Dynamic Optimization of Tunnel Construction

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1. Introduction

Tunnel construction is a critical component of railway engineering, with its quality and efficiency directly affecting the safety and reliability of the entire railway system [1]. Tunnel construction delays can cost millions in railway and urban infrastructure projects. Due to complex geological conditions, advanced technical requirements, and stringent schedule and cost control, tunnel engineering remains a significant challenge. As a typical representative of linear projects, tunnel engineering exhibits significant repetitive characteristics during construction [2,3], such as the cyclic operations involved in tunnel construction. These repetitive processes not only provide some regularity to construction management but also pose higher demands for optimizing construction workflows [4]. Particularly in reverse construction, precise scheduling is crucial for timely completion, along with strict quality and safety measures. Reverse construction refers to a method that incorporates bidirectional construction, where work progresses simultaneously from multiple directions toward a meeting point. This approach is commonly used in tunnel projects to accelerate progress, optimize resource allocation, and reduce overall construction time. However, in tunnel construction, addressing uncertainties such as geological conditions require flexible scheduling, strengthened on-site management, and robust emergency response capabilities. Contractors typically aim to shorten the duration of projects to meet contractual deadlines, avoid delays caused by adverse weather and site conditions, and minimize penalties for contract breaches and delays [5]. To achieve this goal, numerous optimization models have been developed, such as shortening the duration through the optimization of construction continuity [6,7,8]. Optimizing construction continuity is typically beneficial for linear projects [9], including minimizing idle time, stabilizing labor, and maximizing learning curve effect [10,11]. However, strict adherence to work continuity may conflict with the project schedule. Determining an effective schedule in terms of both time and cost is one of the most significant challenges that managers face during the planning process [12]. The optimization objective of this study is to minimize the duration of tunnel construction projects while satisfying constraints such as construction continuity. However, the scheduling of linear projects is dynamic and complex, and static approaches have inherent limitations [13]. Integrating dynamic management with tunnel construction schedule optimization enhances project flexibility and responsiveness, ultimately improving success rates, reducing costs, and maximizing resource utilization [14]. To further strengthen adaptability and efficiency, it is crucial to develop advanced scheduling models that effectively account for challenges such as fluctuating geological conditions and stringent cost control.

The Resource-Constrained Project Scheduling Problem (RCPSP) is widely applied in the management of construction, logistics, and manufacturing industries [3]. In recent years, the scheduling optimization of linear projects has received significant attention from researchers. Research in this field has not only deepened our understanding of complex project management strategies but has also driven the development of relevant algorithms and techniques, with the goal of maximizing project efficiency and benefits in resource-constrained environments. However, current optimization approaches for tunnel construction scheduling are predominantly static, making them ineffective in addressing the various uncertainties and dynamic changes encountered during construction [9]. Moreover, adjustments to the construction schedule on-site are primarily based on manual experience, limiting improvements in construction efficiency and failing to ensure the accuracy and reliability of the schedule. Therefore, exploring more flexible, dynamic, and adaptive tunnel construction scheduling methods, particularly in the context of reverse construction, is crucial for addressing the challenges posed by complex and variable geological conditions, technical difficulties in construction, and stringent requirements for schedule and cost control.

Traditional scheduling methods, such as the Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT), exhibit considerable limitations when applied to linear projects [15]. They fail to accurately represent the location and speed of construction activities, struggle to ensure continuous resource supply, and lack effective spatial representation. Linear scheduling techniques, which serve as core methods for scheduling optimization in linear projects, primarily include Linear Scheduling Method (LSM) and Line-of-Balance (LOB). There are key differences between the two: LOB is event-based [14] and typically does not account for variations in unit scale, whereas LSM is schedule-based. Unlike traditional fixed logic, soft logic refers to a flexible dependency relationship between construction processes, allowing the sequence of activities to be adjusted under specific conditions. LSM clearly illustrates the sequence and progress of construction activities, with particular advantages in handling activities of various types, such as linear, strip, and block activities. Despite the significant advantages of LSM in optimizing linear projects, particularly its ability to comprehensively and accurately reflect complex construction scenarios involving both forward and reverse construction as well as changes in the construction sequence, research on the dynamic optimization of tunnel construction scheduling remains relatively insufficient [11]. This limitation not only restricts the deeper application of LSM in tunnel construction but also hinders the further development of tunnel construction scheduling optimization techniques. Therefore, it is essential to strengthen research on the dynamic optimization of tunnel construction scheduling, exploring strategies and optimization methods for LSM in complex construction scenarios to better meet the practical needs of tunnel construction.

In recent years, scheduling optimization for linear projects has received significant attention in both academia and industry. However, despite its critical role, research on tunnel construction scheduling remains limited, especially in scenarios where construction sequences and directions change. Traditional methods like CPM struggle to accommodate dynamic environmental conditions, resource allocation, and bidirectional construction [15]. While LSM provides advantages in visualizing and optimizing linear projects, its application in tunnel construction is still underdeveloped, limiting its effectiveness in complex, real-world scenarios. Additionally, despite advancements in BIM, AI [16,17], and digital twins [18], their integration into tunnel scheduling remains insufficient. On-site schedule adjustments continue to rely heavily on manual experience, reducing efficiency and accuracy. To address these challenges, we propose a dynamic optimization model that integrates LSM, soft logic within the Work Breakdown Structure (WBS), and the Resource Breakdown Structure (RBS). The key components of this paper are as follows:

(1). A dynamic optimization model for tunnel construction scheduling has been developed by integrating LSM, soft logic, WBS, and RBS methods. This model aims to comprehensively assess the impact of changes occurring during the tunnel construction process on the schedule, particularly variations in construction sequence and direction, thereby providing more precise support for optimizing.
(2). A Mixed-Integer Programming (MIP) model considers complex constraints, including logical, temporal, spatial, resource, and continuity factors, with a focus on reverse construction scenarios. This approach emphasizes the effects of changes in construction sequence and direction for more accurate decision support.
(3). A comparative and empirical study of exact algorithms and Genetic Algorithms (GA) was conducted, highlighting their effectiveness in optimizing tunnel construction schedules. The analysis provides insights into the advantages and limitations of both algorithms in terms of efficiency and adaptability, offering valuable guidance for both practical and academic use.

2. Literature Review

2.1. The Resource-Constrained Project Scheduling Problem for Pepetitive Projects

The RCPSP is a standard problem in project scheduling [19], first introduced by Pritsker et al. in 1969 [20]. With a strong engineering background, it is widely applied in project management, manufacturing, and other fields, playing a significant role in the scheduling optimization of repetitive projects. Construction projects are generally categorized into two types: repetitive and non-repetitive, with tunnels, bridges, and high-rise buildings being typical examples of repetitive projects [21]. As an NP-hard problem, RCPSP has been extensively studied, leading to the development of diverse models and solution methods. RCPSP primarily includes three essential elements: the objective function, activities, and resources. Ding et al. [22] extended the research to address variants of the problem, such as multi-objective, multi-resource, multi-mode, multi-skill, and multi-project scenarios. In repetitive projects, optimization often involves mathematical models that incorporate soft logic [23]. Huang et al. [24] developed such models and demonstrated their advantages in enhancing scheduling flexibility and reducing project duration and costs. The main objective of scheduling optimization is to achieve both duration and cost reductions by improving the continuity of work and resources. The main optimization objectives in the research on RCPSP for repetitive projects currently include the following: (1) project duration [25]; (2) project cost [26]; (3) joint optimization of project duration and cost [27]; (4) weighted sum of project duration, cost, and quality [28,29]; and (5) the optimal trade-off between project duration and work interruptions [30]. Balancing efficiency, quality, and minimizing work interruptions ensures continuity, reduces downtime, and boosts productivity—crucial for the success of repetitive projects, which require consistent performance and optimized resource use.

