Performance-based design of multi-story buildings for a sustainable urban environment: a case study

This paper critically reviews the role of performance-based generative design in fast prototyping of buildings, describes the methodology of an automated generative layout design to produce complete building solutions, and presents a case study of multi-story buildings in urban context. The proposed approach evolves the building design solutions by interacting with the city 3D geometry and evaluates the energy consumption for air-conditioning. The building designs take into consideration urban geometric constraints and objectives, such as alignment with surrounding buildings, urban lot area, and relative and absolute position of the generated elements. During the evaluation process, the urban context is considered for casting shadows and reflecting solar radiation. The case study consists of six alternative 15-story buildings located in the city of São Paulo (Brazil), having commercial areas on the ground floor and two apartments per story on the remaining floors. The results show that, despite having similar apartments in every story, the urban context has a relevant impact on the buildings’ energy performance. The difference between the apartments’ best and worst energy performing stories ranges from 9 % to 12 % (ignoring the outlier story located in the first level), depending on the building solution. The results also show that the most energy efficient apartments’ story is not located in the top or bottom floors, but rather at an intermediate level.


Introduction
The sustainability of the built-environment is a complex subject in which interrelated systems have strong impact on the social, economic, and environment dimensions of the cities [1]. The aspects that influence the building performance are well known and largely studied, such as the form, compactness, orientation, thermophysical properties of the envelope elements, shading mechanisms, and ventilation strategies [2,3], which can be optimized to improve the overall performance [4][5][6][7].
However, buildings should not be evaluated simply as an independent unit but rather understood as part of the urban fabric; surroundings alter the building performance [8] and the building itself influences the urban micro-climate [9].
As the complexity of building design demands knowledge in different fields and large number of alternatives to evaluate, performance-based design methods present themselves as promising tools to assist the building practitioners in the decision-making. These consist of finding novel solutions using building performance as guiding factor [10], where performance is assessed by simulation of a digital model set under predefined conditions [11]. Such generative design methods are "capable of producing concepts and stimulating solutions based on robust and rigorous models of design conditions and performance criteria" [12], where an internal generative logic commands the creation of a range of alternative solutions for the building practitioners to compare and select the ones to be further developed [13]. Besides allowing integration of synthesis and analytic phases of design, these automated and fast-prototyping design methods help to overcome designers' "limitations of knowledge or fixation" and automate tedium design tasks, thus "leaving more time for creative activities, and help reduce errors" [14].
Different approaches have been studied for the last two decades and were used to improve the structural dimensioning of buildings; to determine better building layout and building skin design; to dimension active systems, and to determine the most suitable construction system according to several performance criteria, such as construction and operation costs, energy use and production, indoor air quality, lighting, acoustics, building safety, and sustainability [13,[15][16][17][18][19].
One of the aspects covered by performance-based design methods is the study of the building layout, which consists of finding the best indoor arrangement of rooms such that it satisfies the functional program, as well as the topologic and geometric requirements. The process is commonly referred to as floor planning, space planning, or layout planning. The process of synthesizing design solutions may happen under conflicting objectives and requirements, sometimes vague and fuzzy user preferences. It occurs in the early stages of building design and it is a combinatorial problem in nature, thus the number of potential solutions increases exponentially as the complexity of the building design program grows. As the best performance-based decisions are made in the early stages, it is of outmost importance to explore alternative design solutions during the space planning phase, to help building practitioners to find the most promising designs to be further developed. Associated to each space and dwelling typology, there is information for occupancy, lighting, electric equipment, ventilation, hot water use, etc., therefore it is easy to produce detailed simulation models. Additionally, the urban context is a predominant factor for defining the building design geometry solution and it has a significant influence on the indoor thermal performance of the building due to shadowing and reflection effects of the surroundings [20]. Despite existing several algorithmic approaches to create building layout solutions, usually these are very abstract, missing several of the building designers' objectives and constraints. In the real world, building practitioners take into consideration several requirements and preferences related to urban context, such as building lot, construction area, building boundary, building alignments, visual obstructions, etc.; however, most of them are not considered in the generative design methods.
For instance, Koenig and Schneider [21] reviewed several works using different computational methods to solve buildings' layout problems and none of the analyzed works included surrounding concerns in the form of constraints or preferences to influence the generated designs. Likewise, Dino [22] implemented a 3D layout approach where solutions are generated to fit a provided building forms and then optimized using an evolutionary algorithm. In a later work, Dino andÜçoluk [23] explored the openings contribution to the energy consumption of each 3D building layout.
While Wang et al. [24] implemented an automatic generation of rectangular floor plans by firstly determining the building dual graph and lastly adjusting the final graph by removing isolated rooms and satisfying the user-specified constraints. Duarte [25] used a shape grammar method to generate houses according to a particular architecture style of building layout. Hou and Stouffs [26] developed a shape grammar methodology for generic use, which can be applied to layout planning.
None of these works mentions any concern with the urban context requirements or preferences.
However, Rodrigues et al. [27] carried out some preliminary studies to include them. The authors used a hybrid evolutionary approach to sequentially generate alternative building layouts for each lot in an urban quarter by substituting dummy buildings with fully detailed ones. This approach allowed to take into consideration the impact of the surrounding buildings that were not yet designed. Recently, Rodrigues et al. [28] studied the energy consumption of alternative building geometries with alternative construction systems in an urban quarter in Kuwait. Using an approach that starts at urban scale before reaching the building layout, Hua et al. [29] implemented an integer programming approach to find urban solutions according to several urban criteria; each urban solution has afterwards the building forms filled with space arrangements according to predefined templates. At urban design scale, Nault et al. [30] developed a decision-support tool to compare alternative neighborhood designs according to energy and daylight performance criteria.
Besides being helpful to building design practitioners, performance-based approaches can also be used to study some building performance phenomena, such as the impact of different building indexes [31,32], to compare alternative construction systems [28], and to determine the best thermophysical properties of building elements [33][34][35].
However, the use of simulation in explorative contexts faces some challenges [36], such as the frequent mismatch between the available information and the required simulation model; the development lag between the simulation and the new building technology solutions; simulation outputs can be perceived as non-informative in the decision-making process; and accurate predictions can be time consuming, thus being incompatible with quick feedback. According to Frayssinet et al. [37], simulation of urban context has its own difficulties, such as the estimation of urban buildings' energy demand being more complex at the urban scale due to the amount of information necessary on the built structures; the diversity of occupants' behavior; and the specifications of urban environment, in particular the obstructions from surrounding constructions and surfaces that are specific for each building.
Hence, this paper presents a methodology that integrates some urban design requirements and incorporates other features that are missing or dispersed in the literature, and finally, a case study that exemplifies the potential of such approach. Although the case study consists of evaluating alternative high-rise urban buildings located in São Paulo, Brazil, the methodology is applicable to any building layout design problem and location.

