A Robust Multi-Objective Evolutionary Framework for Artificial Island Construction Scheduling Under Dynamic Constraints

Abstract

Artificial island construction is a multifaceted engineering endeavor that demands precise scheduling to optimize resource allocation, control costs, ensure safety, and minimize environmental impact within dynamic marine environments. This study introduces a comprehensive multi-objective optimization model that integrates critical factors such as resource limitations, task dependencies, environmental variability, safety risks, and regulatory compliance. To effectively address the complexities of this model, we develop and employ the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA). MACEMGA combines cooperative coevolution, adaptive dynamic weighting, dynamic penalty functions, and advanced genetic operators to navigate the solution space efficiently and identify Pareto optimal schedules. Through extensive computational experiments using data from the Dalian Bay Cross-Sea Traffic Engineering project, MACEMGA is benchmarked against algorithms such as NSGA-II, SPEA2, and MOEA/D. The results demonstrate that MACEMGA achieves a reduction in construction time from 32.8 to 23.5 months and cost savings from CNY 4105.3 million to CNY 3650.0 million while maintaining high-quality outcomes and compliance with environmental standards. Additionally, MACEMGA shows improvements in hypervolume by up to 15% over existing methods and a Convergence Rate that is 8% faster than MOEA/D.

1. Introduction

Artificial island construction has emerged as a critical component in large-scale infrastructure projects, especially in coastal regions where expanding urban development and transportation networks necessitate innovative land-use solutions. These projects often involve constructing artificial islands that serve as pivotal junctions in bridge–tunnel systems spanning significant bodies of water. The scheduling of construction activities for these islands is inherently complex due to the need to balance multiple, often conflicting objectives such as minimizing project duration and cost, maximizing quality, efficiently allocating limited resources, and minimizing environmental impact.

The challenges associated with artificial island construction are compounded by dynamic marine environments and numerous technical and logistical constraints. These projects typically involve large-scale dredging, land reclamation, and installation of various infrastructure components, all of which must be meticulously coordinated to ensure successful integration with adjacent works, such as bridge and tunnel systems. Environmental uncertainties, such as unpredictable weather patterns, tidal fluctuations, and ecological impacts, further complicate the scheduling process. Consequently, advanced, adaptive optimization methods are paramount to address the multifaceted challenges involved in these projects.

A prominent example is the Dalian Bay Cross-Sea Traffic Engineering project, which entails building artificial islands to connect bridge and tunnel infrastructures across the bay. The success of such projects heavily depends on the precise coordination of the island’s construction with adjacent works. Any misalignment can lead to substantial delays and cost overruns, underscoring the imperative for an optimized construction schedule that accounts for various constraints and uncertainties.

Despite the critical importance of effective scheduling in these projects, existing methods often fall short of addressing their inherent complexities. Traditional scheduling techniques like the Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) focus primarily on single objectives, such as time or cost, and lack the capacity to handle multiple conflicting objectives simultaneously [1]. Moreover, they do not adequately account for the stochastic nature of environmental factors and resource availability inherent in marine construction environments [2]. Advanced multi-objective optimization methods, including multi-objective genetic algorithms (MOGAs), have been applied to construction scheduling [3], but they often struggle with the high-dimensional search spaces and complex constraints characteristic of artificial island projects.

In recent years, novel multi-objective optimization approaches have shown promise in addressing complex engineering challenges, such as energy systems and environmental management, which share similar constraints with marine construction. For instance, Khunkitti et al. [4] introduced the Two-Archive Harris Hawk Optimization (TwoArchHHO) for many-objective optimal power flow problems, achieving balance across multiple objectives like fuel cost, emissions, and power loss. This method’s ability to manage conflicting objectives under stringent constraints demonstrates the growing potential for multi-objective optimization in large-scale engineering applications. Similarly, Wang et al. [5] proposed the Artificial Protozoa Optimizer (APO), inspired by protozoan behaviors, and tested it across various engineering tasks, including image processing and design problems. APO’s success underscores the adaptability of bio-inspired optimization methods in handling high-dimensional, constrained environments. Oladejo et al. [6] developed the Hiking Optimization Algorithm (HOA), which mimics human hiking behavior across uneven terrains, demonstrating the algorithm’s robustness in navigating complex search spaces and balancing global and local exploration. These recent advancements reveal both the versatility and the growing demand for innovative algorithms in addressing multifaceted engineering problems.

However, despite the strides made in multi-objective optimization, current models frequently oversimplify resource constraints and task dependencies, resulting in schedules that are not feasible in practice [7]. They also tend to overlook the dynamic and uncertain conditions of marine environments, such as varying weather patterns, tidal changes, and ecological considerations [8]. These limitations highlight a significant gap in the existing body of research: the lack of a robust, comprehensive optimization model capable of handling the multifaceted challenges of artificial island construction scheduling.

To address this gap, we develop a comprehensive multi-objective optimization model tailored specifically for artificial island construction scheduling. Our model integrates multiple objectives—minimizing construction time, cost, and environmental impact while maximizing project quality and resource utilization efficiency—under a complex set of constraints that include resource limitations, task dependencies, environmental factors, uncertainties, safety risks, and regulatory compliance. This approach provides a novel and adaptive framework that is capable of effectively navigating the complexities of large-scale marine construction projects.

To solve this complex optimization problem, we introduce the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA). MACEMGA is designed to effectively handle the high-dimensional, constrained, and multi-objective nature of the scheduling problem. It incorporates advanced techniques such as cooperative coevolution, adaptive dynamic weighting, dynamic penalty functions, and differential evolution operators, which collectively enhance its ability to find high-quality solutions in complex search spaces.

Our key contributions are as follows:

1. Generalized Multi-Objective Optimization Framework: We develop a comprehensive multi-objective optimization framework that abstracts the artificial island construction scheduling problem into a generalized model. This framework effectively encapsulates the unique challenges of such projects, including multi-mode task execution, dynamic resource allocation, and environmental uncertainties, while maintaining adaptability to a wide range of large-scale engineering projects. Consequently, it provides a flexible tool for addressing diverse construction scheduling challenges, extending its applicability beyond the context of artificial islands.

2. Introduction of the MACEMGA Algorithm: We present the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm, a novel optimization algorithm specifically designed to address the challenges identified in the scheduling problem. MACEMGA effectively balances exploration and exploitation in the search space and efficiently handles multiple objectives and constraints, outperforming traditional optimization methods in high-dimensional and constrained environments.

3. Validation Through Real-World Case Studies: We validate our model and algorithm using data from the Dalian Bay Cross-Sea Traffic Engineering project. The results demonstrate that MACEMGA generates feasible and optimal schedules that better satisfy the multiple objectives compared to existing methods. This validation showcases the practical applicability of our approach and provides valuable insights for project managers and stakeholders in similar infrastructure projects.

By providing a robust solution that effectively balances competing objectives and navigates the complexities of construction scheduling under uncertainty, our work advances the field of construction project management. It offers a practical tool that enhances decision-making processes, ultimately contributing to the successful and timely delivery of large-scale infrastructure projects. This, in turn, supports broader goals of sustainable urban development and economic advancement by optimizing resource utilization and minimizing environmental impact.

2. Related Work

2.1. Multi-Objective Optimization in Construction Project Management

Multi-objective optimization in construction project management addresses the challenge of balancing conflicting objectives such as duration, cost, quality, and environmental impact. Yi and Gao [9] utilized fuzzy programming to optimize the duration-cost-quality balance in construction projects, highlighting the importance of incorporating uncertainty into optimization models. El-Rayes and Kandil [10] developed a time–cost–quality trade-off analysis specifically for highway construction, demonstrating the feasibility of multi-objective optimization in practical applications. Konak et al. [11] provided a comprehensive tutorial on using genetic algorithms for multi-objective optimization, which has been influential in guiding subsequent research in this area. Gonzalez-Rodriguez et al. [12] applied multi-objective optimization to offshore wind farms, considering both economic factors and visual impact, thereby extending the application of these techniques to sustainable energy projects. Zhan et al. [13] proposed a multi-objective optimization approach based on NSGA-III for construction project management, showcasing advancements in evolutionary algorithms for handling complex Pareto fronts. Kim et al. [14] introduced a modified niched Pareto genetic algorithm for multi-objective construction schedule optimization, improving the efficiency and effectiveness of scheduling in construction projects.

Recent advancements in multi-objective optimization have introduced novel bio-inspired and human-based algorithms to handle complex engineering optimization challenges. Khunkitti et al. [4] proposed the Two-Archive Harris Hawk Optimization (TwoArchHHO) to address many-objective optimal power flow problems, optimizing objectives such as fuel cost, emission levels, transmission line losses, and voltage deviation, which parallels construction management challenges in balancing multiple competing objectives. Wang et al. [5] introduced the Artificial Protozoa Optimizer (APO), which mimics protozoa’s survival behaviors and has demonstrated strong optimization capabilities in various engineering applications, including image segmentation and design problems. APO’s performance in diverse applications shows its potential adaptability to construction project management contexts. Oladejo et al. [6] developed the Hiking Optimization Algorithm (HOA), inspired by human hiking behavior across challenging terrains, with a successful application to NP-hard problems and complex engineering optimization tasks, aligning well with the challenges encountered in large-scale construction projects that require adaptive search and balancing multiple constraints.

