BIM-based Precast Building Optimization PDF

Automation in Construction 155 (2023) 105065 Contents lists available at ScienceDirect Automation in Construction...

Automation in Construction 155 (2023) 105065 Contents lists available at ScienceDirect Automation in Construction journal homepage: www.elsevier.com/locate/autcon BIM-based constructability-aware precast building optimization using optimality criteria and combined non-dominated sorting genetic II and great deluge algorithm (NSGA-II-GD) Weng-Lam Lao a, Mingkai Li a, Billy C.L. Wong a, Vincent J.L. Gan b, *, Jack C.P. Cheng a, * a Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China b Department of the Built Environment, College of Design and Engineering, National University of Singapore, Singapore A R T I C L E I N F O A B S T R A C T Keywords: The popularity of precast concrete in construction is rising due to its capacity for improving building efficiency Precast optimization and quality in comparison to cast-in-situ concrete. While researchers have focused on enhancing constructability Building information modeling factors like logistics and sequencing, the need for developing standardization in this area remains evident. This Constructability paper describes a framework for studying the relationship between construction cost and standardization by NSGA-II Optimality criteria optimizing precast component dimensions and rebar designs. The framework involves using Building Informa­ tion Modeling techniques to extract semantic information from architectural plans, followed by a gradient-based Optimality Criteria method to optimize the sizing variables of precast components. Finally, a hybrid approach called NSGA-II-GD, which combines the Non-dominated Sorting Genetic Algorithm II and Great Deluge Algo­ rithm, is used to optimize the rebar layout design of each precast component. The Results demonstrate that an optimal point exists between construction cost and standardization, particularly for components subjected to similar stresses. Additionally, the NSGA-II-GD algorithm improves the searching efficiency in terms of conver­ gence, computational time, and searching space. these factors. For instance, some researchers have focused on improving installation by identifying the best component assembling sequence 1. Introduction [10,11], while others have sought to ensure resource availability by considering the relationship between production factories and on-site The construction industry has increasingly adopted precast concrete assembly [12–16]. Additionally, there are studies that have attempted due to its improved efficiency and quality, which offers numerous ad­ to improve the logistics and reduce the economic costs of vantages such as enhanced productivity , reduced waste production precast production. Despite standardization being one of the main [2,3], and improved quality control. principles of constructability, little research has been conducted on this Constructability is an important concept in precast design, and constructability factor in the precast design stage, which refers to the use various constructability scoring systems have been developed by gov­ of components with the same dimensions and production process to ernment authorities based on existing research [5,6], such as the reduce the number of mold changes and increase repetition of work, and Buildable Design Appraisal System (BDAS) in Singapore and the thus reduce error occurrence in manual production process and enhance Buildability Evaluation System (BSE(E)) in Hong Kong. These precast productivity. developed assessment models evaluate project constructability system­ Consequently, this paper presents a novel framework for promoting atically, starting from the planning stage to the operation stage, and standardization in precast reinforced concrete design. BIM techniques improve the cost savings, productivity, safety, and coordination of the are adopted to extract information from a pre-defined structural layout project. plan and visualize the results after analysis is completed. Precast Previous researches point out that design attributes such as instal­ structural elements, including columns, beams are optimized according lation, resource availability, economic impacts, and standardization are to their dimensions and rebar detailing. To achieve standardization, the important factors contributing to constructability [5,6]. Numerous at­ multi-objective formulation not only considers construction costs but tempts have been made to optimize precast construction with respect to * Corresponding authors. E-mail addresses: [email protected] (V.J.L. Gan), [email protected] (J.C.P. Cheng). https://doi.org/10.1016/j.autcon.2023.105065 Received 13 April 2023; Received in revised form 15 August 2023; Accepted 18 August 2023 Available online 29 August 2023 0926-5805/© 2023 Elsevier B.V. All rights reserved. W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Abbreviations nyic,j (j = 2,3,4)= number of legs along y direction Subscripts: Other variables: b beam As provided steel area from designed rebar layout c column Amax maximum allowable steel area for a component i component index Areq minimum required steel area for a component j rebar design subsection index (For beams and columns, 1 bib,min , bib,max minimum and maximum beam breadth allowed longitudinal reinforcement; 2,3,4 = transverse bic,min , bic,max minimum and maximum column width allowed reinforcement) dic,min , dic,max minimum and maximum column depth allowed k beam longitudinal rebar type index in different location cc concrete cost per unit volume l lateral constraints index cs steel cost per unit weight cl labor cost per unit time Dimension sizing variables: ci cost coefficient bib breadth of beam e0ic,l , e1ic,l , e2ic,l , e0ib,l , e1ib,l , e2ib,l virtual strain energy coefficient dib depth of beam Gratio group ratio (number of groups of components to the total lib length of beam number of components) bic width of column H building height dic depth of column Nbar number of bars installed inside a component hic height of column Ncomponents total number of components under consideration Ngroups number of groups of components that share identical cross- Rebar layout design variables: sectional dimension and rebar layout design Aib,1 longitudinal rebar layout design for beam Ntie number of ties used to fix the bars nib,1,k number of bars pp rebar placing time dib,1,k diameter of bars pt rebar tying time Aib,j (j = 2,3,4) transverse rebar layout design for beam Ti,c , Ti,b time for rebar installation nib,j (j = 2,3,4) number of legs Vci,c ,and Vci,b volume of concrete excluding the volume of steel dib,j (j = 2,3,4) diameter of bars reinforcement sib,j (j = 2,3,4) spacing of bars v iteration number in OC Aic,1 longitudinal rebar layout design for column Vsi,c , and Vsi,b volume of steel reinforcement dic,1 diameter of corner bars δT total lateral drift nxic,1 number of inner bars along x direction δf , δf− 1 lateral displacements of two adjacent floors dxic,1 diameter of inner bars along x direction σi allowable limit of component stress nyic,1 number of inner bars along y direction σi component stress dyic,1 diameter of inner bars along y direction η parameter to control the convergence speed of OC Aic,j (j = 2,3,4) transverse rebar layout design for column γ density of steel dic,j (j = 2,3,4) diameter of bars λl Lagrangian multipliers sic,j (j = 2,3,4) spacing of bars nxic,j (j = 2,3,4) number of legs along x direction also includes a constructability score, which is defined from the (4) Hybridization of NSGA-II with GD to improve computational ef­ grouping of elements. The Optimality Criteria (OC) method is adopted to ficiency for rebar detailing optimization. optimize the sizing of the elements, while a method called NSGA-II-GD is proposed to optimize the rebar detailing design for each precast In this study, a multi-objective optimization approach is adopted that element. This NSGA-II-GD method is developed by hybridizing the Non- considers