Graph Database and Matrix-Based Intelligent Generation of the Assembly Sequence of Prefabricated Building Components

Abstract

The assembly of prefabricated components is a critical process in prefabricated building construction, influencing both progress and accuracy. However, the assembly sequence planning and optimization (ASPO) of prefabricated components have yet to receive sufficient attention from researchers, and current research has displayed limited automation and poor generalization capabilities. Therefore, this paper proposes a framework for intelligently generating assembly sequences for prefabricated components based on graph databases and matrices. The framework utilizes an adjacency matrix and interference matrix-based modeling method to comprehensively describe the connections and constraint relationships between components, enabling better evaluation of assembly difficulty during optimization. The graph database serves as the central hub for data exchange, facilitating component information storage, automatic querying, and summarization. The obtained assembly sequence and progress plan are fed back into the graph database. To accomplish assembly sequence optimization, a genetic algorithm based on the double-elite strategy is employed. Furthermore, the effectiveness of the proposed framework is validated through an actual engineering case. The results demonstrate that the framework can effectively find an optimal assembly sequence to mitigate the assembly challenge of a prefabricated building.

1. Introduction

With the pursuit of sustainable development, improving construction efficiency, and reducing energy consumption, prefabricated building systems have been recognized as one of the alternative solutions to changing conventional construction methods [1,2,3]. Prefabricated building components are a key element of prefabricated construction systems. These components are manufactured off-site in mechanized plants and then transported to the construction site for assembly instead of manually operated at sites in the traditional cast-in-situ concrete construction [4]. They include various elements such as precast walls, floors, columns, beams, and stairs. Prefabricated construction involves the production, transportation, and installation of prefabricated building components (PBCs) [5]. Furthermore, the lifting and assembly of precast components, such as precast walls and floors, is a prominent feature and an indispensable part of prefabricated building construction [6]. An appropriate assembly sequence not only can accelerate the construction progress and avoid difficulties in component assembly but also provide suggestive instruction for upstream enterprises, such as supply chain and transportation. However, most of the assembly sequence planning is determined by practitioners based on their experience. To this end, assembly sequence planning and optimization (ASPO) of precast components has been an emerging research topic.

Although ASPO in prefabricated construction is still in its infancy, efforts have been made to study the assembly or disassembly sequence planning and optimization (DSPO) in the manufacturing industry, especially for mechanical assembly bodies [7]. From the research in the mechanism domain, to tackle this ASPO or DSPO problem, the first thing is to establish a model, objectively reflecting the real problem, especially the topological relationships between components. The representation of a product structure directly affects the efficiency of an assembly sequence search [8]. Graph-based modeling [9,10], AND/OR graph modeling [11,12], Petri net (PN) modeling [13,14], and matrix-based modeling [15,16] methods are frequently adopted to describe mechanical products. However, previous studies on ASPO of prefabricated components omitted the topological relationships when modeling, although they considered the influence of interference relationships between components on assembly sequence.

In a word, there are three limitations or difficulties to be overcome in current research. (1) When establishing the model, there is a need to consider and describe the components’ topological relationships comprehensively. (2) The current process lacks the ability to automatically extract and summarize the information on components, including their basic properties and, crucially, their topological relationships. (3) The absence of a well-considered selection process for intelligent algorithms poses a challenge in determining whether a high-quality solution can be obtained within a reasonable time frame.

Hence, this research aims to tackle the challenge of attaining a high-quality assembly sequence for prefabricated building components while effectively addressing the identified limitations above. To achieve this objective, a comprehensive framework is proposed. Firstly, a matrix-based modeling method is applied to express the topological relationships between components, where an adjacency matrix and interference matrix are adopted. Secondly, a graph database is employed to store the massive amounts of information in the BIM model, particularly the spatial topological relationships of components. This not only makes information retrieval and interaction convenient but also lays the data foundation for the matrix-based modeling method. And the subsequent construction information can also be extended into the graph database. Last but not least, a genetic algorithm is chosen from multiple intelligent algorithms to optimize the assembly sequence, improved by the elite reservation strategy and elite crossover strategy. The assembly sequence generated by a scientific and reasonable methodology framework on the mathematical and logical foundation might be more reliable than the one determined solely by the construction practitioners’ experience.

In this study, the establishment of a mathematical model for intelligent assembly sequence generation based on matrices is presented in Section 3. Section 4 highlights the role of the graph database as the information interaction center for component and construction information transmission. Section 5 elaborates on how the genetic algorithm intelligently optimizes the assembly sequence. In Section 6, a case study assesses the feasibility and rationality of the proposed framework while also comprehensively comparing the performance of the overall optimization method and the grouping optimization method.

