Development of an Integrated Optimization Model for Container Relocation and Truck Appointment Scheduling (TAS)

Abstract

Background: The rise in global throughput has created major challenges for container terminals and depots, especially in managing limited storage space and unsynchronized truck pickup schedules from different companies. These conditions complicate the organization, retrieval, and relocation of containers, especially when target containers are located in the middle of stacks. This study aims to develop an integrated optimization model that combines Truck Appointment Scheduling (TAS) and Restricted Block Relocation Problem (RBRP) to minimize container relocations, coordinate pickup schedules, and improve operational efficiency. Methods: An integer programming model is formulated to integrate relocation and scheduling decisions by considering practical operational constraints, including maximum stack capacity, queue length, crane movement capacity, relocation validity, prevention of redundant movements, and efficient slot utilization. Two solution schemes are evaluated, simultaneous RPRP-TAS approaches and sequential TAS-then-RBRP approaches, to minimize relocations. Results: The results show that the simultaneous approach produces fewer relocation and more stable than the sequential approach. Sensitivity analysis also confirms that the simultaneous scheme is more robust to variations in the number of containers and crane movement capacity, while maintaining comparable pickup time-shift performance. Conclusions: The simultaneous integration of RBRP and TAS provides coordinated, practical, and efficient decision support for depot and terminal operations.

1. Introduction

1.1. Background

Over 80% of global trade volume is transported via maritime routes, making it the predominant method of goods transportation in international trade. Between 2013 and 2023, global container throughput exhibited a positive growth trend, with a compound annual growth rate (CAGR) of 2.7%, calculated based on data reported by UNCTAD [1]. In 2023, global ports collectively handled 858 million twenty-foot equivalent units (TEUs). The growth rate of container throughput at ports globally slowed to approximately 0.5% in both 2022 and 2023, representing the lowest growth rate in the past decade, excluding the significant decline in 2020 due to the COVID-19 pandemic. The long-term trajectory indicates a consistent increase in container volumes, with global throughput rising by 33% over the past ten years, from 641 million TEUs in 2013. This increase of 217 million TEUs is equivalent to the total container volume managed by developed countries in 2013 [1], as shown in Figure 1.

Indonesia Maritime Gateway has also experienced similar growth. The data presented refers specifically to ports managed by Indonesia Port Corporation (IPC) as the main port operator in Indonesia, with container traffic reaching 17.7 million TEUs in 2023, marking a 3% year-on-year increase. This growth emphasizes the essential role of the logistics and container shipping sectors in the national and global economy [2].

The increasing volume of global and Indonesian container shipments presents significant challenges for container terminals and depots, particularly in managing limited storage space. One of Indonesia Logistics company, that is also a subsidiary of Indonesia Port Corporation (IPC), as a container depot handling export–import containers, faces the issue of container accumulation with varying pickup schedules. This accumulation occurs because limited storage space often fails to accommodate the continuously rising number of containers, hindering the processes of sorting, retrieval, and relocation of containers. The situation becomes even more complicated at the end of the month when shipment surges lead to mismatches in the stacking of containers according to the schedules of the principal container lines. As a result, delays, increased waiting times, high operational costs, and difficulties in relocating containers located in the middle of the pile occur. To improve operational efficiency, optimizing the container relocation process and scheduling truck appointment slots (TASs) are required to reduce costs and enhance the system.

The container relocation process at depots and terminals is fundamentally similar, as both serve as key nodes in the global supply chain to ensure the smooth distribution of goods between countries. At depots, loaded containers are temporarily stacked before being transported to terminals, while at terminals, containers are also temporarily stored before being delivered to their final destinations.

However, both facilities face logistical challenges such as limited storage space and time constraints due to increasing container volumes. Therefore, minimizing relocation time and costs is essential to improve operational efficiency and reduce waste. Efficient management of space and time also accelerates container turnover and supports the smooth flow of goods within the global supply chain. The similarities in the operational processes and the focus of the research scope are illustrated in Figure 2.

Ref. [3] reveals that avoiding unproductive container relocations can reduce fuel consumption by up to 40%. This issue is known as the Block Relocation Problem (BRP), which pertains to the management of container retrievals that require the rearrangement of stacks. The primary goal of BRP is to retrieve target items in a specific order while minimizing the number of relocations required to access them. BRP is divided into two categories, i.e., Restricted Block Relocation Problem (RBRP) and Unrestricted Block Relocation Problem (UBRP). In RBRP, relocation is only allowed for items that obstruct access to the target item, whereas UBRP permits the relocation of any item, including voluntary movements. A more appropriate approach to this problem is the RBRP, which is better suited for cases with limited storage space and container retrieval from stacks [4]. Ref. [4] also states that with the right approach, the efficiency of container management can be improved. Relocation operations (reshuffling) refer to the movement of a block from one stack to another, while retrieval operations involve the movement of a block from a stack out of the system [5]. However, manual (traditional) methods often fail to adapt to the dynamic nature of container relocation management, especially in BRP scenarios [6]. The formulation of BRP using Integer Programming models has been discussed in previous studies [7]. Container relocation management and truck appointment scheduling are two interconnected issues that need to be addressed simultaneously to achieve maximum efficiency [8]. Inefficiencies in the container relocation process can lead to delays, which can be minimized by implementing a more effective retrieval scheduling system [9]. Studies related to RBRP, which formulate the number of time periods equal to the number of items retrieved and tighten the relocation constraints, show potential for improving efficiency [5]. However, to date, no research has integrated truck appointment scheduling with container relocation management (RBRP). The integration of these two aspects is crucial for improving operational efficiency at terminals or depots, reducing truck waiting times, and optimizing the number of relocations using a model that remains underexplored in the related literature.

1.2. Problem Statement

One of the main challenges in container depot management is the Block Relocation Problem (BRP), which occurs when containers required for retrieval are blocked by other containers in the stack. This situation forces depot managers to relocate containers, leading to increased movements, time, and operational costs for repositioning the containers into accessible positions. These frequent relocations not only reduce operational efficiency but also limit the available storage capacity, causing congestion and bottlenecks in the depot area. This issue becomes even more complex during peak periods, such as mid or end of the month, when the volume of containers increases sharply. Inefficient container handling during these times may result in shipment delays and higher logistics costs. Additionally, the lack of coordination in managing relocations contributes to uncertainty in truck waiting times, which further increases the workload, incurs additional costs, and heightens the potential for errors in container management.

One solution that may help mitigate this problem is the implementation of Truck Appointment Scheduling (TAS). The TAS system allows for coordinated and scheduled truck arrivals, which can reduce congestion in the depot area, minimize truck waiting times, and improve container movement efficiency. However, while TAS can alleviate congestion and improve coordination, the BRP remains a major challenge, particularly concerning storage space limitations and the fluctuating volume of containers.

The complexity of container depot management is further exacerbated by the limitations of available storage space. The limited space often cannot accommodate the continuously increasing number of containers, leading to congestion and difficulties in accessing containers that are blocked in the middle of the stack. This worsens relocation and transfer times, disrupting the overall depot operational system. In fact, poorly organized container stacking can result in irregularities in truck pickup schedules, exacerbating relocation issues and increasing waiting times. Furthermore, when containers cannot be arranged according to the principal container line, the process of relocation becomes more complicated and time-consuming. These challenges not only impact operational efficiency but also increase logistics costs, which in turn affect a company’s competitiveness in the highly competitive global logistics industry. Therefore, it is crucial to develop solutions that integrate container relocation management with truck arrival scheduling through more advanced optimization systems.