Methodology
The methodology consists of retrieving the urban geometry to specify the site specifications, defining the building design specifications, generating alternative building solutions, and evaluating each solution. The flowchart of the approach is depicted in Fig. 1. Fig. 1. Flowchart of methodology for building generation with performance evaluation. Four main stages are depicted; the outputs of the site and the design specification stages are the inputs for the algorithm, which proceeds to the buildings' generation and performance evaluations stages.

Site and design specifications
In the initial phase, the limits of the urban surroundings are defined and the 3D objects created, represented by their boundary limits (BREP), which consist of connected surfaces that separate the solid from the non-solid parts. In this case study, the urban shape was downloaded from the Open Street Map website [38], but it can be obtained from other websites, existing city models, or manually created. The information is grouped into adjacent buildings, vegetation, and terrain; the terrain is vertically and horizontally split per story level and per grid (with a defined size), when the terrain is not plane. For each of these urban elements or group of elements, three metadata fields are added, which indicate if the building must be included in the generation process, if the building simulation evaluation is to be carried out, and finally, if the building should be visible. The reason for the inclusion of this metadata is, on one hand, to prevent the computation burden when i) calculating the overlapping of objects with the buildings that are not near the construction lot and do not have any impact in the generation process; and ii) determining the shadows overlapping by the polygon clipping algorithm phase of the dynamic simulation. On the other hand, this allows to simulate future scenarios when the adjacent buildings are not yet constructed and the user wishes to avoid openings facing them, or to calculate their influence on the building performance. Additionally, the user can also indicate the building boundary for each story level and the building alignment requirements -the urban geometry, building boundary, and building alignments are imported to the generative design method as a data exchange file. After the urban context specifications, the user defines the design program specification (see section 3), which includes information on the rooms, openings, occupation, artificial lighting, equipment, HVAC systems, and renewable energy systems. Most of this information is stored in a database as default data.

