2.1. Optimizing Appointment Quota Plans in TASs

TASs play a crucial role in minimizing congestion and waiting time for trucks at terminals. However, a primary challenge faced by these systems is optimizing the number of appointment quotas for per time slot, as this directly influences the arrival patterns of container trucks, thereby shaping TAS performance. In recent years, various approaches have emerged to address this issue, broadly falling into three categories: queuing theory-based approaches, mathematical modeling–based approaches, and simulation-based approaches.

2.2. Optimization Objectives in TAS

Reducing carbon emissions from container trucks is a critical objective in fostering the sustainable development of green ports. As ports play a central role in global supply chains, addressing environmental concerns has become increasingly important. Previous studies have demonstrated a strong positive correlation between container truck exhaust emissions and their turnaround time within ports [24]. Longer turnaround times result in prolonged idling, increased fuel consumption, and higher emissions, highlighting the urgency of minimizing truck turnaround time as a pivotal strategy for achieving greener port operations [2]. In this context, TASs are instrumental in reducing total turnaround times by effectively managing the distribution of truck arrivals. Through optimized scheduling, the TAS can alleviate congestion, improve operational efficiency, and indirectly contribute to a significant reduction in emissions. Consequently, a substantial body of research has focused on using turnaround time as the primary objective function when designing and optimizing appointment quota allocation plans [9, 25]. By aligning truck arrivals with the operational capacity of ports, these models aim to streamline processes and reduce delays, directly supporting environmental sustainability goals.

Furthermore, within the framework of the TAS, the reassignment of appointment quotas adds a layer of complexity, particularly in terms of its impact on truck companies. Adjustments to the appointment quota plan may require container trucks to modify their preferred arrival time to comply with the maximum quota constraints. Such changes can disrupt schedules, increase operational costs, and potentially lead to inefficiencies for truck companies. To address this issue, optimization models must make a balance between minimizing turnaround times and reducing the disruption caused by these adjustments. To achieve this balance, many studies have incorporated penalty costs into their optimization frameworks [3]. These penalty costs represent the impact of changes in a truck’s originally preferred arrival time compared to its newly assigned time slot. Controlling such discrepancies ensures that the interests of truck companies are considered, fostering better cooperation between port operators and logistics providers. Therefore, considering this in TAS optimization models is critical for achieving a harmonious balance between environmental goals, operational efficiency, and stakeholder satisfaction.

2.3. Addressing Uncertainty in Port Operations

Uncertainty is an intrinsic characteristic of port operations, stemming from their highly dynamic and complex nature. The unpredictability of factors such as travel time, equipment handling time, and operation schedules poses significant challenges to achieving operational efficiency. To address this, previous studies have often sought to model the uncertainty of travel and operation time through specific probability distributions, such as normal, exponential, or Erlang distributions [4, 5, 26]. These studies have provided valuable insights into the variability of port operations, offering foundational approaches for incorporating uncertainty into decision-making frameworks [21]. While these approaches have laid the groundwork for understanding and addressing uncertainty, the more they attempt to reflect the real operational process, the more refined the modeling needs to be, leading to increased complexity, and thus, a greater number of parameters. Many of these parameters are still based on assumptions or limited data, and such parameters may not capture the full complexity of actual port environments, potentially leading to model inaccuracies.

The advent of advanced information technologies has revolutionized the way uncertainty is approached in port operations. With the increasing availability of large volumes of real-time data from port systems, data-driven methods are now emerging as powerful tools to bridge the gap between data availability and effective decision-makings. These approaches harness the richness of real-world data to improve the accuracy and reliability of models, offering a more nuanced understanding of operational uncertainty. For instance, Li et al. [8] utilized real data to model yard crane service time, demonstrating that they follow an Erlang distribution. This study highlighted the potential of real data to enhance the realism of operational models. Nevertheless, these approaches primarily leverage real-world data to refine the estimation of parameters that previously relied on theoretical assumptions. However, they remain confined to the original modeling framework and do not reduce the complexity of the optimization model.