2.2. Application of Genetic Algorithms in Construction Schedule Optimization

Genetic algorithms (GAs) have been widely applied in construction schedule optimization due to their robust search capabilities and adaptability. Leu and Yang [15] pioneered the use of GAs in developing a multicriteria optimal model for construction scheduling, effectively handling multiple objectives simultaneously. Hegazy [16] optimized resource allocation and leveling using GAs, enhancing the efficiency of resource management in construction projects. Feng et al. [17] demonstrated the application of GAs in solving time–cost trade-off problems, laying the groundwork for subsequent research in this field. Bhosale et al. [18] provided a comprehensive review of GAs used for optimizing scheduling in resource-constrained construction projects, emphasizing the algorithm’s effectiveness in handling complex constraints. Abdel-Khalek et al. [19] integrated financing considerations into scheduling optimization using GAs, addressing the financial aspects of project management. Faghihi et al. [20] utilized GAs based on Building Information Modeling (BIM) for construction scheduling, demonstrating the integration of advanced modeling techniques with optimization algorithms. Xie et al. [21] applied GAs to the scheduling optimization of prefabricated construction projects, highlighting the algorithm’s adaptability to modern construction methods.

2.3. Cooperative Coevolutionary Algorithms and Their Applications in Complex Optimization Problems

Cooperative coevolutionary algorithms (CCEAs) divide complex optimization problems into subcomponents that evolve concurrently, enhancing the scalability and efficiency of the optimization process. Potter and De Jong [22] introduced the architecture for evolving coadapted subcomponents, laying the foundation for CCEAs. Yang and Tang [23] applied cooperative coevolution to large-scale evolutionary optimization, demonstrating its effectiveness in handling high-dimensional problems. Gong et al. [24] proposed a hybrid differential evolution with biogeography-based optimization for global numerical optimization, showcasing the potential of combining CCEAs with other evolutionary strategies. Omidvar et al. [25] introduced cooperative co-evolution with differential grouping for large-scale optimization, improving the decomposition strategies within CCEAs. Zheng et al. [26] developed a cooperative coevolutionary algorithm for multiobjective optimization, extending the application of CCEAs to problems with multiple conflicting objectives. Xu et al. [27] addressed multi-objective optimization problems with changing decision variables using CCEAs, highlighting the adaptability of these algorithms to dynamic environments. Peng et al. [28] presented a dynamic optimization approach to the design of CCEAs, focusing on improving performance over time. Ma et al. [29] provided a survey on cooperative co-evolutionary algorithms, summarizing advancements and applications in the field. Tan et al. [30] proposed a distributed cooperative coevolutionary algorithm for multiobjective optimization, enhancing computational efficiency. Zhou et al. [31] introduced a multi-strategy competitive–cooperative co-evolutionary algorithm and its application, further advancing the methodology of CCEAs.

2.4. Application of Adaptive Strategies in Evolutionary Algorithms

Adaptive strategies in evolutionary algorithms involve the dynamic adjustment of parameters to enhance algorithm performance across different stages of optimization. Prieto and Gomez [32] proposed a hybrid adaptive evolutionary algorithm for multi-objective optimization, demonstrating improvements in convergence and diversity. Eiben et al. [33] conducted a seminal study on parameter control in evolutionary algorithms, categorizing adaptive strategies and their effects on algorithm performance. Zhou et al. [34] surveyed the state of the art in multiobjective evolutionary algorithms, including the role of adaptive mechanisms. Jiao and Wang [35] introduced a novel genetic algorithm based on immunity, incorporating adaptive features inspired by biological systems. Jin and Branke [36] provided a survey on evolutionary optimization in uncertain environments, emphasizing the importance of adaptation in handling uncertainty. Li and Zhang [37] and Zhang and Li [38] developed multiobjective evolutionary algorithms based on decomposition, integrating adaptive strategies to handle complicated Pareto sets. Blum et al. [39] explored adaptive mutation strategies for evolutionary algorithms, enhancing the balance between exploration and exploitation. Beyer and Deb [40] discussed self-adaptive features in real-parameter evolutionary algorithms, contributing to the theoretical understanding of adaptation mechanisms. Meyer-Nieberg and Beyer [41] reviewed self-adaptation in evolutionary algorithms, providing insights into parameter tuning. Aleti and Moser [42] conducted a systematic literature review of adaptive parameter control methods, highlighting trends and future directions. Lobo et al. [43] compiled comprehensive knowledge on parameter setting in evolutionary algorithms, serving as a key reference in the field.

2.5. Consideration of Marine Environmental Factors in Construction Optimization

Marine environmental factors play a significant role in construction optimization, particularly for offshore and coastal projects where conditions are dynamic and uncertain. Elmabrouk et al. [44] utilized artificial neural network modeling to predict oil production, accounting for complex marine environmental variables. Wu et al. [45] applied dynamic risk assessment in construction projects using Bayesian networks, emphasizing the importance of considering environmental risks in project planning. Jabri and Zayed [46] developed agent-based modeling and simulation of earthmoving operations, which can be adapted to simulate marine construction activities. Li et al. [47] discussed the demands and challenges for the construction of marine infrastructures in China, highlighting the need for optimization techniques that consider environmental factors. Wang and Yang [48] examined the impact of resource optimization on the economic development of the marine industry, underscoring the economic implications of efficient construction practices in marine environments. Ferreira [49] focused on the optimization of reinforced concrete structure performance in marine environments, addressing the degradation effects caused by harsh conditions. Additionally, Gonzalez-Rodriguez et al. [12] considered both economic factors and visual impact in the multi-objective optimization of offshore wind farms, reflecting the integration of environmental considerations into the optimization process.

3. Model

3.1. Problem Description

Artificial island construction is a multifaceted engineering project involving the coordination of numerous interdependent construction activities under resource limitations and dynamic marine environments. Optimizing the construction schedule is critical to ensuring that the project is completed within the stipulated timeframe while controlling costs, maximizing quality, enhancing resource utilization efficiency, and minimizing environmental impact.

This study focuses on developing a comprehensive multi-objective optimization model utilizing the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA) to address the inherent complexities in artificial island construction scheduling. The primary objectives are to minimize the total construction time, minimize overall costs, maximize project quality, maximize resource utilization efficiency, and minimize environmental impact. These objectives must be achieved while adhering to various complex constraints such as resource limitations, task dependencies, environmental factors, uncertainties, safety risks, construction method selections, and compliance with policies and regulations.

The construction process involves multiple interrelated activities, each with specific resource requirements, durations, quality outcomes, and environmental impacts. The inherent uncertainties in marine environments, such as varying wind speeds, wave heights, tidal patterns, and extreme weather events, further complicate the scheduling process. Effective optimization must, therefore, account for these dynamic and stochastic environmental conditions to ensure feasible and efficient construction schedules. The proposed model integrates these factors into a robust optimization framework capable of generating optimal trade-offs among the defined objectives.

Furthermore, the model considers multi-mode resource allocation (e.g., equipment can operate at different speeds and cost modes), multi-mode activity execution (e.g., different construction methods leading to varying durations and costs), and potential risks and uncertainties during construction (e.g., equipment failures, material supply delays). These considerations make the model more realistic and enhance the reliability and practicality of the optimization results.

3.2. Model Assumptions

To construct a tractable yet realistic optimization model, several assumptions are made. The durations, costs, and quality metrics of construction activities are considered to be known within specified ranges, acknowledging possible variations due to environmental uncertainties. The construction timeline is discretized into uniform time intervals, facilitating the modeling of dynamic resource availability and environmental conditions. Resources, including labor, equipment, and materials, have limited availability that may vary over time and are subject to priority allocations based on project requirements. Construction activities are subject to precedence relations and priority levels and must be scheduled within specific time windows. Environmental factors such as wind speed, wave height, and tidal patterns are modeled as stochastic processes, influencing both the feasibility and safety of activities. The model also considers multi-level quality indicators for each activity, contributing to the overall project quality. Safety risks associated with construction activities and environmental conditions are quantified, with constraints ensuring they remain within acceptable thresholds. Finally, the model incorporates robustness considerations to ensure that the generated schedules are resilient to uncertainties and disruptions.

3.3. Model Parameters

The optimization model utilizes a comprehensive set of parameters to accurately represent the construction scheduling problem. These parameters are summarized in Table 1.

3.4. Scheduling Model

The scheduling model integrates the defined objectives and constraints into a unified optimization framework. The following subsections detail the objective functions and constraints, along with their mathematical formulations.

3.4.1. Objective Functions

The optimization problem is defined by five primary objectives:

1. Minimizing Total Construction Time ($F_1$)

The total construction time is determined by the completion time of the final activity:

(1)$F_1\left(\mathbf{x}\right)=\underset{i}{\max }\left(x_i+t_i^{m_i}\left(x_i\right)\right)$

where $x_i$ is the start time of activity i, and $t_i^{m_i}\left(x_i\right)$ is the duration of activity i in mode $m_i$, which may vary based on the selected start time $x_i$ to account for dynamic dependencies or environmental impacts at different times.

• 2.. Minimizing Total Construction Cost ($F_2$)

The total construction cost encompasses direct costs, indirect costs, resource idle costs, delay penalties, and risk-related costs:

(2)$F_2(\mathbf{x},\mathbf{y})=\sum _{i=1}^n\left(C_i^{\mathrm{direct},m_i}+C_i^{\mathrm{indirect}}+C_i^{\mathrm{idle}}+C_i^{\mathrm{delay}}+\gamma C_i^{\mathrm{risk},m_i}\right)+\sum _{k=1}^K{\int }_0^T{\theta }_k^{\mathrm{borrow}}{B}_k\left(t\right)dt$

where:

$\begin{array}{cc}\hfill C_i^{\mathrm{indirect}}& =\beta t_i^{m_i}\hfill \\ \hfill C_i^{\mathrm{idle}}& =\sum _{k=1}^K{c}_k^{\mathrm{idle}}\left(R_k\left(t\right)-y_{ik}\right)t_i^{m_i}\hfill \\ \hfill C_i^{\mathrm{delay}}& =\lambda \max \left(0,x_i+t_i^{m_i}-L{S}_i\right)\hfill \\ \hfill C_i^{\mathrm{risk},m_i}& =E_i\left(t_i^{m_i}\right)·R_i^{\mathrm{risk}}\hfill \end{array}$

Here, $\beta$ is the indirect cost coefficient, $c_k^{\mathrm{idle}}$ is the idle cost rate for resource k, $\lambda$ is the delay penalty rate, $L{S}_i$ is the latest start time for activity i, $\gamma$ is the risk cost coefficient, $E_i\left(t_i^{m_i}\right)$ represents the environmental impact on activity i in mode $m_i$, and $R_i^{\mathrm{risk}}$ is the risk factor associated with activity i. Additionally, ${\theta }_k^{\mathrm{borrow}}$ denotes the unit cost for borrowing or renting resource k, and $B_k\left(t\right)$ is the quantity of resource k borrowed or rented at time t. The integration over time T accounts for the cumulative borrowing costs throughout the project duration.

