Berth Allocation and Quay Crane Scheduling in Port Operations: A Systematic Review

Abstract

Container terminals are facing significant challenges in meeting the increasing demands for volume and throughput, with limited space often presenting as a critical constraint. Key areas of concern at the quayside include the berth allocation problem, the quay crane assignment, and the scheduling problem. Effectively managing these issues is essential for optimizing port operations; failure to do so can lead to substantial operational and economic ramifications, ultimately affecting competitiveness within the global shipping industry. Optimization models, encompassing both mathematical frameworks and metaheuristic approaches, offer promising solutions. Additionally, the application of machine learning and reinforcement learning enables real-time solutions, while robust optimization and stochastic models present effective strategies, particularly in scenarios involving uncertainties. This study expands upon earlier foundational analyses of berth allocation, quay crane assignment, and scheduling issues, which have laid the groundwork for port optimization. Recent developments in uncertainty management, automation, real-time decision-making approaches, and environmentally sustainable objectives have prompted this review of the literature from 2015 to 2024, exploring emerging challenges and opportunities in container terminal operations. Recent research has increasingly shifted toward integrated approaches and the utilization of continuous berthing for better wharf utilization. Additionally, emerging trends, such as sustainability and green infrastructure in port operations, and policy trade-offs are gaining traction. In this review, we critically analyze and discuss various aspects, including spatial and temporal attributes, crane handling, sustainability, model formulation, policy trade-offs, solution approaches, and model performance evaluation, drawing on a review of 94 papers published between 2015 and 2024.

1. Introduction

Efficient operations at container ports rely on several interconnected factors, which can be categorized into three main areas: hinterland, yard, and quayside activities. The hinterland includes the rail and roadside of the terminal. The yard activities involve coordinating trucking schedules and allocating containers to suitable spaces. The quayside includes the berths, which involve allocating berths to the vessel, and vessel mooring, as well as ship-to-shore cranes, which involve assigning and scheduling quay cranes to the vessel. The quayside operations can be further categorized into the berth allocation problem (BAP), the quay crane assignment problem (QCAP), and the quay crane scheduling problem (QCSP). QCAP and QCSP focus on the efficient use of quay cranes. QCAP determines which cranes should be allocated to which vessels while considering speed, availability, and other crane-related constraints. After the cranes have been assigned to the vessels, the next challenge is to effectively schedule the operations of these cranes, which is known as QCSP. In QCSP, the objective is to first decide on the assignment of cranes to berths and then establish the start time of tasks assigned to a specific crane, taking into account the physical limitations of the cranes and safety regulations while minimizing the idling time of cranes and total service time.

Recent research has underscored substantial advancements in addressing BAP in African ports, with particular emphasis on progress in Morocco and Tunisia. The Scopus search engine yields 40 pertinent publications for the African continent. Notably, 30 of these publications pertain to Morocco, centering on the bulk port of Jorf Lasfar, the largest ore port in Africa [1]. Furthermore, nine publications focus on the Tunisian port of Rades, while one publication pertains to Algeria. These publications span the period from 2013 to 2024. The headway in resolving the berth allocation problem in African ports, specifically in Morocco and Tunisia, has been characterized by the development and implementation of advanced mathematical models [2]. These endeavors have yielded substantial enhancements in port efficiency and productivity, showcasing the potential for further progress in this domain. African ports encounter challenges such as deteriorating infrastructure, congestion, and subpar performance, impeding their competitiveness in the global arena. Notably, while Morocco and Tunisia, as constituents of North Africa, have made notable strides in addressing these challenges, there are no publications addressing the berth allocation problem for Sub-Saharan African ports. The inadequate port infrastructure in Sub-Saharan Africa, particularly in the West African region, poses obstacles to economic advancement [2].

The durban container terminal (DCT) is a crucial player in the global container shipping industry, handling 65% of South Africa’s container shipments. It serves as a central point for containerized cargo from the Indian Ocean Islands, the Middle East, the Far East, and Australia. It also acts as a gateway to neighboring nations, including Malawi, Zambia, Zimbabwe, and other landlocked territories. With fourteen berths designed to handle up to 2.9 million TEUs (twenty-foot equivalent unit) of containers annually, the terminal faces challenges accommodating modern vessels due to space constraints and berth limitations. Despite these challenges, DCT is committed to improving its throughput without expanding its physical footprint. Customers and vessel owners with operating contracts at DCT are guaranteed berth slots in advance, but any delays beyond agreed departure times can lead to congestion and backlogs. Currently, Transnet (a South African rail, port, and pipeline company) is employing conventional methods such as simulation techniques and queuing theory to address BAP, QCAP, and QCSP, while the industry is exploring heuristic, metaheuristic, and evolutionary algorithms to tackle these issues.

The berth allocation problem has been the focus of extensive research for many years, initially centered on container ports. Interest in the BAP emerged in the early 1990s, coinciding with the surge in global trade volumes and the need for ports to optimize berthing space. Early research by Hoffarth and Voß [3] and Lim [4] addressed berth planning as an optimization problem. Additional insights into related port terminal problems can be found in the works of Meersmans and Dekker [5], Vis and De Koster [6], Steenken et al. [7], Vacca et al. [8], and Stahlbock and Voß [9]. These papers focus on operations research, optimization, and berth planning at container terminals. The field expanded rapidly, with significant contributions from Park and Kim [10], who were pioneers in integrating berth allocation and quay crane (QC) assignment. As the closely related nature of these problems became apparent, research into QCSP evolved alongside BAP and QCAP. Over time, there has been a growing emphasis on addressing these problems in an integrated manner, recognizing that solving BAP, QCAP, and QCSP in isolation may not always lead to optimal terminal performance. The work by Bierwirth and Meisel [11,12] exemplifies this trend, considering crane performance as a factor in berth allocation.

The berth allocation problem poses a significant challenge as it falls under the category of NP-hard problems. This implies that solving it in polynomial time is unlikely, unless an algorithm capable of solving NP problems in general can be devised. To tackle BAP, QCAP, and QCSP, various approaches have been suggested in the literature, with a focus on optimizing the allocation of berthing spaces and the assignment and scheduling of quay cranes to minimize service time and enhance efficiency.

The emergence of green port policies, influenced by the International Marine Organization aiming to limit the sulfur content in the fuel oil used by vessels and the European Green Deal’s goal of achieving net-zero emissions, has made sustainability a vital consideration in BAP, QCAP, and QCSP. Building on the 2015 survey by Bierwirth and Meisel, which classified models based on spatial and temporal attributes, recent reviews by Raeesi et al. [13], Naeem et al. [14], Aslam et al. [15], Xie and Ambrosino [16] and Rodrigues and Agra [17] have further expanded the field to encompass uncertainty management, automation, real-time decision-making approaches, and eco-friendly objectives. This paper contributes to the literature by synthesizing research published between 2015 and 2024. It presents compelling arguments through discussions on recent developments to address gaps in green infrastructure, sustainability, robust optimization, policy trade-offs, and uncertainty, along with trends and research gaps in the integration of BAP, QCAP, and QCSP, as well as their applications in the existing literature.

The main objective of this paper is to present an updated review of recent research on berth allocation, quay crane allocation, and scheduling in container terminals since 2015. It categorizes over 94 innovative approaches, identifies gaps in the literature through a comprehensive understanding of the current state of BAP, QCAP, and QCSP in existing publications, and addresses common challenges. Additionally, it highlights potential areas for future research. The specific goals of this review are as follows:

1.

Offering a comprehensive summary of the current status of berth allocation and quay crane assignment and scheduling for container terminals, analyzing the berth layout, vessels arrivals, crane handling, sustainability and green infrastructure in port operations, objective function, model formulation, policy trade-offs, solution approach and model performance evaluation associated with different model types.

2.

Providing a detailed discussion on current issues associated with models applied to berth allocation, quay crane assignment, and scheduling.

3.

Outlining the outcomes and identifying future research directions in relation to berth allocation, quay crane assignment, and scheduling.

The remainder of this paper is structured as follows. Section 2 presents the review methodology utilized in this study. Section 3 provides a comprehensive and critical assessment of the current state. Section 4, along with an analysis of the challenges and future research directions, concerns BAP, QCAP, and QCSP. Finally, the conclusions of this review are presented in Section 5.