In contrast, Sun et al. [27] used terminal gate data to establish a stochastic regression relationship between truck arrivals and total turnaround time at a terminal. This relationship was incorporated into a robust optimization (RO) model to develop an appointment quota plan for the TAS. The main contribution of this study lies in proposing a novel data-driven modeling framework distinct from traditional methods. Instead of simply utilizing the real-world data to fine-tune parameters within the traditional framework, it integrates insights derived from the terminal gate data directly into the optimization process, significantly simplifying the model. While this approach marked significant progress, it leveraged only a portion of the information contained in the uncertainty set derived from terminal gate data. As a result, the model did not fully exploit the data’s richness, potentially limiting its scope and effectiveness.

2.4. Limitations in Previous Studies and Research Motivations of This Study

Traditional methods for optimizing truck appointment quota plan in TASs, such as queuing theory, mathematical optimization, and simulation-based approaches, often rely on simplifying assumptions about model parameters. For instance, truck arrivals are typically modeled using beta or Poisson distributions, often neglecting the impact of port congestion. Nonstationary queuing theory is frequently used to estimate queue lengths, assuming yard crane service times follow an Erlang distribution. In addition, many studies consider truck turnaround time within the terminal as predetermined values, which oversimplify the complexities of real-world port operations and undermine model accuracy. As port environments grow increasingly complex, these methods require even more assumptions, further limiting their applicability. In addition, uncertainties such as travel and operation time are often inadequately addressed in these studies, depending on rigid probability distributions or assumptions that fail to reflect the variability and complexity inherent in actual port environments.

Advancements in Information and Communication Technology (ICT) have facilitated the use of data mining technology to improve TAS’s performance, offering benefits on two levels. At the foundational level, data analysis results capture characteristics and uncertainties in truck arrivals and operations, enabling more accurate calibration of model parameters that previously relied on assumptions in traditional modeling frameworks. At a higher level, real-world data insights are directly incorporated into the optimization model for appointment quota plans. Unlike traditional methods that attempt to model every detail of truck operations, this approach maintains the integrity of the “black box”, focusing only on the input–output relationship. By avoiding assumptions for each parameter, the model is simplified, significantly reducing the computational complexity. However, despite the vast amount of real-world data available from ports, most studies have only utilized a small portion of this information. For example, while Sun et al. [27] used terminal gate data to model the relationship between truck arrivals and turnaround time, the full potential of the data within uncertainty sets remains underexplored in their RO model, limiting its effectiveness and scope.

This study aims to bridge these gaps by developing a data-driven optimization framework that fully leverages available terminal gate data through a DRO model. Unlike traditional RO methods, such as those used by Sun et al. [27], and stochastic optimization (SO) methods, the DRO does not rely on a single predefined probability distribution. Instead, it considers a range of potential distributions within an uncertainty set, maximizing the value of information within these sets. This flexibility allows the model to more effectively capture the inherent variability of truck arrivals in port operations. As a result, the DRO strikes a balance between robustness (resilience to uncertainty) and optimality (achieving the best solution), leading to appointment quota plans that are both adaptable to uncertainty and highly efficient in performance.

3.1. Problem Description

A TAS typically divides a day into multiple time slots, each spanning one or more hours. The primary goal of implementing a TAS at a port is to manage and regulate the arrival patterns of trucks within these time slots using an appointment quota plan. This plan establishes the maximum number of appointment quotas for each time slot, thereby capping the number of truck arrivals and distributing them more evenly throughout the day. To enter the port, each truck must secure an appointment for a specific time slot in advance. Consequently, designing an optimal appointment quota plan is crucial to ensure the effectiveness and efficiency of a TAS. Accordingly, the primary objective of this study is to determine the optimal allocation of appointment quotas for each time slot. To accomplish this, two subproblems must be addressed.

First, the underlying logic of how an appointment quota plan affects port operations is as follows: The plan sets a cap on the maximum number of truck arrivals for each time slot. Trucks schedule specific appointments and arrive at the port according to these quota constraints. This, in turn, affects the actual distribution of truck arrivals, which directly impacts the operational efficiency of the port. This efficiency is measured through indicators such as the turnaround time for trucks at the port. Therefore, from an optimization perspective, indicators such as the total truck turnaround time at the port should be minimized to identify the optimal distribution of truck arrivals. This optimal distribution can then serve as a guide for designing an effective appointment quota plan. To achieve this, the first step is to establish a clear relationship between the distribution of truck arrivals and port operational efficiency, using the insights derived from real-world data.