construction cost and constructability, with the integration of dominated Sorting Genetic Algorithm II (NSGA-II) and the Great Deluge BIM to facilitate the analysis process. It can serve as a reference for Algorithm (GDA) to improve the efficiency of the solution search. To future research and development of evaluation systems related to stan­ showcase the proposed framework, an illustrative example of a five- dardization concerns for precast and prefabricated components. story reinforced concrete structure plan is presented. The results show that there is an optimal point between construction cost and con­ 2. Literature review structability score. The same problem is applied to compare the pro­ posed NSGA-II-GD, standard NSGA-II, and weighted GA, demonstrating 2.1. Previous precast studies in different constructability aspects that NSGA-II-GD shows a significant improvement in the time required for a converged result. Numerous studies have been conducted to improve the construct­ The novelties of this research include: ability of precast construction, with each study addressing a different aspect of the production process. For instance, to improve the logistic of (1) Development of a novel BIM-based framework for optimizing the precast production, Chen conducted an overall profit optimization structural design of precast elements. by considering costs associated with mold purchasing, internal and (2) Formulation of an optimization strategy that takes into account external storage, transportation, and production; to make precast construction costs and constructability for precast elements to installation more efficient, researchers have attempted to find the best achieve standardization. assembling sequence on-site using meta-heuristic algorithm, aided by (3) Integration of OC with NSGA-II-GD for element sizing and rebar BIM [10,11]; to make precast construction more cost-effective, Albu­ detailing optimization. querque optimized the precast concrete floor design using Genetic Algorithm to incorporate all construction costs, including 2 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 manufacturing, transportation, and erection costs; to ensure resource by incorporate BIM into the design process to achieve error-free fabri­ availability, researchers have formulated the process of precast cation and erection for precast concrete elements. Jiang et al. component fabrication to assembly as various forms of multi-echelon enhanced the speed of collaborative decision-making for planning and supply chain network problems, with different emphasis such as scheduling by developing a digital twin platform for real-time moni­ reducing fixed and variable costs, fulfilling customer demands, mini­ toring of precast components on-site. mizing required time, and satisfying environmental and sustainable The integration of BIM into precast design is essential to the analysis considerations [12–16]. of constructability factors and structural design. BIM can access the se­ Although numerous research efforts have been made to improve mantic information of a facility in a 3D model, which serves as the precast construction in various constructability aspects, few studies have foundation of simulation and optimization. Nevertheless, the analysis focused on precast design with respect to standardization. As such, this results can then be reimported into the existing building information for study aims to address this gap in the literature by proposing an approach downstream processing, such as automatic shop drawing creation and that utilizes the concept of standardization to optimize precast compo­ visualization. nent dimensioning and rebar detailing. 3. Methodology 2.2. Multi-objective optimization in reinforced concrete design 3.1. Proposed BIM-based framework for precast component sizing and The optimization of both dimension sizing and rebar detailing in rebar layout design optimization structural design presents a challenge due to its high space and time complexity. To tackle this issue, gradient-based approaches and meta- The proposed framework aims to optimize the design of precast el­ heuristic algorithms have been proven effective when dealing with ements by considering constructability through the integration of BIM large design spaces [19,20]. For instance, a hybrid genetic algorithm and an optimization strategy. Shown in Fig. 1, the framework consists of and particle swarm optimization was utilized to optimize frame design, two parts: (1) extraction of building model information, and (2) struc­ taking into account component dimensions and longitudinal reinforce­ tural optimization of precast element design by integrating the OC ment [21,22]. However, this method is limited to single objective method for element dimensions and a proposed NSGA-II-GD method for problems and only provides the steel ratio, excluding rebar detailing. rebar layout design. The designer can select a practical solution from Kashani faced the same limitation when solving a similar problem several design options obtained from the Pareto front resulting from the regarding reinforced concrete cantilever retaining walls, but showed optimization strategy. The resulting optimized precast element sizing that NSGA-II outperformed several other meta-heuristic algorithms in and rebar layout design can then be applied to downstream processes, solving the same problem. Mangal et al. employed a hybrid GA such as auto-generation of shop drawings and visualization. method to identify the optimal bar number and diameter combination for columns and beam structures. However, the resulting solution only 3.1.1. Extracting information from the building model shows a single rebar layer with varying rebar diameters, which may not The process of information extraction begins with a predefined be favorable during installation. Leyva et al. constructed a database architectural plan, which serves as the basis for extracting topology and with combinations of element dimension and rebar detailing using bi­ semantic information, such as load cases, supporting conditions, and nary coding for elements such as slabs, columns, beams and walls, and materials, directly from the BIM model. Different types of precast optimize them using NSGA-II; Li et al. [26,27] adopted a similar data­ components exhibit unique structural behavior and must be designed base for reinforcement layout only by real number coding instead of separately. Using the extracted building information, components such binary coding, and solve the problem using a GA enhanced Hooke and as columns, beams are classified into distinct groups. To account for Jeeves method, they then developed a graph neural network to improve differing design criteria and constraints, each type of component is the generation speed of the optimized design; Xu et al. used a assigned specific variables to define its dimensions and rebar detailing. transformed number range to represent the rebar layout, and obtained The variable definitions will be discussed in detail in Section 3.2.2. results by using neighborhood field optimization algorithm and artificial potential field. However, these researches either overly simplified the 3.1.2. Structural optimization of precast design problem or only focus on the rebar layout and clashes without consid­ After grouping precast elements based on their semantic informa­ ering component dimensions. On the other hand, Gan et al. utilized tion, optimization strategies are applied through structural analysis. The a gradient-based optimality criteria - genetic algorithm (OC-GA) to resulting output is a series of optimized precast element sizing and rebar demonstrate the optimization of component topology and dimension layout designs obtained from the Pareto front. This optimization process sizing, considering structural design at a wider level without further seeks to maximize the constructability score while minimizing con­ detailing of steel reinforcement. struction costs, and a detailed formulation of the objective