2. Review Methodology

This study employs a systematic literature review methodology to structure the review process. It follows the reporting checklist established by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA). To identify relevant publications, databases such as Scopus, Web of Science, and Google Scholar were utilized, focusing on works published from 2015 to 2024 that pertain to the BAP, QCAP, and QCSP models in the context of container port terminals. The Scopus database enabled a thorough systematic review, while Web of Science and Google Scholar were utilized to identify potentially overlooked literature not indexed in Scopus. Scopus was chosen as the primary database due to its extensive coverage of engineering and logistics journals, as well as its substantial collection of indexed, peer-reviewed publications pertinent to container terminal operations. Google Scholar was utilized as a supplementary database to capture gray literature, conference papers, and additional articles. However, some of the identified records required evaluation from their respective publishing journals to ensure quality and relevance. A subsequent search in Web of Science served as an additional resource, as it offers unique peer-reviewed content, particularly from various international sources. In addition to conducting database searches, we implemented citation tracing, both backward and forward, to discover influential or recent publications that may not have surfaced through keyword-based searches alone. The identified records were further evaluated based on their publication source to maintain a standard of academic integrity and relevance.

This timeframe is considered sufficient to provide a thorough overview of advancements in BAP, QCAP, and QCSP research. This study builds upon the foundational surveys conducted by Bierwirth and Meisel [11,12], which underscored the necessity for comprehensive comparisons of models related to BAP, QCAP, and QCSP. It also extends the contributions of Raeesi et al. [13], Naeem et al. [14], Aslam et al. [15], and Rodrigues and Agra [17], who have further progressed the field by addressing uncertainty management, automation, and environmentally sustainable objectives. Consequently, this review aims to update and expand the literature from 2015 to 2024.

The process of identifying relevant publications involved several key steps. An extensive search was conducted using the keywords “Berth Allocation,” “Quay Crane Assignment,” and “Quay Crane Scheduling” in the titles, abstracts, and keywords. This was formulated as “TITLE-ABS-KEY ((berth AND allocation) AND (quay AND crane AND assignment OR quay AND crane AND allocation OR quay AND crane AND scheduling)) AND PUBYEAR > 2014 AND PUBYEAR < 2025,” which yielded 78 records in Scopus. A subsequent search for additional sources on Google Scholar produced 81 records, which were further refined by concentrating on relevant articles, while excluding those that were repetitive. In addition, a complementary search conducted in Web of Science yielded 128 records, broadening the dataset with peer-reviewed content.

Concerning the filtering criteria, the papers included in this review have been peer-reviewed and published in reputable journals ranked in SCImago. It is important to emphasize that no filters were applied based on country; studies from all continents were taken into account. From the 287 records identified through the data sources, articles that did not focus on mathematical models, metaheuristics, heuristics, machine learning, stochastic models, or robust optimization were excluded. This review specifically targeted papers related to container terminals. After eliminating duplicate records, 161 full-text articles were assessed for eligibility. One author conducted the eligibility assessment by meticulously screening the full texts of the remaining papers, with two authors collaborating to review and resolve any conflicts. Ultimately, 67 papers without peer review status were excluded, leaving a total of 94 studies included in the systematic literature review. Figure 1 illustrates the flowchart of the study selection process for the systematic review.

This review examines 94 papers sourced from international scientific journals, book chapters, and conference proceedings, as outlined in

Table 1. Summary of reviewed papers on berth allocation and quay crane scheduling.

