Process Optimization in Sea Ports: Integrating Sustainability and Efficiency Through a Novel Mathematical Model

Abstract

Ports are essential nodes in global trade, linking maritime and land transport. As maritime logistics increasingly drive global supply chains, optimizing port operations has become vital for enhancing economic efficiency and environmental sustainability. This study presents a Mixed Integer Linear Programming (MILP) model to address inefficiencies in berth allocation and stevedoring processes at the Port of Leixões, Portugal. By integrating real operational data, the model reduces vessel waiting times by 47.56% (from 8.1 to 4.2 h) and operational delays by 37.39% (from 11.5 to 7.2 h). These optimizations also result in a 41.85% reduction in greenhouse gas emissions per ship, aligning with global emissions regulations and promoting sustainable port management. The model’s innovations include real-time data integration and a holistic resource allocation approach to mitigate congestion and inefficiencies. Key findings demonstrate the model’s potential to streamline operations and minimize environmental impacts. These advancements align economic efficiency with environmental sustainability, addressing global emissions regulations. However, the study acknowledges limitations, such as excluding unpredictable factors like weather conditions and equipment failures. Future research should explore dynamic variables, such as weather conditions and mechanical failures, and expand the model’s applicability to other seaports worldwide.

1. Introduction

Optimizing stevedoring processes in maritime container parks is essential to guarantee the efficiency and safety of freight transport operations at sea. Stevedoring is the process of loading and unloading goods in ships [1]. It is one of the most critical steps in maritime transport as it directly affects the transport capacity of ships, the safety of goods, and operational efficiency [2]. Optimizing stevedoring processes involves choosing the best arrangement of containers on the ship, considering characteristics and international safety regulations to maximize cargo capacity and minimize the risk of damage or loss [3]. Furthermore, it is important to ensure that stevedoring processes are carried out efficiently and safely, using appropriate equipment and procedures and ensuring the quality of operations.

The global maritime industry is facing increasing challenges due to the increased demand for international shipping and tighter environmental restrictions [4]. Seaports, as the focal point of this industry, are crucial to the efficiency of the global supply chain. However, traditional seaport operations often face issues such as congestion, significant delays, and high pollutant emissions. These problems not only increase operating costs but also negatively impact the environment and the effectiveness of global trade [5].

The efficiency of operations in container parks is an increasingly relevant topic in the context of supply chain management, particularly in the optimization of transport and logistics, considering the growing importance of this equipment for international trade and the transport of goods [6]. Container parks are facilities where freight transport containers are stored, sorted, inspected, repaired, and reconditioned before being sent to other destinations. Efficiency in these spaces is crucial as it allows the optimization of available space and speed in carrying out operations, increases loading and unloading capacities, and reduces waiting times. Furthermore, the good management of container parks can contribute to reducing operational costs and improving the quality of services provided [7].

However, efficiency in these spaces still needs to improve due to the complexity of operations, limitations of space and available resources, and the need to ensure the safety and integrity of stored goods. It is also important to mention the logistical and movement restrictions due to the physical and regulatory limits of the seaport, as well as the handling capacity of the equipment. To achieve high efficiency, it is essential that container parks are well-planned and managed, considering their characteristics and user needs. Advanced technologies, such as warehouse management and container tracking systems, can greatly help [8].

In the context of increasing global trade and environmental concerns, optimizing the efficiency of seaport operations has become a critical challenge. The need to reduce ship waiting times and improve resource allocation while minimizing environmental impacts requires innovative approaches that integrate logistical and sustainability factors. This study focuses on developing a Mixed Integer Linear Programming (MILP) model aimed at addressing two key operational issues in seaports: the Berth Allocation Problem (BAP) and the Quay Crane Scheduling Problem (QCSP). By optimizing these components, the model seeks to enhance overall operational efficiency, reducing delays and improving the management of available resources, ultimately contributing to more competitive and sustainable port operations.

The main objective of this research is to demonstrate how the application of the MILP model can significantly reduce both the waiting and operational times of ships in port, thus optimizing the flow of maritime traffic. In addition, the study examines the environmental impact of these optimizations, with a focus on reducing greenhouse gas emissions and promoting sustainable management practices in ports. The results of this study provide practical and theoretical insights, offering a foundation for future research and practical applications in global ports. This dual focus on operational efficiency and environmental sustainability establishes the relevance and necessity of the proposed model within the modern logistics and maritime industries.

This article is organized into six sections. The current section introduces the context. Section 2 presents a literature review. Section 3 presents the developed MILP optimization model to solve the problem. Section 4 presents the model validation and results comparison using real data from the seaport of Leixões to verify its effectiveness in improving port operations. Section 5 presents a discussion concerning the study, and the main outcomes are presented in Section 6, along with a proposal for future research.

2. Literature Review

Efficient port operations are essential for the global supply chain as they directly influence trade flow and economic performance. Over the past decades, researchers have extensively studied optimization models for addressing operational challenges in ports, particularly focusing on the Berth Allocation Problem (BAP) and Quay Crane Assignment Problem (QCAP). These models aim to minimize ship turnaround times and operational costs while maximizing resource utilization. However, the increasing complexity of port operations, driven by growing trade volumes and environmental regulations, has highlighted the need for more integrated and sustainable optimization approaches. This section explores the existing literature on port operation challenges, including standalone and combined optimization models, and emphasizes the emerging focus on sustainability and its integration with operational efficiency.

2.1. Berth Allocation and Quay Crane Assignment Problem

The scheduling of cranes to ships is a crucial factor for efficient terminal operations [9,10]. The first articles published in the literature concerning the QCAP date back to 1989, when Daganzo [11] presented the problem of scheduling dock cranes in seaports with the main objective of reducing the time it takes for a ship that is docking at a pier to receive an assigned berth and providing a specific number of cranes to be assigned to each ship, thus minimizing the total waiting cost for all ships. This author divided the problem into two situations: a static situation and a dynamic one. To resolve the static situation, the author formulated a Mixed Integer Programming (MIP) problem that obtained results very close to the optimal, but this only worked with a small number of ships when considering that the pier could only accommodate a fixed number of ships and only one crane could operate on each compartment. The dynamic problem was studied similarly, but the ships arriving at the docks were also considered. However, if the ships started to line up, the wording became more complex and was not displayed. The study by Ref. [12] transforms the BAP into a restricted version of a two-dimensional packing problem and briefly represents the problem in a graph model. With a special focus on limiting crane arms from crossings, Zhu and Lim [13] approached the situation by minimizing the work completion time and formulating an integer programming (IP) model limited by the size of the instances used. For medium-sized cases, the results obtained through the Branch-and-Bound (BnB) algorithm surpass the performance of the exact IP mathematical model in CPLEX v22.1.0 Software. They use the Simulated Annealing (SA) algorithm for larger instances that originate optimal or very good solutions.

Abdel-Basset et al. [14] formulated the BAP and QCAP problems through a Genetic Algorithm (GA), which does need to consider the relationship between the operating time and the number of cranes. Rodrigues and Agra [9] provide a mathematical model for solving BACAP as well as two heuristic methods that allow solving the problem for instances of practical size (between 20 and 40 ships), the Squeaky Wheel Optimization heuristic (SWO) and the Tabu Search (TS) heuristic. The results show that the use of these heuristics provides good quality solutions with a shorter processing time, below the limited 10 h.

Since the berthing time is directly related to the number of cranes operating simultaneously on the ship, Liang et al. [15] presented a study that considers BAP and QCAP simultaneously, with the objective of minimizing the sum of operating times, waiting times, and the delay time for each ship. The authors chose to use the GA, combining it with heuristics, thus creating a hybridization concept capable of obtaining better results. Golias et al. [16] addressed the problem of scheduling berthing based on the arrival of ships within pre-established limits compared to the fixed times traditionally used for ship arrivals. The problem was formulated as a Mixed Integer Programming (MIP) optimization problem and a GA-based heuristic was used to bring the result closer to reality.