Currently, methods for solving the RCPSP include exact algorithms, heuristic algorithms, and metaheuristic algorithms. Exact algorithms mainly solve the problem through mathematical programming, such as branch-and-bound algorithms [31], cutting-plane methods [32], integer programming [33], and dynamic programming. Shafahi et al. [34] proposed a MIP model that considers project selection and scheduling with segmented implementation. Exact algorithms guarantee optimal solutions but are computationally expensive and impractical for large-scale problems due to their high complexity. For large-scale scheduling optimization problems, researchers often employ heuristic and metaheuristic algorithms [35], such as GA and Particle Swarm Optimization (PSO) [36,37]. These methods can obtain optimal or near-optimal solutions in a relatively short time [38]. Among these, GA are widely used due to their versatility and simplicity [39]. Long et al. [40] developed a GA-based approach to accelerate construction progress by optimizing work interruptions and overtime, thereby reducing project duration. Laszczyk et al. [41] combined the classical NSGA-II with the multi-skill resource-constrained project scheduling problem, using a weighted method to calculate the objectives. Hu et al. [42] developed a refined multi-objective, multi-mode, multi-team NSGA-III optimization model for multi-objective repetitive project scheduling problems with complex constraints, considering both renewable and non-renewable resource constraints. Furthermore, Yao et al. [43] designed an NSGA-II algorithm to solve the Pareto front of the trade-off model between carbon emissions, duration, and cost under a new scheduling strategy, and successfully validated its effectiveness through case studies. Heuristic algorithms provide fast, practical solutions but may not achieve optimality and can get stuck in local optima. Metaheuristic algorithms offer flexible, near-optimal solutions but require parameter tuning and may struggle with convergence in highly constrained, large-scale problems.

2.2. Linear Project Scheduling Optimization Based on the Linear Scheduling Method

Linear scheduling techniques, including LSM and LOB, are the primary methods for optimizing linear project scheduling. LSM has proven uniquely advantageous for linear projects [44,45]. Originally developed for highway construction optimization [46], LSM graphically represents spatial progress over time, illustrating trends in construction speed. This visualization has become an effective tool for continuous and focused construction schedule management. The Critical Activity Path (CAP) denotes the critical path in construction plans developed using LSM. Activities are categorized as critical, non-critical, or partially critical based on their placement on the CAP. Within the LSM planning framework, repetitive tasks throughout the project lifecycle are depicted as lines with constant or variable slopes, while activities maintaining a fixed state or completing within a defined time period are shown as blocks or bar charts. This visual representation facilitates an intuitive understanding of activity continuity, frequency, and changes in these attributes [47].

LSM exhibits pronounced engineering characteristics and has been widely applied in linear project scheduling research. Georgy [48] utilized LSM and GA to develop a resource scheduling optimization model for linear projects, reducing resource fluctuations by optimizing buffers and construction speed. While GA enhances resource scheduling by optimizing buffers and construction speed, it may struggle with premature convergence. Altuwaim et al. [49] proposed a two-stage optimization model based on LSM, focusing on optimizing construction time and interruptions. García-Nieves et al. [50] integrated site conditions to construct an LSM-based linear programming model that accounts for multi-mode, multi-worker, multi-shift, sub-activity levels, and continuity, achieving a trade-off between construction time and cost while providing a more refined solution for linear project scheduling. Linear programming models provide precise optimization but may struggle with highly complex constraints. Wang et al. [51] incorporated LSM with new constraints, such as mileage and resource limitations, to establish a floating-point-based resource leveling model for linear projects. Yu et al. [52] investigated the effects of reverse construction activities in linear project scheduling under scenarios involving multiple segments, teams, modes, and construction directions. Rahman et al. [53] integrated LSM reflecting labor allocation with Monte Carlo simulations and a multi-objective optimization model, effectively minimizing idle time at offsite workstations and project completion time. Monte Carlo simulations improve risk assessment but require extensive computational resources. In summary, LSM, as a critical tool in linear project scheduling research, continues to drive advancements and innovations in the field.

2.3. The Dynamic Scheduling of Projects

Scheduling optimization for construction projects can be divided into the planning and implementation phases [9]. However, both internal factors during project implementation and external factors such as natural environments and economic issues can cause fluctuations in the original construction schedule [54]. For example, encountering geological faults during tunnel construction may necessitate halting excavation work. Therefore, scheduling optimization should not be confined to static solutions but should instead adopt dynamic approaches. Scheduling can be categorized into reactive and proactive approaches [55], with proactive and reactive scheduling strategies being the most extensively studied methods for handling projects with stochastic durations [56]. Ning et al. [57] proposed proactive and reactive project scheduling problems, developing an optimization model that incorporates time buffers into baseline schedules. Reactive scheduling refers to adjustments made during project execution to respond quickly to disruptions and changes [58]. Fahmy et al. [59] defined dynamic scheduling as an ongoing, dynamic process that updates, evaluates, and modifies schedules based on the selected scheduling framework and predefined rescheduling strategies, rules, and techniques. Research in the field of dynamic scheduling is showing increasing diversification and interdisciplinary integration. Some scholars classify dynamic scheduling into reactive and proactive-reactive categories [60,61,62], while others combine informatization with reactive scheduling to develop dynamic scheduling systems that adapt to complex and rapidly changing data-processing environments [63].

Recent studies have increasingly explored the use of reinforcement learning (RL) in dynamic scheduling, especially in environments where uncertainty and real-time adjustments are prevalent [64]. Alongside these advancements, there has been growing interest in using optimization algorithms to tackle reactive scheduling models, with the goal of maximizing or minimizing objective functions while adhering to resource and precedence constraints. These approaches include exact algorithms, heuristic algorithms, and metaheuristic optimization algorithms. El-Rayes et al. [65] proposed an automated optimization model combining dynamic programming, scheduling algorithms, and interruption algorithms to automatically generate disruptions during the dynamic scheduling process. To mitigate the combined effects of uncertainty and disruptions during project execution, Rahman et al. [66] proposed a real-time reactive scheduling method, followed by the adoption of a metaheuristic method named IGFBIS and IGFBID to address the issue. Hegazy et al. [67] developed a decentralized optimization framework for repetitive projects, determining the optimal combination of workforce, construction methods, interruption times, and activity locations while considering current progress. Satic et al. [68] tested the performance of dynamic programming algorithms, genetic algorithms, optimal reactive baseline algorithms, and priority-based heuristic algorithms, concluding that these methods are suitable only for instances with low uncertainty. Peng et al. [69] applied a discrete-time Markov decision process model to the reactive scheduling phase. They embedded a lookup table method into reinforcement learning to enable dynamic scheduling and passive adjustment of schedules during project execution using benchmarks.

2.4. Research Gap

The development of RCPSP in repetitive projects has shown a trend toward increasing complexity, incorporating multi-objective, multi-mode, multi-resource, and multi-skill considerations. Logical constraints, continuity constraints, and resource constraints are the primary limitations in the scheduling optimization process. However, research on tunnel construction schedule optimization under resource constraints is relatively sparse. Many restrictive factors, such as logical relationships, construction direction, and work continuity constraints, still require more in-depth exploration and analysis, especially the connection between construction schedule optimization and soft logic. This situation presents more stringent and urgent requirements for the design and application of optimization algorithms. These challenges are reflected not only in the number of research outcomes but also in the fact that existing studies often fail to capture the unique and complex nature of tunnel construction. Therefore, there is an increased demand for optimization algorithms that can adapt to complex construction environments while ensuring these algorithms accurately reflect the practical realities of tunnel construction.

Compared to traditional construction planning and scheduling methods, LSM has demonstrated unique advantages in optimizing the scheduling of linear projects, particularly in handling specific project details. However, its application still requires further research. Despite LSM’s excellent performance in linear project scheduling optimization, it is often limited by the assumption of fixed logical relationships between project activities, without adequately considering dynamic changes in construction sequences. This limitation restricts its broader applicability in practical scenarios. Furthermore, there is currently a lack of systematic frameworks or models that effectively address the logical relationships in reverse construction scenarios of tunnel engineering. To enhance the understanding and optimization of complex tunnel construction processes, further in-depth research is required. Therefore, to overcome this limitation and develop more effective and feasible scheduling, it is essential to combine LSM with other advanced theories or methods, which can address the complexity and uncertainty challenges of the logical relationships between activities.

The scheduling of linear projects is inherently complex and dynamic. However, the current management and optimization techniques are primarily based on static or deterministic principles, highlighting the mismatch between industry practices and existing technologies. Compared to static scheduling, research on dynamic scheduling is scarce. However, dynamic scheduling can better address various unforeseen changes and disruptions in the construction process, enabling timely adjustments to the schedule and minimizing resource waste, thereby ensuring the smooth progress of linear projects. Therefore, strengthening the research and application of dynamic scheduling strategies in tunnel engineering is of crucial importance for enhancing the flexibility and adaptability of linear project management, as well as driving technological innovation in the industry.