Reference	Berth Layout	Arrival Process	Objective	Solution Approach	Handling of Crane
Correcher et al. [18]	Continuous	Dynamic	Min service time, setup costs	MILP	Variable
Iris et al. [19]	Continuous	Static	Min service time	MILP	Fixed
Iris and Lam [20]	Continuous	Stochastic	Min costs	Robust MILP	Variable,QCSP
Sun et al. [21]	Continuous	Stochastic	Min costs (carbon tax)	Robust MILP	Variable
Abou Kasm et al. [22]	Continuous	Dynamic	Min flow time, tardiness	MILP + Gurobi	Variable
Agra and Oliveira [23]	Discrete	Dynamic	Max QC value, min housekeeping	MIP + Tabu	Variable,QCSP
Aljasmi et al. [24]	Hybrid	Static	Min service time	GA-based heuristic	Variable
Alnaqbi et al. [25]	Hybrid	Dynamic	Min waiting/handling time	GA	Variable
Al-Refaie and Abedalqader [26]	Discrete	Stochastic	Max emergent service	Three-stage opt	Fixed,QCSP
Al-Refaie and Abedalqader [27]	Discrete	Stochastic	Max emergent service, min disruption	Three-stage opt	Variable,QCSP
Cahyono et al. [28]	Discrete	Dynamic	Min handling/waiting costs	DES + MPC	Variable
Cahyono et al. [29]	Discrete	Dynamic	Min handling/waiting costs	DES + MPC	Variable
Cao et al. [30]	Continuous	Dynamic	Min costs	Chaotic Quantum SSA	Variable,QCSP
Cereser et al. [31]	Continuous	Dynamic	Min total cost	GRASP	Variable
Chargui et al. [32]	Discrete	Dynamic	Min departure time	MILP + VNS	Variable
Chargui et al. [33]	Continuous	Stochastic	Min worst-case costs	Robust decomposition	Fixed
Correcher and Alvarez-Valdes [34]	Continuous	Static	Min service time	BRKGA	Variable
Dai et al. [35]	Continuous	Stochastic	Min turnaround time, setup costs	MILP + RL	Variable,QCSP
De Oliveira et al. [36]	Discrete	Static	Min completion time, QC movements	NSGA-II	Variable
Dhingra et al. [37]	Discrete	Stochastic	Estimate handling time	Stochastic (CTMC)	Variable,QCSP
El-boghdadly et al. [38]	Continuous	Dynamic	Min service time	GP	Variable,QCSP
El-Boghdadly et al. [39]	Continuous	Dynamic	Min service time	GP	Variable
Expósito-Izquiero et al. [40]	Continuous	Stochastic	Min service time, max robustness	Fuzzy MILP	Variable
Fatemi-Anaraki et al. [41]	Continuous	Dynamic	Min service time	Matheuristic (GA + ACO)	Fixed
Fu and Cai [42]	Continuous	Dynamic	Min service time	GA	Variable
Han et al. [43]	Continuous	Dynamic	Min service time	Branch-and-price	Variable
Hoseini et al. [44]	Hybrid	Stochastic	Min service time, costs	Simulation-based EA	Variable
Idris and Zainuddin [45]	Continuous	Stochastic	Min service/waiting time	MO meta-heuristic	Variable
Iris et al. [46]	Continuous	Dynamic	Min service time	ALNS	Variable
Iris and Lam [47]	Continuous	Stochastic	Min energy costs	MILP + Heuristic	Variable,QCSP
Jiachen and Guoyou [48]	Continuous	Stochastic	Min turnaround time	RTPSO	Variable
Jiang et al. [49]	Continuous	Stochastic	Min carbon cost, delays	Robust MILP + ALNS	Variable
Jiao et al. [50]	Continuous	Dynamic	Min turnaround time	ILP + GA/PSO/SA	Variable
Lalita and Murthy [51]	Hybrid	Static/Dynamic	Min service time, costs	Compact ILP	Fixed
Lalla-Ruiz et al. [52]	Continuous	Dynamic	Min service time	MBO	Variable
Li and Li [53]	Continuous	Dynamic	Min service time	Quantum Annealing	Variable,QCSP
Li et al. [54]	Continuous	Stochastic	Min turnaround time	Stochastic ILP + SAA	Variable
Liu et al. [55]	Discrete	Dynamic	Min operational cost	MIP + GA	Variable
Liu et al. [56]	Continuous	Stochastic	Min turnaround time (disruptions)	MIP + GA-SA	Variable
Liu et al. [57]	Continuous	Stochastic	Min departure delays	Robust + CCG	Variable
Ma et al. [58]	Continuous	Dynamic	Max efficiency	MILP + Heuristic	Variable,QCSP
Ma et al. [59]	Continuous	Dynamic	Min service time	GA-based heuristic	Variable
Malekahmadi et al. [60]	Continuous	Dynamic	Min turnaround time	RTPSO	Variable
Nishimura [61]	Discrete	Stochastic	Min waiting time, AGV use	PSO	Variable
Niu et al. [62]	Continuous	Dynamic	Min time, cost, deviation	MBCO	Variable
Olteanu et al. [63]	Discrete	Dynamic	Min completion time	GA	Variable
Pan et al. [64]	Continuous	Dynamic	Min makespan	Online heuristic	Variable
Ran et al. [65]	Continuous	Stochastic	Min departure time (worst-case)	Two-stage robust + CCG
Ren et al. [66]	Continuous	Dynamic	Min turnaround time	GA	Variable,QCSP
Ren et al. [67]	Continuous	Dynamic	Min time loss	MILP	Variable
Rodrigues and Agra [68]	Continuous	Stochastic	Min worst-case completion	Two-stage robust + MOSA	Fixed
Salhi et al. [69]	Continuous	Stochastic	Min service time, costs	MIP + EA	Variable
Samrout et al. [70]	Continuous	Dynamic	Min service time, transshipment costs	NSGA-III	Fixed
Shang et al. [71]	Continuous	Stochastic	Min delays	Robust MILP	Variable
Tang et al. [72]	Discrete	Dynamic	Min service time	Discretization + Heuristic	Fixed,QCSP
Tang et al. [73]	Continuous	Dynamic	Min weighted delay	GA	Variable
Tavakkoli-Moghaddam et al. [74]	Continuous	Dynamic	Min turnaround time	MIP + Mb-MOKA	Variable
Türkoğulları et al. [75]	Continuous	Dynamic	Min service time, setup costs	MILP	Variable
Venturini et al. [76]	Continuous	Stochastic	Min emissions, costs	MILP	Variable,QCSP
Wang et al. [77]	Continuous	Dynamic	Min costs (carbon tax)	MILP	Variable,QCSP
Woo et al. [78]	Continuous	Dynamic	Min delays	Data-driven heuristic	Variable
Wu et al. [79]	Abstract	Deterministic	Min makespan	Approx alg	Variable
Xiang et al. [80]	Discrete	Dynamic	Min service time	MILP	Fixed,QCSP
Xiang et al. [81]	Discrete	Stochastic	Min turnaround time	MIP + Heuristic	Variable,QCSP
Xingchi et al. [82]	Continuous	Stochastic	Min delays	Robust MILP	Variable
Yang et al. [83]	Discrete	Stochastic	Min recovery costs	Heuristic/meta-heuristic	Variable
Yin et al. [84]	Continuous	Dynamic	Min service time	MILP + GA	Fixed
Yu et al. [85]	Continuous	Stochastic	Min emissions, delays	MILP + GA	Variable
Yu et al. [86]	Continuous	Dynamic	Min conflict, delays	GA	Variable
Yu et al. [87]	Continuous	Dynamic	Min service time, fuel use	Spatiotemporal + BB	Variable
Yu et al. [88]	Continuous	Stochastic	Min delays, max safety	MO-GA + SA	Variable
Yuping et al. [89]	Continuous	Stochastic	Max fairness	Fairness heuristic	Variable,QCSP
Zhao et al. [90]	Continuous	Stochastic	Min time, cost, emissions	MO-GA	Variable
Zheng et al. [91]	Hybrid	Dynamic	Min makespan	Online alg (4/3-comp)	Fixed
Zheng et al. [92]	Continuous	Dynamic	Min turnaround time	ILP + GA/HPSO/HSA	Fixed
Zhong et al. [93]	Continuous	Dynamic	Min time, cost, deviation	MORHCO	Variable
Lu et al. [94]	Continuous	Dynamic	Min waiting/handling time	MILP + heuristic	Variable,QCSP
He et al. [95]	Continuous	Dynamic	Min cost, max efficiency	MILP	Variable
He [96]	Continuous	Dynamic	Min time/energy consumption	Multi-objective optimization	Variable
Tasoglu and Yildiz [97]	Discrete	Stochastic	Min total cost/time	Simulated annealing	Variable,QCSP
Li et al. [98]	Continuous	Dynamic	Min service time, max coverage	Heuristic	Variable
Fazli et al. [99]	Discrete	Dynamic	Min makespan	Red Deer Algorithm	Variable
Nourmohammadzadeh and Vo [100]	Continuous	Stochastic	Min cost/time, max robustness	Robust multi-objective	Variable,QCSP
Chu et al. [101]	Continuous	Stochastic	Min total cost, max preference	MILP + heuristic	Variable
Li et al. [102]	Discrete/Continuous	Dynamic	Min disruption time	Real-time heuristic	Variable
Wu and Zhu [103]	Continuous	Stochastic	Min delay, max reliability	Hybrid ALNS	Variable
Tan and He [104]	Continuous	Stochastic	Min emissions/time, max sustainability	Proactive-reactive model	Variable
Lu and Lu [105]	Continuous	Dynamic	Min carbon emissions/time	Multi-objective GA	Variable
Zhang et al. [106]	Discrete	Static	Min cost/time, max utilization	Bi-objective optimization	Fixed, time-invariant
Hsu et al. [107]	Discrete/Continuous	Dynamic	Min total time	Hybrid GA	Variable
Xiang and Liu [108]	Continuous	Stochastic	Min cost/time, max robustness	Almost robust optimization	Variable
Wen et al. [109]	Continuous	Dynamic	Min energy/time, max efficiency	Optimization model	Variable
Jiang et al. [110]	Continuous	Dynamic	Min carbon emissions/time	Collaborative optimization	Variable
Cai et al. [111]	Discrete	Stochastic	Min total time, max robustness	Robust scheduling	Variable

. The berth allocation problem (BAP), quay crane allocation problem (QCAP), and quay crane scheduling problem (QCSP) are addressed through mathematical models, including mixed-integer linear programming, and are enhanced by metaheuristics, machine learning, and robust optimization techniques. These methodologies aim to tackle the relevant issues and improve the overall performance of the models, taking into account berth layouts, arrival processes, objectives, and crane handling.

3. Results

3.1. Current Status of BAP, QCAP and QCSP

This section presents the findings of the systematic review, starting with an overview of publication trends and distribution. The collected studies were classified by publication type, namely journal articles, book chapters, and conference papers, and analyzed over time and by geographic region to better understand the evolution and scope of research activity in this field. Figure 2 illustrates the annual publication trends for BAP, QCAP, and QCSP, indicating that the highest number of papers was published in 2016. Figure 2 demonstrates that articles consistently comprise the majority of outputs throughout the examined period, while contributions from book chapters and conference papers remain minimal. This strong focus on peer-reviewed articles bolsters confidence in the dataset’s reliability, as these sources undergo stringent editorial oversight, thereby adhering to high standards of scientific rigor.

The geographical distribution of the 94 reviewed articles shows a notable concentration of research originating from China, which can be attributed to its dominance in global trade, the implementation of government-supported green port policies, and its leadership in port optimization research, as presented in Figure 3. Overall, Asia accounts for the largest share of contributions, with 14 countries represented. Europe follows, contributing publications from 11 countries, highlighting its significant role in the field, particularly among technologically advanced nations such as the Netherlands and Germany. In contrast, research output from North America, South America, and Oceania remains relatively limited. Notably, there were no studies identified from African countries in this review, which underscores the earlier observation regarding the scarcity of container port research in Sub-Saharan Africa, despite some advancements in bulk port development in North Africa. This geographic imbalance emphasizes the need for future research to focus on container port optimization in underrepresented regions, especially in Sub-Saharan Africa.

The problems of berth allocation, quay crane assignment, and scheduling are interconnected facets of quayside operations that are vital for optimizing terminal performance. The berth plays a critical role in the docking of vessels, and efficient allocation is key to minimizing congestion and idle times. The assignment of berths directly influences the quay cranes that service the vessels; inadequate allocation can result in crane interference and operational delays. Once vessels are assigned to specific berths, crane operations, which include scheduling and sequencing, are contingent on berth placement and handling priorities. Furthermore, this process can have environmental and economic impacts and social implications that affect the safety of workers and nearby communities, necessitating strict adherence to regulations and policies.

Several critical inputs delineate the essential steps involved in the development and implementation of the BAP, QCAP, and QCSP model. Figure 4 presents a conceptual overview of the key elements necessary for evaluating the current status of BAP, QCAP, and QCSP. As depicted in Figure 4, foundational inputs such as spatial, temporal, handling, and environmental elements directly impact the complexity, constraints, and optimization objectives of the model, making them essential for the efficient design of port operations. They define the physical and operational constraints relevant to berth allocation decisions.