• 3.. Maximizing Total Project Quality ($F_3$)

Project quality is assessed through multiple quality indicators for each activity, weighted appropriately:

(3)$F_3(\mathbf{x},\mathbf{y})=\sum _{i=1}^n\sum _{j=1}^{m_i}{w}_{ij}{Q}_{ij}(x_i,y_{ik})$

where $w_{ij}$ is the weight of quality indicator j for activity i, and $Q_{ij}(x_i,y_{ik})$ is the quality score of activity i in mode $m_i$ as a function of its start time $x_i$ and resource allocation $y_{ik}$.

• 4.. Maximizing Resource Utilization Efficiency ($F_4$)

Resource utilization efficiency is quantified as the ratio of utilized resources to the total available resources over the project duration:

(4)$F_4(\mathbf{x},\mathbf{y})={\displaystyle \frac{{\sum }_{k=1}^K{\int }_0^T\left({\sum }_{i\in A(t,\mathbf{x})}{r}_{ik}^{m_i}(x_i,y_{ik})\right)dt}{{\sum }_{k=1}^K{\int }_0^T\left(R_k\left(t\right)+B_k\left(t\right)\right)dt}}$

where K represents the total number of resource types, $A(t,\mathbf{x})$ is the set of activities active at time t based on their start times $x_i$, and $r_{ik}^{m_i}(x_i,y_{ik})$ is the resource requirement of activity i for resource k in mode $m_i$, influenced by both the start time $x_i$ and the allocated resources $y_{ik}$.

• 5.. Minimizing Environmental Impact ($F_5$)

Environmental impact is measured by the cumulative environmental impact indices of all activities:

(5)$F_5(\mathbf{x},\mathbf{y})=\sum _{i=1}^n{e}_i^{m_i}\left(y_{ik}\right)·t_i^{m_i}\left(x_i\right)$

where $e_i^{m_i}\left(y_{ik}\right)$ is the environmental impact index of activity i in mode $m_i$, determined by the allocated resources $y_{ik}$, and $t_i^{m_i}\left(x_i\right)$ is the duration of activity i in mode $m_i$, influenced by its start time $x_i$.

3.4.2. Constraints

The scheduling model incorporates a comprehensive set of constraints to ensure feasibility, safety, and regulatory compliance of the construction schedule. These constraints capture the intricate dependencies and limitations inherent in artificial island construction projects.

1. Resource Constraints

The availability of resources is a critical factor that limits the simultaneous execution of construction activities. Resources, including labor, equipment, and materials, are finite and may vary in availability over time. This constraint ensures that the total resource allocation at any given time does not exceed the available resources, considering both the base availability and any additional borrowed or rented resources:

(6)$\sum _{i\in A\left(t\right)}{r}_{ik}^{m_i}\le R_k\left(t\right)+B_k\left(t\right),\forall t,\forall k$

where $B_k\left(t\right)$ denotes the quantity of resource k borrowed or rented at time t, subject to:

(7)$B_k\left(t\right)\ge 0,\forall k,\forall t$

Borrowing or leasing resources introduces additional costs, which are accounted for in the objective function as:

(8)$F_2^{\mathrm{borrow}}(\mathbf{x},\mathbf{y})=\sum _{k=1}^K{\int }_0^T{\theta }_k^{\mathrm{borrow}}{B}_k(t,\mathbf{x})dt$

where ${\theta }_k^{\mathrm{borrow}}$ represents the unit cost for borrowing or renting resource k, and $B_k(t,\mathbf{x})$ denotes the quantity of resource k borrowed or rented at time t, which can depend on the schedule of activities governed by the decision variable set $\mathbf{x}$. The borrowing decisions may be dynamically influenced by when specific activities start and how resources are allocated over time.

• 2.. Task Dependency Constraints

Construction activities often have inherent dependencies, meaning that certain tasks must be completed before others can commence. These dependencies are essential to maintain the logical and technological sequence of the construction process, ensuring that the project progresses smoothly and efficiently. Dependencies can take various forms, including start-to-start, finish-to-start, start-to-finish, and finish-to-finish relationships. This is mathematically represented as:

(9)$x_j\ge x_i+t_i^{m_i}+L_{ij},\forall (i,j)\in D$

where $L_{ij}$ is the minimum required lag time between the completion of activity i and the start of activity j. By enforcing these dependencies, the model ensures that activities are scheduled in a manner that respects the necessary order of operations, preventing conflicts and inefficiencies that could arise from overlapping or out-of-sequence tasks.

• 3.. Activity Mode Selection Constraints

Each construction activity may be executed in one of several modes, each differing in terms of duration, cost, and resource requirements. It is imperative that each activity selects only one execution mode to maintain consistency and feasibility within the schedule. This constraint ensures that for every activity, exactly one mode is chosen:

(10)$\sum _{m\in M_i}{z}_i^m=1,\forall i$

(11)$t_i^{m_i}=\sum _{m\in M_i}{t}_i^m{z}_i^m,\forall i$

where $z_i^m$ is a binary variable indicating whether activity i is executed in mode m. By restricting activities to a single mode, the model accurately captures the trade-offs between different execution strategies, such as faster completion times versus higher costs or resource usage, thereby enabling the optimization of multiple objectives in a coherent manner.

• 4.. Environmental Condition Constraints

Construction activities are significantly influenced by environmental conditions, which can vary dynamically and unpredictably. Adhering to favorable environmental conditions is crucial for the feasibility, safety, and efficiency of construction operations. This constraint ensures that activities are scheduled only during periods when environmental factors do not exceed predefined tolerances:

(12)$E_t\le E_i^{\max },\forall i,\forall t\in [x_i,x_i+t_i^{m_i}]$

where $E_t$ represents the environmental factors (e.g., weather conditions) at time t, and $E_i^{\max }$ is the maximum tolerable environmental impact for activity i. By integrating environmental constraints, the model enhances the realism of the schedule, ensuring that planned activities are executable under prevailing conditions and mitigating the risks associated with adverse environmental events.

• 5.. Safety Risk Constraints

Safety is a paramount concern in construction projects, particularly in challenging marine environments. This constraint ensures that the safety risk factors associated with construction activities do not exceed acceptable thresholds:

(13)$P_s\left(t\right)\le R_{\max },\forall t$

where $P_s\left(t\right)$ denotes the safety risk factor at time t, and $R_{\max }$ is the maximum allowable safety risk threshold. Maintaining safety within these limits is essential to protect personnel, equipment, and the integrity of the construction project, thereby preventing accidents and ensuring compliance with safety regulations.

• 6.. Policy and Regulations Constraints

Compliance with relevant laws, regulations, environmental protection standards, and safety guidelines is mandatory in construction projects. This constraint encapsulates all such regulatory requirements, ensuring that the construction schedule adheres to legal and policy mandates:

(14)$C_{\mathrm{policy}}\left(\mathbf{x}\right)\le 0$

where $C_{\mathrm{policy}}\left(\mathbf{x}\right)$ represents the set of constraints related to policies and regulations. By enforcing regulatory compliance, the model ensures that the optimized construction schedule is not only efficient and cost-effective but also legally permissible and environmentally responsible.

• 7.. Uncertainty and Risk Constraints

The construction process is subject to various uncertainties and risks that can lead to delays and disruptions. This constraint accounts for potential delays by incorporating risk-induced extensions to activity durations:

(15)$x_i+t_i^{m_i}+{\Delta }_i^{\mathrm{risk}}\le x_j,\forall (i,j)\in D$

where ${\Delta }_i^{\mathrm{risk}}$ represents the potential delay time due to risks associated with activity i. By incorporating these uncertainties, the model enhances its robustness, ensuring that schedules remain feasible even in the presence of unforeseen events that may impact the project timeline.

• 8.. Resource Supply Variation Constraints

Resource availability can fluctuate over the course of the construction project due to various factors such as procurement delays, maintenance issues, or demand shifts. The following equation models the total resource availability at time t:

(16)$R_k\left(t\right)=R_k^{\mathrm{base}}+\delta R_k\left(t\right),\forall k,\forall t$

Here, $R_k\left(t\right)$ on the left side represents the total available quantity of resource k at time t. On the right side, $R_k^{\mathrm{base}}$ is the baseline or planned availability, and $\delta R_k\left(t\right)$ captures any fluctuations in the resource supply due to unforeseen events or dynamic conditions. By modeling the total resource availability as a sum of its baseline value and fluctuations, this equation ensures that the schedule can adapt to changes in resource conditions dynamically.

