecting buildings. Figure 10. Interior images of the current state of San Francisco parish church (Popayán, Colombia). 1926 A. J. PRIETO ET AL. Figure 11. Exterior images of current state of Nuestra Señora de la Encarnación parish church (Popayán, Colombia). Figure 12. Interior images of current state of Nuestra Señora de la Encarnación parish church (Popayán, Colombia). 1927 Disclosure statement No potential conflict of interest was reported by the authors. Funding This paper has been supported and based on the methodology developed by the Research Project Art-Risk, a RETOS project of Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER), code BIA2015- 64878-R (MINECO/FEDER, UE), and the International Cooperation Project code UPO-03 funded by Consejería de Fomento, Infraestructuras y Ordenación del Territorio, Junta de Andalucía. This paper was also supported by the project ANID - FONDECYT Nº11190554 (Chile). ORCID Andres J. Prieto http://orcid.org/0000-0003-0604-0364 Rocio Ortiz http://orcid.org/0000-0002-0249-8821 J. M. Macías-Bernal http://orcid.org/0000-0001-6073-9745 Pilar Ortiz http://orcid.org/0000-0002-8547-7531 References Babuška, R. 1998.Fuzzy modelling for control. Boston: Kluwer Academic Publishers. 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