Building generation
With the site and design specifications as input data, the generative design method starts in the third phase. The building designs are created using a newer version of the Evolutionary Program for the Space Allocation Program (EPSAP) algorithm, presented in its earlier version in Refs. [39][40][41], which produces alternative space arrangements according to the user preferences and requirements.
The new and updated floor plan representation scheme (depicted in Fig. 2) includes negative spaces (spaces that are considered voids but have connectivity and dimensional requirements), different roof types (depicted in Fig. 3: flat, gable, butterfly, mono-pitched, and other types), stairs that adjust the landing step dimensions to fit with neighboring spaces, individual story boundaries, and a newer opening frame type for garage spaces. Besides the new representation scheme, the current version uses an enlarged set of 18 penalty functions (described in Appendix A), nine of which are new: the layout alignment function, the layout fill construction and gross areas function, the story gross area function, the circulation space area function, the space fixed-position function, the space relative importance function, the opening accessibility function, the opening dimension function, and the opening fixed position function (the remaining ones are: the layout construction and gross area limits function, the space connectivity/adjacency function, the space overlap function, the space location function, the space dimensions function, the compactness function, the space overflow function, the opening overlap function, and the opening orientation function). The representation scheme includes new elements, such as negative spaces, element alignments, free position of interior openings, different types of openings' frame, and stairs can now have exterior openings. When the user specifies a number of repetitions of a particular story, such story is repeated in the end of the search process, thus reducing the computation time. In the end, dynamic simulation is carried out to perform energy evaluations of the generated solutions [42,43].
The EPSAP algorithm consists of a two-stage approach that has an Evolution Strategy (ES) framework, where the mutation operation is replaced by a Stochastic Hill Climbing (SHC) method.
The ES selection mechanism picks up for the next generation the individuals with a better fitness   object is moved to a new position and if it then overlaps with another object, the latter adapts its shape or will shift to accommodate the new position of the former object.

Performance evaluation
When the generation process concludes, the last phase initiates to evaluate the best generated designs' solutions (the ones with the lowest cost function value formulated in Eq. (A.8) in Appendix A). The energy consumed in each space is aggregated according to the corresponding commercial space or apartment. The buildings' geometry, construction system, HVAC systems, energy production systems, hot water equipment, internal gains, lighting controls, and natural ventilation are parsed as input data to the EnergyPlus software (version 8.9.0), which, together with the weather data, calculates the energy consumption required for air-conditioning in each thermal zone (each space is a thermal zone).
Due to the consideration of urban surroundings, the shadow calculation method selected in EnergyPlus is the 'Average Over Days in Frequency' method, which performs the shadowing calculations (sun position, etc.) over a selected daytime period, in order to speed up the calculations.
The calculations are performed for every 20 days throughout a weather run period; an average solar position is chosen and the solar factors (such as sunlit areas of surfaces) remain the same for that period of days.
Due to the shadowing algorithm, the number of shadows in a figure may grow quite large even with fairly reasonable looking structures; thus, allowing for too few figures in shadow overlaps may not result in accurate calculations [44]. Therefore, a maximum of 10 000 000 figures in shadow overlap calculations is chosen due to the great number of building surfaces in the current work.
Regarding the solar distribution, the option 'Full Exterior with Reflections' is selected in En-ergyPlus. In this case, shadow patterns on exterior surfaces caused by detached shading, wings, overhangs, and exterior surfaces of all zones are computed, as well as shadowing by window and door reveals. The solar radiation beam entering the zone is assumed to fall entirely on the floor, where it is absorbed according to the floor's solar absorptance. Any radiation reflected by the floor is added to the transmitted diffuse radiation, which is assumed to be uniformly distributed on all interior surfaces. The zone heat balance is then applied at each surface and on the zone's air with the absorbed radiation being treated as an influx on the surface [44]. Relative to the simulation time step, the used value is 15 min.

