Marine container terminal (MCT) plays a key role in the marine intelligent transportation system and international logistics system. However, the efficiency of resource scheduling significantly influences the operation performance of MCT. To solve the practical resource scheduling problem (RSP) in MCT efficiently, this paper has contributions to both the problem model and the algorithm design. Firstly, in the problem model, different from most of the existing studies that only consider scheduling part of the resources in MCT, we propose a unified mathematical model for formulating an integrated RSP. The new integrated RSP model allocates and schedules multiple MCT resources simultaneously by taking the total cost minimization as the objective. Secondly, in the algorithm design, a preselectionbased ant colony system (PACS) approach is proposed based on graphic structure solution representation and a preselection strategy. On the one hand, as the RSP can be formulated as the shortest path problem on the directed complete graph, the graphic structure is proposed to represent the solution encoding to consider multiple constraints and multiple factors of the RSP, which effectively avoids the generation of infeasible solutions. On the other hand, the preselection strategy aims to reduce the computational burden of PACS and to fast obtain a higherquality solution. To evaluate the performance of the proposed novel PACS in solving the new integrated RSP model, a set of test cases with different sizes is conducted. Experimental results and comparisons show the effectiveness and efficiency of the PACS algorithm, which can significantly outperform other stateoftheart algorithms.
With the accelerated development in the world economy and trade, the importance of marine container terminals (MCT) rapidly increases due to the rise in imported and exported goods. According to the review of maritime transport by the United Nations Conference on Trade and Development (UNCTAD), the total volumes of international seaborne trade have reached 163 million twentyfoot equivalent units (TEUs) in 2022 [
Resources of MCT mainly include berths, quay cranes (QCs), vehicles, yard cranes (YCs), and human workers. According to the container terminal layout of Qingdao port as shown in
There are three main steps in the entire process of MCT operations [
First, the available berth and berthing time are assigned to each ship that arrives at the MCT. Concerning the berth allocation problem (BAP) in RSP, many different models and algorithms have been proposed. Jauhar et al. [
Second, loading and unloading operations should be carried out. The container is unloaded or loaded from a ship by a QC in this step (i.e., QC scheduling problem), then moved to the assigned yard by a vehicle or vice versa (i.e., vehicle scheduling problem). Considering the relevance of critical equipment and resources of MCT in the terminal operations process, QCs allocation is carried out together with berths allocation. Multiple berthing positions and multiple QCs were integrated into an allocation problem model in [
Third, when the vehicle arrives yard, the container is picked up or put down from the yard by a YC (i.e., YC scheduling problem). Cao et al. [
Most of the aforementioned studies only study a part of the RSP. That is, scheduling only a part of the resources. However, the MCT manager needs to plan the entire operation process so that global efficiency can be obtained. Therefore, all the resources, including berths, QCs, YCs, human workers, and vehicles should be considered together and integrated. However, each part of the RSP is NPhard, like the BAP, the QC scheduling problem, the vehicle scheduling problem, the YC scheduling problem, and the human workers scheduling problem. In this sense, the integrated RSP would be much more complex. Fortunately, all these scheduling problems share a common objective of reducing operation costs. For example, unreasonable berths allocation can lead to unreasonable waiting times for the ships, causing a higher waiting cost. Moreover, the cost also heavily relates to the handling speed of the operation in the scheduling. If more equipment or resources (i.e., QCs, YCs, human workers, and vehicles) are used, the handling speed will be faster, leading to shorter waiting time but causing a higher handling cost per unit time. Motivated by this, in this paper, we first establish a novel integrated RSP model and set the total cost minimization as the optimization objective of the integrated RSP model by combining the waiting cost and handling cost, which can consider all the resources or equipment together in the integrated RSP model.
Metaheuristic algorithms are suitable for solving NPhard RSPrelated problems, such as PSO [
Ant colony system (ACS), derived from ant colony optimization (ACO), is a metaheuristic algorithm that is more effective and efficient in dealing with discrete optimization problems [
Based on the above, the main contributions of this paper come from both the problem model aspect and the algorithm design aspect, which are summarized as follows:
1. In the problem model aspect, different from other models that only consider scheduling part of the resources in MCT, due to the strong interrelation in the different equipment resources of MCT, we establish a novel integrated RSP model that combines multiple resources (including berths, QCs, YCs, human workers, and vehicles) together in a uniform model. This way, all the resources at the MCT can be considered together at the same abstract level and be allocated and scheduled simultaneously to minimize the total cost.
2. In the algorithm design aspect, the PACS approach is proposed based on graphic structure solution representation and a preselection strategy. Firstly, in order to avoid generating infeasible solutions, the graphic structure is proposed to encode the solution and the solution construction process imitates the idea of finding the shortest path method based on the graphic structure. This graphical structure can avoid the same ship node being repeatedly selected. Secondly, aiming to meet the realtime requirement of the RSP model and reduce the search space to further improve the performance, the preselection strategy is proposed to accelerate the optimization speed of PACS.
The rest of this paper is organized as follows. In
The mathematical symbol representation and formula of the RSP model are given as follows:
1. Set  
Set of ships,  

