2.1 Transportation planning and truck scheduling

Transport planning and truck scheduling are important research directions in the field of transportation. In recent years, researchers have proposed many new models and methods in these areas to optimize the efficiency of traffic networks and truck transportation, aiming to reduce travel time, minimize congestion, optimize resource allocation, and improve overall transportation system performance. The latest research has made significant contributions in areas such as route optimization, traffic flow management, demand forecasting, and intelligent transportation systems.

Firstly, in the aspect of traffic network optimization, researchers aim to minimize traffic congestion and reduce travel time by optimizing the topology and traffic flow of the network. The latest research utilizes various optimization techniques such as integer programming, linear programming, and graph theory to optimize specific components of the traffic network, including traffic signal timings, intersection design, and road network layout [8–10]. Secondly, in terms of path planning and route selection, researchers have developed efficient algorithms to find optimal driving paths and routes considering factors such as traffic conditions, road capacity, real-time data, and optimization criteria like fuel consumption or carbon emissions. These algorithms aim to minimize driving time, distance, and environmental impact. Furthermore, the latest research focuses on multimodal path planning, which involves integrating different modes of transportation, including walking, cycling, and public transportation, in route planning [11, 12]. For truck scheduling and delivery route optimization, researchers are committed to developing efficient truck scheduling algorithms to optimize truck itineraries and delivery routes, considering factors such as truck capacity, load restrictions, delivery windows, and transportation costs. These algorithms aim to minimize transport time, costs, and empty mileage. Recent research also explores truck sharing and consider multi-objective optimization approaches to improve logistics efficiency and reduce carbon emissions [10, 13, 14]. With the development of intelligent transportation systems, researchers are increasingly interested in real-time vehicle path tracking and dynamic scheduling. Utilizing technologies such as GPS, mobile internet, and sensor data to monitor truck positions and traffic conditions in real-time. Based on this information, real-time scheduling and path planning capabilities are enabled [15, 16].

In conclusion, the latest research in traffic planning and truck scheduling aims to improve traffic efficiency, reduce congestion, and optimize logistics efficiency. By developing efficient algorithms and models, the latest research significantly contributes to the sustainable development of urban transportation and logistics, promoting environmental sustainability and enhancing the overall quality of transportation and logistics systems.

2.2 Transporter scheduling

Transporter scheduling is a related research area in the field of truck scheduling and is also the research problem of this paper. Therefore, in this section, a comprehensive review of research in the field of transporter scheduling will be conducted, focusing on key methodologies, algorithms, and optimization techniques. The transporter transportation scheduling at shipyards addresses the simultaneous pickup and delivery and time windows. This problem aims to determines the optimal transportation time and means of transportation between pick-up and delivery locations while considering requirements such as capacity, priority, and time window constraints [17].

Many scholars have researched the problem of single transporter transportation at shipyards. Kim and Joo [18] assumed that the transportation demand of each block is predetermined in a static condition. Under concurrent machine scheduling conditions, considering factors such as order setting time and work process limitations, Park and Seo [19] studied the transporter scheduling and path problem at shipyards. Joo and Kim [20] proposed a method for determining the transit time and method of task blocks from the origin yard while considering delivery constraints. Liu et al. [21] considered a task block transportation storage problem, incorporating constraints related to block time windows and movement costs. Roh and Cha [22] focused on minimizing travel distance without loading transport transporters and avoiding interference between them. They developed a block transportation scheduling system using a hybrid optimization algorithm, which was applied to an actual block transportation scheduling problem in a shipyard. To accurately simulate the uncertainty and dynamics of block transportation, Chen and Tian [23] studied a method of multi-type block storage yards at shipyard. By projecting the status of ship blocks’ transfers and temporary storages, they capture the aforementioned uncertainty and dynamics. Finally, Kwon and Lee [24] solved the problem of spatial scheduling of large assembly blocks, with objectives such as minimizing makespan and achieving load balancing.

It is apparent that the researchers’ primary goal was to minimize overall completion time, expenses, or delays, with relatively less attention paid to considering the non-value-added time of the transportation personnel in their studies.