2. Literature Review

2.1. Product Representation and Modeling Methods in Manufacturing

The mainstream product representation and modeling approaches of ASPO and DSPO in manufacturing can be categorized into four varieties: graph-based modeling, AND/OR graph modeling, Petri net (PN) modeling, and matrix-based modeling methods, respectively. The graph-based modeling is a method that utilizes a directed or undirected graph composed of nodes and edges to depict the precedence relationship among components, mainly based on graph theory [8]. Tseng et al. [17] implemented a liaison graph to express the connection relationships between parts of a fixed support holder. Zhang et al. [18] employed the disassembly hybrid graph to describe the hierarchical information and constraint relations between the product components. Different from the graph-based modeling, the AND/OR graph is applied to graphically represent all possible disassembly sequences of the given disassembly operations or components/parts [12]. Tian et al. [19] used the AND/OR graph to handle a novel dual-objective disassembly problem from economic and energy-efficient perspectives, and the effectiveness has been validated. The Petri net is a graphical and mathematical technique suitable for modeling and analyzing information processing systems [20]. With an initial state called the initial marking, the directed graph consists of two kinds of nodes, called places and transitions, which are connected via arcs [21]. Tang and Grochowski [14] combined the disassembly Petri Net with a hybrid Bayesian network to model the disassembly process and predict the outcome of each disassembly action.

Generally, graph-based product descriptions need to be converted into matrix-based representations for computer processing [8]. The matrices depict the characteristics of a product from three perspectives: precedence relationships, connection relationships, and interference relationships among components. Smith et al. [22] adopted five matrices: a contact constraint matrix for components, a motion constraint matrix for components, a contact constraint matrix for fasteners, a motion constraint matrix for fasteners, and a projection matrix for components. Gulivindala et al. [23] put up with an omnidirectional geometrical feasibility matrix to represent the conditions in the actual disassembly environment. Wang et al. [24] proposed the ‘assembly matrix’ as a fundamental tool to interpret a physical assembly and then derived three new matrices: a contact matrix, a space interference matrix, and a relation matrix.

Compared with a mechanical product, the ASPO of prefabricated components is less complex due to the topological relationships among prefabricated components being solely formed by connection without precedence. An unreasonable assembly sequence might increase installation difficulty for precast components, but for a mechanical product, it may cause assembly failure and rework. Therefore, with appropriate simplification and modification, these modeling methods are competent to deal with the assembly in prefabricated construction. By contrasting the four modeling methods, the matrix-based modeling method is the most computer-friendly and more suitable for prefabricated components.

2.2. ASPO of Prefabricated Components

Wang et al. [7], Huang et al. [25], and Liu et al. [26] have studied the issue of ASPO for prefabricated components successively. Wang et al. [6] proposed a new method called the BIM-IGA-based ASPO method, which used BIM for parametric modeling, used IGA to search for an optimal assembly sequence, and then used BIM again for visual simulation to further test the assembly sequence. Combining the lifting path planning with assembly scheduling in a two-in-one approach, Huang et al. [25] drew upon a BIM4D timeframe and a Muti-level Elitist Genetic Algorithm (MEGA). Liu et al. [26] proposed a new simulated annealing genetic algorithm to optimize the assembly sequence, with the research object of four kinds of standardized prefabricated outer panel envelope panels. And mechanical simulation technology is innovatively used to simulate the assembly of building components.

In terms of mathematical modeling to reflect the ASPO problem, they coincidentally chose the penalty function method to solve this multi-objective optimization problem. Wang et al. [7] selected the weight, occupied space, and interference between the components’ three attributes to evaluate the quality of an assembly sequence. On the basis of Wang et al.’s research, Huang et al. [25] added a new index: path interference. Except for the weight of components and the interference between components, Liu et al. [26] put the quantity and process installation time of components into consideration.

When considering the impact of interference relationships, Wang et al. [7] and Liu et al. [26] adopted a formula to calculate it quantitatively, as shown in Equation (1). Nevertheless, the authors did not mention how to determine these values, which makes the formula challenging to apply in practice. Huang et al. [25] defined the logical conditions that should not be violated as far as possible to avoid any assembly difficulty by presenting six cases of layout interference, which is effective for a specific engineering case. Table 1 shows the comparison and summary of previous research.

$$C={\left(c_k\right)}_{1\times m}={\left(\frac{z_k}{z_{k0}}\right)}_{1\times m},k=1,2,\dots ,m.$$

(1)

where array C stores the interfered components and c_k is any penalty value of the array C. z_k represents the installation difficulty of the k-th component in the C under interference condition. z_k0 represents the installation difficulty of the k-th component in the C under normal condition. m represents the number of interfered components.

In a word, though previous works considered the impact of interference relationships on assembly sequence planning, there is yet to be one mathematical model describing the topological relationships of components precisely and comprehensively. And there is no method with a strong generalization ability to calculate the influence of interference relationship with friendliness to the computer. Except for these, they manually summarized the information of components into a table, which is laborious when the scale is enormous.

Table 1. Comparing previous research on prefabricated components assembly sequence issues.

