Optimization of Precast Concrete Production with a Differential Evolutionary Algorithm

Abstract

This study investigates the limitations of existing models in optimizing equipment resource allocation for the large-scale production of precast concrete components in highway engineering. There are abundant investigations on scheduling models of precast concrete components. However, there is a scientific problem that previous models often overlooked the interruptibility of specific processes and the possibility of performing tasks outside of regular working hours, leading to suboptimal resource utilization. To address this limitation, an improved differential evolution (DE) algorithm was developed, which incorporates an adaptive mutation operator and a dual mutation strategy to enhance population diversity and accelerate convergence speed. The proposed optimization model significantly reduced equipment resource consumption. In a real-world case study, the model achieved an 11.11% reduction in project duration and a 21.4% increase in production capacity under the same resource configuration. The improved DE algorithm demonstrated superior performance in maintaining population diversity and accelerating convergence. These findings provide a scientifically grounded approach for enhancing productivity and resource efficiency in prefabricated construction, with potential applications extending beyond highway projects.

1. Introduction

A nation’s economic activity is based on its transportation system, and the ease of access to transportation has a direct impact on people’s quality of life. More significantly, highways foster social cohesion, facilitate contact and interactions between cities and villages, and connect them all. The Outline for Building a Strong Transport Country, released in 2019 by China’s State Council and the CPC Central Committee, stated that China should essentially be transformed into a strong transport nation by 2035 [1]. The Chinese government has suggested building a state-of-the-art, comprehensive, three-dimensional transportation network, as well as planning and building facilities for civil aviation, water transportation, railroads, and roads, in order to ensure that China becomes a transportation powerhouse. Furthermore, China is leading the world in highway infrastructure, with a total length of 5.35 million kilometers by the end of 2022—an increase of 1.12 million kilometers in just 10 years. Of those highways, 177,000 km are expressways that are open to traffic [2].

In highway engineering, the manufacturing of precast concrete components is not only essential for fulfilling construction specifications, but it is also a crucial component in enhancing productivity and security. The possible advantages of prefabrication for construction quality, efficiency, and environmental impact are demonstrated by its cleaner construction, high degree of industrialization, and low energy consumption [3,4]. There has been a surge in research on precast concrete plants around the world [5,6,7]. Recent optimization studies demonstrate significant progress across multiple sectors: Tuluca et al. developed genetic algorithm-based scheduling optimization for precast component production, achieving notable efficiency improvements in building construction applications [8]. For large-scale nuclear power constructions, Lu et al. established multi-objective optimization models addressing complex technical and economic constraints in concrete production [9]. Wang et al. implemented advanced deep learning systems for quality control in energy infrastructure projects, demonstrating substantial enhancements in product consistency and reliability [10]. Even more important, however, is the fact that modular construction and the assembly of prefabricated components can significantly reduce construction time [11]. The research fervor for optimizing various aspects of precast components has been high [12,13,14]. That said, the mishandling of production scheduling of modular components is a primary cause of poor project performance, leading to cost overruns, schedule delays, and quality problems [15]. The production optimization of large prefabricated components of highway bridges still needs to be improved, and that goal has strong support from the state. Optimization of processes, line layouts, and standard work procedures can reduce lead times [16]. For instance, the production process of prefabricated parts is frequently interrupted by unforeseen events, such as emergency orders and mechanical failures, but the optimization of large prefabricated component scheduling is the most significant issue at hand. Although the manufacturing cycle of prefabricated components typically is brief, the size and complexity of the components necessitate a more complex production process, which further complicates the traditional workshop production mode.

Prefabricated elements of conventional buildings have been the subject of the majority of early studies. The manufacturing of large concrete components has increased due to the rapid development of highway engineering, and workshop scheduling optimization remains challenging because of the low levels of automation and mechanization involved. The precast components of highway engineering are larger, span a greater distance, and require stricter production lines than the precast components of construction engineering do, and these features complicate the production process, in addition to restricting the current methodology for equipment resource allocation. Furthermore, because certain processes, such as steel binding and template installation, can be interrupted, researchers are turning their attention to the flexibility of production scheduling.