function will be presented in Section 3.2.4. To obtain the sizing and rebar design, 2.3. Building information modeling (BIM) application in precast separate procedures are followed as outlined below: construction (A) Element sizing optimization by OC Despite the increased application of BIM in precast construction in recent years, its adoption in the structural design of precast is still Using the semantic information, component dimensions are opti­ limited. Various studies have explored the used of BIM to improve mized through a gradient-based OC. The optimization process exclu­ precast productivity. Nath et al. demonstrated productivity sively minimizes material costs, such as concrete and steel improvement with the automatic generation of shop drawings depend­ reinforcement costs, while adhering to constraints that include total ing on BIM-based parametric precast element that is collaboratively drift, inter-story drift, and component strength. Following optimization, shared among all parties. Barkokébas et al. optimized labor and precast components of beams, columns are further categorized into resource allocation in the modular manufacturing process by extracting different sub-groups if they share the same dimensions. information from a constructed model and database. Li et al. improved precast delivery management by introducing an Internet of (B) Rebar layout optimization by NSGA-II-GD algorithm Things-enabled platform based on BIM and radio frequency identifica­ tion technology to trace precast elements and facilitate progress in­ Within each component sub-group, the rebar layout of each spection in a shared platform. Kaner et al. increased design quality component is simultaneously optimized using the proposed NSGA-II-GD 3 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 1. Framework of structural optimization for precast components. algorithm. The final outcome of this procedure is a set of rebar layouts Of the three elements of standardization, the repeated grid layout, for each precast component within the same sub-group, which mini­ structural plan, and floor height are typically governed by the archi­ mizes cost for different constructability scores. A higher constructability tectural plan, while the other two aspects can be controlled during the score indicates larger number of identical components which share the component design stage. This study extended the constructability same dimensions and rebar layout design, thereby increasing the level of scoring method developed by BSE(E) to evaluate the relationship be­ standardization in the precast design. Ultimately, the designer must tween standardization and construction cost. determine a balance between construction cost and the desired level of standardization in the precast design when selecting the final result. 3.2.2. Variables for precast component To account for the diverse structural behaviors exhibited by beams and columns separate variables are employed to describe their di­ 3.2. Formulation of optimal design mensions and rebar layout designs. The optimization only considers precast components manufactured inside the factory, and therefore, 3.2.1. Constructability consideration in precast component design certain parts of the components are simplified and ignored during the Previous studies on precast construction have focused on improving optimization process. All variables used in the calculations are presented constructability during fabrication, transportation, and assembly stages. in Fig. 2, with independent variables highlighted. However, little research has been conducted on improving construct­ The sizing variables specify the dimensions of each component. As ability during the design stage of precast components. Therefore, this the component topology is predetermined by the designer, only the study aims to enhance the constructability of precast components by cross-sectional variables are included as independent variables during considering standardization. According to the 3S principle which the optimization. Although rectangular cross-sections are used for simplicity BDAS and the BSE(E) are depended on, standardization in pre­ in beams and columns in this study, they can be extended to handle cast construction includes the following elements: atypical shapes, such as L and T-shaped beams and circular columns. The rebar layout design for each component type and section is Repeated grid layout, structural plan, and floor height to facilitate represented by design layout vectors. These variables are initially pre­ faster on-site assembly of precast components sented in low-level form, including the number, diameter, and spacing of Repeated sizing of structural elements or external cladding to reduce the rebar layout. They are then translated into high-level variables the number of mold changes in the factory through a database during the optimization process, as illustrated in Standard reinforcement detailing of similar size, span, and loading Section 3.2.3. components to increase the speed of production and installation 4 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 2. Definition of variables for (a) beams and (b) columns. Since this study only focuses on precast components manufactured the section measured. Designers can incorporate multiple rebar di­ inside the factory, the design of precast beams is limited to their use in ameters and layers by extending the length of the layout vector. The such settings. In some cases, the top longitudinal reinforcements of design criteria for transverse reinforcements in column components vary precast beams may need to be assembled on-site to provide sufficient depending on the stress changes throughout the column. Therefore, anchorage or connections with other components and cast-in-situ with similar to beams, column rebar designs are also classified into three the top floor panel. Therefore, the precast beam component is designed sections. with only bottom longitudinal reinforcement to resist bending moments, along with transverse reinforcement to counteract shear forces. The 3.2.3. Transformation of layout vector for optimization reinforcement design for beam components, except for cantilever Merely specifying low-level variables such as number, spacing, and beams, is typically divided into three sections due to shear and moment diameter is inadequate to fully describe the rebar layout scenario in envelops. As the bottom longitudinal reinforcement ends inside the same precast concrete construction. Structural engineers often need to component, the same layout is used throughout the entire beam consider a combination of multiple layers and diameters for longitudinal component. The beam longitudinal layout is defined as Aib,1 = [nib,1,1, layouts in columns and beams to conserve material while satisfying in­ dib,1,1, nib,1,2, dib,1,2, …, nib,1,k, dib,1,k], where k is the number of different ternal force requirements. However, the number of variables required to rebar arrangement specified by the designer for the same section. For describe the rebar layout increases exponentially with each additional example, assuming that within the same project, the vector [4, 20, 1, 16, rebar generation rule. To address this issue, high-level variables are 0, 0], which describes the use of two different diameters in a single layer, utilized to represent the rebar layout instead of using a variable-length and the vector [3,20,0,0,2,20], which describes the use of two layers in layout vector. This approach helps to reduce the complexity of the the rebar design, both of them are permissible as shown in Fig. 2a. In this optimization problem and enables a more efficient optimization process. project, k is equal to 3, indicating that there are 3 different rebar ar­ To represent the diverse rebar layout designs in precast reinforced rangements available for the beam rebar design. The transverse rein­ concrete components, databases are constructed, mapping all existing forcement in beams, however, may vary according