Black-box simulation-based optimization for berth allocation in multipurpose ports: a case study in Colombia Open Access

For centuries, ports have played a pivotal role in global trade, serving as gateways for the exchange of goods and people between nations and driving the world economy. With the advancement of globalization, their significance has only grown, facilitating the seamless flow of commodities and services across borders. The importance of efficient port management has been highlighted in recent studies. For instance, the growing complexities of global supply chains have necessitated advanced decision-support systems for port operations (Elmi et al., 2023). These systems are essential to ensure competitiveness and adaptability in the rapidly evolving maritime sector. Moreover, the need for robust sustainability practices in port operations has gained prominence, as ports are increasingly scrutinized for their environmental impact (Li et al., 2023).

Consequently, effective port management has become indispensable, demanding specialized knowledge and expertise in overseeing cargo processing, vessel traffic management, security, and infrastructure maintenance. Proficient port management can result in enhanced efficiency, cost reduction, increased revenue, and a steadfast commitment to safety and environmental conservation.

One significant challenge in port management is the Berth Allocation Problem (BAP). This involves assigning berths to ships to maximize port efficiency and minimize scheduling costs. The BAP is complex due to constraints like berth capacity, time restrictions, specific vessel needs, safety considerations, and resource limitations. Solving the BAP is essential for improving port efficiency, reducing operational expenses, and enhancing service quality in the logistics and maritime sectors.

Simulation-based optimization (SBO) is a method that uses simulation models to evaluate different decision-making strategies in complex systems. The goal is to find the best combination of parameters to optimize a specific performance metric. SBO is particularly useful for systems too complex for analytical (Fowler et al., 2023) solutions and those with stochastic or dynamic features. It employs methods like metaheuristics, reinforcement learning, and stochastic approximation to iteratively test configurations and find optimal solutions. A subset of SBO is black-box simulation-based optimization, where the internal workings of the simulation model are inaccessible or too complex to understand. These models are treated as “black boxes” that produce outputs based on inputs, without revealing their inner workings. This is common with proprietary models or highly intricate simulations. Black-box optimization relies on iterative testing and feedback methods, such as evolutionary algorithms or surrogate models, to approximate input-output relationships and guide the search for optimal solutions (Gosavi, 2005).

To evaluate port performance, crucial metrics such as cargo volume, vessel calls, container throughput, and turnaround time come into play. These metrics offer invaluable insights into productivity and efficiency and allow for benchmarking a port’s performance against its peers. Notably, in 2020, the Port of Shanghai handled over 43 million TEUs, solidifying its status as the world’s largest container port and showcasing its exceptional ability to manage high cargo volumes efficiently (Shi et al., 2020), Overall, ports remain indispensable to the global economy, and effective port management is vital to ensure the uninterrupted flow of goods and services. Solving the Berth Allocation Problem emerges as a key strategy to enhance port efficiency, reduce costs, and elevate service quality.

Colombia, strategically located for global trade, holds immense potential for the study of berth allocation, particularly in its Caribbean coast ports. The nation’s logistics performance has made significant strides, improving its global ranking from 94th in 2016 to 58th out of 160 nations in the 2018 Logistics Performance Index. Additionally, the Colombian government has invested substantially in port infrastructure, particularly along the Caribbean coast. Ports like Barranquilla and Cartagena, among the largest on the Caribbean coast, have seen significant capacity and efficiency enhancements. The Port of Cartagena, serving as a crucial trade route between the United States and South America, presents an ideal location for the development and testing of simulation-based optimization strategies. Colombia’s burgeoning logistics performance, substantial port infrastructure investments, and strategic Caribbean coast location collectively establish a promising avenue for addressing the Berth Allocation Problem.

Port terminals are the linchpin of global trade, facilitating the transportation and storage of goods. The efficient operation of these terminals is paramount for the seamless flow of goods and the integrity of global supply chains. However, port terminals confront a multitude of challenges spanning from environmental concerns to operational inefficiencies. Categorizing these challenges is essential for devising effective solutions. Research in port terminal management typically falls into three broad categories. The first, and most extensively studied, revolves around environmental challenges, including efforts to reduce carbon emissions, enhance water treatment, and optimize fuel usage (Zhen et al., 2019). The second and third categories center on operational and strategic planning within port operations. Operational planning concerns the effective management of existing resources and the optimization of workflows, while strategic planning delves into administrative management and future projections (Burns, 2018). These categories are intertwined with overarching themes like Industry 4.0 and Artificial Intelligence, which increasingly influence operational processes. This literature review specifically delves into the third category, focusing on the strategic planning of port operations (De la Peña Zarzuelo et al., 2020).

