Production Sequencing and Layout Optimization of Precast Concrete Components under Mold Resource Constraints

Abstract

Precast concrete components have attracted a lot of attention due to their efficient production on off-site production lines. However, in the precast component production process, unreasonable production sequence and mold layout will reduce production efficiency and affect the workload balance between each process. Due to the multi-species and small-lot production characteristics of precast concrete components, the number of molds corresponding to each precast concrete component is generally limited. In this paper, a production sequence and layout optimization model for assembling precast concrete components under a limited number of molds is proposed, aiming to improve the comprehensive utilization efficiency of the mold tables and balance the workload between each production process of precast components. In order to obtain a better production sequence and a richer combination of mold layout schemes, a multi-objective teaching-learning-based optimization algorithm based on the Pareto dominance relation is developed, and an enhancement mechanism is embedded in the proposed algorithm. To verify the superior performance of the enhanced teaching-learning-based optimization algorithm in improving the comprehensive utilization efficiency of the mold tables and balancing the workload between various processes, three different sizes of precast concrete component production cases are designed. The research results show that the proposed model and optimization algorithm can help production managers to efficiently formulate more reasonable precast component production sequence and layout schemes, especially for those enterprises that are struggling to improve the efficiency of precast concrete component production.

1. Introduction

Prefabricated buildings are buildings assembled on construction sites using precast concrete components (PC), which is not only conducive to the saving of resources and energy, reduction of construction pollution, enhancement of labor productivity and quality and safety level, but also conducive to the promotion of the development of the traditional construction industry with the deep integration of information technology and industrialization, as compared with the traditional on-site pouring construction method [1]. The Chinese government strongly encourages the development of prefabricated buildings; for example, the assembly rate of prefabricated buildings in Sichuan Province is targeted to reach more than 50% of buildings, accounting for 40% of new buildings in 2025. However, since the development of prefabricated buildings in China is still in the preliminary stage, a complete design standard system for prefabricated buildings has not yet been established, and there are various forms and types of components. A survey shows that prefabricated buildings cost about 300 to 500 yuan per square meter higher than traditional cast-in-place buildings [2], which is very unfavorable for the application and promotion of prefabricated buildings, thus arousing great concern and interest from government authorities and researchers. PC production is an important part of prefabricated buildings, and its production organization and production efficiency also directly affect the construction progress and total cost of prefabricated buildings [3,4,5].

Given the sustained high demand for PC in the construction industry and the increasingly fierce market competition, optimizing the production sequence and mold layout of PC has become more and more critical and urgent. The production process of PC heavily relies on mold technology, exhibiting distinct, discrete manufacturing characteristics [6]. A reasonable PC production sequence directly determines the operational efficiency and overall performance of the entire production line. The core of the mold layout strategy lies in maximizing the space utilization of the mold table through precise production sequence planning and layout design within the limited workspace of the mold table, which in turn promotes the significant improvement of the overall production efficiency [7].

Faced with the practical challenges of limited mold resources, diverse PC product types, and small production batches, how to accurately formulate the production sequence and layout of the program has become a complex and challenging task. This requires us to ensure the efficient use of the mold table while also balancing the relative workload between each mold table to optimize the overall production process and improve the efficiency of resource utilization [8,9]. Therefore, in-depth exploration and implementation of PC production sequencing and mold layout optimization strategy are of great significance to enhance the competitiveness and market adaptability of PC production enterprises. The main contributions of this paper are as follows:

• The production sequence and layout optimization of PC under mold resource constraints model for the limited quantity of PC molds is established while considering the production spacing layout requirements between the molds and the four sides of the mold table, as well as between molds so that the constructed PSLO-PC-MRC model is more in line with the actual production scenarios.
• In the PSLO-PC-MRC model, both consider the highest comprehensive utilization of the mold tables required to complete the order. At the same time, considering that the utilization rate of each mold table fluctuates as little as possible, the PC production line has a better rhythm.
• A multi-objective teaching–learning-based optimization with an embedded enhancement mechanism is developed to solve the PSLO-PC-MRC, and case studies are used to validate the algorithm’s effectiveness in determining the optimization of the PC production sequencing and layout schemes.

The rest of the paper is organized as follows. Section 3 describes the problem and constructs a mixed-integer planning model for PSLO-PC-MRC. Then, a multi-objective enhanced teaching–learning-based optimization method for solving PSLO-PC-MRC is described in detail in Section 4. Section 5 presents three different PC production sequencing and layout optimization cases to validate the performance of the proposed model and algorithm. The last section summarizes and indicates possible future research directions.

2. Related Works

2.1. PC Production Process and Production Methods

PC production mainly includes six processes [10]. (1) Mold assembly. After the mold and mold table are cleaned up, the mold corresponding to the PC to be produced is selected and installed according to the delineated position. (2) Reinforcing steel and embedded parts installation. The reinforcement mesh, truss bars, and embedded parts (e.g., electrical conduits and boxes) are lifted or placed into the pre-designed positions of the mold. (3) Concrete casting. The well-mixed concrete is poured into the molds and is spread smoothly and evenly. (4) PC curing. Either natural or steam curing is used, and in production, PC is generally placed in a curing chamber filled with saturated steam to accelerate concrete hardening in a higher temperature and humidity environment. (5) Demolding. When the concrete strength of the PC reaches the specified threshold range, it can be demolded and lifted. (6) PC finishing and repair. After demolding, repair non-stressed cracks on the PC surface and damaged exterior finishing materials, if required [11]. As far as PC production organization is concerned, it currently consists mainly of fixed mold table production (also known as stationary system) and flow mold table production (also known as traveling system) [12]. No matter what production organization method is adopted, the main process is also the six mentioned above. The difference is that in fixed mold table production, the mold table is fixed and immobile, and the workers, steel, concrete, and other materials carry out various operations around the mold table. It has a wide range of applications and strong universality, and it is suitable for producing various standardized and non-standard-shaped PCs [13]. In the flow mold table production, which is the research object of this paper, the production cycle time is generally relatively fixed, the assembly line is divided into several workstations, and a fixed number of operators and tools are configured in each workstation according to the operation content of the process, and each workstation is only fixed to complete a one or a small number of processes, indicating a high degree of specialization. During production, the mold tables are driven by power to move between the workstations. This production method has high efficiency, low production cost, and stable PC quality, which is especially suitable for mass production of PCs [14,15].