The results show that this scheduling policy, in which the terminal operator decides the arrival time of ships, can benefit both the shipping company and the terminal operator. Bierwirth and Meisel [17] prepared a review of all investigations carried out to date, which proved that determining and classifying the maneuvering time of ships is a difficult but crucial requisite for planning coastal operations and for the efficiency of supply chains. The authors also concluded that the growing interest in integrating models has been gaining popularity, as they are capable of providing balanced solutions in relation to the sub-problems developed. Zhang et al. [18] understand that the problem is formulated as a MIP programming model, and a sub-gradient optimization algorithm is constructed for its resolution, which takes into account the coverage ranges of quay cranes and only allows limited adjustments during the loading and unloading. This study highlights the algorithm’s effectiveness when dealing with real problems and suggests incorporating uncertain factors in the model to bring solutions closer to reality.

2.2. Combined Optimization Models

Tasoglu and Yildiz [19] developed a MIP model that addresses the issues of simultaneously assigning and scheduling cranes (QCAP and QCSP), taking into account uncertainties and delays related to the arrival of the ship, as well as the maneuvering time of the pier cranes. In this model, ships arrive dynamically and randomly, and cranes can operate at different berths even before finishing the previously assigned ship. Rodrigues and Agra [9] presented a search procedure based on a GA that was applied to assist this method, which proactively generates both mooring and crane schedules. In the article [20], the BAP is modeled as a parallel programming problem with inclusive processing set constraints, where the assignment of ships to berths is limited by the water depth and tidal conditions, where the objective is to minimize the total weighted duration of ship servicing, considering the unit waiting on the costs of the ship.

In contrast, Chang et al. [21] first developed a dynamic objective programming model based on the rolling-horizon technique for formulating BAP and QCAP. Next, a Hybrid Parallel Genetic Algorithm (HPGA) was developed, combining a genetic algorithm with a heuristic, resulting in practical and effective solutions to the problems in question. In the article proposed by Barros et al. [22], the authors proposed a model of Linear Programming (LP) to solve BAP in bulk cargo seaports with stock-level restrictions. As in the results presented, it was difficult to make an accurate prediction of the time required, and the SA algorithm was proposed as a valid alternative, producing robust results with a satisfactory processing time in 90% of the tests (instances) performed. Regarding the literature published in recent years, Raa et al. [23] once again simultaneously addressed BAP and QCAP and took into account new characteristics and restrictions, such as ship priorities, preferred berthing locations, and maneuvering time. The authors used MILP and solved it in time intervals using the Rolling Horizon Heuristic (RHH) since solving this problem is fundamental and applied several times a day in a real seaport. One year later, Yang et al. [24] presented a model that promotes the interaction between BAP and QCAP with the development of an Evolutionary Algorithm (EA) consisting of three loops. The first loops solve the BAP and QCAP subproblems based on a GA, and a third external loop involves the first two with the objective of finding an approximate solution based on these. In Song et al. [25], the BAP and QCSP problems are formulated through a bi-level programming (BLP) approach; BAP is considered the higher-level problem, and QCSP is the lower-level problem. A GA algorithm and BnB method are subsequently applied to minimize the total waiting time and improve the solution.

In the article proposed by Chen et al. [26], a new method called the Combinatorial Benders Cuts (CBC) algorithm was developed, which revealed that the proposed approach is more efficient than the Branch-and-Cut (BnC) algorithm incorporated in CPLEX Software. According to the approach by Kaveshgar et al. [27], the authors chose to develop a GA, where they used a new approach to define chromosomes (the solution representation) to reduce the number of decision variables and introduced new procedures to calculate lower and upper limits more accurately for the decision variables. Experimental results show that the model provides faster solutions for larger problems than the best-known solutions.

In contrast to other studies that seek to develop more complex search algorithms, in the article by Chen et al. [26], the objective is to propose a more concise mathematical formulation for the QCSP based on a unidirectional grouping, which can be simply solved by a conventional solver. Regarding the positioning conditions of quay cranes and non-crossing restrictions, in the article by Diabat and Theodorou [28], a GA is developed to solve the QCASP, and it is concluded that this algorithm is a successful approach, producing near-optimal solutions for small and medium-sized problems, with extremely time-efficient performances for all problem sizes.

The study by Ref. [29] approaches the problem through two distinct formulations in order to deal with the uncertainty of ship operating times. From a stochastic point of view, a stochastic programming formulation capable of coping with different probability distributions for operating time deviations is presented. Furthermore, a robust formulation is proposed that applies to situations with limited information about probability distributions. The results obtained in this study show that the robust formulation can quickly get an almost optimal solution for the stochastic model, also presenting the advantage of limiting the worst possible results.

Furthermore, the instances demonstrated the efficiency and effectiveness of the proposed meta-heuristic algorithms compared to the CPLEX Software solver. On the other hand, Diabat and Theodorou [28] developed a model for the mathematical formulation of the QCASP problem and a Lagrangian Relaxation (RL) heuristic to solve the problem. This analysis concludes that problems with small and medium instances are solved quickly; however, for larger problems, their performance could be better in terms of limits and computational time increases due to the extreme increase in variables. Thus, this heuristic was considered a successful approach to QCASP, especially when applied to small and medium problem instances. Hu et al. [30] address the resolution of the BQCAP, considering the ship’s fuel consumption and emissions, where the inverse proportionality between the seaport’s operating cost and fuel consumption is highlighted. Furthermore, the impact of the optimal number of cranes on ship emissions while moored at the pier is analyzed.

In 2015, Bierwirth and Meisel [31] continued Bierwirth and Meisel’s [17] literature review that covered research up to 2009. In this article, the authors use MILP to determine the sequence of ship unloading operations a crane will perform to minimize operating times. Despite considering restrictions that increase the complexity of the problem, such as bidirectional cranes and the ability to move between docks before completing the allocated tasks, the resulting solution is a simple and accurate solution since the objective function aims to minimize the relative difference between the container load and the number of docks.

Türkoǧullari et al. [32] proposed a model for integrating the BAP, QCAP, and QCSP problems. First, they formulated a MILP problem that provides an exact solution for the positions and docking times of ships, along with the scheduling of cranes for the ships’ staying at the docks. They then used the BnB method in a decomposition sequence where the problem is solved in time intervals in a master problem. Then, a search for the optimal solution for scheduling the cranes was carried out by resolving subproblems. With this approach, they also concluded that the subproblems generated are NP-complete since they are conceived through reducing an NP-hard problem. In the same year, Liu et al. [33], in their article, divided the BAP and QCSP problems into two phases in order to build a model that deals with the constant disturbances that occur in a container terminal, such as, for example, the breakdown of cranes in the middle of scheduling formulated previously. Thus, in the first phase, they used the MIP-based Relax-and-Fix Algorithm (MRF) method to obtain optimal solutions for the BAP. In the second phase, they applied a dynamic programming technique to solve the QCSP, unlike Ref. [34], who, in the same year, functionally integrated the BAP, QCAP, and QCSP problems into a feedback loop structure. In this way, it was possible to consider time-varying dock assignments, bringing the solution closer to reality. In Ref. [35], a new MIP formulation of the QCSP was proposed that considered ship stability constraints. Due to the complexity of the problem, a GA was proposed, and the computational results validate the model in small problems and highlight the performance of the developed model.

From the perspective of Mauri et al. [36], the authors found a solution for the BAP through the formulation of an Adaptive Large Neighborhood Search (ALNS) heuristic where ships are represented by rectangles to be allocated in a space–time area, respecting temporal and overlapping restrictions. The results indicate that the ALNS heuristic can generate high-quality solutions, and compared with other competing heuristics, they demonstrate statistically significant improvements. In the article written by Qin et al. [37], the authors showed that Constraint Programming (CP) tends to be superior to IP models in several cases and that the creation of a hybrid CP/IP procedure is a simple way to increase the optimality of the solution. Then, Agra and Oliveira [38] presented a mathematical model based on integrating BAP and QCASP problems. First, a model is built based on the Relative Position Formulation (RPF) for the dock assignment parameters. Then, a new model is introduced to avoid big-M restrictions included in the RPF and results from a discretization of time and space variables. It also features an RHH heuristic and a BnC approach to help solve large instance problems. However, this methodology needs to consider some important aspects, such as the storage capacity of containers and the uncertainties associated with certain events. Zhang et al. [39] extend the formulation of the QCSP, taking into account ship stability constraints. The authors propose a Bicriteria Evolutionary Algorithm (BiEA) to find promising solutions and introduce the sliding-window heuristic to correct schedules that do not meet scheduling constraints.