2.3 Cooperative transportation

So-called multiple transporters transport, in other words, it is a way to transport heavy or large blocks in cooperation with other transporters, and such transportation requires the cooperation of transporters. When two transporters cooperate to load the same cargo, they must maintain the same transport route and transport speed. Several researchers have studied different cooperative transport problems. In the problem formulation of aircraft, Ioachim et al. [25] introduced a novelty sort of constraints, related to synchronization with schedules, and an optimal solution is proposed. Li et al. [26] investigated a simultaneous scheduling problem with the goal of achieving accurate delivery with minimal cost in assembly and air freight for the consumer electronics supply chain. To coordinate the different activities in a supply chain solution, Zandieh and Molla-Alizadeh-Zavardehi [27] studied the synchronization between production and air transport scheduling with the help of a mathematical model. Salazar-Aguilar et al. [28] introduced an arc-shaped synchronous routing problem applicable to snow clearing operations. El Hachemi et al. [29] presented a synchronization problem between routing and scheduling, specifically applied to forestry and the same type as the log truck scheduling problem. In some cases, Reinhardt et al. [30] considered the transportation of passengers as a multimodal problem with simultaneous constraints. Salazar-Aguilar et al. [31] addressed a synchronous routing problem for arcs and nodes, inspired by practical applications in road marking operations. Lee et al. [32] developed a supply chain model that integrates supplier multi-buyer inventory with transportation synchronization. For the scheduling of transporters in assembly areas at shipyards, N.-R. Tao et al. [33] proposed a metaheuristic algorithm considering constraints such as task block order and transportation cooperation. Jiang et al. [17] created a heuristic optimization algorithm to tackle the challenge of scheduling eco-friendly transportation with multiple vehicles and a single cargo in shipyards.

However, few scholars have taken into account the dynamic uncertainty in the scheduling process when addressing cooperative transportation scheduling problems.

2.4 DT

Research referring to DT has also inspired numerous studies in recent years. Dynamic scheduling methods are crucial for modern manufacturing systems due to the unpredictable nature of events that can occur during production. Examples of unforeseen circumstances in a production system include task insertions, order cancellations, workers’ absenteeism, and machine and equipment breakdowns [34]. In response, researchers have proposed various approaches to improve scheduling accuracy and efficiency. For instance, Zhang et al. [35] presented a DT model with five dimensions for workshop settings and investigated the techniques for predicting machine availability, detecting interference, and evaluating performance using this model. To overcome the limitations of traditional scheduling methods, Villalonga et al. [36] proposed a new framework that utilizes DT aggregation. Zhang et al. [37] devised a two-tier distributed architecture for dynamic shop floor scheduling, integrating DT scheduling agents for shop floor and multiple service units. These innovative approaches demonstrate the potential of DT technology to enhance manufacturing scheduling processes. Neto et al. [38] have developed a decision support system that utilizes real-time opportunities, such as supply shortages, temporary machine idleness, and equipment breakdowns, to minimize overall production penalties while scheduling preventive maintenance interventions. Rao Pabolu and Shrivastava [39] proposed an optimal dynamic solution for job rotation of fatigued workers on assembly lines, utilizing machine learning-based DT. Frantzén et al. [40] developed a new approach to optimization and data analysis using genetic programming heuristics and simulations running on DT. In terms of process simulation and production scheduling in the food processing industry, Koulouris et al. [41] discuss the application of integrated process and DT models. H. Zhang et al. [42] presented a multi-layered modelling framework that spans from the unit level to the system level, addressing the challenge of implementing model building from different perspectives on temporal and spatial scales. Z. Liu et al. [43] developed an intelligent scheduling approach utilizing DTs and super networks to generate process plans quickly and efficiently. Zhao et al. [44] proposed a DT-based approach to multi-crane scheduling and crane quantity selection.

Based on the above studies, we can conclude that DT can be effectively used to predict the dynamic behavior and scheduling of multiple transportation vehicles, thereby better improving shipbuilding efficiency. With the use of DT technology, dynamic uncertainties in transporter movements can be predicted and evaluated in advance. As a result, we present a study on how the scheduling of multiple transporters can be improved, for example, by optimizing routes, reducing idle time, or minimizing conflicts, using this technology.

3.1 Problem description

There are many sections between ship blocks, and each section has to go through multiple processes such as processing and cutting. These processes need to be completed separately in different yards. Therefore, transporters play a crucial role in shipyards scheduling. In the actual transportation process of blocks, due to other uncertain constraints such as irregular length, width and height, large volume and weight, and the height of the road, it is likely that the transporters cannot pass through the obstructed road section or affect its transportation scheduling efficiency. This represents a typical multi-vehicle-one-cargo dynamic scheduling problem. The problem of multiple transporters of block transportation scheduling at shipyards can be described as follows.