The performance and efficiency of port terminals and operational processes are a constant focus for improvement. Various authors, such as (Lim et al., 2019), emphasize the importance of integrating environmental management, social responsibility, and operational efficiency as pillars for port sustainability and performance. Multipurpose ports, given their irregularity and variability in cargo types, operation times, and characteristics, pose one of the most complex management challenges. Nonetheless, researchers have dedicated years to identifying optimal management approaches (Agerschou et al., 1983; Keceli, 2016; Yu et al., 2023).

The Berth Allocation Problem (BAP), a well-established issue in maritime logistics, involves the allocation of berths to incoming ships to maximize port efficiency. This problem is exceptionally challenging due to its multi-faceted nature, making it complex to assign berths to arriving ships based on factors such as arrival time, vessel size, and cargo requirements. Early discussions on the BAP can be traced back to De Weille and Ray (1974), who pondered how far a port should extend its berths to accommodate sporadic ship arrivals. Over time, the BAP has been the subject of extensive research proposals, ranging from heuristic algorithms to mathematical programming techniques. Lai and Shih (1992) introduced the term “berth allocation problem” and developed a heuristic algorithm motivated by the goal of optimizing terminal usage in Hong Kong’s HIT terminal. Brown et al. (1994) employed an integer programming model for vessel-to-berth assignments, considering real-world constraints, discrete berthing positions, and vessel berth movements. As technology advanced, researchers leveraged more sophisticated methods such as tabu search and machine learning, along with optimization techniques. Brown et al. (1994) and Gkolias (2007) specifically highlighted the use of these techniques in solving the BAP. Simulation, over time, emerged as one of the most popular methods for precise strategic and operational planning, aided by its accessibility and shorter response times (Dragović et al., 2016). While ports handle various types of cargo, containerization has gained prominence, with Bierwirth and Meisel (2010) gathering pertinent research on this subject.

Recent advancements in BAP highlight its evolving complexity and significance. Tang et al. (2022) propose resilience-focused strategies to mitigate disruptions in berth allocation, enhancing adaptability in dynamic port environments. Mnasri and Alrashidi (2021) introduce a dynamic, discrete BAP model that extends ship arrival flexibility, facilitating more robust temporal planning. In multi-port scenarios, Martin-Iradi et al. (2022) employ cooperative game theory and exact optimization to minimize total vessel time in port, including waiting and handling costs. Guo et al. (2023) build on this approach with column generation methods, emphasizing inter-terminal cooperation to improve operational efficiency. Bouzekri et al. (2023) expand the BAP framework by integrating ship stability considerations and conveyor routing in bulk ports. Sustainability has also become a critical focus; Jiang et al. (2022) explored the integration of carbon emission constraints into berth allocation and vessel scheduling models, aligning port operations with environmental objectives. From a strategic perspective, Ursavas (2022) investigates priority-based berth allocation policies, highlighting agreements between shipping lines and port authorities to enhance operational decision-making. Methodologically, Wawrzyniak et al. (2020) examined efficient algorithms for large-scale BAPs, optimizing mooring and cargo handling costs to support scalable solutions. These contributions collectively advance the state of the art in port operation management.

In recent years, black-box simulation-based optimization (SBO) has found extensive applications in maritime logistics, particularly in optimizing operations within ports and container terminals. Studies from 2020 onwards have demonstrated the effectiveness of these methodologies in addressing complex logistical challenges. For instance, Tao et al. (2023) tackled the issue of internal truck sharing at multi-terminal container ports by combining a genetic algorithm with a rolling-horizon-based simulation model to minimize overflow workload and transfer costs. Similarly, Evangelista et al. (2022) employed a particle swarm optimization approach to make operational and tactical decisions for a logistics network comprising a port and a dry port, integrating simulation models for performance evaluation. Layeb et al. (2018) addressed scheduling problems and stochastic service network design in multimodal freight transportation by utilizing OptQuest’s meta-heuristic algorithms in conjunction with Arena-based simulation models to optimize performance. These cases exemplify the growing trend of leveraging black-box SBO techniques to enhance decision-making processes and operational efficiencies in maritime logistics.