In addition to operational efficiency, container terminal operations are increasingly evaluated in terms of sustainability and environmental impact. Inefficient container handling and prolonged truck waiting times contribute to higher fuel consumption and increased carbon emissions in port operations [10]. Therefore, optimizing container relocation and truck scheduling not only improves operational performance but also supports efforts to reduce environmental impact in port and logistics systems.

1.3. Objectives and Significance

This study aims to develop an optimization model that integrates solutions to the Restricted Block Relocation Problem (RBRP) and Truck Appointment Scheduling (TAS), with a primary focus on minimizing the number of container relocations. Consequently, this research will explore existing deficiencies and create a more efficient approach to managing storage space and container movement. Through the development of a model that optimizes the outcomes of relocation and scheduling processes, the study seeks to generate a container relocation scheme that aligns with an optimal pickup schedule, while ensuring more efficient and well-scheduled relocation management.

The proposed solution is highly relevant given the significant challenges faced by many container depots and terminals in managing high fluctuations in container volumes, coupled with limited storage capacity. By leveraging this model, it is expected that more efficient depot management can be achieved, reducing truck waiting times and maximizing the use of limited storage capacity. This research will also assess and select problem-solving schemes that can be implemented in real-world scenarios, considering factors such as the number of relocations and operational efficiency, thereby contributing significantly to improving operational effectiveness and efficiency, as well as reducing high logistics costs in the port and container terminal industries.

1.4. Research Questions

This study formulates research questions to examine the integration between the Restricted Block Relocation Problem (RBRP) and Truck Appointment Scheduling (TAS), aiming to improve operational efficiency and minimize container relocations. To provide a clear focus, this study addresses the following main research question:

“How container relocation and truck scheduling can be jointly optimized to minimize relocations and improve operational efficiency under practical constraints?”

This main research question is further elaborated into three specific research questions.

• RQ 1: How can an optimization model be developed for the integration of RBRP with TAS to minimize the number of container relocations?
• RQ 2: What are the outcomes of optimizing relocations and truck appointment scheduling, both in terms of time efficiency and the number of relocations, after the modeling process?
• RQ 3: How can a solution method be selected for implementing this model in real-world scenarios, considering both the number of relocations and time efficiency in operational performance?

1.5. Scope and Limitations

In this study, the scope and limitations of the problem are as follows: First, the model considers a single-column block configuration to enable a more controlled and structured analysis of relocation decisions. Second, the number of containers, stacks, and tiers is based on data collected from the initial position, with no containers added to the block column during the operational process. Third, the study does not explicitly incorporate multi-block interactions or crane assignment decisions, as the primary focus is on a single block column to minimize relocations and on the integration of relocation decisions with truck appointment scheduling. Fourth, the study assumes that containers in the stacking yard will be fully retrieved (complete retrieval), with a unique retrieval priority for each container.

2. Related Works

2.1. Block Relocation Problem

The classical definition of the Block Relocation Problem (BRP) involves retrieving all items according to their departure time with minimal relocation. The Block Relocation Problem (BRP) itself refers to the unloading of goods from storage in a specific order, with the shortest possible sequence of movements. During the retrieval process, decisions may need to be made to move items that obstruct the target item. The objective of BRP is to retrieve all items according to their departure time while minimizing the number of relocations [11].

The Block Relocation Problem (BRP) can be divided into two categories: Restricted BRP (RBRP) and Unrestricted BRP (UBRP). RBRP only allows the relocation of items that obstruct the target item, whereas UBRP permits the movement of any items, including voluntary movements. Although UBRP may result in fewer relocations by sacrificing a much larger search space, this approach has the potential to increase operational costs. Therefore, a more appropriate approach to address this problem is through the use of RBRP, which is better suited for cases with smaller storage spaces and the retrieval of containers from stacks [4].

The application of the Block Relocation Problem (BRP) in container depot logistics systems aims to optimize the process of retrieving and relocating containers at port terminals or container depots. In this context, the Block Relocation Problem refers to the reorganization of containers arranged either vertically or horizontally. This issue arises when containers that need to be retrieved are not at the top of the stack, necessitating multiple moves to create space for their retrieval. The management of storage space and container relocation becomes a critical aspect of logistics operations, particularly in ports or container depots with high volumes. The application of the Block Relocation Problem facilitates more efficient planning and management of containers by minimizing the number of relocations required. The block column in the Block Relocation Problem (BRP) case, consisting of stacks, tiers, and empty slots, is illustrated in Figure 3.

In the context of the Block Relocation Problem (BRP), several key terms are commonly used. A block refers to a unit of container stored within a storage area or bay, which must be retrieved according to a predefined order. Within a block, there is a block column, which is a vertical arrangement of containers designed to maximize space utilization and facilitate efficient loading and unloading. A stack consists of a vertical pile of blocks within a column, with limited capacity, and can only be accessed from the top. The bay is the storage area that contains multiple stacks, organized to ensure efficient retrieval of blocks. Each tier represents a height level within a stack, where the bottom block is typically the first to be stored and retrieved, unless a block above it must be moved first. Lastly, a slot is the position within a tier of the stack for storing a block, with its capacity determined by the number of available slots. These terms collectively help in understanding and addressing the logistical challenges posed by the Block Relocation Problem in container depots.

The Block Relocation Problem (BRP) has been addressed using various solution approaches in the literature. These approaches can generally be classified into three categories. First, exact optimization methods, such as integer programming and branch-and-bound algorithms, aim to obtain optimal solutions but are often limited by computational complexity when applied to large-scale problems. Second, heuristic methods are widely used to generate near-optimal solutions with lower computational effort, making them suitable for real-time applications. Third, metaheuristics approaches, including genetic algorithms, tabu search, and simulated annealing, have been developed to balance solution quality and computational efficiency, particularly for large and complex instances of BRP.

In this study, an exact optimization approach based on integer programming is adopted. This approach enables a precise formulation of decision variables and constraints while ensuring optimality of the solution, which is essential for capturing the operational characteristics of container relocation problems.

2.2. Truck Appointment Scheduling

Truck appointment scheduling refers to a system implemented by container terminals or depots to manage truck arrivals and schedule appointments by dividing truck arrivals into specific time windows. This system aims to reduce congestion, minimize waiting times, and enhance operational productivity by preventing excessive truck queues both inside and outside the stacking area. Research by [12] indicates that the use of a Truck Appointment System (TAS) can reduce truck waiting times, which leads to shorter truck turnaround times and mitigates congestion in the stacking area. Truck turnaround time is the sum of truck waiting time and the time taken for the truck to be serviced [13].

Truck appointment scheduling, in the context of the Block Relocation Problem, not only involves truck arrival times but also requires coordination with container relocation and retrieval operations in the stacking yard. This coordination includes planning for container pickup and the capacity of equipment such as cranes. Research by [14] demonstrates that efficient scheduling can reduce truck waiting times and improve the utilization of stacking yard capacity. Coordination between truck arrival schedules and the container relocation process is one of the key challenges in truck appointment scheduling. Often, when trucks arrive on time according to the scheduled appointment, the container intended for retrieval is not located at the top of the stack, necessitating the relocation of other containers to access the desired one. Poor coordination between arrival times and container relocation can result in reduced productivity and increased truck waiting times. It is found in [15] that the truck waiting times can be reduced by aligning truck arrival schedules with container management movements, by developing a truck appointment scheduling model involving container relocation. Later on, we will refer to this research in [15] to validate our proposed model, and it is referred to as BRP-AS.