Set of available berths at MCT,  

Set of available QCs at every berth, 

Set of available YCs at yard,  

Set of available vehicles,  

Set of available workers,  

Set of handling speeds, 

2. Index  
Index of berth, 

Index of worker, 

Index of handling speed, 

Index of QC, 

Index of vehicle, 

Index of ship, 

Index of YC, 

3. The known variables  
Number of workers required for every QC  
Number of workers required for every YC  
Number of workers required for every vehicle  
Speed of a vehicle (TEUs/hour), 

Cost of a vehicle per hour (USD/hour), 

Speed of a QC (TEUs/hour), 

Cost of a QC per hour (USD/hour), 

Speed of a YC (TEUs/hour), 

Cost of a YC per hour (USD/hour), 

Cost of a worker per hour (USD/hour), 

Arrival time of Ship 

Number of containers to be loaded and unloaded on Ship 

The waiting cost per hour (USD/hour)  
4. Variables related to solving the handling speed  
The number of handling speeds  
Value of the 

Cost per hour for the 

Number of QCs assigned for the 

Number of vehicles assigned for the 

Number of YCs assigned for the 

Number of workers assigned for the 

5. Variables related to the objective function  
x_{sbh}  = 1 if Ship 
The waiting cost of Ship 

The waiting time of Ship 

The handling cost of Ship 

The handling time of Ship 

The 

The starting time of service for Ship 

The finishing time of service for Ship 
The RSP is a problem for the unified scheduling of equipment and resources at MCT by the terminal manager. The scheduled equipment and resources include:
1. Available discrete berth
2. Available QCs at every berth,
3. Available YCs at every yard,
Specifically, since multiple berths are corresponding to one yard area, the total handling speed of YCs needs to cope with the simultaneous operation of QCs at all berths. Meanwhile, the quantity of YCs will not exceed that of QCs too much. Otherwise, some YCs will be idle during unloading or make the QCs too busy to meet the requirements of all containers during loading. Furthermore, a YC can only serve one berth at a time.
4. Available vehicles. The number of vehicles used by per handling speed is determined by the number of QCs and YCs. Using more QCs and YCs requires more vehicles, causing higher cost. Only one operation (loading or unloading) is performed at a time. Correspondingly, each vehicle also carries out only one operation at a time. For example, the vehicle takes the container from the ship to the yard and returns to the berth empty, or loads the container from the yard to the berth with containers and returns to the yard empty. As shown in
5. Available workers, the number of workers is sufficient. We set each equipment is operated by only one worker. The total number of workers is the number of equipment served for Ship
For a better understanding of the operation flow in the whole MCT,
Considering the actual situation of MCT, the assumptions can be summarized as follows: (1) Ships cannot berth at multiple berths at the same time and cannot be served at multiple handling speeds. (2) Every ship continues to work at a selected handling speed until the loading and unloading are done. (3) All the equipment keeps the same running speed.
We integrate multiple equipmentdependent parameters into one metric, called handling speed. The faster handling speed often needs more equipment (i.e., QCs, YCs, human workers, and vehicles) also brings the higher cost. Therefore, the values of each handling speed are computed according to the quantity of partially used equipment (
The value of
Since QCs in discrete berths cannot be used in different berths, the upper limit of
In Line 5 of Algorithm 1, the number of vehicles assigned for the
The total number of workers served for the
Combined with the above calculation results, in Line 7 of Algorithm 1, the value of the
We compare the speed of QCs (
Then, the cost per hour of the
The quantity of handling speed
In the proposed RSP model, the calculation of handling speed needs to meet the following constraint:
Constraint
Objective:
Based on the components mentioned above, the objective of the integrated RSP model is to minimize the total cost, which includes waiting cost and handling cost of all ships, shown in function
PACS is used to solve RSP, which is to find an optimal berthing scheme with the minimum cost. The objective function of RSP is defined in