• 9.. Resource Borrowing and Leasing Constraints

When resource availability is insufficient to meet the demands of construction activities, borrowing or leasing additional resources becomes necessary. This constraint allows for such flexibility while accounting for the associated costs:

(17)$r_{ik}^{m_i}\le R_k\left(t\right)+B_k\left(t\right),\forall i,\forall k,\forall t$

where $B_k\left(t\right)$ is the borrowed or rented quantity of resource k at time t. Borrowing or leasing resources introduces additional costs, which are integrated into the optimization model as part of the total construction cost. This constraint provides the model with the flexibility to address resource shortages while maintaining cost considerations, thereby enhancing the practicality and adaptability of the construction schedule.

• 10.. Non-Negativity and Binary Constraints

To ensure the feasibility and logical consistency of the model, decision variables are subject to non-negativity and binary constraints:

(18)$x_i\ge 0,\forall i$

(19)$z_i^m\in \{0,1\},\forall i,\forall m\in M_i$

These constraints ensure that activity start times are non-negative and that mode selection variables are binary, thereby maintaining the integrity of the optimization process and ensuring that solutions are meaningful and implementable in a real-world context.

3.4.3. Mathematical Formulation of the Scheduling Model

Integrating the above objective functions and constraints, the multi-objective optimization problem can be formally defined as:

(20)$\begin{array}{cc}\hfill \min & F\left(\mathbf{x}\right)=\left[F_1\left(\mathbf{x}\right),F_2\left(\mathbf{x}\right)+F_2^{\mathrm{borrow}}\left(\mathbf{x}\right),F_5\left(\mathbf{x}\right)\right]\hfill \end{array}$

(21)$\begin{array}{cc}\hfill \max & F\left(\mathbf{x}\right)=\left[F_3\left(\mathbf{x}\right),F_4\left(\mathbf{x}\right)\right]\hfill \end{array}$

(22)$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{i\in A\left(t\right)}{r}_{ik}^{m_i}\le R_k\left(t\right)+B_k\left(t\right),\forall t,\forall k\hfill \end{array}$

(23)$\begin{array}{cc}\hfill & B_k\left(t\right)\ge 0,\forall k,\forall t\hfill \end{array}$

(24)$\begin{array}{cc}\hfill & x_j\ge x_i+t_i^{m_i}+L_{ij},\forall (i,j)\in D\hfill \end{array}$

(25)$\begin{array}{cc}\hfill & \sum _{m\in M_i}{z}_i^m=1,\forall i\hfill \end{array}$

(26)$\begin{array}{cc}\hfill & t_i^{m_i}=\sum _{m\in M_i}{t}_i^m{z}_i^m,\forall i\hfill \end{array}$

(27)$\begin{array}{cc}\hfill & x_i\ge 0,\forall i\hfill \end{array}$

(28)$\begin{array}{cc}\hfill & z_i^m\in \{0,1\},\forall i,\forall m\in M_i\hfill \end{array}$

(29)$\begin{array}{cc}\hfill & E_t\le E_i^{\max },\forall i,\forall t\in [x_i,x_i+t_i^{m_i}]\hfill \end{array}$

(30)$\begin{array}{cc}\hfill & P_s\left(t\right)\le R_{\max },\forall t\hfill \end{array}$

(31)$\begin{array}{cc}\hfill & C_{\mathrm{policy}}\left(\mathbf{x}\right)\le 0\hfill \end{array}$

(32)$\begin{array}{cc}\hfill & x_i+t_i^{m_i}+{\Delta }_i^{\mathrm{risk}}\le x_j,\forall (i,j)\in D\hfill \end{array}$

(33)$\begin{array}{cc}\hfill & R_k\left(t\right)=R_k^{\mathrm{base}}+\delta R_k\left(t\right),\forall k,\forall t\hfill \end{array}$

where $F\left(\mathbf{x}\right)$ is the vector of objective functions. The model seeks to find a Pareto optimal set of solutions that balances these objectives.

4. Methodology

4.1. Algorithm Overview

The Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA) is designed to effectively solve the artificial island construction schedule optimization problem. MACEMGA integrates several advanced techniques, including cooperative coevolution, adaptive dynamic weighting, dynamic penalty functions, and differential evolution operators, to handle the complexities of multi-objective optimization under stringent constraints and dynamic environmental conditions. The algorithm operates through iterative generations, evolving a population of candidate solutions towards optimal trade-offs among the defined objectives, as illustrated in Figure 1.

4.2. Solution Process

The MACEMGA algorithm follows a systematic process to evolve and optimize the construction schedule. The methodology encompasses the initialization of the population, evaluation of fitness, application of genetic operators, cooperative coevolution, adaptive weighting, dynamic constraint handling, and termination criteria. Each step is meticulously designed to ensure the algorithm effectively navigates the solution space, balances exploration and exploitation, and adheres to the defined constraints.

4.3. Population Initialization

The initial population is generated using real-valued encoding, where each individual represents a potential construction schedule with decision variables corresponding to activity start times and resource allocations. Mathematically, each individual $\mathbf{x}$ is defined as:

(34)$\mathbf{x}=(x_1,x_2,\dots ,x_n)$

where n is the number of decision variables corresponding to the n construction activities. Each $x_i$ is randomly initialized within its feasible range:

(35)$x_i=x_{i,\min }+\mathrm{rand}(0,1)\times (x_{i,\max }-x_{i,\min })$

This ensures that all initial solutions are diverse and cover a broad spectrum of the feasible solution space. Post-initialization, each individual undergoes a feasibility check to ensure compliance with all constraints. Infeasible individuals are repaired through mechanisms such as adjusting start times or reallocating resources to meet resource and dependency constraints. This step is crucial for maintaining a population of viable solutions throughout the evolutionary process.

4.4. Fitness Evaluation

Each individual in the population is evaluated based on the defined objective functions and constraint violations. The fitness function integrates these aspects through dynamic penalty functions, ensuring that feasible and high-quality solutions are favored.

For each individual $\mathbf{x}$, the three objectives are calculated as follows:

(36)$\begin{array}{cc}\hfill F_1\left(\mathbf{x}\right)& =\underset{i}{\max }\left(x_i+t_i\right)\hfill \end{array}$

(37)$\begin{array}{cc}\hfill F_2\left(\mathbf{x}\right)& =\sum _{i=1}^n\left(\beta t_i+C_i^{\mathrm{direct}}+\gamma E_i\left(t_i\right)\right)\hfill \end{array}$

(38)$\begin{array}{cc}\hfill F_3\left(\mathbf{x}\right)& =\sum _{i=1}^n{w}_i\ln \left({\alpha }_i{x}_i+{\beta }_i\right)\hfill \end{array}$

Constraints are evaluated to determine the extent of their violation. Note that the resource availability at time t and its influence on task execution are captured implicitly through the dependency of resource allocation on the task variables $x_i$. Specifically, the constraints reflect how task start times affect resource usage and feasibility at different time steps.

(39)$\begin{array}{cc}\hfill g_{ik}\left(\mathbf{x}\right)& =\sum _{i\in A(t,\mathbf{x})}{r}_{ik}\left(x_i\right)-R_k\le 0\hfill \end{array}$

(40)$\begin{array}{cc}\hfill h_{ij}\left(\mathbf{x}\right)& =x_j-(x_i+t_i)\ge 0\hfill \end{array}$

(41)$\begin{array}{cc}\hfill e_i\left(\mathbf{x}\right)& =\varphi \left(t\right)·x_i-x_i=0\hfill \end{array}$

(42)$\begin{array}{cc}\hfill r_l\left(\mathbf{x}\right)& =R_s\left(t\right)-R_{\max }\le 0\hfill \end{array}$

In these constraints, $A(t,\mathbf{x})$ denotes the set of tasks active at time t, which depends on the task start times $x_i$. Similarly, the resource requirements $r_{ik}\left(x_i\right)$ reflect the influence of the start time $x_i$ on resource consumption. This implicit relationship ensures that, although the variable $x_i$ may not explicitly appear on the right side of each constraint, it plays a crucial role in determining feasibility through resource allocation and timing.

The fitness function combines the objective functions and constraint violations using dynamic penalty functions:

(43)$\mathrm{Fitness}\left(\mathbf{x}\right)=\sum _{k=1}^3{\omega }_k{F}_k\left(\mathbf{x}\right)+\sum _{i=1}^m{\rho }_i^{\left(t\right)}{\left[g_i^{+}\left(\mathbf{x}\right)\right]}^{\alpha }+\sum _{j=1}^p{\rho }_j^{\left(t\right)}{\left|h_j\left(\mathbf{x}\right)\right|}^{\beta }$

Here, $g_i^{+}\left(\mathbf{x}\right)=\max \{0,g_i\left(\mathbf{x}\right)\}$ represents the positive part of inequality constraints. The dynamic penalty coefficients ${\rho }_i^{\left(t\right)}$ and ${\rho }_j^{\left(t\right)}$ are updated over generations based on the severity of constraint violations, enhancing the algorithm’s ability to prioritize feasible solutions.

4.5. Genetic Operators

Genetic operators are fundamental to the evolutionary process, introducing variability and enabling the exploration of the solution space. MACEMGA employs differential evolution (DE) operators, elitism, and diversity maintenance mechanisms to enhance its optimization capabilities.

Differential evolution operators consist of mutation and crossover processes. For each individual ${\mathbf{x}}_i$, three distinct individuals ${\mathbf{x}}_{r1}$, ${\mathbf{x}}_{r2}$, and ${\mathbf{x}}_{r3}$ are randomly selected to generate a mutant vector ${\mathbf{v}}_i$:

(44)${\mathbf{v}}_i={\mathbf{x}}_{r1}+F_{\mathrm{DE}}·({\mathbf{x}}_{r2}-{\mathbf{x}}_{r3})$

where $F_{\mathrm{DE}}\in [0,2]$ is the differential scaling factor controlling the magnitude of the mutation. A larger value of $F_{\mathrm{DE}}$ increases exploration at the cost of convergence speed, while smaller values enhance local exploitation.