Regarding Abou Kasm et al. [40], the study addressed the QCSP of the following year by introducing minimum safety distance restrictions between cranes and no crossing for a single ship, which they called QCSP-NS. By formulating the problem in IP, the authors propose a two-step approach, starting by introducing a technique called Container Division and Combination (CDC) and then reformulating the problem as a partition problem and running the Branch and Price (BnP) algorithm to find the closest solution to the optimum. Considering the integration of the three problems [28], the authors present a new mathematical formulation that takes into account different operational policies of quay cranes, such as allowing or not allowing pre-emption of quay tasks in QCSP and the static or dynamic allocation of cranes in QCAP. Commercial software was used to resolve this new formulation.

In the article performed by Kizilay et al. [41], a series of CP models are proposed to solve the Integrated Port Container Terminal problem (IPCTP). The IPCTP consists of integrating the issues previously described in the review, the allocation and scheduling of quay cranes, their location, and the scheduling of trucks that move containers within the seaport terminal. In this follow-up, Malekahmadi et al. [42] presented an integer programming model for scheduling integrated problems, BAPQCASP, considering the impact of water depth on ship berthing. Results in a reasonably short time led the authors to develop a Random-Topology Particle Swarm Optimization (RTPSO) algorithm. The following year, Skaf et al. [43] compared the results obtained through an Exact Enumerative Algorithm (EEA) and a GA with a real case in solving the QCYTSP (Quay Crane Yard Truck Scheduling Problem), considering a single quay crane and several transport trucks to the park, based on the MILP formulation. They developed a model for the integrated BAPQCASP problem where they considered that ship arrival times may suffer from delays and uncertainties. That same year, Rodrigues and Agra [9] proposed a two-phase mathematical model, where berthing decisions are first-phase decisions and crane assignments are second-phase decisions. The authors follow a decomposition algorithm to divide the problem into a main problem and a separation problem, and to help solve large instances, and the decomposition algorithm is combined with the Rolling Horizon algorithm.

Finally, the study by Ref. [44] addresses a robust model tool for scheduling maritime container terminal operations, taking into account possible ship arrival scenarios and crane availability. Several solution methods are proposed and compared, including Multi-Objective-Simulated Annealing (MOSA) and Pareto-Simulated Annealing (PSA).

2.3. Sustainability in Port Operations

Sustainability in port operations has become a pressing issue, driven by the need to balance economic growth with environmental responsibility. Ports are key nodes in global trade but are also significant contributors to greenhouse gas emissions and marine pollution [45]. Measures such as reducing ship idling times, optimizing berth allocations, and adopting cleaner energy sources can significantly lower the carbon footprint of maritime activities [46]. Technological advancements, such as automated port equipment and digital twins for real-time monitoring, are enabling a shift towards greener operations. However, these efforts require alignment with international environmental regulations, such as the International Maritime Organization’s (IMO) decarbonization goals, which aim to reduce shipping emissions by 50% by 2050. Sustainable practices not only mitigate environmental damage but also improve operational efficiency, creating a win-win scenario for both economic and ecological objectives [47].

In addition to reducing emissions, ports must address broader sustainability challenges, including waste management, water quality, and the preservation of marine ecosystems [48]. Frequent port operations can lead to the discharge of harmful chemicals and wastewater into nearby waters, adversely affecting biodiversity and local fishing industries. To counter these effects, ports are increasingly investing in eco-friendly initiatives, such as wastewater treatment facilities and shore-side electricity, which allow ships to turn off their engines while docked [49]. Furthermore, incorporating sustainability into port planning ensures long-term resilience against climate change impacts, such as rising sea levels and extreme weather events [50].

2.4. Summary and Research Gap Observations

The reviewed literature demonstrates significant advancements in optimizing port operations, focusing on BAP and QCSP and their integration. These studies employ various methods, including MILP, genetic algorithms, and heuristic approaches, achieving improvements in operational efficiency and resource utilization. Recent research highlights the increasing importance of incorporating sustainability metrics, such as emission reductions and eco-friendly practices, into port management strategies. However, most models focus on isolated aspects of operations or make static assumptions that overlook real-world complexities.

Despite the significant advances in optimization models applied to port operations, none of the existing approaches fully meet the specific needs of the Port of Leixões to integrate sustainability with operational efficiency in an optimized manner. Most of the current models either focus on a single aspect of port operations or make assumptions that overlook the real-world variability of daily operations, such as equipment failures, demand fluctuations, and adverse environmental conditions. Additionally, few models comprehensively address the environmental impact of operations, which is increasingly critical due to stricter emissions regulations. Therefore, it is necessary to develop a new model that not only optimizes operations but also integrates environmental sustainability, vessel waiting times, and operational processes in a more holistic manner, considering more realistic and dynamic variables. This analysis highlights the gap between existing models and modern demands.

This study builds on these gaps by introducing an optimization model that holistically integrates sustainability metrics with operational efficiency. Unlike prior approaches that typically address single objectives—either optimizing operational performance or reducing environmental impact—this model bridges the gap by addressing both aspects simultaneously. By incorporating vessel waiting times, emissions, and operational variability, the proposed framework aligns with the specific needs of modern seaports, such as the Port of Leixões. This dual focus not only addresses the weaknesses in existing models but also contributes a practical and robust solution for balancing economic and environmental objectives in port operations.

3. Optimization Model Proposal

Addressing the main problematic situation, this section presents a mathematical model to optimize the assignment of container-maneuvering tasks to ships and cranes. The model aims to minimize the total waiting time of ships in seaports, considering multiple operational and logistical constraints. Decisions on the allocation of ships to cradles, the order in which operations are executed, the use of cranes, the division of cranes between ships, and the calculation of maneuvering times are carefully handled using interlocking decision variables and constraints. The following assumptions were adopted:

■

Each ship is only allocated to one berth;

■

Each cradle has a fixed number of cranes;

■

The number of cranes operating simultaneously on a ship influences the ship’s operating time;

■

Cranes may not leave ships until they have completed their tasks nor enter a ship that has already begun operations;

■

Only two cranes can operate on a ship at the same time.

• Consider:

■

N: Number of Ships

■

B: Number of Cribs

■

U: Number of Cranes

• Sets:

■

I = {1,2,…, N}

■

J = {1,2,…, B}

■

G = {1.2,…, U}

• Parameters:

■

COut_i: Number of containers to be unloaded from the Ship i, ∀i ∈ I.

■

CIn_i: Number of containers to be loaded from the Ship i, ∀i ∈ I.

■

TimeByCrane_j: Time required, in minutes, to maneuver a container in the Cradle j, ∀i ∈ J.

■

Arrival_i: Ship’s Arrival Date i, ∀i ∈ I.

■

Length_i: Ship Length i, ∀i ∈ I.

■

LengthCradle_j: Berth Length j, ∀j ∈ J.

■

TimeCradle_i: Average berthing/uncradling time, in minutes, of the ship i, ∀i ∈ I.

The model incorporates a set of parameters that capture the critical elements of port operations and provide the necessary inputs for the formulation of constraints and the objective function. These parameters reflect the characteristics of the ships, berths, and operational activities, ensuring that the model aligns with real-world port conditions. Specifically, the parameters include information about the number of containers to be loaded and unloaded for each ship, the time required for cranes, ship arrival dates, and the dimensions of both ships and berths. Additionally, average berthing and uncradling times are considered to account for the variability in operational efficiency across different vessels.

Decision variables:

■

Begin_i: Decision variables start date from Ship i, ∀i ∈ I.