3. Methodology

3.1. Problem Formulation and Planning

3.1.1. Problem Description

The duration is often one of the most critical objectives in construction projects. The construction schedule is typically adjusted based on the project timeline, and when delays occur due to various factors, project managers usually need to adjust resource supply, construction sequencing, and other aspects to accelerate progress. In tunnel construction, the development of auxiliary tunnels is an effective method for accelerating the construction schedule, as illustrated in Figure 1. By excavating cross passages and creating additional working faces, multiple construction tasks can be carried out simultaneously, thus significantly reducing the overall project duration. The parallel adits are connected to the main tunnel through cross passages, with each newly opened cross passage providing a new working face for the main tunnel. This construction method can improve work efficiency. However, due to the requirement for construction continuity, each additional working face can only be switched after a construction team completes a continuous section. Therefore, determining how to reasonably arrange the construction sequence and adjust resource allocation becomes a critical factor in optimizing the construction schedule.

As shown in Figure 1, D represents the minimum distance constraint between the newly added working face and the nearest working face in the same direction. The distance d between the two working faces should be greater than the minimum distance constraint.

The situation at the construction site and the limiting factors are not fixed, so managers must adopt flexible strategies based on continuously updated data to adapt to various changes. While the general key routes of the entire tunnel project are mostly fixed, the specific construction plan will evolve with the progress of construction. However, as construction advances, non-critical paths may become critical paths, requiring managers to dynamically adjust the supply of resources such as labor, materials, equipment, and funds. Based on the actual situation on-site, adjustments to the construction sequence and resource allocation can be made flexibly.

Following the basic constraints of construction logic, continuity, time, space, and resources, determining the optimal location and timing for opening additional working faces to shorten the project duration is a core issue in construction management at this stage. It is especially important to note that the optimization of the tunnel construction schedule should adhere to the expected completion time constraint to ensure timely completion. During construction, distance constraints are often regarded as crucial standards to ensure safety, efficiency, and regulatory compliance. However, it is important to recognize that these distance constraints are not fixed and can be adjusted based on the specific conditions and actual needs of the construction site. Additionally, the construction schedule must account for limitations in the supply of labor, materials, machinery, and other resources to ensure efficient resource utilization and prevent delays caused by resource shortages.

3.1.2. Dynamic Optimization Model for Tunnel Construction Scheduling

Tunnel construction projects are often accompanied by significant uncertainty due to variable geological conditions, complex construction environments, and high potential risks. This uncertainty causes frequent changes in project conditions during construction, posing challenges to construction schedule and resource allocation. If the existing construction teams are working at maximum speed and still cannot meet the established project timeline, scheduling optimization becomes necessary. This optimization can be achieved by adjusting construction team scheduling strategies or increasing the number of construction teams, effectively shortening the project duration and enhancing the scientific management and efficiency of construction progress. LSM is a scheduling method used to plan and manage linear projects and is widely applied in this context. Although previous research has made important contributions in this area, most studies have not adequately considered the dynamic changes in construction sequence and resource supply, nor have they developed a comprehensive dynamic optimization model for targeted adjustments. Based on this, we integrate dynamic scheduling for construction, the LSM method, soft logic relationships, and the WBS and RBS techniques to propose the dynamic optimization model for tunnel construction scheduling (TCSDOM), as illustrated in Figure 2.

WBS can decompose the project into construction units and construction activities [70]. Construction units are the basic spatial divisions of the construction process, while construction activities are the specific tasks performed within these units. There are hierarchical and logical relationships between the two. Construction activities are carried out within the scope of the construction units, and activities in different units need to meet sequencing logic, resource allocation, and spatiotemporal constraints, while avoiding interference with each other. The division of construction units and the arrangement of activities directly affect the construction schedule, resource utilization, and overall efficiency. Scientifically optimizing the relationship between the two is key to achieving rational and efficient construction scheduling. RBS mainly includes labor, machinery, materials, and construction methods, with labor and equipment forming construction teams. The optimization scheduling of a project must meet various constraints, including logical, continuity, temporal, spatial, and resource constraints. If the optimization results meet the project requirements, a new schedule should be developed based on these results. If not, the schedule can be revised by increasing the number of construction teams or making other adjustments until the project meets the requirements. WBS and RBS are typically used together. In tunnel construction, the project can first be decomposed spatially into construction units, and then activities can be detailed according to the process flow, while considering the coordination of time and resources to ensure the scientific and rational construction organization.

Tunnel projects exhibit significant linear characteristics, with tasks typically distributed along the tunnel axis and progressing segment by segment. LSM combines time and space, visually representing the position of each construction unit in terms of both time and space, as well as the interrelationships between different construction processes. A key feature of such projects is that they involve relatively few processes but a large number of construction units. Therefore, the division of construction units and the sequence in which they are arranged significantly impact the overall project duration and cost. In tunnel engineering, the excavation of auxiliary tunnels provides flexibility for the main tunnel construction, resulting in an adjustable construction sequence. The adjustable nature of construction units can be reflected in LSM charts, allowing tasks to be performed in parallel or rearranged based on actual requirements. This characteristic provides the possibility for optimizing construction schedules and resource allocation, helping to improve construction efficiency and reduce project delays under complex engineering conditions.

3.2. Dynamic Optimization Rules

The dynamic optimization rules for tunnel construction scheduling involve multidimensional coordination and management, primarily focusing on five aspects: logical, temporal, spatial, resource, and work continuity constraints, as shown in Figure 2. Additionally, target changes must remain within defined limits. Logical constraints ensure that the sequence of operations meets technical requirements and safety standards, preventing conflicts or redundancy in the processes. Temporal constraints require that construction tasks be completed within a specific time frame, with dynamic adjustments to the schedule to accommodate critical paths and task dependencies. Spatial constraints address the limitations of the construction site, requiring careful planning of equipment layout, working face divisions, and so on. Resource constraints emphasize the availability and efficient allocation of labor, machinery, materials, and other resources, preventing wastage and shortages. Continuity constraints focus on maintaining the consistency and stability of construction activities, avoiding sudden interruptions or unnecessary delays during the process. Together, these constraints work in coordination, providing a systematic framework for dynamic optimization in construction.

3.2.1. Parameters

The parameters referenced by the dynamic optimization rules are shown in Table 1.

3.2.2. Objectives

LSM represents both the temporal and spatial dimensions simultaneously on a single chart. As shown in Equation (1), the start time of the entire construction schedule is defined as the start time of the construction activity without any predecessors, which is marked as 0 in the established coordinate system.

(1) $S_{i, j_{i}^{S}} = 0$

The duration refers to the longest time required for the completion of all construction tasks, which is determined by the time span of the latest completion date among all tasks, as presented in Equation (2).

(2) $M i n T = m a x (E_{i, j})$

3.2.3. Constraints

(1). Logical constraints

The logical relationships between construction activities refer to the dependencies of various construction tasks in terms of time and resources, which determine the sequence and coordination of different activities. Common logical relationships include finish-to-start (FS), start-to-start (SS), finish-to-finish (FF), and start-to-finish (SF). If activity $a$ is the predecessor of activity $b$ , these logical relationships can be represented by corresponding equations to reflect their time dependencies and sequential relationships.

a.. The FS relationship

The subsequent construction activity can only commence once the preceding activity has been officially declared complete, as stated in Equation (3).

(3) $E_{a, j} \leq S_{b, j}, \forall a, b \in N$

b.. The SS relationship

The subsequent activity can only commence after the preceding activity has started, as demonstrated in Equation (4).

(4) $S_{a, j} \leq S_{b, j}, \forall a, b \in N$

c.. The FF relationship

The subsequent construction activity can only end after the preceding construction activity has finished, as illustrated in Equation (5).

(5) $E_{a, j} \leq E_{b, j}, \forall a, b \in N$

d.. The SF relationship

The subsequent construction activity can only end after the preceding construction activity has started, as depicted in Equation (6).

(6) $S_{a, j} \leq E_{b, j}, \forall a, b \in N$

In construction practice, although the LSM method primarily focuses on sequential relationships, it also allows for parallel activities, enabling multiple construction tasks to be carried out simultaneously, provided they do not interfere with each other.

(2). Temporal constraints

The definition of temporal constraints depends largely on the nature of the construction units and the activities executed within them. Linear projects primarily categorize construction activities into three main types: linear, strip, and block activities. It is worth noting that strip activities, due to their unit span of zero, can be considered a special type of block activity and categorized together for discussion. Consider an example where construction activity $e$ precedes $f$ , and construction unit $g$ comes before $h$ . Taking the minimum temporal constraint at the junction between construction units $g$ and $h$ as an example, this constraint can be specifically described using the following equation.

a.. As shown in Figure 3, when both activities $e$ and $f$ are linear activities, the temporal constraints are shown in Equations (7)–(10).