There exist a variety of transportation tasks (indexed by i,j∈{1,2,…,n},i ≠ j) based on production requirements; The transportation of blocks within shipyards, from the starting yard to the destination yard, is performed using special vehicles called transporters, which are indexed by t ∈T = {1,2,…,t}. Various types of transporters generally have a capacity ranging from 200-300 tons, with some capable of carrying up to 500 tons. Block transportation primarily relies on heavy-duty transporters for movements between the work yard and storage yard. However, there are limited resources of transporters available within the shipyards. Currently, the maximum load-bearing capacity of an existing shipyard transporter is 450 tons. When a single block exceeds this capacity, it is referred to as a large block and requires multiple transporters transportation.

The principle of multiple transporters transportation in the block yard is presented in Fig 1. The transportation includes information such as task, block ID, block weight capacity, starting yard code, destination yard code, and time window. The flatbeds are characterized by their car number and load capacity.

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Fig 1. The principle of multiple transporters transportation.

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The underlying assumptions for this question are as follows.

1. The transporters cannot be aborted during a block transporter mission.
2. The weight of the large blocks must not exceed the combined load capacity of the two transporters.
3. Both transporters must maintain the same speed during cooperative transport.
4. Each block has a time window and all blocks must be released before they can be transported.
5. Block beyond the release time or expected time is subject to penalty.
6. Task blocks and constraint blocks are in the task sequence of the transporters and their sequencing needs to be satisfied when scheduling.
7. The extraction and stacking of blocks are an important part of segmented transport and cannot be ignored.
8. Transporters maintain a consistent speed regardless of their load status.
9. The analysis focuses on road access conditions within the block yard and does not consider external road conditions.

The assumptions in this paper are mainly based on two articles, such as [29, 33]. Based on the above assumptions, this paper will address the sequence of transport tasks and the best transport routes for the transporters, with the starting execution time of every block.

3.2 Model formulation

The sets, parameters and decision variables associated with the model are defined in Table 1.

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Table 1. Notation.

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The scheduling problem at hand can be formulated as a mixed integer linear problem with the objectives: (1)

Objective (1) aims to minimize the total transportation time for the transporters, excluding any non-value-added time. The time consists of three parts: the unloaded transport time of one task block and another neighboring task block by the transporter, and the transporter the unloaded transport time including departure and back to parking area, and the not working time of the transporters ahead of time window. In this paper, the weighting coefficients α and β mentioned here represent the balance between the empty running time and waiting time of the transporter when performing tasks. Considering the actual situation of the shipyard, there may be differences in the no-load time and waiting time of the transporter during different stages of the transportation process, and therefore they need to be adjusted through weighting coefficients. In this study, we determined that both α and β are 1 by references [17, 18, 20, 22] and repeated experiments.

The constraints of this paper can be categorized as assignment constraints, which include constraints (2) to (9).

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

The constraint (2) guarantees that the normal block is only one transporter required for transportation, and then the large block requires two transporters synchronization transportation. The constraint (3) indicates that the time window of block. The constraint (4) is the close time of the park area. The constraint (5) represents that the start time of the park area. The weight of block must not exceed the load capacity of the transporters which is represented by constraint (6). The constraint (7) states that the large block can only be executed when both transporters have arrived at the starting yard. In constraint (8), each task block is assigned to only one transporter, ensuring that it appears once. Constraint (9) enforces the same condition for all task blocks.

The assignment constraints are constraint (10) to constraint (14).

(10)

(11)

(12)

(13)

(14)

The constraint (10) shows that each transporter only has a first task. The constraint (11) shows that each transporter just has a final task. Constraint (12) defines the start time of the next block based on the upper limit of the previous task block’s end time () for the corresponding transporter. Constraint (13) prohibits the repetition of consecutive task blocks. Constraint (14) defines the feasible range of values for the decision variables, taking into account the specific constraints and requirements of the problem.

4.1 DT framework

Building upon the work of Tao et al. [45] in DT-driven product design, manufacturing, and service, our study has developed a DT-based framework for transporter scheduling that addresses dynamic uncertainty, as illustrated in Fig 2. The framework consists of five essential components: a physical block yard, a virtual block yard, a twin data center, system support services, and the connections and interactions among these elements.

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Fig 2. DT-based architecture of block transportation.