to the stresses in layouts to an index. These databases contain additional information different sections and thus three sub-sections are used to define the regarding the provided area from the layout (As), and the minimum and transverse reinforcement. maximum dimensions allowed in each component. The dimension range The precast column component primarily experiences axial forces is calculated from the rebar layout based on spacing constraints stated in that remain relatively constant along the entire height of the component, the code of practice. These additional information serves as the and thus, a single longitudinal reinforcement layout is sufficient. An constraints for filtering infeasible rebar layout, as illustrated in Fig. 3. example of a column's longitudinal reinforcement layout is shown in The first condition is to ensure that the provided steel area As of the Fig. 2b, which is represented by the layout vector Aic,1 = [dic,1, nxic,1, designed rebar layout falls within the available range between the dxic,1, nyic,1, dyic,1], consisting of four corner bars and inner bars along minimum required (Areq) and maximum (Amax) allowable steel area, the x and y directions. The longitudinal reinforcement is assumed to which is unique to each precast element and based on their subjected have a symmetric layout, with only the number of bars along one side of internal forces. The second condition is the dimension constraints set by 5 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 3. Database establishment and layout filtering process for rebar layout design variables. the designer. Any sizes outside the dimension ranges are considered dimension standardization is borrowed and extended to evaluate not infeasible and must be filtered out. The remaining layouts after the only the type of cross-section but also the rebar layout. The construct­ filtering process are potentially feasible rebar layouts, which are then ability score is evaluated by combining the standardization of compo­ mapped to a continuous real number stream, serving as the domain of nent dimension and rebar layout design and is formulated as follow: the variable in the optimization procedure. / Gratio = Ngroups Ncomponents (1) 3.2.4. Objective function score = 1, Gratio < 0.15 A) First objective: maximizing constructability score 0.7 − Gratio score = , 0.15 < Gratio < 0.7 0.55 Both the BDAS in Singapore and the BSE(E) in Hong Kong are developed based on the constructability factors identified by previous score = 0, Gratio > 0.7 (2) studies. These systems provide a comprehensive evaluation of the level of constructability from the planning stage to the operation stage of a where Gratio is the group ratio; Ngroups is the number of groups of com­ project, encompassing various aspects of construction. Standardization ponents that are identical, which have the same cross-sectional dimen­ of different structural components, including their dimensions and rebar sion and rebar layout design; Ncomponents is the total number of detailing design, is included as part of the scoring system in both BDAS components having the same cross-sectional dimension, score is the and BSE(E). constructability score. In BDAS, standardization of sizing is calculated based on the per­ centage of coverage of the three most common sizes of a particular B) Second objective: minimizing construction cost component category. The constructability score is assigned discretely if the coverage exceeds a certain threshold. However, BDAS does not The construction cost can be broken down into two main compo­ include a section for evaluating the rebar layout design, and the discrete nents: material and labor costs. The material cost encompasses the cost scoring method makes it difficult to analyze the relationship between of concrete and steel reinforcement. By minimizing the material cost, it standardization and construction cost. is possible to save on materials while ensuring the structural integrity of In BSE(E), the standardization level of the dimension of columns, the component. Meanwhile, the labor cost covers the cost of rebar beams, and slabs is evaluated based on the ratio between the types of placement and tying. While there are machines available to assist with dimension and the total number of components. The score contains a the prefabrication process of precast components, such as mold place­ continuous linear value when the ratio is within 15% to 70%. On the ment, concreting, de-molding, and transportation, the rebar assembly other hand, the standardization of rebar detailing is evaluated based on procedures are not yet fully automated and require human intervention. several aspects, including workers' experience, which is assigned a While machines for the automatic cutting and bending of steel re­ discrete value. inforcements are available, robots for placing and tying rebars are still In this study, the method established by BSE(E) for component under development. Current automatic placement and tying machines 6 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 are mainly used for the construction of decks of bridges or for large in- ( ) δf − δf − situ casting, and may not be as efficient in the production of smaller (9) 1 ≤ df , f = 1, 2, …, F hf components like precast panels. According to research by Li et al. , including the labor cost in the optimization process can result in σi ≤ σi (10) less congested rebar designs, facilitating rebar clash avoidance and reducing the needs for reworking. Bb,l ≤ bib ≤ Bb,u , Db,l ≤ dib ≤ Db,u , Note that, though the rebar prefabrication procedure involves cut­ ting, bending, transporting, placing, and tying the rebar, cutting and Bc,l ≤ bic ≤ Bc,u , Dc,l ≤ dic ≤ Dc,u , (11) bending are assumed to be performed automatically in the precasting factory, while transportation depends on project-specific factors like the where δT is the total lateral drift at the top floor, and H is the building distance between the storage location and the production area. Thus, height; δf and δf − 1 are the lateral displacements of two adjacent floors, only the rebar placing and tying costs are taken into account as the labor and hf is the corresponding floor height; σi and σ i represents the indi­ cost in this study's optimization process. vidual element stress and its allowable limit; l and u denote the lower Thus, the objective of minimizing construction cost is formulated as and upper limits of each dimension variable, respectively. follows: The rebar design level constraints comprise those stipulated in the design code, and take into account ductility, workability, congestion, Cost = Cc + Cs + Cl (3) and strength by restricting the distribution of rebar within the compo­ nent. These constraints encompass various factors, including but not Iw ∑ ∑ Ic ∑ Ib ∑ J Cc = ( ( ) ( )) Vci,c bic , dic , Aic,j + Vci,b bib , dib , Aib,j × cc (4) limited to the minimum number of rebar placements, the allowable iw ic ib j spacing range for between bars, and the smallest diameter permitted. The principal constraint of the rebar design is the element strength, Iw ∑ ∑ Ic ∑ Ib ∑ J ( ( ) ( )) which is governed by the steel area in different sections of the compo­ Cs = Vsi,c Aic,j + Vsi,b Aib,j × γ × cs (5) nents, as depicted as follows: iw ic ib j Areq ≤ As ≤ Amax (12) Iw ∑ ∑ Ic ∑ Ib ∑ J ( ( ) ( )) Cl = Ti,c Aic,j + Ti,b Aib,j × cl (6) whereAreq is the required steel area according to the subjected stress or iw ic j the minimum steel requirement, whichever is bigger; Amax is the ib I ∑ ∑ J ( ) ( ) maximum allowable steel area; As is the provided steel area from the Nbar Ai,j Ntie Ai,j Ti = + (7) chosen layout which is derived from the layout vector. i j pp pt where Cc , Cs , and Cl are the cost of concrete, steel and labor (in HKD); 3.3. Integrate OC and NSGA-II-GD for precast building optimization Vci,c ,and Vci,b are the volume of concrete excluding the volume of steel reinforcement (in m3); Vsi,c , and Vsi,b are the volume of steel reinforce­ 3.3.1. Two-stage optimization for