2.2. PC Production Scheduling

Since Warssawski first proposed the PC production planning and scheduling problem, numerous scholars have made significant efforts and attempts [16]. Subsequently, Hu and his partners abstracted and proposed a basic scheduling model for PC production based on flow shop, which paved the way for subsequent research on PC off-site production planning and scheduling [4,8,17,18]. Generally, standard types of PCs are produced based on inventory, while non-standard types of PCs are produced based on orders. The production scheduling problem of PCs in a flow shop in a hybrid MTO-MTS environment and the manufacturing scheduling mode of distributed multi-PC factories have been investigated to enhance delivery timeliness [10,19]. With the uninterruptible nature of the two processes of PC casting and curing, the blocking scheduling problem between multiple mold tables on the assembly line is investigated to make the scheduling model more compatible with the production reality [11]. The PC production dynamic scheduling model, which is characterized by uncertainty in construction project progress, has reduced the probability of delayed delivery to a certain extent [1,15,20]. A full supply chain production scheduling model considering mold manufacturing, PC storage, and transportation processes has been proposed, ensuring a more accurate calculation of PC off-site completion time and on-site delivery time [21]. A three-level supply chain management model with random constraints for off-site construction has been proposed, aiming to meet the needs of construction sites while minimizing total costs [22]. Recently, a production redundancy scheduling optimization model developed to cope with unstable production interruptions in PCs has achieved the minimization of interruption losses in PC production [23]. A PC production scheduling model with sequentially dependent due dates is proposed, and the total weighted earliness and tardiness are minimized using integrated differential evolution simulation [24]. The optimization objectives of PC production planning and scheduling mainly include minimizing the makespan [25,26], production cost [27,28], inventory cost [29], and penalty costs incurred due to delayed or early delivery [30,31]. Reducing bottlenecks and construction delays [32], maximizing profitability and supply security [33], etc. Production scheduling is an NP-hard problem [21], and there are roughly three types of optimization methods commonly used: exact methods, heuristic methods, and meta-heuristic algorithms. In the exact methods, the mathematical analysis method [16], mixed-integer linear programming [10], branch and bound [34] are mostly used in LINGO [35] and CPLEX [36]. Palmer, Gupta, Campbell Dudek Smith, rapid access, and other methods were developed to reduce the makespan of PCs [21]. However, as the number of jobs increased, the exact methods and heuristics became overwhelming, and meta-heuristics were proposed, mainly consisting of swarm intelligence evolutionary algorithms represented by genetic algorithms [37], simulated annealing [38], etc. In recent years, simulation [39], reinforcement learning, and deep reinforcement learning-based algorithms [25,40] have also been successfully applied in PC production scheduling.

2.3. The Role of Molds in PC Production

During a production cycle, the PCs move unidirectionally between workstations on the mold table by the line. As the first PC production process, researchers have carefully studied mold assembly due to its complexities. Tharmmaphornphilas and Sareinpithak studied the problem of a PC production process containing identical parallel molds with shared resources to minimize the total product cost [36]. Lim studied the optimization of production scheduling and yard design for PC in-site production under the constraint of mold tables [3]. BENJAORAN et al. and Yang et al. investigated the impact of the availability of customized molds on PC processing time [41,42]. Hu incorporated mold planning into flow shop scheduling optimization and investigated the balanced use of molds during PC production to reduce excessive mold manufacturing waste triggered by peak mold usage demand [8].

2.4. Research Gap

In terms of the layout of the PCs on the mold table, there is only a little literature related to it. Zheng et al. introduced group technology into the allocation of PCs on the mold tables to optimize the scheduling problem in mixed-flow PC production [14,26]. However, only a 0–1 random approach was used to arrange PCs onto the mold tables without paying attention to the geometric and positional relationship between them, resulting in the inability to provide a solution for the mold assembly of PCs, which is a certain gap with the actual PC production. Wang et al. overcame the above shortcomings by proposing a linear optimization model that takes into account constraints such as the order demand, the size of the mold table, and the positional relationship between the PCs and the mold table, which can maximize the average utilization of the mold tables used in the mold assembly process [7]. But there are two shortcomings. On the one hand, it ignored the fact that the quantity of molds for each type of PC is in limited supply due to the higher manufacturing cost. On the other hand, it is assumed that the arrangement of PCs on the mold table is limited, and the quantity is unknown, resulting in poor practical applicability. Finally, the production line balancing should be considered when PCs are placed on the mold tables [43]. For example, two adjacent mold tables on the production line, if the projected area of the PC molds on them varies greatly after placement, the working time of the operators will fluctuate greatly in the subsequent workstations such as reinforcing steel and embedded parts installation and concrete casting, which will lead to wasted time during the waiting process for the PCs to be processed, increase the production cycle. Secondly, an unbalanced PC production line can result in the overloading of some workers and underloading of others. When some workers are busy with PC operations, other workers may not have enough work to tackle, wasting human resources. In addition, an unbalanced production line may lead to poor collaboration among workers, causing dissatisfaction among operators, thus affecting overall work efficiency. Therefore, we propose the optimization of the production sequence and layout of precast concrete components under mold resource constraints (PSLO-PC-MRC) to compensate for the shortcomings of the above research.

3. Model of PSLO-PC-MRC

3.1. Problem Description

On the PC production line, there are a series of mold tables, and all six production processes of the PC are completed on the mold tables. Figure 1 shows a part of the production line. At the mold assembly workstation, a total of eight PCs of five different sizes are placed on two front-to-back adjacent mold tables. Not only are these PCs not allowed to overlap each other, but they also need to be spaced apart from each other to allow the operators to work on them. It can be intuitively seen that the area where the PCs are placed is the effective utilization area of the mold tables, while the other areas are wasted areas. The larger the projected area of PCs on the mold table, the higher the utilization rate. Improving the comprehensive utilization rate of the mold table can greatly save amortization costs and energy consumption, especially for the higher energy-consuming curing process. Under the limited capacity of the curing kiln, more PCs can be heated in the same curing cycle.