Case study
The generated buildings are located in São Paulo, Brazil (−46.608°latitude and −23. The vertical circulations serve all stories and provide access to the roof.

Design specifications
The design program is for a 15-story building, with the residential stories (L 2···14 ) corresponding to a single floor plan repeated 13 times. The ground floor story (L 1 ) has a 4.00 m height (C lh ) and is dedicated to commerce, while the remaining stories (L 2 to L 15 ) are 3.00 m-high and include two apartments per story (see Table 1  The first-floor level (L 1 ) comprises a shop (S 1 ), two offices (S 2 and S 3 ), and a coffeehouse (S 4 ).
In the residential stories (L 2 to L 14 ), there are one three-bedroom apartment (Apartment T3) Table 2 lists all the specified requirements. For example, the living room (S 8 ) is of function type Living (C sf ), with relative importance of Max (C ri ), repeated on every story from L 2 to L 14 (C sl , C su ), having a minimum side dimension of 3.40 m (C ss ), and side limits of 1.7 for the smaller side (C ssr ), and 2.0 for the larger side (C slr ) of the room.
Each room may have exterior openings (windows or doors). For instance, Office A (S 2 ) has Table 2. Rooms' geometry and topologic specifications.
2.0 C sn -name, C sf -function, C ri -relative importance, C sl and C su -served lower and upper stories, C ss -minimum side, C ssr and C slr -space small side and large side ratios an opening (Oe 2 ) of type Window/Door (C oet ), with 3.0 m width (C oew ), 3.4 m height (C oeh ), elevated 0.0 m from the floor (C oev ), and preferable orientation South or West (C oeo ). Table 3 lists all exterior openings in the design program per room (C os ). Table 3. Geometry specifications of exterior openings. .00 -C os -space, C oet -opening type, C oew -minimum width, C oeh -minimum height, C oev -vertical position, C oeo -orientation Besides exterior openings, the rooms may have adjacency or connectivity requirements; e.g., the Oi 1 is an adjacency between the rooms S 1 and S 5 , while the interior opening Oi 3 is of type Door (C oit ), with 0.9 m width (C oiw ), 2.0 m height (C oih ), and 0.0 m elevation from the floor (C oiv ), which connects room S 5 (C oia ) to room S 6 (C oib ). Table 4 lists all the interior openings in the building.
The compactness of the building is controlled using clusters (see Table 5). The cluster G c 3 defines the rooms in Apartment T3 and cluster G c 4 in Apartment T2.
The building alignments are defined by two linear requirements for the cluster G a 1 , which is defined by the residential and vertical circulation spaces. The first requirement sets a horizontal Table 4. Interior openings geometry and topologic specifications.  Table 5. Rooms' compactness specifications.  Table 6 summarizes the alignment specifications.

