Optimizing Stack-Yard Positioning in Full Shoreline Loading Operations

Abstract

Loading operations are a crucial part of container terminal activities and play a key role in influencing shoreline operation efficiency. To overcome the challenge of mismatched local ship decisions and global yard decisions during single-vessel operations, which often result in conflicts related to container retrieval in the yard, a novel intelligent decision-making model for stack-yard positioning in full shoreline loading operations is proposed. This model seeks to optimize the balance between yard operation instructions and quay crane operation instructions. An enhanced Constrained Optimization Genetic Algorithms-Greedy Randomized Adaptive Search (COGA-GRASP) algorithm is introduced to tackle this decision-making issue, and it is applied to identify the most optimal bay configuration for full shoreline loading operations. The proposed model’s effectiveness is validated through testing and solution outcomes.

1. Introduction

Loading operations are a critical component of container terminal operations, directly influencing the efficiency of shoreline activities. As a dynamic process, loading operations involve continuous changes in the yard state due to tasks such as loading, unloading, container gathering, pickup, and relocation. Simultaneously, the pending tasks of vessels evolve, increasing the computational complexity of decision-making. To maintain operational continuity, instructions must be activated promptly to avoid stagnation in yard or shoreline operations. Therefore, developing intelligent decision-making models and algorithms that balance computational speed and solution quality is essential for optimizing full shoreline loading operations.

Currently, as shown in Figure 1, container terminal loading operations typically adopt a single-ship decision-making approach, where the containers to be loaded onto the vessel are distributed throughout the entire yard. This creates a mismatch between the local decision-making for the vessel and the global decision-making for the yard, leading to insufficient overall control, which can negatively impact loading efficiency.

To address the issue of the mismatch between local and global strategies, this paper conducts a systematic analysis of terminal loading operations and proposes the concept of full-quay container ship loading decision-making. As shown in Figure 2, it considers the vessel at berth as a unified whole, thereby aligning the global decision-making for the quay and the global decision-making for the yard. However, to implement full-quay loading decision-making, the first challenge is to address the selection of operation locations within the yard for all handling routes. In the loading process, there are multiple available operation locations for each route, which are unevenly distributed across the yard. This requires collaborative computation across all handling routes to select the appropriate yard positions for each. As the number of handling routes increases, the computational difficulty also rises.

To address the critical issue of misalignment between local operation strategies and global resource scheduling in port logistics systems, this study conducts systematic modeling analysis on container terminal handling operations based on operations research and combinatorial optimization theories. By constructing a multi-objective optimization model, we innovatively propose the theoretical framework of “Full-quay Collaborative Loading Decision-making”. This framework transcends the limitations of traditional segmented scheduling models by considering the berthed vessel cluster as a dynamic operational entity with spatiotemporal coupling characteristics. The research focuses on overcoming the collaborative optimization challenges between terminal front operations and rear yard resources, achieving integrated optimization of terminal-wide scheduling decisions and yard space allocation strategies through a planning model. At the algorithmic implementation level, this paper specifically addresses the approximation of optimal solutions for the NP-Hard problem of yard operation location selection. Notably, when concurrent operation routes n increase exponentially with growing port throughput, the solution space dimension expands drastically to posing severe challenges to traditional optimization algorithms.

Based on the above analysis, this paper proposes a full-quay container ship loading decision-making model for yard location selection, which is solved using the improved COGA and GRASP algorithms. By applying the above algorithm, the efficiency of the container terminal’s quay crane loading operation has increased by about 2%, and the yard operation have become smoother. The contributions of this paper are as follows:

• This paper develops a full-quay container ship loading yard location selection model to dynamically address the location selection problem during the loading process.
• By integrating the COGA and GRASP algorithms, an improved COGA-GRASP algorithm is designed to enhance both solution efficiency and quality.
• Through comparative analysis of different weightings and algorithms, the robustness of the proposed algorithm is validated.

Section 2 reviews previous research and provides a comparative analysis. Section 3 develops the full-quay container ship loading yard location selection model. Section 4 presents the improved algorithm design for solving the problem. Section 5 validates the model and algorithm’s effectiveness through comparisons of different weightings and solution algorithms. Section 6 concludes the paper and proposes directions for future research.

2. Literature Review

To enhance the efficiency of container terminal loading operations, several researchers have investigated the optimization of operation instruction sequences [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Some scholars have proposed optimization models that integrate quay cranes, yard cranes, and horizontal transportation for improved operational efficiency [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. Its research objectives include operational efficiency and consumption, and the research scenarios include traditional terminals and U-shaped terminals.

2.1. Instruction Optimization

Wen et al. [1] investigated the scheduling problem of container terminal loading operations, modeling and solving the problem by considering the impact of the quay crane spreader trajectory on the operation sequence.

Zhou et al. [2] addressed a combined optimization problem for continuous loading and unloading operations on a floating cargo handling platform. While the research does not directly focus on the container terminal loading process, the established goal is to minimize vessel turnaround time using a Markov decision process model. Additionally, a tabu-based tree search algorithm was developed and improved to control the depth and breadth of the local tree structure, which remains valuable for reference.

Rahman et al. [3] investigated the optimization of port container handling, integrating dual-cycle crane operations and dockyard rehandling minimization. A unified model was proposed, encompassing two operations: one for unloading and loading containers via quay cranes, and another for reducing rehandling at the dock during loading. A hybrid genetic algorithm (GA), QCDC-DR-GA, composed of one-dimensional and two-dimensional GAs, was developed and validated for solving the problem.

Yan et al. [4] proposed a deep reinforcement learning algorithm employing directed search for the CRP problem, where a deep neural network is trained by DRL to learn container repositioning strategies. Directed search is integrated into the algorithm to improve the quality of the solution and hence the efficiency of the loading operation.

Zhang et al. [5] develops a data-driven model to predict the Probabilistic Container Repositioning Problem (PCRP) for the overturning of containers during ship loading, and proposes an adaptive Monte Carlo tree search algorithm. The expected number of container repositionings is minimized by integrating a new heuristic: local safety and global flexibility.