The PSLO-PC-MRC problem is similar to, but more complex than, the 2D bin packing problem because the 2D bin packing problem only needs to consider the size constraint of the bin [44], while the PSLO-PC-MRC problem not only needs to consider the size of the external bin (mold table), but also needs to cyclically layout the items (PCs) waiting for production due to the limited number of molds, and the spacing distance of the molds has to be considered. As shown in

Table 1. Order information containing 14 pieces of PCs (P14).

PC	Mold Quantity	Order Demand	Size/mm
			Length	Width	Thickness
1	2	4	2170	1960	60
2	1	2	2470	1100	60
3	2	3	2570	1580	60
4	3	5	4670	2700	60

, the number of molds for each type of PC is less than the order demands. Therefore, the PSLO-PC-MRC problem can be described as how to allocate PCs to the mold tables in a reasonable manner under limited mold resources to maximize the comprehensive utilization rate of the mold tables while balancing the utilization rate of each mold table to improve the balance of the PC production line.

3.2. Mathematical Model of PSLO-PC-MRC

3.2.1. Notations

$i,j:$ PC type index, $i,j\in I$.

$l_i,w_i:$ the length and width of component $i$,$i\in I$.

$I:$ PC types, $I=\left\{\mathrm{1,2},3,\cdots ,N_c\right\}$.

$d_i:$ demand quantity of component $i$, $d_i\in N^{+}$.

$Q_i:$ mold quantity of component $i$,$Q_i\in N^{+}$.

$k:$ mold table index, $k\in K$.

$L,W:$ length and width of the mold table, $L=\mathrm{10,000} \mathrm{m}\mathrm{m},$ and $W=4000 \mathrm{m}\mathrm{m}$.

$K:$ mold table numbers, $K=\left\{\mathrm{1,2},3,\cdots ,N_m\right\}$.

$d:$ layout spacing between any two PCs, $d=300 \mathrm{m}\mathrm{m}$.

${\eta }_k:$ utilization rate of mold table $k$.

${(x}_i,y_i):$ positioning coordinates of the lower left corner of component $i$.

$s:$ layout sequence index, $s\in S$.

$S:$ total number of layout sequences,$S=\left\{\mathrm{1,2},3,\cdots ,N_s\right\}$.

$P_{is}:$ the number of $i$ type molds used in the $s$ layout sequence.

$r_i:$ if PC $i$ is located vertically, then $r_i=1$; horizontally, then $r_i=0$.

$UR:$ average utilization rate of mold tables.

$SI:$ workload smoothness index of all mold tables.

3.2.2. Objectives

$$\mathit{max}UR={\displaystyle \frac{{\sum }_{i=1}^{N_c}{d_i·l}_i·w_i}{N_m·L·W}}\times 100$$

(1)

$$\mathit{min}SI=\sqrt{{\sum }_{k\in K}{({\eta }_k-max\left\{{\eta }_k\right\})}^2/N_m}$$

(2)

Two objectives will be optimized in the PSLO-PC-MRC. Equation (1) represents the average utilization rate (UR) of all the mold tables when all PCs are arranged. The utilization rate of each mold table is obtained by the ratio of the rectangular projected area of all PCs on it (not including layout spacing distance) to the area of the mold table, and a higher UR means higher production capacity. Among all the mold tables, some have higher utilization rates, and some have lower utilization rates. Different utilization rates mean that the operating time of the mold tables in the same process is different. If the utilization rate of all the mold tables fluctuates greatly, it will have a large negative impact on the balance of the production line, and even the process content cannot be completed within the specified cycle time. Equation (2) is formulated to express the influence of the utilization rate of all mold tables on the balance of the production line, which is represented by the smoothness index (SI), the smaller SI is more favored, and the ideal value is 0.

For example, there are 5 PCs on the mold table 1# and 3 PCs on the mold table 2# in Figure 1. ${\eta }_1$ of the mold table 1# is equal to the ratio of the area of these 5 PCs to the area of the mold table, and similarly, ${\eta }_2$ is the ratio of the area of the 3 PCs to the area of the mold table. Then UR is the average of ${\eta }_1$ and ${\eta }_2$, and SI can be obtained by bringing ${\eta }_1$ and ${\eta }_2$ into Equation (2).

3.2.3. Constraints

$$0\le P_{is}\le Q_i, \forall i\in I,s\in S$$

(3)

$${\sum }_{s=1}^{N_s}{P}_{is}=d_i, \forall i\in I$$

(4)

$$d/2\le x_i\le L-\left(1-r_i\right)·l_i-{r_i·w}_i-d/2, \forall i\in I$$

(5)

$$d/2\le y_i\le W-\left(1-r_i\right)·w_i-{r_i·l}_i-d/2, \forall i\in I$$

(6)

$$x_i+\left(1-r_i\right)·l_i+{r_i·w}_i+d\le x_j, \forall i,j\in I$$

(7)

$$x_j+\left(1-r_j\right)·l_j+{r_j·w}_j+d\le x_i, \forall i,j\in I$$

(8)

$$y_i+\left(1-r_i\right)·w_i+{r_i·l}_i+d\le y_j, \forall i,j\in I$$

(9)

$$y_j+\left(1-r_j\right)·w_j+{r_j·l}_j+d\le y_i, \forall i,j\in I$$

(10)

$$r_i=0 or 1, \forall i\in I$$

(11)

$$r_j=0 or 1, \forall j\in I$$

(12)

$$d_i\in N^{+}, \forall i\in I$$

(13)

$$Q_i\in N^{+}, \forall i\in I$$

(14)

Since the molds are used cyclically, for a certain type of PCs, the total number of molds participating in each layout will not be greater than the number originally prepared, which can be limited by Formula (3). Similarly, for any specific type of PC in the whole layout scheme, the total number of molds involved should be consistent with its demand, and the constraint can be described as Equation (4).