(7) $d_{f} \times S_{f, h} - d_{e} \times S_{e, h} \geq I n t {m i n}_{e, f}$

(8) $(1 - d_{f}) \times S_{f, g} - (1 - d_{e}) \times S_{e, g} \geq I n t {m i n}_{e, f}$

(9) $d_{f} \times S_{f, h} - (1 - d_{e}) \times S_{e, g} \geq I n t {m i n}_{e, f}$

(10) $(1 - d_{f}) \times S_{f, g} - d_{e} \times S_{e, h} \geq I n t {m i n}_{e, f}$

The above equations can be transformed into Equations (11) and (12):

(11) $d_{f} \times E_{f, j} + (1 - d_{f}) \times S_{f, j} - d_{e} \times E_{e, j} - (1 - d_{e}) \times S_{e, j} \geq I n t {m i n}_{e, f}$

(12) $d_{f} \times E_{f, j} + (1 - d_{f}) \times S_{f, j} - d_{e} \times E_{e, j} - (1 - d_{e}) \times S_{e, j} \leq {I n t m a x}_{e, f}$

b.. As shown in Figure 4, when activity $e$ is a linear activity and activity $f$ is a strip or block activity, the temporal constraints are shown in Equations (13) and (14).

(13) $S_{f, h} - d_{e} \times S_{e, h} \geq I n t {m i n}_{e, f}$

(14) $S_{f, h} - (1 - d_{e}) \times S_{e, g} \geq I n t {m i n}_{e, f}$

The above equations can be transformed into Equations (15) and (16):

(15) $S_{f, j} + - d_{e} \times E_{e, j} - (1 - d_{e}) \times S_{e, j} \geq I n t {m i n}_{e, f}$

(16) $S_{f, j} + - d_{e} \times E_{e, j} - (1 - d_{e}) \times S_{e, j} \leq {I n t m a x}_{e, f}$

c.. As shown in Figure 5, when activity $e$ is a strip or block activity and activity $f$ is a linear activity, the temporal constraints are shown in Equations (17) and (18).

(17) $d_{f} \times S_{f, h} - S_{e, h} \geq I n t {m i n}_{e, f}$

(18) $(1 - d_{f}) \times S_{f, g} - \times S_{e, h} \geq I n t {m i n}_{e, f}$

The above equations can be transformed into Equations (19) and (20):

(19) $d_{f} \times E_{f, j} + (1 - d_{f}) \times S_{f, j} - E_{e, j} \geq I n t {m i n}_{e, f}$

(20) $d_{f} \times E_{f, j} + (1 - d_{f}) \times S_{f, j} - E_{e, j} \leq {I n t m a x}_{e, f}$

d.. As shown in Figure 6, when both activities $e$ and $f$ are strip or block activities, the temporal constraints are shown in Equations (21) and (22).

(21) $S_{f, j} - E_{e, j} \geq I n t {m i n}_{e, f}$

(22) $S_{f, j} - E_{e, j} \leq {I n t m a x}_{e, f}$

Furthermore, the latest completion time for all activities must fall within the project duration specified in the contract or before.

(23) $m a x (E_{i, j}) \leq T_{c}$

(3). Spatial constraints

In construction, strictly adhering to construction processes and technical standards often means that activities will face significant spatial limitations. Spatial resources, as core elements ensuring the smooth progress of construction activities, are particularly crucial in actual construction projects. Completing tasks efficiently in limited space places high demands on construction efficiency and quality. These spatial constraints not only directly affect construction speed but also interact with temporal factors, creating a complex and dynamic relationship. On one hand, spatial limitations restrict the scope and speed of construction, directly impacting the project schedule. On the other hand, the progress of construction can lead to changes in temporal constraints, further complicating spatial constraints. In certain specific situations, spatial constraints and temporal constraints may even transform into one another. This section briefly mentions spatial constraints as a background factor to help the reader better understand the subsequent content, without delving into a detailed discussion.

(4). Continuity constraints

In the same construction activity, a construction team is responsible for taking over and coordinating a series of adjacent and sequential construction units within the same section, all of which belong to the same working face. To ensure safety and operational efficiency, a minimum distance between working faces is maintained, preventing excessive stress concentration that could lead to structural instability while also optimizing resource allocation and reducing worksite interference. This relationship is defined in Equations (24)–(26).

(24) $y_{j, f_{i}} \leq (x_{j, f_{i}} - x_{j, f_{i} + 1}) (x_{j, f_{i}} - 1) M$

(25) $y_{j, f_{i}} \geq x_{j, f_{i} + 1} - x_{j, f_{i}}$

(26) $\sum_{f_{i} = 1}^{F_{i}} y_{j, f_{i}} \leq 1$

If $g$ and $h$ are the starting positions of two working faces in activity $i$ , and $h$ is located after $g$ , then a certain continuity constraint must be satisfied between the two points. As shown in Equation (27), the distance between the additional working face and the original working face must satisfy the continuity constraint before construction activities on the additional working face can begin. It is important to note that we can treat this distance constraint as a flexible constraint, which can be adjusted based on actual circumstances and specific conditions. By skillfully utilizing the flexibility of this soft constraint, we can optimize the tunnel construction schedule in a more detailed and efficient manner.

(27) $e_{g} e_{h} \times (({D i s}_{i, h} - {D i s}_{i, g}) - {e v}_{i, j} (S_{h} - S_{g})) \geq D i s, \forall g, h \in j$

By applying linearization to the previously mentioned equations, we successfully derived the results shown in Equations (28) and (29).

(28) $e_{g} e_{h} \times (({D i s}_{i, h} - {D i s}_{i, g}) - {e v}_{i, j} (S_{h} - S_{g})) \leq e_{g} e_{h} M, \forall g, h \in j$

(29) $e_{g} e_{h} (({D i s}_{i, h} - {D i s}_{i, g}) - {e v}_{i, j} (S_{h} - S_{g})) \geq D i s + (e_{g} e_{h} - 1) M, \forall g, h \in j$

Among all activities, only one construction team is allowed to open and operate from the uniquely designated position of each additional working face, with strict prohibition on multiple construction teams entering the same working face from different positions to work concurrently. This rule is specifically reflected in Equation (30).

(30) $\sum_{f_{i} = 1}^{F_{i}} \sum_{j = 1}^{J} y_{j, f_{i}} = 1$

Additionally, if the additional working face of activity $i$ , opened at position $f_{i}$ , starts construction from unit $j$ , it must switch within construction unit $j$ , as shown in Equation (31).

(31) $\sum_{f_{i} = 1}^{F_{i}} y_{j, f_{i}} = e_{j}, \forall \in c$

The duration of activity $i$ on the additional working face in construction unit $j$ is given in detail by Equation (32).

(32) $D_{j} = \sum_{j = 1}^{J} \sum_{f_{i} = 1}^{F_{i}} x_{j, f_{i}} D_{f_{i}}$

(5). Resource constraints

As shown in Figure 7, the resources required for projects can be divided into two categories: renewable resources and non-renewable resources [71]. Non-renewable resources, such as steel bars, cement, and sand, are numerous and cannot be reused. Labor and machinery are renewable resources. They are key components of construction teams, and there is a close interrelationship between the two.

The renewable resources referred to in the text mainly refer to construction teams. Typically, the number of construction teams is fixed, or it can be adjusted slightly. If the scheduling optimization plan based on the originally allocated number of construction teams and other resources does not meet the project’s requirements, consideration may be given to increasing the number of construction teams to accelerate the construction speed.

a.. Material constraints

During the construction process, the total consumption of non-renewable resources must be controlled within the available total supply of those resources, as shown in Equation (33).

(33) $\sum_{j = 1}^{J} \sum_{i = 1}^{N} \sum_{p = 1}^{P_{i}} r_{i, p, n k} \times D_{i, j} \leq R_{n k}$

For renewable resources required for construction, the quantity of resources consumed by construction activities per unit of time should not exceed the maximum available supply of that resource, as expressed in Equation (34).

(34) $\sum_{j = 1}^{J} \sum_{i = 1}^{N} \sum_{p = 1}^{P_{i}} r_{i, p, t, k} \times x_{j, f_{i}} \times e_{j} \leq R_{k}$

b.. Construction team constraints

If construction activity $i$ is selected to begin working face construction from unit $j$ , it must be ensured that at least one construction team is assigned to carry out the relevant activities in unit $j$ , as expressed in Equation (35).

