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Outbound container storage location assignment problem (OCSLAP) could be defined as how a series of outbound containers should be stacked in the yard according to certain assignment rules so that the outbound process could be facilitated. Considering the NP-hard nature of OCSLAP, a novel particle swarm optimization (PSO) method is proposed. The contributions of this paper could be outlined as follows: First, a neighborhood-based mutation operator is introduced to enrich the diversity of the population to strengthen the exploitation ability of the proposed algorithm. Second, a mechanism to transform the infeasible solutions into feasible ones through the lowest stack principle is proposed. Then, in the case of trapping into the local solution in the search process, an intermediate disturbance strategy is implemented to quickly jump out of the local solution, thereby enhancing the global search capability. Finally, numerical experiments have been done and the results indicate that the proposed algorithm achieves a better performance in solving OCSLAP.

Scientific storage location assignment (SLA) is a key factor to improve port efficiency. Economic globalization, transport containerization, container ship scale-up, and port automation in the past 4 decades have resulted in a significance restructuring of global container ports. Many major international container ports have been booming; e.g., the top three global container ports of Shanghai (China), Singapore (Singapore), and Shenzhen (China) handled 42.01 million TEUs, 33.6 million TEUs, and 25.74 million TEUs, respectively, in 2018, and enjoyed annual increase of 4.4%, 8.7%, and 2.1% accordingly. At the same time, many other container ports have been facing grand challenges; e.g., port of Kaohsiung handled 10.45 million TEUs with an annual increase of mere 1.7%, while port of Hamburg handled 9 million TEUs with an annual increase of -0.8% in 2018. The slow global economy recovery since the recess of 2008, the ever-increasing international trade protectionism, and disputes render fiercer container port competition. As a pivotal component of any container port, the storage has been attracting more attention recently, since the storage operations, especially storage location assignment, not only streamline the quay operations, the horizontal transport operations, and the gate operations, but also affect directly the overall efficiency and profitability of the port. Therefore, storage location assignment mechanisms that cater to the need of extra-large container vessels and automated container ports are in great demand.

Outbound container storage location assignment (OCSLA) is the preparation work before shipment, which is closely related to the loading efficiency and waiting time of vessels in port. In general, outbound containers usually arrive at the storage yard within three to seven days before the ship arrives at the port. And the load planners will arrange the corresponding storage location according to quantity, size, weight, cargo type, and ship destinations of containers. However, due to the long time span of outbound containers entering into the storage yard and the uncertainty of arriving container quantity distribution, it is more difficult to allocate the detailed location of outbound containers. Unreasonable location assignment will lead to an increase in the rate of container reshuffle and in waiting time for quay crane, thus reducing the loading efficiency of vessels and resulting in huge pressure on outbound operations. So, it is of practical significance to solve the problem of OCSLA in a scientific way.

Outbound container storage location assignment problem (OCSLAP) is a classic constraint combination problem that has been proven to be an NP-hard problem [

To our knowledge, when PSO pursues the global optimal solution, it tends to ignore some local optimal solutions, which leads to fast convergence [

The main contributions of this article could be summarized as follows:

(1) A new PSO algorithm is developed to solve OCSLAP, in which a neighborhood-based mutation operator and an intermediate disturbance strategy are introduced to balance the local exploitation and global exploration ability.

(2) Infeasible solutions are transformed into feasible ones through a repairing strategy in which all the particles are repeatedly initialized or constantly adjusted according to the constraints until all the constraints are satisfied during initialization. At the same time, the lowest stack principle is introduced into the repairing strategy to balance the number of containers among different bays.

(3) A series of small and medium scale OCSLAPs are solved to illustrate the effectiveness of the proposed algorithm. And results also prove that the proposed PSO algorithm performs better than GA and traditional PSO.

The rest of this paper is organized as follows. A systematic review of OCSLAP and particle swarm optimization methodology in the literature is provided in Section

In recent years, OCSLAP has attracted considerable practical and academic attention due to a series of challenges such as the uncertainty of loading sequences, irregular distribution of outbound containers, temporary changes to the stowage plan, and the ever-increasing size of container ships.