■

Departure_i: Ship’s Departure Date i, from the port, ∀i ∈ I.

■

$X_{ij}=\left\{\begin{array}{c}1, if the ship is allocated in the berth j\\ 0,otherwise\end{array}, \right.\forall i\in I,\forall j\in J$.

■

${Ordering}_{i’j}=\left\{\begin{array}{c}1, if the ship is allocated after the shi{p}^{\prime }, in the berth j\\ 0,otherwise\end{array}, \right.\forall i \in I, \forall i^{\prime } \in I, \forall j \in J.$

■

Number of Ship Containers i, maneuvering in the berth j, ∀i ∈ I, ∀j ∈ J.

■

${Crane}_{gij}=\left\{\begin{array}{c}1, if the crane operates the ship i, in the berth j\\ 0, otherwise\end{array}, \right.\forall g \in G, \forall i \in I, \forall j \in J.$

■

Number of containers maneuverer by the crane g, in the ship i, in the berth j, ∀g ∈ G, ∀i ∈ I, ∀j ∈ J.

■

Time of operation of the ship i, by the berth j, ∀i ∈ I, ∀j ∈ J.

■

Number of the cranes operating simultaneously in the ship i, in the berth j, ∀i ∈ I, ∀j ∈ J.

The decision variables define the operational choices that need to be made within the optimization model to achieve the objectives. These variables capture the temporal, spatial, and resource allocation aspects of port operations, providing a comprehensive representation of the decision-making process. Specifically, the variables include the start and departure dates of each ship, which determine the scheduling and order of operations. Binary variables are used to indicate berth allocation for ships and the sequencing of their operations within the same berth. Additional variables account for the number of containers, the assignment of cranes to specific ships and berths, and the number of cranes operating simultaneously.

The mathematical model presented defines a complex optimization problem related to the management of seaport terminals. Then, a detailed analysis of each constraint will be conducted, assessing its relevance to the problem at hand.

Objective Function:

$$\mathrm{m}\mathrm{i}\mathrm{n}\sum _{i\in I}\left(Begi{n}_i-Arriva{l}_i\right)$$

(1)

Subject to:

$$\sum _{j\in J}{X}_{ij}=1, \forall i\in I$$

(2)

$$Orderin{g}_{ij}=X_{ij}, \forall i, i^{\prime }\in I,\forall j\in J$$

(3)

$$\sum _{i^{\prime }\in I}Orderin{g}_{ij}=1, \forall i\in I,\forall j\in J$$

(4)

$$\sum _{i^{\prime }\in I}Orderin{g}_{ij}\le 1, \forall i\in I,\forall j\in J$$

(5)

$$\begin{array}{c}\left(Begi{n}_i\le Begi{n}_{i^{\prime }}\right)\bigwedge \left({Departure}_i\ge {Departure}_{i^{\prime }}\bigwedge {Departure}_i\ge {Begin}_{i^{\prime }}\right)\vee \\ \begin{array}{c}\left(Begi{n}_i\le Begi{n}_{i^{\prime }}\right)\bigwedge \left({Departure}_i\ge {Departure}_{i^{\prime }}\bigwedge {Departure}_i\ge {Begin}_{i^{\prime }}\right)\Rightarrow \\ Lengt{h}_i+Lengt{h}_{i^{\prime }}\le LengthCradl{e}_j,\forall i,i^{\prime }\in I,\forall j\in J\end{array}\end{array}$$

(6)

Begin_i ≥ Arrival_i + TimeCrane_i, ∀i ∈ I

(7)

Departure_i ≥ Begin_i + T_ij + TimeCrane_i, ∀i ∈ I, ∀j ∈ J

(8)

Ordering_ij = 1 ⇒ Begin_i ≥ Begin, ∀i, i ∈ I, ∀j ∈ J

(9)

Y_ij = (COut_i + CIn_i) × X_ij, ∀_i ∈ I, ∀j ∈ J

(10)

$$\sum _{g\in G}{Crane}_{gij}\ge 1, \forall i\in I,\forall j\in J$$

(11)

$$\sum _{g\in G}{NG}_{gij}={ Y}_{ij}, \forall i\in I,\forall j\in J$$

(12)

$$S_{ij}=\sum _{g\in G}{Crane}_{gij}, \forall i\in I,\forall j\in J$$

(13)

T_i1 = max_g∈G NG_gij × TimeByCrane_j, ∀i ∈ I, ∀j ∈ J

(14)

Begin_i ≥ 0, ∀i ∈ I

(15)

Departure_i ≥ 0, ∀_i ∈ I

(16)

X_ij ∈ {0,1}, ∀i ∈ I, ∀j ∈ J

(17)

Ordering_ij ∈ {0,1}, ∀i ∈ I, ∀i^′ ∈ I, ∀j ∈ J

(18)

Y_ij ≥ 0, ∀i ∈ I, ∀j ∈ J

(19)

Crane_gij ∈ {0,1}, ∀g ∈ G, ∀i ∈ I, ∀j ∈ J

(20)

NG_gij ≥ 0, ∀g ∈ G, ∀i ∈ I, ∀j ∈ J

(21)

T_ij ≥ 0, ∀i ∈ I, ∀j ∈ J

(22)

S_ij ≥ 0, ∀i ∈ I, ∀j ∈ J

(23)

Equation (1) defines the objective function of the problem that aims to minimize the sum of waiting times between the arrival and start of operations for all ships, reflecting the intention to reduce delays in seaport operations. Constraints (2) and (3) ensure that each ship is allocated to a single berth, preventing the duplication of allocations, ensuring an equitable distribution of ships in the available berths and a relationship of precedence between the allocation of ships and berths, ensuring that the order of arrival of ships is respected, preventing newer ships from being allocated before those that arrived earlier. Constraint (4) ensures that each ship has only one berthing order, and Constraint (5) limits the number of orders from the first ship for each berth. Constraint (6) ensures that the lengths of ships with the same order, i.e., with simultaneous operations, do not exceed the total length of the berth in which they are allocated. Constraints (7) and (8) control the time of the start and departure of operations concerning the arrival and end of operations, respecting the temporal logic of seaport activities, and Constraint (9) establishes the relationship between the berthing order of ships and their start and departure times, ensuring the temporal order of operations.

Constraint (10) calculates the number of containers maneuvered based on the allocation of ships to berths, providing an essential metric for optimizing operations. Constraint (11) ensures the availability of cranes in berths, respecting the minimum need for cranes to operate ships. Constraint (12) controls the number of containers moved by cranes at each terminal, ensuring that operations are efficient and distributed among the best number of cranes. Constraint (13) determines the number of cranes to act simultaneously on ships in the north and south terminals, directly impacting the total operating time of each ship. Constraint (14) calculates the total maneuver time of containers in each terminal based on the number of containers maneuvered by the cranes and the operating time of the cranes in each terminal.

Constraint (15) ensures that the start time of operations (Begin_i) for each $shi{p}_i$ is non-negative. This restriction reflects the practical requirement that operations cannot commence before time zero, establishing a logical boundary for the scheduling process. Similarly, Constraint (16) ensures that the departure time (Departure_i) for each $shi{p}_i$ is also non-negative, aligning with the temporal structure of the optimization model.

Constraint (17) enforces that the variable X_ij, representing whether a $shi{p}_i$ is allocated to a $bert{h}_j$, is binary. This ensures that each ship is either assigned to a specific berth or not, preventing partial or invalid allocations. Constraint (18) also enforces a binary condition for the variable Ordering_ij, which defines the sequencing of operations for ships ii and i at berth j. This guarantees that the order of operations is unambiguously defined.

Constraint (19) requires that the number of containers maneuvered (Y_ij) for a $shi{p}_i$ at a $bert{h}_j$ is non-negative, reflecting the physical impossibility of handling a negative quantity of containers. Similarly, Constraint (20) ensures that the variable Crane_gij, representing whether crane gg operates on a $shi{p}_i$ at a $bert{h}_j$, is binary, ensuring clarity in crane allocation.