With the rapid development of assembly construction technology, the production of precast concrete components has attracted a great deal of attention. The problems of precast concrete production can be reduced to the problem of assembly line production [17]. The production scheduling problem in regard to prefabricated-component production plants has been extensively studied by scholars both domestically and internationally [14,15,18,19]. In order to optimally minimize the prefabricated-component production completion time, scholars began by studying the traditional flow shop sequencing problem [20] and proposed the idea of production shop scheduling [21]. Following these publications, the optimization problem gained prominence in the literature. For the issue of qualified productivity and timely delivery of prefabricated components, production scheduling is crucial [22], but it becomes challenging to schedule such manufacturing optimally. A parallel rescheduling strategy for multiple precast concrete production lines to minimize schedule changes and prevent major material rescheduling while managing production contingencies [23] has been explored, as has defining an existing prefabrication workflow to identify measures for increased labor time utilization rates [24]. The transportation of prefabricated components with a delivery time window has been established, and an optimal scheduling method to balance delivery timeliness and transportation has been sought [25]. Static scheduling was the main emphasis of early research, but dynamic scheduling is desperately needed to address the numerous interruptions that arise in real-world scenarios [26]. Furthermore, many studies have not considered the actual production environment [27,28], which limits research advancement, with only a few scholars having considered this factor [29]. Furthermore, by employing a novel mathematical model based on the permutation flow shop scheduling problem and solving it using an iterated greedy algorithm, the system performance is enhanced while considering shift work constraints [30]. A dynamic multi-objective green shop scheduling problem, optimizing total completion time, energy consumption, and carbon emission under uncertain order insertion [31]. A bi-population deep Q-network was developed to solve the distributed flexible job shop scheduling problem with transportation under a PC manufacturing environment [32].

As is shown in

Table 1. Literature related to production scheduling.

No.	Authors	Methodology	Limitations to Consider in Scheduling	Research Topics
1	[21]	Easy, correct algorithms	-	Investigation of optimal two- and three-stage production scheduling strategies, including setup times
2	[40]	GA	-	Model of flow shop sequencing, to ensure on-time delivery of PCs
3	[35]	GA and branch-and-bound method	Labor and inventory	Optimization of precast production costs subject to internal resource constraints
4	[30]	GA	Production stations	Proposal of a decision support system to optimize production plans
5	[33]	Mixed integer linear programming model	Molds	Proposal of an integrated prefabrication configuration and component grouping approach for resource optimization in precast production
6	[28]	GA	Molds	Proposal of an optimized flow shop scheduling method for multiple production lines in precast production
7	[26]	GA	-	Consideration of demand variability using a pure optimization model
8	[6]	Whale optimization algorithm	-	Investigation of flow shop optimization methods for hybrid make-to-order and make-to-stock production in precast concrete component manufacturing

Note: GA = genetic algorithm.

, many studies have assumed that production resources are not limited, and they have not considered the impact of limited resources on the production cycle, which is inconsistent with the actual real-world situation. Therefore, in order to address this shortcoming, some researchers have imposed maximum restrictions on the use of molds to optimize production plans [33,34]. In addition, there are constraints on factory workers and cranes [20]—internal resource constraints such as inventory, labor, and workspace [35], as well as constraints on buffer zones between production stations—that are considered in studies’ improved models [4,36]. At the same time, the characteristics of prefabricated components in highway projects have great limitations in terms of production resources. It is believed that production scheduling is a nondeterministic polynomial (NP) problem [17,36]. Thus, researchers have studied the heuristic method in order to quickly arrive at the best answer, and it has been used in the study of prefabricated-component production scheduling [7]. Among heuristic algorithms, intelligent optimization algorithms primarily have consisted of differential evolution algorithms, ant colony optimization algorithms, genetic algorithms (GAs), and tabu search algorithms [37,38,39]. Genetic algorithms have been found to perform better in prefabrication scheduling [17].

Notwithstanding these advancements, a discernible gap persists in the current body of knowledge. Conventional production scheduling models often fail to adequately incorporate two critical characteristics of large-scale precast plants: the interruptibility of specific processes and the potential for multi-shift or off-hour operations to enhance production continuity. This oversight limits the practical applicability and optimization potential of existing models in real-world, large-scale production environments for highway components, where such factors are paramount for efficient resource utilization.