Construction system
The building has strong inertia with current material properties. Its construction elements and respective properties are presented in Table 7. These meet the Brazilian legal limits for the thermal transmittance, and follow the specifications of exterior opaque elements presented in [45] (electric energy consumption optimization study regarding the performance of solar protection systems in a passive building in Brazil).   Each apartment is considered as a single-family dwelling occupied by three (T2) or four (T3) people. The commercial spaces -coffeehouse, shop, office 1 and office 2 -are occupied by a maximum of thirty, four, three and five people at the peak of occupancy, respectively. The shop space represents a small family shop (e.g., cellphone store, insurance store, barber shop), usually found in these residential neighborhoods, which are characterized by local commerce in the ground floor and by a small number of clients and employees -thus the small occupancy level considered.
The occupancy patterns in the different spaces throughout the day (for workdays and weekends) are depicted in Fig. 5. The internal heat gains due to occupancy are also related to the maximum number of people per zone and the respective activity level, which are presented in Table 8.  The internal gains due to electric lights are defined by the maximum design lighting level for each zone type, as presented in Table 9, and the corresponding usage schedules, depicted in Fig. 6, according to RTQ-R [46] and RTQ-C [47].
The methodology proposed in Annex I of RTQ-R is used as a basis for the shading profile of the window openings. The method aims to assist the sizing of solar protection devices, regarding the shading variable of the envelope performance equation for spaces that do not present shutters,  and that are shaded by overhanging, balcony or horizontal brise soleil elements [46]. Hence, it should be made clear that the referred methodology does not present a window shading standard, serving here only to assist in defining the shading profile of the window openings. Thus, according to the RTQ-R, small window openings (less than 25 % of the floor area) should be shaded when the outdoor temperature exceeds T n 1 +3 • C and the incident solar radiation surpasses 600 W · m −2 ; and large window openings (above 25 % of the floor area) should be shaded when the outdoor temperature exceeds T n + 3 • C or the incident solar radiation surpasses 600 W · m −2 . However, when selecting two shading setpoint types (temperature and radiation) in EnergyPlus, it allows to consider them only paired, not independent from each other. Therefore, and since geometry restrictions prevent glazing areas above 25 % of the floor area, only the first operation mode is considered: shading whenever T n + 3 • C and 600 W · m −2 . Moreover, EnergyPlus also does not allow for variable setpoints. For that reason, an annual average T n value of 27.24 • C was computed, which is considered in this work. Hence, whenever the outdoor temperature exceeds 30.24 • C and the incident solar radiation exceeds 600 W · m −2 , the window shadings are activated. The shadings are assumed to be PVC roller shutters in all the apartments' windows and internal cloth shades are considered in the offices' windows. The coffeehouse and the shop are assumed not having window shadings.
The internal heat gains due to electric equipment depend on the maximum design wattage levels of the equipment in each zone. A constant value of 1.5 W · m −2 is considered in the apartments' living rooms, as pointed in RTQ-R [46]. For kitchen zones, a constant value of 2.0 W · m −2 is assumed [45]. Regarding the commercial spaces, the electric equipment heat gains depend on the maximum design wattage levels (Table 10), which are based on the building zone typology and the appliances typically found in each space [48], and the respective usage schedules (Fig. 7). These schedules are based on the zones' typology and occupancy.  Concerning natural ventilation, RTQ-R [46] refers that all dwelling spaces with ventilation openings must be modeled in the natural ventilation mode, operating with the following specifications: from 9:00 to 21:00, and only when the temperature inside is above 20.0 • C and above the outdoor temperature. Accordingly, a natural ventilation (wind and stack) object is applied to all apartments' windows, with the above defined specifications. The minimum opening level of each window depends on its own type. According to the Annex II of RTQ-R [46], a 45 % opening is considered for sliding windows (circulation areas, living rooms, bedrooms and kitchens) and 90 % for pivoted windows (bathrooms). Regarding the commercial spaces, a nominal 1.5 ACH was considered in the coffeehouse and in the shop, due to the constant entering and exiting of clients, with ventilation profiles equivalent to the light schedules defined for these spaces (Fig. 6), as lighting is considered to be on during all working hours.
Regarding climatization specifications, the living room and the bedrooms are the only apartments' spaces where heating and cooling must be considered (unitary split system air-conditioning) [46]. and each equipment has a 70 % fan efficiency, 90 % motor efficiency and runs continuously between 21:00 and 9:00. Accordingly, the corresponding heating/cooling availability schedules for each zone are equivalent to the occupation schedules defined for the respective zones (Fig. 5), within the referred time boundaries.
Regarding the commercial spaces, and following RTQ-C [47], they are heated/cooled with equipment (unitary split system air-conditioning) that supply hot/cold air with an 11.0 • C difference in relation to the zone's setpoint (24.0 • C for cooling and 22.0 • C for heating); the cooling and heating equipment's coefficient of performance are 3.28 and 3, respectively; and each equipment's fan has a 65 % efficiency, 250 Pa static pressure and runs continuously. The corresponding heating/cooling availability schedules for each zone are equivalent to the lighting schedules defined for the respective zones (Fig. 6), as lighting is considered to be turned on during all working hours in these spaces.