Constraint (21) establishes that the number of containers maneuvered by crane g on a $shi{p}_i$ at a $bert{h}_j$ (NG_gij) is non-negative, reinforcing operational realism. Constraint (22) ensures that the total operation time (T_ij) for a $shi{p}_i$ at a $bert{h}_j$ is non-negative, consistent with the time required for loading and unloading operations. Finally, Constraint (23) ensures that the number of cranes operating simultaneously on a $shi{p}_i$ at a $bert{h}_j$ (S_ij) is non-negative, aligning with the operational reality of crane usage in seaport logistics.

The mathematical model comprehensively and rigorously addresses the management of seaport operations, considering the arrival dates of ships and the number and characteristics of cranes available in the respective terminals, aiming to minimize waiting times and improve efficiency in seaport operations.

4. Case Study

Maritime transport is an important means of travel involving ships and other vessels to transport people, materials, and goods across rivers, lakes, seas, or oceans. This type of transport plays a vital role in the global economy, and, according to Ref. [51], in Portugal, this type of transport is responsible for 58.5% of all imported goods, accounting for a total of 34.7 million tons. Regarding exports, it is responsible for 50.0% of the total volume, a total of 17.9 million tons displaced, in 2021. Maritime transport is generally divided into two main categories: commercial transport and passenger transport. Commercial transport involves the movement of goods, such as raw materials and final products, while passenger transport refers to the transport of people, including tourists and migrants.

In this scenario, it is possible to list several advantages when it comes to the use of maritime transport, essentially in terms of sustainability, such as

■

Capacity: Ships can transport large amounts of cargo;

■

Cost-effective: Maritime transport is generally less expensive than other modes of transport, especially for long distances.

■

Fuel efficiency: Ships are more fuel efficient than planes and trucks, resulting in lower fuel costs per distance traveled;

■

Low polluting: Maritime transport produces greenhouse gases greenhouse, as road and air transport;

■

Flexibility: Ships can travel to a wide range of destinations.

However, despite its advantages, it also has some disadvantages:

■

Low Speed: Despite their efficiency, ships are slower than planes and trains, which makes them less suitable for time-sensitive cargo;

■

Dependence on port terminals: Maritime transport depends on port terminals for loading and unloading goods, which can influence shipping times;

■

The complexity of the supply: Maritime transport involves a complex chain, as it is dependent on a large number of factors, is conditioned by the different regulations of different countries, is susceptible to delays due to adverse weather factors, and involves a complex logistics and distribution system.

Maritime transport constitutes a vital part of the global transport system and plays a large role in the world economy [9]. This makes it a fundamental topic of discussion and analysis with a view to ensuring its safety and efficiency, contributing to sustainable development. This section will describe the detailed structure of the problem as well as the various parameters and considerations that will serve as the basis for the developed mathematical formulation to obtain optimized solutions.

The problem in question aims to optimize the allocation of ships to berths, distribute the work of maneuvering containers between the cranes assigned to each ship, and consider restrictions such as the ship’s length, crane availability, and space in the shipping terminals anchoring. This complex challenge is especially relevant to the Leixões Container Terminal (TLC) context, where operational efficiency and the correct use of resources are of utmost importance. Through a detailed mathematical formulation, it is possible to seek to find a solution that minimizes the total waiting time for ships at the seaport, i.e., the time elapsed from the arrival of the ship at the seaport until the moment the loading/unloading operations are initiated. Reducing this waiting interval is extremely important for seaport sustainability as ships wait.

They continue to consume fuel and emit greenhouse gases that harm the environment. Furthermore, it promotes more effective and fluid container handling, significantly benefiting port management and ships. This problem aligns with the constant search for sustainability and operational efficiency in contemporary seaports. As the maritime industry faces increasing demands for productivity and reduced environmental impact, optimizing stevedoring processes is emerging as a critical component in achieving these goals. Minimizing ship waiting times reduces pollutant emissions and improves the use of port resources, maximizing container handling capacity. Faced with this challenging scenario, developing and applying mathematical models and implementing efficient operational strategies contribute to more sustainable, economical, and environmentally friendly seaport management. In the specific context of the Leixões Container Terminal, which plays a vital role in regional and international logistics chains, optimizing stevedoring processes and reducing ship waiting times can boost not only the terminal’s competitiveness but also the economic and environmental vitality of the region in which it is located.

4.1. Description and Results Analysis

Developing a MILP model with resources from the software IBM ILOG CPLEX Software Optimization Studio Version 22.1.1 was necessary to carry out the validation and practical application of the model formulated earlier. Its tool is known for its effectiveness in solving complex optimization problems and is widely used in several areas, including logistics and port management. The MILP model will be used in the assessment. To implement the MILP, it is possible to simulate several hypothetical scenarios and carry out detailed analyses of resource allocations of seaport resources, considering specific environmental restrictions and conditions of the port in question.

The mathematical formulation, including the objective function and constraints, was structured and input into CPLEX through its interface, while MS EXCEL^® v16.0 software was used to manage and organize the input data and parameters. MS EXCEL^® spreadsheets were designed to store key data, such as ship arrival times, berth lengths, crane capacities, and operational requirements. This setup allowed for user-friendly data management and ensured flexibility in adapting the model to different problem instances. The CPLEX engine was then configured to read the MS EXCEL^® files, process the data, and solve the MILP model.

During the implementation, Excel served as a practical tool for preprocessing data and storing results, facilitating the interpretation and visualization of outputs. Decision variables, such as ship schedules, berth assignments, and crane usage, were exported back to MS EXCEL^® software for post-processing and analysis. The integration between MS EXCEL^® software and CPLEX was streamlined using built-in data import/export functionalities, ensuring accuracy and ease of use. By combining MS EXCEL^® software accessibility with the CPLEX computational engine, the model achieved optimal or near-optimal solutions while respecting all operational constraints.

Based on the solutions obtained for the model, it will be possible to compare them with real data from the seaport of Leixões, allowing an accurate assessment of efficient seaport operations about ship waiting times. This approach offers insights to improve port management, identify areas for improvement, and make decisions based on real data, contributing to a more effective and efficient functioning of the seaport.

All ships docked during January 2024 will be used as data at the Terminal of Containers from Leixões. These data encompass periods of ship arrival, the number of containers to load and download, the length of the ships, the operating times, and the terminal to which they were assigned, as shown in Figure 1.

Analyzing Figure 1, it is possible to observe dispersed values that may distort the conclusions. In this case, the existence of higher values, usually caused by unusual events, originates from bigger average in-docking times. As a result, ship numbers 4, 8, 14, 19, 29, 53, 60, and 105 will be removed, and a new average will be calculated, as illustrated in

Table 1. Excerpt of the data of the ships—TCL.

Ship At the	IN	OUT	LOA	Arrived	Start Oops	End Oops	Sailed	Terminal
1	233	28	200	01/01/2024 07:00	02/01/2024 08:24	02/01/2024 15:07	02/01/2024 17:35	South
2	92	128	139	01/01/2024 07:39	02/01/2024 08:34	02/01/2024 16:55	02/01/2024 18:06	South
3	60	205	135	01/01/2024 15:35	02/01/2024 08:26	02/01/2024 18:50	02/01/2024 22:38	North
4	0	171	135	01/01/2024 18:02	01/06/2024 08:45	01/06/2024 15:37	01/06/2024 18:15	North
103	780	473	211	01/31/2024 06:21	01/31/2024 08:28	02/01/2024 08:52	02/01/2024 11:27	South
104	59	59	210	01/31/2024 07:15	02/01/2024 01:25	02/01/2024 09:53	02/01/2024 10:42	North
105	130	297	200	01/31/2024 11:30	02/01/2024 14:51	02/01/2024 23:44	02/02/2024 00:55	South

and Figure 2.