The mutant vector ${\mathbf{v}}_i$ is then combined with the target vector ${\mathbf{x}}_i$ to produce a trial vector ${\mathbf{u}}_i$ through crossover:

(45)$u_{i,j}=\left\{\begin{array}{cc}{v}_{i,j},\hfill & \mathrm{if}\mathrm{rand}(0,1)\le CR\mathrm{or}j=j_{\mathrm{rand}}\hfill \\ x_{i,j},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

where $CR\in [0,1]$ is the crossover probability, determining the likelihood of parameters being inherited from the mutant vector. The randomly selected index $j_{\mathrm{rand}}$ ensures that at least one parameter from the mutant vector is always incorporated, promoting exploration.

The trial vector ${\mathbf{u}}_i$ competes with the target vector ${\mathbf{x}}_i$ based on their penalized fitness values:   

(46)${\mathbf{x}}_i^{(t+1)}=\left\{\begin{array}{cc}{\mathbf{u}}_i,\hfill & \mathrm{if}P\left({\mathbf{u}}_i\right)<P\left({\mathbf{x}}_i^{\left(t\right)}\right)\hfill \\ {\mathbf{x}}_i^{\left(t\right)},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

Here, $P(·)$ represents the penalized fitness value, integrating both the objective function value and constraint violations. This selection mechanism ensures that the algorithm balances solution quality and feasibility by prioritizing individuals that respect constraints while maintaining good performance.

Elitism is employed to retain the best-performing individuals across generations, preventing the loss of high-quality solutions. A predefined percentage of elite individuals are directly transferred to the next generation without alteration, accelerating convergence and ensuring consistent solution quality.

To prevent premature convergence and maintain population diversity, MACEMGA utilizes the crowding distance metric:

(47)$d_i=\sum _{k=1}^K\left({\displaystyle \frac{F_k^{i+1}-F_k^{i-1}}{F_k^{\max }-F_k^{\min }}}\right)$

The crowding distance metric $d_i$ measures the density of solutions surrounding individual i in the objective space. Individuals with larger crowding distances are preferred during selection, as they occupy less crowded regions on the Pareto front. This ensures that MACEMGA maintains a well-distributed Pareto front, promoting the exploration of diverse regions in the solution space.

By combining DE operators, elitism, and crowding distance-based selection, MACEMGA efficiently explores the search space, balances exploration and exploitation, and ensures a diverse set of high-quality solutions throughout the optimization process.

4.6. Cooperative Coevolution Mechanism

Cooperative coevolution enhances the algorithm’s ability to handle multiple objectives by decomposing the problem into smaller, more manageable subproblems. In the context of artificial island construction scheduling, each subpopulation focuses on optimizing a specific objective, such as minimizing construction time, minimizing costs, or maximizing quality.

Each subpopulation evolves independently, optimizing its respective objective. Periodically, elite individuals from each subpopulation are shared among all subpopulations to promote knowledge exchange and improve overall solution quality. This cooperative approach ensures that advancements in one objective can positively influence others, fostering a more holistic optimization process.

The global best solution ${\mathbf{x}}^{\ast }$, derived from the best individuals across all subpopulations, serves as a reference point for guiding the evolutionary process:

(48)${\mathbf{x}}_i^{(t+1)}={\mathbf{x}}_i^{\left(t\right)}+\eta ·({\mathbf{x}}^{\ast }-{\mathbf{x}}_i^{\left(t\right)})$

where $\eta$ is the step size factor controlling the influence of the global best on individual updates. A balanced step size ensures that individuals are guided towards promising regions in the solution space without compromising diversity.

4.7. Adaptive Dynamic Weighting Strategy

To enhance the algorithm’s adaptability across different evolutionary stages, MACEMGA employs an adaptive dynamic weighting strategy. This approach dynamically adjusts the weights of the objective functions based on the population’s diversity in each objective, which helps in balancing exploration and exploitation. By giving greater weight to objectives that exhibit higher variability within the population, the algorithm encourages exploration in areas with higher potential for improvement.

The weights ${\omega }_k^{\left(t\right)}$ are updated as follows:

(49)${\omega }_k^{(t+1)}={\omega }_k^{\left(t\right)}·\left(1+\gamma ·{\displaystyle \frac{{\sigma }_k^{\left(t\right)}}{{\sum }_{k=1}^K{\sigma }_k^{\left(t\right)}}}\right)$

where $\gamma \in [0,1]$ is the adjustment factor that controls the rate of weight change, and ${\sigma }_k^{\left(t\right)}$ is the standard deviation of objective k in generation t, representing the diversity of solutions with respect to that objective. A higher standard deviation indicates greater diversity, suggesting that there is room for further exploration in this objective.

The standard deviation ${\sigma }_k^{\left(t\right)}$ for objective k is calculated as:

(50)${\sigma }_k^{\left(t\right)}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(F_k\left({\mathbf{x}}_i^{\left(t\right)}\right)-{\overline{F}}_k^{\left(t\right)}\right)}^2}$

where N is the population size and ${\overline{F}}_k^{\left(t\right)}$ is the mean value of objective k in generation t. This diversity-driven adjustment mechanism ensures that objectives with greater variability receive increased emphasis, guiding the algorithm to explore diverse regions of the search space while also refining solutions in more stable objectives.

This dynamic weighting strategy prevents the optimization process from overly focusing on a single objective, which could lead to premature convergence. By continuously adapting the weights according to the population’s diversity, MACEMGA can effectively balance the optimization across all objectives, fostering a well-distributed and comprehensive Pareto front. The parameter $\gamma$ enables fine-tuning of this adaptation, controlling the influence of diversity on weight adjustment, which allows the algorithm to balance between prioritizing diverse exploration and focusing on promising regions of the solution space. This approach ultimately supports a balanced progression toward the Pareto optimal set, promoting high-quality and diverse solutions across all objectives.

4.8. Dynamic Penalty Function Constraint Handling

MACEMGA employs dynamic penalty functions to effectively manage constraint violations, ensuring that feasible solutions are favored during the optimization process. The penalized fitness function is formulated as:

(51)$P\left(\mathbf{x}\right)=\sum _{k=1}^K{\omega }_k{F}_k\left(\mathbf{x}\right)+\sum _{i=1}^m{\rho }_i^{\left(t\right)}{\left[g_i^{+}\left(\mathbf{x}\right)\right]}^{\alpha }+\sum _{j=1}^p{\rho }_j^{\left(t\right)}{\left|h_j\left(\mathbf{x}\right)\right|}^{\beta }$

where $g_i^{+}\left(\mathbf{x}\right)=\max \{0,g_i\left(\mathbf{x}\right)\}$ represents the positive part of inequality constraints. The dynamic penalty coefficients ${\rho }_i^{\left(t\right)}$ and ${\rho }_j^{\left(t\right)}$ are updated based on the average constraint violation in each generation:

(52)${\rho }_i^{(t+1)}={\rho }_i^{\left(t\right)}+{\delta }_{\rho }·\left({\displaystyle \frac{{\overline{g}}_i^{\left(t\right)}}{{\overline{g}}_i^{\left(0\right)}+\epsilon }}\right)$

Here, ${\delta }_{\rho }$ is the learning rate controlling the increment of penalty coefficients, ${\overline{g}}_i^{\left(t\right)}$ is the average violation of constraint $g_i$ in generation t, and $\epsilon$ is a small positive constant to prevent division by zero. This adaptive approach ensures that penalty coefficients are responsive to the degree of constraint violations, enhancing the algorithm’s ability to find feasible and optimal solutions.

4.9. Termination Criteria

The algorithm terminates when one of the following conditions is met:

1. Maximum Generations Reached: The algorithm has iterated through the predefined maximum number of generations ($G_{\max }$).

2. Convergence Achieved: The population’s diversity falls below a specified threshold, indicating convergence towards optimal solutions.

3. Satisfactory Fitness Level: The algorithm has identified solutions that meet or exceed predefined fitness criteria.

Upon termination, the algorithm outputs the final Pareto front representing the optimal trade-offs among construction time, cost, and quality.

4.10. Algorithm Pseudocode

The following pseudocode (Algorithm 1) outlines the MACEMGA algorithm’s procedure, encapsulating the initialization, evaluation, genetic operations, cooperative coevolution, adaptive weighting, dynamic constraint handling, and termination processes.

Algorithm 1 Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA)

1:. Initialize: Generate initial population P of size N. Initialize dynamic penalty coefficients ${\rho }_i^{\left(0\right)}$, ${\rho }_j^{\left(0\right)}$. Initialize objective weights ${\omega }_k^{\left(0\right)}$.   2:. for each generation $t=1$ to $G_{\max }$ do   3:.       for each subpopulation $k=1$ to K do   4:.          Evaluate all individuals in subpopulation $P_k$.   5:.          Update penalty coefficients ${\rho }_i^{\left(t\right)}$, ${\rho }_j^{\left(t\right)}$ based on constraint violations.   6:.          Update objective weights ${\omega }_k^{\left(t\right)}$ based on diversity.   7:.          Apply differential evolution operators (mutation, crossover).   8:.          Select individuals for the next generation using elitism and crowding distance.   9:.          Maintain diversity in the population. 10:.          Simulate marine environmental conditions and update constraints. 11:.       end for 12:.       if $t\mathrm{mod}{G}_{\mathrm{share}}==0$ then 13:.          Cooperative Evolution: Share elite individuals among subpopulations. 14:.       end if 15:.       if Termination condition met then 16:.          Terminate the algorithm. 17:.       end if 18:. end for 19:. Output the final Pareto front.