Parreño-Torres et al. [6] developed two exact solution methods to minimize crane operation time. One method was an integer linear model, and the other a branch-and-bound algorithm. These methods introduced new upper and lower bounds, dominance criteria, and heuristic programs, providing optimal solutions for problems of practical scale.

Xia et al. [7] proposed a Double Deep Q-Network (DDQN) with priority experience replay to address the container terminal loading sequence problem. In this approach, simulations with varying complexity are introduced to model the loading process. The proposed PERDDQN, trained on real-world data, can solve the loading sequence problem in seconds. The results show that the algorithm is highly efficient and versatile.

Cai et al. [8] designed a container rehandling algorithm for ship loading based on inverse reinforcement learning, aiming to minimize the number of container relocations in the yard and the waiting time of horizontal transportation vehicles at the quay.

Xia et al. [9], based on sequential decision theory, decomposed the ship loading process into container retrieval and instruction sequencing decisions, solving each sequentially.

Yue et al. [10] introduced a dynamic scheduling process to solve the AGV scheduling and path planning problem, in which the scheduling scheme determines the starting and ending nodes of the path, and the selection of the paths between nodes affects the scheduling of subsequent AGVs. Under the loading and unloading time constraints, a two-stage mixed integer optimization model is proposed with the objective of minimizing the AGV transportation cost. A dynamic optimization algorithm including an improved rule-based heuristic algorithm and a dynamic optimization algorithm integrating Dijkstra’s algorithm and Q-Learning algorithm are designed to solve the optimal AGV scheduling and path schemes. A novel conflict avoidance strategy based on graph theory is also proposed to reduce the probability of path conflicts among AGVs.

Fibrianto et al. [11], in the context of automated container terminals equipped with overhead shuttle cranes (OS), proposed a job scheduling model and heuristic methods. By categorizing each operation as a main operation, pre-unloading, or re-unloading task, the model aims to reduce total delay times for flatbeds and external trucks. The Operation Sequencing Job Scheduling Problem (OSJSPS) is formulated as a Mixed Integer Programming (MIP) model, which simultaneously sequences a set of jobs and explores the possibility of separating them into pre-sorting and re-sorting tasks. A two-stage genetic algorithm (TGA) is proposed, incorporating two local improvement procedures: an iterative local search and an opportunistic task separation procedure. Case studies validate the effectiveness of the model.

Oladugba et al. [12] proposed a novel algorithm based on the Modified Johnson Algorithm (JA) for the job instruction scheduling problem of a Twin Automatic Stacking Crane (Twin-ASC), which employs a Mixed Integer Programming (MIP) model for solving the Twin-ASC scheduling problem in order to minimize the scheduling time span.

Liu et al. [13] considering the fluctuation of the arrival time of external trucks and the change in the loading and unloading volume of yard cranes, a stochastic programming model is established for the yard crane scheduling problem, and a customized genetic algorithm is used to find the optimal scheme with good adaptability to uncertain scenarios.

Yu et al. [14] built an integer model for the problem of space allocation for incoming containers at an automated container terminal with the goal of minimizing the waiting time for container trucks, and developed a genetic algorithm embedded in simulation to solve it.

Yang et al. [15] considered the integration and optimization of yard resources in light of the uncertainties associated with container demand, establishing a stochastic optimization model to enhance the robustness of daily yard management. He designed a harmony search algorithm based on a two-stage decomposition strategy to solve the proposed model. To accelerate the solving speed, a scoring method was introduced to generate initial space allocation plans, and a tabu list was added to improve the quality of solutions.

Gulić et al. [16] studied the two-dimensional, static, offline, and constrained container repositioning problem in actual-sized yard bays for the box-changing issue. He proposed a new method using a Genetic Algorithm (GA) to solve the container relocation problem, aiming to minimize the number of container relocations.

Yang et al. [17] addressed the scheduling problem of electric AGVs from the perspective of energy conservation and emission reduction by proposing a speed control strategy that takes into account the port traffic environment. Additionally, the charging capacity of the port was discretized to simulate the limited processing capability of the port and avoid congestion at battery swapping stations. To minimize the delay costs and carbon emissions associated with AGV operations, a mixed-integer programming model was established. This model optimizes job efficiency and carbon emissions by assigning and prioritizing container transportation tasks and AGV battery swapping tasks. An improved genetic algorithm-based method was designed, incorporating a greedy strategy to obtain better initial solutions, while simulated annealing was used for population selection to prevent the algorithm from becoming trapped in local optima. Furthermore, an adaptive adjustment strategy for crossover and mutation probabilities was employed to enhance the convergence of the algorithm.

Xu et al. [18] developed a mixed-integer programming model for the integrated scheduling of quay cranes, AGVs, and dual cantilever rail cranes during the unloading process at a U-shaped automated container terminal. The model’s objectives are to minimize the loading and unloading times of various tasks and avoid AGV conflicts. It integrates discrete event dynamics models and continuous time dynamics models. Additionally, an improved Genetic Seagull Optimization Algorithm (GSOA) was designed to solve this complex optimization problem.

Wang et al. [19] studied the scheduling and routing problem of Automated Guided Vehicles (AGVs) with multiple bidirectional paths to generate conflict-free routes. A mixed-integer programming (MIP) model was established to minimize the makespan, i.e., the completion time of all operations. A customized branch-and-bound (B&B) algorithm was developed to solve the problem within a reasonable time frame. The B&B process was used to obtain job assignments and job sequences, while a heuristic algorithm was proposed to generate conflict-free routes.

2.2. Scheduling Optimization

Lu Y et al. [20] proposed an improved genetic algorithm (GA)-based integrated optimization method for automated container terminal scheduling. The method aims to minimize the total operation time of container loading and unloading tasks at automated terminals by constructing a three-stage integrated optimization model. An improved genetic algorithm, named PGA (Probabilistic Genetic Algorithm), was developed to solve the model, which alters the population distribution to accelerate convergence. The constructed model is of significant reference value for determining the optimal allocation of QC (quay cranes), AGV (automated guided vehicles), and ASC (automated stacking cranes) in automated container operations.

Niu et al. [21] presented a multi-objective integrated container terminal scheduling problem, considering berth allocation, yard crane assignment, and horizontal transportation. The objective is to shorten ship operation time and reduce quay crane operation costs. He proposed a multi-objective bacterial colony optimization algorithm (MOBCO) incorporating concepts such as multiple populations, topology, personal optimal, and global optimal. The algorithm was verified, and it demonstrated superior performance, especially for large-scale container terminals.