However, the relationship between truck arrival distribution and operational efficiency is often uncertain rather than deterministic. In addition, while this relationship can be utilized to determine the optimal number of truck arrivals for each time slot by minimizing efficiency metrics such as total turnaround time, the next challenge is to translate these optimal truck arrival numbers into appropriate upper bounds for appointment quotas in the plan. Therefore, a typical Min–Max optimization model is employed to achieve the optimal upper bounds. Given the stochastic nature of this relationship and the Min–Max optimization framework, the focus shifts to developing a DRO model to address the challenge, offering a comparison with traditional RO models.

Overall, the decision variable in the optimization model is the number of appointment quotas allocated for each time slot, which collectively form the appointment quota plan. The objective of the model is to minimize the total turnaround time of trucks at the port. To establish the connection between the appointment quota plan and the total turnaround time, it is assumed that the truck turnaround time at a specific time slot is primarily influenced by the number of truck arrivals during that time slot. However, this relationship may be not deterministic but rather stochastic, and it needs to be further explored and validated in this study. Therefore, the optimization process must account for this uncertainty and adopt an appropriate modeling framework. Moreover, while the shortest total truck turnaround time serves as the optimal objective function from the port’s perspective, reflecting its overall decision-making interests, the study also incorporates a constraint to control the deviations between the original quota plan and the new plan. This ensures that the truck arrival distribution can only be adjusted within a certain range, thereby balancing the rights of the truck companies. The specific details of the other constraints and assumptions are presented in the model establishment section.

3.2. The Overall Theoretical Framework

4.1. Data Processing and Mining

4.2. The Establishment of a Traditional RO Model for Appointment Quota Optimization

4.3. The Establishment of a DRO Model for Appointment Quota Optimization

Although the traditional RO model employing the Min–Max approach provides a robust and effective solution for the designing appointment quota plans, it is overly conservative. This conservatism arises because the RO model focuses solely on optimizing under the worst-case scenario for the trucks’ total turnaround time. Specifically, in relation to equation (7), if the worst-case scenario occurs, the number of trucks arriving during each time slot must reach the upper limit of the appointment quotas. Simultaneously, the variable ε_j, representing the influence of other factors on trucks’ total turnaround time at the port, must take the highest value (max $$) observed historically. While the former condition is relatively more likely to be met in real-world scenarios, the latter often leads to an overestimation of the uncertainty. Consequently, the RO model fails to capitalize on opportunities for more flexible and dynamic adjustments under less extreme conditions.

In this context, the DRO model addresses the limitations of the traditional RO approach by shifting from a worst-case scenario focus to a probabilistic framework that captures a range of plausible scenarios. Using terminal gate dataset information, the DRO model constructs a distributional uncertainty set that contains potential probability distributions, each representing different uncertain scenarios and aligning closely with the prior probability distribution of ε_j. By considering the likelihood and variability of these scenarios, the DRO model facilitates dynamic adjustments, striking a balance between conservatism and adaptability. This ensures robust performance against disruptions while maintaining operational efficiency. With its data-driven foundation, the DRO model provides practical, real-world solutions that are more flexible and effective than the overly cautious outcomes of the RO approach. The formulation of the DRO model is as follows, accompanied by a detailed explanation of how it differs from the traditional RO model.

The equations (5), (8)–(10) are constraints, as same as those in the RO model.

The additional notations introduced in the DRO model, compared to the RO model, are summarized in Table 3, while the remaining notations retain the same definitions as provided in Table 2. Building on the objective function of the RO model, as shown in Equations (11), (12) presents the objective function for the DRO model. This formulation also employs a Min–Max approach, aiming to minimize the total turnaround time of trucks at the port. The key distinction lies in the treatment of ε_j, which represents the impact of external factors such as weather, mechanical failures, and other disruptions on truck turnaround time. Unlike the RO model, which directly considers the extreme impact of ε_j, leading to its overly conservative nature, the DRO model adopts a more balanced approach, mitigating this conservatism by accounting for a range of plausible scenarios rather than single worst-case scenario. As specifically reflected in equation (12), the DRO model departs from the RO model’s reliance on the historically observed worst-case values of ε_j. Instead, the DRO model focuses on the expected values of ε_j across a range of different but possible scenarios, denoted as E(ε_j). The model then evaluates the worst-case scenario for E(ε_j), incorporating it into the optimization. This approach allows the DRO model to strike a balance between robustness and flexibility by taking in to account of a broader distribution of potential impacts rather than fixating on a single extreme value.