Author	Selected Parameters	Optimization Algorithm	Information Summary Method	Contributions	Limitations
Wang et al. [7]	Weight, occupied space, interference between components	Improved genetic algorithm (IGA)	Manually	The first to study ASPO of precast components; establish an equation to calculate interference penalty.	The calculation process of interference penalty is not explicit.
Huang et al. [25]	Weight, occupied space, layout interference, path interference	Multi-level elitist genetic algorithm (MEGA)	Manually	Take the assembly scheduling and lifting path planning two issues in a two-in-one approach.	The six layout cases are difficult to extend to other application cases.
Liu et al. [26]	Weight, quantity, interference between components, process installation time	Simulated annealing genetic algorithm	Manually	Introduce the evaluation index of mechanical assembly process and mechanical simulation technology.	Only consider standardized prefabricated outer panel envelope panels.

Table 1. Comparing previous research on prefabricated components assembly sequence issues.

Author	Selected Parameters	Optimization Algorithm	Information Summary Method	Contributions	Limitations
Wang et al. [7]	Weight, occupied space, interference between components	Improved genetic algorithm (IGA)	Manually	The first to study ASPO of precast components; establish an equation to calculate interference penalty.	The calculation process of interference penalty is not explicit.
Huang et al. [25]	Weight, occupied space, layout interference, path interference	Multi-level elitist genetic algorithm (MEGA)	Manually	Take the assembly scheduling and lifting path planning two issues in a two-in-one approach.	The six layout cases are difficult to extend to other application cases.
Liu et al. [26]	Weight, quantity, interference between components, process installation time	Simulated annealing genetic algorithm	Manually	Introduce the evaluation index of mechanical assembly process and mechanical simulation technology.	Only consider standardized prefabricated outer panel envelope panels.

2.3. BIM, IFC and Graph Database

BIM (Building Information Model) is an internationally defined term that refers to a “shared digital representation of physical and functional characteristics of any built object which forms a reliable basis for decisions” [27]. Generally, data-sharing issues in BIM projects are resolved using a standardized file exchange format, namely, Industry Foundation Classes (IFCs) [28]. However, the rigid and complex hierarchical structure of the IFC schema poses difficulties in manually extracting building information, as it requires a deep understanding of the IFC object model [19].

Graphs, consisting of a finite set of nodes and edges, provide a homogeneous way to encode spatial and non-spatial information of different types, which makes them a suitable representation for complex relationships among building elements and data in Building Information Models [20]. Converting BIMs based on the IFC standard into a graph-based information retrieval model can significantly facilitate exploring and analyzing highly connected BIM data [21]. The graph database is a database that stores information in the form of a graph. Compared with other types of database systems, it is more effective in representing relationships. In contrast, interconnected information full of relationships is a feature of IFC data, whereas IFC data can be thought of as an implicit graph consisting of interconnected objects [29].

Vicedo et al. [30] thought that the increase in the number of publications on Enterprise Resource Planning Systems (ERPs) and Critical Success Factors (CSFs) is due to the continuous digitalization of the business world and the need for quality data in order to make faster decisions. The aforementioned reason can similarly account for the advancement of graph database applications in the field of civil engineering. Aided by the IfcWebServer, a web-based server application that supports the management and sharing of IFC (Industry Foundation Classes) models, Ismail et al. [31] extracted the building’s topological structure and attributes of its components from IFC models, stored the information in a property graph database, and then the required knowledge was extracted by querying it. To tackle the challenge of integrating monitoring data with building information models, Gradišar and Dolenc [32] employed a graph-based database management system and the IFC standard to support the need for interoperability and collaborative work. Zhu et al. [29] applied the graph-theory-based graph database technology to reveal hidden relationships within building information, enabling a complete conversion of IFC data into labeled property graphs. Dong et al. [33] used a graph database to store and twin the IFC-based information model. This graph database-based analysis of the model automatically recognized the engineering case’s critical path information, delay analysis information, and schedule network analysis information.

From the above, it can be concluded that previous research has provided a mature solution to transform the information from the IFC model to the graph database. However, the subsequent application of the graph database is not well-developed.

3. Matrix-Based Establishment of Mathematical Model

Based on the adjacency matrix and interference matrix, a precise and comprehensive description of spatial topological relationships between components can be achieved. Subsequently, the optimization function is proposed with the help of the penalty function method to solve this multi-objective problem. By leveraging the Critical Success Factors (CSFs) [30], we select the three types of factors of weight, occupied space, and interference relationships to evaluate the quality of an assembly sequence. And the detailed calculation methods corresponding to the three penalties are given below.

3.1. Matrix-Based Description of Topological Relationships between Components

3.1.1. Adjacency Matrix

As previously mentioned, significant progress has been made in the field of assembly and disassembly sequence planning for mechanics with well-established theories. This paper introduced the component-fastener graph [34] and adjacency matrix as auxiliary tools to describe the connections between components.

Component-fastener graph G = (V, E) represents the components of a product and their relationships. The vertices V = {v₁, v₂, v₃, …, v_n} represent the components, where n is the number of components. Meanwhile, the edges E = {e₁, e₂, e₃, …, e_m}, where m is the number of edges, represent the relationships between components. If two components v_i and v_j (i ≠ j) are assembled, then (v_i, v_j) ∈ E; otherwise, (v_i, v_j) ∉ E. If i = j, then (v_i, v_j) ∉ E.