Port terminals, particularly multipurpose terminals, confront unique operating issues due to the variety of cargo kinds, vessel characteristics, and variable demand. However, there is a significant deficit in research that focuses on the optimization of berth allocation for multipurpose ports in Latin America, particularly in areas such as Colombia’s Magdalena River. While much of the existing literature focuses on container ports or specific operational limitations, there is a scarcity of research that incorporates these issues into a unified optimization framework for multipurpose terminals in such regions. This study aims to address the existing research gap by exploring the Berth Allocation Problem (BAP) within the context of multipurpose terminals in Latin America. The objective is to provide new insights into effective port management under the complex operational conditions characteristic of this environment. In recent years, the export landscape in Latin America has shifted towards the transportation of liquid goods, such as crude oil and its derivatives, in addition to bulk commodities like coal, copper, gold, and agricultural products such as soybeans and corn (ECLAC, n.d.). This transformation has prompted the development of specialized port infrastructure tailored to accommodate diverse cargo types, underscoring the increasing significance of berth allocation and optimization research across different port environments.

This study focuses on a case in Colombia, a developing South American country known for exporting fuel products like oil and coal, as well as agricultural goods such as fruits, vegetables, and flowers (e bananas, avocados, and roses). On the other hand, the country’s imports consist of technologically advanced products such as wheat and cereals and industrial items like rods and sheets. Colombia’s major ports are concentrated in three main regions, strategically located to access both the Pacific Ocean and the Caribbean Sea.

The primary movement occurs in Buenaventura, in the southern part along the Pacific Ocean, which mainly handles containerized cargo. Conversely, the ports in the northern regions, along the Caribbean Sea, are focused on the mining sector, particularly coal. Meanwhile, ports in between, such as Cartagena and Barranquilla (the focus of this study), are equipped with multipurpose terminals capable of handling various types of cargo, including cruise ships, vehicles, and other commodities. A study conducted by Vega et al. (2019) provides more insights into Colombia’s port dynamics.

The Magdalena River plays a crucial role in Colombia’s transportation network, linking the country’s interior to the coast. However, the river’s navigability is a significant concern due to its natural shallowness and susceptibility to sediment buildup. This poses a challenge for large ships to navigate securely, compounded by riverbank erosion.

In 2022, the average depth of the Magdalena River in the port of Barranquilla was only 6.2 m, falling short of the 9-m minimum depth requirement for large ships. Consequently, Colombia’s economy has suffered, hampering the movement of goods via the river. In response, the Colombian government approved a $100 million dredging plan for the Magdalena River to address this issue, with the aim of increasing the river’s average depth to eight meters within five years. The plan involves dredging 15 million cubic meters of sediment, equivalent to almost 1.5 million trucks' worth of sediment (Knight, n.d.; Solano et al., 2023).

Over the next five years, the dredging operation will be continuous, targeting regions with significant sediment accumulation, especially near ports and bends in the river. Simultaneously, the government has launched a program to restore riverbanks, including the construction of containment barriers, tree planting, and reforestation efforts. The Colombian government anticipates that these projects will significantly benefit the country’s economy. The expected increase in the river’s depth is projected to enhance its transit capacity by 40%, potentially leading to cost reductions of up to $200 million per year in transportation. Additionally, a deeper river will make shipping safer, potentially attracting more investment in the maritime and port sectors, thereby creating new job opportunities and promoting overall development (Altamar et al., 2025).

The successful execution of the Magdalena River dredging and riverbank restoration plan relies heavily on the effective implementation of the proposed measures. It is imperative for the Colombian government to ensure sustainable and efficient dredging practices, along with the effectiveness of the riverbank restoration program. Successfully executing this strategy could elevate the Magdalena River’s significance as a crucial transit route for Colombia, ultimately contributing to the country’s economic growth. The increase in depth will accommodate larger ships, leading to increased port capacity, reduced congestion, and improved berth allocation efficiency (Inter-American Development Bank, n.d.).

Specifically, the deeper river will facilitate:

• (1)

  Meeting the growing demand for cargo.

• (2)

  Reducing transportation costs.

• (3)

  Attracting new investments in ports.

The specific terminal under consideration is a multipurpose facility located near the Magdalena River, currently operating four berths dedicated to multipurpose operations. While some of these berths specialize in handling specific types of merchandise, the terminal is capable of accommodating various types of vessels, including barges, general cargo ships, container carriers, liquids, and bulk carriers of different sizes. The analysis indicates an average turnaround time of approximately 35 h between ship arrivals, with an average unloading time of 50 h. Although certain berths specialize in specific cargo types, they can handle others as well. While the terminal has existing criteria for prioritizing berth assignments, the varying types of arriving vessels and changing arrival trends necessitate a review and potential update of these criteria for optimal management.