When using PACS to solve the RSP, the solution representation should be defined. Herein, a novel graphic structure for solution representation is proposed. Based on the solution representation, the optimization process is described as follows. Firstly, the ant randomly chooses a node as the starting point. Then, the ant chooses the next node according to the biased exploration based on the pheromone and heuristic information between nodes. A complete solution path is constructed by repeating this process until all the ship berthing schemes are determined. After that, local pheromones updating rule is executed to reduce the probability of other ants choosing the same path, where the pheromone on the paths visited by the ant is weakened by the local pheromone updating rule. When all ants construct their solutions in this generation, these solutions are compared with the historically best solution to obtain a new historically best solution. The pheromone is enhanced on the edges of the new historically best solution, to accelerate the convergence speed of the algorithm. The process will be repeated until the terminal condition is met.
In the following parts, the graphic structure solution representation, initialization state configurations, state transition rule, and pheromone updating rules are described in details, with the complete PACS algorithm in the end.
The RSP is similar to TSP. The goal of TSP is to find the shortest path for a traveler to go through all the cities without repetition. Therefore, RSP can also be constructed into a graphic structure similar to a city map. The goal of RSP is to find the shortest path which is an optimal berthing scheme for every ship, including berthing berth, handling speed, and berthing sequence.
The RSP can be modeled into a digraph with 
Initialization operation includes computing handling speed, initialing pheromones, and determining the starting node. The preselection strategy is proposed in initialization operation to reduce the computational burden and to fast obtain a higher quality solution.
In the initialization stage of PACS algorithm, RSP determines the handling speed by considering multiple resources (QCs, YCs, human workers, and vehicles). These values of variables related to handling speed (i.e.,
The quantity of vehicles and workers are determined by the quantity of QCs and YCs, shown in
We first proposed the preselection strategy based on first come first served (FCFS) to determine the initial pheromone value
where
In this step, the preselection mechanism is used to determine the starting node of every ant (
In PACS algorithm, pheromone and heuristic information are used to guide forward behavior of ants in the state transition rule. Heuristic information is introduced as follows:
In PACS algorithm, each ant from node
In PACS algorithm, there are two kinds of pheromone updating rules, which are the local pheromone updating rule and the global pheromone updating rule. The local pheromone updating rule is used to reduce pheromones of visited edges to avoid the following ants choosing the same edges. The local pheromone is updated when each ant constructs a complete solution, shown as:
When every generation is complete, the global pheromone update is carried out on the edge of the bestsofar solution. The pheromone is increased as:
These two pheromone updating rules act in different roles and complement with each other. The local update is more beneficial to expand the selection direction of ants. The global pheromone updating rule makes ants gradually converge to the best solution and speeds up the convergence of the algorithm.
The complete PACS algorithm is shown in
Step 1: Initialize the handling speed and the pheromone, shown in the subflowchart on the right side of
Step 2: Set the first ship by preselection strategy for ant.
Step 3: Construct the whole solution by state transition rule as
Step 4: When all ants find the solution in the current generation, we calculate a more appropriate handling speed for the current optimal solution, similar to Step 1, and obtain a new optimal solution.
Step 5: Perform the global pheromone updating as
Step 6: Termination check. If the maximal generations
According to the complete PACS algorithm shown in
In this section, we conducted experiments with different sizes of tests to verify the performance of PACS. The code of the PACS algorithm is implemented in Visual C++ on a PC with Dell Intel(R) Corei7 and 8.0 GB RAM.
References in scheduling application literature [
Symbol  Value  Symbol  Value 

 
{2, 4, 6}  12  
6  10  
54  60  
 
180  20  
 
270  40  
5000  20  
MNQ, MNY, MNV  1  50 
According to the aboveknown data, the handling speed is calculated by formula in
1  2  3  10  15  120  950 
2  4  5  17  26  200  1650 
3  6  9  30  45  360  2850 
The parameter configurations are based on experimental verification of
Herein, we select several stateoftheart algorithms to compare with PACS, including SFLA (2023) [
In test cases, the number of berths available is set to 2 berths, 4 berths, and 6 berths. The arrival time of ships is randomly generated by the exponential distribution with a mean of 2 h (i.e.,
In small size test case (Case 1–Case 15), the number of ships changing from 16 to 20, and the number of berths changed from 2 to 6. The average fitness of the objective function of the RSP model is shown in
Case   
 