Fasteners can be neglected for the assembly of prefabricated wall components, and the component-fastener graph can be simplified as the assembly linking graph [35] which is similar to the liaison graph [36]. Although assembly linking graph is intuitionistic and comprehensible, it is more convenient for the computer to process the information if the graph is converted into an adjacency matrix. Let M = [m_ij] be G’s adjacency matrix. The adjacency matrix M is defined as follows:

$$M=\left. [\begin{array}{cccc}{m}_{11}& m_{12}& \cdots & m_{1n}\\ m_{21}& m_{22}& \cdots & m_{2n}\\ ⋮& ⋮& \cdots & ⋮\\ m_{n1}& m_{n2}& \cdots & m_{nn}\end{array}\right]$$

(2)

where

$$m_{ij}=\left\{\begin{array}{l}1, \mathrm{if} \mathrm{component} i \mathrm{is} \mathrm{connected} \mathrm{to} \mathrm{component} j\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$$

Taking a group of three walls as an example, Figure 1 displays the corresponding linking graph and adjacency matrix of the three walls.

3.1.2. Interference Matrix

The interference matrix is a tool to express the layout interference and degrees of freedom during assembly in a matrix format. Shuang et al. [37] proposed a method to generate a homologous moving wedge with the interference matrix. Firstly, they defined the problem: let P = {p₁, p₂, p₃, …, p_n} denote the set of parts of an assembly A. Then they defined a set of valid assembly directions for each pair of (p_i, p_j) as a moving wedge [38] of p_i to p_j, denoted by MW(p_i, p_j) = {d_+x, d_-x, d_+y, d_-y, d_+z, d_-z}, where:

$$d_{*}=\left\{\begin{array}{l}1, \mathrm{if} p_i \mathrm{can} \mathrm{be} \mathrm{assembled} \mathrm{together} \mathrm{in} \mathrm{direction} *\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$$

(3)

when assembling prefabricated wall components, the interference along the directions of +z and −z can be ignored. A hidden premise of MW(p_i, p_j) is that component j is assembled before component i, and component j is connected to component i. If component i and component j have no connection relationships, the elements in MW(p_i, p_j) are all zero. From a computer-friendly perspective, dividing the moving wedge into four interference matrices corresponding to four directions is better.

Table 2. The moving wedge table of components in Figure 1.

	pj
pi		1	2	3
	1		0, 1, 1, 1
	2	1, 0, 1, 1		0, 1, 1, 1
	3		1, 0, 1, 1

and Figure 2, respectively, list the moving wedge and interference matrices of the example in Figure 1.

Based on the classified discussion of the relative position between two components, such as the coordinate, length, and extension direction of walls, the interference matrix is easy to obtain.

3.2. Optimization Function

According to practical engineering experience, it can be determined that walls possess more complex spatial topological relationships than other prefabricated components, which have the most significant impact on the overall assembly task duration. So this paper focuses on determining the assembly sequence of a group of wall components within a subprocess rather than considering all wall components in an engineering project.

Optimizing the assembly sequence for prefabricated components fundamentally involves identifying the optimal order for a group of components. Based on the previous study [7,25], this study utilizes a three-pronged approach to assess the quality of the assembly sequence in terms of weight, occupied space, and interference relationships. In order to address this multi-objective sequence optimization problem, the penalty function method is employed to establish the objective function. This is achieved by assigning relative weight values to each sub-problem and combining them linearly. And the final optimization function is represented by Equation (4).

$$P_{total}=\kappa {\displaystyle \sum P_{weight}+}\lambda {\displaystyle \sum P_{space}+}\mu {\displaystyle \sum P_{layout}}$$

(4)

where P_weight, P_space, and P_layout represent the weight penalty, occupied space penalty, and interference relationship penalty, respectively. κ, λ, and μ are the relative weight values reflecting the importance of the penalty. P_total is the total penalty value.

3.3. Calculation of Weight Penalty

The weight penalty principle states that the component with the larger weight should be assembled preferentially. If this principle is violated, a weight penalty will be given. All the weight penalties are stored in matrix W, whose matrix element is w_ij.

If component i is assembled before component j, w_ij will be calculated as Equation (5) [7,25,26]. As indicated in Equation (6), the total weight penalty (∑P_weight) for a specific sequence can be calculated by aggregating the penalty values (P_weight) corresponding to all the components in the sequence.

$$w_{ij}=\left\{\begin{array}{l}weigh{t}_j/weigh{t}_i,\mathrm{if} weigh{t}_j>weigh{t}_i\\ 0 \mathrm{if} weigh{t}_j\le weigh{t}_i \end{array}\right.$$

(5)

$${\displaystyle \sum P_{weight}={\displaystyle \sum _{j=1}^n{(P_{weight})}_j}}={\displaystyle \sum _{j=1}^n({\displaystyle \sum _{i=1}^j{w}_{ij}})}$$

(6)

where weight_i and weight_j represent the weight of the ith and jth components, respectively. (P_weight)_j represents the weight penalty value of the jth component in a sequence.

The parameter n in Equation (6) is the total of the components.