As a result, this study created a model for optimized resource allocation of big precast concrete equipment that takes into account the “interruptibility” of certain processes, as well as the features of large precast components in highway engineering. This study used a differential evolution (DE) algorithm that is population-based and was first proposed by Storn and Price in 1997. Since its proposal, it has been widely used in many fields, ranging from engineering optimization, image processing, and machine learning [41,42,43] to wireless power transmission [44], finance [45,46], medicine [47], and energy [48,49]. Many scholars have also improved the algorithm, which has been applied to the shop floor scheduling problem [50,51,52]. The algorithm uses disparities across populations to solve continuous optimization issues. The DE’s attractive features include robustness to parameters, minimal space complexity, strong algorithm performance, and ease of use.

The selection of the DE algorithm is well-justified by its proven effectiveness in handling complex optimization problems. Babu et al. demonstrated DE’s superiority over the Non-dominated Sorting Genetic Algorithm in chemical synthesis optimization [53]; Xu et al. developed an adaptive DE variant with multiple mutation strategies for dynamic optimization scenarios [54]; and Eduardo et al. confirmed DE’s capability in handling high-dimensional optimization tasks through big data clustering applications [55]. These studies collectively validate DE’s advantages in continuous search spaces, adaptive mutation mechanisms, and robust performance—making it particularly suitable for the equipment resource optimization problem addressed in this study.

Furthermore, the ongoing refinement of DE algorithms continues to yield significant advancements in both mutation strategies and parameter adaptation mechanisms. For instance, Cui et al. proposed an adaptive DE based on archive reuse [56], while Bu et al. developed a multi-strategy framework utilizing adaptive hash clustering [57]. Innovations also include leveraging local binary patterns for parameter adaptation in multimodal optimization [58], designing deeply-informed mutation strategies to escape local optima [59], and adapting mutation operators based on fitness landscape analysis [60]. Further enhancing the paradigm, researchers have effectively hybridized DE with other optimizers [61], devised dual-stage self-adaptive mechanisms with ensemble strategies [62]. These developments collectively underscore the dynamic evolution and robust capabilities of adaptive DE in solving complex, high-dimensional problems.

The rest of this article is structured as follows. First, the following section provides a literature review. In Section 3, titled “Materials and Methods,” the characteristics, production process, and production line optimization of precast concrete components for highway engineering are introduced. Then, the optimization model is built and solved. Finally, the study’s findings are discussed, the conclusions that can be drawn from the findings are summarized, and their implications are noted. The overall research framework of this paper is illustrated in Figure 1.

2. Materials and Methods

2.1. Analysis of the Production for Precast Concrete Components for Highway Project

2.1.1. Flow Shop Production for Precast Concrete Components

Depending on the production method, precast production can be categorized into two types: flow shop production and fixed location production, with flow shop production being more efficient [34]. Flow shop production is divided into five processes: molding, the placing of rebars and embedded parts, casting, curing, and mold stripping. Each process is carried out on stationary machinery, with a mobile pallet moving the production components on the assembly line to the next process.

The production process of precast concrete components is different from the traditional industrial flow production process and needs to consider the work hours of the workers, as well as the continuity of the process. Moreover, the process of the traditional flow shop is generally not interruptible, whereas some processes for precast concrete components can be interrupted—such as reinforcement tying, moldboard installation, and stripping. Hence, when these processes cannot be completed during work hours, the remaining parts can be continued the next day without interruption. In addition, some processes must be started during working hours.

Unlike the highly automated manufacturing industry, the intelligent assembly line production of precast components cannot be completely separated from manual labor. For example, in the steam curing process, after demolding, the components need to be moved to the steam curing area through the manual control of the steel pedestal, thus requiring that the start time of steam curing be within the working time of workers. However, the process of steam curing is automatically controlled through intelligent monitoring within the control system, so the process of curing can be carried out at any time. These production characteristics must be reflected in the modeling of production for precast concrete components.

The complexity of the production process for precast concrete components leads to a wide range of machinery and equipment, most of which is expensive. It is therefore necessary to optimize the equipment resourcing plan in a scientific and reasonable manner in order to reduce the costs of the precast plant as much as possible and to increase the utilization rate of the equipment.

2.1.2. Layout of Production Lines

According to the precast yard construction mode, the production mode for precast concrete bridge components can be divided into a conventional production line, a fixed mold table production line, and a circular production line. In comparison with a fixed table production line, the circular production lines are conducive to saving production space, integrating various processes, and enhancing production efficiency.