The problem formulation stage of constructing a BAP model entails defining the mathematical and conceptual framework that governs the allocation of berths to vessels at a port. This formulation clarifies the decision variables, objectives, and constraints while also addressing policy trade-offs. It encapsulates both the physical and operational characteristics of the port and balances competing objectives, all while ensuring compliance with regulations in the modeling of the port system. This step is vital, as it transforms the real-world problem into a solvable optimization model. The primary purpose of the model is to achieve optimal BAP performance. Therefore, selecting an appropriate solution approach is crucial for effectively addressing the specific BAP issue at hand.

The berth layout pertains to the spatial arrangement and design of berths in a port where vessels dock for loading and unloading containers. This layout can be classified into three types: discrete, continuous, and hybrid. A discrete layout is distinguished by individual berths that accommodate only one vessel at a time. It consists of distinct berthing segments within defined physical constraints, allowing a vessel to moor and be serviced by port equipment. Each designated service position is intended for one vessel at any given moment.

In contrast, a continuous berth layout treats the berths as a single, uninterrupted length, permitting multiple vessels to utilize the space simultaneously. In this arrangement, a vessel can moor anywhere along the designated area, with the allocation of berth space based on unique requirements rather than being divided into separate segments. The hybrid layout combines elements of both discrete and continuous berths. It features a wharf divided into sets of distinct berthing positions while also allowing a single vessel to occupy multiple segments simultaneously.

Bierwirth and Meisel [11,12] observed that although the influence of berthing positions on handling times has been extensively studied in discrete layouts, it has garnered less attention in continuous layouts. Research addressing hybrid berth layouts is notably scarce in the existing literature. This lack of studies can be attributed to the complexity involved in managing a combination of discrete and continuous berth allocations. Hybrid berth layouts are primarily employed in ports that handle both containerized and bulk cargo, which adds to the complexity of mixed cargo management. Most research in berth allocation problems has concentrated on container ports. Figure 5 illustrates various berth layouts, while Figure 6a indicates the number of papers reviewed for different berth layouts, revealing that most studies concentrate on continuous berth layouts, as they lead to greater utilization of berth wharves compared to discrete layouts.

The primary vessel arrival process, also referred to as temporal, can be classified into three main categories: static, dynamic, and stochastic. In static scenarios, all vessels are already docked in the port, with predetermined arrival times that facilitate planned berth allocation. In contrast, dynamic situations involve unpredictable arrivals, which require real-time flexibility to accommodate changes such as delays or early dockings. Stochastic arrivals denote the random and unpredictable nature of vessel entries into the terminal.

As seen in Figure 6b, there is a notable scarcity of studies focusing on static vessel arrivals in BAP compared to dynamic and stochastic arrivals. Stochastic models represent vessel arrivals as random variables, adding a layer of complexity to their development. They necessitate detailed historical data for establishing probability distributions, which many ports may not possess, rendering these models less practical. Conversely, dynamic models allow for real-time adjustments to vessel schedules, aligning more closely with the needs of port operators who are managing daily disruptions. These models integrate effectively with the real-time data systems used in port management. As a result, researchers tend to prioritize dynamic rescheduling issues related to operational variability rather than focusing solely on probabilistic uncertainty. Most studies do not utilize static arrival scenarios due to their overly simplistic nature. The static arrival problem involves discrete slots with complete information available in advance, particularly in port terminal operations that rely on fixed resources. This approach lacks the flexibility and adaptability to accommodate changes.

Handling time refers to the duration a vessel remains at a berth to complete its cargo operations, typically measured from the commencement of operations (after mooring) to the conclusion of loading and unloading activities. This time span is influenced by the number of quay cranes assigned, making it a crucial element in QCAP and QCSP. Given their interconnected nature, it is essential to address them either simultaneously or in a coordinated manner to optimize port operations.

Sustainability in port operations entails managing and optimizing processes to reduce environmental impact. This includes waste management and safeguarding the port environment by lowering carbon emissions through efficient crane usage and the adoption of renewable energy sources. Additionally, it focuses on improving economic efficiency via investments in green infrastructure, optimizing berth utilization, and minimizing idle time. Achieving these goals requires adherence to environmental regulations while ensuring a safe working environment for local residents and port workers alike.

This paper will examine both discrete and continuous berth layouts, as they align more closely with the practical and theoretical requirements of port operators. Static models tend to be overly simplistic and lack adaptability to changes. Hybrid models are often used for mixed cargo in bulk ports; however, this topic will not be included in our review since our focus is specifically on container terminals.

3.1.1. Inputs and Configuration for BAP, QCAP, and QCSP

An analysis of the literature selected for review, encompassing the period from 2015 to 2024, on the berth allocation problem and quay crane assignment or scheduling revealed that 19 studies concentrated on discrete issues, while 67 focused on continuous problems. The majority of these studies modeled continuous berth layouts, enabling flexible vessel positioning to optimize quay utilization in alignment with contemporary port operations. Figure 7a,b illustrate the number of studies employing discrete berth layouts compared to those utilizing continuous berth layouts, highlighting the varying handling of cranes.

The application of discrete berth allocation, incorporating dynamic variants, has been explored by several researchers, including Chargui et al. [32], Agra and Oliveira [23], Olteanu et al. [63], Liu et al. [55], Cahyono et al. [29], Tang et al. [72], Cahyono et al. [28], and Xiang et al. [80], in order to address operational challenges in sea ports. Exceptions include works by Al-Refaie and Abedalqader [27], Nishimura [61], Dhingra et al. [37], Yang et al. [83], Xiang et al. [81] and Al-Refaie and Abedalqader [26], who utilized discrete berth layouts with stochastic arrival processes.

Notably, Jiang et al. [49] and Zhao et al. [90] distinguish themselves by modeling multiple terminals or quays, which enhances their applicability to large ports. Meanwhile, Yin et al. [84] implemented a unique U-shaped continuous berth layout with asynchronous vessel access to optimize quay utilization. Additionally, Zheng et al. [92], Lu et al. [94], Malekahmadi et al. [60], and Jiao et al. [50] addressed berth allocation considering the effects of tidal influences, thereby introducing environmental complexities. Finally, issues related to uncertainty factors in berth allocation have been examined by Chargui et al. [33], Rodrigues and Agra [17], Ran et al. [65], Zhang et al. [112], and Jiang et al. [113]. Pan et al. [64] and Cahyono et al. [29] developed a model for real-time adaptability and responsiveness, while Rodrigues and Agra [17] focused on addressing arrival uncertainty through robust two-stage approaches.

The handling time of a vessel is determined by the assignment of quay cranes. The assignment of quay cranes can be variable or fixed. The variable quay crane assignment allows adjustments to the crane allocation to a vessel to optimize the utilization of released cranes. The assignments can further be time-invariant or time-variant. In a time-invariant assignment, the assigned cranes do not change during the vessel’s service. Several factors in the process include environmental impact, worker variation, energy efficiency, and possible disruptions. There is a clear preference for variable and time-variant assignments in both discrete and continuous berth allocation, highlighting the necessity for adaptability in contemporary container terminals.

The following authors used fixed time-invariant assignments: Fatemi-Anaraki et al. [41], Samrout et al. [70], Yin et al. [84], Zheng et al. [91], Chargui et al. [33], Lalita and Murthy [51], Rodrigues and Agra [68], Zheng et al. [92], and Iris et al. [19], while Al-Refaie and Abedalqader [27], Agra and Oliveira [23], Dhingra et al. [37], Yuping et al. [89], Dai et al. [35], Xiang et al. [81], Tang et al. [72], Cao et al. [30], Wang et al. [77], Ma et al. [58], Xiang et al. [80], Li and Li [53], Ren et al. [66], Chargui et al. [32], El-boghdadly et al. [38], Iris and Lam [47], Iris and Lam [20], Venturini et al. [76], Xie and Ambrosino [16], and Iris and Lam [47] have all contributed to the schedule discussion. Fu and Cai [42] and Ma et al. [58] discuss the coordination of berth allocation with quay crane scheduling, with the emphasis being toward finer-scale scheduling.

The articles by Cai et al. [111], Li et al. [102], and Yin et al. [84] focus on innovations in crane handling aimed at advancing operational efficiency. Cai et al. [111] explores the implementation of double-cycling modes, which enable cranes to conduct both loading and unloading in a single trip, thereby minimizing idle time. This approach incorporates robust scheduling techniques to effectively manage uncertainties in operation times, particularly enhancing efficiency in U-shaped terminals. Meanwhile, Li et al. [102] presents adaptive crane scheduling strategies designed to quickly recover from disruptions in real time. This method allows for adjustments in assignments and schedules in response to unexpected events such as crane failures or vessel delays, ensuring continued operational efficiency amid uncertainties. Additionally, Yin et al. [84] discusses automated crane scheduling within a U-shaped port layout, utilizing spatio-temporal coordination to boost throughput. Their approach automates crane assignments based on real-time data to effectively handle complex port geometries and asynchronous operations.