Figure 2 visualizes the layout of two PCs on the mold table, and the relevant symbol explanations will be elaborated in detail in the third section. When the length direction of the PC is parallel to the length direction of the mold table, then $r_i$= 0, and $r_{i }$= 1 when they are perpendicular. At the same time, considering the protection of the PC, the minimum edge distance between the PC and the mold table is defined as $d/2$, where $d$ is the layout spacing distance between any two PCs. Therefore, the position coordinate constraints of PC (i) on the mold table in Figure 2 are shown in

Table 2. Position constraint of a single PC on the mold table.

Directions	X-Dimensional Constraint		Y-Dimensional Constraint
	${\mathit{r}}_{\mathit{i}}$ = 0	${\mathit{r}}_{\mathit{i}}$ = 1	${\mathit{r}}_{\mathit{i}}$ = 0	${\mathit{r}}_{\mathit{i}}$ = 1
Coordinate constraints	$\left\{\begin{array}{c}d/2\le x_i\\ x_i+l_i\le L-d/2\end{array}\right.$	$\left\{\begin{array}{c}d/2\le x_i\\ x_i+w_i\le L-d/2\end{array}\right.$	$\left\{\begin{array}{c}d/2\le y_i\\ y_i+w_i\le W-d/2\end{array}\right.$	$\begin{array}{c}d/2\le y_i\\ y_i+l_i\le W-d/2\end{array}$
Results	$d/2\le x_i\le L-\left(1-r_i\right)·l_i-{r_i·w}_i-d/2$		$d/2\le y_i\le W-\left(1-r_i\right)·w_i-{r_i·l}_i-d/2$

, and Equations (5) and (6) limit the position of any PC not to exceed the size of the mold table.

Equations (7)–(10) are the relative position constraints of any two PCs; that is, each PC does not overlap the other, and the layout spacing constraint is also satisfied at the same time. The four equations are a set of redundant constraints because the relative position of any two PCs can only be one or two of the up-down relationship and the left-right relationship. Using the principle similar to Equations (5) and (6), for example, consider PC $i$ on the right side of Figure 1 as PC $j$; when PC $i$ is just located on the left side of PC $j$, then only constraint (7) is satisfied. When PC $i$ is located on the lower left side of PC $j$, then constraints (7) and (9) are both satisfied. The other mutual positional relationships between any two PCs are also the same. Equations (13) and (14) ensure that the demand quantity of any type of PC and the quantity of molds are positive integers.

4. Enhanced Teaching–Learning-Based Optimization for PSLO-PC-MRC

The teaching–learning-based optimization (TLBO) algorithm was recently developed by Rao [45] to optimize mechanical design problems. The basic idea of TLBO comes from the influence of teachers’ work on students; that is, teachers’ teaching level will affect students’ academic performance. Therefore, the process of TLBO is divided into two stages. The first stage is the teaching stage, where students learn primarily from their teachers. Then comes the learning stage, where students learn from each other. Compared with the famous swarm intelligence optimization algorithms such as genetic algorithm (GA) [37], particle swarm algorithm (PSO), and simulated annealing (SA), TLBO has special advantages, mainly reflected in the following aspects:

• Careful adjustment of control parameters such as crossover probability and mutation probability is required in GA, and the selection of these parameters directly affects the performance and effectiveness of the algorithm. TLBO greatly simplifies this process, mainly relying on two basic parameters: population size and iteration times, reducing the complexity of parameter adjustment.
• High interaction between particles is relied on in PSO to find the optimal solutions, but this mechanism sometimes leads to premature convergence of the particle swarm to a local optimal solution, especially in complex multimodal problems. The TLBO algorithm achieves an effective balance between global and local search while maintaining population diversity through its unique teaching and learning mechanisms.
• SA jumps out of local optimal by simulating the probability changes during the physical annealing process, but its search process is essentially random, which may result in slow convergence speed and unstable results. In contrast, the TLBO algorithm is a deterministic search algorithm with clear and controllable search steps, which can find high-quality solutions in a shorter period.

However, basic TLBO is mostly used to solve single-objective continuous optimization problems [46,47], and no effective application has been found in the field of discrete multi-objective precast component layout optimization. Therefore, a discrete multi-objective enhanced teaching–learning-based optimization (ETLBO) based on the Pareto optimization mechanism [48] is proposed, which can effectively expand the global optimization space of the population, strengthen the local search ability, and improve the convergence speed and solution accuracy of basic TLBO. The encoding, decoding, teacher process, learner process, and enhanced process are detailed in Section 4.1, Section 4.2, Section 4.3, Section 4.4 and Section 4.5, respectively, and the steps of the ETLBO algorithm are described in Figure 3.

4.1. Encoding Scheme

Based on the reality of the PSLO-PC-MRC problem, sequence encoding is employed, and the encoding consists of two sequence vectors. The first is the sequence of mold types, which indicates the types of molds that are recycled during the production of precast components. In order to ensure the feasibility of the mold type sequence, the actual number of molds participating in each production layout should not exceed the number of its preparations. Therefore, the mold types in the first sequence vector are in the form of segments, with each segment indicating that all molds complete one production layout, and the number of segments is determined by the number of precast components produced and the number of molds. The second one is the mold orientation sequence, where the placement orientation of each mold can be randomly generated between 0 and 1. It should be noted that the same type of mold can have different orientations when it is placed, but the geometric requirements of being placed on the mold table should be met. Otherwise, it should be adjusted. Figure 4 depicts a feasible coding scheme for P14 in Table 1, where two production layouts are required to complete the required number of PCs, with all the molds used in the first production layout and only part of the molds used in the second layout to meet the production requirements. Mold #2, with its longer side, is placed parallel to the long side of the mold table, while in the second layout, they are perpendicular to each other.

4.2. Decoding Strategy

Decoding is converting the encoded individuals in the ETLBO algorithm back to specific solutions in the original problem space. The decoding strategy in this paper is to use the lowest horizontal line algorithm to determine how to place the PC molds on the mold table [49]. The core idea is to first initialize the set of horizontal lines, and then compare the width of the PC mold to be placed with the width of the lowest horizontal line in the set, and place it if the lowest horizontal line can be satisfied, otherwise raise the height of the lowest horizontal line to continue to determine until the lowest horizontal line meets the width of the PC mold and the production layout spacing requirement.

Figure 5 shows a decoding scheme for the individual encoded in Figure 4 using the following steps.