This data filtering process allows for obtaining a more representative view of the weather Mooring Medium, eliminating atypical phenomena and highlighting consistent trends. In this form, the value of the time average in docking was adjusted for 360 min, which is equivalent to 6 h, representing a value frequently observed. It should be noted that this value includes the docking and undocking of the ship, which corresponds to the time from the ship’s entry at the seaport until the anchoring at the pier to which it was allocated. It is from the time of the unanchoring until the exit from the ship of seaport.

Table 2. Parameters TCL.

Parameters TCL
Number in Pier	2
Length Pier North (meters)	360
Length Pier South (meters)	540
Number in Cranes North	2
Number in Cranes South	4
Time per Container North (minutes/container)	2.5
Time per Container South (minutes/container)	2
Time Average in Docking It is Unberthing (minutes)	360

contains all the previously mentioned parameters, which will be based on the model MILP, for which it is possible to obtain results concerning the reality of the seaport in Leixões.

The introduction of information regarding the arrival times of ships, together with their length and parameters previously defined by the TCL, is made through the connection of a model developed by MS EXCEL^® software.

Once the calculations are executed, CPLEX Software determines the optimal solution. This solution is designed to minimize ship waiting times at the seaport of Leixões. These results are then meticulously transcribed to the same MS EXCEL^® file mentioned earlier. This process ensures that the results are not only easily comprehensible but also ready for in-depth analysis, instilling confidence in their accuracy.

Table 3. Terminal and crane results obtained for the first 10 Ships.

	Terminal		Cranes North			Cranes South
Ship Number	REAL	CPLEX SOFTWARE	1	2	1	2	3	4
1	South	South	-	-	-	-	130	131
2	South	North	110	110	-	-	-	-
3	North	South	-	-	133	132	-	-
4	North	North	171	-	-	-	-	-
5	North	South	-	-	-	-	80	81
6	South	South	-	-	-	-	130	129
7	North	North	79	79	-	-	-	-
8	South	South	-	-	186	185	-	-
9	South	North	176	-	-	-	-	-
10	North	South	-	-	-	-	273	273

presents the results, corresponding to the first ten ships to be awarded at the TCL, where it is possible to see the pier in which they were allocated. It is the number of containers maneuvered by cranes; they were associated with them. For example, the ship identified as number one was allocated to the South terminal. The cranes were assigned to cranes number three and number four, which maneuvered a total of 130 and 131 containers, respectively.

Table 4. Results of dates obtained for the 10 first Ships.

		Start Ops		Waiting Time (Hours)		End Ops		Sailed		Operation Time (min)
Ship Number	Arrived	REAL	CPLEX SOFTWARE	REAL	CPLEX SOFTWARE	REAL	CPLEX SOFTWARE	REAL	CPLEX SOFTWARE	REAL	CPLEX SOFTWARE
1	01/01/2024 07:00	02/01/2024 08:24	01/01/2024 10:00 am	25.4	3	02/01/2024 15:07	01/01/2024 14:22	02/01/2024 17:35	01/01/2024 18:35	403	262
2	01/01/2024 07:39	02/01/2024 08:34	01/01/2024 10:39	24.9	3	02/01/2024 16:55	01/01/2024 15:14	02/01/2024 18:06	01/01/2024 18:14	501	275
3	01/01/2024 15:35	02/01/2024 08:26	01/01/2024 18:35	16.9	3	02/01/2024 18:50	01/01/2024 23:01	02/01/2024 22:38	02/01/2024 02:01	624	266
4	01/01/2024 18:02	01/06/2024 08:45	01/01/2024 21:02	110.7	3	01/06/2024 15:37	02/01/2024 04:09	01/06/2024 18:15	02/01/2024 07:09	412	427.5
5	01/01/2024 22:15	02/01/2024 08:27	02/01/2024 02:01	10.2	3.8	02/01/2024 15:33	02/01/2024 04:43	02/01/2024 16:15	02/01/2024 07:43	426	162
6	02/01/2024 04:50	02/01/2024 22:50	02/01/2024 07:50	18	3	03/01/2024 10:41	02/01/2024 12:10	03/01/2024 11:40	02/01/2024 15:10	711	260
7	02/01/2024 10:15	02/01/2024 19:03	02/01/2024 13:15	8.8	3	03/01/2024 11:22	02/01/2024 16:32	03/01/2024 12:10	02/01/2024 19:32	979	197.5
8	02/01/2024 11:20	03/01/2024 08:40	02/01/2024 15:10	21.3	3.8	03/01/2024 19:49	02/01/2024 21:22	03/01/2024 22:46	03/01/2024 00:22	669	372
9	02/01/2024 14:55	02/01/2024 19:40	02/01/2024 19:32	4.7	4.6	03/01/2024 02:12	03/01/2024 02:52	03/01/2024 06:48	03/01/2024 05:52	392	440
10	02/01/2024 19:16	03/01/2024 13:40	03/01/2024 00:22	18.4	5.1	01/04/2024 16:03	03/01/2024 09:28	01/04/2024 17:25	03/01/2024 12:28	1583	546

presents the results regarding the operation’s start and end times as well as waiting times. It includes the date the ship set sail, which is the TCL, so there are as many data at the contextual level as in the simulation of CPLEX Software.

As can be concluded from the instance presented previously, the model demonstrated significant improvements compared to the real scenario. These improvements not only translated into substantial reductions in ship waiting times but also improved operational efficiency. However, to expand it and analyze all ships docked at TCL in January 2024, accounting for a total of 105 ships, it becomes possible to check if those results are consistent on one set of wider scenarios.

The CPLEX Software model can maintain or improve these gains with efficiency in a more realistic environment. It is also important to highlight that this analysis will allow for a better understanding of how seaport operations can be optimized regarding resource allocation and waiting times. It is efficient and operational, considering one sample more representative. It is crucial to determine the viability of the practical application of the CPLEX Software model in real-world situations where complexity and operational dynamics are inevitable.

Through all results, it was possible to build Figure 3 and Figure 4 with information regarding the waiting time and operating time of the ships allocated, respectively.

In a first analysis, it is possible to observe very high values that can distort the conclusions. These abnormal values can result from events such as breakdowns or the maintenance of cranes, accidents, or even registration errors in collecting data. In this way, ship number four will be removed from the waiting time analysis and ship numbers 73 and 96 from the operating time analysis, as exposed in Figure 5 and Figure 6.

By analyzing the graphs, it is possible to see that the developed mathematical model presented significant improvements regarding the real scenario, clearly highlighting that the implementation of CPLEX Software induced a positive impact on the ships’ waiting and operating times at the seaport of Leixões. The graphics reveal tendencies that indicate consistent reductions in waiting times throughout the period of analysis, reinforcing the model’s effectiveness in optimizing seaport operations. These visual representations provide a more accessible understanding and intuitive analysis of the results, complementing the quantitative analyses. In Figure 7, it is possible to clearly see the improvements resulting from the developed model of average waiting times and average operating times for each ship.

Thus, the results obtained by the implementation of the model in the selected case study are highly encouraging. Data analysis revealed a significant reduction in the average waiting time of ships at the seaport of Leixões, from 8.1 h to 4.2 h. These values represent a notable improvement of 47.56%, pointing out a more efficient allocation of resources and a substantial reduction in downtime. In addition, average operation times also showed a remarkable decrease, going from 11.5 h to 7.2 h, which leads to an improvement of 37.39%.

In this work, a careful analysis of real instances was conducted, differentiating it from the common approach using randomly generated instances. For example, in the articles by Han et al. [52] and Zhen [53], distributions are used to generate sets in ships. The type of approach adopted in this study allows for obtaining more relevant solutions applicable to the reality in question, serving as a comparison and enabling a more precise assessment of the practical implications of the results obtained and promoting decision-making.

However, the results obtained in instances simulated under ideal conditions, where no delays are considered, are a novel contribution to the field. Despite the undeniable improvement in the use of this model in optimization mathematics in the seaport industry context, unforeseen events such as crane failures, operator availability, and meteorological factors are not taken into account in the simulation. These factors, as discussed in the articles by Rodrigues and Agra [9] and Ursavas and Zhu [13], have the potential to substantially impact the performance of the developed model. Therefore, it is crucial to recognize that these results provide a general vision of the model’s performance under ideal conditions and that adaptation to these unforeseen events is a crucial consideration in the implementation of these solutions.