Sustainability in Port Operations

The reviewed papers indicate that sustainability in port operations is an emerging yet underexplored theme in berth and crane scheduling research, with only a limited number of studies directly addressing sustainability. Approximately 50% of the studies focus exclusively on efficiency, cost, or scheduling without considering environmental or social factors. Chargui et al. [33] explicitly optimizes energy consumption in quay crane allocation and scheduling, achieving sustainability by reducing power usage through an exact decomposition algorithm. Iris and Lam [47] emphasizes the integration of renewable energy and the use of smart grids for sustainable energy management amidst uncertainty in operations planning at seaports.

Jiang et al. [49] incorporates carbon costs into the objective function when developing a model for multi-terminal berth and quay crane scheduling, thereby encouraging low-emission operations. Similarly, Wang et al. [77] examines the effects of carbon taxation in joint berth allocation and quay crane assignment, promoting emission reductions under varying carbon prices. Sun et al. [21] also integrates a carbon tax incentive into weekly berth and quay crane planning. Additionally, Venturini et al. [76] focuses on optimizing vessel speeds in multi-port operations to minimize emissions. Other articles contribute to sustainability indirectly by enhancing efficiency, reducing idle times, or optimizing resource usage, which ultimately lowers energy consumption and emissions.

Both Yu et al. [87] and Malekahmadi et al. [60] also have an indirect impact on sustainability within port operations. Yu et al. [87] optimizes quay crane coverage to reduce fuel consumption, thereby indirectly lowering emissions, while Malekahmadi et al. [60] optimizes tidal operations, which helps to minimize environmentally disruptive practices. Studies employing multi-objective optimization or robust models could potentially include sustainability as part of the constraints.

Green Infrastructure in port operations

A growing body of research, including studies by Yu et al. [85], Iris and Lam [47], Jiang et al. [49], Jiang et al. [110], and Zheng et al. [92], highlights an increasing focus on the integration of green infrastructure in port operations. This trend emphasizes the necessity of incorporating renewable energy, shore power, and tidal optimization to support broader sustainability objectives. In this context, green infrastructure encompasses the deployment of environmentally sustainable technologies and systems such as shore power, renewable energy sources, and low-emission equipment aimed at reducing the ecological impact of port activities.

Yu et al. [85] examines the application of green technologies, including shore power, electric cranes, and energy-efficient systems, in berth and crane assignments, directly addressing how green infrastructure can be implemented in operational decision-making. Similarly, Iris and Lam [47] integrates smart grids and renewable sources such as solar and wind energy to enhance berth and crane scheduling amid energy supply uncertainty, underscoring the critical role of renewables as foundational elements of green infrastructure.

Building on this, Jiang et al. [49] incorporates shore-side electricity and carbon cost considerations into multi-terminal scheduling, effectively connecting carbon-conscious operational planning with sustainable technologies. Jiang et al. [110] further progresses this integration by employing low-carbon technologies and energy-efficient systems within a collaborative berth and crane scheduling framework, reinforcing the significance of infrastructure in sustainable port management. Finally, Zheng et al. [92] introduces tidal influence as a form of green infrastructure, leveraging natural cycles to improve energy-efficient scheduling and minimize environmental impact. This approach aligns port operations with ecological rhythms, promoting long-term sustainability.

3.1.2. BAP, QCAP, and QCSP Mathematical Model Formulation

Various formulations have been utilized in the literature to address berth allocation problems, each differing based on certain assumptions, decision variables, and constraints. In the context of the discrete berth allocation problem, numerous formulations typically use decision variables to assign vessels to specific berths and to establish precedence relations among vessels docked at each berth Golias et al. [114]. One of the foundational model formulations that has inspired significant research is that presented by [115,116,117]. Throughout the literature, researchers have adapted these formulations to better align with the specific operational characteristics of their individual ports. Notably, significant advancements have been made in the mathematical modeling of berth allocation, quay crane assignment, and scheduling problems. Robust and stochastic formulations have been developed to address uncertainties such as vessel arrival times, crane breakdowns, and environmental factors. For example, Ran et al. [65] and Rodrigues and Agra [68] utilize uncertainty sets to ensure feasibility under worst-case scenarios. Furthermore, modern models now incorporate multiple objectives, balancing traditional goals with sustainability targets. These mathematical models effectively integrate elements of green infrastructure, such as shore power, renewable energy, and tidal impacts, often including carbon cost constraints to support sustainability trends and enhance applicability to contemporary port regulations. In the following, we outline the sets, parameters, and variables necessary to formulate the discrete dynamic berth allocation problem. The model establishes a policy trade-off aimed at simultaneously minimizing waiting times, service durations, and crane setup frequencies through a multi-objective optimization framework. See Box 1.

Box 1. Summary of the sets, decision variables, and parameters used in the mathematical model.

Sets

• i: $(i=1,2,\dots ,I)\in S$ set of vessels.
• j: $(j=1,2,\dots ,J)\in B$ set of berths.
• k: $(k=1,2,\dots ,K)\in Q$ set of quay cranes.
• t: Time index $(t=1,2,\dots ,T)$.

Parameters

• $A_i$: Arrival time of vessel i.
• $L_i$: Length of vessel i.
• $N_j$: Maximum number of quay cranes available at berth j.
• $H_{ijk}$: Handling time for vessel i when quay crane k is assigned at berth j.
• $B{L}_j$: Length of berth j.
• $D_i$: Water depth of vessel i.
• $d_i$: Draft of vessel i.
• $g_{ik}$: Cost of setting crane k for vessel i.
• $C_i^e$: Desired departure time of vessel i.
• $g_i$: Penalty for exceeding the desired departure time of vessel i.

Decision variables

• $$x_{ij}=\left\{\begin{array}{cc}1\hfill & \mathrm{vessel}i\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{berth}j\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
• $$y_{ki}=\left\{\begin{array}{cc}1\hfill & \mathrm{quay}\mathrm{crane}k\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{vessel}i\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
• $$z_{kt}=\left\{\begin{array}{cc}1\hfill & \mathrm{quay}\mathrm{crane}k\mathrm{is}\mathrm{used}\mathrm{at}\mathrm{time}t\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
• $s_i$: Start time of service for vessel i.
• $c_i$: Departure time of service for vessel i.
• $w_i$: Waiting time of vessel i.

In tackling both the berth allocation problem and the quay crane allocation and scheduling problem, most models have been developed to optimize terminal operational efficiency by minimizing waiting times, handling times, penalties for delays, berthing operating times, departure operating times, and overall time loss. Additionally, these models aim to reduce the total costs incurred by both the port and the vessel owners, particularly in relation to service times within the berth allocation framework of the port container terminal. The overall cost encompasses waiting costs, departure delay costs, handling costs, deviation costs, tardiness costs, distance costs, and various penalty costs. Recently, there has been a growing trend toward modeling multi-objectives by incorporating additional goals, such as minimizing disruptions, improving housekeeping, and optimizing quay crane movements. Researchers such as Al-Refaie and Abedalqader [27], Agra and Oliveira [23], and De Oliveira et al. [36] focus on developing multi-objective models to address berth allocation issues, aiming to minimize additional factors such as disruptions, quay crane movements, and housekeeping activities. Additionally, De Oliveira et al. [36], Olteanu et al. [63], Xiang et al. [81], Tang et al. [72], Xiang et al. [80], and Chargui et al. [32] emphasize the importance of minimizing time-related elements, including waiting time, completion time, and service time. The intended purposes for which these models were developed are illustrated in Figure 8, including the proportion of various objectives in relation to the total number of papers reviewed.

The primary objective of this model formulation in the current paper is to minimize the total turnaround time, which encompasses waiting time, service time, the cost penalties associated with exceeding the desired departure time, and the duration required for crane setup. This can be represented in the objective function detailed below:

$$\mathrm{Minimize}:\alpha \sum _{i\in S}{w}_i+\beta \sum _{i\in S}(c_i-s_i)+\gamma \sum _{i\in S}max\left\{0,c_i-C_i^e\right\}·g_i+\kappa \sum _{i\in S}\sum _{k\in Q}{g}_{ik}{y}_{ik}$$

(1)

where:

• $\alpha$ is the weight of waiting time,
• $\beta$ is the weight of service time,
• $\gamma$ is the weight of the penalty cost for exceeding the desired departure time,
• $\kappa$ is the weight of crane setup cost.