Step 1: Initialize the set of horizontal lines and set the bottom boundary line on one side of the mold table as the current lowest horizontal line.

Step 2: According to the width of the PC mold to be arranged, select the current lowest horizontal line from the collection of horizontal lines, and if there are multiple lines available at the same time, select the leftmost horizontal line.

Step 3: Judge whether the width of the current lowest horizontal line meets the width of the PC mold to be arranged; if it meets, then place the PC mold on the leftmost side of the current lowest horizontal line, update the collection of horizontal lines and go to Step 5.

Step 4: If it does not meet, compare the two horizontal lines on both sides of the current lowest horizontal line and set the lower one as the current lowest horizontal line if the heights of the two horizontal lines are different. Otherwise, select the left horizontal line, then go to Step 3.

Step 5: Determine whether all the PC molds in the encoded individual are placed; if yes, the decoding process ends. Otherwise, go to Step 2.

4.3. The Teaching Process

In multi-objective optimization, a set of solutions is called Pareto-optimal if they are not dominated by other solutions and are not dominated by each other. Therefore, the optimal solution to the PSLO-PC-MRC problem is a set of Pareto-optimal solutions. For each individual in the ETLBO algorithm population, the first stage of learning is completed through the teacher, and the teacher’s knowledge is richer, more comprehensive, and more inspiring to the students. Therefore, in the teaching process, each student exchanges information with the teacher separately to improve the individual student’s knowledge, and the teacher can randomly choose one of the Pareto-optimal solutions. Here, the method of sequence cross-mapping between two points is used, and all of the encoded individuals are performed during the sequence cross-swap. After completing the crossover, the newly generated sequences of mold types and mold directions need to be repaired and reconstructed.

Figure 6 depicts one kind of teacher process for P14. First, two crossing points are randomly selected on the encoding individuals of the two production layouts. Here, positions 5 and 10 are chosen, with the body part between them being the part to be crossed. Then, the body of the current individual is interchanged with the body part of the teacher (including the mold type and orientation). After crossing, it is found that the current coding individual may have two problems: one is that the mold type sequence no longer meets the production requirements, and the other is that the mold placement direction may not meet the size requirements of the mold table. For example, molds 1 and 2 need to be used 3 and 2 times, respectively, in the first production layout, while their preparation quantities are only 2 and 1, which violates the rationality of actual production. Therefore, it needs to be adjusted according to the mold quantity constraint. Taking the first production layout as an example, the uncrossed part can still use molds 3 and 4 two times each, which is then randomly reorganized and arranged as 4-4-3-3. Then, it was found that the placement direction of mold 4 needed to be adjusted from 1 to 0 for the second time. After the above teacher stage, the newly generated individual of the current individual becomes completely feasible.

4.4. The Learning Process

In the real teaching process, based on the teacher’s teaching process, students may not fully grasp the knowledge imparted by the teacher, and students will also learn from each other. However, in contrast to the knowledge taught by the teacher, the knowledge that students learn from each other is partial. Therefore, in the learning process, each student and another randomly selected student also use the two-point cross-mapping method. The difference is that the crossover operation is completed in either one production layout, and the newly produced encoding individual does not need to be repaired and reconstructed after the crossover.

Figure 7 shows two scenarios of the learner process, where the crossover operation can be performed in two production layouts, and the ETLBO algorithm only needs to randomly select one of them.

4.5. The Enhanced Process

The TLBO algorithm proposed by Rao [50] will start a new cycle after performing the teacher process and the learner process. According to the learning experience of real students, in addition to the above two learning methods (mainly manifested in the understanding and mastery of knowledge), there are also processes of knowledge forgetting and self-consolidation learning. Therefore, the enhanced process of the ETLBO algorithm is mainly reflected in two aspects. On the one hand, the forgetting of knowledge leads to a discrepancy between the acquired knowledge and the objectively existing knowledge, and this process is realized by the swap of partial encoding information. On the other hand, the forgotten parts of the knowledge can be compensated by self-consolidated learning. Therefore, for an individual with poor encoding effect (also known as fitness), the probability shown in Equation (15) can be used to accept if the prob is less than the fixed probability of 0.2, hoping that this part of knowledge will be migrated in the future to produce more valuable knowledge, where m represents the number of objectives to be optimized, S and S’ represent two individuals before and after the enhanced process, and MaxIter and CurIter indicate the maximum and current iterations. Through the enhanced process, the local search capability of the ETLBO algorithm can be effectively improved. Figure 8 depicts the swap process of the mold type and orientation sequences.

$$prob={\prod }_{t=1}^{m}exp(-{\displaystyle \frac{f_t\left(S^{\prime }\right)-f_t\left(S\right)}{MaxIter-CurIter}})$$

(15)

4.6. Pareto Nondominated Solutions

The objective of PSLO-PC-MRC is to simultaneously maximize UR and minimize SI, which is a multi-objective optimization problem. Therefore, Pareto-dominated solutions are introduced to compare and evaluate the relationship between the advantages and disadvantages of different solutions. A solution P is said to be Pareto non-dominated by another solution Q if P is non-inferior in all objectives and superior in at least one objective to Q. A solution is said to be Pareto nondominated if any other solution in all objectives does not dominate it [51]. To ensure consistency in the direction of optimization, we take the mold table’s idle rate (IR) as the minimization objective, where IR = 100 − UR.

5. Experimental Verification and Discussion

The proposed ETLBO is developed using Matlab R2021a, running on the Windows 11 operating system with a 3.10 GHz Intel (R) Core (TM) i5-10500 processor and 8 GB of RAM. We will use three case studies of different scales to verify that the proposed model and ETLBO can provide effective solutions for the PSLO-PC-MRC problem.

It should be pointed out that in the existing literature on PC production scheduling and management, we have not yet found information on the specific dimensions of PC and related molds. Therefore, the small and medium-scale cases mainly come from the atlas of truss-reinforced concrete laminated slabs published by the China Institute of Building Standard Design and Research Co., Ltd in Beijing, China. Meanwhile, assuming that the order demand dates for each type of problem are the same.