(35) $\sum_{f_{i} = 1}^{F_{i}} \sum_{j = 1}^{J} y_{j, f_{i}} = e_{j}$

The number of construction teams that can be simultaneously engaged in all construction activities must not exceed the specified maximum limit, as shown in Equation (36).

(36) $\sum_{j = 1}^{J} \sum_{f_{i} = 1}^{F_{i}} x_{j, f_{i}} \times {w t}_{t} \leq {W T}_{i}$

Considering the specific tasks and construction efficiency of each construction team, the total work completed by all construction teams must exactly equal the total planned workload for the entire project, as shown in Equation (37).

(37) $\sum_{j = 1}^{J} \sum_{i = 1}^{N} \sum_{f_{i} = 1}^{F_{i}} x_{j, f_{i}} \times Q_{i, j} = Q_{t o t a l}$

4. Algorithm Design

4.1. Exact Algorithm

Exact algorithms guarantee the optimal or precise solution to a problem within a finite time, ensuring that no potential solutions are overlooked and providing correct results. These algorithms rely on rigorous mathematical principles and logical reasoning, making them ideal for fields requiring high accuracy. However, the dynamic optimization of tunnel construction scheduling in reverse construction scenarios, is an NP-hard problem. This means that, due to its inherent complexity, the computational time required to solve the problem can grow exponentially with problem size. As exact algorithms typically explore or compute all possible solutions, especially for large-scale problems, the computational complexity increases significantly, resulting in long runtimes. In order to solve the mixed-integer programming (MIP) equations presented above, we selected the Gurobi solver as the core tool and developed the corresponding solution program using Python 3.12.7.

The cutting plane method is preferred over branch-and-bound and dynamic programming because it efficiently refines the feasible region and handles complex constraints. It better represents dependencies between construction teams and work surfaces, ensuring work continuity and feasibility under distance constraints. Therefore, this study adopts the cutting plane method to ensure work continuity, optimize construction team management, and maintain feasibility under distance constraints. The specific steps for optimizing the tunnel construction schedule using an exact algorithm are outlined as follows:

(1). Organize the data to create input tables that meet the calculation requirements.
(2). Based on input parameters such as the forecasted date, current progress, and others, generate the initial data required for computation. The input parameters include the contract-specified construction duration, the current progress of the main tunnel, the current progress of the parallel adit, the construction direction, continuity constraints, and so on.
(3). Determine whether the distance between the current progress of the parallel adit and the main tunnel exceeds the continuity distance constraint. If the actual ahead distance of the parallel adit is greater than or equal to the set distance, a cross passage for the main tunnel construction can be created at this location. An additional working face is opened at the current mileage, and the start time of the main tunnel’s additional working face, as well as the progress of both the main tunnel and the parallel adit, are updated. If the actual ahead distance of the parallel adit is less than the set distance, the cross passage cannot be set for the main tunnel construction, and the process proceeds to step (4).
(4). Move the date forward and update relevant data such as the date, main tunnel progress, parallel adit progress, and construction direction.
(5). Loop through steps (3) and (4) until one of the following stopping criteria is met: the contract-specified construction duration, the maximum allowable mileage, or any other predefined operational limit, such as resource constraints.

4.2. Genetic Algorithm

Like other metaheuristic algorithms, GAs incorporate stochastic mechanisms during the optimization process to prevent premature convergence and enhance solution diversity. GAs are highly effective for discrete optimization, as they can handle non-convex, non-differentiable objective functions without relying on gradient information. Their population-based search strategy reduces the risk of getting trapped in local optima, making them particularly suitable for combinatorial problems where the solution space expands exponentially. GAs naturally accommodate discrete variables, such as those in tunnel construction scheduling, and offer flexible constraint handling through penalty functions or repair mechanisms. Their stochastic nature ensures a well-balanced exploration and exploitation of the search space, making them a powerful tool for solving complex discrete optimization problems in engineering and scheduling. As a result, these algorithms can generate diverse tunnel construction schedules during optimization. Due to this characteristic, the computational models used in traditional exact algorithms can no longer be directly applied as objective functions for optimization in GA, requiring adjustments or replacements. To address this, this study designs a virtual construction algorithm that simulates real-world construction elements, including the main tunnel, parallel adit, cross passages, inclined shaft, surrounding rock conditions, construction teams, and working faces, as shown in Algorithm 1. The algorithm dynamically adjusts construction speed based on varying rock grades and fully considers conflicts among a fixed number of construction teams, as well as bidirectional construction after cross-passage excavation. The virtual construction algorithm operates by: Calculating the advancement speed of working faces based on the geological conditions of the surrounding rock; Determining whether the distance between newly added work surfaces and existing ones meets the requirements for opening additional working faces; Assessing whether construction teams can be assigned to a working face; Verifying whether different working faces are interconnected; Checking whether the entire project has been completed. Additionally, the algorithm performs daily checks on constraints related to logic, time, space, continuity, and resources. If any violations of these constraints are detected, a penalty value is added to the final computation results. This mechanism effectively prevents the generation of solutions that fail to meet the specified constraints.

Algorithm 1 Virtual construction algorithm
Input: tunnel construction schedule
1	Initial duration (d = 0) and flag for project completion (flag = True)
2	While flag:
3		Excavation
4		Assign work teams
5		Establish interconnections
6		Open working faces according to schedule
7		Check project completion
8		If project finish:
9			d = d + 1
10		else:
11			Flag = False
12		End
13	End
Output: d

In the virtual construction algorithm, the main challenge is handling the diversity of working face interconnections based on different construction schedules. We have designed a process to assess the interconnection status between two working faces. First, if the construction directions are the same, the two working faces are connected end to end. If the rear team’s tail can open an additional working face, they can start reverse tunneling from the rear, saving time by avoiding cross-passage excavation. If not, the rear team will complete its task and wait for further instructions. If the construction directions differ, the teams will be performing counter-excavation. If neither team can open an additional working face, they will pause and wait for new instructions, signaling that the project is nearing completion. If both teams can open additional working faces, they will start reverse tunneling. If only one team can open an additional working face, that team will proceed while the other pauses. As the project progresses, the working faces will eventually connect. After receiving the construction schedule, the virtual algorithm calculates the corresponding project duration based on the given conditions. This process forms the basis for constructing an objective function in the genetic algorithm, which outputs the optimized construction duration.

The tunnel construction schedule optimization plan in the GA is encoded as an N × 2 array, where N represents the number of additional working faces, as shown in Figure 8. The first column indicates whether an additional working face is added: 0 means no additional working face is opened at that location, and 1 means an additional working face is opened at that location. The second column indicates the construction direction of the additional working face: 1 represents forward construction, 0 represents reverse construction, and 2 represents bidirectional construction. After the tunnel construction schedule is input into the virtual excavation algorithm, the rows where the first column is 0 are removed. Then, based on the project information, the corresponding mileage of the additional working face locations is matched to determine whether the construction of the additional working face can proceed.

The proposed GA optimization method is shown in Figure 9 and Algorithm 2. First, the basic parameters of the GA are set, including population size, chromosome length (number of additional working faces, N), crossover probability, and mutation probability. The tunnel construction schedule is initialized based on these parameters. The developed virtual construction algorithm is used as the objective function (fitness function) to calculate the fitness of different populations. In the GA, while hard constraints, representing real-life necessities, must always be satisfied, other constraints are treated as soft. If a soft constraint is violated, the result is assigned an infinite value (inf), which is considered unbounded and will be discarded during the optimization process. Since the virtual construction algorithm is time-consuming, Python multithreading is employed to significantly reduce the computation time for evaluating population fitness. The project duration calculated by the virtual construction algorithm serves as the fitness value. Next, a selection operation is performed to retain individuals with higher fitness. The crossover operation is performed using a uniform crossover approach, where each gene (row) is independently swapped between two parent solutions with a predefined probability, ensuring a diverse combination of genetic material while maintaining feasibility. Additionally, a one-point or two-point crossover strategy can be applied by selecting random crossover points and exchanging corresponding segments between parents. The mutation operation introduces random changes to prevent premature convergence and enhance exploration. Specifically, for mutation, a small probability is assigned to flip the values in the first column (0 to 1 or 1 to 0), while the values in the second column are randomly reassigned among the three possible construction directions (0, 1, or 2). These operations ensure that the genetic algorithm effectively searches for an optimal tunnel construction schedule while preserving feasible and practical solutions. After several iterations, the algorithm terminates when the population’s fitness converges or when the preset stopping conditions are met. The individual with the highest fitness is then output as the best tunnel construction schedule.