Multipurpose terminals, equipped with specialized handling machinery like cranes, forklifts, and conveyor belts, play a vital role in handling a diverse range of cargoes, including containers, break bulk cargo, and bulk cargo. Upon vessel arrival at the terminal, cargo is unloaded using specialized equipment and then moved to designated storage yards or warehouses for temporary storage. The storage method depends on the cargo type; containers, for instance, can be stacked in container yards, while general cargo may be stored in warehouses or open storage yards, and bulk cargo can be stockpiled in designated areas.

Managing various cargo types presents one of the most significant challenges for multipurpose terminals. Containers, break bulk cargo, and bulk cargo each come with their specific requirements, including different lifting techniques, stowage methods, and storage conditions. Coordinating and optimizing the flow of these distinct cargo types can be complex, demanding effective planning, coordination, and efficient utilization of handling equipment. Moreover, ensuring effective coordination among different stakeholders, including vessel operators, cargo owners, port authorities, and terminal operators, through robust communication and collaboration, is essential for ensuring seamless operations at multipurpose terminals. Well-coordinated cargo movement and vessel scheduling are critical for avoiding delays and optimizing terminal throughput.

Our research tackles this challenge by employing a three-phase methodology, with Simulation-based Optimization (SBO) playing a central role, as shown in Figure 1.

In the initial phase, the focus is on standardizing and characterizing the essential operations related to ship berthing assignment. The emphasis here is on capturing intricate details to closely replicate real-world dynamics. This phase involves a meticulous characterization of the entire journey of a ship, starting from the moment it has the option to arrive at the terminal until its departure from the berth. Extensive data are collected during this phase, encompassing aspects such as time intervals between ship arrivals based on cargo type, options for berthing based on cargo characteristics, considerations related to ship length, loading times, waiting times, and associated cargo handling costs. The goal is to amass comprehensive information that enables the accurate emulation of the complex decision-making process involved in berthing assignment, accounting for all relevant factors affecting the process.

The second phase involves the creation of a dynamic simulation model designed to accurately replicate the current operational reality of the terminal. Utilizing the data and insights acquired during the first phase, this simulation model is comprehensive and dynamic. It is equipped to make informed decisions regarding whether a ship should be granted entry into the terminal. Factors such as arrival time, dock type, departure details, and other pertinent information are considered in determining the optimal berthing assignments for incoming ships. Additionally, the simulation model incorporates mechanisms to assess costs, timeframes, and the impact of existing constraints within the terminal. This enables a thorough analysis of the berthing assignment process, accounting for various restrictions and limitations that influence terminal operations. The objective is to create a robust simulation model capable of faithfully emulating the intricate decision-making process and operational dynamics of the terminal, facilitating precise evaluation and optimization of berthing assignments.

The third and final phase leverages Simulation-based Optimization (SBO) techniques to identify optimal decisions that can lead to tangible enhancements in the terminal’s performance indicators. Here, the simulation model developed in phase 2 plays a central role.

SBO is an iterative process that combines simulation modeling with optimization algorithms. We will utilize the data gathered in phase 1 to define the objective function within the SBO framework. This function will quantify the desired outcome of berth allocation, such as maximizing revenue or minimizing waiting times. Additionally, the constraints faced by the terminal, such as berth availability and equipment limitations, will be incorporated as constraints within the optimization algorithms. The SBO process works by iteratively running the simulation model with different berth allocation strategies. The performance of each strategy is evaluated based on the objective function defined earlier. This evaluation allows the optimization algorithms to identify promising allocation strategies and refine them further. The process continues iteratively until an optimal berth allocation strategy is identified, one that delivers significant improvements compared to traditional methods.

These three phases collectively contribute to the development of a comprehensive and robust solution that faithfully replicates the real-world dynamics of the berthing assignment process. They enable precise evaluation and optimization of terminal operations. The research findings and recommendations hold the potential to drive tangible improvements in the terminal’s performance indicators, ultimately contributing to enhanced operational efficiency and decision-making in the berthing assignment process.

In port management, grasping key components such as entities, attributes, variables, and resources is essential for developing an effective simulation model to tackle complex challenges like the Berth Allocation Problem (BAP). The duration and priority of cargo unloading are influenced by cargo type, demonstrating the model’s ability to adapt to various cargo categories. Simulation-based optimization (SBO) offers a flexible approach to accommodate factors such as cargo types, berth availability, and operational constraints. Unlike rigid mathematical models, simulation captures real-world interactions more realistically, making it more suitable for exploring different scenarios and improving operational efficiency without being constrained by fixed mathematical formulations.