In the context of this study, an average of 25 min was spent analyzing each set of 10 optimized ships. This is in line with the results obtained in other works, such as those by Diabat and Theodorou [28] and Kaveshgar et al. [27], where GA was developed and often achieved superior performance, completing all tasks within a smaller time frame.

In summary, the results provide an exemplary view of the efficiency of CPLEX Software in optimizing waiting times for ships, in this case, taking the seaport of Leixões as a case study for validation of the model. The allocation of terminals and cranes demonstrated significant improvements in the real scenario, highlighting the capacity of the CPLEX Software to find solutions that minimize delays and improve operational efficiency. In addition, waiting for times and ship departure dates have been considerably optimized, demonstrating the usefulness of the optimization approach. It is important to highlight that the CPLEX Software has effectively optimized waiting times and operations in the seaport of Leixões, with samples operating in a controlled and idealized environment. This model does not consider many of the challenges and unforeseen events that may occur in the real context of operating a seaport, such as breakdowns in cranes, technical problems with ships, and the availability of human resources. Other limitations of the results involve using the time average in docking in each ship. It is important to consider all available resources.

4.2. Environmental Impact

In the context of the maritime industry, the search for operational improvement and reduction in the environmental impact are crucial priorities. In this part, an analysis is carried out on the connection between the consumption of fuel and the emissions of pollutants gases. Understanding this relationship is essential to minimize the environmental impact and optimize resource usage as well.

After the results obtained from the improvements in waiting and operating times, it is clear that the practices in optimization performance are very important in reducing the environmental impact from the maritime industry, being able to originate significant decreases in the consumption of fuel and emissions in pollutants gases, both during navigation and when they are moored in seaports. These reductions are fundamental to mitigating several harmful environmental factors, benefiting seaport operations in terms of efficiency and significantly contributing to the preservation of the environment surrounding them [54].

In this explored scenario, the optimizations of these lawsuits are directly connected with the following environmental factors:

■

Greenhouse Gas Emissions: Decreasing waiting and operating times results in a direct reduction in greenhouse gas emissions, such as carbon dioxide carbon (CO₂), nitrogen oxides (NO_x), and sulfur oxides (SO_x).

■

Air Pollution: Reducing waiting and operating times reduces the number of pollutants released into the atmosphere, which improves air quality in seaport areas and nearby, benefiting not only seaport workers but also local communities;

■

Water Quality: Frequent and prolonged seaport operations can lead to dumping wastewater and chemicals into the sea. Reducing the time of permanence of ships in seaports contributes to preserving the quality of water, avoiding contamination;

■

Impact on Marine Ecosystems: Reducing ships’ time at berth can minimize disturbance to marine ecosystems near seaports. This is especially important in sensitive areas, like estuaries and reefs of corals;

■

Noise and Sound Disturbance: Ships in operation and seaport activities can be significant noise sources. Reducing waiting and operating times reduces the period in which the heavy engines and machinery are in operation, resulting in less noise pollution in the surrounding areas, which is beneficial for life navy and your local inhabitants;

■

Consumption of Natural Resources: A more efficient seaport operation requires fewer resources, such as fuel and water.

Reducing waiting and operating periods contributes to sustainability in the maritime transport sector. This approach responds to environmental concerns and promotes compliance with regulations and environmental standards, which become more rigorous. Accurate calculation or prediction of ship emissions represents a big challenge due to the complexity of variables involved in this context. However, Hu et al. [30] describe Equation (24), which can be used to calculate emissions from ships while they are in seaports.

$${ME}_i={\sum }_n{PO}_I\times {AC}_i\times {LF}_i\times {EF}_{i,n}\times {FCF}_n\times {EN}_i$$

(24)

where

■

$M{E}_i$, represents the total emissions of ship i while it is docked.

■

$P{O}_i$, is the nominal power of the motor of a ship.

■

$A{C}_i$, corresponds to the average time activity of each assistant motor on the ship.

■

$L{F}_i$, is the rate in charge (the relationship between the power average used during normal operations and the maximum nominal power).

■

$E{F}_{i,n}$, represents the factor issuance of pollutant n for the ship.

■

$FC{F}_n$, means the correction factor of the fuel for the pollutant.

■

$E{N}_i$, corresponds to the number of engines in a ship.

The presented system allows for calculating the emissions in pollutants gases, such as carbon dioxide (CO₂), nitrogen oxides (NO_x), and sulfur oxides (SO_x), while the ship is docked based on waiting times.

Table 5. Reference values regarding the Emission Factors. Adapted from Hu et al., 2014 [30].

Factor in Emission	CO2	NOx	SOx
Emissions (g/kWh)	683	13	12.3
Factors in Correction	1	0.948	0.04

presents the reference values of the factors in emission that will be used.

To calculate the emission values, it will parameterize one fictitious ship based on the most common characteristics of ships since the diversity of ships existing increases the difficulty of creating instances in comparison trustworthy. Thus, the Motor Nominal Power was defined as an average value of 250 horsepower (hp), a critical factor reflecting the diversity of ship classes and their engines. The relationship of Engine Average Load LF was set to 0.5, representing the average load under which the engines operate during activities, where the ship’s engine operates at half of its maximum capacity during operations. The Number of Engines was set at four as an average, offering a comprehensive overview considering various ship types. The sum of the waiting time and the operating time will also be used to define the Average Activity Time, accounting for a total of 19.6 h for the actual average values and 11.4 h for the average values obtained in the CPLEX Software.

The combination of these parameters becomes essential to accurately estimate pollutant emissions during seaport operations, taking into account the results of the waiting and operating times previously calculated, and to understand the true impact of improving these times on the sustainability of a seaport, as expressed by (25) and (26).

ME iREAL = 250 × 0.746 × (8.1 + 11.5) × 0.5 × ((683 × 1) + (13 × 0.948) + (12.3 × 0.04)) × 4 = 5086971.61 g = 5.09 tons

(25)

ME iCPLEX = 250 × 0.746 × (4.2 + 7.2) × 0.5 × ((683 × 1) + (13 × 0.948) + (12.3 × 0.04)) × 4 = 2958748.80 g = 2.96 tons

(26)

Comparing the results in (25) and (26), the difference between average emissions for each ship in a real context and in the solution obtained by CPLEX Software is notable. On average, a ship, in all its operations, releases a total of 5.09 tons of greenhouse gases into the atmosphere. This value is substantially lower, 2.96 tons, in the solution generated by CPLEX Software using the MILP model developed. This means that, with the optimization of operations and the consequent reduction in waiting times for ships in seaport, it is possible to avoid the emission, in this case, of 2.13 tons per ship of harmful gases for the environment, a reduction of 41.85%.

This difference emphasizes the complexity of accurately estimating pollutant gas emissions in seaport environments and the need for sustainable approaches. Furthermore, it highlights the importance of considering variables that directly affect emissions, such as coordination of seaport operations and waiting time for ships. These factors play a significant role in reducing emissions with severe environmental impact [48].

In this context, it is clear that the search for sustainability in the maritime industry is fundamental to reducing greenhouse gas emissions and promoting more efficient operating practices. Implementing optimization strategies, such as the one proposed in this study, not only benefits environmental significance but also contributes to the economy of resources. It promotes a more sustainable future for the maritime transport industry.

The results of the proposed MILP model reveal substantial operational improvements, particularly in reducing vessel waiting times and optimizing resource allocation. The reduction in average vessel waiting times by 47.56% (from 8.1 to 4.2 h) and operational delays by 37.39% (from 11.5 to 7.2 h) underscores the model’s capability to significantly enhance port efficiency. Such improvements directly translate into better vessel turnaround times, enabling the port to accommodate more ships and reduce congestion. Moreover, the 41.85% reduction in greenhouse gas emissions per ship aligns with global sustainability goals, positioning the port as a leader in environmentally responsible maritime logistics.