In [

Different from the literature of Hu et al. [

However, many literatures have also proved that PSO has a disadvantage of fast convergence [

Executing neighborhood-based mutation operator can strengthen local exploitation abilities, but it also has the disadvantage of trapping into local solutions if the neighborhood radius is not chosen properly. Therefore, measures need to be taken to jump out of local solutions within a certain probability to achieve global optimality, which is good for maintaining the balance between local exploitation and global exploration abilities. Meanwhile, the existing literatures have also proposed various kinds of disturbance strategies. In [

In conclusion, OCSLAP has been endowed with new meaning in the circumstance of container ship scale-up and container port automation, and research on PSO with neighbourhood-based mutation operator and intermediate disturbance strategy in OCSLAP domain is still in its infancy stage. Therefore, it is of both theoretical and practical significance to research on the proposed algorithm for OCSLAP.

Storage operation of outbound containers refers to the detailed location assignment according to a series of rules before loading onto ships. Outbound containers enter from the gate or crossdocking ship and are transported directly by the special vehicle to container yard before ship’s arrival. At the same time, the load planners will also arrange the corresponding location according to quantity, weight information, and ship destinations of outbound containers. In general, the stowage plan should not be disturbed at will as it will cause a large number of reshuffles, resulting in huge operation costs. So it is of great practical significance to solve OCSLAP by scientific approaches for the sake of port operation efficiency and competitiveness.

In a container yard, one block consists of many bays and one bay contains a series of stacks. Assume that a batch of outbound containers will be assigned to specific locations in two bays. As is shown in Figure

Detailed description of OCSLAP.

In order to speed up the loading process, outbound containers are usually transported directly to the yard for location selection. First, the specific position in one bay determines the moving cost of the internal truck between in the yard side and quayside. Therefore, the first goal of this problem is to minimize the travel distance of the internal truck. Second, the order of picking up outbound containers is opposite to the stacking sequence, which greatly affects the loading efficiency of vessels. In the preparation for loading shipment, to obtain a better stacking strategy, it is necessary to reduce the number of reshuffles so that the containers can be loaded in the original loading order. So, the second goal of this problem is to minimize the number of reshuffles. Finally, when the detailed location of outbound containers is arranged, it is necessary to ensure that a series of equipment such as gantry cranes and trucks is protected from traffic congestion. To this end, the third goal is to maintain the balance of the number of containers among bays as uniformly as possible.

Mathematical models in this paper are established on the following assumptions:

(1) Outbound containers and inbound containers are not mixed up in the storage location assignment process. No matter the traditional Asian port layout where the queue line is parallel to the storage yard or the European port layout where the queue line is perpendicular to the storage yard which is adopted by most automated container ports, containers of different features are usually stored separately, e.g., empty and heavy containers, inbound and outbound containers, and twenty feet and forty feet containers. Therefore, the first assumption is valid even in automated container ports.

(2) Only one size of outbound containers (i.e., Forty Equivalent Unit, FEU) is considered. This assumption is based on the observation that containers of the same size are usually stored in the same bay and containers of different size in different bays. To facilitate the theoretical research, only FEU is considered in this paper.

(3) Container ship stowage plan is known in advance and does not change arbitrarily. In practice, the stowage plan is influenced by many factors like custom inspection quantity and speed and ship harboring process. Hence it is not fully determined until the ship is securely harbored in the port and the custom is completely cleared. However, more than 80-90% of the containers usually arrive in the port storage yard by then. Therefore, it does not hurt to assume that the stowage plan is known and determined in advance.

(4) The weight levels of outbound containers are known in advance before shipments and the arrival sequence of outbound containers is generated randomly which is in accordance with the practical statistical data analysis results of Guangdong-Hong Kong-Macao Greater Bay Area container ports.

(5) The incoming outbound containers satisfy the first-in first-stacking discipline and the initial status of bays is empty.

(6) To ensure the stability of the vessel at the time of loading (the heavy container is loaded on the bottom layer as much as possible to lower the vessel’s center of gravity), the container that is usually stacked on the storage yard should follow the principle that the heavier container is on the upper locations and the lighter container in the lower. If this principle is violated, reshuffles will occur, as manifested in Figure

The number of reshuffles produced when outbound containers are assigned.

Notations used in this model are listed as follows.

Slot(

M A sufficiently large positive number (i.e., M ≫ 0)

The OCSLAP model can be formulated as follows:

Objective (

Particle swarm optimization (PSO), derived from the study of bird flocking, is a population-based stochastic search method proposed by Kennedy et al. [_{1} and_{2} represent two randomly generated numbers in the range

As is well known, classical PSO is prone to premature convergence for the lack of population diversity [

Based on the idea of differential evolution (DE), a neighborhood-based mutation operator strategy is obtained and introduced into PSO algorithm in order to achieve the purpose of enhancing population diversity, in which the local neighborhood of each particle could be found through a circular topology.