SBO integrates simulation with optimization techniques to address the BAP. It begins by modeling the port’s operational dynamics, including entities such as ships, with attributes like cargo type, quantity, arrival time, and unloading duration. The simulation reflects the random nature of ship arrivals and cargo types, mirroring real-world variability. SBO uses this model to simulate port operations, assessing berth availability and waiting times while applying optimization algorithms. By dynamically adjusting parameters and constraints, SBO explores various “what-if” scenarios, ensuring that the optimization remains grounded in realistic conditions. This approach prioritizes ships based on cargo type, assigns berths accordingly, and calculates profitability and efficiency metrics. The result is a flexible, adaptive model that enhances port efficiency and profitability by minimizing ship waiting times and optimizing berth utilization, ultimately leading to better operational decision-making.

The simulation model structure includes:

Entities:

• (1)

  Ships: These represent the vessels arriving at the port with various types of cargo.

Attributes of Entities:

• (1)

  Cargo Type: Each entity (ship) is associated with a cargo type, which categorizes the type of goods it is carrying. This attribute plays a crucial role in determining berth allocation priorities and unloading durations, as different cargo types may require varying handling procedures.

• (2)

  Cargo Quantity: This attribute specifies the quantity or volume of cargo carried by each ship, influencing the time required for unloading.

• (3)

  Ship Length: Ship length is an attribute that affects berth compatibility and assignment feasibility. Longer ships may require specific berths to accommodate them.

• (4)

  Arrival Time: Arrival time indicates when each ship reaches the port, contributing to the simulation’s temporal dynamics.

• (5)

  Priority: Priority is assigned based on cargo type, determining the order in which ships are allocated berths. Priority levels are influenced by cargo type and potentially other factors, such as historical data or operational criteria.

• (6)

  Duration of Unloading: The duration of unloading is the time it takes to offload cargo from a ship. This attribute varies depending on the cargo type and other factors.

• (7)

  Profit: Profit is associated with each cargo type and is used to calculate revenue based on the unloaded cargo. Different cargo types may yield varying profits, adding complexity to the model.

Variables:

• (1)

  Berth Priorities (X_ij): These are positive integer variables representing the priority of assigning cargo type i to berth j.

• (2)

  Cargo Type Priorities (Ai): These are positive integer variables indicating the priority of each cargo type i.

Resources:

• (1)

  Berths: These represent the available docking areas in the port for ships to unload their cargo.

State Variables:

• (1)

  Time in Waiting List: This records the time a ship spends on the waiting list before being assigned a berth.

• (2)

  Maximum Waiting Time: A time threshold (24 h) is established, after which a ship is considered diverted if it is not assigned a berth.

Results and Metrics:

• (1)

  Revenues: These represent the economic benefits derived from berth allocation and cargo unloading.

• (2)

  Average Berth Utilization: Indicates the average utilization of berths over a specified time period.

• (3)

  Average Number of Ships in the Waiting List: Shows the average number of ships waiting before being assigned a berth.

• (4)

  Percentage of Time Berths Occupied: Indicates the percentage of time berths are utilized compared to the total available time.

These components are integral to the simulation model developed to address the Berth Allocation Problem (BAP) within the context of port management. The model utilizes these entities, attributes, variables, and resources to replicate the operational dynamics of the port and evaluate different scenarios with the goal of maximizing revenue and improving port operational efficiency.

In order to create a model capable of simulating the assignment of docks, we needed to analyze three key aspects. The first aspect focused on gathering crucial information and making decisions before assessing dock availability. This involved examining various aspects of arriving ships, such as the time intervals between cargo arrivals and cargo types, and utilizing goodness-of-fit tests to establish the probability distribution for each cargo type. Additionally, historical data related to each ship was studied to facilitate informed decision-making. Once ship attributes, including length, cargo type, and quantity, were defined, a priority ranking was assigned.

The primary objective of this study is to assess the established policies of the company, particularly those concerning the allocation of berths within a port. To gain a deeper understanding of this allocation process, let’s delve into the steps involved in the simulation model.