Despite the extensive development of solution approaches for BRP, many existing studies treat container relocation as an isolated problem without explicitly considering the dynamics of truck arrivals. In practical container depot operations, however, container retrieval activities are closely linked to truck appointment schedules. Poor synchronization between these two aspects can lead to increased relocation operations, longer truck waiting times, and reduced yard efficiency.

Therefore, this study focuses on integrating Truck Appointment Scheduling (TAS) with the Restricted Block Relocation Problem (RBRP) within a unified optimization framework. The use of an exact optimization approach allows for a structured and consistent representation of both scheduling and relocation decisions, enabling more accurate coordination between truck arrivals and container handling processes.

2.3. Previous Works

The Block Relocation Problem (BRP) was first introduced as an optimization problem aimed at determining the retrieval sequence of containers while minimizing the number of relocation moves [16]. Their study demonstrated that containers blocked by other containers require reshuffling operations before retrieval, making relocation minimization a critical factor in improving container yard operational efficiency. The main contribution of this study was the introduction of the BRP model along with the Expected Number of Additional Relocations (ENARs) heuristics to estimate additional relocations during the retrieval process. Although the heuristics were capable of generating feasible solutions, it was unable to guarantee optimality, particularly for large-scale and complex instances. Since then, BRP has evolved into one of the major research topics in container terminal optimization due to its close relationship with crane utilization, service time, yard productivity, and operational costs.

In its early development, BRP research mainly focused on heuristic approaches to obtain operationally feasible solutions. Following the ENAR heuristics proposed in [16,17], the Corridor Method was developed to improve reshuffling efficiency through a local-search approach [17]. Although this method reduced the number of relocations compared to previous heuristics, its performance strongly depended on the initial container stack configuration and did not always produce optimal solutions. Subsequently, the PR (Priority Rule) and PU (Priority Update) heuristics for both restricted and unrestricted BRP were introduced in [18]. These approaches improved heuristic solution quality; however, the resulting solutions remained highly dependent on the priority rules employed. In addition, several studies began applying metaheuristics approaches such as Genetic Algorithms (GAs), Simulated Annealing (SA), and Ant Colony Optimization (ACO) to address large-scale BRP instances. While metaheuristics were able to generate high-quality solutions for complex problems, they could not guarantee optimality and were highly sensitive to algorithmic parameter settings.

The limitations of heuristic approaches subsequently motivated the development of exact optimization methods. An integer programming-based formulation through the Multi-Relocation Integer Programming (MRIP) model was developed in [19]. This study represented an important milestone in BRP evolution because it enabled the problem to be mathematically formulated and solved systematically to obtain optimal solutions. Nevertheless, the formulation generated a very large number of variables and constraints, making it less efficient for large-scale applications. As BRP research evolved, four major categories, namely mathematical formulations, heuristics, metaheuristics, and tree search-based methods, were classified as BRP solution approaches [4]. This classification indicates that BRP research has evolved not only toward achieving optimal solutions but also toward improving the capability of models to represent increasingly complex terminal operational conditions.

A major advancement occurred when the BRP-I and BRP-II formulations were introduced in [17]. BRP-I was developed for the Unrestricted Block Relocation Problem (UBRP), whereas BRP-II became the foundation of the Restricted Block Relocation Problem (RBRP), in which only containers directly blocking the target retrieval are allowed to be relocated. The RBRP approach is considered more realistic because it better reflects practical operational constraints in container terminals, such as crane capacity, yard space limitations, and service requirements. Although BRP-II reduced the solution search space compared to BRP-I, the formulation still suffered from weaknesses related to model complexity and the consistency of several mathematical constraints. It is found that the BRP-II formulation still contained redundant variables and required further refinement to improve model efficiency [20]. Similarly, it is also found that several constraint structures in BRP-II led to inefficient search processes in certain cases [21].

To address these limitations, subsequent studies focused on improving mathematical formulations and enhancing model scalability. Therefore, preprocessing and constraint-tightening techniques were developed to reduce model size and improve solution efficiency [22]. Their study showed that preprocessing was able to reduce 65–89% of model variables, significantly improving model performance. However, the approach still faced limitations when applied to very large-scale instances. Subsequently, a relationship representation through the CRP-I model to reduce the number of variables used in previous formulations was developed in [23]. Although this approach improved model efficiency, the formulation still focused primarily on static operational environments.

Further developments concentrated on more efficient search algorithms. A Branch-and-Cut approach to improve the performance of solving large-scale RBRP instances was proposed by [5]. This approach was able to obtain optimal solutions with better performance than previous exact formulations. However, algorithm performance still deteriorated significantly as the number of stacks and containers increased. Later, an iterative exact optimization approach based on relocation sequence combinations to solve instances involving up to 100 items optimally was developed in [24]. Although this method demonstrated substantial performance improvements, the model still assumed relatively deterministic terminal operations and did not consider truck arrival uncertainty or real-time operational dynamics.

As container terminal operations became increasingly complex, BRP research evolved from static models toward more realistic and dynamic environments. Ref. [8] developed the Block Relocation Problem with Time Windows, which incorporates container retrieval time constraints. Their study demonstrated that retrieval sequences are not always fixed because they are influenced by vehicle arrival times. However, the model still assumed deterministic operational conditions. Subsequently, a Stochastic Block Relocation Problem with Flexible Service Policies, which considers service time uncertainty and truck waiting times, was developed in [25]. The main contribution of this study was the introduction of stochastic aspects into BRP, making the model more realistic. Nevertheless, the research still focused primarily on container relocation optimization without simultaneously integrating other terminal operational decisions.

On the other hand, developments in container terminal operations have also encouraged the implementation of Truck Appointment Scheduling (TAS) systems to reduce gate congestion and truck waiting times through predefined truck arrival time slots. However, most TAS studies primarily focus on vehicle traffic optimization and gate service efficiency without considering their impact on container relocation activities in the yard area. Conversely, traditional RBRP models generally ignore truck arrival schedules and assume fixed retrieval sequences. As a result, the relocation solutions generated are often difficult to implement in dynamic and real-time terminal operations.

The relationship between BRP and appointment scheduling was further investigated by [15] through the Block Relocation Problem with Appointment Scheduling model. Their study demonstrated that truck arrival schedules directly influence retrieval sequences and the number of container relocations. The main contribution of this study was the introduction of the integration between relocation decisions and truck scheduling in container terminal operations. Nevertheless, the proposed approach remained sequential, where scheduling and relocation decisions were solved separately. Such an approach may lead to suboptimal solutions because decisions made in earlier stages can restrict optimization flexibility in subsequent stages.

Based on the evolution of previous studies, there remains a need for an optimization model capable of simultaneously integrating the Restricted Block Relocation Problem (RBRP) and Truck Appointment Scheduling (TAS), so that relocation and scheduling decisions can be optimized within a more realistic and efficient framework. Therefore, this study develops an integrated optimization model that connects container relocation decisions and truck appointment scheduling within a simultaneous optimization framework to improve the operational efficiency of container terminals.

This section presents a structured review of previous studies related to the Block Relocation Problem (BRP) and Truck Appointment Scheduling (TAS). The literature was collected from major academic databases, including Scopus, Web of Science, and Google Scholar, to ensure comprehensive and relevant research coverage. The search focused on publications from 2010 to 2025, covering both foundational studies and recent advancements in container relocation optimization and terminal scheduling operations. The primary keywords used in the search included “block relocation problem,” “restricted block relocation problem,” “container relocation,” and “truck appointment scheduling.”