In order to optimize production, this study used an integrated circular production line that was centered on the process flow route, and each process facility was arranged along the production flow direction, with stations for each process. The centerpiece of the construction process was the mobile cart, which was capable of moving around. The process moved through the hydraulic system to drive the mobile cart sequentially to different operating zones, in order to achieve reinforcement processing, reinforcement assembly, steel mold installation, concrete pouring and vibration, moldboard hydraulic stripping, steam curing, tensioning and grouting, and other mobile operations on the assembly line, and then it returned to the starting point through the return track, thus following a circular loop of the circular production line (Figure 2).

The disadvantage of an integrated circular production line is that the production space is generally linear and the layout is fixed. In addition, the production equipment corresponds to a specific product, with high replacement costs being required when changing product types, so the degree of flexibility is relatively low. The advantages of the integrated circular production line are the orderly organization of each step of the production process in a linear space, the optimization of logistics and process paths, and the separation of production functions—which manifests in the relative independence of each process—ease of management, ease of carrying out specialized work, and good compatibility with the degree of automation, as well as a higher degree of automation. For large prefabricated T-beams, the dimensions and models are more standardized. Moreover, the large span, large cross-section size, reinforcement, and beam arrangement are more standardized, which is conducive to standardized and automated construction and has high levels of compatibility with the circular production line, thus making the adoption of the circular production line arrangement appropriate.

2.1.3. Production Characteristics of Precast Concrete Components for Highway Projects

In contrast to traditional precast components, highway project precast components are huge in size, reaching 25 m or more in length. This is the result of the characteristics of highway projects, which require precast beams and slabs to meet larger spans and carry more loads than ordinary civil buildings must support. In fact, in order to satisfy the load requirements, the precast components of highway projects must be configured with complex steel reinforcement. Although some automatic welding machines are able to apply reinforcing grids, in practice, this process is still heavily manual because of the intricacy of internal reinforcement. The complexity of the reinforcement also affects the vibrating process. According to an onsite survey, the automatic vibrating system and manual vibrating coexist.

Larger dimensions and higher load-carrying requirements make the precast component process for highway projects more complex than that for ordinary precast components. There are a total of seven processes: reinforcement processing (M1), reinforcement assembling (M2), steel moldboard installation (M3), concrete pouring and vibrating (M4), moldboard hydraulic stripping (M5), steam curing (M6), and tensioning and grouting (M7) (Figure 3). The typical cycle time for these processes to produce one precast concrete component lasts for 2 h, 3 h, 1.5 h, 14 h, 16 h and 1 h, respectively.

In the past, these highway construction components could only be manufactured at the construction site or in the open air. Now, however, with the increasing industrialization of construction, these components can also be fabricated on assembly lines, but the challenges that come with that progress are also evident. The production pallets, steel moldboards, and other machinery need to be customized for the huge size of the components, and the investment cost for each line is huge. Because of their massive size, such components cannot be turned or lifted on the production line; they can only move forward or backward on a set course. This implies that all lines must be straight and parallel, which increases the footprint of the production workshop. Therefore, it is important to determine the number of sets of machines at the planning stage of the workshop, to enable a configuration of the production line that controls production costs.

2.2. Optimization Model for the Allocation of Production Equipment for Precast Concrete Components for Highway Projects

2.2.1. Basic Assumptions for the Allocation Model of Production Equipment

Working from the characteristics of the production process of large precast concrete components, several assumptions were made for the allocation model of production equipment, as follows

(a)

A machine is capable of processing only one part of a component at a time.

(b)

There is no priority in the processing of different components.

(c)

No other component process can be inserted while the current component is being processed.

(d)

Once a process is started, it must be completed before the next process can be carried out.

(e)

The processing sequence of components must follow the production flow.

(f)

No consideration is given to unexpected situations, such as machinery breakdown and maintenance.

(g)

The adjustment time for machinery and equipment and the transportation time of components between adjacent processes are included in the processing time of the corresponding process.

The model’s parameters were then set, as follows:

N denotes the set of precast component groups, N = {1, 2, 3, …, i, …, n}, where n is the total number of precast component groups and i is produced as the ith component.

M denotes the set of processes, M = {1, 2, 3, …, j, …, m}, where m is the total number of processes and j is the jth process of production.

U_j denotes the sets of machines for the jth process, U_j = {1, 2, 3, …, k_j, …, u_j}, where u_j is the total number of sets of machines in the jth process and k_j is the kth machine.