In Figure 2, we present an example of a priority table used for dock assignment. This table dictates the sequence in which ships are allocated berths based on their cargo type. Ships carrying cargo with higher priority receive preference. For instance, if berth one is occupied, the ship would be assigned to berth two. If berth two is also unavailable, the ship would proceed to berth three, and so forth. It’s worth noting that certain berths may be inaccessible to specific types of ships, as demonstrated in Figure 2. For instance, if the cargo type is 5, the ship would initially check the availability of berth 3. If berth 3 is occupied, the ship would then proceed to berth 5, and if that berth is also occupied, it would ultimately move to berth 1. The creation of this priority table involved the expertise of the company’s port administrators, who leveraged their knowledge and industry insights to craft it. These three pivotal factors—cargo type, berth availability, and priority sequence—are essential for ensuring the efficient allocation of berths, which is critical for the seamless operation of the port.

The subsequent phase entails verifying berth availability, assessing whether there is a waiting period, and recording the duration of each ship’s stay. Upon the arrival of ships at the port, each ship is assigned berth priorities. The system checks the availability of the priority berth first. If it’s unavailable, the model considers berths with lower priorities. In cases where no available berths are found, the ship may be placed on hold. However, if any of the priority berths become available, the ship may be required to utilize the hold option. The time spent in the hold is logged in the model’s statistics, enabling us to calculate the characteristics of ships that go unserved due to berth shortages. If a ship remains unserved for 24 h, it is assumed that it will never be serviced at that terminal, and it is subsequently removed from the model. Currently, if any of the ship’s optional berths become available, the ship can enter immediately, depending on the priorities of other waiting ships. As shown in Figure 3, this explains the method for attending ships by the type of cargo.

In the final phase, once a ship has been allocated a berth, it is considered occupied. If another vessel requests the same berth during the scheduled operation time of the first ship, the operation of the first ship may be delayed. During this phase, all performance indicators measuring operational efficiency are updated. Upon completing its operation, the ship is released from the system, and the dock becomes available for assignment to another ship. By following these steps and decision-making processes within the simulation model, the company can optimize berth allocation and enhance overall port operational efficiency.

The performance indicators of the system are linked to the efficient utilization of resources, including the average utilization of berths and the revenue generated from their use. Among these indicators, the parameter considered for the optimization model’s objective function is the average berth utilization and the revenue derived from berth usage. The variables associated with this model represent the adjustments in berth priorities for each cargo type, according to the table in Figure 2.

The mathematical structure of the model is as follows:

Sets:

• (1)

  i: Cargo types {1, 2, 3, …, I}

• (2)

  j: Company berths {1, 2, 3, …, J}

Parameters:

• (1)

  P_i: Profits per hour unloading the cargo type i

• (2)

  M_i,: Maximum number of options of berth to select

• (3)

  U_ij: Binary variable that represents it is possible to unload cargo type i at berth j, 0 otherwise.

Variables:

• (1)

  X_ij: Positive integer variable that represents the priority associated with discharge of the cargo type i at berth j

• (2)

  A_i: Positive integer variable that represents the priority for each cargo type i to be processed.

• (3)

  OP: Over hall Profit

Constraints:

• (1)

  $1≤Xij≤Mij∀i,j$  This constraint restricts the value of $Xij$ to be between 1 and $Mij$ (inclusive), ensuring that the priority variable $Xij$ is within the valid range of berth options for cargo type i.

• (2)

  $1≤Ai≤I∀i$   This constraint restricts the value of Ai to be between 1 and I (inclusive), ensuring that the priority variable Ai is within the valid range of cargo types.

Objective:

Where

• (1)

  OP Expected profit, calculated as the average outcome from multiple simulation runs

• (2)

  $E[.]$ Expected value, representing the mean profit across all simulation iterations.

• (3)

  $Tij(ξ)$ Actual handling time for cargo type i at berth j, which depends on:

  • •

    Decision integer variables Ai and $Xij$

  • •

    Random conditions (ξ), such as

    • –

      Ship arrival times.

    • –

      Loading/unloading durations.

    • –

      Berth availability.

In this optimization model, it is essential to highlight the uncertainties involved, such as the time between ship arrivals and the quantity and type of ships docking at the terminal. These factors are subject to randomness and follow specific distributions identified through data analysis, meaning that each simulation run produces different results. The model’s constraints are designed to assign integer values as priorities to various cargo types, ensuring that these priorities fluctuate according to cargo type and berth availability. The model is classified as a Mixed-Integer Nonlinear Programming (MINLP) model because it combines integer variables, such as berth and cargo priorities (Ai and X_ij), with continuous elements like operational times and costs. Additionally, the objective function (overall profit) is not explicitly defined but emerges from a black-box simulation, which introduces nonlinearity and depends on complex interactions such as vessel arrivals, cargo variability, and berth constraints. These characteristics, along with the use of heuristic optimization methods to handle the model’s stochastic and non-differentiable nature, solidify its classification as MINLP. We must consider that the data used to construct the simulation model is derived from operational estimates, and as the simulation process evolves, the system often exhibits nonlinear behavior.