E(ε_j) is obtained by considering the prior values of ε_j and the associated probabilities of these values occurring, as illustrated in equation (13). Here, O_j is denotes as the dataset of $$, which represents the prior values of ε_j, obtained from historical regression residuals at time slot j. The values inside O_j are sorted in ascending order and then divided into K intervals with equally spaced values. $$ is defined in this study as the average value of $$ from the kth interval of dataset O_j, k = 1, 2, 3, …, K; this value is used to denote the expected value of ε_j when it falls within the kth interval. $$ represents the probability of ε_j falling within the kth interval of dataset O_j, considering all the potential distributions.

However, $$ is always unknown and must be estimated based on the prior probability of ε_j, denoted as $$. To construct the uncertainty set for $$, this study employs the discrepancy-based method given in [30] to measure the difference between $$ and $$, denoted as $$, and constrains the potential probability $$ to be close to the prior probability $$ within a maximum discrepancy ρ_j, as shown in equation (14). In this study, the Φ divergence function proposed in [28] is adopted to define $$, where ϕ(x) is a known function determined by different definitions of Φ divergence, as specified in equation (15). Moreover, the maximum discrepancy ρ_j is calculated by using equation (16), where B denotes the sampling size, $$ represents the Chi-square distribution, d_j is the dimension of $$, and α is the confidence level of the uncertainty set [29]. Finally, equations (17) and (18) constrain that the probability cannot be negative and that the sum of probabilities across all intervals must equal 1. The remaining constraints are consistent with those defined in the RO model.

4.4. The Establishment of the Lagrange Dual Model for the DRO Model

The large-scale nature of dataset U_j introduces an infinite number of potential scenarios for $$, making the proposed DRO model a semi-infinite programming (SIP) problem. As a result, directly obtaining the optimal solution to the proposed DRO model becomes a significant challenge. The study in [30] emphasized the limitations of traditional methods, such as interior point techniques, in solving SIP problems, even when these problems exhibit convexity. To address this, it becomes essential to reformulate the SIP problem into a more manageable form. This study employed the Lagrange dual method to transform this SIP problem containing uncertain parameters, into a dual problem characterized by deterministic parameters [31]. The reformulation process is described as follows.

As a result, the dual function g(λ_j, η_j) establishes an upper bound for $$. As demonstrated in the study by Ben-Tal et al. (10), there must exist a pair of λ_j, η_j that satisfies $$. This implies that by solving the minimization for g(λ_j, η_j), the solution for $$ can be obtained. Therefore, the original problem can be converted into the dual model, as demonstrated below.

The equations (14)–(19) are constraints, as same as those in the original model.

Equation (26) defines the conjugate function ϕ^∗(y), where the original Φ-divergence function is characterized as a variance distance function, denoted by ϕ(x) = |x − 1|. Meanwhile, equation (27) imposes constraints on the variable’s range within the conjugate function, thereby specifying the corresponding range of uncertainty for ε_j in the original problem. Equations (29) and (30) present the range of λ_j and η_j.

By the principle of Lagrangian duality, the solution of the dual model is equivalent to that of the original problem. Moreover, the dual model is clearly an integer programming problem, making it well suited for direct and efficient resolution using mathematical solvers.

5.1. The Terminal Gate Data Used in This Study

The terminal gate data used in this study were collected from the container terminal at Yan port in China. Yan port is a significant port in southern China, with a container terminal throughput of nearly 13 million TEU in 2019. The container terminal operates 24 h a day, and the dataset used in this study encompassed the terminal gate data from March 1st to July 31th, 2019. The dataset consisted of over 2.3 million records.

5.2. The Results of the Regression Relationship

Using the terminal gate data, the number of truck arrivals and their corresponding total turnaround times for each hour over a 4-month period were analyzed, as detailed in Section 4.2. This analysis yielded a dataset of 2928 sample pairs. These samples were utilized to train a regression model designed to predict truck turnaround time at the port based on the number of truck arrivals within a specified time slot. The model was trained on data from the first 3 months, with the final month’s data reserved for performance evaluation.