3.4. Calculation of Occupied Space Penalty

The principle regarding the occupied space penalty is that the component with the larger occupied space should be assembled preferentially. Account that components with a larger occupied space need more steps and motions when hoisting and are more prone to collision with installed components. Therefore, it is a safer and more efficient approach to assemble the larger components first. A violation of this principle incurs an occupied space penalty. These penalties are recorded in matrix S, with its elements represented as s_ij.

If component i is assembled before component j, s_ij will be calculated as Equation (7). The total space penalty (∑P_space) of a certain sequence can be obtained by summing up the penalty values (P_space) of all the components in the sequence, as indicated in Equation (8).

$$s_{ij}=\left\{\begin{array}{l}spac{e}_j/spac{e}_i,\mathrm{if} spac{e}_j>spac{e}_i\\ 0 \mathrm{if} spac{e}_j\le spac{e}_i \end{array}\right.$$

(7)

$${\displaystyle \sum P_{space}={\displaystyle \sum _{j=1}^n{(P_{space})}_j}}={\displaystyle \sum _{j=1}^n({\displaystyle \sum _{i=1}^j{s}_{ij}})}$$

(8)

where space_i and space_j represent the occupied space of the i-th and j-th components, respectively. (P_space)_j represents the space penalty value of the j-th component in a sequence.

The parameter n in Equation (8) represents the total number of the components.

3.5. Calculation of Interference Relationship Penalty

To address the calculation of the interference relationship penalty, this paper utilizes an adjacency matrix and an interference matrix to describe the connections and constraints between components. The complexity of assembly is evaluated based on the number of components connected to the current ones and the degree of freedom for movement. By implementing appropriate amplification coefficients, we can ultimately determine the interference relationship penalty.

The adjacency matrix and interference matrix are static and not influenced by the assembly sequence of prefabricated components. However, the interference penalty is a dynamic figure that depends on different assembly sequences. There are two decisive factors for the interference penalty: the number m of components connected with the target component and assembled before the target component and the number n of free-to-move directions considering the layout interference brought by m components.

(1)

m: the number of components connected with the target component and assembled before the target component

The value of m can be easily obtained from the adjacency matrix when an assembly sequence is given. If m is equal to 0 or 1, then there is no significant difficulty in assembling the target component. When m is equal to or greater than 2, the interference penalty needs to be calculated, and m will influence the value of n by restricting the maximum of n. It is evident from engineering experience that the difficulty of assembly increases as m increases.

(2)

n: the number of free-to-move directions considering the layout interference brought by m components

The number of free-to-move directions n depends on the value of m, and different circumstances need to be considered for each value of n.

Table 3. Examples of different combinations of m and n.

	n	0	1	2	2	3
m
2
				±y	−x, +y	−x, ±y
3
			+y	±y	−x, +y	−x, ±y
4
		/	+y	±y	−x, +y	−x, ±y

lists some examples of different combinations of m and n with the free-to-move directions of target components. The difficulty of assembly increases with the decrease of n.

As mentioned above, the interference penalty changes dynamically with different assembly sequences. For a given assembly sequence π_k, the assembly wedge AW(p_i) can be obtained through Equation (9).

$$AW(p_i)={\displaystyle \underset{j}{\cap }MW(p_i,p_j)}$$

(9)

where p_j is the component assembled before p_i with the given assembly sequence π_k. The elements in AW(p_i) are q₁, q₂, q₃, and q₄, which correspond to four directions, and n is the sum of q₁ to q₄.

(3)

Calculation formula of interference penalty

As depicted in Equation (10), the interference relationship penalty of the j-th component (P_layout)_j is computed by multiplying the amplification factor s with the previously introduced m. Equation (11) demonstrates that the sum of the P_layout values for all elements in the assembly sequence yields the total value (∑P_layout) of that sequence. The optimal assembly sequence is obtained by eliminating the assembly sequences with high assembly difficulty.

$$(P_{layout}{)}_j=m\times s$$

(10)

$${\displaystyle \sum P_{layout}={\displaystyle \sum _{j=1}^n{(P_{layout})}_j}}$$

(11)

where s represents the amplification factor to distinguish the assembly difficulty of different values of n. Here, n is the number of free-to-move directions considering the layout interference brought by m components. m is the number of components connected with the target component and assembled before the target component.

The parameter n in Equation (11) represents the total number of the components.

4. Graph Database-Based Information Interaction

The graph data-based information interaction part functions as the data preprocessor and result updater, shown in Figure 3. The input of the preprocessor is the BIM, and the output is a set of consolidated information of components’ attributes, laying the data foundation for the subsequent optimization. The result updater can add the construction micro-schedule and assembly sequence to the graph database, realizing the information update and completion.

The IFC file, directly exported from Revit, excludes the information related to the construction schedule. With the help of the xBIM toolkit, we can modify the original IFC file by implementing the construction information extension. Then a set of rules is established to parse and twin the IFC file into the graph database, which is essential to reveal the IFC files’ complicated inner relationships in an intuitive graph and increase data interoperability for further requests [39] The required information can be extracted, summarized, and stored in arrays by customizing query statements. There are three types of information, one pertaining to components’ geometric data, including the ID, length, width, height, and true volume. Figure 4 shows an example of the information query process. The second type concerns the direction information of components, including the ID, coordinates, and extension direction. The third type describes the connection relationships between components by noting all the connected component pairs. As shown in Figure 5, the expression of connection relationships in Neo4j accurately twins the connected components in the BIM model.