precast dimensioning and rebar layout ment (in m3); Ti,c , Ti,b are the time for rebar installation (in min); cc is the design concrete cost per unit volume (in HKD/m3), cs is the steel cost per unit In this study, we propose a two-stage optimization approach for weight (in HKD/kg), cl is the labor cost per unit time (HKD/min); γ is the optimizing both component sizing and rebar layout design, which density of steel (in kg/m3), Nbar is the number of bars installed inside a simultaneously considers constructability and construction cost. The component, Ntie is the number of ties used to fix the bars, pp is the rebar first stage utilizes the gradient-based OC algorithm, which is suitable for placing time (in bar/min), pt is the rebar tying time (in tie/min); b and c solving objective functions with a large number of variables and non- denotes the beam and column components, i and j refers to the linear constraints, to iteratively adjust the dimensions of components component index and rebar design subsection index respectively, I and J by minimizing only the construction cost. In the second stage, we refers to the total number of component and total number of subsection employ the proposed NSGA-II-GD algorithm, a hybridization of the inside the component respectively. NSGA-II algorithm and the GDA, to optimize the rebar layout design for each sub-group of components that share the same type and dimensions, 3.2.5. Design constraints while considering both the constructability score and construction cost In structural optimization, some constraints are defined by the de­ objectives. signers and others provided according to the design codes. Designers The NSGA-II algorithm is a well-known meta-heuristic algorithm determine the material availability such as the dimension range of that is popular in solving multi-objective optimization problems, while components, or the rebar diameter used, while the design codes cover the GDA is an algorithm that converges quickly for high-dimensional the precast reinforced concrete strength, workability, serviceability, and search spaces with few tuning parameters. We selected the GDA for durability requirements. As the optimization process is divided into two the hybridization due to its high success rate in finding a feasible solu­ stages, the constraints of the problem are also divided into two levels: tion. Details of the selection reasons are presented in Section 4.4.1. The component level and rebar design level. rebar layout design problem is complex and involves many variables, The component level constraints contain the consideration of the which results in an extensive solution space that encompasses vast whole building's integrity such as lateral drift and element-wise re­ infeasible regions. By integrating the GDA, we ensure that the search is quirements such as component strength. Following, Eq. (8) describes the restricted to the feasible region, which eliminates unnecessary explo­ maximum allowable serviceability total lateral drift ratio dH of the ration in the infeasible regions and enhances search efficiency. whole building, Eq. (9) defines the upper limit of lateral inter-story drift df , Eq. (10) sets the component strength constraints, and Eq. (11) rep­ 3.3.2. Gradient-based OC resents the designer-imposed constraints on the allowable dimensions The optimization of component dimensions is a crucial task in the for the project. design of a building's structural system. Buildings are subjected to both vertical loading, including self-weight and utility loads, and horizontal δT ≤ dH (8) forces such as wind loading. Consequently, the sizing of structural H members must satisfy strength and lateral drift serviceability constraints simultaneously. To achieve this, a gradient-based OC method is 7 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 employed to optimize the dimensions of precast components based on a lateral drift constraints; ci is the cost coefficient of each component i pre-defined structural layout plan. The derivation of the explicit respectively; λl denotes the Lagrangian multipliers of the lth constraint. formulation, virtual strain energy coefficient, unconstrained function, and recursive relationship is based on the research conducted by Chan 3.3.3. NSGA-II-GD. The following summary presents the key points of this derivation. Fig. 4 illustrates the procedure for the OC method used in this study. A) NSGA-II-GD optimization procedure To obtain the gradient for iterative variable updates, explicit formula­ tions of lateral drift constraints are necessary, which are dependent on Fig. 5 depicts the NSGA-II-GD optimization procedure. Initially, the dimension variables, including the breadth (bib) and depth (dib) of random layout variables from the filtered database are assigned to each beams and the width (bic) and depth (dic) of columns. The lateral drift component to form the initial population. The GDA is then utilized to constraints can be explicitly represented using the virtual work princi­ refine the layout index and ensure that the initial population only con­ ple, as shown by tains feasible solutions. This involves two modes of GDA, which will be ( ) ( ) expounded upon in Section 3.3.3C). From the feasible population, a ∑Ic ∑ Ib dl = e0ic,l + e1ic,l + e2ic,l + e0ib,l + e1ib,l + e2ib,l ≤ dl (13) mating process is applied to generate offspring and the generating op­ ic bic dic bic dic3 b3ic dic ib bib dib bib dib3 b3ib dib erators are chosen through testing and fine-tuning. This starts with a binary tournament selection where two parents are chosen, and the where e0ic,l , e1ic,l , e2ic,l , e0ib,l , e1ib,l , and e2ib,l are the virtual strain energy parent with a smaller front rank or higher crowding distance of the same coefficient calculated from internal moments and forces of the precast rank is selected. Subsequently, a two-point crossover operator is used to components under actual loading and virtual loading; l is the lateral combine different parts of the selected parents, which are then subjected constraints index. to a random resetting mutation operator to produce offspring. All Transforming the design formulation into an unconstrained function infeasible children will be rectified by the GDA operation. After that, the that integrates the objective function of minimizing construction cost generated offspring are merged with their parents and proceed to non- and the drift constraints can be achieved by applying Lagrangian mul­ dominated sorting and crowding distance sorting. Individuals with tipliers. Afterwards, the gradient can be computed by taking the partial larger front ranks or smaller crowding distances are filtered, until the derivatives of the unconstrained function with respect to each dimen­ number of individual in the new population equals the initial popula­ sion variable. Using the gradient, the recursive relation of each variable tion. This process is repeated until the maximum number of iterations, can be formed to determine the optimal value of the dimension vari­ which serves as the stopping criterion, is attained. ables. Eqs. (14)–(15) present the recursive formulation of the breadth and width (bib and bic), and depth (dib and dic) of beams and columns, B) GDA operator respectively: { ( ( ) ) } A feasible solution is one that does not violate any constraints. 1 ∑ L λl e0i,l e1i,l 3e2i,l Constraints are formulated such that when there is no violation, the (bi )v+1 = (bi )v × 1 + + + − 1 (14) η l ci b2i di2 b2i di4 b4i di2 value returned is zero, and the more a solution violates a constraint, the v { ( ( ) ) } higher the returned value. The search space of a problem with a high 1 ∑ L λl e0i,l 3e1i,l e2i,l degree of complexity is large, and most solutions are located in the (di )v+1 = (di )v × 1 + + + − 1 (15) η k ci b2i di2 b2i di4 b4i di2 infeasible region, with constraint values higher than zero. When using a single metaheuristic search algorithm, the algorithm first explores so­ v wherev and v + 1 signify the iteration number; η is a parameter to lutions by minimizing the value of the constraints before optimizing the controls the pace of convergence of the recursive relationship; l is the objective function. However, applying GDA can eliminate the explora­ index of lateral drift constraints and L represents the total number of tion of the infeasible region. The operational procedure of the GDA operator is depicted in the Fig. 4. Optimization of dimension variables using Optimality Critiera method. 