In Figure

Neighborhood of

When the optimal solution obtained by using this strategy is close to the global optimal solution, it displays that the algorithm has a smaller jump and better stability. However, if the radius of the particle is not properly selected, it is easy to fall into the local optimal solution. Therefore, it is extremely necessary to introduce an intermediate disturbance strategy to change the position of the particles so that they could quickly jump out of the local solution, thereby obtaining the global optimal solution.

To conquer the premature convergence of PSO and to accelerate the convergence of particles, an intermediate disturbance strategy is introduced. The intermediate disturbance strategy was proposed by Grime [

In Eq. (

Coding is the key factor affecting PSO algorithm. It could be a string of integers or real numbers. Considering the NP-hard combinatorial nature of OCSLAP, stack-based integer encoding representation is adopted in discrete search space. In this encoding scheme, a solution for the problem of assigning containers to locations is represented by an array which length equals the number of containers.

Take seven containers, three stacks, and three tiers, for instance. In Table

A coding example.

Container number | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

Stack number | 1 | 2 | 3 | 3 | 2 | 1 | 3 |

Particle value | 0.82 | 1.56 | 2.44 | 2.71 | 1.32 | 0.38 | 2.74 |

In this article, initial solution is generated randomly. However, there exists a series of infeasible solution, such as

In reality, the lowest stack principle [

The flowchart of the proposed algorithm is presented in Figure

The flowchart of the proposed algorithm.

Set parameters such as

Infeasible solutions are adjusted by repairing strategy.

Compute fitness function (_{i} and

The particle velocity

Calculate

Define the particle position

First, keep the boundary value of the particle position; then perform repairing strategy and compute the fitness values; finally, update the personal best_{i} and the global best

If it satisfies the stop criterion, the algorithm terminates; otherwise go to Step

In order to assess the performance of the proposed algorithm, a series of benchmark functions are tested, as detailed in Table _{k} [

Benchmark optimization functions.

Name | Function formula | Domain | |
---|---|---|---|

Rosenbrock | | | 0 |

Ackley | | | 0 |

Griewank | | | 0 |

Alpine | | | 0 |

Rastrigin | | | 0 |

The results are displayed in Table

The mean and standard deviations of the function values for 30-D problems.

Function | TPA | CPSO-H_{6} | LPSO | FIPS | CLPSO | APSO | |
---|---|---|---|---|---|---|---|

| Mean | | 3.23e+03 | 6.71e+01 | 3.32e+01 | 2.76e+01 | 3.05e+02 |

Std | | 1.64e+04 | 4.00e+01 | 2.12e+01 | 2.14e+01 | 5.29e+02 | |

| Mean | | 4.81e-05 | 1.62e-05 | 5.41e-07 | 2.95e-09 | 9.01e-02 |

Std | | 3.09e-05 | 1.18e-05 | 2.32e-07 | 1.12e-09 | 4.98e-02 | |

| Mean | | 8.06e-02 | 1.79e-02 | 4.77e-04 | 4.27e-09 | 3.38e-01 |

Std | | 5.91e-02 | 1.55e-02 | 1.95e-03 | 1.41e-08 | 2.35e-01 | |

| Mean | | 1.65e-01 | 2.22e-03 | 6.64e-03 | 5.99e-04 | 8.51e-03 |

Std | | 8.08e-01 | 4.06e-03 | 1.63e-03 | 3.63e-04 | 9.75e-03 | |

| Mean | | 5.25e+01 | 4.46e+01 | 8.45e+01 | 3.32e-02 | 1.17e+00 |

Std | | 2.08e+01 | 1.31e+01 | 1.16e+01 | 1.85e-01 | 8.61e-01 |

Note: TPA stands for the proposed algorithm. “Mean” is the mean of the solutions. “Standard deviation” is abbreviated as “Std”. The numbers in italic denote the best solution.

In metaheuristic algorithms, parameter setting is necessary, for it is a critical factor that matters to the convergence and stability of algorithms. The control parameters of PSO are inertia weight

Since the larger weight is conducive to jumping out of the local minimum value and facilitating global search and the smaller weight is more conducive to strengthening the local search of the current region and facilitating the convergence of the algorithm, this study adopts the linear weight (_{max}_{max-}_{min})_{max}=0.9,_{max}=0.4) to balance the global search capability and local exploitation ability. Other parameters used in the experiments are listed in Table

Results of cases with different parameters.