Luo et al. [22] integrated the scheduling of automated container terminal handling equipment with the goal of minimizing ship berth time. He constructed a mixed integer programming (MIP) model, which can optimize small-scale problems using existing solvers. For large-scale problems, an adaptive heuristic algorithm was developed to adjust the parameters of the genetic algorithm (GA) based on observed performance, aiming to approximate the optimal solution.

Li et al. [23] utilized a double-cycle container loading and unloading strategy to coordinate quay cranes, yard cranes, and horizontal transport equipment. He proposed a mixed integer linear programming (MILP) model to minimize the operation time for multiple devices. Additionally, an enhanced Benders decomposition algorithm was designed to solve the problem, along with several acceleration measures to speed up convergence.

Yin et al. [24] studied the coordinated scheduling problem of quay cranes and shuttle vehicles, which involves decisions on assigning tasks from the apron to the quay crane (QC), ordering QC operations, allocating tasks to shuttle vehicles (SV), and ordering operations for the SV. A mixed integer linear programming (MILP) model was developed to minimize operation time. The relationship and interaction between four constraint sections (berth scheduling QC, task ordering by QC, task scheduling for SV, and apron buffer capacity constraint per berth) were analyzed, leading to the development of three rules: berth position rule, near-to-far rule, and full-buffer rule. Finally, sequence insertion, greedy insertion, and an improved genetic algorithm were proposed to solve medium and large-scale cases.

Yue et al. [25] addressed the configuration and scheduling problem of dual 40-foot dual-trolley quay cranes (QC) and automated guided vehicles (AGVs) in container terminals. He proposed a two-stage bi-objective mixed integer programming model to maximize customer satisfaction, minimize AGV idle time, and reduce QC delay time. In the proposed model, the first stage’s fragmentation function was linearized by adding special ordered set constraints, and the GUROBI solver version 9.5 was used. An improved Non-Dominated Sorting Genetic Algorithm III (NSGA-III) was applied to efficiently solve the second stage. Numerical experiments validated the model and algorithm’s effectiveness.

Yu et al. [26] investigated the potential yard space conflicts during the ship loading process and, to minimize conflicts, established a loading cluster selection model. A more flexible allocation strategy was derived to organize the yard space for outbound containers. A bi-objective model was constructed, considering both transportation distance and the balance between loading and unloading across blocks. Additionally, a model was developed to minimize all possible loading and unloading scenarios in the export yard during the loading process.

Zheng et al. [27] addressed the coordinated scheduling problem of quay cranes, yard cranes, and horizontal transportation equipment in container terminal operations. A mixed integer linear programming (MILP) model was constructed, and an improved three-dimensional chromosome representation was designed based on the genetic algorithm (GA) to solve the problem.

He et al. [28] studied the berth allocation and quay crane assignment problem, considering factors such as QC driver costs, diurnal operational efficiency, and performance-based wages. The study analyzed how factors related to QC drivers influence vessel schedules and formulated the objective function to include QC driver costs. A hybrid integer programming model with an accelerated algorithm was developed to address the proposed problem, and a metaheuristic framework incorporating a three-stage algorithm was introduced to solve the problem.

Xu et al. [29] proposed an integrated scheduling optimization model based on mixed integer programming to analyze the dual-trolley quay cranes, conflict-free automated guided vehicles (AGVs), and dual cantilever cranes in U-shaped automated container handling modes. Unlike previous studies, which primarily focused on traditional container terminal coordination, this model is tailored to automated systems. A super-heuristic genetic algorithm based on reinforcement learning was applied to solve the model. The proposed algorithm was compared with dual-layer genetic algorithms, adaptive genetic algorithms, hybrid genetic algorithms, and cuckoo search algorithms to validate its effectiveness.

Jonker et al. [30] considering QCs, automated guided vehicles and shore bridges simultaneously during ship loading, a Hybrid Flow Shop (HFS) approach is used to construct a synergistic schedule, which is solved by a simulated annealing algorithm.

Lu [31] took maximizing the efficiency of automated container terminals as his goal and loading and unloading time as his research object. On the premise of balancing the operating efficiency of each container ship at the berth, he established a comprehensive optimization scheduling model for automated container terminals with proportional fairness priority by setting priority parameters.

Gao et al. [32] established a mathematical models for multi-AGV dispatching, and a QL-CNP algorithm is proposed to tackle the multi-AGV dispatching problem (MADP). The distribution of traffic load is balanced for multiple AGVs performing tasks in the road network. The proposed model is validated using a Gurobi solver with a small experiment. Then, QL-CNP is used to conduct experiments with different sizes.

Cahyono et al. [33] uses a finite state machine (FSM) framework to dynamically model the integrated (end-to-end) operations of a container terminal, where each state machine is represented by a discrete event system (DES). The hybrid model includes the operations of quay cranes, internal trucks and yard cranes, as well as the storage location selection of container yards and ship slots. Using the integrated container terminal hybrid model, we propose a model prediction algorithm to obtain a near-optimal solution to the integrated terminal operation problem.

Shouwen et al. [34] established a bi-level programming model for the integrated scheduling problem with the objective of minimizing the completion time of all containers. He designed two bi-level optimization algorithms based on conflict resolution strategies: a Bi-level Adaptive Genetic Algorithm based on Conflict Resolution Strategy (CRS-BAGA) and a Bi-level Genetic Algorithm based on Conflict Resolution Strategy (CRS-BGA). Finally, numerical experiments were conducted to verify the effectiveness of the two optimization algorithms.

Hsu et al. [35] addressing the collaborative scheduling problem of quay cranes, yard cranes, and container trucks, proposed a simulation-based optimization framework considering vessel stowage factors. This framework employs four heuristic/metaheuristic algorithms: Sorting by Bay (SBB), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Multi-Group Particle Swarm Optimization (MGPSO). It simultaneously handles the Yard Crane Scheduling Problem (YCSP), Yard Truck Scheduling Problem (YTSP), and Quay Crane Scheduling Problem (QCSP) for export containers, while taking operational constraints into account.