From this objective, we derive constraints related to berth allocation, crane assignment, and scheduling, as outlined below:

1.

Berth Allocation Constraints

(a)

Each vessel is assigned to exactly one berth:

$$\sum _{j\in B}{x}_{ij}=1,\forall i\in S.$$

(2)

(b)

Vessel assignments must not exceed berth length:

$$x_{ij}·L_i\le B{L}_j,\forall j\in B,\forall i\in S.$$

(3)

(c)

Vessel its serviced after its arrival:

$$s_i\ge A_i,\forall i\in S.$$

(4)

(d)

Vessel draft can not be more than water depth:

$$x_{ij}.d_i\le D_j,\forall i\in S,\forall j\in B.$$

(5)

2.

Quay Crane Allocation Constraints

(a)

Each crane can work on one vessel at a time:

$$\sum _{i\in S}{y}_{ki}\le 1,\forall k\in Q.$$

(6)

(b)

The number of quay cranes allocated to a vessel cannot exceed the maximum available at its assigned berth:

$$\sum _{k\in Q}{y}_{ki}\le N_j·x_{ij},\forall i,j.$$

(7)

(c)

Quay cranes are only allocated to vessels assigned to a berth:

$$y_{ki}\le \sum _{j\in B}{x}_{ij},\forall i\in S,k\in Q.$$

(8)

3.

Quay Crane Scheduling Constraints

(a)

Quay cranes allocated to adjacent vessels should not interfere:

$$y_{ik}+y_{\overline{i}k}\le 1,\forall k\in Q,\mathrm{if}\mathrm{vessels}i,\overline{i}\mathrm{are}\mathrm{adjacent}.$$

(9)

(b)

Quay crane must work the entire period until the end of service:

$$\sum _{t=1}^T{z}_{kt}\ge (c_i-s_i)y_{ik},\forall i\in S,k\in Q.$$

(10)

The nature of the mathematical formulation we are addressing is a mixed integer linear programming model. This problem is challenging to solve, as it is classified as NP-hard; as the problem size increases or with extremely large instances, the computational complexity grows exponentially. For smaller instances, the mixed integer linear programming model can successfully solve the problem. However, for larger instances, solution approaches such as metaheuristics or heuristics prove to be effective and can significantly enhance the model’s performance.

Next, we will explore the formulations of continuous layouts for the papers currently under review. While the discrete berth allocation problem can be interpreted as a machine scheduling issue, the continuous berth allocation problem can be modeled as a two-dimensional rectangle packing problem within a time-space diagram. In this representation, the service provided to a vessel is illustrated as a rectangle, with the horizontal axis denoting quay length and the vertical axis representing the scheduling time horizon for vessel operations [117]. Lim [4] was a pioneer in applying graph theory to the continuous berth allocation problem, simplifying it into a two-dimensional packing challenge. An example of a time-space diagram is shown in Figure 9.

In the following, we will outline the sets, parameters, and variables necessary to formulate the continuous dynamic berth allocation problem with QCAP and QCSP. See Box 2.

Box 2. Summary of the sets, decision variables, and parameters used in the mathematical model.

Sets

• i: $(i=1,2,\dots ,I)\in S$ set of vessels.
• q: $(q=q_1,q_2,\dots ,q_N)\in Q$ set of quay cranes.
• t: time index $(t=1,2,\dots ,T)$.

Parameters

• $A_i$: Arrival time of vessel i.
• $L_i$: Length of vessel i.
• N: Total number of quay cranes available.
• $H_i$: Handling time for vessel i.
• P: Total length of the quay.

Decision variables

• $x_i$: Starting berth position of vessel i.
• $q_{it}$: Number of quay cranes assigned to vessel i is at time t.
• $$y_{qit}=\left\{\begin{array}{cc}1\hfill & \mathrm{quay}\mathrm{crane}q\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{vessel}i\mathrm{at}\mathrm{time}t\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
• $s_i$: Start time of service for vessel i.
• $c_i$: Departure time of service for vessel i.
• $w_i$: Waiting time of vessel i.
• $N_i$: Maximum number of quay cranes for vessel i.
• $N_t$: Maximum number of quay cranes available at time t.

The objectives for BAP in relation to QCAP and QCSP for continuous problems are similar to those of discrete problems. Most models focus on minimizing service time, waiting time, turnaround time, or completion time. A trend that has emerged for continuous problems, which was not prominent in discrete problems, is the inclusion of sustainability elements such as emissions, carbon taxes, safety, and energy costs, often within multi-objective models.

Research by Expósito-Izquiero et al. [40], Yu et al. [88], Yuping et al. [89], Yu et al. [85], Iris and Lam [47], Sun et al. [21], Venturini et al. [76], and Iris and Lam [47] addresses continuous problems with stochastic arrivals, incorporating sustainability elements like efficiency, fairness, energy costs, emissions, carbon costs, and safety. Additionally, Yu et al. [86], Wang et al. [77], and Ma et al. [58] focused on addressing sustainability elements for dynamic arrivals. As illustrated in Figure 10, the remaining papers primarily utilized operational objectives that align closely with those found in discrete problems.

The objective of this formulation for the continuous dynamic berth allocation problem with QCAP and QCSP is to minimize the waiting time, handling time (service time), and time required for crane setup. This can be expressed in the objective function outlined below:

$$\mathrm{Minimize}:\sum _{i\in S}\left[\alpha (s_i-A_i)+\beta \left(c_i-s_i\right)+\gamma \sum _{t=1}^T\sum _{q\in Q}{y}_{qit}\right]$$

(11)

where

• $\alpha$ is the weight of waiting time;
• $\beta$ is the weight of service time;
• $\gamma$ is the weight of the crane setup.

From this objective, we state constraints related to berth allocation, crane assignment, and scheduling, as outlined below:

1.

Berth Allocation Constraints

(a)

Vessel must be fully accommodated within the quay:

$$0\le x_i\le P-L_i,\forall i.$$

(12)

(b)

Vessel is serviced after its arrival:

$$s_i\ge A_i,\forall i.$$

(13)

2.

Quay Crane Allocation Constraints (QCAP and QCSP)

(a)

Each vessel can be assigned a limited number of cranes:

$$\sum _{t=1}^T{q}_{it}\le N_i,\forall i.$$

(14)

(b)

At any time t, the total number of cranes assigned across all vessels cannot exceed the maximum available:

$$\sum _{i\in S}{q}_{it}\le N_t,\forall t.$$

(15)

(c)

Quay cranes can be assigned to only one vessel at a time:

$$\sum _{i\in S}{y}_{qit}\le 1,\forall q,\forall t.$$

(16)

The mathematical formulation we are addressing is a mixed integer linear programming model, similar to the discrete problem.

BAP, QCAP, and QCSP Policy trade-offs

The discussion in the articles regarding policy trade-offs in port operations centers on finding the right balance among competing objectives, such as efficiency, cost, sustainability, and fairness, particularly in berth allocation, quay crane assignment, and scheduling. When optimization efforts in one area (such as reducing service time) conflict with another (like minimizing emissions or ensuring safety), policy trade-offs become necessary, requiring careful compromises.

There exists a policy trade-off in port operations between maximizing service efficiency by minimizing service time and maximizing throughput, and minimizing environmental impact. This trade-off becomes particularly pronounced when functional objectives conflict with sustainability goals. To address these challenges, recent research has proposed models that strike a balance between efficiency and environmental considerations. For instance, Chargui et al. [33] presents an efficient exact decomposition algorithm that takes into account energy efficiency and uncertainty to reconcile operational speed with environmental objectives. Similarly, Jiang et al. [49] explores the Multi-Terminal Berth and Quay Crane Joint Scheduling, aiming to balance carbon cost reduction with throughput efficiency. Additionally, Wang et al. [77] examines joint berth allocation and quay crane assignment under various carbon tax policies, illustrating the impact of carbon compliance on both efficiency and costs.