$x(i_j,k_j)$ indicates whether the jth process of the ith component is performed on the $k_j^{th}$ kj set of machines.

$C_{ij}$ denotes the completion time of the jth process of the ith component.

$t_{ij}$ denotes the processing time of the jth process of the ith component.

Hw denotes the hours worked in a day.

$D_{ij}$ indicates the number of days worked at a time.

$FT$ indicates the set processing period.

2.2.2. Optimization Model for the Allocation of Production Equipment

The issue to be optimized for allocating the production equipment of the prefabrication plant can be expressed as follows. There are n components to be processed in the prefabrication plant flow shop, with m processes for each component. The jth process of the ith component has a set of optional processing machines U_j to determine the optimal allocation of equipment resources, in order to meet the requirements of the duration and the number of components to be processed. The objective is to determine the optimal allocation of equipment resources under the conditions of meeting the schedule and the number of components to be processed.

The objective function was:

$$min Z=\sum _{j=1}^m{u}_j$$

(1)

The objective function is stated in Equation (1). The term $\sum _{j=1}^m{u}_j$ is the sum of the number of sets of machines for all processes, and its minimization allowed for the minimization of the number of sets of machines for each process.

The constraints were as follows:

(1)

Constraints of the production machinery:

$$x(i_j,k_j)=\left\{\begin{array}{ll}1,& \mathrm{t}\mathrm{h}\mathrm{e} j^{\mathrm{t}\mathrm{h}} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} i^{\mathrm{t}\mathrm{h}} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d} \mathrm{o}\mathrm{n} k_j^{\mathrm{t}\mathrm{h}} \mathrm{s}\mathrm{e}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\\ 0,& \mathrm{o}\mathrm{r}\end{array}\right.$$

(2)

$$\sum _{k=1}^{u_j}x(i_j;k_j)=1$$

(3)

Equation (2) is a description of the state of the set of machines and production information. Equation (3) indicates that a process of a component can only be performed on one set of machines and only once.

(2)

Constraints of the completion time

$$T_{ij}=\mathit{max}\{C_{i-1 j},C_{i j-1}\}+t_j,$$

(4)

$$D_{ij}=int\{T_{ij}/24\}$$

(5)

Equation (4) calculates the completion time without considering the working time, which is the start time plus the processing time. Equation (5) utilizes a forward rounding function to calculate the number of days in which the completion time is located.

$$C_{ij}=\left\{\begin{array}{ll}{T}_{ij},& if{ T}_{ij}<24D_{ij}+H_w\\ T_{ij}+H_w,& if{ T}_{ij}\ge 24D_{ij}+H_w\end{array}\right. \forall j\in \mathrm{1,2},\mathrm{3,5}$$

(6)

$$C_{ij}=\left\{\begin{array}{ll}{T}_{ij},& if{ T}_{ij}<24D_{ij}+H_w\\ 24D_{ij}+t_j,& if T_{ij}\ge 24D_{ij}+H_w\end{array}\right. \forall j\in \mathrm{4,7}$$

(7)

$$C_{ij}=\left\{\begin{array}{ll}{T}_{ij},& if T_{ij}<24D_{ij}+H_w\\ 24(D_{ij}+1),& if 24D+H_w\le T_{ij}<24(D_{ij}+1)\hspace{1em}\\ T_{ij},& if 24D_{ij}+H_w\le T_{ij}<24(D_{ij}+1)\end{array}\right.\forall j\in 6$$

(8)

Equations (6)–(8) depict the constraints on the completion times of individual processes. Processes were categorized into three types, on the basis of whether they were interruptible and could only be performed during working hours.

(1)

A the process that is interruptible and can only be performed during working hours: processing (M1), reinforcement assembling (M2), steel moldboard installation (M3), and moldboard hydraulic stripping (M5). Production tasks that are not completed during working hours are scheduled to continue the next day, and this is formulated in Equation (6).

(2)

A process that is not interruptible and can only be performed during working hours: concrete pouring and vibrating (M4) and grouting (M7). Production tasks that are not completed during working hours will not be scheduled on the initial day, but will be scheduled for the next day in terms of starting production. This situation is described in Equation (7).