Climate location
The weather data used in the dynamic simulation is the one available in the EnergyPlus weather data webpage for São Paulo, Brazil, and it is classified as a mild humid subtropical climate with hot summer and no dry season (Cfa type according to the Köppen classification [49]).

Results and discussion
The EPSAP algorithm ran a single time, which took a runtime of 1 h and 32 min, using 20 threads in parallel computing in a ten-core 3.   When stores are located directly under the apartments, such as, for example, Apartments T3 on story L 2 in buildings FPD 324, FPD 73, and FPD 319, the energy consumption for heating is similar to the ones placed on the upper stories (see Fig. 11). It is also possible to observe in Fig. 11 that all apartments in FPD 163 and FPD 73, and Apartment T3 in FPD 58, FPD 149, and FPD 319 have high heating needs and low cooling needs, while Apartment T3 in FPD 324 has high cooling needs. When comparing all buildings, the best story performance presents a difference of 11.7 % between buildings (L 7 in FPD 163 and L 10 in FPD 149).
In Fig. 10, the energy consumption for air-conditioning in each apartment is labeled with a colored gradient from green (4 kW · h · m −2 ) to red (14 kW · h · m −2 ). The building designs that have Apartment T2 facing West are the ones with the highest difference between stories (FPD 58, FPD 149, and FPD 319) and with Apartments T2 having the lowest heating needs.
FPD 324 is the one that has the lowest heating demand and the highest cooling demand for Apartment T3, independently of the story level. As West-oriented facades are the least shadowed by the surroundings, the apartments facing that orientation are the ones with lowest heating energy consumption. The same is valid for Apartment T3 in FPD 73, albeit to a lesser extent, as it is placed Southwest, thus being partly self-shadowed during most of the day.
The smaller Apartments T2 have the lowest energy demand in all design solutions, regardless of the story on which they are located (see Tables 11 and 12). However, and due to the amount of energy consumed, it is Apartment T3 that defines the overall performance ranking of the building, except for FPD 324 and FPD 73.  Fig. 10. The six generated buildings with the 3D simulation model perspective view on the left and two apartments' stories on the right. The top floor plan indicates the story with the worst energy performance, while the bottom one shows the story with the lowest energy consumption; the story number is marked over the stair space; the annual energy demand per floor area for air-conditioning is depicted by the colored gradient from green (4 kW · h · m −2 ) to red (14 kW · h · m −2 ). Floor plans are not in scale. Fig. 11. Apartments' heating and cooling energy consumption per residential story (L2 to LL14). The cooling energy is marked as black bar while the heating energy is marked as white bar. Table 12. Annual total energy, cooling energy, and heating energy consumption for air-conditioning per floor area. Story and apartment lowest energy consumption are marked in bold font for each building. 8.734 kW · h · m −2 , FPD 319 9.140 kW · h · m −2 , FPD 324 9.257 kW · h · m −2 , FPD 73 9.451 kW · h · m −2 , and FPD 163 9.481 kW · h · m −2 (see Table 12).