8 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 5. NSGA-II-GD optimization procedure. pseudocode in Fig. 6. Given that no conflicts arise among individual group that share the same dimension. Then, a GDA operation is per­ components' constraints, GDA is implemented separately for each formed to determine a feasible solution for the fake component. The component. At the outset, GDA generates a random population, result from the fake component is then propagated to other real com­ whereupon it calculates the fitness of each individual. A “water level” is ponents within the same group. Another GDA operation is initiated to initiated at the average constraint value of the initial population and amend the rebar layout inside the real components if the real compo­ gradually descends in static intervals. The individual with the minimum nents having the rebar layout from the fake component violates any fitness value is chosen as the current individual. Then, the current in­ constraints. Consequently, the initial population predominantly consists dividual is compared with the best recorded individual. If the current of solutions with the same rebar layout design, and the search subse­ individual has a lower fitness value than the best individual, it becomes quently expands to include designs falling outside the main group, the next individual, whereas, if the current individual is worse than the provided there is a potential cost saving. The resulting solutions are best individual, it replaces the next individual only if its fitness value is typically located in regions with a higher constructability score, signi­ lower than the “water level”. The next population is generated based on fying a greater degree of standardization. the next individual. This process continues to iterate until the water level drops to a certain threshold or the constraint value reaches zero, 4. Illustrative example and discussion signifying the attainment of a feasible solution. 4.1. Example: 5-story reinforced concrete structure C) Two modes of initial population generation: random mode and grouped mode To demonstrate the optimization framework and the correlation between standardization level and construction cost, a 5-story building The first mode of initial population generation is the random mode, is used as a demonstrative example. An arbitrary lateral load is which generates the rebar layout of each component in a random employed as a practical lateral load cannot demonstrate the dependency manner. The optimization process starts with components having mostly of the serviceability lateral drift constraint in a short building. Addi­ dissimilar layouts and then moves towards identifying groups of com­ tionally, in a tall building scenario, it is difficult to observe how the rebar ponents with similar layouts. This search provides results in areas with layout of each component changes according to different construct­ lower constructability scores, which indicates a lower level of ability scores. Therefore, the lateral wind load used in this example is standardization. 100 times higher than the actual loading. Only one load case is The second mode of initial population generation is the grouped considered for the demonstration purpose. The structural layout and the mode, whose operation procedure is depicted in Fig. 7. Firstly, a fake 3D architectural model of the building are depicted in Fig. 8a, while the component is constructed that is assumed to be subjected to the mini­ design attributes are listed in Table 1. mum and maximum internal forces from all components within the same The precast design process begins with an architectural model, which 9 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 6. pseudocode of GDA optimization process. is then transformed into a structural layout using Revit. Lateral wind withstand the anticipated forces and moments. Consequently, these re­ load and component topology information are extracted from Revit and bars are also considered in the calculation of construction costs. imported into Robot Structural Analysis (RSA). The dimension optimi­ The optimized 3D building is displayed in Fig. 9 showing the di­ zation procedure using the OC algorithm is performed through a linkage mensions of column only. All beams possess an identical depth and between RSA and Excel. RSA is used to obtain the internal forces of each width of 400x300mm. In addition to the stress distribution constraint in element during structural analysis, while Excel is used to compute the Eq. 10, an additional constraint is imposed during the dimension opti­ virtual strain energy coefficient and update the component sizing in mization process. This constraint requires that lower components have each iteration. The rebar layout optimization using NSGA-II-GD is per­ larger dimensions than upper components, in accordance with industry formed in Matlab, where the required features, including available practice. The total number of component sub-groups are summarized in reinforcement area ranges and component dimensions, are obtained Table 2, where each sub-group includes elements with the same value in from RSA. The engineers then design the appropriate dimensions and all three dimensions. There are 3, and 14 sub-groups for beams and rebar layout for each component, which is then reimported into Revit to columns respectively, and there are a total of 30 sub-groups. It is noted update the building information before handing over the design to the that most sub-groups have only one or two members as they are sub­ precast fabricators for manufacturing. jected to distinct stresses. However, some sub-groups comprise >10 In accordance with the predetermined structural layout plan, as members, mostly the smallest possible sections as they experience illustrated in Fig. 8b, information such as elements' locations, loadings similar stresses. For instance, the sub-group of beams with dimensions of and support conditions are extracted, and the structural components are 400 × 300 × 6000 mm consists of 43 members. classified into beams, columns. Slabs are not taken into account in this Each of the 30 component sub-groups undergoes the NSGA-II-GD study, as they are not sensitive to lateral drift. Shear walls are considered procedure to establish the relationship between construction cost and cast-in-situ and are not involved in the consideration of this study. constructability score while optimizing the rebar layout design. Prior to Variables are assigned to each component based on its category. Uti­ the procedure, several parameters need to be fine-tuned to ensure lizing the OC algorithm, the material cost of the building is minimized maximum efficiency. For the mating process, the crossover rate is fixed by optimizing the sizing variables. Constraints such as strength at 0.75, while the mutation rate is adjusted to 0.25. The population and constraint and lateral drift serviceability requirements are derived from generation depend on the number of variables defined in each compo­ structural analysis results based on the extracted semantic information. nent sub-group, and are determined through a trial-and-error process According to the study proposed by Guerrero et al. , the behavior until a converged solution is reached. The population is approximately of precast joints is comparable to that of monolithic cast-in-situ joints, 10 times the number of variables, while the generation requirements with precast joints often exhibiting higher strength. The same joint range from 200 to 300. In addition, the GDA operator necessitates configuration employed for cast-in-situ joints is adopted during struc­ parameter tuning. The population is set at 20, and the water level de­ tural analysis, and a more conservative result is ensured. creases statically with a step value of 0.001. The primary constraints are To enhance the structural performance and integrity of the joints in obtained from the stresses calculated based on the structural analysis, precast elements, additional rebars are incorporated at the joint loca­ whereby stresses acting on the component are translated to the neces­ tions. These extra rebars are necessary to enable the joints to effectively sary steel areas or steel ratio. Other constraints, such as those stipulated 10 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 7. Illustration of grouped mode for initial population generation. in the Hong Kong code of practice, are also integrated. The results furnish a sequence of rebar layouts that meet the steel area requirement while achieving varying constructability scores. 