This model is considered a black-box simulation-based optimization because the internal mechanisms generating the results are not transparent or easily interpretable. For example, the overall profit (OP) in the mathematical model can only be quantified through simulation runs, relying on random variables like arrival time, cargo type, operation time, and dock availability based on previously serviced ships and evaluated priorities The complexity of interactions within the simulation, including random ship arrivals and dynamic operational conditions, means the precise calculation processes remain hidden. Instead, the optimization relies on output data from numerous simulation runs, treated as a “black box,” to refine and improve berth allocation strategies.

The simulation was conducted using a system capable of accurately modeling port operations, including berth availability verification, assessment of waiting times, and tracking the duration of each ship’s stay. The optimization process within this simulation utilized advanced techniques such as evolutionary algorithms, tabu search, and scatter search. Furthermore, the simulation-based optimization technique permitted the study of many “what-if” possibilities by dynamically altering parameters and restrictions, ensuring that the optimization remained practical and operationally grounded. This approach helped identify practically feasible berth allocation strategies, improving port efficiency and profitability by effectively managing berth priorities and minimizing waiting times.

The initial model run utilized the company’s existing operational parameters, considering the established priorities. A thorough analysis was conducted through 60 replications, and the results were averaged. This rigorous validation process confirmed that the model’s output indicators exhibited no statistically significant differences when compared to the actual performance indicators observed in the operational environment, achieving a high confidence level of 95%.

Subsequently, optimization models were deployed to automate the discovery of optimal parameters for maximizing the company’s revenue. A systematic approach was adopted to dynamically adjust the model’s parameters and evaluate their performance across various scenarios, treating each parameter adjustment as an independent scenario. To enhance the model’s versatility, it was customized to accommodate scenarios with varying replication numbers, facilitating the computation of scenario-specific average performance values. The replication count was automatically calibrated to ensure minimal variance at a confidence level of 95%, with a minimum requirement of 30 replications.

A comprehensive analysis was conducted, considering integer variables, to explore numerous potential scenarios in the study. Approximately 1,60,000 scenarios were evaluated as part of the detailed assessment. To assist in decision-making, a real-time optimization model was employed, evaluating around 300 different scenarios starting from the current state. The search terminated when it was determined that achieving a solution surpassing the maximum expected gain was not feasible. The result obtained, as presented in Figure 4, showcased a significant revenue improvement of 6.22% compared to the current outcome. When examining the results graphically, it was noted that less than 9% of the scenarios exhibited inferior performance relative to the current result, partly attributable to the optimization evolutionary algorithm integrated into the software.

Analyzing the best-performing scenario generated by the model, certain patterns emerge that lead to an intriguing outcome. For instance, it involves assigning a specific berth for a certain type of ship carrying a particular product to the terminal, giving all priority scenarios the same berth, regardless of the possibility of being serviced in other berths. Similarly, it includes multiple ships with different types of cargo having equal priority, even though they could be assigned differently. When evaluating the improvement itself and comparing it to the current scenario and the worst-case scenario recorded by the model, it becomes evident how changes in priorities enhance the company’s performance indicators.

Figure 5 presents a comparison of the number of diverted ships across three scenarios: the current configuration, the worst-case scenario, and the optimized scenario. The optimized scenario demonstrates a nearly 50% reduction in ship diversions compared to the current configuration, with the median value of diverted ships declining from X to Y. This improvement can be attributed to enhanced berth allocation priorities that minimize waiting times. Notably, the distribution of results in the optimized scenario shows less variance, reflecting more consistent performance under diverse conditions It’s important to note that all scenarios were conducted using the same replications, and in this case, the deviation among the three scenarios is nearly identical, adding further validity to the results. Additionally, this model considers the possibility of ships waiting a maximum of 24 h from their arrival to be attended. If the waiting time exceeds 24 h, the ship is considered diverted. Figure 6 displays a graph depicting the average number of ships in the queue by delay times for each scenario.

Figure 6 illustrates a trade-off between reducing the total number of diverted ships and the average delay times for those serviced. The optimized scenario increases the frequency of 12–24-h delays from 5 to over 20 annually, a consequence of prioritizing berth assignment to minimize overall ship diversions. These results highlight the model’s focus on maximizing terminal capacity while accommodating a greater number of vessels, which, despite increasing some delays, maintains operational efficiency.