(3)

A process that is not interruptible and is allowed to take place during non-working hours: steam curing (M6). Whereas the next process needs to be carried out during work hours, the end time for a production task completed during non-working hours will be set as the start time of the next day. The end time for a production task completed during work hours remains the same. Equation (8) expresses this case (Figure 4).

$${\mathit{C}}_{\mathit{n}\mathit{m}}\le \mathit{F}\mathit{T}$$

(9)

Equation (9) is a constraint on the overall time. Here, ${\mathrm{C}}_{nm}$ is the completion time of the last process of the last component (i.e., the final completion time), which must be inferior to the time limit.

2.3. Improved Differential Evolution for the Optimization Model

The mathematical optimization model above shows that the resource allocation optimization model for mechanical equipment is not a typical linear programming model and that it is challenging to find a workable solution that uses normal optimization techniques. For practical application, the intelligent optimization method has a strong adaptive ability for solving complex and challenging optimization problems, even though it cannot guarantee the optimality of the solution, which the conventional optimization method can. In order to solve the model, therefore, this research used the differential evolutionary algorithm. An algorithm flowchart is shown in Figure 5.

2.3.1. Encoding

In this study, process-based integer coding was used. The coding took into account six aspects: the number of components, number of processes, machinery and equipment numbers, the time for processing, the process start time, and the process end time. R is a 7n × 6 matrix, and Column 1 is the component number, with its value being an integer from 1 to n. Each integer occurs seven times and each component needs to go through seven processes to be considered to have completed processing, which is mathematically expressed as Equation (10):

$$R=\left[\begin{array}{ccccc}1& 1& 1& 2& \dots \\ 1& 2& 2& 3& \dots \\ 1& 3& 1& 1& \dots \\ 1& 4& 1& 1.5& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮\\ n& 7& 2& 16& \dots \end{array}\right]$$

(10)

2.3.2. Population Initialization

The population initialization process included setting relevant parameters, such as the population size (NP), individual dimension (D), maximum iteration number (G_max), initial variation operator (F₀), and crossover operator (CR). The initial population is then formed by randomly creating NP individuals with dimension D.

2.3.3. Mutation Operations

In order to attain a faster convergence speed, as well as to maintain the diversity of the population, this study adopted the adaptive mutation operator, basing the approach on the combination of the DE/rand/1/bin mutation strategy and the DE/best/1/bin mutation strategy. The adaptive mutation operator was larger at the start of the iteration, and the primary mutation strategy, DE/rand/1/bin, could guarantee population diversity in order to prevent premature convergence. As the number of iterations increased, the mutation operator decreased and the rate of convergence increased.

$$F_G=F_0\cdot 2^{\mathit{exp}\left(1-G_{max}/(G_{min}+1-G)\right)}$$

(11)

$$\begin{array}{l}{v}_{1,G}=\left\{\begin{array}{ll}{x}_{r1,G}+F_G·\left(x_{r2,G}-x_{r3,G}\right),& rand\left(\mathrm{0,1}\right)\le F_G\\ x_{best}+F_G·\left(x_{r1,G}-x_{r2,G}\right),& otherwise\end{array}\right.\end{array}$$

(12)

Equation (12), where i = 1, 2, 3, …, n, r1, r2, r3 ∈ {1, NP} and i ≠ r1, r2, r3, rand(0, 1) ∈ (0, 1), and xbest is denoted as the best individual in the current population. The adaptive mechanism in Equation (11) ensures a smooth transition from global exploration to local refinement by exponentially decreasing the mutation operator FG as evolution progresses. This design promotes diversity early in the search while enabling precise convergence near optimal solutions later. Equation (12) implements a dual-strategy approach that stochastically alternates between the exploratory DE/rand/1 strategy and the exploitative DE/best/1 strategy. This balanced combination effectively navigates the complex solution space of equipment allocation.

2.3.4. Crossover Operation

To retain some information from the original individual while updating it, the target individual underwent a mutation and created a new, mutated individual. This mutated individual was then cross-operated with the target individual to facilitate the exchange of information. The target individual x_i,G and its variant individual v_i,G underwent a crossover operation, as described by Equation (13), to produce the test individual u_i,G.

$$u_{i,G}^j=\left\{\begin{array}{ll}{\nu }_{i,G}^j,& if \left(rand\left(\mathrm{0,1}\right)\le CR or j=rand\left(D\right)\right)\\ x_{i,G}^j,& otherwise\end{array}\right.$$

(13)

Equation (13), where i = 1, 2, 3, …, n, j = 1, 2, 3, …, D, CR ∈ (0, 1), and rand(D) is a [1, D] random integer, prevents invalid crossover operations and guarantees that there is at least one dimension of (rand(D)) information coming from the variant individuals.