Gong et al. [36] proposed a hybrid multi-AGV scheduling algorithm aimed at minimizing energy consumption in the Automated Container Terminal (ACT) yard and the total completion time of AGVs. This framework models the AGV scheduling process as a Markov Decision Process (MDP). Furthermore, he introduced a novel scheduling algorithm named Multi-Agent Deep Deterministic Policy Gradient for AGV Scheduling (MDAS), based on Multi-Agent Deep Deterministic Policy Gradient (MADDPG), to facilitate online real-time scheduling decisions.

Xin et al. [37], with the primary goal of energy savings for quay cranes and AGVs, established a Mixed-Integer Nonlinear Programming (MINLP) model. He developed an efficient Genetic Algorithm (GA) incorporating a weighted strategy to solve the problem. Additionally, he devised an ϵ-constraint algorithm to analyze the Pareto frontier.

Wang et al. [38] investigates an integrated optimization problem concerning berth allocation, quay crane assignment, and truck deployment in container terminals, with a specific focus on incorporating quay crane maintenance into the integrated models. A nonlinear integer programming model is proposed, and then a number of equivalent or relaxed models are developed to ease the model. To solve the model on large-scale instances, a SWO-GA is developed to provide a solution.

Shen et al. [39], from the point of AGV road blocking, the scheduling mode of group operation area is proposed. In order to minimize the maximum completion time of AGV, an AGV scheduling optimization model is established considering the interference constraints and AGV congestion in the actual operation of the terminal. Hybrid Genetic Algorithm with Fuzzy Logic Controller (HGA-FLC) is used to simulate the behavior of AGVs, and different scale examples are designed to solve the problem.

Fan et al. [40] designed a novel relocation strategy and established a collaborative optimization model to address the yard allocation for inbound and outbound containers and the scheduling of dual automated stacking cranes at automated container terminals. The model considers constraints such as safety distances and buffer zone capacities, aiming to minimize the total completion time of all storage requests for outbound containers and the total reshuffling time during retrieval, according to the priority defined by early retrieval and early loading rules for inbound and outbound containers. A hybrid genetic algorithm based on variable neighborhood search was developed to solve the model.

Jian et al. [41] addressed AGV optimization issues by considering different congestion scenarios along the routes, employing symmetric triangular fuzzy numbers to describe the distribution of AGV operation times. With the objectives of minimizing quay crane delay risk and reducing AGV operation time, a multi-objective scheduling optimization model was established. An improved genetic algorithm was designed to solve this model. The effectiveness of the model and algorithm was validated by comparing the results of AGV scheduling and container storage optimization models under fixed congestion coefficients across different instance scales.

Liu et al. [42] focused on the integrated scheduling of horizontal transportation equipment and handling equipment, as well as AGV path planning, based on a newly constructed U-shaped yard layout at a sea-rail intermodal container terminal. A bi-level programming model was established with the objectives of minimizing the maximum completion time and total waiting time for both AGVs and trucks. The model aims to derive the scheduling plan for yard cranes and determine the specific running trajectories of AGVs. Additionally, a bi-level genetic algorithm based on spatiotemporal greedy strategies was designed to handle task scheduling and conflict resolution.

Hsu et al. [43] proposes a hybrid model, termed WOA + PSO, which combines the Whale Optimization Algorithm (WOA) and Particle Swarm Optimization (PSO) as a novel approach to address the integrated scheduling problem of automated quay cranes (AQCs), automated lift vehicles (ALVs), and automated stacking cranes (ASCs) in an ACT.

Fontes et al. [44] formulates a mixed-integer linear programming (MILP) model and proposes a bi-objective multi-population biased random key genetic algorithm (mp-BRKGA) for the joint scheduling of quay cranes and speed adjustable vehicles in container terminals considering the dual-cycling strategy.

Hsu et al. [45] tackled the joint scheduling problem of yard cranes and container trucks by applying and comparing three optimization algorithms: Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Subgroup PSO (SGPSO). The performance and effectiveness of each algorithm were evaluated to determine the most suitable approach for optimizing the scheduling of yard cranes and container trucks.

2.3. Summary

Table 1. Literature summary and analysis.

Algorithm Type	Literature	Key Features
Genetic Algorithm (GA) and Variants	[1,3,8,11,12,13,14,16,17,18,20,22,23,25,27,29,31,34,37,38,39,40,41,44]	Includes standard GA, hybrid GA (e.g., embedded tabu search), adaptive GA, multi-objective GA (e.g., NSGA-III), etc.
Swarm Intelligence Algorithms	[21] (MOBCO), [30] (Simulated Annealing), [35] (PSO/MGPSO), [36] (MADDPG), [43] (WOA+PSO), [45] (GA/PSO/SGPSO)	Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), Bacterial Colony Optimization (BCO), etc., often used for multi-objective problems.
Deep Reinforcement Learning (DRL)	[4,7,36]	Combines deep learning and reinforcement learning for dynamic decision-making (e.g., container loading sequences, AGV scheduling).
Tree Search Algorithms	[2] (tabu-based tree search), [5] (Monte Carlo Tree Search)	Tree-structured search strategies with heuristic rules or probabilistic extensions.
Exact Algorithms and Decomposition Methods	[6] (Branch-and-Bound)], [19] (Branch-and-Bound), [23] (Benders Decomposition), [28] (Mixed-Integer Programming)]	Suitable for small-scale problems or model decomposition (e.g., accelerated convergence).
Hybrid Heuristic Algorithms	[10] (Dijkstra + Q-Learning), [15] (tabu list), [24] (sequence insertion+ greedy insertion), [33] (Model Predictive Control), [42] (Spatiotemporal Greedy Strategy)	Integrates rules, graph theory, or simulation optimization to address complex constraints (e.g., path conflicts, dynamic scheduling).
other	[9] (sequential decision), [26] (not explicitly mentioned), [32] (Gurobi solver)

summarizes the reviewed literature. In ship loading operations, many scholars focus on single-ship or single-quay crane as the research objective, often using completion time minimization as the objective function. There is relatively little consideration given to yard operation conflicts or workload balancing, which hampers the ability to effectively ensure the overall loading efficiency of container terminals. In terms of the choice of solution algorithm, scholars use GA and its deformation, multi-objective solution algorithm, deep learning and other methods to solve problems.

