Multi-zone optimisation of high-rise buildings using artificial intelligence for sustainable metropolises. Part 2 Optimisation problems, algorithms, results, and method validation

High-rise building optimisation is becoming increasingly relevant owing to global population growth and ur-banisation trends. Previous studies have demonstrated the potential of high-rise optimisation but have been focused on the use of the parameters of single floors for the entire design; thus, the differences related to the impact of the dense surroundings are not taken into consideration. Part 1 of this study presents a multi-zone optimisation (MUZO) methodology and surrogate models (SMs), which provide a swift and accurate prediction for the entire building design; hence, the SMs can be used for optimisation processes. Owing to the high number of parameters involved in the design process, the optimisation task remains challenging. This paper presents how MUZO can cope with an enormous number of parameters to optimise the entire design of high-rise buildings using three algorithms with an adaptive penalty function. Two design scenarios are considered for quad-grid and diagrid shading devices, glazing type, and building-shape parameters using the setup, and the SMs developed in part 1. The optimisation part of the MUZO methodology reported satisfactory results for spatial daylight autonomy and annual sunlight exposure by meeting the Leadership in Energy and Environmental Design standards in 19 of 20 optimisation problems. To validate the impact of the methodology, optimised designs were compared with 8748 and 5832 typical quad-grid and diagrid scenarios, respectively, using the same design parameters for all floor levels. The findings indicate that the MUZO methodology provides significant improvements in the optimisation of high-rise buildings in dense urban areas.


Introduction
The demand for high-rise buildings is increasing in metropolises owing to population growth and urbanisation trends (Ali and Al-Kodmany, 2012). For realising sustainable urban areas, sustainable high-rise buildings should be one of the topics under investigation because they consume a significant amount of energy owing to their excessively large size (Ali and Armstrong, 2008). Designing a sustainable high-rise building is a complex task because the process involves various types of design parameters that affect multiple performance aspects. Rafiei and Adeli (2016) presented robust optimisation algorithms and neural dynamic models for investigating sustainable highrise alternatives to cope with this complexity. The previous works mentioned in part 1 showed that optimisation algorithms and machine learning techniques have been widely used for designing sustainable high-rise buildings over the last two decades. However, in none of these studies, were the various floor levels considered as separate design problems, which is crucial for improving the overall performance of high-rise buildings (Wood, 2007). Using the same design parameters for the entire high-rise design is a limited approach because the performance of the building varies between the ground and sky floor levels in dense urban areas. Optimising the design of an entire high-rise building is challenging as the simulations require expensive computational time, and the optimisation process needs to cope with an enormous number of design parameters. The use of multi-zone optimisation (MUZO) methodology is proposed to divide high-rise buildings into subdivisions (zones) to be considered as separate problems using artificial intelligence methods to address both aspects. Part 1 of the study is focused on solving computationally expensive simulations of each zone using surrogate models (SMs). Part 2 deals with the optimisation challenge, wherein each zone is considered as a design problem using algorithms belonging to different optimisation domains. In parts 1 and 2 of the MUZO study, quad-grid and diagrid scenarios with the shading device, glazing type, and building-shape parameters were used to demonstrate the proposed methodology.
This study is focused on optimising the entire design of high-rise buildings for quad-grid and diagrid scenarios using the 40 SMs developed in part 1. The performance aspects of the study take into consideration the two daylight metrics of Leadership in Energy and Environmental Design (LEED) v4.1., namely, the spatial daylight autonomy (sDA) and annual sunlight exposure (ASE). The optimisation process uses phase 3 of the MUZO methodology for single-objective constrained formulation with three algorithms: self-adaptive differential evolution with an ensemble of mutation strategies (jEDE) in the Optimus plug-in (Cubukcuoglu et al., 2019), radial basis function optimisation (RbfOpt), and covariance matrix adaptation with evolution strategy (CMA-ES) in the Opossum plug-in (Wortmann, 2017b). In addition, an adaptive penalty function, called the near-feasibility threshold (NFT) (Coit and Smith, 1996;Smith and Coit, 1997), is used for each optimisation algorithm in the Grasshopper 3D algorithmic modelling environment (GH) (Rutten, 2015). The paper reports the optimisation results of 20 problems for two scenarios, which comprise 260 and 220 design parameters, respectively, with the aforementioned algorithms for five replications. Part 2 of the study also validates the significance of the proposed methodology by presenting a comparison of the performances of the optimised high-rise designs and typical high-rise scenarios generated by the same design parameters for all the floor levels. The optimisation results and validation of the method show that the MUZO methodology can play a significant role in investigating sustainable high-rise alternatives in metropolises. The rest of this paper is structured as follows: Section 2 presents the state of the art for sDA and ASE optimisation, Section 3 introduces the optimisation problems and algorithms of this paper, Section 4 reports the optimisation results, Section 5 presents the validation of the MUZO methodology, Section 6 discusses the importance and potential of MUZO with surrogate-based design optimisation, and Section 7 presents the conclusions of this paper.