5.2.1. The Training of the Regression

To train the regression model, a scatter diagram (Figure 2) was initially created to visualize the relationship between the two variables, aiding in the identification of the potential function type for the regression.

As shown in Figure 2, the relationship between the variables was observed to approximately follow a quadratic function. Consequently, EViews 10 software was used to conduct a quadratic regression analysis, with the results presented in Table 4. Notably, since the turnaround time is expected to be zero when no trucks arrive at the terminal, the constant term in the regression model was set to zero.

Table 4. The fitting results of the regression in the training dataset.

Variable	Coefficient	Std. error	t-statistic	Prob.
$$	0.014243	0.000329	43.24278	0.00
Nj	30.34224	0.284702	106.5756	0.00
R2		0.9612
MAPE		0.0755

Table 3 displays the goodness-of-fits of the regression function, all of which satisfied the requirements for significance testing. The determination coefficient R² was 0.9612, surpassing the threshold of 0.95, indicating a strong fit of the regression. In addition, the mean absolute percentage error (MAPE) was 0.0755, less than 0.1. These results confirmed that the regression model fitted well with the data, supporting that f(N_j) was a quadratic function of N_j. Thus, the relationship between T_j and N_j was described as shown in the following equation, where ε_j denoted the residual, capturing the influence of the other factors on truck turnaround time at the port and the regression errors.

$$ ()

5.2.2. The Testing of the Obtained Regression Function

The performance of the function on the test set is presented in Figure 3, with an R² value of 0.9573 and a MAPE value of 0.070. This indicated that the proposed regression function exhibited good generalization and could be utilized in the proposed optimization model.

5.3. Obtaining the Prior Distribution of the Residuals

To assess the assumption of independence between the distributions of residuals across time slots, this study investigates the prior distributions of residuals for each appointment time slot throughout the 4-month period. The aim is to ascertain whether the distribution of residuals is consistent across different time slots. As depicted in Figure 4, the distributions of residual $$ exhibit variation across each time slot. In addition, the correlations between any two distributions are investigated by analyzing the relationship in the probability of specific residuals occurring within both distributions. The results confirm the validity of the assumption about distinct residual distributions across different time slots, as shown in Figure 5.

In addition, to provide a quantitative description of the distributions of $$ for each time slot, the range of values $$ was divided into equally spaced intervals. In this study, the number of intervals, denoted as K, was set to 20. This choice was made after testing various values, as K = 20 ensures that each interval contains prior values in the majority of distributions. The number of values falling within each interval was then counted, enabling the computation of the prior probability $$. Moreover, the number of intervals that contained data were recorded to determine the dimensions of $$ in each time slot. Finally, ρ_j, the maximum discrepancy between $$ and $$ was calculated using equation (16), with a 95% confidence level. The distribution information of $$ in each time slot is presented in Table 5, which are used to estimate $$ and solve the DRO model.

5.4. The Model Results

5.4.1. Model Preparation

The data collected from the terminal gates of the port during the first week of July 2019 were utilized. The average number of truck arrivals during specific time slots in that week was calculated to create an initial appointment quota plan. This initial plan was then optimized using the proposed DRO model. The total number of daily appointment quotas was determined based on this initial plan.

In addition, in the model’s constraints, the value of integer M should be discussed and defined, ensuring that the adjustments to the allocation of quotas do not exceed this limit, thus preventing significant changes to the initial plan. This study sets the integer M equal to the total daily appointment quota multiplied by a positive weighting factor β, as shown in the following equation:

$$ ()

This means that, for instance, when β = 0.5, the constraint ensures that the total number of adjusted quotas does not exceed half of the total daily appointment quotas. To identify a suitable value for M, the effect of the weighting factor β on the DRO model solution is analyzed by iteratively calculating the total turnaround time for trucks under each solution across a range of β values. The results are depicted in Figure 6. It can be observed that as β increases, the total turnaround time decreases. This result is attributed to the fact that as β increases, the constraints on quota transfers become more relaxed, optimizing the system toward reducing the trucks’ total turnaround time. However, this relaxation also leads to more significant changes in quota transfers compared to the initial plan, which could substantially affect the operational schedules of specific truck companies. Moreover, it is found that when β exceeds 0.355, the constraint on quota transfers loses its effectiveness completely. Thus, to ensure the efficacy of this constraint, we set β = 0.3 in this study.