This study aims to address these gaps by proposing an intelligent decision-making model and algorithm for full shoreline loading operations, ensuring balanced yard and quay crane operations while minimizing conflicts and delays. The principal conclusion highlights the model’s effectiveness in improving terminal efficiency and reducing operational bottlenecks.

3. Mathematical Model

3.1. Assumptions

The following assumptions are made to simplify the model while maintaining its practical relevance: (1) All equipment operates normally and is fully attended during production operations. (2) The target bays for quay crane operations are known during production. (3) The unloading plan for vessels has been confirmed. (4) The stowage plan for vessels has been confirmed and will not be revised. (5) The distance between the quay crane’s operating position and the target bays in the yard is known. (6) Only loading and unloading operations are considered in the decision-making process. (7) The current loading and unloading instructions for each yard crane are known.

3.2. Symbol Specifications

3.2.1. Dimensions

The following dimensions (

Table 2. Dimensions.

Symbol	Description
$Q$	$\mathrm{Set} \mathrm{of} \mathrm{quay} \mathrm{cranes} \mathrm{operating} \mathrm{normally} \mathrm{during} \mathrm{production}, q,q^{\prime }ϵ Q$
$Y$	$\mathrm{Set} \mathrm{of} \mathrm{yard} \mathrm{cranes} \mathrm{operating} \mathrm{normally} \mathrm{during} \mathrm{production}, y,y^{\prime } ϵY$
$C$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{containers} \mathrm{requiring} \mathrm{operations} \mathrm{on} \mathrm{the} \mathrm{berthed} \mathrm{vessel}, c,c^{\prime }ϵ C$
$V$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{vessel} \mathrm{bays} \mathrm{requiring} \mathrm{operations} (\mathrm{categorized} \mathrm{into} \mathrm{deck} \mathrm{bays} \mathrm{and} \mathrm{hold} \mathrm{bays}), v,v^{\prime }ϵ V$;
$B$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{yard} \mathrm{bays} \mathrm{requiring} \mathrm{operations}, b,b^{\prime }ϵ B$
$CH$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{yard} \mathrm{channels} \mathrm{requiring} \mathrm{operations}, ch,{ch}^{\prime }ϵ CH$

) are defined to describe the key components of the model.

3.2.2. Model Parameter

The key parameters of the model are listed in

Table 3. Model parameters.

Symbol	Description
${WT}_q$	$\mathrm{Working} \mathrm{state} \mathrm{of} \mathrm{quay} \mathrm{crane} q$ (−1: unloading, 0: simultaneous, 1: loading)
$W_{qv}$	$\mathrm{Binary} \mathrm{variable}: \mathrm{whether} \mathrm{quay} \mathrm{crane} q$
${CT}_c$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{containers} \mathrm{requiring} \mathrm{operations} \mathrm{on} \mathrm{the} \mathrm{berthed} \mathrm{vessel}, c,c^{\prime }ϵ C$
${BL}_{cb}$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{vessel} \mathrm{bays} \mathrm{requiring} \mathrm{operations} (\mathrm{categorized} \mathrm{into} \mathrm{deck} \mathrm{bays} \mathrm{and} \mathrm{hold} \mathrm{bays}), v,v^{\prime }ϵ V$;
${VL}_{cv}$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{yard} \mathrm{bays} \mathrm{requiring} \mathrm{operations}, b,b^{\prime }ϵ B$
${CL}_{bch}$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{yard} \mathrm{channels} \mathrm{requiring} \mathrm{operations}, ch,{ch}^{\prime }ϵ CH$
${VB}_{c{c}^{\prime }}$	$\mathrm{Binary} \mathrm{variable}: \mathrm{whether} \mathrm{containers} c$ $\mathrm{and} c^{\prime }$ belong to the same vessel bay
${HL}_{ch}$	$\mathrm{Number} \mathrm{of} \mathrm{loading} \mathrm{instructions} \mathrm{in} \mathrm{channel} ch$
${HD}_{ch}$	$\mathrm{Number} \mathrm{of} \mathrm{unloading} \mathrm{instructions} \mathrm{in} \mathrm{channel} ch$
${HL}_q$	$\mathrm{Number} \mathrm{of} \mathrm{loading} \mathrm{instructions} \mathrm{for} \mathrm{quay} \mathrm{crane} q$
${HD}_q$	$\mathrm{Number} \mathrm{of} \mathrm{unloading} \mathrm{instructions} \mathrm{for} \mathrm{quay} \mathrm{crane} q$
${len}_c$	Total number of containers requiring operations
${len}_q$	Total number of quay cranes
${len}_{ch}$	Total number of channels
$T_{c{c}^{\prime }}$	$\mathrm{Binary} \mathrm{variable}: \mathrm{whether} \mathrm{container} c$ $\mathrm{is} \mathrm{above} \mathrm{container} c^{\prime }$ in the same column
${CH}_{b{b}^{\prime }}$	$\mathrm{Binary} \mathrm{variable}: \mathrm{whether} \mathrm{yard} \mathrm{bays} b$$\mathrm{and} b^{\prime }$ belong to the same channel

, including variables related to quay cranes, yard cranes, containers, vessel bays, yard bays, and channels.

3.2.3. Decision Variables

$x_{qc}$: A binary decision variable indicating whether quay crane $q$ selects container $c$, A value of 0 means no, and 1 means yes.