This literature review mainly considers peer-reviewed journal articles and selected conference proceedings that contribute to the development of mathematical models, optimization methods, and integrated logistics systems. Through this review, previous studies are analyzed comprehensively to identify the evolution of existing approaches and to map unresolved research gaps. The analysis particularly focuses on the relationship between container relocation decisions and truck appointment scheduling, especially concerning the objective function of minimizing container relocations within a single block column. Accordingly, this review provides a basis for integrated terminal optimization. The position of this research compared with previous studies is summarized in

Table 1. Position of the research.

Aspect	[7]	[20]	[22]	[26]	[5]	[15]	[27]	(This Research)
Objective Function
Minimizing container relocations	✓	✓	✓	✓	✓	✓	✓	✓
Set
Set	Container, stack, tier, time	Container, stack, tier, time	Container, stack, tier, time	Container, stack, tier, time, group	Container, stack, tier, time	Container, stack, tier, time, time windows, queue, stage	Container, stack, tier, number of configurations	Container, stack, tier, time, time windows, queue, stage
Parameter
Parameter	Number of containers, stack, time	Number of containers, stack, tier, initial position, time	Number of containers, stack, tier, picking order, time, maximum tier capacity	Number of containers, stack, tier, initial position, time	Number of containers, stack, initial position time, maximum tier capacity	Number of containers, stack, tier, initial position, time, pickup time, queue length, time delta	Number of containers, stacks, tiers	Number of containers, stack, tier, initial position, time, pickup time, queue length, time delta, maximum tier capacity
Decision Variables
Container position	Binary	Binary	Binary	Binary	Binary	Binary	Binary	Binary
Containers are moved from	Binary	Binary	-	-	-	Binary	-	Binary
The container was moved to	Binary	Binary	Binary	Binary	Binary	Binary	Binary	Binary
Container taken	Binary	Binary	Binary	Binary	-	Binary	Binary	Binary
Constraint
Pickup time						✓		✓
Maximum queue						✓		✓
Crane capacity						✓		✓
Initial position		✓	✓	✓	✓	✓	✓	✓
Relocation sequence	✓	✓	✓	✓	✓	✓	✓	✓
Relocation movement	✓	✓	✓	✓	✓	✓	✓	✓
Containerpickup	✓	✓	✓	✓	✓	✓	✓	✓
Maximum Height			✓		✓			✓
Restricted relocation	✓	✓	✓	✓	✓			✓
Non-redundant retrieval					✓			✓
Method	Integer Linear Programming	Branch-and-Bound	Mixed-Integer Linear Programming	Branch-and-Bound	Integer Programming	Integer Programming	Mixed-Integer Linear Programming	Integer Programming
Object	Container Terminal	Terminal, Depot, WH Container	Container Terminal	Container Terminal	Container Terminal	Container Terminal	Container Terminal	Container Depot

Note: The symbol “✓” indicates that the corresponding aspect is considered or included in the study.

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This study addresses a gap in the literature by integrating the Restricted Block Relocation Problem (RBRP) with Truck Appointment Scheduling (TAS), which are typically studied separately. The proposed model incorporates practical constraints such as relocation validity, avoidance of redundant movements, efficient slot utilization, and controlled time deviations. Furthermore, the study compares simultaneous and sequential solution approaches, providing insights into their effectiveness and applicability. These contributions offer a more comprehensive and realistic approach to container logistics optimization compared to existing studies.

3. Research Method

The primary reference model in this study is the model proposed by [15], which focuses on optimizing container relocation with the objective of minimizing the number of relocations required while adhering to truck appointments. This reference model has several limitations that need to be addressed in practical applications. The mapping of the shortcomings of the primary reference model [15] and the approaches used to address these limitations with research steps, as proposed by [5,27], is shown in Figure 4.

Following Figure 4, the methodology systematically transforms the reference model into an enhanced formulation addressing identified limitations while ensuring clarity, consistency, and practical applicability. The model reduces redundancy, maintains consistent notation and constraints to better integrate relocation and scheduling decisions. Illustrative examples are included to validate the model and demonstrate its feasibility, ensuring both theoretical soundness and practical relevance.

4. Mathematical Modeling

4.1. Mathematical Model

This mathematical model development combines the primary reference model [15] with additional reference models [5,27] as well as field conditions. In this development, a maximum stack capacity constraint is introduced, limiting the height of the container stack in accordance with operational conditions. Additionally, the container relocation constraint is tightened by ensuring the validity of the relocation, preventing redundant movements, and ensuring correct slot filling of container retrieval and relocation. Therefore, this model can address container scheduling and relocation issues in a more comprehensive manner.

The development of this model is divided into two optimization approach schemes. The explanations of these two schemes are as follows:

• Scheme First (Scheme-1) is a simultaneous integration of the RBRP and TAS (RBRPAS). This approach considers constraints that were previously not addressed in the primary reference model. It is expected that this scheme will provide an optimal solution by combining both problems into a single, integrated optimization process.
• Scheme Second (Scheme-2) is a sequential solution approach, where the Truck Appointment Scheduling (TAS) will be optimized first, followed by the optimization of container relocation (RBRP) based on the results of the scheduling optimization. This scheme also takes into account constraints not considered in the primary reference model, with the aim of improving the accuracy of the optimization results.

Figure 5 illustrates the problem-solving framework using both the simultaneous and sequential approaches.

This section presents the integer programming model developed to integrate the Restricted Block Relocation Problem (RBRP) with Truck Appointment Scheduling (TAS). The model aims to minimize container relocations while ensuring feasible and coordinated scheduling decisions under practical operational constraints.

The formulation includes sets, parameters, and decision variables representing container positions, stack configurations, and truck schedules. The objective function minimizes relocations, while the constraints ensure relocation feasibility, prevent redundant movements, enforce capacity limits, and maintain consistency between relocation and scheduling decisions.

The mathematical model notation for the first scheme (Scheme-1), which adopts a simultaneous optimization approach and considers constraints that were previously unaddressed in the primary reference model, is presented in

Table 2. Notation model of Scheme-1.

Notation	Description	Unit
Set
N	Set of containers N = {1, 2, …, n}	-
C	Set of stacks C = {1, 2, …, c}	-
H	Set of tiers H = {1, 2, …, h}	-
T	Set of time windows T = {1, 2, …, t}	-
K	Set of movement stages K = {1, 2, …, g}	-
Index
i	Container index i ∈ N	-
s	Stack index s ∈ C	-
r	Tier index r ∈ H	-
t	Time window index t ∈ T	-
k	Movement stage index k ∈ K	-
Decision Variables
$u_{isrk}^t$	1 if container i occupies slot (s, r) during stage k of time window t. 0 otherwise	Binary
$x_{isrk}^t$	1 if container i is moved from slot (s, r) during stage k of time window t. 0 otherwise	Binary
$y_{isrk}^t$	1 if container i is moved to slot (s, r) during stage k of time window t. 0 otherwise	Binary
$v_{isrk}^t$	1 if container i is retrieved from slot (s, r) during stage k of time window t. 0 otherwise	Binary
Parameter
N	Number of containers	Unit
C	Number of stacks	Unit
H	Number of tiers	Unit
T	Time windows	Hour
G	Number of stages or crane capacity, maximum number of container movements within a time window	Unit/Hour
$P_i$	Desired retrieval time range for container i	Hour
$I_{isr}$	Initial position of container i in slot (s, r) at time t	-
L	Maximum queue length	Unit
δ	Maximum allowed time shift for container retrieval	Hour
h	Maximum stack capacity (slots/tiers)	Unit

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The mathematical model notation for the first scheme (Scheme-1), which adopts a simultaneous optimization approach, includes the addition of constraints in this model, specifically from Equations (15)–(20), is presented in

Table 3. Mathematical model of Scheme-1.