Conclusion
Automated floor plan design generation methods usually ignore the urban context in their generation and performance evaluation. The approach presented in this paper includes the urban surroundings to limit the building lot, to define the building boundary, to prevent inadequate openings orientation, and to define building alignments with adjacent buildings. Besides including urban elements (adjacent buildings, vegetation, and terrain) during the buildings' generation phase, these are also considered during the performance assessment using dynamic simulation.
The The results demonstrated that the EPSAP algorithm was able to produce distinctive and alternative solutions for this high-rise urban building scenario. The results also showed that either the buildings' performance or the story and apartment performance (despite stories having equal geometry and construction) vary significantly, with intermediate stories having the best energy performance. Also, the size and position of each apartment (and also the window orientation) have a strong impact in the building performance. The best story performance difference between buildings can reach 11.7 % of energy consumption per floor area. The results also illustrated that the energy consumption for air-conditioning between stories in the same building ranges from 9 % to 12 % relatively to the best performing one (ignoring the first story, which is an outlier; otherwise the difference would range from 22 % to 44 %). Therefore, this approach can be a useful tool for building designers to explore and test solutions in fast prototyping tasks of early design stages.
This new approach is originally intended to have a set of urban and building requirements and preferences in a single automated tool. Its uniqueness is also a contribution to fast prototyping of buildings, not only in building design scenarios, but also as a research tool to produce datasets of building designs for the statistical analysis of specific building phenomena. Despite the demonstrated capabilities of the presented approach, future developments and tests are required. Some of the developments are the inclusion of alternative requirements (e.g., alignment with building A or alignment with building B), occupation of underground levels, definition of walkthroughs in the urban lot, etc., and testing in denser urban environments and in renovation scenarios. Additionally, a methodology to consider the interactions of the generated building with the micro urban climate to improve the indoor thermal performance of the building will also be implemented.

Data availability
The dataset related to the six IDF files of 15-story buildings in São Paulo can be found at     Each floor (F , Eq. (A.5)) is a rectangle defined by its top-left corner x and y coordinates (f x and f y, respectively), width (f w), and depth (f d).
The exterior opening (Oe, Eq. (A.6)) is a vector of values where oet defines the opening type (oet = 0 is a void, oet = 1 a door, oet = 2 a window, and oet = 3 a gate), oeo is the opening orientation (oeo = 0 is oriented to North, oeo = 1 to East, oeo = 2 to South, and oeo = 3 to West), oea is the space floor rectangle number that the opening belongs to (N f oea is the total number of floor objects in space S oea ), oew is the opening width, oeh is the opening height, oev is the opening vertical position, and oep is the relative position in the exterior wall of the oea floor rectangle side (eoo).
When one of the spaces is a negative space, the interior opening is treated as an exterior opening (Oe). Similarly, this object is a vector where oit defines the opening type (oit = 0 is a void, oit = 1 a door, and oit = 2 a window), oia is the space floor rectangle number that the opening connects from (N f oia is the total number of floor objects in space S oia ), oib is the space floor rectangle number that the opening connects to (N f oib is the total number of floor objects in space S oib ), oiw is the opening width, oih is the opening height, oiv is the opening vertical position, and oip is the relative position in the interior wall that is common to both spaces (S oia and S oib ). If the opening width (oiw) is set to zero, this object is treated as a simple adjacency between those two spaces.
Oi n oit, oia, oib, oiw, oih, oiv, oip ; 1 n N io , The individuals' fitness is assessed according to several groups of objectives. Each group has its penalty function (f c ), which is multiplied by its corresponding weight (w c ). In total, there are 18 functions summed in a weighted cost function to be minimized (see Eq. (A.8)). Each function is triggered when its weight is different from 0 and the user constrains, necessary for that function to work, are defined. Eq. (A.9) defines this penalty function. The third penalty function (Eq. (A.14), f 3 , weight w 3 ) determines the individual gross and construction areas of the floor plan (f ga and f ca , respectively), and if these are below the maximum limit of gross and construction areas (C ga and C ca , respectively), the square root of the difference of each area is summed. This function is only activated when the user specifies at least one of these areas' limits. The goal is to maximize the building construction areas.  The ninth penalty function (Eq. (A.22), f 9 , weight w 9 ) aims to reduce the size of circulation spaces (C sf k ). When those spaces are corridors or halls, the minimum admissible space size (C ss k ) is subtracted with the space's side dimensions and the differences are summed up. In the case of being stairs or elevators, the dimensions of the space are only summed.
, if S k vertical circulation the window-to-floor ratio of the space (C os l ) is used to determine the minimum window width necessary. The difference is then calculated if the window width is smaller than this value. f 18 (Oi n ) = f ioc (Oi n , C oia n , C oib n , C oiw n ) (A.37)