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Based on the above framework and model, the DT-based framework for multiple transporters of block transportation scheduling optimization at shipyards are fully utilized. The objective is to achieve dynamic scheduling of multiple transporters by integrating real-time data and historical data generated from cooperative blocks transportations. Additionally, the framework drives relevant services within the system through a data service platform to effectively manage uncertainties in the scheduling process of transporters.

To facilitate the comparison, we utilize Unity and Plant-simulation to construct a 3D twin space, accurately replicating the real scene on a 1:1 scale, as shown in Fig 3. We construct 3D DT models of task blocks, yards, and transporters to create a comprehensive virtual model that incorporates actual transportation data from the shipyard. This ensures the verification of virtual-real consistency.

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Fig 3. The process of a DT-based multiple transporters scheduling.

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4.2 Approach for scheduling transporters based on GA

While minimizing redundant trips by transporters can assist in uncertainty management, modifying task sequences and assignments may introduce trade-offs between efficiency and potential contingencies. In the case of t transporters and i tasks, there can exist a large number of possible scenarios tⁱ, making multiple transporters scheduling a challenging NP-hard problem. To address this challenge, various researchers have employed different methods, such as branch-and-check (Diefenbach et al. [46]; Li and Zhou [47]) and heuristic methods (Yin et al. [48]; Sistig and Sauer [49]). Among these methods, GA has demonstrated good performance and robustness in scheduling problems (Xu et al. [50]; Wu and Pang [51]; He et al. [52]). Therefore, GA was employed as the transporter scheduling method in this study.

The flowchart illustrating the integration of GA and the simulation process is presented in Fig 4.

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Fig 4. The process of a DT-based multiple transporters scheduling.

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4.2.1 Chromosome representation.

The first step is to define the encoding scheme. A critical aspect in GA design is the representation of chromosomes, which plays a vital role in achieving effective solutions. In this study, a three-dimensional coding scheme is utilized to represent the chromosomes, enabling the generation of an initial feasible solution, as depicted in Fig 5. Fig 5 displays the chromosome representation, where the first row corresponds to block numbers (40 blocks in total). The second and third rows indicate the number of transporters(six transporters in total), facilitating the encoding scheme. Number of transporters is listed in the second and third rows of the chromosome, with a total of six transporter.

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Fig 5. Three-dimensional coding and the associated initial feasible solution.

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4.2.4 Crossover and mutation operation.

Offspring are generated using a single-point mutation method, which involves altering a single gene within the chromosome. A crossover operation example is displayed in Fig 6. Once the selection operation is complete, parent individuals randomly to produce offspring. The crossover probability, which determines the likelihood of genetic material exchange between parental chromosomes, is determined by their crossover rate.

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Fig 6. Example of single-point crossover operations.

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Chromosome mutation in GA serves a distinct purpose from crossover operations, introducing diversity and exploring new solutions. While sharing some similarities with crossover, chromosome mutation plays a separate role in the optimization process. The exchange mutation method involves two steps, as shown in Fig 7. Firstly, generating an index that corresponds to a specific block within the parent chromosome. And secondly, obtaining the new generation chromosome by swapping the indexed block with another block based on its chromosome value.

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Fig 7. Example of swap mutation operations.

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5.1 Case study and data generation

In this study, we assume that there are task blocks and transporters. In addition, we obtained detailed data from a shipyard for 40 groups of block tasks, including specific information such as starting yard, time window and load bearing, respectively, as shown in Table 2. In this paper, we refer to the blocks whose weight exceeds the load-bearing capacity of a single transporter truck as large block, e.g., the 9th block, the 18th block, the 28th block, and the 38th block in Table 2 are all large blocks.

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Table 2. Notation.

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Regarding the transporters, the number of transporters and their carrying capacity are shown in Table 3. And when the transporters are traveling unloaded, the speed is 3 m/s; when the transporters are loaded, the speed is 6 m/s.

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Table 3. The data information for transporters.

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In addition to this, we have specified the distances between the starting yards in Table 1, which is shown in Table 4. It is worth noting that these experiments were conducted on advanced computers equipped with an Intel Core i5 processor, 4GB memory, MATLAB software, and CPLEX software, ensuring the highest level of accuracy and reliability.

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Table 4. The distance information between yards.