Numerous studies have explored the intricate trade-offs involved in berth and quay crane assignment at container terminals, focusing on cost, efficiency, and operational dynamics. Chargui et al. [32] examines the optimization of berth scheduling, berth assignment, and quay crane scheduling, considering labor and truck deployment costs alongside operational efficiency. Acknowledging the time-dependent nature of terminal operations, Türkoğulları et al. [75] analyzes time-variant quay crane assignment and scheduling, emphasizing the balance between setup time efficiency and setup costs in optimal berth assignment. Similarly, Correcher et al. [18] investigates strategies to mitigate setup costs while enhancing crane travel efficiency in the berth and quay crane assignment problem. A methodological innovation in berth assignment is introduced by Dai et al. [35], who applies reinforcement learning to optimize vessel turnaround speed and setup costs simultaneously. Additionally, Cao et al. [30] utilizes the Chaotic Sparrow Search Algorithm (SSA) for optimizing berth-quay crane allocation, prioritizing economic efficiency alongside maximizing operational effectiveness.

Efforts to optimize seaport operations increasingly focus on integrating various subsystems such as energy management, berth planning, and crane scheduling while balancing the efficiency of individual components with the overall performance of the system. For instance, Iris and Lam [47] tackles the challenge of incorporating renewable energy amidst uncertainty by utilizing smart grids to manage energy resources in seaports. This strategy prioritizes sustainable energy usage, even if it means sacrificing some operational flexibility. In the realm of weekly berth and quay crane planning, Sun et al. [21] highlights the necessity for recoverable robustness in light of carbon taxation, striving to balance schedule stability with environmental compliance.

Beyond energy and emissions considerations, the integration of physical operations introduces additional trade-offs. Alnaqbi et al. [25] explores the interplay between quay crane scheduling and a dynamic-hybrid berth allocation problem, emphasizing the inherent tension between maximizing berth efficiency and managing the complexities of integrated scheduling. Similarly, Fatemi-Anaraki et al. [41] proposes a mathematical model designed for the simultaneous scheduling of waterways, berths, and quay cranes, acknowledging that achieving cohesive integration may lead to reduced waterway efficiency. Utilizing a Hybrid Flow Shop framework, Tavakkoli-Moghaddam et al. [74] investigates how the integration of waterway scheduling, berth allocation, and crane assignment necessitates a careful balance between global coordination and local performance.

In the context of tidal ports, integration brings forth additional constraints. Liu et al. [55] develops a mathematical model aimed at optimizing overall performance across various components, trading off certain individual efficiencies to facilitate integrated decision-making. Likewise, Lu et al. [94] illustrates how tidal efficiency may be compromised to allow for effective integration of quay crane assignment and berth allocation.

Policy decisions in seaport operations frequently involve trade-offs between maximizing throughput and ensuring fair distribution of resources among vessels or terminals. These trade-offs underscore the tension between operational efficiency and equity. For example, Yuping et al. [89] introduced a continuous berth allocation algorithm aimed at striking a balance between efficiency and equitable berth distribution, thereby enhancing fairness in service. Similarly, Tang et al. [73] explored the impact of service priority within the Integrated continuous berth allocation and quay crane assignment problem amid port congestion, demonstrating how prioritization strategies influence both service equity and operational performance.

While many port studies inherently involve trade-offs, some focus solely on optimizing a singular objective without explicitly accounting for these trade-offs. For example, Al-Refaie and Abedalqader [26] examines the optimum assignment and scheduling of quay cranes at container terminals, prioritizing assignment efficiency while neglecting competing goals. Similarly, El-Boghdadly et al. [39] centers its attention on heuristic performance, employing evolving local search techniques to address the integrated berth allocation problem alongside quay crane assignment, but overlooks broader operational trade-offs. Another illustration can be found in Expósito-Izquiero et al. [40], who tackles seaside port logistics through fuzzy optimization problems, emphasizing the robustness of quay crane and berthing schedules without considering the explicit trade-offs involved between competing objectives.

3.1.3. BAP, QCAP, and QCSP Solution Approach

A thorough literature review has identified various model types for addressing BAP, QCAP, and QCSP. There is a notable trend toward the use of hybrid algorithms that combine exact methods, such as mixed integer linear programming, with metaheuristics like genetic algorithms and simulated annealing to efficiently solve integrated BAP, QCAP, and QCSP models. Solution methodologies are increasingly incorporating robust optimization and stochastic methods to manage uncertainties related to vessel arrivals, crane failures, and energy variability. Metaheuristic techniques, including genetic algorithms, particle swarm optimization, and red deer algorithms, have gained popularity, often enhanced with local search or evolutionary strategies to improve convergence. The emerging application of machine learning and reinforcement learning is facilitating adaptive, real-time solutions, particularly for disruption recovery and dynamic scheduling, aligning with advancements in modern smart port technologies. Additionally, solution approaches are placing greater emphasis on multi-objective optimization by employing techniques such as weighted sums, Pareto optimization, or bacterial colony optimization to balance time, cost, and sustainability objectives. Heuristic methods, including variable neighborhood search, memetic algorithms, and decomposition strategies, are also being utilized to address the computational complexity associated with integrated models.

To address the BAP, QCAP, and QCSP issues, most researchers develop a mathematical model to analyze the problem, which is complemented by metaheuristic, heuristic, machine learning, stochastic, and robust optimization approaches to identify optimal solutions. From the reviewed articles, the majority of researchers employed metaheuristic approaches to solve and identify optimal solutions for both discrete and continuous problems, as illustrated in Figure 11. Mathematical models, such as mixed-integer linear programming, are commonly utilized for model formulation and effectively solving problems with smaller instances. Other heuristics include algorithms such as the chaotic quantum social spider algorithm and the greedy randomized adaptive search procedure.

Genetic algorithms are increasingly recognized as sophisticated models for addressing the berth allocation problem. Furthermore, there is a growing array of enhanced genetic algorithms that incorporate local search techniques and exact methods, including tabu search, neighborhood search algorithms, biased random-key genetic algorithms, non-dominated sorting genetic algorithms, and three-level genetic algorithms.

In the academic literature, advanced techniques such as particle swarm optimization (PSO) and simulated annealing (SA) have emerged as state-of-the-art methods for effectively addressing the BAP. Recent studies indicate a growing preference for hybrid approaches to tackle berth allocation challenges, moving away from reliance on individual algorithms like GA and PSO. Multi-objective optimization models have also been introduced in recent studies, including multi-objective genetic algorithms and multi-objective constraint handling. Enhanced algorithms, along with hybrid and multi-objective approaches, have demonstrated superior performance compared to traditional algorithms.

Dai et al. [35] utilize a reinforcement learning model to address real-time uncertainties such as unpredictable arrivals and delays, optimizing quay crane setup times. This is an emerging technique developed to solve real-time optimization challenges. While it requires extensive training, it effectively assists in dynamically setting up times. Similarly, Ran et al. [65], Rodrigues and Agra [68], and Chargui et al. [32] employed robust optimization models, which are highly effective in managing uncertainty, though they tend to be computationally intensive.

Rodrigues and Agra [68] examined berth allocation and quay crane scheduling while accounting for vessel arrival time uncertainty. A robust mixed integer program with two stages is used in their methodology. To tackle this problem, an exact decomposition algorithm is proposed, consisting of a master problem and a separation problem. Additionally, new scenario reduction and warm start techniques to improve the algorithm’s performance for large instances are introduced.

Chargui et al. [32] introduced a unique method to address scheduling and allocation issues in container terminals. The objective of the project is to minimize the total departure times for all serviced vessels by using a mixed-integer linear programming model. Additionally, the authors proposed an efficient technique for determining a lower bound and recommend the use of a heuristic alongside variable neighborhood search (VNS) to address practical scenarios involving large-scale data. The algorithm’s performance surpassed that of a commercial solver when dealing with real-time datasets, prompting the adoption of the recommended solutions into a decision-support system for a multinational company managing a container terminal.

In their study, Malekahmadi et al. [60] proposed an integer programming model and a meta-heuristic approach called the random topology particle swarm optimization algorithm to tackle extensive cases. The results indicate that the random topology PSO outperforms existing techniques in terms of computational time and accuracy. The proposed methodology also considers safe crane distance when assigning and scheduling tasks to reduce the overall waiting time for vessels.

In their work, Türkoğulları et al. [75] developed a new mixed-integer linear program to solve the integration of berth allocation, quay crane assignment, and scheduling simultaneously. The quay crane scheduling problem was described as a minimum-cost network flow problem permitting arc crossing. The study proposed a branch-and-bound technique to create a crane schedule devoid of any arc crossings. In addition, it was discovered that the branch-and-bound approach outperforms the shortest-path approach when it comes to handling the quay crane scheduling problem.