4.2. Relationship between construction cost and constructability score This section presents the correlation between construction cost and constructability score for all three types of components by showcasing the results of the component sub-groups containing >10 members. Fig. 10a is constructed by merging solutions from five results of random mode and five results of grouped mode for beams with dimensions of 400 × 300 × 6000 mm. The NSGA-II-GD is a stochastic algorithm that provides different results that are close to the global minimum. The proposed random and grouped mode initial individual generation techniques allowed searches to focus on different constructability score Fig. 8. (a) 5-story 3D architectural model and (b) structural layout plan in the regions. For each operation, there is an increasing construction cost illustrative example. trend with an increasing constructability score. However, when the re­ sults are merged, a curved trend is observed instead. In Fig. 10a, the construction cost decreases drastically from 84,412 HKD to 89,021 HKD, 11 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Table 1 reverses at the optimal point. The results show that, for components Design attributes of the 5-story reinforced concrete structure. subjected to similar stresses, such as beams and columns, a collective Design attribute Value design approach is favored. This is because there exists an optimal design that minimizes the construction cost while promoting standard­ Typical Beam Dimension 750 mm × 600 mm Typical Column Dimension 600 mm × 600 mm ization. However, for components that endure diverse loading, indi­ Beam Breadth Range 300 mm to 800 mm vidual design may be more favored. This is because the diversity of rebar Beam Depth Range 400 mm to 1000 mm layouts leads to a decrease in construction costs for such components. Column Width and Depth Range 300 mm to 1000 mm Fig. 13 illustrates how the random mode and grouped mode indi­ Concrete Strength 45 MPa High-yield Rebar Grade 500 MPa vidual initialization techniques affect the final results provided by the Mild Steel Rebar Grade 250 MPa NSGA-II-GD. The random mode focuses on searches in low construct­ Longitudinal Rebar Diameter 12, 16, 20, 25, 32, 40 mm ability score regions, while the grouped mode focuses on searches in Transverse Rebar Diameter 8, 10, 12, 16, 20 mm high constructability score regions. However, the random mode has Concrete Costs 771.9 HKD/m3 better convergence than the grouped mode, while the grouped mode has High-Yield Rebar Cost 5075 HKD/t Mild Steel Rebar Cost 5961 HKD/t a wider searching scope. Therefore, merging the results from both modes Labor cost 4420 HKD/day (assume two workers) can provide results that are good enough for both convergence and Arbitrary lateral wind load 100 x real lateral load space. Load combination 1.4 x self weight + 1.4 x arbitrary lateral wind Table 3 presents a comparison of construction costs across various load Upper limit of topmost lateral Building height / 500 constructability scores. The results indicate that the overall cost differ­ drift ential between the minimum score and optimal point is negligible, with Upper limit of inter-story drift Story height / 400 an average difference of 0.64% for beams and 0.83% for columns. This shows that if components are designed independently, it may not yield the best construction cost. However, the construction cost increases but after reaching the optimal constructability score of 0.777, the con­ significantly for the maximum constructability score, exceeding the struction cost shows a slight increasing trend rather than an expected optimal point by 6.84% and 9.55% for beams and columns, respectively. decreasing trend to 85,122 HKD. The rationale behind this scenario is These results indicated that fully emphasis on constructability leads to illustrated by Fig. 11. After reaching the optimized point, any attempt to increase in cost. Therefore, a balance between standardization and include another layout inside the group results in an increasing con­ construction cost should be established to achieve the two goals. struction cost rather than a decreasing cost. The optimal point usually occurs when all longitudinal layouts are optimized to their possible state, and another additional layout will attempt to include a changing transverse layout that has less impact but an increase on the construc­ Table 2 tion cost. Grouping of components with identical dimensions in beams and columns. A similar trend is observed in all three types of components. Fig. 12 Type Beam Column Total shows the results obtained by permutating the solutions from all component groups for beams and columns. The pareto fronts are high­ Number of components 66 45 111 Number of sub-groups 3 14 17 lighted. Noted that for columns and beams, the Pareto front search Fig. 9. Columns sizing after sizing optimization. 12 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 10. Aggregated Pareto front from rebar detailing optimization by NSGA-II-GD for (a) beam group 400 × 300 × 6000 mm and (b) column group 300 × 300 × 3700 mm. Fig. 11. Rebar detailing of beam group 400 × 300 × 9000 mm according to different constructability scores. Fig. 12. Permutation results of rebar design optimization of all component groups in (a) beams and (b) columns. 4.3. Cost distribution in precast components concrete, reinforcement, and labor costs, across beams, columns. Notably, the concrete cost accounts for approximately half of the rein­ Fig. 14 provides a cost portion analysis of construction costs, namely forcement cost across all three components. However, the proportion 13 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 13. Final result performance comparison between random and grouped mode individual generation technique for beam group with dimensions 400 × 300 × 9000 mm. Table 3 Analysis of cost and constructability score relationship in extreme points in (a) beams, and (b) colums. (a) Beam – cost and constructability score relationship Optimal score Minimum constructability score Maximum constructability score Cost(HKD) Cost(HKD) Cost difference(HKD) Cost difference(%) Cost(HKD) Cost difference(HKD) Cost difference(%) Construction cost 119,986 120,758 772 0.64% 128,191 8205 6.84% Concrete cost 33,861 33,872 11 0.03% 33,812 − 49 − 0.14% Reinforcement cost 48,432 49,210 778 1.61% 55,227 6795 14.03% Labor cost 37,097 37,677 580 1.56% 39,152 2055 5.54% (b) Column – cost and constructability score relationship Optimal score Minimum constructability score Maximum constructability score Cost(HKD) Cost(HKD) Cost difference(HKD) Cost difference(%) Cost(HKD) Cost difference(HKD) Cost difference(%) Construction cost 85,311 86,019 708 0.83% 93,460 8149 9.55% Concrete cost 25,025 25,066 41 0.16% 24,973 − 52 − 0.21% Reinforcement cost 45,929 46,550 621 1.35% 52,262 6333 13.79% Labor cost 14,129 14,404 275 1.95% 16,225 2096 14.83% taken up by labor costs varies significantly between components. For Each algorithm was executed 600 times to search for the minimum example, labor costs represent 31% of the total construction cost for constraint value of a single beam component, and the results are pre­ precast beams, whereas they only account for 17% in columns. This sented in Table 4. The average value represents the average constraint discrepancy may be attributed to the small component size and highly value, while the average time indicates the average time required to congested rebar detailing associated with precast beam units. search for the optimal value for one single component. The success rate was calculated as the number of feasible solutions (zero constraint so­ lutions) over the total number of operations. The time per valid solution 4.4. Algorithm comparison was calculated as the success rate over the average time. The great deluge algorithm demonstrated the highest success rate of 99.84%. 