Algorithm 2 GA with virtual construction algorithm
Input: maximum iterations(M)
1	Initial times of iteration (m = 1), flag for project completion (flag = True)
2	While m < M:
3		Parallel Fitness Calculation with virtual construction algorithm
4		Select top-performing individuals (elites) based on fitness
5		Crossover
6		Mutation
7		Combine elites, crossover offspring, and mutated individuals
		m = m + 1
13	End
Output: top-performing individual

4.3. Model Verification

To verify the accuracy of the constructed model and the effectiveness of the algorithm, this study is based on a real tunnel project case. The data used in the study were systematically organized according to a set of reasonable preset conditions. The main content is as follows:

The main tunnel has a total length of 9225 m, with a parallel adit of 9141 m and an inclined shaft of 1905 m. The parallel adit starts 84 m after the starting point of the main tunnel and ends at the same location as the main tunnel. The inclined shaft intersects the parallel adit at 5090 m from the starting point of the main tunnel. A total of 19 cross passages are set between the parallel adit and the main tunnel, each approximately 42 m in length. The distance of each cross passage from the main tunnel is shown in Table 2. The entire project was originally planned to take 2400 days, and 1033 days had already passed when the inclined shaft reached the parallel adit. There are 12 construction teams on-site, and the continuity distance between the new and existing working faces is constrained to about 300 m. However, as construction progressed, a significant deviation emerged between the predicted rock mass classification based on geological survey data and the actual geological conditions encountered at the construction site. This change directly led to a noticeable deviation between the planned construction schedule and the actual situation on-site, challenging the original construction rhythm and arrangements. Therefore, it is necessary to scientifically optimize and adjust the construction schedule based on the current situation. After the change in the rock mass classification, the construction speed for the main tunnel, parallel adit, and cross passages is shown in Table 3.

In solving the aforementioned case, we incorporated TCSDOM and utilized Python 3.12.7 to implement two different algorithmic strategies to ensure the accuracy and reliability of the results. First, we applied an exact algorithm based on rigorous mathematical derivation and computational rules, capable of directly determining the optimal solution to the problem. Subsequently, to verify the correctness of the results obtained by the algorithm and explore potential improvements in its efficiency, we employed GA as a heuristic search method. In this case, after sufficient iterations and optimized parameter settings, the results obtained by GA were highly consistent with those of the exact algorithm, yielding an optimal project duration of 2219 days. Furthermore, the average computation time for optimization using Python 3.12.7 over 10 iterations for each method is as follows: the exact algorithm required 3.9 s, while the GA took 352.1 s. Both computation times were within an acceptable range.

This finding not only validated the accuracy of the computed results but also demonstrated that the GA, when properly configured, can efficiently and accurately approximate the optimal solution to complex problems. Furthermore, when speed variations are excessively frequent or complex, the exact algorithm may face significant difficulties and fail to effectively handle such highly dynamic optimization problems. However, as a more flexible and adaptive method, metaheuristic algorithms can continue performing optimization calculations even in these complex environments.

From Figure 10, it can be observed that the construction teams, using the parallel adit, access the main tunnel via Cross Passage 20, Cross Passage 22, Cross Passage 25, and Cross Passage 28 to implement the shortest construction schedule plan. Through Cross Passage 20, the construction team enters the main tunnel and begins forward construction after accessing the additional working face; through Cross Passage 22, the team accesses the main tunnel and begins reverse construction from the additional working face; through Cross Passage 25, the team enters the main tunnel and conducts construction in both directions from the additional working face; through Cross Passage 28, the team enters the main tunnel and starts reverse construction from the additional working face. The final interconnection of the parallel adit is achieved in 2108 days, and the main tunnel interconnection is completed in 2219 days, making the latest completion date 2219 days. Compared to the original schedule, the optimized timeline shortens the project by 181 days, demonstrating not only the effectiveness and practicality of TCSDOM but also highlighting the importance of adopting scientific methods and advanced technologies in project management to enhance efficiency and accelerate progress.

The optimized construction plan reduces the number of construction teams from the original 12 to 9, with the specific construction arrangement shown in Figure 11. There is a direct correlation between the changes in construction speed and the slope of the chart; changes in construction speed are reflected in the slope of the chart, showing corresponding trends. This phenomenon suggests that the actual variations in construction progress can be understood through the analysis of the chart’s slope. By implementing an optimization strategy for construction operations, we successfully reduced the number of construction teams. This innovative solution not only ensures the shortest construction duration but also achieves cost optimization. This optimized schedule not only shortens the construction period, reduces project costs, and improves resource utilization but also provides greater buffer space for potential future delays. As a result, it demonstrates significant value and practical relevance in both academic research and real-world applications.

4.4. Algorithm Comparison

Building on the comparison of actual engineering cases, we further compared GA, PSO, Ant Colony Optimization (ACO), and Deep Q-Network (DQN) to validate the effectiveness of the proposed model. GA is suitable for complex combinatorial optimization problems, offering global search capabilities but with slower convergence. PSO has high computational efficiency but limited handling of discrete variables. ACO performs well in path planning and combinatorial optimization but requires significant computational resources. DQN is well-suited for dynamic optimization and better adapts to complex construction environments. Given the discrete nature of the engineering problem, the reinforcement learning method adopts DQN combined with a virtual excavation algorithm for optimization. The model comparison results are shown in Figure 12.

As shown in Figure 12, GA achieves the best optimization results, with a final project duration of 2219 days. PSO and ACO algorithms, both of which are metaheuristic methods, yield mixed results, with final durations of 2401 days and 2350 days, respectively. This may be due to the fact that PSO and ACO are more suitable for continuous optimization problems, making them less effective for the discrete problem in this study. DQN algorithm also produces a final project duration of 2219 days. However, its optimization efficiency is significantly lower, as its early-stage performance is clearly inferior to the other three algorithms. Although DQN has the advantage of training a reusable model, this benefit is not particularly evident in practical engineering applications.

Based on these results, GA demonstrates the best performance in optimizing tunnel construction scheduling. Its strong global search capability and adaptability to discrete constraints make it a more suitable choice for complex tunneling projects compared to the other tested algorithms.

5. Case Study

5.1. Case Description

In the context of rapid advancements in infrastructure development, tunnel engineering plays a crucial role as an essential connection between urban areas and nature, with a strong focus on construction quality and efficiency. This paper presents a detailed analysis of a specific tunnel project, exploring how construction schedules can be optimized through GA, even in the face of complex geological conditions and stringent project deadlines. The case study used is based on field research and data collection, as described below:

The main tunnel of the project is a single-bore, double-track tunnel, which includes a parallel adit and an inclined shaft that were specially planned and constructed, as shown in Figure 13. Both the main tunnel and the parallel adit have the same starting and ending mileage, with both having a length of 9255 m. The inclined shaft is 3705 m long, and it intersects the parallel adit 5090 m from the main tunnel’s starting point. A total of 21 cross passages are set between the main tunnel and the parallel adit, each approximately 42 m in length. The distance of each cross passage from the starting mileage of the main tunnel is shown in Table 4. The construction of the main tunnel entrance began first. The construction of the parallel adit entrance started 28 days after the main tunnel entrance, the construction of the main tunnel exit began 166 days after the main tunnel entrance, and the construction of the parallel adit exit began 57 days after the main tunnel entrance. When the inclined shaft reaches the parallel adit, 708 days have passed since the start of the tunnel construction. The construction site has 12 available construction teams, and the continuity constraint is about 300 m. The construction speed of the tunnel is a dynamically changing parameter significantly affected by the rock mass classification. The changes in rock mass classification are shown in Figure 14. During tunnel excavation, the construction speed varies in response to differences in the rock mass classification, and the construction speeds for various tunnel sections are shown in Table 5. In the original construction schedule, the completion time for the main tunnel was planned to be 1650 days, with the completion date set for day 1650.

5.2. Calculation Results and Analysis

Through optimization calculations, the latest results of the dynamic optimization of the tunnel construction schedule are shown in Figure 15. We have identified an efficient construction path, where the construction teams traverse the parallel adit and proceed to Cross Passages 7, 10, 13, 16, and 20, using these cross passages as access routes to the main tunnel. This construction schedule maximizes efficiency, enabling the entire project to be completed smoothly and efficiently within 1451 days.