Finally, Figure 7 illustrates how berth utilization is distributed among different scenarios. It considers how the scenario yielding better profit allocates berth usage differently among the available berths in the terminal. It also emphasizes that Berth 5 has distinct characteristics due to its length and the type of cargo it can accommodate. However, the improved scenario proposes increased utilization of berths 3, 4, and 5. It’s also worth noting that not all berths are in similar conditions. One distinguishing factor is the distance to the exit, with some berths being closer than others. Although this has a minor impact on the speed at which a ship is serviced, it was not considered in this research if all berths had similar conditions.

To validate the proposed simulation-based optimization model, we conducted a comparison between the simulation results and actual port data. The model was run through 60 replications under different scenarios, and its outcomes were evaluated for consistency and accuracy. The results showed that the Best Scenario, which achieved a 6.22% increase in port revenue, aligned closely with real-world operational data, confirming the model’s effectiveness. Statistical analysis of the output revealed no significant differences between the optimized results and the terminal’s actual performance, with a confidence level of 95%. Although the model demonstrated strong predictive capabilities, further validation could include incorporating additional dynamic factors, such as fluctuating ship arrival rates and unexpected berth availability. Overall, the validation confirms that the optimization model is reliable and can be effectively applied to real-world port management scenarios.

This study addressed the Berth Allocation Problem (BAP), a critical challenge in port management, through a black-box simulation-based optimization (SBO) framework. By combining dynamic simulation modeling with advanced optimization techniques, the methodology tackles the complexities of berth allocation under real-world constraints. Focusing a multipurpose terminal on the Magdalena River in Colombia provides unique insights into the interplay between local operational challenges and advanced optimization methods.

The SBO approach proved particularly effective for this study due to its ability to model the inherent stochasticity of port operations. By iteratively simulating real-world conditions, such as vessel arrivals, cargo variability, and berth constraints, the model captured the complex dynamics of berth allocation. This iterative framework enabled the integration of metaheuristics, such as evolutionary algorithms, to identify optimal allocation strategies that maximize revenue and minimize waiting times. The results underscored the adaptability of SBO, which successfully navigated the trade-offs between resource utilization and service quality, demonstrating a 6.22% increase in revenue and a 50% reduction in ship diversions under the optimized scenario. A deeper analysis of the methodology highlighted the strengths of the three-phase approach adopted in this study: (1) the detailed characterization of vessel operations, (2) the development of a dynamic simulation model tailored to local conditions, and (3) the application of optimization algorithms within the SBO framework. This structured process not only replicated the operational complexities of the terminal but also provided actionable insights for enhancing berth allocation policies.

The local context of the Magdalena River added layers of complexity to the analysis. The river’s navigability challenges, such as sedimentation and depth limitations, directly influenced berth operations and vessel scheduling. The case study demonstrated how infrastructure constraints and government interventions, such as dredging projects, shape the operational landscape of Colombian ports. The study’s findings underscore the importance of aligning local infrastructural realities with advanced optimization techniques to achieve sustainable operational improvements.

This research makes a significant theoretical and practical contribution to port management. By leveraging SBO to model and optimize berth allocation, it demonstrates the transformative potential of data-driven methodologies in enhancing operational efficiency, safety, and environmental sustainability. The tailored methodology ensures relevance to the unique challenges of multipurpose terminals, particularly in developing economies like Colombia, where ports handle diverse cargo types under variable conditions.

However, certain limitations remain. First, the model assumes homogeneity across berths, overlooking berth-specific attributes such as proximity to operational areas. Second, static prioritization rules, while effective in simulation, may not fully capture real-time decision-making needs. Lastly, the study’s focus on local optimization does not consider broader supply chain dynamics or regional interdependencies.

Future research should address these limitations by

• (1)

  Integrating berth-specific characteristics and dynamic prioritization algorithms.

• (2)

  Exploring the integration of berth allocation with other port operations, such as crane scheduling and yard management.

• (3)

  Expanding the scope to include multi-port systems, enabling cooperative strategies and regional optimization.

In conclusion, this study highlights the applicability of black-box SBO in solving complex logistical challenges, particularly within the constraints of developing economies. By focusing on the Magdalena River’s unique context, the research bridges the gap between advanced optimization methods and practical, context-sensitive solutions. These findings reinforce the importance of integrating local insights with cutting-edge methodologies, providing a robust framework for future studies and practical applications in port management.