FCFS  SFLA  IVGGA  RCMGA  AEA  ACS  PACS 

1  16  2  2,217,860.8  269,880.8  269,304.7  270,516.7  269,786.1  281,759.4  
2  16  4  260,580.8  144,934.6  144,868.7  145,255.4  144,775.6  157,227.6  
3  16  6  479,485.0  188,221.1  188,202.1  188,974.4  188,617.8  196,888.9  
4  17  2  1,763,605.8  199,900.7  199,569.4  200,961.9  198,893.0  199,506.7  
5  17  4  747,566.3  183,623.6  184,173.8  185,489.1  183,314.6  188,168.8  
6  17  6  185,895.8  169,093.1  176,044.4  
7  18  2  2,911,067.1  554,682.6  555,402.4  561,251.3  553,827.6  568,319.0  
8  18  4  340,239.2  184,593.3  184,124.7  185,086.7  183,818.1  191,476.1  
9  18  6  323,599.2  186,240.8  186,228.1  187,464.7  186,458.6  188,052.5  
10  19  2  1,588,260.8  181,983.4  181,191.5  184,210.1  181,951.9  182,149.2  
11  19  4  613,572.9  180,672.7  180,053.5  181,494.7  179,949.7  180,743.8  
12  19  6  334,255.4  197,952.7  197,850.5  198,977.8  198,939.1  210,651.8  
13  20  2  3,160,860  468,262.5  469,043.1  471,632.2  466,325.8  512,702.8  
14  20  4  1,433,283.8  261,165.4  260,321.3  263,266.8  263,057.4  298,308.5  
15  20  6  488,711.7  232,993.1  233,661.7  235,272.4  233,825.3  267,146.8 
The experimental results in
Case   
FCFS  SFLA  IVGGA  RCMGA  AEA  ACS  PACS  

16  30  2  6,425,621.3  620,573.5  617,909.8  625,789.8  617,854.5  695,242.3  
17  35  2  12,765,945.8  1,391,697.3  1,382,568.3  1,397,414.3  1,385,468.3  1,654,584.3  
18  40  2  14,432,001.7  1,384,087.6  1,388,406.1  1,409,158.1  1,396,486.6  1.587,367.1  
19  45  2  21,089,600  2,856,796.1  2,759,488.4  2,979,559.7  2,795,367.4  3,252,322.4  
20  50  2  25,036,730  3,226,693.7  3,105,652.6  3.300,453.7  3,158,548.6  3,447,362.1  
21  55  2  3,3048,447.5  5,606,679.3  5,549,237.4  5,747.280.9  5,587,568.1  6,184,327.5  
22  60  2  35,897,784.6  6,052,925.4  5,983,762.3  6,275,098.3  6,075,548.3  6,051,322.4  
23  65  2  32,200,345.8  1,895,698.4  1,856,992.1  1,890,119.6  1,958,364.1  2,264,235.4  
24  70  2  37,520,856.3  2,769,048.8  2,596,470.8  2,969,353.8  2,665,845.3  3,123,887.6  
25  75  2  51,687,475.8  4,509,635.7  4,398,256.4  4,481,258.2  4,348,231.5  4,856,681.1 
The parameters in the PACS algorithm include
The investigation begins with the parameter
The next parameter tested is
Then parameters
Finally, parameter
The efficiency of resource scheduling directly determines the operation performance in MCT. To solve the resources scheduling problem (RSP) in MCT effectively, this paper proposes an efficient ACS based on a preselection scheme (PACS), which contributes to both the problem model and algorithm design. In the problem model, different from other researches that only consider scheduling part of the resources in MCT, we designed a complete model of RSP by integrating multiple equipmentdependent parameters into one metric (i.e., handling speed). In this way, the RSP is modeled to find a scheduling scheme including berth allocation, berth time selection, and handling speed selection for every ship to minimize the total cost. In the algorithm design, the graphic structure is introduced to guarantee the feasibility of the solution. In addition, the preselection strategy is used to accelerate the convergence speed of the PACS algorithm. A large number of experimental results validate the effectiveness and efficiency of the proposed PACS algorithm, which can significantly outperform other stateoftheart algorithms.
Although PACS achieves satisfying results, some parameters like pheromone decay parameter
None.
This research was supported in part by the National Key Research and Development Program of China under Grant 2022YFB3305303; in part by the National Natural Science Foundations of China (NSFC) under Grant 62106055, in part by the Guangdong Natural Science Foundation under Grant 2022A1515011825, and in part by the Guangzhou Science and Technology Planning Project under Grants 2023A04J0388 and 2023A03J0662.
The authors confirm contribution to the paper as follows: Study conception and design: Rong Wang; data collection: Xinxin Xu, Nankun Mu; analysis and interpretation of results: Zijia Wang, Fei Ji; draft manuscript preparation: Rong Wang. All authors reviewed the results and approved the final version of the manuscript.
The data are contained within the article.
The authors declare that they have no conflicts of interest to report regarding the present study.