State of the art for sDA and ASE optimisation
This section presents the previous optimisation studies for the sDA and ASE daylight metrics of LEED within the performative computational architecture (PCA) framework in two subsections: one presenting conventional optimisation and the other computational optimisation. Conventional methods comprise an analysis of the predefined design parameters, whereas computational methods involve the use of optimisation algorithms while automating the PCA framework to investigate the best design performance. Subsequently, the novelty of this study is summarised.

Conventional optimisation
Over the last decade, sDA and ASE metrics have been used to investigate daylight performance and visual comfort for various building functions. An early study was focused on a classroom case with the use of three optimisation approaches while using the optical properties and size of a south-facing window (Kazanasmaz et al., 2016). Owing to the classroom requirements, the authors maximised sDA 500/50% to evaluate an illuminance level of 500 lx with respect to ASE 1000,250h . In the case of a hospital-patient room, in two studies, the window blinds were optimised by shaping the slats and the configuration of external sun-breakers on south-oriented windows to maximise sDA 300/50% subject to ASE 1000,250h (Sherif et al., 2016;Wagdy et al., 2017). In the case of office spaces, in three studies, sDA 300/50% was maximised subject to an ASE 1000,250h less than 10% as a preferable result, and between 10% and 20% as an acceptable limit for various design parameters, i.e., solar screens, 3D tessellation, fixed/dynamic shading devices, and surface reflectance (Fathy et al., 2017;Giostra et al., 2019;Palarino and Piderit, 2020). The general approach of these studies was to maximise sDA 300/ 50% , with the exception of one study, owing to the educational requirements (Kazanasmaz et al., 2016). The ASE 1000,250h was generally considered as less than 10% as a comfort limit, while two studies considered the results of less than 20% as acceptable solutions (Giostra et al., 2019;Palarino and Piderit, 2020). In addition, in the aforementioned studies, a limited number of design alternatives that might be related to conventional optimisation techniques were examined. Consequently, none of these studies were focused on optimising the daylight performance for the design of entire buildings, such as high-rise buildings.