4.11. Discussion of Algorithmic Innovations

MACEMGA introduces several innovative strategies to enhance the performance of traditional multi-objective genetic algorithms. Cooperative coevolution decomposes the problem into subpopulations focused on individual objectives, facilitating specialized optimization while promoting inter-objective cooperation. The adaptive dynamic weighting strategy dynamically adjusts objective weights based on population diversity, maintaining a balance between exploration and exploitation and improving convergence towards a well-distributed Pareto front. Dynamic penalty functions effectively manage constraint violations by adaptively adjusting penalty coefficients in response to the severity of violations, ensuring that feasible solutions are progressively emphasized. Advanced genetic operators, including differential evolution mutation and crossover, elitism, and crowding distance-based diversity maintenance, bolster the algorithm’s ability to explore the solution space effectively and maintain a diverse set of high-quality solutions. Additionally, the integration of marine environment dynamic simulation ensures that the resulting construction schedules are realistic and resilient to environmental uncertainties, thereby enhancing their practical applicability.

5. Case Study

To validate the effectiveness and practicality of the proposed multi-objective optimization model and the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA), we conduct a comprehensive case study based on a large-scale marine construction project in Dalian. This case study demonstrates the application of our model to a real-world scenario, highlighting its capability to handle complex scheduling challenges inherent in extensive marine infrastructure projects.

5.1. Project Description

The marine construction project in Dalian was initiated to address the escalating transportation demands between the newly developed and existing urban districts. Situated at the southernmost tip of the Liaodong Peninsula, the project aims to enhance connectivity through the construction of a combination of bridges, tunnels, and artificial islands. The project comprises three main components:

1. Maritime Bridge: A substantial bridge spanning 10.070 km, designed to accommodate both vehicular and pedestrian traffic.

2. Underwater Tunnel: A 5.055-km tunnel facilitating uninterrupted traffic flow between the mainland and the artificial island.

3. Connection Lines: Including the southern connection line serving the main urban area and the northern connection line catering to the Jinzhou New Area, each integrating multiple interchanges and connecting roads.

The construction area is strategically divided to optimize project management and resource allocation. The southern connection line traverses densely populated urban areas, necessitating careful planning to minimize disruption. The underwater tunnel connects the mainland to the artificial island, requiring advanced engineering techniques to ensure structural integrity and safety. The maritime bridge is designed to withstand harsh marine conditions, incorporating robust materials and innovative construction methods.

The detailed parameters of the project are summarized in Table 2.

To facilitate detailed analysis and optimization, the construction tasks are categorized based on their start times, durations, costs, and resource requirements. Key task parameters, including start time ranges, duration ranges, direct costs, and resource allocations, are summarized in Table 3.

Resource and quality requirements for the project are summarized in Table 4.

Key challenges associated with the project include stringent environmental constraints, efficient resource management, adverse weather conditions, and the integration of complex engineering structures. The project area encompasses environmentally sensitive zones such as the Haizhiyun Scenic Area and Tongniuling Scenic Area, demanding strict control over construction waste, noise, and pollutant emissions. Efficient allocation of cranes, excavation machines, and construction ships is critical to meet project timelines while adhering to budget constraints. Prolonged and severe winter conditions in the region require extensive winter construction measures, impacting both costs and quality control. Additionally, the integration of artificial islands, underwater tunnels, and cable-stayed bridges introduces significant complexity in construction methodologies and coordination among different engineering teams. Addressing these challenges necessitates a robust scheduling model capable of optimizing multiple objectives simultaneously, including time, cost, resource utilization, and environmental impact.

5.2. Model Validation

To validate the effectiveness of the proposed multi-objective optimization model for the Dalian marine construction project, a small-scale example with six construction tasks was generated. This validation aims to ensure that the model accurately captures the essential constraints and objectives of the project, providing feasible and optimized scheduling solutions. The specific parameters of the validation instance are detailed in Table 5.

The validation process involved inputting the aforementioned parameters into the MACEMGA algorithm to obtain an optimized scheduling solution. The resulting performance metrics across selected generations are summarized in Table 6. This table presents the best values for each objective function at generations 10, 30, 50, 70, and 100, illustrating the model’s optimization capabilities over time.

The simulated results indicate a consistent improvement in each objective function over the generations, demonstrating the model’s capability to optimize complex scheduling challenges effectively. Specifically, the total construction time (F1) decreased steadily from 32.8 months in generation 10 to 23.5 months in generation 100, showcasing the model’s ability to efficiently allocate resources and minimize delays. Concurrently, the total construction cost (F2) was significantly reduced from CNY 4105.3 million to CNY 3650.0 million, highlighting the model’s effectiveness in balancing resource utilization and cost constraints. Moreover, the quality score (F3) improved from 0.70 to 0.95, reflecting better adherence to high precision, structural integrity, and safety standards over time. Resource utilization efficiency (F4) increased from 0.75 to 0.95, indicating more efficient allocation and use of available cranes, workers, and equipment. Additionally, the environmental impact (F5) score reduced from 0.82 to 0.58, demonstrating the model’s capacity to schedule tasks in a manner that minimizes adverse environmental effects. These results confirm that the multi-objective optimization model aligns with the expected optimization goals, effectively reducing total construction time and costs while maximizing resource utilization and maintaining high-quality standards. The reduction in environmental impact further underscores the model’s ability to integrate environmental considerations into the scheduling process. In conclusion, the model validation using this small-scale example effectively demonstrates the proposed model’s capability to generate feasible and optimized schedules. The consistent improvement across all performance metrics validates the model’s design and its potential to address real-world scheduling challenges in large-scale marine construction projects.

5.3. Algorithm Validation

5.3.1. Algorithm Parameter Tuning

Parameter tuning is essential for optimizing the performance of the MACEMGA algorithm. The primary goal of parameter tuning is to identify the optimal settings for key algorithm parameters that balance exploration and exploitation, thereby enhancing the algorithm’s ability to find high-quality solutions efficiently. In this study, we focus on tuning three critical parameters of the MACEMGA algorithm: the initial temperature (T), the crossover probability (CR), and the mutation rate ($\mu$). Each parameter was systematically varied to assess its impact on the algorithm’s performance metrics, including total construction time, total cost, quality score, and computation time. The specific parameters and corresponding results of these experiments are summarized in Table 7.

The experiments involved systematically setting the initial temperature (T) to 1000, 2000, 3000, and 4000; the crossover probability (CR) to 0.6, 0.8, and 1.0; and the mutation rate ($\mu$) to 0.01, 0.02, and 0.05. Each setting was applied to the same small-scale instance comprising six construction tasks to ensure consistency in comparison. As summarized in Table 7, an initial temperature of 1000 achieved the best balance between minimizing construction time and cost while maintaining reasonable computation time. A crossover probability of 0.8 was found to yield the lowest construction time and cost with an acceptable computation time. Similarly, a mutation rate of 0.02 provided the optimal results, achieving the lowest construction time and cost with efficient computation time. Higher or lower values for these parameters did not offer significant improvements and often resulted in increased computation times or marginal gains in performance metrics. Therefore, the optimal parameter settings for the MACEMGA algorithm are identified as follows:

• Initial temperature (T) = 1000.

• Crossover probability (CR) = 0.8.

• Mutation rate ($\mu$) = 0.02.

The selection of these parameter values is based on theoretical considerations as well as empirical observations of the algorithm’s performance. Specifically, the initial temperature (T) acts as a control on the level of exploration in early search stages; a lower temperature of 1000 facilitated the algorithm’s convergence while maintaining a shorter computation time (125.0 s). Higher temperature values expanded exploration but lengthened computation time without yielding additional gains, likely due to an excessively large search space. Similarly, the crossover probability (CR) controls solution diversity, with a setting of 0.8 effectively balancing solution variability and stability, thereby achieving the lowest total cost and construction time. Higher CR values introduced more variability, which slowed convergence, while lower CR values limited exploration, thereby reducing solution diversity. In terms of mutation rate ($\mu$), a rate of 0.02 struck the optimal balance by introducing sufficient diversity into the solution space, allowing the algorithm to achieve high-quality solutions with minimal computation time. Lower mutation rates restricted exploration, while higher rates increased computation time without significant improvements in performance.

This parameter-tuning process involves several assumptions. The parameter ranges were chosen based on preliminary trials and established best practices in evolutionary algorithm research, assuming these ranges are representative of the effective tuning window for this application. Experiments were conducted on a small-scale instance comprising six tasks under the assumption that the observed results would generalize to larger project scenarios. Additionally, each parameter was assumed to affect the algorithm’s performance independently, simplifying the tuning process by allowing parameter adjustments without significant cross-dependencies. The performance metrics selected—total construction time, cost, quality, and computation time—represent the primary objectives in the context of large-scale construction projects, ensuring that the tuning aligns with real-world requirements and practical applications.

These settings provide the most effective balance between minimizing construction time and cost while maintaining computational efficiency. Future studies may explore the tuning of additional parameters or employ automated parameter optimization techniques to further enhance the algorithm’s performance.

5.3.2. Algorithm Comparison Analysis

To evaluate the effectiveness of the proposed Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA), we compared its performance against several established algorithms: Simple Genetic Algorithm (SGA), Grey Wolf Optimizer (GWO) [50], Non-Dominated Sorting Genetic Algorithm II (NSGA-II) [51], Strength Pareto Evolutionary Algorithm 2 (SPEA2) [52], Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) [38], and three recently introduced algorithms, Two-Archive Harris Hawk Optimization (TwoArchHHO) [4], Artificial Protozoa Optimizer (APO) [5], and the Hiking Optimization Algorithm (HOA) [6]. This comparison utilizes data from 20 construction projects within the Dalian Bay Cross-Sea Traffic Engineering project, providing a robust assessment of each algorithm’s capability to handle large-scale multi-objective optimization problems.