On account that the IFC standards have a specialized expression to describe the construction schedule, the initial extension of the IFC file can ensure that the construction information update in Neo4j still obeys the semantic structures of IFC standards, shown in Figure 6. The result updater fills the scheduling information in the corresponding nodes and edges, such as IfcTask and IfcTaskTime.

5. Optimization Algorithm—Genetic Algorithm Based on Double-Elite Strategy

5.1. The Selection and Comparison of Optimization Algorithm

From the mathematical perspective, optimizing a prefabricated components assembly sequence is an unconstrained multi-objective optimization problem. From another perspective, it is to seek the optimal plan of integer sorting. The meta-heuristic algorithm performs excellently in the assembly sequence, such as ant colony optimization [40,41], particle swarm optimization [42,43,44], and genetic algorithm [23,35,38,45].

Genetic Algorithm (GA), characterized by a gene–chromosome structure, is an expert in dealing with integer sorting problems. A gene on the chromosome can represent an integer, and the position of the gene on the chromosome can stand for the order of an integer in a group of numbers, eliminating the process of encoding and decoding. Therefore, this paper chose GA as the optimization algorithm.

5.2. Genetic Algorithm Based on Double-Elite Strategy

To improve the efficiency of finding the optimal individual, modifications to the traditional genetic algorithm are necessary. One effective method is to implement an elite strategy, including the elite reservation strategy and the elite crossover strategy. An elite individual has the highest fitness during the population evolution and shares partially excellent genes [46].

The elite reservation strategy involves directly reserving the individual with the best fitness for the next generation, thereby bypassing the genetic operations of the current generation. This approach prevents the excellent genes of the current generation from being damaged during crossover and variation, yet this may cause the result to get stuck in a locally optimal solution. The elite crossover strategy is making the elite cross with every individual in the current population according to a given probability [47]. It allows the excellent genes to be possessed by more offspring individuals, while the elite individual is not fully preserved.

To improve the performance of the genetic algorithm, this paper adopted a double-elite strategy, as detailed in Figure 7. This approach involves using both elite reservation and elite crossover strategies simultaneously. Specifically, the best individual in the current generation is directly reserved for the next generation while also being allowed to mate with every individual in the population. This approach balances the advantages and disadvantages of the two strategies, resulting in a more effective genetic algorithm.

5.2.1. Initialization of Population

Selecting the appropriate encoding method is crucial during initialization. Standard encoding methods include binary, real, and gray codes. To simplify the encoding process, this study employed integer encoding to directly obtain the biological characteristics based on the genotype.

In this study, a chromosome represents a feasible solution, with the number of genes in the chromosome equal to the sum of prefabricated components. The ID of each component is similar to an individual’s identification card, which typically consists of an integer of at least five digits, causing unnecessary workload during sequencing. Therefore, this study sorted the components in ascending order based on their ID numbers and assigned serial numbers to replace the component’s inherent ID as the new ID. Figure 8 depicts a chromosome that adopts integer encoding and represents an assembly sequence.

5.2.2. Fitness Function

Fitness is a primary target to assess individual performance and serves as the foundation for the genetic manipulation of superiors to win and inferiors to be eliminated. This paper applies a penalty function to construct an unstrained optimization function based on the original function and constraint conditions. The optimization function aims to minimize the P_total, and thus the individual with the highest fitness should be the one with the minimum P_total. The fitness function of the assembly sequence is shown in Equation (3).

5.2.3. Selection of Genetic Operators

In genetic algorithm optimization, it is crucial to determine the appropriate genetic operators in advance. The selection, crossover, and mutation operators are the critical genetic operators that must be chosen carefully. In this paper, the authors utilized roulette wheel selection, partially matched crossover, and displacement mutation to optimize the integer sequencing problem.

Roulette wheel selection is a widely used strategy that calculates the probability of each individual appearing in the next generation and selects individuals randomly based on this probability. To mitigate the randomness of roulette wheel selection and ensure that the optimal individual in the current generation is not overlooked, an elite reservation strategy is also implemented. And we choose partially matched crossover (PMC) and displacement mutation to address the integer sequencing optimization problem. They enable us to ensure that the new individuals generated are still feasible solutions, where all integers are within the specified range and cannot be repeated.

In practice, four key factors still need to be determined for the genetic algorithm: population size, iteration times, crossover rate, and mutation rate.

6. Case Study

The BIM-based and graph database-based framework has been validated through an engineering case of prefabricated buildings. The selected project is a residential community in Shanghai, China, which includes several residential buildings. The research focused on a group of wall components in the 3# building, which has 16 floors, with a standard construction area of 593.04 m² and a total height of 49.2 m, as depicted in Figure 9. The complete set of wall components on a particular floor is shown in Figure 10. Considering the division of construction flow sections and the symmetrical layout of building components, a group of wall components from Figure 10 has been chosen as the research object. Figure 11 presents the BIM of the research object.