Mathematical Equations	Equation Number
Objective Function
$Min{\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k=1}^G}{\displaystyle \sum _{t=1}^T}{y}_{isrk}^t$	(1)
Constraints
$\left|p_i-{\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k=1}^G}{\displaystyle \sum _{t=1}^T}t{v}_{isrk}^t\right|\le \delta ,\forall i\in \left\{1,\dots ,n\right\}$	(2)
${\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k=1}^G}{\displaystyle \sum _{t=1}^T}{v}_{isrk}^t\le L,\forall t\in \left\{1,\dots ,t\right\}$	(3)
${\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k=1}^G}{\displaystyle \sum _{t=1}^T}{v}_{isrk}^t+{\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{x}_{isrk}^t\le 1$ $\forall k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(4)
${\displaystyle \sum _{i=1}^N}{x}_{isrk}^t\le {\displaystyle \sum _{i=1}^N}\left(u_{isrk}^t-u_{is\left(r+1\right)k}^t\right)$ $\forall s\in \left\{1\dots ,c\right\},r\in \left\{1,\dots ,h-1\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(5)
${\displaystyle \sum _{s^{\prime }=1,s^{\prime }\ne s}^C}{\displaystyle \sum _{r=1}^H}{x}_{isrk}^t\ge {\displaystyle \sum _{r=1}^H}{x}_{isrk}^t$ $\forall i\in \left\{1,\dots ,n\right\},s\in \left\{1,\dots ,c\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(6)
${\displaystyle \sum _{i=1}^N}{v}_{isrk}^t+{\displaystyle \sum _{i=1}^N}{y}_{isrk}^t+{\displaystyle \sum _{i=1}^N}{x}_{isrk}^t\le 1$ $\forall s\in \left\{1,\dots ,c\right\},r\in \left\{1,\dots ,h\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(7)
$u_{isr1}^1=I_{isr},\forall i\in \left\{1,\dots ,n\right\},s\in \left\{1,\dots ,c\right\},r\in \left\{1,\dots ,h\right\}$	(8)
$u_{isrk+1}^t=u_{isrk}^t+y_{isrk}^t-x_{isrk}^t-v_{isrk}^t$ $\forall i\in \left\{1,\dots ,n\right\},s\in \left\{1,\dots ,c\right\},r\in \left\{1,\dots ,h\right\},$ $k\in \left\{1,\dots ,g-1\right\},t\in \left\{1,\dots ,t\right\}$	(9)
$u_{isr1}^t=u_{isrG}^{t-1}+y_{isrG}^{t-1}-x_{isrG}^{t-1}-v_{isrG}^{t-1}$ $\forall i\in \left\{1,\dots ,n\right\},s\in \left\{1,\dots ,c\right\},$ $r\in \left\{1,\dots ,h\right\},t\in \left\{2,\dots ,t\right\}$	(10)
${\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k^{\prime }=k+1}^G}{u}_{isr{k}^{\prime }}^t+{\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k^{\prime }=1}^G}{\displaystyle \sum _{t^{\prime }=t+1}^T}{u}_{isr{k}^{\prime }}^{t^{\prime }}$ $\le G·T\left(1-{\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{v}_{isrk}^t\right)$ $\forall i\in \left\{1,\dots ,n\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(11)
${\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{\displaystyle \sum _{k=1}^G}{\displaystyle \sum _{t=1}^T}{v}_{isrk}^t=1,\forall i\in \left\{1,\dots ,n\right\}$	(12)
${\displaystyle \sum _{i=1}^N}{u}_{isrk}^t\le 1$ $\forall i\in \left\{1,\dots ,c\right\},r\in \left\{1,\dots ,h\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(13)
${\displaystyle \sum _{s=1}^C}{\displaystyle \sum _{r=1}^H}{u}_{isrk}^t\le 1,\forall i\in \left\{1,\dots ,n\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(14)
${\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{r=1}^H}{u}_{isrk}^t\le H,\forall s\in \left\{1,\dots ,c\right\},k\in \left\{1,\dots ,g\right\},t\in \left\{1,\dots ,t\right\}$	(15)
$x_{i,s,r,k,t}\le u_{i,s,r,k,t}$	(16)
$u_{i,s,r,k,t+1}\le 1-x_{i,s,r,k,t}$	(17)
$x_{i,s,r,k,t}-y_{i,s,r,k+1,t+1}\le 1$	(18)
${\displaystyle \sum _{i\in N}}{u}_{isrk}\ge {\displaystyle \sum _{i\in N}}{u}_{is\left(r+1\right)k}$ $\forall s\in \left\{1,\dots ,c\right\},\forall r\in \left\{1,\dots ,h-1\right\},\forall k\in \left\{1,\dots ,g\right\},\forall t\in \left\{1,\dots ,t\right\}$	(19)
$\begin{array}{c}\hfill u_{isrk}^t, x_{isrk}^t, y_{isrk}^t \mathrm{and} v_{isrk}^t\in \left\{0,1\right\}\\ \forall i\in \left\{1,\dots ,n\right\},s\in \left\{1,\dots ,c\right\},r\in \left\{1,\dots ,h\right\},k\in \left\{1,\dots ,g\right\},\hfill \\ t\in \left\{1,\dots ,t\right\}\hfill \end{array}$	(20)

.

The constraints can be grouped based on their functional roles to improve clarity and interpretation of the model. These include the following:

• Retrieval scheduling and coordination constraints: Equations (2)–(4), which regulate retrieval time windows, the number of retrieved containers, and crane operations.
• Stacking and retrieval sequence constraints: Equations (5)–(10), which enforce the LIFO principle, restrict feasible relocation positions, prevent access conflicts, and ensure consistency of container positions over time.
• Relocation feasibility and movement constraints: Equations (11), (12), and (18), which limit crane capacity, ensure each container is retrieved only once, and prevent redundant relocation movements.
• Capacity and structural constraints: Equations (13), (14), (19) and (20), which enforce slot occupancy, stack capacity limits, and proper stacking structure.
• Linking constraints between relocation and scheduling decisions: Equations (16) and (17), which ensure consistency between container movements and scheduling decisions.

The description of the equation above is as follows. The minimization of relocation aims to reduce unnecessary movement of containers Equation (1). Constraints include limiting the time for container retrieval within the desired time window Equation (2), setting a maximum number of containers that can be retrieved during each time window Equation (3), and controlling crane movement at each relocation stage Equation (4). The relocation rules follow the Last In, First Out (LIFO) principle based on the vertical position of containers within the stack Equation (5) and restrict where containers can be moved Equation (6). Other constraints prevent access conflicts at physical locations Equation (7) and ensure that the initial container placement aligns with field data Equation (8). Each change in container position must be accurately recorded Equation (9), with the initial configuration at time t derived from the final configuration at time t − 1 Equation (10). Container movements during each time window must not exceed crane capacity Equation Equation (11), and each container can only be retrieved once during the time period T Equation (12). Furthermore, each slot (s, r) can only contain one container at any given time t and stage k Equation (13), with the stack capacity not to be exceeded Equation (14). Container transfer can only occur if it aligns with the correct position at the relevant stage and time Equations (16) and (17). After relocation, containers cannot immediately return to their previous positions Equation (18). Finally, if a slot in a stack column is filled, all slots beneath it must also be filled, preventing empty slots Equation (19) and binary variables Equation (20).