The commonly used GA, SA, and TLBO algorithms without enhanced strategy in PC production scheduling were used for performance comparison. These algorithms have different calculation parameters and termination conditions. The calculation parameters involved in GA, TLBO, and ETLBO all include population size, while GA also includes crossover and mutation probabilities. SA includes initial temperature, Markov chain length, and cooling rate. All these parameters were determined using the analysis of variance method in the literature [51], based on equivalent scale cases in the literature [37,38]. To ensure fairness, all algorithms are run independently five times, with a single run time of $N_c·\sum d_i·Q_i$ seconds, after which all algorithms automatically terminate.

5.1. Small-Scale Case

This case is from Table 1, with 14 PCs of four types and only eight molds. Therefore, the population size was set to 60, and the crossover and mutation probabilities of GA were set to 0.7 and 0.3, respectively. The initial temperature, Markov chain length, and cooling rate were set to 500, 10, and 0.90, respectively.

After the execution, all algorithms obtained more than 20 Pareto-nondominated solutions, and

Table 3. Production sequence and the layout schemes of the small-scale case.

Method	NO.	IR	SI	Production Sequence	Layout
SA	1	51.16	5.02	{2-4-4-1-3-3-4-1} → {2-4-4-1-3-1}	[2 4 1][4 3 3][4 1] → [2 4 1][4 3 1]
GA	1	51.16	5.02	{1-4-2-3-3-4-1-4} → {1-4-2-3-4-1}	[1 4 2][3 3 4][4 1] → [1 4 2][3 4 1]
TLBO	1	51.16	5.02	{2-1-4-3-3-4-1-4} → {2-1-4-3-4-1}	[2 1 4][ 3 3 4][4 1] → [2 1 4][3 4 1]
ETLBO	1	51.16	5.02	{1-2-4-3-3-4-4-1} → {1-2-4-3-4-1}	[1 2 4][3 3 4][4 1] → [1 2 3][3 4 1]
	2	51.16	5.02	{3-3-4-2-4-4-1-1} → {3-4-2-4-1-1}	[3 3 4][2 4 1][4 1] → [3 4 1][2 4 1]

only lists the most representative production sequence and layout schemes. All four algorithms require molds to be rotated on five mold tables to complete the production order, and their comprehensive IR and SI are 51.16% and 5.02, respectively. However, the first three algorithms only obtained one utilization scenario for the mold tables, with UR values of 48.95%, 51.83%, 42.16%, 48.95%, and 52.31% for the five mold tables, respectively. But ETLBO also obtained another type, with UR values of 51.83%, 48.95%, 42.16%, 52.31%, and 48.95%. This preliminarily indicates that in small-scale situations, ETLBO can discover more Pareto optimal solutions with slightly better performance than the first three methods.

5.2. Medium-Scale Case

The data for the medium-sized case is shown in

Table 4. Medium-scale case with 59 pieces of PCs (P59).

PC	Mold Quantity	Order Demand	Size/mm
			Length	Width	Thickness
1	1	2	1790	1500	60
2	2	4	2500	2200	60
3	1	2	2900	1250	60
4	2	3	2900	1820	60
5	2	3	3000	1300	60
6	3	6	3000	1620	60
7	1	2	3000	1910	60
8	2	3	3200	1620	60
9	3	5	3200	2270	60
10	1	2	3600	1820	60
11	1	2	3600	2020	60
12	3	6	3600	2030	60
13	2	3	3600	2300	60
14	2	4	3600	2260	60
15	1	2	3700	1200	60
16	1	2	3900	1960	60
17	2	3	3900	2020	60
18	2	3	4000	1990	60
19	1	2	4000	1410	60

, with 19 types of PCs, a total order demand of 59 pieces, and 33 prepared molds. Similar to the above, the population size was set to 120, and the crossover and mutation probabilities for GA were set to 0.8 and 0.2. The initial temperature, Markov chain length, and cooling rate were set to 800, 20, and 0.95, respectively. 11 Pareto nondominated solutions were obtained in total, and

Table 5. Production sequence and the layout schemes of the medium-scale case.