2.3.5. Select Operation

The test individuals and target individuals were selected via one-to-one comparison through the objective function, leaving the individuals with smaller objective values to form a new population in order to complete an iterative search. The target individual x_i,G and the test individual u_i,G were selected, as shown in Equation (14), to obtain the next generation target individual x_i,G+1.

$$x_{i,G+1}=\left\{\begin{array}{ll}{u}_{i,G},& if\left(F\left(u_{i,G}\right)\le F\left(x_{i,G}\right)\right)\\ x_{i,G},& otherwise\end{array}\right.$$

(14)

where i = 1, 2, 3, …, n, F(·) is the objective function.

3. Case Study

3.1. Case Profile

In order to verify the feasibility and effectiveness of the model and algorithm, this study selected the T-beam production line of a precast factory in Anhui Province as an example with which to carry out its research on the optimization of equipment resource allocation. This case was an exploratory case study [63,64] with a precast factory that was noted as “A precast factory”. The construction area for “A precast factory” was then divided into a light T-beam-prefabrication area; a box culvert; an arch-culvert prefabrication area; a round-pipe, covered-beam prefabrication area; and a steel processing yard. The factory was primarily responsible for the production of prefabricated components such as lightweight T-beams, cover beams, box culverts, arch culverts, round pipe culverts, and small prefabricated components. In this case, the integrated circular production line mentioned above was used to take full advantage of its high degree of automation. The production order for A prefabrication factory was for a 25-m lightweight T-beam with 1920 pieces and five production lines, with a planned construction period of 18 months. Therefore, as discussed in this section, 1920 T-beams were simulated for equipment configuration optimization. The process included primarily seven processes: reinforcement processing, reinforcement assembling, steel moldboard installation, concrete pouring and vibrating, moldboard hydraulic stripping, steam curing and tensioning, and grouting. The processing times for each process were [2, 3, 1, 1.5, 14, 16, 1]. The work hours of the workshop workers were 8:00 to 12:00 and 14:00 to 18:00, with a single break. break.

3.2. Simulation Analysis

3.2.1. Experimental Preparation

The MATLAB2021 coding software platform was used in this study to implement the equipment resource allocation optimization simulation. The computer used had a CPU main frequency of 3.0 GHz, 16 GB of memory, and Windows 10 installed. A total of 25 distinct simulations were run, with 100 iterations serving as the termination criterion.

3.2.2. Experimental Results and Analysis

The differential evolution algorithm iteration curve is shown in Figure 6. The optimal allocation of equipment resources was [2, 2, 1, 2, 5, 5, 1], which was basically the same as the original proposed program. However, when the construction period was changed to 18.5 months, the optimal allocation became [1, 2, 1, 1, 4, 4, 1], and the number of equipment resources needed was significantly reduced, thus confirming the feasibility and effectiveness of the model and algorithm. Efficient optimization of equipment resource allocation was crucial for increasing the use of equipment in order to lower production costs and boost the business’s economic efficiency.

In order to further verify the validity of the model and algorithm, the schedule, the number of components, and the processing time of the process were adjusted, making it possible to determine the optimal plan for the configuration of the number of resources in terms of equipment, as shown in Appendix A. The results indicated that when the equipment configuration program number was [2, 2, 1, 2, 5, 5, 1], and the processing time for each process [2, 3, 1, 1.5, 14, 16, 1] was reduced by two months, the required schedule was only 16 months, which is 11.11% better than the original program of 18 months.

In consideration of the adjustments of the number of components, the results showed that when using a unified configuration scheme for the number of pieces of equipment, the processing time, and the duration of the work, the maximum number of components produced by the configuration scheme [2, 2, 1, 2, 5, 5, 1] that could be satisfied in the duration of 18 months was 2330, which was an increase of 21.4% compared with the enterprise’s original number of components produced.