In Scheme-2, the optimization model is performed separately. The process begins with the TAS optimization, which aims to determine the optimal truck arrival schedule based on the time window proposed by the trucking company. Subsequently, the RBRP optimization model will process the optimal truck arrival schedule generated by the TAS optimization. At this stage, the RBRP model will further optimize the truck arrival schedule produced by TAS. This step is necessary because the truck arrival times scheduled by TAS optimization often conflict with the time and capacity required for container relocation, leading to an infeasible model. Evidence of the conflicting container management outcomes from TAS optimization can be seen in Figure 6.

The illustration in Figure 6 depicts a scenario of container relocation management that conflicts with the results of TAS optimization. Container 3, which is scheduled for pickup during Time Window 1, is obstructed by another container, requiring relocation that causes delays, longer waiting times, and lateness in the subsequent container pickup. This renders the management model infeasible and reduces operational efficiency. Therefore, an adjustment to the truck arrival times is necessary so that the relocation can utilize the available capacity, which can be addressed through the constraint on permissible arrival time shifts Equation (2).

Figure 6. Evidence of container management scenario conflicts in the stacking yard. Note: Green arrows indicate relocation movements, red arrows indicate retrieval movements, black arrows indicate movement directions within the stack, and the black circle highlights the conflicting container.

Figure 6. Evidence of container management scenario conflicts in the stacking yard. Note: Green arrows indicate relocation movements, red arrows indicate retrieval movements, black arrows indicate movement directions within the stack, and the black circle highlights the conflicting container.

The notation for the TAS mathematical model in the second scheme (Scheme-2), which utilizes a sequential optimization approach, is presented in

Table 4. Notation TAS model of Scheme-2.

Notation	Description	Unit
Set
N	Set of containers N = {1, 2, …, n}	-
T	Set of time windows T = {1, 2, …, t}	-
Index
i	Container index i ∈ n	-
t	Time index t ∈ t	-
Decision Variables
ai,t	1 if container i is scheduled in time window t. 0 otherwise	Binary
Parameter
N	Number of containers	Unit
T	Time Window	Hour
Q	The maximum capacity of containers that can be scheduled within each time window	Unit
$P_i$	The desired time range for the pickup of container i	Hour

:

Below are the equations for the development of the mathematical model for TAS Scheme-2, is presented in

Table 5. Mathematical TAS model of Scheme-2.

Mathematical Equations	Equation Number
Objective Function
$Min Z={\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{t=Mintimei}^{Maxtimei}}{P}_i·a_{i,t}$	(21)
Constraints
${\displaystyle \sum _{t=Mintimei}^{Maxtimei}}{a}_{i,t}=1,\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in N$	(22)
${\displaystyle \sum _{i=1}^N}{a}_{i,t}\le Q,\hspace{1em}\hspace{1em}\hspace{1em}\forall t\in T$	(23)
$a_{i,t}\in \left\{0,1\right\} \forall i\in \left\{1,\dots ,n\right\},t\in \left\{1,\dots ,t\right\}$	(24)

.

The description of the equation above is as follows. The constraints in the container scheduling system aim to minimize the retrieval time Equation (21). Each container is scheduled only once within a specific preference time window Equation (22), and the number of containers scheduled in each time window must not exceed the predetermined maximum capacity Equation (23). Additionally, there are constraints related to binary variables that need to be considered in this scheduling process Equation (24).

The optimization model for RBRP in Scheme-2 adopts the same mathematical structure as the integrated RBRPAS model in Scheme-1. The similarity in the formulation arises because both schemes are based on the same representation of container relocation decisions.

The main difference lies in the treatment of truck arrival times. In Scheme-1, truck arrival times are based on predefined time windows provided by the trucking company. In contrast, in Scheme-2, truck arrival times are first optimized through the TAS model and subsequently used as input for the RBRP optimization.

To avoid redundancy, the formulation of RBRP in Scheme-2 follows the same structural framework as in Scheme-1, with differences only in the input parameters related to truck scheduling. In particular, Equation (2) is adapted to represent the constraint that limits the allowable deviation between the actual container retrieval time and the scheduled retrieval time P_i, which is determined from the TAS optimization results. This constraint specifically regulates how much the actual handling time of container i can deviate from the predetermined time P_i, thereby ensuring consistency between the TAS outcomes and the RBRP optimization.

The problem-solving process aims to optimize the relocation of containers and the scheduling of truck arrivals at the container depot to enhance operational efficiency. The flowchart of the optimal solution in Scheme-1 and Scheme-2 illustrates the steps involved in finding the solution, as shown in Figure 7 and Figure 8.

4.2. Illustrative Numerical Example

To improve the clarity and comprehensibility of the proposed optimization model, a small numerical example is presented. This example illustrates how the model determines container relocation and scheduling decisions under simplified conditions.

Consider a container yard consisting of three stacks (C = 3) and a maximum of three tiers (H = 3). A total of four containers (N = 4) are stored in the yard with a predefined retrieval sequence. The initial configuration of the containers is shown as follows:

• Stack 1: Container 2 (bottom), Container 1 (top)
• Stack 2: Container 3
• Stack 3: Container 4

The retrieval order is defined as follows: Container 1 → Container 2 → Container 3 → Container 4.

In this configuration, Container 1 can be retrieved directly since it is located at the top of Stack 1. However, retrieving Container 2 requires a relocation operation because it is blocked by Container 1.

According to the model constraints, Container 1 must first be retrieved before accessing Container 2. If Container 2 were required earlier, Container 1 would need to be relocated to another stack with available capacity, in accordance with the relocation feasibility constraints.

For illustration, suppose that Container 1 is relocated to Stack 3, resulting in the following configuration:

• Stack 1: Container 2
• Stack 2: Container 3
• Stack 3: Container 4, Container 1

After the relocation, Container 2 becomes accessible and can be retrieved. The model ensures that such relocation decisions minimize unnecessary movements while respecting stack capacity, non-overlapping constraints, and retrieval priorities.

In the context of truck appointment scheduling, assume that each container is associated with a preferred time window. The model determines the optimal retrieval schedule by aligning container availability with truck arrival times while minimizing deviations from the desired schedule.

This simplified example demonstrates how the proposed model coordinates container relocation and scheduling decisions, ensuring both feasibility and efficiency in operations.

5. Results and Discussion

5.1. The Results of the Computational Solution

The results of the model development computation show that the best solution found was 1 solution by Scheme-1. The comparison of the computational results for each scheme is summarized in

Table 6. Computation results for each scheme.

Objective Function	Scheme-1 Solution	Scheme-2 Solution
Total container relocations	One Relocation	Two Relocations

.

Based on the model development results obtained using Scheme-1 and Scheme-2 approaches, Scheme-2 produces one relocation, whereas Scheme-2 results in two relocations. This difference indicates that simultaneous and sequential solution approaches lead to different outcomes. The simultaneous approach optimizes the process simultaneously, meaning that the container transfer and pickup processes directly consider the truck arrival scheduling, which allows the simultaneous optimization to minimize the number of relocations needed. In contrast, the sequential optimization approach performs the optimization process separately, meaning the TAS result is already optimized in advance. As a result, the subsequent RBRP optimization stage is based on the truck arrival time that has already been determined. The minimal number of relocations will optimize the container flow and ensure its movement is efficient in terms of both time and cost.

5.2. The Results of the Optimal Truck Arrival Time

The truck arrival schedule is derived from the analysis of optimal container pickup scenarios. The scenario refers to the planned sequence of container management events based on the optimization results. This is done to ensure and support the smooth distribution process of containers efficiently within the planned time frame. The truck arrival schedule for Scheme-1 can be seen in the

Table 7. Excerpt of optimal truck arrival schedule Scheme-1.