Computational optimisation
Optimisation algorithms have been widely used to cope with the complexity of the design problem while investigating desirable sDA and ASE results for various building functions. An early example was focused on an office space to maximise sDA 300/50% subject to an ASE 1000,250h of less than 10% using a genetic algorithm (GA) in the Galapagos plug-in of GH while considering a single-objective formulation for kaleidocycle typology (Wagdy et al., 2015). In addition to daylight, Vera et al. (2017) addressed single objective constrained optimisation to minimise the total energy usage subject to an sDA 300/50% greater than 50% and ASE 2000,400h less than 20% for exterior fenestration systems of office spaces using particle swarm optimisation with the Hooke-Jeeves algorithm in GenOpt. Another example of combining performance aspects into one fitness function was examined by Yi et al. (2018) to maximise sDA 300/50% and minimise ASE 1000,250h and daylight glare probability (DGP) for auxetic structures with advanced daylight control systems in an office space using a GA in the Galapagos. As an alternative to singleobjective constrained formulation, Tabadkani et al. (2018) and Mangkuto et al. (2018) maximised |sDA − ASE| subject to an sDA 300/50% greater than 50% and 75%, and ASE 1000,250h less than 10% for sunresponsive skin and light shelf design in office and hospital spaces using a GA in the Galapagos and Octopus plug-ins. In the case of multiobjective optimisation, Yi (2019) maximised the sDA 300/50% and minimised ASE 1000,250h with an aesthetic perception objective function using non-dominated sorting genetic algorithm II for a hotel building. Pilechiha et al. (2020) also considered the quality of the view from office windows in the optimisation of the sDA 300/50% , ASE 1000,250h , and energy usage intensity while considering weighted summation and the HypE algorithm in the Octopus plug-in. As an alternative to multi-objective optimisation, Mangkuto et al. (2019) identified the simulation results for an office space with full factorial analysis of the internal shading devices to explore the non-dominated solutions while maximising sDA 300/50% and minimising ASE 1000,250h and DGP >0.21 subject to an sDA greater than 55%, ASE less than 10%, and DGP less than 50%. Five of these studies comprised the consideration of a single objective, whereas others used multi-objective and weighted summation formulations. Three studies utilised static penalty functions that might limit the search ability during the optimisation process. Finally, none of the reviewed studies consisted of a comparison of the results of different optimisation algorithms using various initial populations (replications) for the entire design of the building.

Novelty of this paper
This study is focused on the optimisation of an entire high-rise building for the quad-grid and diagrid scenarios through phase 3 of the MUZO methodology, which is based on the use of multiple algorithms with replications for each optimisation task owing to the no free lunch (NFL) theorem (Wolpert and Macready, 1997). Because of the computational burden of optimising the entire design, the high-rise building is divided into 10 subdivisions (zones), which correspond to 10 design problems starting from the first zone (Z1) at the ground level until the tenth zone (Z10) at the sky level ( Fig. 1). Forty SMs, and the high-rise setup, which were developed in part 1 of this study, were used to optimise the sDA and ASE metrics based on the simulation results obtained for the second and fifth floors in each zone. The quad-grid scenario comprises 2.893399115e+28 design alternatives with 26 parameters, whereas this number is 3.054543465e+23 for the diagrid scenario with 22 parameters in one zone. For each optimisation task, phase 3 of the MUZO methodology is considered by employing the jEDE, RbfOpt, CMA-ES algorithms, and NFT adaptive penalty function with five replications, which suggests a decision-making process using 15 optimisation results. Consequently, this paper reports on the optimised high-rise buildings after a total of 300 optimisation runs is complete, using 260 parameters for the quad-grid, and 220 parameters for diagrid, and it validates the impact of the proposed methodology by comparing the optimised scenarios with the typical high-rise scenarios. Thus, part 2 of the study not only deals with the optimisation of the entire design of high-rise buildings for the performance metrics under study, but also addresses 20 complex design problems, each having an enormous number of design alternatives in the optimisation search space, owing to the involvement of multiple design parameters.

Optimisation problems and algorithms
This section explains the problem formulation and algorithms used in each optimisation process. The first subsection explains the singleobjective constrained formulation, whereas the subsequent subsections present the RbfOpt, CMA-ES, and jEDE algorithms with applications in the architecture domain. Finally, the NFT describes the adaptive penalty function for constraint handling.