A higher monthly production demand resulted in a lower average quantity of equipment needed and a lower cost; in addition, a larger production scale resulted in a somewhat lower average cost of equipment inputs. The models and algorithms presented in this paper can also solve for the duration and the number of components produced, serve to obtain the maximum production demand that can be met by a certain equipment resource allocation scheme, and establish the optimal schedule.

4. Discussion

The case study results demonstrate the effectiveness of the proposed methodology through multiple quantifiable outcomes. Under identical resource configurations, the optimization model achieved an 11.11% reduction in project duration while maintaining the same production scale of 1920 T-beams. Concurrently, the model enabled a 21.4% increase in production capacity—from 1920 to 2330 components—within the original 18-month project timeline. These improvements, systematically validated through the scenario analyses presented in Appendix A.1, Appendix A.2 and Appendix A.3, demonstrate the model’s robust capability to enhance resource utilization efficiency while satisfying complex production constraints.

The consistent performance of the improved Differential Evolution algorithm stems from its carefully designed architecture. The integration of an adaptive mutation operator with a dual mutation strategy (DE/rand/1/bin and DE/best/1/bin) created a dynamic balance between exploration and exploitation throughout the optimization process. This methodological innovation enabled the algorithm to maintain population diversity while achieving accelerated convergence, as empirically demonstrated by the stable iteration curve in Figure 5. The algorithm’s reliability in addressing the high-dimensional equipment allocation challenge is further evidenced by its consistent convergence to feasible solutions across all tested scenarios in Appendix A.1, Appendix A.2 and Appendix A.3.

Compared to conventional approaches in precast production scheduling, the proposed model represents a substantial methodological advancement through its systematic incorporation of two critical production characteristics: process interruptibility and worktime constraints. These features address fundamental aspects of precast operations that are typically oversimplified or neglected in existing methodologies. By accurately representing these production realities within a mathematical optimization framework, our approach provides more practical and operationally relevant solutions for large-scale component manufacturing. This is particularly valuable in highway projects, where the demonstrated improvements in operational efficiency directly translate to enhanced project viability and substantial economic benefits, as quantified in our case study.

5. Conclusions

This study addresses the optimization of equipment resource allocation for the large-scale production of precast concrete components in highway engineering. The conclusions are structured as follows:

5.1. Scientific Results

The primary scientific contributions to construction science are threefold. First, a realistic production line model was successfully developed, incorporating the interruptibility of specific processes and the feasibility of non-working-hour operations, which addresses limitations in conventional scheduling models. Second, a constrained optimization framework was formulated to minimize equipment resources while adhering to process continuity, working-hour restrictions, and project deadlines. Third, an improved differential evolution algorithm was implemented and validated, featuring an adaptive mutation operator and a dual mutation strategy that significantly enhanced population diversity and accelerated convergence speed.

5.2. Applied Engineering Results

The practical engineering value for real-world road projects was demonstrated through a case study. The proposed model and algorithm achieved an 11.11% reduction in project duration and a 21.4% increase in production capacity under the same resource configuration. The optimal equipment allocation scheme effectively reduced resource consumption while meeting production targets, confirming the feasibility and effectiveness of the approach for actual prefabrication factories.

5.3. Limitations and Risks

Despite the achievements, several limitations and risks warrant attention. The model assumes equal priority among components, no transportation time, fixed working hours, and neglects overtime scenarios during schedule compression, which might reduce the real-world applicability. Seasonal climatic variations affecting curing and stripping times are not accounted for, potentially impacting accuracy. The optimization focuses on resource quantity without considering equipment costs, which is critical for economic objectives. Furthermore, real-world uncertainties such as machinery breakdowns, maintenance, and supply chain fluctuations were not incorporated, posing risks to implementation stability.

5.4. Future Prospects

Future research can be pursued in both scientific and applied areas. Scientifically, the model can be extended by integrating cost objectives and climate-adaptive time parameters. It is suggested to compare the proposed algorithm with other algorithms in terms of accuracy or efficiency. The algorithm’s performance can be further refined by integrating the strengths of other intelligent optimization algorithms, aiming to enhance its robustness, convergence speed, and population diversity. From an applied perspective, the framework can be adapted for dynamic rescheduling to manage production disruptions and scaled to other large-scale prefabrication projects, including stadiums and rail systems, to broaden its industrial impact. It is also suggested to consider machinery breakdown and maintenance time and transportation time among different procedures of precast concrete production in order to simulate the actual practices.