Block Column	Truck Arrival Time	Container
1	0	Container 9 picked up at Time Window 0
1	1	Container 14 picked up at Time Window 1
1	1	Container 16 picked up at Time Window 1
1	1	Container 17 picked up at Time Window 1
1	1	Container 18 picked up at Time Window 1
1	2	Container 2 picked up at Time Window 2
1	2	Container 11 picked up at Time Window 2
1	2	Container 12 picked up at Time Window 2
1	2	Container 13 picked up at Time Window 2
1	3	Container 4 picked up at Time Window 3
1	3	Container 5 picked up at Time Window 3
1	3	Container 6 picked up at Time Window 3
1	3	Container 8 picked up at Time Window 3
1	4	Container 3 picked up at Time Window 4
1	4	Container 7 picked up at Time Window 4
1	4	Container 15 picked up at Time Window 4
1	5	Container 0 picked up at Time Window 5
1	5	Container 1 picked up at Time Window 5
1	5	Container 10 picked up at Time Window 5

.

The truck arrival schedule for Scheme-2 can be seen in the

Table 8. Excerpt of optimal truck arrival schedule Scheme-2.

Block Column	Truck Arrival Time	Container
1	0	Container 2 picked up at Time Window 0
1	0	Container 6 picked up at Time Window 4
1	1	Container 0 picked up at Time Window 6
1	1	Container 1 picked up at Time Window 5
1	1	Container 5 picked up at Time Window 3
1	2	Container 8 picked up at Time Window 4
1	2	Container 10 picked up at Time Window 5
1	3	Container 4 picked up at Time Window 3
1	3	Container 9 picked up at Time Window 1
1	3	Container 16 picked up at Time Window 2
1	4	Container 3 picked up at Time Window 3
1	4	Container 7 picked up at Time Window 4
1	4	Container 15 picked up at Time Window 5
1	5	Container 12 picked up at Time Window 4
1	5	Container 13 picked up at Time Window 3
1	5	Container 14 picked up at Time Window 2
1	6	Container 11 picked up at Time Window 5
1	6	Container 17 picked up at Time Window 2
1	6	Container 18 picked up at Time Window 1

.

5.3. The Results of the Container Management Scenario

The container management scenario in the stacking yard is formulated based on the decision variables $u_{isrk}^t$, $x_{isrk}^t$, $y_{isrk}^t$, and $v_{isrk}^t$, are defined to represent container positions, retrieval decisions, and relocation movements across stacks, tiers, time periods, and movement stages. These are processed through computation to determine the best management strategy. Below is an excerpt of the container management scenario for Scheme-1 and Scheme-2 in the stacking yard, as shown in Figure 9.

5.4. Comparison Analysis of Container Pickup Time Shifts

The subsequent analysis focuses on the time shift for container pickup, considering both the expected pickup time set by the trucking company and the optimal container pickup time that minimizes the number of relocations required. The desired container pickup time by the trucking company is known in advance and will be optimized through various schemes that have been designed. Picking up containers earlier or later than the desired time can also impact operational efficiency, particularly in terms of relocation management. To support this efficiency, the allowable time shift has been established within the model parameters, with a constraint of two time windows. The comparison chart of the optimal container pickup time shifts for Scheme-1 with the desired pickup time set by the trucking company can be seen in Figure 10.

The comparison chart of the optimal container pickup time shifts for Scheme-2 with the desired pickup time set by the trucking company can be seen in Figure 11.

Based on the results of the t-test analysis for Scheme-1 and Scheme-2, it can be concluded that there is no significant difference between the two schemes. This is evidenced by the Sig. value of 0.855, which is greater than 0.05. The t-Test results can be seen in Figure 12.

The implication of the results from the entire t-Test analysis is that there is no significant difference in the deviation in container pickup times between the two schemes.

5.5. Analysis of Crane Stage or Capacity

The stage or crane capacity (Equation (4)) refers to the maximum number of movements that the crane can perform to manage containers within a single time window. Crane capacity efficiency can be achieved by maximizing crane movements within a time window, thereby reducing the number of container relocations.

Based on the optimization results as shown in Figure 13, the number of crane movements or stages performed within a single time window for each scheme has reached the maximum capacity without exceeding the limit set in the model parameters, which is 4 (four) stages or movements within one time window.

5.6. Occupancy Rate Analysis

Occupancy rate is a measure that indicates the percentage of capacity of a facility that has been utilized. The occupancy rate is used to assess the efficiency of space or capacity utilization in the stacking yard. The occupancy rate is calculated by comparing the number of occupied units (containers) with the total number of available units. The occupancy rate chart can be seen in Figure 14.

It is known that with seven stacks, five tiers, and 13 containers, the occupancy rate is only around 37.14%. However, as the number of containers increases, this percentage also rises. At 25 containers, the occupancy rate reaches 71.43%, indicating a relatively high occupancy of the stacking area, leaving an allowance of 28.57% for container movement. This analysis illustrates that container stacking management must be optimized to prevent overloading and difficulties in container retrieval or relocation, especially as the number of containers continues to increase.

5.7. Sensitivity Analysis

Sensitivity analysis is a technique used to evaluate the changes in input parameters and their impact on the results or outputs produced by a model. This technique serves to identify which variables have a significant effect on the model being used.

In this study, sensitivity analysis was conducted to test and assess the impact of parameter variations on the results obtained from the optimization model. The parameters used include stack capacity (C = 7), number of tiers (H = 5), number of time windows (T = 8), number of movement stages (G = 4), maximum queue length (L = 4), and the allowed time shift limit (Delta = 2). Variations were then made in the number of containers (N), with variations in N = 15, N = 17, N = 19, N = 21, N = 23, and N = 25. Sensitivity testing was applied to the two schemes that had been defined, namely Scheme-1 and Scheme-2. The comparison results of the variation in the number of containers in the stacking yard and the number of container relocations can be seen in

Table 9. Comparison of sensitivity results for the number of containers.

Parameter	C = 7, H = 5, T = 8, G = 4, L = 4, Delta = 2
	N = 13	N = 15	N = 17	N = 19	N = 21	N = 23	N = 25
Scheme-1	1	1	1	1	2	3	3
Scheme-2	1	2	2	2	2	3	3
Percentage Change	−31.58%	−21.05%	−10.53%	0%	10.53%	21.05%	31.58%

.

Based on the trade-off chart for Scheme-1 and Scheme-2 as shown in Figure 15, it can be concluded that the simultaneous optimization (Scheme-1) approach tends to be stable, with minimal changes in the number of relocations as the value of N increases, and it stabilizes again when the number of N reaches 23. Data analysis shows that for N = 13 to N = 19, the number of relocations is 1. At N = 21, the number of relocations increases to 2, marking a 100% increase. Subsequently, from N = 23 to N = 25, the number of relocations increases to 3, showing a 50% increase compared to the previous value at N = 21. In Scheme-2, it can be observed that in the sequential optimization approach, the number of relocations tends to increase as the value of N increases.

The sensitivity analysis of the number of crane movements or stages within a single time window in relation to the number of relocations generated is conducted to assess the impact of parameter variations. In this sensitivity analysis, the parameters used include stack capacity (C = 7), number of tiers (H = 5), number of time windows (T = 8), number of containers (N = 25), maximum queue length (L = 4), and the allowed time shift limit (Delta = 2). Furthermore, variations were made in the number of crane movements or stages, with variations in G = 3, G = 4, N = 5, and N = 6. The comparison results of the variation in the number of movements or stages and the number of container relocations can be seen in

Table 10. Comparison of sensitivity results for the number of movements or stages.