Problem formulation
The Illuminating Engineering Society (IES) recommends a minimum sDA 300/50% of 55% with a maximum ASE 1000,250h of 10% as desirable daylight with acceptable comfort (IES, 2013). However, the LEED standards acknowledge design proposals with two points, i.e., while the sDA 300/50% is greater than 55% and ASE 1000,250h is less than 10% for regularly occupied floor areas. When reaching a minimum of 75% of sDA 300/50% with 10% of ASE 1000,250h , the design is acknowledged with three points. Considering the formulations of previous studies and the recommendation of the IES and the LEED standards, in this study, a single-objective constrained optimisation is considered for each design problem as max : sDA 300/50% X = (x 1 , x 2 , ..., x n ) and X ∈ S subject to : ASE 1000/250h ⩽ASE bound (1) where n is the number of design parameters in each zone for both quadgrid and diagrid scenarios, S is the entire search space of one zone, and ASE bound is the maximum limit for direct sunlight. The state of the art shows that the ASE results can be related by more than 10% to the design of the shading devices. Because the sufficiency of shading devices is unexplored at the beginning of the optimisation processes, an adaptive ASE boundary is considered in each zone as 10% if sDA 300/50% ⩾55% 20% if sDA 300/50% ⩽55% and ASE 1000/250h > 10% 30% if sDA 300/50% ⩽55% and ASE 1000/250h > 20% (2) where ASE bound increases by 10% when the sDA result is less than 55%. This approach is considered in both quad-grid and diagrid scenarios to optimise the sDA and ASE metrics using the SMs. The optimisation task starts from Z1 and ends at Z10. After the best parameter set is determined in one zone for each algorithm, the optimisation process of the next zone is started. The parameters presented in part 1 of the MUZO study are also used herein (Appendix A). The supplementary material presents the predictive models with learning scores of 40 SMs that were used during the optimisation process.

Radial basis function optimisation
RbfOpt is a model-based algorithm used for solving computationally expensive problems and was recently presented by Costa and Nannicini (2018). For the unknown cost function, the algorithm constructs and iteratively refines an approximation model with sampled points. Compared to the existing open-source model-based algorithms available, RbfOpt provides two main contributions: an efficient method for automatic model selection using a cross-validation scheme, and an approach to exploit noisy but faster function evaluations. Opossum provides the RbfOpt algorithm to be used in architectural design optimisation as an open-source plug-in developed for GH (Wortmann, 2017b). RbfOpt in Opossum has been widely used for various design problems, i.e., daylight and glare problems (Wortmann, 2017a), optimal viewing angle in stadium design (Zargar and Alaghmandan, 2019), structural optimisation (Ilunga and Leitão, 2018), urban design (Wortmann and Natanian, 2020), and optimisation problems focused on building energy (Waibel et al., 2019). In this study, the optimisation process uses the default RbfOpt parameters while running the algorithm through Opossum v2.0.0.

Covariance matrix adaptation with evolution strategy
CMA-ES is a well-known optimisation algorithm in the evolutionary computation (EC) domain proposed by (Hansen, 2006;Hansen et al., 2003;Hansen and Ostermeier, 2001). One of its most powerful features is that the search space can be increased or decreased in the next iteration based on the results of every solution. The algorithm uses this procedure for the multivariate normal distribution parameters (mean and sigma) and for the entire covariance matrix that belongs to the decision variable space. Opossum v1.7.0 provides a CMA-ES algorithm for design optimisation in the architecture domain as an open-source plug-in for GH. Recently, this algorithm has been used for various design problems, e.g., Waibel et al. (2019) optimised building energy problems while reporting promising results with a large evaluation budget, Zhang et al. (2020) focused on aerodynamic shape optimisation problems, and Fortich Mora (2020) used CMA-ES for the design problem of sustainable high-rise buildings. The optimisation process in this study comprises the use of Opossum v2.0.0, while considering the default features of the CMA-ES algorithm.
3.4. Self-adaptive differential evolution with ensemble of mutation strategies jEDE is a hybrid algorithm that belongs to the EC domain using differential evolution (Storn and Price, 1997), self-adaptive strategy (Brest et al., 2006), and an ensemble of mutation strategies (Mallipeddi et al., 2011). The purpose of the algorithm is to cope with highdimensional problems in the domain of architectural design optimisation. The algorithm comprises a self-adaptive approach that converges to different directions with various rates of mutation and crossover operators. Moreover, with the ensemble idea, jEDE also selects the best mutation strategy for every dimension among predefined operators during the optimisation process. Therefore, the algorithm can adapt its search behaviour to different problems. The first application of jEDE, which is provided by Optimus v1.0.0 as an open-source plug-in for GH, was used for 30D CEC 2005 benchmark problems and a 70D structural design problem (Cubukcuoglu et al., 2019). The algorithm presented promising results as compared with particle swarm optimisation, genetic algorithm, and RbfOpt. In addition, recent publications have demonstrated the potential of jEDE in solving a 20D problem of daylight (Ekici et al., 2019b) and the optimisation of sustainable high-rise building design focused on daylight, comfort, and energy use intensity aspects with SMs (Fortich Mora, 2020). The optimisation process in this study comprises the use of the default parameters of Optimus v1.0.2 for the jEDE algorithm.