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5.2 Parameter tuning

After completing the data set preparation in the preceding section, this study aims to conduct experimental investigations and research on parameter tuning in the current section to improve the effectiveness of metaheuristic algorithms in solving complex optimization problems. To achieve this goal, we utilize the Taguchi experimental method to fine-tune algorithmic parameters and reduce the number of iterations required. It is worth noting that this methodology has been employed in numerous prior investigations, as demonstrated by [53, 54]. Within this framework, the signal-to-noise ratio (S/N) is leveraged to optimize the response variable [55, 56], which is particularly suitable for mathematical models that aim to minimize the objective function (OF), as indicated by the following formula: (15)

In this study, the GA algorithm involves the adjustments of three parameters, including population size(nPop), crossover rate(Pc), and mutation rate(Pm). Similarly, the SA algorithm also has three parameters that need to be adjusted, including the number of sub-iterations (N), initial temperature (t₀), and temperature reduction rate (T). Based on the literature of [53, 54], we have selected three potential values for each parameter, as shown in Table 5.

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Table 5. Candidate values of the parameters of our algorithms.

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For both algorithms, the Taguchi method determined that the optimal choice was an L9 orthogonal array, requiring only 9 experiments for each test problem. After running each algorithm 9 times for each test problem, we used relative percent deviation (RPD) from the Taguchi method to convert the optimal value of the response metric to a normalized value. RPD is defined as follows: (16)

In this study, we used Min_sol to represent the optimal solution and Alg_sol to represent the output result of the algorithm. It is worth noting that a smaller value of this metric indicates better performance. After defining the RPD for each test problem and computing the average RPD for all test problems, we determined the optimal values for each algorithm, which are presented in Table 6 based on the average RPD graph for each parameter.

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Table 6. Tuned values of parameters of each algorithm.

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5.3 Comparisons with different task sizes

To facilitate a comparison between the two approaches, we integrate distinct task blocks using various numbers (B = {20,30,40,50,60}) and numbers of transporters (T = {4,5,6,}). Essentially, the GA is utilized to tackle the model, and subsequently, the outcomes are cross-checked with the exact solution obtained through the CPLEX software to verify the effectiveness of the proposed approach in this study.

We will compare the optimal values solved by the GA with the software-accurate values of CPLEX in the following points: the optimization model, min objective values and the transportation efficiency of transporters, penalty value and CPU time are calculated. Furthermore, the transportation efficiency is calculated as the ratio of the transporters’ load transport time to the total transport time, expressed as γ. The penalty value resulting from delays and number of transporters, represented as a proportion of the objective function value, is denoted by δ.

Table 7 presents the solution results and computational performance solved by the GA and obtained using the CPLEX software. From the above table we can see that the solution time for CPLEX software increases sharply, the more the number of tasks and transports increases. In the actual solution process, when the number of tasks is less than or equal to 20, CPLEX software will quickly find the optimal exact solution. But when the number exceeds 20, the GA algorithm outperforms CPLEX, verifying that the GA algorithm has good results in practical large-scale applications.

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Table 7. Results about different task blocks and transporters between CPLEX and GA.

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GA has limitations in some sense that there is no guarantee that an optimal solution will be found. We can conclude from Table 7 that when the number of task segments is small, the difference between the exact solution and the optimal solution obtained by CPLEX is small, which can be allowed. At the same time, the GA can also quickly obtain better optimal solutions in large-scale quantities, which verifies the effectiveness of the method proposed by this paper in practical applications.

5.4 Comparisons with simulated annealing algorithms

Drawing on the numerical experimental data that we have prepared and comparisons that we have made using different task scale methods, our focus is to evaluate the GA that we have proposed in contrast to the simulated annealing algorithm, and subsequently, analyzing and comparing the results. Convergence analysis and result comparison will be carried out to achieve this objective. We have discussed the details of parameter tuning in subsection 4.2.

To better compare the effectiveness of the GA, this section initially compares the convergence speeds of this algorithm and the simulated annealing algorithm from the convergence’s viewpoint, by iterating both algorithms for 1000 times. The GA has a slower convergence speed compared to the simulated annealing algorithm, however, the results obtained using the GA are better. Kindly find the details in Fig 8.

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Fig 8. Comparison of algorithm convergence process.

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After analyzing the convergence of the two algorithms as discussed earlier, we now compare their scheduling results using the same iteration count of 1000 times while incrementing the number of block tasks from 20, 30, 40, 50, to 60, respectively. The results are then compared in terms of total non-value-added time, waiting time, no-load time, and postponement time.