4.4.1. Rationale for selecting GDA to improve the searching efficiency for Hence, it was selected to hybridize with NSGA-II to enhance the NSGA-II searching efficiency. To select an approach to improve the search efficiency of NSGA-II, six algorithms were considered in the selection procedure, namely 4.4.2. Comparative study of NSGA-II-GD to standard NSGA-II and simple hill climbing, stochastic hill climbing, late acceptance hill weighted GA climbing, simulated annealing, GDA, and threshold accepting. Sensi­ The standard NSGA-II and weighted GA are both modified versions tivity analysis was conducted to tune the parameters of all algorithms to of the GA algorithm that are capable of solving multi-objective their maximum efficiency before comparing them to other algorithms. 14 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Fig. 14. Construction cost distribution analysis for (a) beam, (b) column. Fig. 15 illustrates the aggregated Pareto front for each of the algo­ Table 4 rithms for beams and columns. For all three component categories, Performance comparison of feasible solution searching. NSGA-II-GD demonstrates better convergence than the other two algo­ Average Average Average Success Time per rithms, providing solutions with lower construction costs at the same value iteration time (s) rate (%) valid constructability score. Furthermore, the aggregated results of the two solution (s) modes of NSGA-II-GD not only include solutions with low construct­ Simple hill 0.05847 475.4 0.00870 49.00% 0.01776 ability scores but also those in high score region. In contrast, the solu­ climbing Stochastic hill 0.06510 292.0 0.22636 43.87% 0.56379 tions provided by both the standard NSGA-II and the weighted GA are climbing concentrated only concentrate in the region of low constructability score Late 0.00308 1774.3 0.01557 93.71% 0.01683 region. For instance, in the beam results, the constructability score acceptance ranges provided by the standard NSGA-II and weighted GA are 0.419 to hill 0.777 and 0.281 to 0.722, respectively, whereas the range obtained from climbing Simulated 0.00898 868.3 0.00718 87.42% 0.00837 NSGA-II-GD is from 0.446 to 1. Although the standard NSGA-II and annealing weighted GA can calculate the results in the extremely low construct­ Great deluge 0.00007 94.2 0.02967 99.84% 0.02974 ability score regions, the solution range they provide falls into the sub- Threshold 0.00177 11.9 0.01841 96.83% 0.01922 optimal region, which is not within the region in interest in finding the accepting relationship between standardization level and construction cost. Table 5 presents the computational time required by the three al­ optimization problems. Since the principles of these algorithms are gorithms in different scenarios. Each scenario was executed 10 times, similar to those of the proposed NSGA-II-GD, the same parameter set­ and the average computational time was recorded. The first row displays tings of a crossover probability of 0.75 and a mutation probability of the time required when all three algorithms use the same population and 0.25, are used. generation settings. Due to the additional GDA operator integrated into Fig. 15. Comparison of aggregated permutating Pareto front between NSGA-II-GD, standard NSGA-II and GDA for (a) beam and (b) column. 15 W.-L. Lao et al. Automation in Construction 155 (2023) 105065 Table 5 computational time, searching space, and convergence. The developed Computation time comparison of NSGA-II-GD, standard NSGA-II and weighted grouped mode demonstrates that a different initial individual generation GA under different scenarios. can achieve a wider searching space, providing more insight into the NSGA-II-GD NSGA-II-GD Standard Weighted same problem. (random (grouped NSGA-II GA Although the results demonstrate the relationship between stan­ mode) mode) dardization and construction cost, this study has several limitations. Beam: 400x300x3000mm, number of variables = 24, number of components = 11 (6 Firstly, the formulation does not account for logistics costs associated with different required areas) with precast components, such as transportation and assembly, which Time (s) are difficult to quantify and highly project-specific. This limitation may (Same parameter: population = 300, 81.639 208.021 58.907 45.949 lead to atomistic results since the precast fabrication process is more Generation = 200) complex than the fabrication of structural components on-site. Future Converged 100 100 300 300 studies can incorporate these precast-related factors to enhance the population optimization of precast design and provide a more holistic perspective. Converged generation 100 100 200 200 Secondly, only the precast manufacturing process inside the factory is Time (s) 26.081 61.623 58.907 45.949 considered, and components are simplified to focus on the demonstra­ Beam: 400x300x9000mm, number of variables = 40, number of components = 10 (10 tion between the relationship between constructability and construction with different required areas) cost. Therefore, the top longitudinal reinforcement of the beam is Converged 300 300 1200 1500 ignored. Only simple geometries such as rectangular beams and columns population Converged are studied, however this approach could be extended to various shapes, 200 200 200 500 such as T-and L-shaped beams or circular columns. Thirdly, labor costs generation Time (s) 100.321 593.603 184.420 394.2 may change in the future as the precast industry adopts more automa­ Beam: 400x300x6000mm, number of variables = 152, number of components = 45 tion for prefabrication. To account for this, machine-related costs, rather (38 with different areas) than manual operation costs, should be introduced. Converged 1500 1500 – – population Converged generation 400 400 – – Declaration of Competing Interest Time (s) 1469.679 1748.090 – – The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the NSGA-II-GD algorithm, the time required for obtaining a solution is the work reported in this paper. higher than that of the standard NSGA-II and weighted GA when sharing the same parameter setting. However, both NSGA-II and weighted GA Data availability need a higher population and generation setting to provide a converged solution. Therefore, the actual time required for a converged solution The authors are unable or have chosen not to specify which data has provided by the random mode of NSGA-II-GD to provide a converged been used. solution is half that required by the other two algorithms. When the number of variables is very large, standard NSGA-II and weighted GA Acknowledgement are unable to provide a converged solution, and the total computational time is thus not recorded in the table. The authors would like to acknowledge the support by the Hong In all, the results show that NSGA-II-GD outperforms the other two Kong Construction Industry Council, Grant No. CIC19EG03. Any opin­ algorithms in terms of convergence, searching region, and computa­ ions and findings are those of the authors, and do not necessarily reflect tional time. 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