The reformulated DRO dual model, being an integer programming problem, was solved using IBM ILGO CPLEX Optimization Studio¹, commonly referred to as CPLEX. The model comprises 1236 variables and 1226 constraints, and CPLEX solved it in approximately 0.3 s. As illustrated in Figure 7, the results emphasize the differences between the initial and optimized plans. The optimization process redistributed appointment quotas, shifting them from peak hours to idle hours, resulting in a more balanced allocation of quotas.

6.1. Generating Truck Arrival Pattern Based on the Appointment Quota Plan

To evaluate the efficiency of the optimized appointment quota plan in minimizing the overall turnaround time for container trucks at the port, the truck arrival pattern is simulated based on the optimized plan. The simulation assumes that truck arrivals follow the appointment outcomes determined for each time slot, with the expectation that all trucks adhere to their scheduled times, excluding any potential delays.

6.2. Calculating the Total Expense of Trucks Based on Their Arrival Pattern

6.3. Validating the Effectiveness of the Proposed DRO Model

The actual count of truck arrivals for each time slot during the first week of July 2019 is extracted from the terminal gate records. These figures are then used to represent the number of appointment requests for each time slot on the corresponding days. Following the guidelines outlined in Section 6.1 and applying the appointment quota plans shown in Figure 7, the potential truck arrival patterns are generated for both the initial and optimized appointment quota plans, as illustrated in Figure 8.

The results demonstrated that implementing the optimized plan effectively achieved a more evenly distributed pattern of truck arrivals on a daily basis. Specifically, between July 1st and 5th, the optimized plan successfully redistributed trucks that typically arrived during peak hours to adjacent time slots with fewer incoming trucks, thereby alleviating congestion during these peak periods. However, on July 6th and July 7th, the count of appointment requests was lower than the number of appointment quotas designated for the majority of time slots, leading to no changes in the arrival pattern by the optimized plan.

Then, the overall time expenses are calculated based on their arrival patterns under both the initial and optimized plans. To account for the uncertain relationship between truck arrivals and their total turnaround time at the port, stochastic scenarios are constructed and simulated, as outlined in Section 6.2. In addition, to assess the impact of sampling size on the overall time expenses, sampling sizes are varied between 10 and 1000 in intervals of 50. For each sampling size, 1000 iterations of the stochastic experiments are conducted, and the overall time expenses for each of the seven days in that specific week are calculated. The average overall time expenses, along with their range for each sampling size, are presented in Figure 9.

The optimized plan outperforms the initial plan in terms of average overall time expenses, demonstrating the significant optimization effect and advantages of the proposed DRO model in real stochastic environments. As the sampling size increases, the uncertainty regarding the relationship between truck arrival patterns and their total turnaround time decreases. Both the initial and optimized plans show reduced volatility in overall time expenses with larger sampling sizes. However, the sampling size does not significantly affect the average overall time expenses. The average overall time expenses based on the optimized plan remain relatively stable and yield favorable results from the outset. This suggests that even with a small sample size, the DRO model is able to quickly capture the stochastic characteristics of uncertain parameters and efficiently obtain a relatively optimal solution for minimizing overall time expenses.

6.4. The Performance Comparison Between RO and DRO Model

This study evaluates the effectiveness of the proposed DRO model by comparing its results with those from the traditional RO model. First, the RO model is used to optimize the initial plan, producing a new optimized version. Then, the truck arrival patterns are simulated based on this new plan, and the overall time expenses are calculated using a process similar to the one for the DRO model. The results, shown in Figure 10, highlight the significant impact of sampling size on model solutions and emphasize the superior performance of the proposed DRO model compared to the traditional RO model. Specifically, the DRO model leads to a reduced average total truck turnaround time and lower overall time expenses.

In addition, in scenarios with the same sampling size, indicating an identical level of uncertainty, the appointment quota plan generated by the RO model exhibits greater robustness and lower volatility in overall time expenses than the DRO model. This is because the RO model is inherently more conservative, focusing on cost minimization in the worst-case scenario. In contrast, the DRO model combines the advantages of both stochastic and RO to achieve better performance, at the cost of some robustness. These findings, also supported by the results in Figure 10, demonstrate that the proposed DRO model, which effectively integrates terminal gate data, can optimize the appointment quota plan of a TAS, significantly improving port efficiency.