Parameter	N = 19, C = 7, H = 5, T = 8, L = G, Delta = 2
	G = 2	G = 3	G = 4	G = 5	G = 6
Scheme-1	Infeasible	1	1	1	1
Scheme-2	Infeasible	3	2	3	2
Percentage Change	−50.00%	−25.00%	0.00%	25.00%	50.00%

.

Based on the trade-off chart for Scheme-1 and Scheme-2 as shown in Figure 16, it can be concluded that in Scheme-1, the simultaneous optimization approach tends to be stable, with the number of relocations remaining constant as the number of G increases. This occurs because, in Scheme-1, the TAS problem is solved simultaneously with the RBRP, allowing both processes to be optimized concurrently. The stable and consistent number of relocations indicates that simultaneous optimization can maintain system performance, avoid resource waste, and reduce imbalance in operational processes. In Scheme-2, the sequential optimization approach is less stable, with fluctuating relocations as the number of G increases. This happens because, in Scheme-2, the TAS problem is solved separately from the RBRP, leading to the optimization of each process independently. The trade-off results between crane movements (stages) and the number of container relocations show that sequential optimization can lead to resource wastage and imbalance in operational processes.

5.8. Model Comparison

To evaluate the performance and consistency of the proposed model, a comparison study was conducted against the reference model developed by [15]. The comparison was performed using identical operational parameters to ensure that both models were evaluated under equivalent operational conditions. The experiment involved 15 containers distributed across six stacks with a maximum stack height of three tiers. The operational planning horizon was set to 6 h with four operational stages per hour. In addition, the maximum queue length was limited to four units, while the allowable scheduling time-shift parameter (δ) was set to 1 h. These parameter settings were selected to represent controlled operational conditions for evaluating the consistency and performance of the proposed model. The comparison results are presented in the

Table 11. Model validation solution results.

Objective Function	BRP-AS Model Solution in [15]	Scheme-1 Model Solution
Total container relocations	Three relocations	Three relocations

.

Both the reference model and the proposed Scheme-1 model produced the same objective function value, namely, three container relocations. This result indicates that the proposed model is mathematically consistent and capable of reproducing the relocation performance generated by the previous model. Therefore, the validation confirms the correctness of the developed formulation and demonstrates that the integration mechanism introduced in this study does not reduce solution quality in terms of relocation minimization.

Although both models produced the same number of relocations, the proposed model provides several important extensions compared to the model developed by [15]. The previous model primarily focused on the relationship between relocation activities and truck appointment scheduling using a sequential decision structure. In contrast, the proposed model integrates the Restricted Block Relocation Problem (RBRP) and Truck Appointment Scheduling (TAS) simultaneously within a unified optimization framework. This integration enables relocation and scheduling decisions to be evaluated concurrently rather than independently.

Furthermore, the developed model incorporates additional operational considerations that were not explicitly represented in the previous model, including retrieval sequence coordination, queue length limitations, and scheduling synchronization. These additional constraints allow the proposed model to represent container terminal and depot operations more realistically, where relocation decisions are strongly influenced by truck arrival schedules and yard congestion conditions.

From an operational perspective, the simultaneous optimization approach developed in this study provides a more coordinated decision-making mechanism compared to conventional sequential approaches. The integrated optimization framework improves operational consistency and reduces the potential for inefficient reshuffling movements caused by disconnected scheduling and relocation decisions. Therefore, although the relocation results are numerically identical, the proposed model offers a more realistic and comprehensive representation of terminal operations.

Overall, the comparison results demonstrate that the proposed model not only maintains the relocation performance achieved by the reference model but also extends previous research toward a more integrated and operationally applicable optimization framework for container terminal and depot operations.

5.9. Discussion of Results and Contributions

The results obtained in this study align with previous research on the Block Relocation Problem (BRP), which emphasizes minimizing the number of relocations to improve operational efficiency. However, most existing research still treats BRP as a stand-alone problem and has not explicitly considered detailed operational constraints or their interactions with truck arrival schedules.

Compared to these approaches, the proposed model in this study introduces an integrated framework that simultaneously considers container relocation and Truck Appointment Scheduling (TAS). In addition, the model incorporates several practical constraints that are often not fully addressed in the existing literature, including ensuring the validity of relocation movements, preventing redundant container movements, enforcing proper utilization of empty slots, and accounting for both retrieval and relocation priorities.

By incorporating these additional constraints, the proposed model is able to generate more realistic and operationally feasible solutions. The results show that the model can generate optimal relocation decisions while maintaining minimal allowable time deviation from the scheduled truck arrivals. This level of coordination between relocation and scheduling decisions has not been explicitly considered in previous research.

Furthermore, the findings indicate that the simultaneous approach provides more stable and efficient results compared to the sequential approach. This is consistent with previous studies that highlight the importance of integrated optimization in logistics systems. However, this study extends the literature by demonstrating that the inclusion of detailed operational constraints can further enhance solution quality and applicability in real-world container depot operations.

Overall, the results of this study confirm that integrating relocation optimization with truck scheduling, combined with realistic operational constraints, leads to improved system performance and provides a meaningful contribution to the existing literature.

6. Conclusions and Recommendations

This study develops an integrated optimization model that combines the Restricted Block Relocation Problem (RBRP) with Truck Appointment Scheduling (TAS) to minimize container relocations and improve operational efficiency in container depot management. The proposed model incorporates several practical constraints, including stack capacity limits, relocation validity, prevention of redundant movements, proper utilization of empty slots, and controlled time deviations, resulting in a more realistic and applicable optimization framework.

The findings of this study directly address the research questions proposed at the beginning of the study. First, in relation to RQ1, this study successfully develops an integrated optimization model that combines RBRP and TAS, enabling more efficient and well-coordinated container relocation management.

Second, addressing RQ2, the results show that the simultaneous approach (Scheme-1) outperforms the sequential approach (Scheme-2) in minimizing relocations, with greater stability and robustness under varying conditions, while the sequential approach is more sensitive to fluctuations. However, the t-test results indicate no significant difference (p = 0.855) in pickup time deviations, suggesting similar time-shift performance. Therefore, the advantage of the simultaneous approach lies in improving relocation efficiency rather than pickup time performance.

Third, with respect to RQ3, the selection of an appropriate solution method for real-world implementation depends on its ability to balance relocation efficiency and operational time. The results indicate that the simultaneous approach is more effective and efficient in optimizing both aspects, making it highly suitable for practical application in container depots and similar logistics systems.

From a theoretical perspective, this study contributes to the existing literature by extending conventional BRP models into an integrated framework that incorporates truck appointment scheduling and additional operational constraints. The findings demonstrate that ensuring relocation validity, preventing redundant movements, enforcing proper slot utilization, and considering retrieval and relocation priorities can lead to optimal solutions with minimal allowable time deviation—an aspect that has not been fully addressed in many previous studies.

From a practical perspective, the proposed model provides valuable decision support for container depot and terminal operators. It can be used to reduce unnecessary relocation operations, minimize truck waiting times, and improve the utilization of limited storage space. The simultaneous optimization approach, in particular, offers a more stable and efficient solution, making it more applicable under real-world operational conditions.

Based on these findings, this study recommends that future research explore more about container pickup priority, complex operational scenarios, such as multi-block container management and crane assignment strategies. In addition, future studies should consider uncertainty in truck arrival times and operational capacities, as well as the application of heuristics and metaheuristics approaches to address large-scale computational challenges and improve solution efficiency.

This study recommends future research to explore multi-block interactions with a crane assignment scheme, priority pickup containers considering uncertainty in truck arrival times and capacity, and utilizing heuristics and metaheuristics methods to address large-scale computational challenges, providing faster and more efficient solutions.