Method	NO.	IR	SI	Production Sequence	Mold Tables’ UR
SA	1	34.56	16.39	{14-12-12-18-17-6-7-13-16-8-9-17-5-2-6-14-12-2-4-13-11-8-9-4-15-6-9-3-18-19-5-10-1} → {14-12-12-18-17-6-7-13-16-8-9-5-2-6-14-12-2-4-11-9-15-6-3-19-10-1}	{78.78-74.46-67.10-51.05-68.00-9.30-63.62-42.99-75.74-74.46-65.24-61.31-57.37-66.82}
	2	38.92	8.98	{14-12-11-1-13-14-12-13-16-10-15-9-17-5-2-8-9-18-9-8-18-7-17-6-2-6-12-5-6-4-4-3-19} → {14-12-11-1-13-14-12-16-10-15-9-17-5-2-8-9-18-7-6-2-6-12-6-4-3-19}	{63.50-66.47-66.99-61.65-62.97-8.88-64.10-60.74-56.79-58.85-56.19-62.39-52.56-56.79-57.37}
	3	42.74	4.14	{13-6-5-9-14-12-9-4-2-18-7-6-6-11-17-8-2-17-8-18-4-1-9-12-16-12-10-19-3-13-14-5-15} → {13-6-5-9-14-12-9-4-2-18-7-6-6-11-17-8-2-1-12-16-12-10-19-3-14-15}	{60.76-56.77-59.00-57.62-53.12-3.70-58.69-58.23-57.42-60.76-56.77-59.00-58.76-55.47-55.19-54.99}
GA	1	34.56	12.18	{18-8-8-12-2-19-5-12-7-4-9-6-17-16-17-6-4-11-3-9-13-18-12-13-14-10-9-6-15-5-1-2-14} → {18-8-12-2-19-5-12-7-4-9-6-17-16-6-11-3-9-13-12-14-10-6-15-1-2-14}	{60.15-74.03-63.31-50.96-60.34-5.99-68.52-62.00-65.46-67.91-70.02-64.62-63.71-69.21}
	2	38.92	9.72	{14-16-4-6-17-13-5-10-12-9-9-2-6-9-15-17-6-4-18-7-8-11-2-12-19-12-3-13-8-5-14-1-18} → {14-16-4-6-17-13-5-10-12-9-9-2-6-15-6-18-7-8-11-2-12-19-12-3-14-1}	{64.80-68.42-63.80-67.22-63.31-0.94-66.90-59.91-64.80-63.65-57.36-64.17-45.80-48.28-56.88}
	3	42.74	5.33	{6-6-17-9-18-2-8-9-8-9-17-5-5-6-2-18-15-7-4-11-4-1-16-14-12-12-14-12-10-19-13-13-3} → {6-6-17-9-18-2-8-9-5-6-2-15-7-4-11-1-16-14-12-12-14-12-10-19-13-3}	{62.16-59.57-55.82-56.70-55.04-3.63-57.72-56.88-57.78-56.96-59.96-56.76-59.80-55.23-56.88-55.35}
TLBO	1	34.56	13.06	{7-6-6-12-17-18-8-9-9-12-15-2-13-18-3-5-16-13-17-9-5-4-12-10-6-2-8-4-14-19-14-11-1} → {7-6-6-12-17-18-8-9-9-12-15-2-13-3-5-16-4-12-10-6-2-14-19-14-11-1}	{76.59-67.51-70.23-62.50-62.52-1.05-66.03-58.86-76.59-60.88-61.17-62.07-67.61-72.61}
	2	38.92	8.28	{17-13-6-14-11-15-8-18-9-5-9-2-4-13-2-4-8-12-18-6-12-3-6-14-9-5-12-17-7-10-16-19-1} → {17-13-6-14-11-15-8-18-9-5-9-2-4-2-12-6-12-3-6-14-12-7-10-16-19-1}	{65.51-67.78-61.56-60.61-63.52-5.89-66.52-63.92-65.51-59.37-56.22-58.97-51.63-54.99-54.25}
	3	42.74	3.43	{4-5-18-7-14-17-14-6-5-13-12-2-12-9-3-17-9-6-2-8-12-8-9-10-4-18-16-11-6-15-19-13-1} → {4-5-18-7-14-17-14-6-13-12-2-12-9-3-9-6-2-8-12-10-16-11-6-15-19-1}	{66.10-63.42-69.06-63.14-63.20-3.14-69.24-57.99-64.06-67.32-64.29-69.15-68.34-67.77}
ETLBO	1	34.56	4.92	{4-5-18-7-14-17-14-6-5-13-12-2-12-9-3-17-9-6-2-8-12-8-9-10-4-18-16-11-6-15-19-13-1} → {4-5-18-7-14-17-14-6-13-12-2-12-9-3-9-6-2-8-12-10-16-11-6-15-19-1}	{66.10-63.42-69.06-63.14-63.20-3.14-69.24-57.99-64.06-67.32-64.29-69.15-68.34-67.77}
	2	42.74	3.43	{13-11-12-7-3-18-16-19-17-18-6-2-5-6-12-17-6-2-5-13-12-8-4-8-4-15-9-9-14-1-9-14-10-} → {13-11-12-7-3-18-16-19-17-6-2-5-6-12-6-2-12-8-4-15-9-9-14-1-14-10}	{60.09-58.85-59.86-58.46-55.52-3.59-57.39-56.66-54.88-60.09-57.80-55.47-57.75-58.18-56.77-54.88}

lists the Pareto-nondominated solutions corresponding to each algorithm. Overall, all the algorithms have obtained acceptable solutions, but upon closer examination, it was found that the first solution obtained by ETLBO can dominate all solutions except for the solution with IR = 42.74% and SI = 3.43.

Choosing hypervolume (HV) as the convergence evaluation index [52], we tracked the change of HV of the four algorithms from the beginning to the end of the run, as shown in Figure 9a. It can be intuitively seen that the quality of the Pareto optimal solutions obtained by each algorithm gradually improves as the algorithms run, and the various algorithms start to converge when it comes to 3/5 to 4/5 times the total time. From the HV convergence index, GA and SA perform equally well on the P59 problem, TLBO performs relatively well and obtains a nondominated solution, while ETLBO performs the best, further demonstrating the superior performance of the proposed ETLBO algorithm.

Figure 9b–d shows the obtained mode table utilization scenarios for SA, GA, TLBO, and ETLBO, respectively. Compared with the other three algorithms, ETLBO can always obtain solutions with higher mold utilization and balanced mold utilization. Meanwhile, the difference between the two Pareto optimal solutions obtained by ETLBO is mainly reflected in the fact that the utilization rate of the mold table corresponding to ETLBO1 is higher than that of ETLBO2. In contrast, the fluctuation of the utilization rate is reversed.

5.3. Large-Scale Case

The case is from a PC production line in Ya’an, Sichuan Province, China. The case data is shown in

Table 6. Large-scale case with 192 pieces of composite floor slabs (P192).

PC	Mold Quantity	Order Quantity	Size/mm
			Length	Width	Thickness
1	2	4	1500	1260	60
2	1	4	1920	1220	60
3	1	3	2170	1020	60
4	1	3	2320	2120	60
5	1	2	2420	2420	60
6	2	5	2520	1200	60
7	2	7	2520	1500	60
8	2	4	2520	1190	60
9	1	2	2520	2520	60
10	2	5	2520	1860	60
11	1	2	2620	2060	60
12	2	4	2820	960	60
13	2	5	2820	1800	60
14	2	4	2920	2360	60
15	3	8	2920	1800	60
16	1	2	2920	2420	60
17	2	7	3120	1200	60
18	3	5	3120	2000	60
19	3	8	3120	1710	60
20	2	4	3310	1320	60
21	2	6	3420	1260	60
22	2	6	3420	1560	60
23	2	4	3420	2000	60
24	3	5	3420	1220	60
25	2	8	3520	1620	60
26	2	8	3520	2360	60
27	2	6	3720	1200	60
28	2	3	3720	1500	60
29	2	3	4020	1200	60
30	2	7	4020	1500	60
31	2	8	4320	1260	60
32	2	5	4320	1560	60
33	2	4	4320	2620	60
34	2	5	4620	1200	60
35	1	4	4620	2160	60
36	3	5	4920	1760	60
37	2	4	5220	2160	60
38	3	5	5520	960	60
39	1	3	5520	1920	60
40	1	3	5820	2160	60
41	1	2	5920	1920	60

, which contains 41 types and 192 pieces of PC composite floor slabs. Data such as concrete strength, reinforcement content, weight, and location to be installed, which are not very relevant to the objectives, are no longer considered.