Near feasibility threshold constraint handling
In previous studies mentioned in Section 2, single-objective constrained optimisation is considered as a problem formulation for the ASE and sDA metrics according to the LEED and IES standards. The general approach of these studies was to consider the ASE as a constant penalty function to be embedded in the sDA fitness function. In this method, the result of the fitness function (sDA) is multiplied with a constant value if the solution of the constraint function (ASE) is in the infeasible region. Previous studies have also discussed that the ASE results could be related to the sufficiency of the shading devices by more than 10%. Another reason for this outcome may be related to the limited search ability of the constant penalty functions. In the case of challenging constraint problems, Mallipeddi and Suganthan (2010) emphasised the importance of using advanced constraint-handling approaches. Therefore, in this study, the NFT adaptive penalty function is taken into consideration (Coit and Smith, 1996), which is an advanced version of the constant penalty function. The approach of the NFT is to define a threshold distance from a feasible region and to encourage the search within this region and the NFT neighbourhood while discouraging the search beyond that threshold. Eqs. (3) and (4) explain the penalised fitness function f p (x) using the NFT as where f(x) is the fitness function; v(x) is the violation; α and λ are userdefined positive parameters taken as 2 and 0.04, respectively, NFT 0 is the upper bound of the NFT taken as 0.1; and g is the generation or iteration number. The optimisation process of RbfOpt, CMA-ES, and jEDE takes into consideration the NFT approach to obtain a reasonable comparison between algorithms for each problem. The Optimus plug-in v1.0.2 provides an open-source NFT module that can work with other optimisation plug-ins in GH.