For intelligent assembly process planning, grouping optimization is a method that has been employed previously. Qian et al. [48] constructed an automatic assembly system by dividing the assembly planning into within-group planning and between-group planning. In this case, there are 34 prefabricated wall components with varying thicknesses. Considering that components’ building or structural functions vary according to their thickness, components may be delivered to the site in batches according to their thickness. So this paper employed an overall optimization method and a grouping optimization method, dividing 34 wall components into three subgroups according to their thickness. We assume that the sequence between the subgroups is as follows: the 270 mm subgroup is first, followed by the 200 mm subgroup, and lastly, the 100 mm subgroup. This ordering of subgroups is considered relatively ideal because the initial assembly of the 270 mm exterior wall panels facilitates the subsequent erection of scaffoldings while also adhering to optimization principles prioritizing the assembly of larger, heavier components first. The results and performances of the two methods are compared below.

6.1. System Specifications

The case study is performed on a PC with a 1.60 GHz Intel® Core(TM) i5-8250U CPU, 8 GB installed memory (RAM), an NVIDIA® GeForce MX150 graphics card, and the Microsoft Windows 10 operating system.

6.2. Realization of Optimization

Considering Section 4 has already elaborated on the details of data preparation, and Figure 4 shows an example of consolidated data, we skip the repeated description of data preparation.

In order to apply the optimization method to a specific project, two sets of parameters need to be determined. The first set of parameters involves the three weight coefficients of the three penalties, as shown in Equation (3). The second set of parameters is the genetic algorithm parameters, which include population size, iteration times, crossover rate, and mutation rate.

6.2.1. Determination of Weight Coefficients in Penalty Function

As mentioned in Equation (3), the weight coefficients κ, λ, and μ correspond to the weight penalty, occupied space penalty, and interference penalty, respectively. The weight coefficient determines the relative importance of each indicator in the optimization process. Since the calculation methods of the three penalties are different, their values may vary significantly.

Considering that the difficulty of the assembly sequence is determined mainly by component connection interference, the optimization rule of interference penalty should be given priority. This means we aim to minimize the interference penalty as much as possible. Therefore, the basic principle for determining the weight coefficient is to optimize the weight penalty and the occupied space penalty on the premise of having minimized the interference penalty. For this reason, the initial value of μ is set slightly higher and then gradually reduced until the value of the component interference penalty reaches its critical state at the minimum. This approach ensures that all three optimization rules play their respective roles effectively.

6.2.2. Determination of Genetic Parameters

The success of the genetic algorithm optimization is dependent on the setting of various parameters, including population size, iteration times, crossover rate, and mutation rate.

The size of the population is positively correlated with the length of the chromosome. However, an appropriate population size should not only ensure the diversity of individual genes but also consider the time cost of program operation.

In this study, iteration times are set as the termination condition of the genetic algorithm, and this parameter dramatically affects the time required to obtain the optimization results. The setting of iteration times should strike a balance between effect and efficiency. If the iteration times are too small, the total penalty value may not reach the minimum before the algorithm terminates, and if it is set too large, the running time cost would be too high.

The crossover rate determines the global search ability of the genetic algorithm, and the mutation rate determines the local search ability of the genetic algorithm. Hence, the appropriate combination rate and mutation rate significantly affect the optimization performance of the genetic algorithm. However, there is yet to be a complete and feasible theory to guide how to determine the combination rate and mutation rate. The most reliable method is to determine the rates by parameter adjustment tests.

Although 34 components are divided into three subgroups according to the thickness, the optimization of these subgroups was not independent and co-occurred. Therefore, when optimizing the assembly sequence of a subgroup of components, the influence of previous subgroups must be taken into consideration. While the calculation of the weight penalty and the occupied space penalty is independent of previous subgroups, the interference penalty is influenced by all the assembled components, especially those in the earlier subgroups. As a result, this penalty requires greater attention. The parameters of the three subgroups and the group with total components are presented in

Table 4. The parameters of subgroups.

Subgroup	Number of Components	Pweight	Pspace	Playout	Population Size	Iteration Times	Crossover Rate	Mutation Rate
270 mm	13	1	1	10	100	500	0.6	0.3
200 mm	15	1	1	20	300	1000	0.6	0.3
100 mm	6	1	1	5	50	300	0.6	0.3
Overall	34	1	1	25	400	1200	0.6	0.3

.

6.3. Optimization Results

Based on set-up parameters, the optimization process can be triggered smoothly. However, due to the possibility of the genetic algorithm getting trapped in local optima, the optimization process should be repeated independently multiple times to obtain the optimum assembly sequence. The iteration course shown in Figure 12 indicates that the total value decreases rapidly in the beginning and then gradually until it becomes stationary in the later stages, which can be considered an indication of iteration convergence.