From a practical standpoint, these results highlight the potential for the model to transform daily port operations. By integrating real-time operational data into decision-making processes, the model equips port authorities with a robust tool to address inefficiencies and respond proactively to challenges such as berth congestion and crane allocation conflicts. For instance, the optimized allocation of resources could lead to cost savings through reduced fuel consumption and overtime labor expenses. Additionally, the enhanced operational fluidity improves the predictability of shipping schedules, fostering stronger relationships with shipping companies and other stakeholders. This dual focus on efficiency and sustainability not only boosts the port’s competitiveness but also sets a benchmark for other ports aiming to balance economic and environmental objectives. Future implementations of the model could incorporate dynamic variables, such as fluctuating demand and unexpected disruptions, to further enhance its adaptability and operational impact.

5. Discussion

This study focused on the development of an innovative MILP model designed to optimize waiting times and ship operations at the seaport of Leixões. The novelty of this model lies in its integrated approach, which not only addresses the traditional logistical inefficiencies and congestion but also incorporates sustainability considerations, a critical factor in modern seaport operations. The choice of Leixões as a case study reflects its strategic importance in the European logistics network, serving as a crucial hub for maritime trade. The model was validated using real-world data from this port, demonstrating its effectiveness in tackling specific seaport challenges while improving both competitiveness and environmental impact. This validation further supports the model’s practical applicability in addressing the complex demands of contemporary maritime operations.

The application of the MILP model has proven to be feasible and effective, significantly reducing the ships’ waiting and operating times. The use of real data from the seaport of Leixões to validate the model reinforced its relevance and practical applicability. The feasibility of the model is also evidenced by its ability to be adjusted to different operational scenarios, which suggests its applicability in other port contexts that face similar challenges of efficiency and resource management.

The main benefits of implementing this model include the optimization of resource allocation, better planning of anchorage spaces, and the reduction in ship idle times, which directly influence the reduction in operating costs. In addition, improved efficiency leads to lower emissions of polluting gases, contributing to more sustainable seaport operations. This positive environmental impact is crucial considering the increasing global regulations for carbon footprint reduction in the transport and logistics sectors.

In comparison to existing models, the proposed MILP model offers significant advantages in both operational efficiency and sustainability. Traditional models often optimize individual components of port operations, neglecting the interdependencies between processes. Unlike these models, the MILP model developed in this study integrates multiple aspects of port operations, including berth allocation, crane scheduling, and environmental impact, into a unified framework. This holistic approach allows for a 47.56% reduction in ship waiting times and a 41.85% reduction in greenhouse gas emissions per ship, which surpasses the performance of many existing models that do not account for environmental sustainability. Additionally, the ability of this model to dynamically adapt to real-world constraints, such as varying ship sizes and operational timelines, highlights its novelty and practical utility. These comparisons demonstrate that the proposed model not only outperforms traditional models in terms of operational efficiency but also aligns with modern sustainability goals, reinforcing its relevance and innovation in addressing current port management challenges.

Future investigations should focus on enhancing the adaptive capacity of the model by incorporating artificial intelligence to predict and mitigate potential delays due to unexpected conditions. Additional studies could also explore the applicability of the model in different types of seaports, considering specific variables such as cargo types and space configurations. In addition, the integration of the model with environmental management systems could extend its benefits beyond operational efficiency, directly addressing the impact of seaport operations on the environment. In a detailed structure, below are some suggestions for future work:

■

Enhancement of the model: Additional investigations can concentrate on improving the MILP optimization model, incorporating variables and parameters that consider unpredictable factors. These variables are external to the seaport operations.

■

Artificial Intelligence Integration: Incorporating artificial intelligence techniques can allow the model to adapt dynamically to real-time conditions, improving its capacity to deal with unforeseen events. The model is continuously optimized for operations.

■

Economic Impact Analysis: In addition to emission reductions, it is important to carry out a more comprehensive analysis of the economic impact of optimizing port operations, considering operational costs and resource savings.

■

Global Sustainability Assessment: Expand the analysis to assess environmental impact, considering the operational efficiency throughout the entire supply chain, including terrestrial transport.

■

Development in Policies: Develop policies and guidelines that encourage the implementation of optimization strategies in seaports in accordance with environmental and economic objectives.

6. Conclusions

In one global scenario, the search for sustainable practices is imperative, with the awareness of environmental regulations becoming stricter. In the case of seaport operations, two components of sustainability need to be cared for: economic and environmental. This work allows us to understand that caring for the first and second is greatly improved. Minimizing the waiting time of the ships in the seaports and reducing the operating time in unloading and loading operations decrease the associated costs and the risks of environmental pollution. Therefore, it becomes crucial that the maritime industry continues investing in optimizing its strategies to promote a more sustainable future.

This work contributes to a critical and effective analysis of optimization challenges related to seaport operations, investigating and detailing these problems, including variables, goals, restrictions, and the methods used for their resolution. This analysis plays a fundamental role in understanding the challenges faced in the operation of seaports, thus allowing the development and validation of mathematical models to support decisions on terminal containers in seaports.

In this context, various mathematical models and heuristics previously developed were analyzed, decision-making support systems for operations in seaport container terminals were evaluated, and a novel optimization model based on MILP was developed to solve the problem. The model was validated using real data from the seaport of Leixões, which demonstrated how optimizing seaport processes could result in significant reductions in pollutant emissions, thus increasing sustainability in the maritime sector.

The results obtained by implementing the MILP optimization model in the selected case study are extremely promising for the instances resolved. A significant reduction in the average waiting time of ships at the seaport of Leixões was achieved, reducing from 8.1 h to 4.2 h, representing a 47.56% improvement. Operations time also reported a significant reduction, going from 11.5 h to 7.2 h, indicating an improvement of 37.39%. This evidence points to an effective allocation of resources. This represents a substantial reduction in inactivity periods. Comparison between results obtained in real contexts and the solution derived from the model MILP developed reveal a notable difference. On average, greenhouse gas emissions per ship during its operations are 5.09 tons. However, applying the MILP model set with CPLEX Software resulted in a significant reduction in that emission, decreasing by 2.96 tons per ship, representing a reduction of 41.85%. This contrast highlights the importance of optimization strategies in mitigating environmentally harmful emissions, highlighting the complexity involved in accurately estimating these emissions in port environments. However, the model developed here does not consider a series of challenges and unpredictable circumstances that may arise during port operations. Elements like crane failures, technical problems on ships, labor availability and levels of operator experience, tidal fluctuations, and meteorological factors can substantially impact the performance in practical situations.

The practical applications of the model extend beyond the Port of Leixões, offering a scalable framework adaptable to other seaports with similar challenges. Its ability to integrate real-time data and account for port-specific constraints provides a robust tool for decision-makers aiming to balance economic and ecological objectives. Additionally, the results offer valuable guidance for policymakers in developing and implementing environmentally sustainable port management practices.

To conclude, the search for sustainability in the maritime industry is crucial, not only for reducing greenhouse gas emissions but also for promoting more efficient and responsible operational practices. Implementing optimization strategies, such as those proposed in this study, results in significant environmental benefits and contributes to saving economic resources, promoting a more sustainable future for the maritime transport industry. These results reinforce the idea that the marine sector can achieve a balance between operational efficiency, responsibility, and respect for the environment.

For future research, one promising avenue is the integration of real-time data, such as live ship tracking and dynamic berth availability, to enhance the model’s adaptability to changing conditions. Suggestions for future research include

■

Dynamic Variables: Incorporate factors like weather, equipment breakdowns, and demand fluctuations into the model.

■

Broader Applications: Test the model across various port settings and operational scales to assess its adaptability.

■

Real-Time Optimization: Explore the integration of real-time data and digital twin technologies to enhance responsiveness and decision-making.

■

Advanced Algorithms: Investigate the inclusion of machine learning techniques for adaptive scheduling and predictive analytics.

■

Sustainability Metrics: Expand the model to include additional environmental impact measures, such as noise pollution and water quality.