3.3. Objective Function

3.3.1. Yard Channel Instruction Balancing

The yard is a core resource in container terminals, and its efficiency directly impacts the overall productivity of the terminal. As a central hub in container logistics, all inbound and outbound containers are temporarily stored in the yard. Uneven distribution of yard operation instructions may lead to congestion in some areas and idle resources in others, hindering container flow. By balancing yard operation instructions, container flow efficiency can be optimized, congestion reduced, and overall operational efficiency improved. The number of loading instructions can be expressed as in Equation (1), and the number of unloading instructions can be expressed as in Equation (2).

$${IL}_{ch}={HL}_{ch}+\sum _c\sum _b\sum _v\sum _q{CT}_c{·BL}_{cb}·{VL}_{cv}·W_{qv}·x_{qc}$$

(1)

$${ID}_{ch}={HD}_{ch}+\sum _c\sum _b\sum _v\sum _q{(1-CT}_c){·BL}_{cb}·{VL}_{cv}·W_{qv}·x_{qc}$$

(2)

For all channels in the yard, the objective is to minimize the difference in instruction quantities between any two channels, with weights assigned to loading and unloading instructions. Let ${\alpha }_1$ and ${\alpha }_2$ represent the weights for loading and unloading instructions, respectively, as shown in Equation (3):

$$\min F_1=\sum _{ch}\sum _{c{h}^{\prime }}\left({\displaystyle \frac{{\alpha }_1\left|I{L}_{ch}-I{L}_{c{h}^{\prime }}\right|+{\alpha }_2\left|I{D}_{ch}-I{D}_{c{h}^{\prime }}\right|}{2}}\right)\hspace{1em}\forall ch,c{h}^{\prime }\in CH,ch\ne c{h}^{\prime }$$

(3)

3.3.2. Quay Crane Instruction Balancing

Quay crane workload balancing is critical for terminal efficiency. Imbalanced workloads can lead to overburdened or idle cranes, disrupting operations and increasing dwell time. Balancing involves ensuring equitable distribution of loading and unloading tasks across all cranes, The number of loading instructions can be expressed as in Equation (4), and the number of unloading instructions can be expressed as in Equation (5).

$${IL}_q={HL}_q+\sum _c\sum _v\sum _q{CT}_c·{VL}_{cv}·W_{qv}·\left(1-\left|{WT}_q\right|\right)·x_{qc}$$

(4)

$${ID}_q={HD}_q+\sum _c\sum _v\sum _q{(1-CT}_c)·{VL}_{cv}·W_{qv}·\left(1-\left|{WT}_q\right|\right)·x_{qc}$$

(5)

The balance of quay crane instructions across the entire shoreline within a unit time can be expressed as:

$$\min F_2=\sum _q\sum _{q^{\prime }}\left({\displaystyle \frac{{\beta }_1\left|L_q-L_{q^{\prime }}\right|+{\beta }_2\left|U_q-U_{q^{\prime }}\right|}{2}}\right)\hspace{1em}\forall q,q^{\prime }\in Q,q\ne q^{\prime }$$

(6)

3.3.3. Model Objective Function

The intelligent decision-making problem for stack-yard positioning in loading operations involves balancing instructions for quay cranes and yard cranes, aiming to determine an optimal container pickup point. This is a multi-objective combinatorial optimization problem. To address this, the two sub-objectives are first normalized to ensure they are on the same scale. Then, a weighted average method is applied to integrate these normalized sub-objectives into a single objective function, providing a unified evaluation criterion for decision-making.

• Normalization of Sub-Objective Functions

The sub-objective $F_1$ representing channel instruction balancing is normalized using Equation (7):

$$F_1^{\prime }={\displaystyle \frac{F_1}{{len}_c·{len}_{ch}}}$$

(7)

The sub-objective $F_2$ representing quay crane instruction balancing, is normalized using Equation (8):

$$F_2^{\prime }={\displaystyle \frac{F_2}{{len}_c·{len}_q}}$$

(8)

2.

Total Objective Function

After normalization, the sub-objective functions $F_1$ and $F_2$ are scaled to the range [0,1]. Let ${\lambda }_1$ and ${\lambda }_2$ denote the weights of the sub-objectives, which can be determined using expert scoring methods. The total objective function is expressed as:

$$minF={\lambda }_1·F_1^{\prime }+{\lambda }_2·F_2^{\prime }$$

(9)

3.4. Constraint Condition

Loading operations must strictly follow the stowage plan to ensure the final loading result aligns with the plan. During operations, containers cannot be picked up unsupported, as expressed in Equation (10).

$$\sum _q\sum _b\sum _{c^{\prime }}{T}_{c{c}^{\prime }}·{{BL}_{cb}·DH}_b·x_{qc}\le 0, \forall c ϵ C$$

(10)

Each container can only be stored in one yard bay, as shown in Equation (11).

$$\sum _b{BL}_{cb}=1, \forall c ϵ C$$

(11)

Each container can only be stowed in one vessel bay, as shown in Equation (12).

$$\sum _v{VL}_{cv}=1, \forall c ϵ C$$

(12)

Each container can only be handled by one quay crane, as shown in Equation (13).

$$\sum _q{x}_{qc}=1, \forall c ϵ C$$

(13)

Each yard bay can only belong to one channel, as shown in Equation (14).

$$\sum _{ch}{CL}_{bch}=1, \forall b ϵ B$$

(14)

The binary variables defined in the model must satisfy the following constraint:

$$x_{qc} ϵ \left\{\mathrm{0,1}\right\}$$

(15)

4. GOGA-GRASP Algorithm

The intelligent decision-making for berth allocation in full-quay container ship stowage is an NP-Hard problem. As the problem size increases, the solution time for conventional algorithms grows exponentially. Given the inherent complexity of this problem, traditional exact solution methods face challenges when addressing large-scale issues. Therefore, to efficiently solve this problem, this paper adopts the Constrained Optimization Genetic Algorithm with Greedy Randomized Adaptive Search Procedures (COGA-GRASP) algorithm. COGA-GRASP is a hybrid algorithm that integrates Genetic Algorithms (GA) and Greedy Randomized Adaptive Search Procedures (GRASP). It is primarily designed to address constrained optimization problems by synergistically combining the global search capabilities of GA with the local search efficiency of GRASP. This dual-mechanism approach ensures that the algorithm not only maintains the global quality of solutions but also rapidly identifies high-quality solutions within local search spaces when tackling optimization problems with constraints.

4.1. Gene Coding

The objective of the intelligent decision-making problem for berth allocation in full-quay container ship stowage is to calculate the optimal container dispatch points. Here, $x_{qc}$ represents whether quay crane $q$ is assigned to work on container $c$.

This paper defines a chromosome as a combination of three gene points: the first gene point represents the quay crane ID, the second gene point represents the position in the yard, and the third gene point represents the number of containers to be handled. This chromosome does not represent the execution order of quay cranes, but rather whether the crane will perform the operation. The specific execution sequence will be determined in the later stages of the problem-solving process.