The optimized three-stage assembly sequence is depicted in Figure 13, with blue, yellow, and pink colors assigned to distinguish between different subgroup components. By aligning the assembly sequence of the three subgroups, the final optimal assembly sequence of 34 wall components is obtained, as indicated in black font in Figure 14. And the red font denotes the assembly sequence result of optimizing the 34 components together.

Table 5. Penalty of three subgroups.

Subgroup	Pweight	Pspace	Playout	Ptotal
270 mm	6.68	12.27	0	18.95
200 mm	26.17	11.07	0	37.24
100 mm	0	0	6.5	32.5
Grouping	170.63	50.09	6.5	383.22
Overall	135.81	135.23	2.1	323.54

summarizes the penalties for the three subgroups and their corresponding grouping and overall optimization methods.

6.4. Discussion

As observed from the iterative curves depicted in Figure 12, the penalty value of the optimal individual in the final population demonstrates significant optimization when compared to the average penalty value of the initial population, which is produced randomly. Notably, there is a substantial 6.5 times reduction in penalty value for the overall grouping method. Furthermore, when optimizing the 100 mm, 200 mm, and 270 mm groups individually, the penalties decrease by factors of 4.6, 6.4, and 3.3, respectively. These results highlight the considerable improvements achieved through the optimization process. Moreover, the results further demonstrate that the optimization process remains uncompromised even with the increase in computational cost resulting from a larger number of building components. In fact, it is observed that the overall optimization method, which involves the highest number of components, exhibits the most significant decrease in penalty value. This finding attests to the robustness of the intelligent algorithm employed in this study, affirming its capability to handle complex scenarios and deliver effective optimization results.

When comparing the performance of the overall optimization and grouping optimization, the following three points can be found: (i.) The grouping or overall optimization methods offer basically the same treatment for the local connection relations, as shown in the two examples in Figure 15; (ii.) The grouping optimization achieves the optimization of the interference penalty value in the grouping state, which may not be globally optimal. But the overall optimization strictly adhered to the optimization criteria of minimizing layout interference as much as possible, achieving the global optimum of the interference penalty value; (iii.) By comparing the total penalties, it can be observed that the overall optimization method outperforms the grouping optimization method, although the overall optimization requires more computational resources.

In general, we recommend prioritizing the overall optimization method because the layout interference between components significantly impacts the assembly difficulty and should be avoided. And this viewpoint guides the entire optimization process. Indeed, the results of grouping optimization are also acceptable when there are situations where overall optimization cannot be realized.

Once the construction schedule is settled, it can be imported into Microsoft Project 2016 to generate a Gantt chart. Navisworks 2020 offers a platform to integrate the construction schedule with the BIM model to create an animated construction process. The schedule information is also utilized to update the information in the graph database, thereby elevating the stored 3D BIM to a 4D model, resulting in a bidirectional flow of information in the graph database. Figure 16 provides visual evidence of this process.

7. Conclusions

This study develops a graph database and matrix-based intelligent generation framework for the assembly sequence of prefabricated components. This framework achieves three objectives: (1) The information interaction center based on the graph database is proposed to address the challenges of automated information query, summary, and retrieval. This center serves as a platform to receive the optimized assembly sequence and corresponding construction schedule. Utilizing the graph database enhances the adequacy of information utilization and provides convenient interactivity. (2) The assembly problem modeling method based on the adjacency matrix and interference matrix is established to describe the connection relationships between components. Moreover, weight, occupied space, and interference relationship are identified as critical success factors determining the assembly sequence of prefabricated wall components. Through this modeling approach, the framework provides a comprehensive understanding of the assembly process, enabling efficient decision-making in generating assembly sequences. (3) The double-elite strategy genetic algorithm improves the performance of the traditional genetic algorithm, making the optimization algorithm more adaptive to deal with the assembly sequence problem. In addition, the case study conducted demonstrates the effectiveness of this framework in intelligently generating assembly sequences and reducing the difficulty of assembly during prefabricated building construction.

In general, the contributions of this framework to the development of construction schedule optimization are threefold. Firstly, this framework verifies the feasibility of transferring the data center from the BIM to the graph database, making information utilization more adequate and interactivity more convenient. Secondly, it offers a powerful tool for inexperienced practitioners to accomplish the assembly task efficiently, smoothly, and safely by providing an intelligently optimized assembly sequence. Furthermore, this research provides a valuable basis for future research on the optimization of autonomous tower crane operations. By integrating the optimized assembly sequence and construction schedule, the framework establishes a solid foundation for the development of intelligent crane control systems, which could potentially increase safety, lower operational costs, and reduce the overall construction time.

Based on the results of this paper, further work may include the following. (1) In this paper, the influence of the sequence of component production and arrival is simplified and idealized. Future work could investigate how to put a component supply chain into consideration to make the construction schedule more practicable and feasible. (2) Although currently all the research has unanimously chosen the genetic algorithm as the optimization algorithm, there is potential to explore the use of machine learning algorithms for assembly sequence optimization and compare the results with traditional optimization algorithms. (3) The framework developed in this study could be expanded to include the scope of modular building construction, which depends primarily on the assembly task.