Results
The optimisation results were obtained using a computer with an Intel Xeon E5-1620 v3 core processor at 3.50 GHz, 16-GB DDR3 memory, and a 512-GB solid-state drive (Fig. 2). As the termination criterion, 10,000 was considered as the maximum number of function evaluations (FES). In the implementation of CMA-ES and RbfOpt, non-populated approaches were considered in the Opossum plug-in. Therefore, 10,000 was set as the maximum FES for CMA-ES and RbfOpt, while 40 population sizes and 250 generations were considered for the population-based jEDE algorithm. During the optimisation process, Opossum automatically stopped the iteration if there was no alteration in the fitness function. Therefore, the computation times of all the algorithms were also recorded. To evaluate the optimisation performance    of RbfOpt, CMA-ES, and jEDE, the following five criteria were considered: maxf(x), and stdf(x), respectively, are the maximum, and standard deviation of the function x for five replications; CPU is the average time in seconds to complete one replication; FES is the total number of function evaluations, and FES/CPU is the number of completed function evaluations in 1 s (Fig. 3). The convergence graphs for the best results among the five replications of each algorithm are presented in Figs. 4 and 5. In addition, Appendix B presents the convergence graphs of all the replications.
In the quad-grid results, jEDE outperformed the other algorithms in six zones, whereas jEDE and CMA-ES yielded the same results in three zones, and CMA-ES outperformed jEDE in one zone. In the case of the LEED scores, jEDE and CMA-ES reached three points in nine zones, while both algorithms reached two points only in Z10. In contrast, RbfOpt reported three points for Z1, two points for Z3, Z4, Z5, and Z7, and sDA results less than 55% in other zones. Hence, the jEDE and CMA-ES could cope with the quad-grid scenario and provided satisfactory results for LEED standards, while the RbfOpt could not achieve the same result owing to the insufficient sDA levels reported for Z2, Z6, Z8, Z9, and Z10. In the diagrid results, the constraint of ASE bound ⩽10 resulted in undesirable sDA solutions in Z10 for all the algorithms. Thus, the boundary was increased by 10% to consider the new constraint function as ASE bound ⩽20. As a result, the jEDE outperformed the other algorithms in seven zones. In two zones, jEDE and CMA-ES yielded the same results, whereas only in one zone, the CMA-ES outperformed the jEDE. In the case of the LEED scores, the jEDE and CMA-ES presented three points in six zones and two points in three zones, whereas the RbfOpt found three points in three zones, two points in three zones, and insufficient results in four zones. Therefore, the jEDE and CMA-ES could cope with the diagrid scenario, while providing satisfactory results for the LEED standards in nine zones and acceptable results (ASE bound ⩽20) in Z10, while the RbfOpt could not present a desirable performance for the entire building owing to the insufficient sDA results reported for Z7, Z8, Z9, and Z10. With respect to the computation time, the RbfOpt and CMA-ES were automatically terminated at a smaller FES than the jEDE. Based on the CPU results, the CMA-ES converged faster than the other algorithms in Z1 of the quad-grid, and Z2 and Z5 of the diagrid scenarios. In all the other problems, the jEDE converged faster than the CMA-ES and RbfOpt with less deviation in computation time despite the higher FES. In contrast, the FES/CPU results suggested that the jEDE could evaluate a single function much faster than the other algorithms.
In the optimised solutions, the results showed that the sDA values diversified in all zones for the both scenarios. For instance, optimised solutions of the lower zones presented a high percentage of sDA because the dense areas in the built environment significantly blocked direct sunlight. Thus, the daylight was controlled using shading devices and considering high-transmittance glazing materials between Z1 and Z3. In the middle zones, it was observed that the sDA values started to vary between Z4 and Z7 owing to the different shading densities and glazing types used. In the higher zones (Z8-Z10), the sDA results were lower than those in the other zones because direct sunlight met with the corresponding floors from all directions (north, south, east, and west). Therefore, either dense use of shading devices or low-transmittance glazing materials were selected, especially in the south and east orientations, to cope with this challenge. In addition, it was observed that a significant building twist would be desirable in the zones between Z8 to Z10 to decrease the impact of direct sunlight as compared with the other zones. The described design differences in the various zones were based on several reasons. Firstly, the density of the surroundings caused various design challenges, i.e., high building density at the ground levels and low density at the sky levels. Therefore, the optimisation algorithms found different design parameters owing to the different surrounding conditions. Secondly, higher zones were dependent on the lower zones because of the rotation and floor-to-floor height parameters. The optimised parameters in the lower zones could negatively affect the higher zones. Nevertheless, desirable solutions were obtained from the results reported after the MUZO optimisation process because the independent rotation and floor-to-floor height parameters could control the performance of each zone.
With a focus on the overall building performance based on the average results of all the zones, Table 1 presents the sDA results for the entire high-rise building. The overall results of the algorithms demonstrated that the jEDE and CMA-ES found a higher sDA in the quad-grid than in the diagrid. However, the RbfOpt presented a superior sDA performance in the diagrid scenario. Consequently, the jEDE presented the best sDA performance, while the CMA-ES presented the second-best performance, and the RbfOpt presented the third best design options. Moreover, based on the results in Figs. 2-5, we can also conclude that the quad-grid shading devices provided better daylight performance within acceptable comfort conditions as compared with the diagrid devices. Fig. 6 presents the best parameters reported after the optimisation process for both scenarios, whereas Figs. 7 and 8 illustrate these parameters in the form of high-rise buildings. The supplementary material presents the results of the optimised building designs.