The construction teams enter the main tunnel via Cross Passage 7, where they begin forward construction after accessing an additional working face; via Cross Passage 10, where they start reverse construction after upon reaching the main tunnel; via Cross Passage 13, where they begin work on both sides after entering the main tunnel; via Cross Passage 16, where they start reverse construction after entering the main tunnel; and via Cross Passage 20, where they begin reverse construction after accessing the main tunnel. The overall project duration is 1451 days, at which point the final interconnection of the main tunnel is completed, while the parallel adit’s interconnection is completed in 1325 days. Compared to the manually adjusted original construction schedule, the interconnection time of the main tunnel was advanced by 159 days. This demonstrates that through precise workload calculation, rational allocation of resources and time, and scientifically organized scheduling, we can ensure smooth and efficient collaboration between forward and reverse construction teams, maximizing construction efficiency and quality. Moreover, compared to traditional methods that rely on manual adjustments based on experience, the optimized schedule derived through TCSDOM significantly shortens the project timeline. It not only improves the accuracy of forecasts but also effectively reduces the project’s overall duration.

As shown in Figure 16, the optimized construction schedule reduces the number of teams from the original 12 to 9, while also rationally planning the teams’ transitions and reallocation. This optimization strategy not only demonstrates the significant impact of process optimization on improving efficiency and resource allocation but also underscores the importance of balancing the three key elements of time, cost, and resources in project management. Specifically, through meticulous planning and dynamic adjustments to the construction schedule, we have not only accelerated the construction progress but also effectively reduced manpower and material consumption, thus lowering construction costs. As a result, the approach shows outstanding performance and broad application potential in both academic theory and engineering practice.

The traditional approach to adjusting the construction schedule on-site primarily relies on manually modifying Excel spreadsheets based on human experience. If the completion time does not meet the requirements, the work arrangements for the construction teams must be adjusted and recalculated repeatedly. This process is time-consuming and labor-intensive, often taking 2–3 days to complete each adjustment. When using TCSDOM to adjust the construction schedule, the data required for optimization decisions is relatively simple and typically does not require significant additional workload. Through automated processes and algorithmic optimization, the optimization calculations can be completed in just a few minutes, enabling timely adjustments to the construction schedule. Compared to traditional manual adjustments based on human experience, using TCSDOM for construction schedule planning and adjustments results in a significant increase in efficiency. This substantial reduction in time is due to the high efficiency of the optimization method in data processing and schedule generation, which not only improves the efficiency of construction planning but also provides managers with a powerful tool to quickly respond to changes on-site.

Furthermore, during the optimization process, we identified a key issue: the construction speed difference between the parallel adit and the main tunnel was not significant. This caused some construction teams to remain idle for extended periods, becoming a major obstacle that severely hindered the project’s progress. To effectively address this challenge, we proposed a strategy involving a series of technical measures and management practices aimed at increasing the construction speed of the parallel adit. The goal is to create a greater distance between the working faces of the parallel adit and the main tunnel. By doing so, we can open as many additional working faces as possible while still adhering to continuity constraints. This approach is designed to minimize team idle time, thereby achieving a reasonable reduction in the overall project duration. The strategy of optimizing the construction speed of parallel adits is particularly beneficial for large-scale, time-sensitive tunnel projects, such as those in high-speed rail. By minimizing idle time and accelerating construction, this strategy helps reduce overall project duration, enhances flexibility, improves resource allocation, and increases efficiency, which is crucial in complex projects with strict timelines and logistical challenges.

6. Conclusions

This study focuses on the dynamic optimization of tunnel construction scheduling in reverse construction scenarios, aiming to enhance efficiency while adhering to predefined constraints. The core objective is to develop dynamic optimization models and intelligent optimization algorithms to improve scheduling decisions and shorten the project duration. The main contributions of this study in this field are as follows:

(1). This study systematically analyzes the key factors affecting the tunnel construction schedule under reverse construction scenarios through the integration of soft logic. It specifically addresses issues such as changes in the construction sequence, the impact of construction direction selection, and the constraints related to work continuity. A method is proposed to shorten the project duration and reduce idle time and delays by opening additional working faces through auxiliary tunnels, including parallel adits, inclined shafts, and cross passages.
(2). By applying the core principles of dynamic scheduling and integrating them with WBS, RBS, soft logic, and LSM, an efficient and highly adaptable dynamic optimization model has been developed: the TCSDOM model. This model addresses the uncertainties and variability inherent in the construction process. By dynamically adjusting construction parameters and strategies, it ensures that the project achieves optimal resource allocation and schedule management, even in complex and fluctuating environments.
(3). A dynamic construction scheduling strategy is proposed using an MIP model optimized by exact algorithms and GA. The model involves a detailed analysis of the logical, temporal, spatial, continuity, and resource constraints between activities, specifically under reverse construction scenarios, and expresses these relationships through mathematical formulas. Both algorithms achieve satisfactory results efficiently, with GA excelling in complex constraint handling. Compared to manual methods, optimization techniques significantly reduce project duration and improve prediction accuracy.

This study dynamically optimizes tunnel construction schedules to enhance efficiency, shorten project duration, optimize resource allocation, and reduce costs, all while ensuring safety. TCSDOM is both theoretically innovative and practically adaptable to various construction conditions. However, despite the significant contributions made in tunnel construction scheduling optimization, certain limitations remain. The study primarily focuses on schedule optimization without fully considering cost, quality, and resource balance, which may constrain overall project benefits. Additionally, the efficiency and stability of the optimization algorithms need improvement in complex environments, potentially affecting performance in large-scale or highly uncertain projects. Future research should integrate BIM and IoT for real-time monitoring to include factors like construction teams’ technical skills, enhancing the model’s practicality. Without considering such factors, the practical implementation of the proposed model may face challenges in accurately reflecting on-site conditions. Addressing these limitations will not only improve adaptability but also make the approach more robust and effective for real-world tunnel construction management.

Author Contributions

Conceptualization, J.W. and Y.L.; methodology, J.W. and X.L.; validation, J.W., X.L., R.Z. and G.W.; formal analysis, G.W. and Y.L.; data curation, J.W. and R.Z.; writing-original draft preparation, J.W.; writing-review and editing, J.W.; visualization, G.W.; supervision, J.W.; project administration, J.W.; funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Schematic of Tunnel Construction.

Figure 2. Schematic of the dynamic optimization model for tunnel construction scheduling.

View Image - Figure 3. Temporal constraint relationships for activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] when both are linear activities: (a) Both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are forward construction; (b) Both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are reverse construction; (c) Activity [Forumla omitted. See PDF.] is reverse construction, and activity [Forumla omitted. See PDF.] is forward construction; (d) Activity [Forumla omitted. See PDF.] is forward construction, and activity [Forumla omitted. See PDF.] is reverse construction.

Figure 3. Temporal constraint relationships for activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] when both are linear activities: (a) Both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are forward construction; (b) Both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are reverse construction; (c) Activity [Forumla omitted. See PDF.] is reverse construction, and activity [Forumla omitted. See PDF.] is forward construction; (d) Activity [Forumla omitted. See PDF.] is forward construction, and activity [Forumla omitted. See PDF.] is reverse construction.

View Image - Figure 4. Temporal constraint relationships for activity [Forumla omitted. See PDF.] as a linear activity and activity [Forumla omitted. See PDF.] as a strip or block activity: (a) Activity [Forumla omitted. See PDF.] is forward construction; (b) Activity [Forumla omitted. See PDF.] is reverse construction.

Figure 4. Temporal constraint relationships for activity [Forumla omitted. See PDF.] as a linear activity and activity [Forumla omitted. See PDF.] as a strip or block activity: (a) Activity [Forumla omitted. See PDF.] is forward construction; (b) Activity [Forumla omitted. See PDF.] is reverse construction.

View Image - Figure 5. Temporal constraint relationships for activity [Forumla omitted. See PDF.] as a strip or block activity and activity [Forumla omitted. See PDF.] as a linear activity: (a) Activity [Forumla omitted. See PDF.] is forward construction; (b) Activity [Forumla omitted. See PDF.] is reverse construction.

Figure 5. Temporal constraint relationships for activity [Forumla omitted. See PDF.] as a strip or block activity and activity [Forumla omitted. See PDF.] as a linear activity: (a) Activity [Forumla omitted. See PDF.] is forward construction; (b) Activity [Forumla omitted. See PDF.] is reverse construction.

View Image - Figure 6. Temporal constraint relationships when both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are strip or block activities.