As shown in Figure 3, the green part of the content forms a gene that indicates quay crane Q01 handles three containers at yard location 1A46. Therefore, this chromosome can represent quay crane Q01 handling three containers at 1A46, quay crane Q01 handling four containers at 2B23, quay crane Q02 handling five containers at 2D34, and quay crane Q03 handling one container at 3C28.

4.2. Initial Solution Generation

The initial solution is one of the key components of a genetic algorithm. The quality and diversity of the initial solution directly impact the algorithm’s output quality and computational efficiency. Currently, there are various strategies for generating the initial solution, among which the greedy–random strategy combines both greediness and randomness. This approach aims to quickly find a good starting solution while satisfying the problem constraints. In this paper, the greedy–random strategy is employed to construct a set of initial solutions, as shown in

Table 4. InitializePopulation.

$S\leftarrow \mathrm{\varnothing }$ ...Function GreedyRandomInitializePopulation()       $s_0$ ← Empirical design       S ← S + $s_0$       While S.size ≤ Initial population size do          n ← GreedyRandomSelection(seed)          If n Satisfy constraint             S ← S + n       End While Return S Function GreedyRandomSelection(seed)       Based on seed, generate a list of potential solutions in a greedy manner       Randomly select a solution from the list Return selected solution

.

4.3. Fitness Value

When using genetic algorithms to solve constrained optimization problems, the design of the fitness function is crucial. The COGA algorithm used in this paper must consider both the objective function value and the constraint conditions when calculating the fitness function. To ensure that the solution satisfies all constraints, a penalty function is typically introduced to penalize solutions that violate the constraints. In this paper, the fitness value is designed using the penalty function method, as shown in Equation (16):

$$R=F+\theta ·constraint$$

(16)

where $\theta$ represents the penalty coefficient matrix and $constraint$ represents the matrix of constraint violations.

4.4. Select, Crossover and Mutation

This paper adopts Tournament Selection (TS) as the selection algorithm. TS randomly samples $m$ individuals from the population and selects the optimal solution based on fitness values. The best individual is then passed on to the next generation.

This paper adopts the Partially Matched Crossover (PMX, as shown in

Table 5. PMX.

Function PMX ()   $P_1,P_2\leftarrow$ random select 2 individual   ${cross}_{start},{cross}_{end}\leftarrow$ random select 2 index   $P_1^{\mathrm{’}},P_2^{\mathrm{’}}\leftarrow change from {cross}_{start} to {cross}_{end}$   ${Child}_1,{Child}_2\leftarrow$ remove duplication    Return ${Child}_1,{Child}_2$

) strategy. PMX ensures that each gene in a chromosome appears only once, preventing duplicate genes, thus guaranteeing that the chromosome satisfies the constraints and is feasible. As shown in Figure 4.

First, a pair of parent chromosomes is randomly selected, and crossover points are chosen. When selecting crossover points, it is important to ensure that gene points are intact (e.g., the gene {Q01, 2B23, 4} cannot be split). The crossover points should be the same for both parent chromosomes. Next, the crossover genes are exchanged, and finally, the feasibility of the child chromosomes is verified to ensure there are no duplicates. Duplicates are determined based on whether the quay crane and yard location are the same, while the number of containers is not used as a criterion for duplication.

In this paper, Self-adaptive Mutation (SAM) is chosen as the mutation strategy. SAM adjusts the mutation parameters based on the algorithm’s evolutionary state or the population characteristics, thereby enhancing the algorithm’s performance. During the mutation process, $\varpi$ gradually decreases with the number of iterations. Initially, the mutation rate is high to allow for extensive exploration. Then, based on the evolutionary progress, the mutation rate is gradually reduced to better exploit the information in the search space. For a given individual $P$, the mutation probability is adjusted according to the fitness value $r$. The larger $r$ is, the lower the mutation rate, increasing the likelihood of retaining that individual. Conversely, the smaller $r$ is, the higher the mutation rate, increasing the probability of searching for a better solution.

5. Experiment

This chapter conducts a case study on the intelligent decision-making problem of stack-yard positioning for full shoreline loading operations to validate the correctness of the model, the effectiveness and applicability of the algorithm, and the practicality of the final results. Through result analysis and algorithm comparison, it is demonstrated that the proposed COGA algorithm with tournament selection and adaptive mutation (COGA-GRASP) outperforms the traditional GA algorithm with roulette wheel selection and single-point mutation in terms of computational efficiency and robustness.

5.1. Experiment Description

The source of the basic data for the test case is with Shanghai Mingdong Container Terminal Company Limited (Shanghai, China), as shown in Figure 5. It consists of three main parts: containers to be loaded and unloaded, existing quay crane loading and unloading instructions and existing channel loading and unloading instructions. A total of 500 General Purpose Containers (GP) are distributed along three channels, each channel contains existing loading and unloading instructions. The types of quay crane operations and the number of quay crane operation instructions are also taken from the actual production data of MingDong Terminal to ensure the validity of the test case. This experiment is coded in Python 3.10 language and runs on a Mac computer with an M3 pro chip and 18 GB of RAM.

5.2. Experiment Analysis

In this paper, multiple computations were performed with different population sizes, iteration counts, and algorithmic strategies. The following are the computational results:

• Using Tournament Selection and Self-adaptive Mutation algorithms, with 100 iterations and population sizes of 50 and 100, the convergence curves are shown in Figure 6. In this example, the fitness function values are calculated with the parameters for loading and unloading instructions in Equations (3) and (6) set to ${\alpha }_1=0.8,{\alpha }_2=0.2,{\mathsf{\beta }}_1=0.8,{\mathsf{\beta }}_2=0.2$ For the total objective function in Equation (9), the parameter values are set to ${\lambda }_1=0.5,{\lambda }_2=0.5$.

From the Figure 6, it can be observed that in this problem, the difference in fitness values between a population size of 50 and 100 is relatively small. The algorithm achieves good convergence around 40 iterations in both cases.

Table 6. Result.