BAP, QCAP, and QCSP Model performance evaluation

The evaluation metrics utilized in the literature for the selected papers addressing the berth allocation, quay crane assignment, and scheduling problem primarily include solution quality, computational time, CPU usage, fitness value, algorithm convergence speed, deviation under uncertainty, and Pareto front quality for multi-objective optimization. Idris and Zainuddin [45] and Zhong et al. [93] employed Pareto front quality to assess the performance of their multi-objective optimization approaches. Heuristic and metaheuristic models, such as genetic algorithms, assessed their convergence rates to determine how quickly the models approached an optimal solution, as noted by Ren et al. [66] and Olteanu et al. [63].

4. Discussion

Limitation/Current Issues

Section 3 illustrates that substantial progress has been made between 2015 and 2024 in developing various models for BAP, QCAP, and QCSP. However, a critical analysis of the current status uncovers limitations and ongoing challenges related to the models designed for optimizing BAP, QCAP, and QCSP. The issues present in discrete problems mirror those found in continuous problems. Below are the listed challenges:

1.

Uncertainty handling

Several studies fall short in adequately addressing uncertainties, including weather variations, mechanical failures, schedule adjustments, vessel congestion, equipment malfunctions, and fluctuations in cargo volume [33,68,81]. While Xiang et al. [81] examine BAP, QCAP, and QCSP under uncertainty, they lack robust techniques to effectively manage uncertainties such as adverse weather conditions and disruptions.

2.

Limited sustainability impact and multi-objective optimizations

Few studies consider multi-objective optimization models for the BAP, QCAP, and QCSP. Many studies tend to overlook environmental, economic, and social factors like emissions, carbon tax, energy, and safety, focusing instead solely on efficiency constraints. Chargui et al. [32], Alnaqbi et al. [25], and El-Boghdadly et al. [39] concentrate exclusively on efficiencies, such as minimizing service time, handling time, and costs, without considering the impact on sustainability. Many models do not effectively integrate green infrastructure, such as shore power and renewable energy, nor do they adequately balance economic and environmental objectives.

3.

Narrow global coverage

Among the selected articles for review, none addressed berth allocation problems within the sub-Saharan African region, indicating a significant research gap in this area. Most studies tend to focus on China, as illustrated in Figure 3.

4.

Real-world data and real-time adaptability

The majority of the studies utilized simulated data to model BAP, QCAP, and QCSP. Very few studies employed real-world data, while others combined both real-world and simulated data. This reliance on simulation has resulted in a lack of benchmark data, raising concerns about the generalizability of the models. Despite advances in adaptive scheduling, many solutions still lack full real-time implementation due to reliance on predefined scenarios or delayed data updates. This limitation hinders responsiveness to sudden disruptions, which is a practical challenge in port management.

5.

Integration

During the review period, most papers focused on the integration of the BAP and QCAP. However, very few addressed the integration with the QCSP, yard operations, or storage. As an example, the study by Al-Refaie and Abedalqader [26], Woo et al. [78], and Chargui et al. [33] did not take into account the integration of the BAP with other related issues in the terminal, such as the yard and container stacking.

Future direction

Based on the current state of BAP, QCAP, and QCSP discussed in Section 3 and the identified issues outlined above, it can be concluded that future efforts should focus on enhancing BAP, QCAP, and QCSP by incorporating the following improvements in subsequent research:

1.

Innovative mechanism to handle uncertainties

Future research should emphasize the introduction of new algorithms, such as reinforcement learning, stochastic models, robust optimization techniques, and hybrid algorithms that combine metaheuristics with exact methods. These advancements aim to tackle uncertainties, disruptions, and sustainability objectives while also accommodating large-scale terminal operations.

2.

Inclusion of Policy trade-off

The majority of the paper did not address policy trade-offs; instead, most studies concentrated on efficiency objectives, such as minimizing completion time, waiting time, and service time, without considering conflicting objectives and regulatory requirements. Future research should incorporate objectives that focus on efficiency while also addressing other important factors, such as sustainability, safety, costs, and other unspecified considerations.

3.

Multi-objective optimization and integration with other port operations

Most current studies aim to minimize total turnaround time. However, in real-world scenarios, various objectives often come into play, such as minimizing waiting time, reducing energy consumption, and maximizing revenue. Future research should concentrate on developing multi-objective optimization models for the BAP that can accommodate these diverse objectives. One possible direction is to extend the model to consider multiple quays instead of a single quay.

4.

Incorporating machine learning and AI techniques

Machine learning and AI techniques, including deep learning and reinforcement learning, have shown promising results in addressing various optimization problems. Future research could aim to apply these techniques to BAP, QCAP, and QCSP to improve the performance and efficiency of algorithms. Rodrigues and Agra [68] emphasize the potential of machine learning approaches, such as neural networks, to enhance predictions of vessel arrival times. Overall, advancing the applications of machine learning could significantly improve berth allocation algorithms. Research should aim to enhance automation in crane scheduling and berth allocation by utilizing the Internet of Things (IoT), artificial intelligence (AI), and machine learning for automated guided vehicles (AGVs). This approach aligns with the increasing trend toward the development of intelligent ports and smart terminal management.

5.

Real-world Data

Future research could focus on developing algorithms that adapt to real-time data and can be tested in real-world scenarios. This would enable dynamic adjustments to berth allocation and crane assignment plans. Additionally, the study could be expanded to include actual data from port terminals to validate the models and enhance their practical application.

5. Conclusions

This review evaluates the advancements related to BAP, QCAP, and QCSP from 2015 to 2024. The key conclusions drawn from the findings can be summarized as follows:

During the period from 2015 to 2024, there has been a predominant use of continuous and dynamic BAP approaches, followed by continuous and stochastic methods. This trend reflects a shift among researchers from discrete to continuous models, with an increasing prevalence of stochastic arrival patterns. Most studies reviewed originate from China, with no papers addressing Sub-Saharan Africa among the collected works, which indicates a limitation in global coverage. However, much of the existing research fails to consider uncertain events, sustainable impacts, and policy trade-offs. Sustainability factors, including environmental, economic variability, and social considerations, have also emerged as significant areas of focus in studies from 2022. In terms of uncertainties, the application of robust optimization and stochastic techniques remains limited when addressing issues such as weather conditions, variable arrival times, equipment failures, and maintenance concerns, primarily due to the high computational demands associated with these robust optimization methods.

There is a notable emergence of innovation in crane handling, such as the implementation of double-cycling mode cranes, adaptive crane scheduling, and automated crane scheduling. Additionally, there is an increased focus on integrating green infrastructure within port operations. This includes the incorporation of renewable energy, shore power, and tidal optimization, all aimed at supporting broader sustainability objectives. The rising application of machine learning and reinforcement learning is enabling adaptive, real-time solutions, particularly for disruption recovery and dynamic scheduling, in line with advancements in modern smart port technologies.

A notable gap exists in the comparison of optimization models, particularly with genetic algorithms, which are the most widely utilized metaheuristic models. Advancements have been made in employing hybrid and enhanced algorithms, as well as introducing innovative techniques such as reinforcement learning. Furthermore, only a limited number of studies have addressed multi-objective optimization to enhance the performance of BAP, QCAP, and QCSP by incorporating objectives like energy consumption, penalty time, and labor efficiency, rather than concentrating solely on total turnaround time. However, most studies have not accounted for policy trade-offs.

Data availability remains a considerable concern, as most studies rely on randomly generated datasets. Observing results that utilize real-world data from ports would be particularly valuable, especially as we move toward real-time applications. Many researchers are shifting toward simultaneous integration rather than employing a two-stage integration approach or treating BAP, QCAP, and QCSP as isolated issues. It would be advantageous to integrate both the water side and the land side to achieve comprehensive optimization of the entire port system.

Future research should focus on innovative algorithms like reinforcement learning and robust optimization to handle uncertainties and sustainability goals in large-scale terminal operations. It is essential to incorporate policy trade-offs that balance efficiency with sustainability, safety, and costs. Additionally, developing multi-objective optimization models that consider various factors such as waiting time and energy consumption while integrating landside operations, is crucial. Leveraging machine learning and AI can enhance algorithm performance and predictive capabilities. Lastly, creating algorithms that adapt to real-time data and utilizing actual port data for validation will improve the practical application of these models.