Figure 6. Temporal constraint relationships when both activities [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are strip or block activities.

Figure 7. Types of resources in projects.

Figure 8. Tunnel construction schedule optimization plan encoding.

Figure 9. Flowchart of the virtual construction algorithm.

Figure 10. Schematic diagram of the tunnel construction schedule optimization plan.

Figure 11. Slope diagram of the tunnel construction schedule optimization plan.

Figure 12. Algorithm comparison results.

Figure 13. Schematic diagram of the tunnel construction arrangement.

Figure 14. Rock mass classification distribution of the tunnel.

Figure 15. Slope diagram of the dynamic optimization for the tunnel construction schedule.

Figure 16. Gantt chart of the dynamic optimization for the tunnel construction schedule.

Table 1

Definition of Parameters.

Index
$N$	Set of activities, where $i \in N$
$N_{L}$	Set of linear activities, where $i \in N_{L}$
$N_{S, B}$	Set of strip and block activities, where $i \in N_{S, B}$
$J$	Set of construction units, where $j \in J$ , $j_{i}^{S}$ is the first unit of the construction activity $i$ , and $j_{i}^{E}$ is the last unit of the construction activity $i$ .
$F_{i}$	Set of additional working face positions for activity $i$ , where $f_{i} \in F_{i}$
$P_{i}$	Set of execution modes for activity $i$ , where ${p \in P}_{i}$
K	Set of renewable resource types, where $k \in K$
NK	Set of non-renewable resource types, where $n k \in N K$
Constant
$T_{c}$	Contractual deadline
$Q_{t o t a l}$	Total construction quantity
$Q_{i, j}$	Construction quantity of activity $i$ in construction unit $j$
$R_{k}$	Maximum supply of renewable resource $k$ per unit time
$R_{n k}$	Total amount of non-renewable resource $n k$
$W T$	Total number of construction teams
${W T}_{i}$	Maximum number of parallel construction teams in activity $i$
$D i s$	Minimum distance between the additional working face and the current working face
$M$	An infinitely large number
${D i s}_{i, f_{i}}$	The distance from the additional working face position $f_{i}$ of activity $i$ to the project start point, where $f_{i} \in J$
$r_{i, p, t, k}$	The amount of renewable resource $k$ required by activity $i$ in each construction unit at time $t$ in execution mode $p$
$r_{i, p, n k}$	The amount of non-renewable resource $n k$ required by activity $i$ in each construction unit per unit of time in execution mode $p$
$S_{i, j}$	The start time of activity $i$ in construction unit $j$
$S_{j}$	The start time of the additional working face opened by activity $i$ in construction unit $j$
$E_{i, j}$	The end time of activity $i$ in construction unit $j$
$E_{j}$	The end time of the additional working face opened by activity $i$ in construction unit $j$
$D_{i, j}$	The time span of activity $i$ in construction unit $j$
$D_{j}$	The time span of the additional working face opened by activity $i$ in construction unit $j$
$D_{f_{i}}$	The time span of the additional working face opened by activity $i$ at position $f_{i}$
${e v}_{i, j}$	The construction speed of activity $i$ in construction unit $j$
$I n t {m i n}_{e, f}$	The minimum time interval between activity $e$ and activity $f$
${I n t m a x}_{e, f}$	The maximum time interval between activity $e$ and activity $f$
${D i s m i n}_{e, f}$	The minimum spatial interval between activity $e$ and activity $f$
${D i s m a x}_{e, f}$	The maximum spatial interval between activity $e$ and activity $f$
Decision variable
$x_{j, f_{i}} = \{\begin{matrix} 1, & The additional working face of activity i being constructed in \\ construction unit j is opened at position f_{i} \\ 0, & Otherwise \end{matrix}$
$d_{f_{i}} = \{\begin{matrix} 0, & The additional working face of activity i at position f_{i}, \\ with the construction direction being reverse \\ 1, & The additional working face of activity i at position f_{i}, \\ with the construction direction being forward \\ 2, & The additional working face of activity i at position f_{i}, \\ with the construction direction being bidirectional \end{matrix}$ ${w t}_{t} = \{\begin{matrix} 1, & Construction team w t is performing the construction activity \\ at time t \\ 0, & Otherwise \end{matrix}$
Indicator variable
$e_{j} = \{\begin{matrix} 1, & Open the additional working face in construction unit j \\ 0, & Otherwise \end{matrix}$ $y_{j, f_{i}} = \{\begin{matrix} 1, & x_{j, f_{i} + 1} > x_{j, f_{i}}, j \in e_{j} \\ 0, & Otherwise \end{matrix}$

Table 2

Cross passage mileage position.

Cross Passage Name	Mileage Position (Meter)
Cross Passage 1	450
Cross Passage 2	935
Cross Passage 3	1370
Cross Passage 4	1880
Cross Passage 5	2390
Cross Passage 6	3182
Cross Passage 7	3290
Cross Passage 8	3714
Cross Passage 9	4190
Cross Passage 10	4640
Cross Passage 11	5090
Cross Passage 12	5530
Cross Passage 13	6010
Cross Passage 14	6548
Cross Passage 15	6920
Cross Passage 16	7360
Cross Passage 17	7800
Cross Passage 18	8240
Cross Passage 19	8685

Table 3

Construction Speed.

Construction Structure Part	Segment Mileage (Meter)	Regression Speed (Meter/Day)
Main tunnel	0–1310	2.03
	1310–5225	1.19
	5225–9225	1.70
Parallel adit	84–1310	1.73
	1310–5225	1.54
	5225–9225	2.47
Cross passage	-	0.7

Table 4

Mileage location of the cross passages of the tunnel.

Cross Passage Name	Mileage Position (Meter)
Cross Passage 1	30
Cross Passage 2	130
Cross Passage 3	615
Cross Passage 4	1100
Cross Passage 5	1535
Cross Passage 6	2045
Cross Passage 7	2555
Cross Passage 8	3347
Cross Passage 9	3455
Cross Passage 10	3879
Cross Passage 11	4355
Cross Passage 12	4805
Cross Passage 13	5255
Cross Passage 14	5695
Cross Passage 15	6175
Cross Passage 16	6713
Cross Passage 17	7085
Cross Passage 18	7525
Cross Passage 19	7965
Cross Passage 20	8405
Cross Passage 21	8850

Table 5

Construction speed of the tunnel.

Construction Structure Part	Rock Mass Rating	Average Construction Speed (Meter/Day)
Main tunnel	I	5
	II	4.67
	III	3.67
	IV	2.33
	V	1.5
Parallel adit	I	5
	II	6.67
	III	5
	IV	3.33
	V	1.83
Cross passage	-	0.7

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Abstract

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Tunnel construction, a critical aspect of railway engineering, is a repetitive process with distinct linear characteristics. While the Linear Scheduling Method (LSM) is widely used for scheduling optimization in linear projects, it struggles to accommodate dynamic construction sequences, reverse construction, and flexible team allocation. Minimizing the project duration is a primary objective in tunnel construction scheduling optimization. To optimize tunnel construction, we propose a duration-shortening method using additional working surfaces, adaptable to multi-segment and multi-team scenarios. A dynamic optimization model is developed for tunnel construction scheduling, integrating LSM, soft logic, Work Breakdown Structure (WBS), and Resource Breakdown Structure (RBS) within a dynamic scheduling framework. This model analyzes logical relationships, work continuity, temporal and spatial constraints, and resource variation, focusing on reverse construction. The Mixed-Integer Programming (MIP) approach is used to build the mathematical model, solved with both exact algorithms and Genetic Algorithms (GA), and implemented in Python 3.12.7. Both algorithms perform well, with the GA excelling at handling complex constraints. Case studies confirm the method’s effectiveness in optimizing durations, devising flexible schedules, and improving efficiency and practicality. This research provides both theoretical insights and practical guidance for tunnel construction scheduling optimization in railway engineering.

Details

Title

Dynamic Optimization of Tunnel Construction Scheduling in a Reverse Construction Scenario

Author

Wei, Jianying¹

; Liu, Yuming¹; Lu, Xiaochun¹

; Zhao, Rong²; Wang, Gan¹

¹ School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China; [email protected] (J.W.); [email protected] (X.L.); [email protected] (G.W.)
² China Academy of Building Research, Beijing 100013, China; [email protected]

First page

168

Publication year

2025

Publication date

2025

Publisher

MDPI AG

e-ISSN

20798954

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.3390/systems13030168

ProQuest document ID

3182195201