Population Size:50			Population Size: 100
Quay INS	Yard INS	Fitness Value	Quay INS	Yard INS	Fitness Value
Q01:21: 19	3B:18:45	0.0958	Q01:20:19	3B:22:45	0.0944
Q02:21: 14	3A:36:20		Q02:20: 14	3A:36: 20
Q03:21: 15	2B:38: 30		Q03:20: 15	2B:38: 30
Q04:21: 15	1A:34:19		Q04:20: 15	1A:22: 19
Q05: 21: 0	2A:42:34		Q05: 20: 0	2A:35: 34
Q06:21: 12	1C:26: 46		Q06:20: 12	1C:26: 46
Q07:21: 15	3C:38: 32		Q07:20: 15	3C:39: 32
Q08:21: 15	2C:29: 24		Q08:20: 15	1B:33: 39
Q09: 21: 4	1B:27: 39		Q09: 20: 4	2C:22: 24
Q10: 21: 16			Q10: 20: 16
Q11: 21: 4			Q11: 20: 4
Q12: 21: 10			Q12: 20: 10
Q13: 21: 6			Q13: 20: 6
Q14: 21: 5			Q14: 20: 5
Q15: 21: 4			Q15: 20: 4

presents the final container handling results and the corresponding fitness values. Specifically, quay crane Q01:21:19 indicates that quay crane Q01 has 21 loading instructions and 19 unloading instructions, while yard 3B:18:45 indicates that yard 3B has 18 loading instructions and 45 unloading instructions. The fitness values are 0.0958 and 0.0944, both of which yield good solutions.

2.

Due to the varying emphasis placed on quay crane balance and yard balance by different container terminals, the importance of loading and unloading instructions differs. Therefore, the parameters ${\alpha }_1,{\alpha }_2,{\lambda }_1,{\lambda }_2$ are set differently. To analyze the impact of different parameters on the algorithm’s convergence, several experiments were conducted, and some of the results are summarized below.

• Experiment 1: The parameters are set as ${\alpha }_1=0.5, {\alpha }_2=0.5, {\mathsf{\beta }}_1=0.5, {\mathsf{\beta }}_2=0.5, {\lambda }_1=0.5, {\lambda }_2=0.5$. The fitness value is 0.11975, as shown in Figure 7a.
• Experiment 2: The parameters are set as ${\alpha }_1=0.3, {\alpha }_2=0.7, {\mathsf{\beta }}_1=0.3, {\mathsf{\beta }}_2=0.7, {\lambda }_1=0.5, {\lambda }_2=0.5$. The fitness value is 0. 08304 as shown in Figure 7b.
• Experiment 3: The parameters are set as ${\alpha }_1=0.5, {\alpha }_2=0.5, {\mathsf{\beta }}_1=0.5, {\mathsf{\beta }}_2=0.5, {\lambda }_1=0.3, {\lambda }_2=0.7$. The fitness value is 0. 07185, as shown in Figure 7c.
• Experiment 4: The parameters are set as ${\alpha }_1=0.5, {\alpha }_2=0.5, {\mathsf{\beta }}_1=0.5, {\mathsf{\beta }}_2=0.5, {\lambda }_1=0.8, {\lambda }_2=0.2$. The fitness value is 0. 07185, as shown in Figure 7d.

Figure 7. Experimental comparison.

Figure 7. Experimental comparison.

From the experimental convergence curves, it can be observed that for different parameter settings, the algorithm demonstrates high convergence, quickly reaching a good solution. Its solving capability is stable, with both feasibility and practicality.

3.

This paper presents a statistical analysis of different strategies and population sizes, as shown in Figure 8. It can be observed that the COGA algorithm, utilizing Tournament Selection (TS), Partially Matched Crossover (PMX), and Self-Adaptive Mutation (SAM), achieves faster solution speeds and higher solution quality. A comparison between the COGA and GA algorithms indicates that COGA offers higher solving efficiency and greater stability, which is of significant importance for practical production.

5.3. Applicationt Analysis

The Mingdong Terminal primarily accommodates scheduled liner services with fixed shipping routes. To conduct a comprehensive benchmark analysis of full-quay vessel loading efficiency and operational capacity, comparative evaluations were implemented through two distinct approaches: the terminal’s proprietary digital twin platform versus conventional loading operations.

As shown in Figure 9, the quay crane productivity metrics reveal three experimental configurations: Dataset 1 (Baseline): Conventional loading operations from 7 to 13 December 2024. Dataset 2 (Field Implementation): Mathematical model-driven operations from 14 to 20 December 2024. Dataset 3 (Digital Twin Simulation): Model-enhanced operations within the digital environment during 7–13 December 2024.

The comparative analysis substantiates the operational efficacy of the proposed model across multiple validation scenarios, with quantified results demonstrating a 2.12% productivity increase in quay crane operations through Digital Twin Implementation (Dataset 3) and a 1.98% efficiency improvement in Physical Terminal Deployment (Dataset 2). This dual-environment verification confirms statistically significant enhancements in both simulated and real-world settings, wherein the digital twin platform’s capabilities yielded marginally superior optimization outcomes compared to physical implementations, thereby systematically validating the algorithm’s cross-domain adaptability and technical robustness.

6. Conclusions

This study investigates the container loading instruction allocation problem within full shoreline operational areas, establishing an intelligent decision-making model through refined optimization objectives and constraint formulations. A novel COGA-GRASP hybrid algorithmic framework is proposed to address this NP-hard combinatorial optimization challenge.

The experimental configuration utilizes an initial dataset comprising 500 pending containers from Shanghai Port’s Mingdong Container Terminal. The methodology incorporates tournament selection mechanisms, partially matched crossover operators, and adaptive mutation strategies to enhance solution quality. Comparative analysis of multiple parameter configurations and fitness evaluations demonstrates the superior solution optimality achieved by COGA-GRASP across operational scenarios.

Benchmark testing against conventional GA implementations reveals COGA-GRASP’s enhanced adaptability to varying dataset scales and problem complexities. The algorithm exhibits robust performance characteristics with rapid solution convergence, maintaining computational efficiency under different operational scales. However, given the real-time dynamic nature of terminal operations and strong interdependency with horizontal transportation systems, future research should prioritize tighter integration with vehicular scheduling models to develop comprehensive loading optimization frameworks.