Validation of the method
The design of high-rise buildings has changed owing to technological improvements, design concerns with environmental impacts, and regulation changes over time (Oldfield et al., 2009). Few buildings appear to be examples of such design concerns in the 21st century, as they comprise various building shapes and façade configurations and a combination of transparent and opaque surfaces. However, the design of high-rises using various design parameters could provide solutions for realising better building performance in dense urban districts, as discussed in this paper. This section presents the potential performance improvement that can be realised in sustainable high-rise buildings in metropolises by comparing the optimised scenarios obtained using the MUZO methodology with typical high-rise scenarios. In the majority of the existing high-rise buildings, the same parameter values are applied to the entire high-rise design (e.g., singular floor-to-floor height, same façade configuration, and a single glazing type). For profound comparisons, various combinations of parameters are defined to develop typical scenarios using the same parameters in the optimisation process. In total, 8748 typical quad-grid and 5832 typical diagrid scenarios were generated using the values listed in Table 2, and Figs. 9 and 10 illustrate several examples of these scenarios. The performance of each typical scenario was calculated for every zone using the same SMs in a short time. The average performance results obtained for all the zones were considered to evaluate the overall building performance for the typical scenarios. In the case of the optimised scenarios, the jEDE results were used for comparison, as they were the best proposed design solutions. Figs. 11 and 12 present comparisons of the quad-grid and diagrid scenarios, respectively. As a result, the MUZO designs exhibited the best performances with an ASE of 9.8% and sDA of 88.7% in the quad-grid scenario and an ASE of 10.5% and sDA of 82.0% in the diagrid scenario. As mentioned in the results section, owing to the insufficient shading performance of diagrid Z10, ASE bound ⩽20 was considered, which resulted in a slightly higher ASE performance than 10%. Ultimately, the overall performances of the typical high-rise scenarios could not provide satisfactory LEED scores, which demonstrates the importance of using the MUZO methodology in dense urban districts.

Discussion
This section presents the discussion based on the optimisation results and the validation of the method explained in the previous sections. Firstly, two discussion topics are addressed: the importance of the MUZO methodology for metropolises, and its potential. Secondly, the ongoing Table 2 Parameter values used for generating typical scenarios.  (1) The importance of the MUZO methodology for future metropolises: The results obtained in this study indicated that the MUZO methodology could present desirable performance outcomes for 20 complex design problems while considering multiple parameters related to the architecture of high-rise buildings. Recent reviews have shown that not only performance aspects related to sustainable buildings but also parameters related to architectural design may present additional complexity during the optimisation process (Attia et al., 2013;Ekici et al., 2019a;Evins, 2013;Touloupaki and Theodosiou, 2017). Therefore, the use of the MUZO methodology may support architects and engineers as they investigate sustainable high-rise scenarios by taking into consideration parameters related to design concerns in the conceptual phase. The results also proved that the performance outcomes on different floor levels of high-rise buildings may be affected in dense urban areas. The main reason for the superior results obtained in the optimised designs proposed by the MUZO methodology was the division of one large design problem into sub-problems (zones). Hence, the optimisation algorithms could determine the best design alternatives for each zone while considering the performances of the various floor levels.
(2) Potential of the MUZO methodology: This study focused on optimising the sDA and ASE daylight metrics of LEED standard to evaluate the sustainability score of high-rise buildings. The MUZO methodology may integrate more performance aspects related to sustainable buildings (e.g., energy consumption, building integrated photovoltaics, and adaptive comfort). In such a case, the formulation of the problem could comprise multiobjective or many-objective optimisation to handle more than  two conflicting performance aspects. In addition, the complexity of the problem can be controlled by varying the number of zones. In this study, ten zones were considered, which is a predefined parameter that can be changed by the decision-maker based on the density of the surroundings. The consideration of fewer zones would limit the number of design decisions for the entire highrise design, while the use of a larger number of zones may increase the complexity and computational burden exponentially. During the optimisation process, 1,095,395 and 1,139,785 FES were considered for the quad-grid and diagrid scenarios, respectively, in order to determine which presents the best performance, and 14,580 FES were considered to evaluate the performance generated in typical high-rise scenarios. If these tasks were based on simulations, which required 4 min to calculate the performance of one design, 17.12 years would be required to complete all these computations. The MUZO methodology provided near-optimal alternatives for 4 days using SMs. Moreover, the aforementioned optimisation tasks were completed in GH using the Optimus and Opossum plug-ins. The flexibility of the proposed methodology allows the use of other digital platforms for optimisation, e.g., Python, C++, and C#, because the predictive models can be defined in another software.