According to our findings, the simulated annealing algorithm’s total non-value-added time, waiting time, no-load time, and postponement time are higher than those of the GA algorithm. The detailed results can be seen in Fig 9.

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Fig 9. Comparison of scheduling results.

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5.5 Comparisons with not DT-based scheduling

To showcase the advantages of employing a DT-based approach for dynamic scheduling of transporters, a shipyard multiple transporters scheduling system is developed, leveraging the unique strengths provided by this approach. To evaluate and compare the benefits and drawbacks of the two scheduling solutions, a comparison experiment is conducted using a practical block transportation task at shipyard. In Fig 10, the results of the transporters and task blocks assignment are presented in the form of a Gantt chart. The computing time and objective function values are summarized in Table 8.

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Fig 10. Gantt chart of the results of transporters and task blocks scheduling.

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Table 8. Optimization of objective function values and mean CPU time results.

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During the scheduling and execution process, a malfunction was detected in transporter 9 by the job process status monitoring service at 261s, necessitating a three -minutes repair period. The multiple transporters scheduling system, leveraging the DT-based approach, employs predictive techniques to anticipate the consequences of the transporter malfunction on the execution plan. Additionally, the system validates whether the disrupted plan adheres to the existing constraints, ensuring that the revised schedule remains feasible and compliant with the requirements. In case it fails to meet the requirements, a re-scheduling will be initiated, involving reassignment of unfinished tasks and confirmation of the new plan in the twin space before replacing the current execution plan. Upon completion of re-scheduling, the final plan is obtained as depicted in Fig 10.

As presented by Fig 11, the DT-based transporters scheduling method reschedules uncompleted block tasks when uncertainty in the scheduling process occurs, i.e., tasks on the faulty transporter are assigned to the earliest completed transporter. By analyzing the results of the DT system for transporter scheduling, the system can respond in a timely manner to uncertain events during yard scheduling operations while ensuring the optimality of its scheduling solutions.

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Fig 11. The ultimate DT-based scheduling solution.

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Based on our analysis of the presented results, several observations can be deduced. When tackling small-scale instances, the CPLEX approach can optimally resolve them within a reasonable computational timeframe compared to GA. However, for large-scale mathematical scenarios, GA can yield improved solutions within an acceptable duration.

And the existence of the DT therefore makes it possible to interact the twin space with the physical space. The interaction between the two triggers timely rescheduling, which reduces the idle time of transporters, increases transport efficiency and thus reduces the maximum make span time. In other words, by the means of the closed-loop and iterative optimization between the service system and the twin space, the parameters in transporter scheduling become more and more precise and the scheduling optimality more and more optimal.

5.6 Sensitivity analyze

In order to verify the validity of the proposed methodology in this paper, a sensitivity analysis of the key parameters is performed in this section. However, since this paper has already investigated the comparison of results in different cases of various block numbers (B = {20,30,40,50,60}) and numbers of transporters (T = {4,5,6}) in subsection 4.2.

Therefore, this subsection discusses the effect of the number of transporters on the objective function from the case of constant task size as well as the effect of the number of block tasks of different sizes on the objective function under the condition of fixed number of flatbed trucks. The details are shown in the following Fig 12.

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Fig 12. Comparison of scheduling results for different number of tasks and number of transporters.

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As we can see from Fig 12, with the increasing number of block tasks, the objective function value is relatively low when the number of transporters is 4, at which point the optimum is achieved. In addition, we go to analyze the transportation efficiency and the percentage of penalty value due to delay, delay and the number of transporters to the objective function for the cases of 4, 5 and 6 transporters, respectively, as shown in Fig 13.

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Fig 13. Corresponding transportation efficiency and penalty values for transporters for different numbers of tasks.

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From Fig 13 we can see that with the increasing number of tasks, when the number of transporters is 6, the transportation efficiency is higher, and its corresponding penalty value due to delay, delay, etc. is lower.

Through the sensitivity analysis above, we can check that the number of flatbed trucks and the task scale in different schemes have a strong impact on the objective function. Therefore, we cannot blindly increase the number of flatbed trucks in pursuit of transport efficiency in the actual operation process of the shipyard. This will instead lead to many other problems. Nor can we reduce the number of flatbed trucks in order to reduce investment, otherwise transportation efficiency cannot be guaranteed. Therefore, in summary, related shipyard management companies can refer to the solution method proposed in this question to make decision-making support when making choices.