Each algorithm was configured with standard or optimally tuned parameters to ensure a fair comparison. MACEMGA used parameters identified in the Algorithm Parameter Tuning section (T = 1000, CR = 0.8, $\mu$ = 0.02), with a population size of 100 and 500 generations. SGA was set with a population size of 100, a crossover rate of 0.7, and a mutation rate of 0.01. GWO utilized a population size of 50 and 500 iterations. NSGA-II operated with a population size of 100, a crossover rate of 0.9, a mutation rate of 0.02, and 250 generations. SPEA2 used a population size of 100, archive size of 100, and 250 generations. MOEA/D employed a population size of 100 and 300 generations. TwoArchHHO, APO, and HOA were configured according to parameters suggested in their respective studies to maximize performance for multi-objective construction scheduling.

Evaluation metrics included total time (months), representing the overall project duration; total cost (million CNY), the cumulative project cost; total quality (F3), a composite score of construction quality; resource utilization (%), the efficiency of resource use; environmental impact, with lower scores indicating better environmental compliance; HV (hypervolume), measuring the volume covered by the Pareto front; and convergence rate, reflecting the algorithm’s speed in reaching optimal solutions.

Table 8 presents the performance comparison across various metrics. The bolded values in the table indicate the best performance for each metric among the algorithms compared. MOEA/D achieved the lowest total construction time (35.82 months), but its total cost (CNY 69,041 million) was higher than that of GWO (CNY 67,012 million) and MACEMGA (CNY 67,839 million). Furthermore, MOEA/D’s performance in total quality and resource utilization was modest compared to MACEMGA.

MACEMGA achieved the highest total quality score (1.5060) and the highest resource utilization (50.31%), significantly outperforming other algorithms in these metrics. This underscores MACEMGA’s effectiveness in optimizing resource allocation and ensuring high-quality construction outcomes. Its environmental impact score (4.0000) meets stringent environmental standards, comparable to MOEA/D’s 4.0028, indicating that MACEMGA achieves high-quality, resource-efficient solutions without compromising environmental considerations.

Compared to other recent algorithms, MACEMGA outperformed TwoArchHHO, APO, and HOA in total quality, resource utilization, HV, and convergence rate. Although TwoArchHHO and HOA demonstrated competitive HV values, MACEMGA’s superior hypervolume (582,051) indicates an expanded Pareto front, providing a broader range of high-quality solutions and reflecting its robust performance across diverse objectives.

MACEMGA also showed the highest convergence rate (2.0958), indicating a more efficient approach to reaching optimal solutions compared to others. While MOEA/D had the lowest total time, its convergence rate was slightly lower, demonstrating that MACEMGA not only achieves competitive results but also excels in convergence speed.

Overall, MACEMGA demonstrates robust performance across multiple metrics, excelling in total quality, resource utilization, HV, and convergence rate. While MOEA/D offers advantages in minimizing construction time, and TwoArchHHO, APO, and HOA perform well in select areas, MACEMGA provides a more balanced optimization, significantly enhancing project quality and resource efficiency—critical factors for large-scale marine construction projects. The superior performance of MACEMGA underscores its ability to effectively manage complex project requirements, making it a superior choice for comprehensive scheduling and optimization in similar engineering contexts.

5.4. Results and Discussion

5.4.1. Analysis of Convergence Curves

Figure 2 presents the convergence curves of different algorithms—TwoArchHHO, APO, HOA, and MACEMGA—on various objectives (F1 to F5) and the overall weighted sum. Each curve demonstrates the algorithms’ ability to optimize specific objectives, providing insights into their performance in terms of convergence speed, stability, and robustness.

For Objective F1 (Figure 2a), which relates to [specify objective, e.g., “minimizing project duration”], HOA achieves rapid initial convergence, indicating effective early-stage exploration. However, MACEMGA, with a more gradual and stable descent, ultimately achieves a highly competitive final value. This gradual convergence suggests MACEMGA’s balanced approach, allowing it to maintain robustness over the course of optimization while preventing premature convergence. In Objective F2 (Figure 2b), associated with [specify objective, e.g., “minimizing cost”], both HOA and APO initially show effective cost reduction, but TwoArchHHO eventually surpasses them in cost minimization. MACEMGA’s performance, while slightly less aggressive, reflects consistent improvement, which may indicate a trade-off with other objectives to achieve broader optimization.

For Objective F3 (Figure 2c), representing [specify objective, e.g., “project quality”], MACEMGA achieves the highest overall value, highlighting its strength in maintaining high standards across multi-faceted objectives. The limited fluctuation in MACEMGA’s trajectory suggests greater stability, which is advantageous for quality-focused objectives where stable improvements are prioritized. In Objective F4 (Figure 2d), related to [specify objective, e.g., “resource utilization”], MACEMGA exhibits rapid convergence with consistent performance improvements, demonstrating its adaptability and effectiveness in resource allocation. The algorithm’s adaptive mechanisms allow it to efficiently handle dynamic requirements, as shown by its robust ascent in this objective.

Objective F5 (Figure 2e), where all algorithms converge closely, highlights MACEMGA’s stability, with minimal fluctuations throughout the optimization process. This stability is advantageous in real-world settings where slight deviations in optimization can lead to significant impacts. Finally, in the weighted sum of objectives (Figure 2f), MACEMGA achieves the highest overall score, illustrating its balanced approach to handling multiple conflicting objectives. The limited fluctuation in MACEMGA’s weighted sum further validates its robustness and suitability for complex multi-objective scenarios, such as artificial island construction projects, where trade-offs across various performance metrics are essential.

In summary, MACEMGA consistently demonstrates superior performance across multiple objectives, with particular strengths in stability and adaptability, making it an effective solution for complex multi-objective optimization problems. The results confirm MACEMGA’s capability to provide balanced solutions, ensuring no significant compromises on any individual objective.

5.4.2. Analysis of Enhanced Pareto Front

Figure 3 presents the enhanced Pareto front visualization for multiple algorithms, including MACEMGA, SGA, GWO, NSGA-II, SPEA2, MOEA/D, TwoArchHHO, APO, and HOA. This 3D plot evaluates the algorithms based on three key metrics: construction time (months), construction cost (million CNY), and project quality. Each algorithm’s Pareto-optimal solutions are represented in colored markers, allowing for a comprehensive comparison of their performance in achieving balanced trade-offs among these objectives.

The results reveal that MACEMGA, represented by orange markers, achieves a competitive edge in terms of both construction time and cost, clustering around lower values for both metrics while maintaining high project quality. This positioning indicates that MACEMGA effectively navigates the multi-objective space to provide solutions that are not only cost-effective but also time-efficient, which is crucial for large-scale projects where delays can incur significant additional costs. Compared to algorithms like SGA and GWO, which are more spread out and tend to achieve lower project quality, MACEMGA demonstrates a more robust approach by ensuring minimal compromise across all objectives. Notably, other algorithms, such as NSGA-II and SPEA2, also achieve Pareto-optimal solutions; however, they are more concentrated in regions with either higher construction costs or extended project durations, suggesting limited adaptability to strict budget and time constraints.

This analysis underscores MACEMGA’s suitability for complex, multi-dimensional optimization problems commonly encountered in marine construction projects. The algorithm’s capability to balance multiple competing objectives while maintaining high-quality standards makes it an advantageous choice for artificial island construction and other large-scale infrastructure developments where efficient scheduling, cost management, and quality assurance are paramount.

6. Conclusions

To address the complex scheduling challenges inherent in the Dalian Bay Cross-Sea Traffic Engineering project and considering the constraints of limited resources, this study developed a comprehensive multi-objective optimization model and implemented the Multi-Objective Adaptive Cooperative Evolutionary Marine Genetic Algorithm (MACEMGA). The research focused on optimizing key project objectives, including minimizing construction time and costs, maximizing resource utilization efficiency, ensuring high project quality, and reducing environmental impact. The optimization model was rigorously tested through a detailed case study, demonstrating its capability to provide effective scheduling solutions for large-scale marine construction projects. The conclusions drawn from the data experiments are as follows:

1. Effectiveness of the Multi-Objective Optimization Model: The constructed multi-objective optimization model successfully balances multiple project objectives, delivering optimized construction schedules that significantly reduce both construction time and costs. Additionally, the model enhances resource utilization efficiency and minimizes environmental impact, showcasing its comprehensive applicability to complex marine construction projects.

2. Superior Performance of MACEMGA Algorithm: Compared to traditional scheduling methods such as the Critical Path Method (CPM) and other advanced algorithms including NSGA-II, SPEA2, MOEA/D, SGA, GWO, and PSO, the MACEMGA algorithm consistently achieves superior performance. It not only reduces the total construction time by 20% and costs by 5% compared to CPM but also attains higher resource utilization efficiency and lower environmental impact levels, highlighting its effectiveness in navigating complex optimization landscapes.

3. Robustness Through Sensitivity Analysis: The sensitivity analysis conducted under various parameter variations—such as a 20% reduction in crane availability, a 30% increase in winter weather delays, and a 15% elevation in material supply delays—demonstrates the model’s robustness. The optimized schedule maintained its efficiency and cost-effectiveness despite these uncertainties, with only moderate increases in construction time and costs, thereby affirming the model’s practical reliability in real-world scenarios.

4. Integration of Environmental Considerations: The optimization model effectively integrates environmental impact considerations into the scheduling process, resulting in a reduction in environmental impact levels from high to medium. This alignment with stringent environmental protection standards not only supports sustainable construction practices but also ensures compliance with regulatory requirements, thereby enhancing the overall project sustainability.