To improve the statistical analysis’s reliability, each algorithm’s parameters were similarly calibrated before the algorithm comparison tests. The population size was 200 for GA, TLBO, and ETLBO, and the crossover and mutation probabilities for GA were set to 0.9 and 0.1, respectively. The initial temperatures, Markov chain lengths, and cooling rates were set to 1000, 30, and 0.98, respectively. Each algorithm was executed 10 times, where 3 Pareto nondominated solutions were obtained for ETLBO, and 2, 6, and 6 Pareto nondominated solutions were obtained for TLBO, GA, and SA, respectively.

By merging the 17 Pareto nondominated solutions, the dominant relationship between them is again determined, and the results are shown in Figure 10. By plotting the Pareto nondominated solutions for each algorithm, it can be roughly divided into four curves. The three Pareto nondominated solutions obtained by the ETLBO algorithm are (32.7%, 8.51), (34.3%, 8.14), and (35.83%, 7.49), respectively. From a macro perspective, the solutions obtained by GA and SA are significantly dominated by the three solutions obtained by ETLBO, and the solutions of SA are also dominated by GA. Compared with ETLBO, although the solutions of TLBO and ETLBO are relatively close, the solution of TLBO is still dominated by ETLBO, further demonstrating the powerful performance of the ETLBO algorithm in large-scale PC production sequencing and layout optimization.

To assist decision-makers in analyzing the advantages and disadvantages of each production sequencing and layout scheme, Figure 11, Figure 12 and Figure 13 present the layout schemes corresponding to the three Pareto-nondominated solutions obtained by ETLBO. The small rectangular blocks labeled with numbers in the figures indicate the identifier and area size of individual PCs. The vertical coordinate indicates the mold table number, and the horizontal coordinate indicates the cumulative projected area of the PCs after they are placed on the mold table, with the maximum value of 40 m² (4000 mm × 10,000 mm) indicating the area of a single mold table. The blank areas on the right side of each figure then indicate areas of the mold table that are not being utilized.

The three scenarios require 41, 42, and 43 mold tables, respectively, with 3–6 PCs on each mold table. The utilization rate of a single mold table ranges from 48.63% to 73.28%, with a difference of about 25%. The reason lies in the differences in mold size and the combination relationship between various PCs. Due to their larger size, large-sized PCs cannot match small-sized PCs, resulting in more wasted space on the mold tables, which leads to a downward trend in the utilization of the mold tables. With the addition of small-sized PCs, the combination opportunities between molds are increased, and the utilization rate of the mold table shows a rising trend. In addition, under the constraint of the number of molds, to complete the order task, the molds need to be cycled four times, which leads to some of the mold tables not continuing to place the PC molds because of an insufficient number of molds, even though it has available space. Together with the influence of the spacing of the mold production layout, it is inevitable that some more space of the mold table will be wasted, which ultimately leads to the average utilization rate of the mold table stabilizing at about 65.7%.

Taking Figure 11 as an example, it consumed 41 mold tables. From the perspective of mold utilization rate, the utilization rate of a single mold table with 6 PCs placed is above 70%. Of course, there are also mold tables that only hold 3 PCs, with a utilization rate of over 70%, such as mold table 4 and mold table 20. The main reason why the utilization rate of these mold tables cannot be further improved is the limitation of mold table space. There are also molds with four pieces of PCs placed on them, such as molds numbered 15, 24, 27, 39, and 40, whose utilization rate is relatively low, and the space utilization rate of mold 40 is even less than 50%. The main reason is that all the PCs placed on the mold table are of large sizes, and there are no PCs of smaller sizes in the subsequent production sequences, which ultimately fails to achieve a better combination of layouts.

From the perspective of PC production line smoothness, the scheme in Figure 11 has the fewest total number of molds, which means a higher utilization rate of mold tables. In contrast, a lower SI in Figure 13 indicates a lower probability of relative idleness and excessive busyness, and the UR and SI of the scheme in Figure 12 are in a moderate state. The decision-makers can choose the specific production sequencing and layout scheme based on their key focus objective.

6. Conclusions

An optimization model for sequencing and layout of assembly precast component production is proposed in this paper. The model mainly considers the neglected mold quantity constraint in practice; that is, under the limited number of molds, the PCs in the order are reasonably and efficiently allocated on the mold tables of the production line. Two objectives, the mold table’s average utilization rate and the production line’s smoothness index, are designed. Then, a multi-objective teaching–learning-based optimization algorithm with embedded enhanced mechanism is proposed, and three cases are designed to verify the effectiveness of the PSLO-PC-MRC model and ETLBO algorithm. The experimental results show that, in small-sized cases, the number of Pareto optimal solutions obtained by ETLBO is twice that of the comparative algorithms mentioned above, and in medium-sized and large-scale cases, the average UR corresponding to ETLBO is 14.28% and 23.46% higher than the comparison algorithm, and the average SI reduction rate is 35.98% and 40.74%, respectively. This proves that the proposed ETLBO has significant advantages in improving the rhythm of the PC production line and enhancing overall mold utilization.

The PC production sequence and layout optimization model and method proposed in this paper can improve the comprehensive utilization efficiency of the mold table and the balance of the workload of each mold table in the PC production process to a certain extent. However, there are some limitations, such as the production schedule after the molds are laid out on the mold table is not taken into account and the production manager’s preference is not taken into account in the algorithm, leading to the need for further manual judgment of which production sequence and layout should be adopted after obtaining the Pareto optimal solutions. In future research, researchers can further improve the PSLO-PC-MRC model, for example, by integrating scheduling and production manager preferences into the model, while also considering how to optimize both the number of molds used and the efficiency of mold tables utilization when the number of molds is unknown. In addition, the hybrid optimization problem of reinforcement learning with the proposed algorithm (or other intelligent algorithms) can be investigated, which are all of great significance in achieving green and efficient production and promoting the development of the prefabricated construction industry.