(3) Surrogate-based optimisation algorithms versus optimisation with
SMs: An ongoing discussion in the literature is focused on the use of either surrogate-based optimisation (e.g., RbfOpt) or SMs with heuristic optimisation algorithms (e.g., this study). While the user can optimise the design task using surrogate-based algorithms when considering a small amount of FES, the overall process still requires a significant amount of time owing to the replication of the optimisation process using simulations. However, decisionmakers can investigate the design problem extensively in a reasonable amount of time using SMs, various algorithms, and replications, but with a prediction error. The accuracy of the SMs can be improved for each design problem, as explained in part 1 of this study; however, achieving zero error is almost impossible. Therefore, we can conclude that surrogate-based optimisation is convenient for small-scale design problems (e.g., office spaces), whereas optimisation with SMs is useful for large-scale design problems (e.g., high-rise buildings).

Conclusion
This paper presents the second part of the MUZO study and is focused on the optimisation problems and algorithms, results, and validation of the method. The results of this study showed that the performance of the entire high-rise building in dense urban districts can be improved by focusing on each zone as a separate design problem, and the optimisation process is explained in this paper. The combination of these approaches with the SMs presented in part 1 allowed us to complete the optimisations of entire high-rise buildings in a short time. The obtained results indicated satisfactory sDA and ASE performances that met the LEED criteria in 19 out of 20 design problems comprising various complexities. Although the jEDE slightly outperformed the CMA-ES algorithm, the RbfOpt presented a lower sDA performance as compared to the other algorithms. This underscores the importance of employing various optimisation algorithms with replications in architectural design optimisation because "the global optimal of each design problem is unexplored". In addition, the validation of the method also demonstrated that the building performance achieved using the MUZO methodology exhibited a remarkable improvement as compared to that of typical high-rise scenarios in dense urban districts. Therefore, the consideration of different parameters for various floor levels may provide significant performance improvements in the design of sustainable high-rise buildings in metropolises.
In conclusion, the relevance of this study is confirmed by the obtained optimisation results and the validation of the presented method. Thus, this study underscores the affect of the use of the MUZO approach for metropolises while dividing high-rise buildings into zones to be considered as separate design problems. The importance of artificial intelligence methods for swift optimisation for determining sustainable high-rise alternatives with the use of a large number of parameters was also demonstrated. In real-world applications, there is a possibility of combining 10 zones into one objective function instead of dealing with 10 separate problems. However, the design process may involve a large number of parameters, such as 260 parameters in the quad-grid and 220 parameters in the diagrid scenarios of this study. Therefore, the domain of architectural design optimisation requires tools and algorithms that can simultaneously cope with more than 200 parameters for highdimensional constrained problems (Chu et al., 2011;Jia et al., 2011). A sensitivity analysis could decrease the total number of design parameters; however, the final design may not reflect all the architectural concerns owing to some variables having been discarded. Another alternative to decrease the overall complexity of the design process could be the consideration of two algorithms that belong to different optimisation domains (e.g., surrogate-based and EC). Finally, in realworld applications, fewer zones may be considered, which would also decrease the computational complexity, based on the density of the urban plot under study.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.