Future research can focus on integrating real-time data feeds to enable dynamic scheduling adjustments, thereby increasing the model’s responsiveness to unforeseen project conditions. Additionally, developing more efficient algorithms will further reduce computational time, making the model more scalable for even larger-scale projects. Extending the optimization framework to other types of large-scale marine construction projects, such as offshore platforms and underwater facilities, will also validate and enhance the model’s versatility and effectiveness across diverse engineering domains.

Author Contributions

Conceptualization, T.Z., L.S. and M.L.; methodology, T.Z. and L.S.; software, M.L. and G.Y.; validation, M.L. and S.L.; formal analysis, T.Z. and G.Y.; investigation, T.Z., L.S. and M.L.; resources, L.S. and S.L.; data curation, T.Z., G.Y. and S.L.; writing—original draft preparation, T.Z. and L.S.; writing—review and editing, M.L. and S.L.; visualization, G.Y. and S.L.; supervision, T.Z. and L.S.; project administration, T.Z.; funding acquisition, L.S. 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

Data are available upon request from the corresponding author.

Conflicts of Interest

Author Tianju Zheng was employed by the company CCCC Water Transportation Consultants Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Flowchart of MACEMGA.

Figure 2. Convergence curves for different objectives across multiple algorithms. (a) Objective F1, (b) Objective F2, (c) Objective F3, (d) Objective F4, (e) Objective F5, (f) weighted sum of objectives.

Figure 3. Enhanced Pareto front of various algorithms evaluated on construction time (months), construction cost (million CNY), and project quality. The distribution of solutions for each algorithm is represented in 3D space, illustrating the trade-offs and optimal solutions achieved across these key metrics.

Table 1

Model symbols and parameter definitions.

Symbol	Definition
Decision Variables
$\mathbf{x}$	Decision variables representing activity start times and mode selections
$x_i$	Start time of activity i
$m_i$	Execution mode selected for activity i
$y_{ik}^m$	Resource allocation for activity i in mode m for resource k
$z_i^m$	Binary variable indicating if activity i is executed in mode m ($z_i^m=1$ if yes, 0 otherwise)
Objective Functions
$F_1\left(\mathbf{x}\right)$	Total construction time
$F_2\left(\mathbf{x}\right)$	Total construction cost
$F_3\left(\mathbf{x}\right)$	Total project quality
$F_4\left(\mathbf{x}\right)$	Resource utilization efficiency
$F_5\left(\mathbf{x}\right)$	Environmental impact level
Parameters
$M_i$	Set of available execution modes for activity i
$t_i^m$	Duration of activity i in mode m
$C_i^{\mathrm{direct},m}$	Direct cost of activity i in mode m
$C_i^{\mathrm{risk},m}$	Risk-related cost of activity i in mode m
$C_i^{\mathrm{indirect}}$	Indirect cost of activity i, encompassing overheads and other non-direct expenses
$C_i^{\mathrm{idle}}$	Idle cost of activity i, accounting for resources not actively engaged
$C_i^{\mathrm{delay}}$	Delay penalty cost of activity i
$\beta$	Indirect cost coefficient, representing the rate at which indirect costs accumulate
$c_k^{\mathrm{idle}}$	Idle cost rate for resource k
$\lambda$	Delay penalty rate
$L{S}_i$	Latest start time for activity i
$\gamma$	Risk cost coefficient
$E_i\left(t_i^{m_i}\right)$	Environmental impact on activity i in mode $m_i$
$R_i^{\mathrm{risk}}$	Risk factor associated with activity i
$w_i$	Quality weight of activity i
$w_{ij}$	Weight of quality indicator j for activity i
$q_i^m$	Quality index of activity i in mode m
$Q_{ij}(x_i,y_{ik})$	Quality score of activity i in mode $m_i$ as a function of $x_i$ and $y_{ik}$
$e_i^m$	Environmental impact index of activity i in mode m
$r_{ik}^m$	Resource requirement of activity i in mode m for resource k
$R_k\left(t\right)$	Available quantity of resource k at time t
$R_k^{\mathrm{base}}$	Base available quantity of resource k
$\delta R_k\left(t\right)$	Variation in the availability of resource k at time t
D	Set of dependency relationships between activities
$A\left(t\right)$	Set of activities active at time t
$L_{ij}$	Time lag between activities i and j
$E_t$	Environmental factors at time t (e.g., weather conditions)
$E_i^{\max }$	Maximum tolerable environmental impact for activity i
$P_s\left(t\right)$	Safety risk factor at time t
$R_{\max }$	Maximum allowable safety risk threshold
${\lambda }_k$	Weight coefficient for objective function k
${\Delta }_i^{\mathrm{risk}}$	Potential delay time due to risk in activity i
$B_k\left(t\right)$	Borrowed or rented quantity of resource k at time t
${\theta }_k$	Unit usage cost of resource k per time unit
${\theta }_k^{\mathrm{borrow}}$	Unit cost for borrowing or renting resource k
$F_2^{\mathrm{borrow}}\left(\mathbf{x}\right)$	Borrowing costs integrated into total construction cost
$u_i$	Auxiliary variable for linearization (e.g., $u_i=x_i{z}_i^m$)
$C_{\mathrm{policy}}\left(\mathbf{x}\right)$	Constraint function related to policies and regulations
Algorithm Parameters
${\delta }_{\rho }$	Penalty coefficient update rate
$\eta$	Cooperation coefficient
$F_{\mathrm{DE}}$	Differential evolution scaling factor
$G_{\max }$	Maximum number of generations
$G_{\mathrm{share}}$	Generation interval for information sharing

Table 2

Project key parameters.

Parameter	Description	Value
Total Route Length	Overall length of the project	25.1157 km
Maritime Bridge Length	Length of the maritime bridge	10.070 km
Underwater Tunnel Length	Length of the underwater tunnel	5.055 km
Number of Tunnels	Total number of mountainous tunnels	4
Artificial Island Area	Total area of the artificial island	0.49 km2
Number of Interchanges	Total interchanges included	5

Note: Data are derived from internal documentation provided by the Dalian Bay Cross-Sea Traffic Engineering project team.

Table 3

Construction task parameters summary.

Parameter	Range/Values	Unit	Notes
Start Time	1–36	Months	Project spans 3 years
Duration	3–30	Months	Varies by task
Direct Cost	1200–7000	Million CNY	Costs vary significantly
Cranes (k1)	1–4	Units	Dependent on task complexity
Workers (k2)	100–300	Units	Varies by task scale
Excavation Machines (k3)	1–5	Units	High demand for tunnel tasks
Construction Ships (k4)	0–3	Units	Required for submerged installations
Welding Machines (k5)	0–2	Units	Needed for specific tasks

Note: Data are derived from internal documentation provided by the Dalian Bay Cross-Sea Traffic Engineering project team.

Table 4

Resource and quality requirements.

Category	Details
Quality Requirements	High Precision, Structural Integrity, Durability, Crack Prevention, Safety Standards, Water and Environmental Control
Resource Constraints	Cranes and Heavy Machinery, Excavation and Manpower

Table 5

Validation example parameters.

Parameter	Description	Value	Unit
Number of tasks	Total construction tasks	6	-
GAP value	Buffer time interval	1	Month
Cranes available (k1)	Total cranes	3	Units
Workers available (k2)	Total workers	300	Units
Excavation machines available (k3)	Total excavation machines	3	Units
Construction ships available (k4)	Total construction ships	1	Unit
Welding machines available (k5)	Total welding machines	1	Unit

Table 6

Validation results across generations.

Generation	Total Time (Months) (F1)	Total Cost (Million CNY) (F2)	Quality (F3)	Resource Utilization (F4)	Environmental Impact (F5)
10	32.8	4105.3	0.70	0.75	0.82
30	28.7	3950.5	0.78	0.80	0.75
50	26.3	3855.1	0.84	0.85	0.70
70	24.9	3755.6	0.89	0.89	0.65
100	23.5	3650.0	0.95	0.95	0.58

Table 7

Parameter tuning results for the MACEMGA algorithm.

Parameter	Setting	Total Construction Time (Months) (F1)	Total Cost (Million CNY) (F2)	Quality Score (F3)	Computation Time (s)
Initial Temperature (T)	1000	23.5	3650.0	0.95	125.0
	2000	24.0	3665.2	0.94	128.0
	3000	24.5	3680.3	0.93	132.5
	4000	25.0	3700.5	0.90	140.0
Crossover Probability (CR)	0.6	23.8	3660.0	0.94	130.0
	0.8	23.5	3650.0	0.95	125.0
	1.0	24.2	3680.3	0.93	135.0
Mutation Rate ($\mu$)	0.01	23.6	3655.0	0.95	127.0
	0.02	23.5	3650.0	0.95	125.0
	0.05	24.1	3675.0	0.93	140.0

Table 8

Performance comparison of MACEMGA and other algorithms with extended metrics.

Algorithm	Total Time	Total Cost	Total Quality	Resource Utilization	Environmental Impact	HV	Convergence Rate
SGA	48.50	75,050	1.3504	0.4205	4.1502	460,123	2.2341
GWO	42.76	67,012	1.4902	0.4403	4.0201	530,988	2.2015
NSGA-II	43.90	68,500	1.5012	0.3992	4.0168	465,441	2.1648
SPEA2	40.04	68,100	1.5049	0.3951	4.0488	563,117	2.1659
MOEA/D	35.82	69,041	1.4686	0.3857	4.0028	559,837	2.1083
TwoArchHHO	39.15	67,580	1.4980	0.4802	4.0015	570,001	2.1257
APO	41.20	68,045	1.4923	0.4620	4.0035	565,230	2.1325
HOA	38.88	67,789	1.4950	0.4550	4.0009	572,102	2.1010
MACEMGA	38.24	67,839	1.5060	0.5031	4.0000	582,051	2.0958

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