In automated container terminals (ACTs), horizontal transport vehicles and yard storage decisions must be planned jointly, but the reference formulation of Hu, Chen, Wang and Zhang (2019) for the joint vehicle dispatching and storage allocation problem assumes fully deterministic timing and omits container stacking sequences and automated yard crane (AYC) scheduling decisions. The authors themselves list three unresolved extensions: robustness of the joint plans under uncertainty (time windows, operation times, travel times), integration of AYC scheduling with container stacking order, and more efficient solution methods at scale. Later literature — most notably the same first author's 2025 deep reinforcement learning framework for real-time joint task assignment and storage allocation with random tasks — has resolved the task-arrival-uncertainty part reactively, while yard-side stochastic models and integrated equipment-scheduling formulations treat the remaining pieces in decoupled, deterministic settings. What remains open is a formulation and solution methodology for the joint vehicle dispatching and storage allocation problem that simultaneously makes stacking sequences and AYC scheduling explicit decision variables and explicitly handles variability in operation and travel times, at realistic instance scales.

Background

An automated container terminal (ACT) moves containers between berth and yard using driverless equipment: automated quay cranes (AQCs) load and discharge vessels, automated yard cranes (AYCs) stack and retrieve containers in yard blocks, and automated guided vehicles (AGVs) or automated lifting vehicles (ALVs) carry containers horizontally — AGVs must be loaded and unloaded by cranes, whereas ALVs can lift and lower containers by themselves. Storage allocation decides in which yard block (and hence which destination for each transport task) each discharged container is placed, and vehicle dispatching decides which vehicle serves which task and in which order. Because the storage choice determines every vehicle's travel distances while vehicle timing must respect crane availability at both quay and yard, the two decisions are strongly coupled, and the joint problem is a large mixed-integer program.

Hu, Chen, Wang and Zhang (2019) formulated two mixed-integer linear programming (MILP) models — one for AGV-based and one for ALV-based operation — minimizing a weighted sum of the number of vehicles used and total vehicle travel time, with task ready times, loading due times, block capacity limits, and (in the AGV model) timing relations to AQCs and AYCs such as fixed completion times for loading tasks and minimum setup-time gaps between tasks handled by the same AYC. To solve realistic instances they proposed a three-stage decomposition: particle swarm optimization (a population-based metaheuristic that iteratively moves a swarm of candidate solutions toward better-performing ones) selects storage locations for unloaded containers, a greedy search assigns tasks to a minimum number of vehicles using a quantitative Degree-of-Conflict measure for task pairs, and each vehicle's route is then solved exactly as a small traveling salesman problem with time windows. In the paper's numerical experiments the heuristic was near-optimal for small cases (validated against CPLEX, a standard exact mixed-integer programming solver, only where exact solving remained possible) and computationally feasible on larger cases where CPLEX was not, and ALVs required fewer vehicles than AGVs.

The formulation is deliberately deterministic: time windows, crane operation times, and vehicle travel times are fixed known parameters; the stacking of containers within a stack (which container sits on which, and hence how many rehandles — extra moves to dig out a buried container — later retrievals incur) is not modeled; and AYC schedules enter only as timing constraints, not as decisions. In their conclusions the authors therefore identify three open directions: (i) formulations that account for uncertain factors such as variability in time windows, operation times, and travel times, so that the robustness of the joint dispatching and storage plans can be guaranteed; (ii) integration of container stacking sequences and AYC scheduling into the joint problem; and (iii) more efficient solution methods for larger, more complex terminals.

Subsequent literature has advanced each direction only partially. On uncertainty, Hu, Liu and Yang (2025) — the same first author — proposed a deep reinforcement learning (DRL) framework — one in which a neural-network policy learns good decisions by trial and error against a reward signal — in which an attention-based actor network (the decision-proposing component of the learned policy) assigns transport tasks to AGVs and a critic-based modification strategy (an evaluation network that scores and corrects the actor's proposals) adjusts storage allocation in real time, explicitly handling random task arrivals and uncertain arrival times for the joint problem; this resolves the task-side uncertainty reactively but does not address variability of operation times and travel times nor provide a priori robust plans. Wang, Lin and Zhen (2023) formulated a two-stage stochastic program — an optimization model in which first-stage decisions are fixed before the uncertain outcome is realized and recourse decisions are optimized afterwards — for storage space allocation, yard crane deployment, and container allocation under container-demand uncertainty, but at the tactical yard-management level, decoupled from operational vehicle dispatching. On the integration side, Yue, Fan and Fan (2023) jointly optimized block allocation with quay crane, yard crane, and AGV scheduling (deterministically), and Xing, Liu, Wang, Chew, Lee and Tan (2023) integrated AGV dispatching with quay and yard crane scheduling plus vehicle speed optimization; neither models stacking sequences within stacks. Stacking itself is studied in separate problems — dynamic yard allocation (He, Xiao, Yu and Zhang, 2024) and yard remarshalling, i.e., pre-emptive container re-stacking to reduce later rehandles (Wang, He, Yu and Guo, 2024) — decoupled from vehicle dispatching. On methods, a 2023 comprehensive review (Naeem, Gheith and Eltawil) maps the integrated-scheduling literature, and reinforcement learning (Zhang et al., 2022; Hu et al., 2025), enhanced logic-based Benders decomposition (an iterative exact method that alternates between a master problem and subproblems, feeding derived cuts back into the master; Xue, He, Zhou, Lee, Wang and Chen, 2026), and adaptive dynamic dispatching policies with robustness tests under breakdowns and outages (Chang, Chen and Xiang, 2026) show that scalable and dynamic solution techniques are available — but they have not been brought to bear on the stacking-integrated, timing-uncertain version of the joint problem.

Problem Statement

Extend the joint vehicle dispatching and storage allocation problem in automated container terminals, as formulated by Hu, Chen, Wang and Zhang (2019), into a single integrated optimization problem in which: (i) the stacking sequence of containers within each yard stack and the scheduling of the automated yard cranes (AYCs) are explicit decision variables coordinated with vehicle dispatching and storage allocation, so that rehandle implications of storage choices and AYC workload are captured rather than reduced to fixed timing constraints; and (ii) variability of the timing parameters named in the source — task time windows, crane operation times, and vehicle travel times — is explicitly represented in the formulation, so that the robustness of the joint dispatching and storage plans under this variability can be guaranteed, going beyond the random-task-arrival setting already handled reactively by Hu, Liu and Yang (2025). Consistent with the source's third open direction — more efficient solution methods for larger, more complex terminals — the integrated formulation must be tractable on instances at least as large as those in the source's numerical experiments.

References

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