Scenario | Parameters | Case 1-2 × 5 × 4 (30) | Case 2-4 × 6 × 4 (72) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

| | | | Avg | Opt | Std | Avg | Opt | Std | |

1 | 50 | 1 | 2 | 200 | 0.05124 | 0.0333 | 0.0045 | 0.0888 | 0.0722 | 0.0053 |

2 | 50 | 2 | 2 | 200 | 0.05078 | 0.0333 | 0.0042 | 0.0881 | 0.0722 | 0.0051 |

3 | 50 | 4 | 1 | 200 | 0.05147 | 0.0333 | 0.0047 | 0.0905 | 0.0722 | 0.0053 |

4 | 50 | 2 | 2 | 100 | 0.05317 | 0.0333 | 0.0054 | 0.0933 | 0.0722 | 0.0055 |

5 | 100 | 2 | 2 | 200 | 0.04997 | 0.0333 | 0.0042 | 0.0852 | 0.0722 | 0.0049 |

6 | 100 | 2 | 2 | 100 | 0.05194 | 0.0467 | 0.0039 | 0.0917 | 0.0806 | 0.0053 |

7 | 100 | 1 | 4 | 200 | 0.05066 | 0.0333 | 0.0043 | 0.0875 | 0.0722 | 0.0050 |

The average values and standard deviation of different scenarios.

In this part, the proposed method is applied to solve a real case, which comes from a small and medium terminal in China. According to the actual investigation of Guangdong-Hong Kong-Macao Greater Bay Area, this paper sets the max stack height to 6 in one bay, and a series of small and medium-sized cases are tested. The results are displayed in Table

Comparison of the results of different approaches.

Bay×stack×tier | Different algorithms | Improved efficiency | |||
---|---|---|---|---|---|

TPA | PSO | GA | (1-TPA/PSO)×100% | (1-TPA/GA)×100% | |

1 × 4 × 4 | | 0.0617 | 0.1333 | 32.42% | 68.72% |

1 × 6 × 4 (20) | | 0.1333 | 0.1667 | 58.29% | 66.41% |

1 × 10 × 6 (55) | | 0.1237 | 0.1534 | 32.66% | 45.70% |

2 × 4 × 3 (22) | | 0.0542 | 0.0752 | 23.06% | 44.55% |

2 × 5 × 4(30) | | 0.0833 | 0.15 | 60.02% | 77.8% |

2 × 6 × 4 (36) | | 0.0694 | 0.125 | 59.94% | 77.76% |

2 × 6 × 4 (44) | | 0.083 | 0.132 | 16.39% | 47.42% |

2 × 6 × 6 (65) | | 0.0944 | 0.1748 | 17.58% | 55.49% |

2 × 6 × 6 (70) | | 0.1561 | 0.191 | 21.72% | 36.02% |

4 × 6 × 4 (72) | | 0.1444 | 0.1722 | 50% | 58.07% |

4 × 6 × 4 (90) | | 0.1723 | 0.2615 | 34.31% | 56.71% |

7 × 5 × 4(105) | | 0.1567 | 0.327 | 45.31% | 73.79% |

7 × 6 × 3 (110) | | 0.1484 | 0.1924 | 40.43% | 54.05% |

7 × 6 × 3 (120) | | 0.1737 | 0.197 | 37.42% | 44.82% |

7 × 6 × 4(126) | | 0.1071 | 0.159 | 52.47% | 67.99% |

Average | 38.8% | 58.35% |

Note: TPA stands for the proposed algorithm. The best values are marked in italic.

The average values and standard deviations obtained using different methods.

Computational time consumed by different approaches.

In Table

Comparative trend graphs of convergence of both general PSO and proposed algorithm with 200 simulation iterations are displayed in Figure

Convergence characteristics of PSO and the proposed algorithm.

A set of experiments employing the proposed algorithm are performed and the detailed results are presented in Table

Experiments results under different scenarios.

Bay×stack×tier (containers) | | | | |
---|---|---|---|---|

2 × 4 × 3 (22) | 33 | 1 | 0 | 0.0417 |

2 × 6 × 4 (36) | 54 | 2 | 0 | 0.0278 |

2 × 6 × 4 (44) | 66 | 5 | 0 | 0.0694 |

2 × 6 × 6 (65) | 97 | 14 | 1 | 0.0778 |

2 × 6 × 6 (70) | 105 | 22 | 0 | 0.1222 |

4 × 6 × 4 (72) | 177 | 13 | 2 | 0.0722 |

4 × 6 × 4 (90) | 219 | 25 | 3 | 0.1132 |

7 × 6 × 3 (110) | 416 | 24 | 4 | 0.0884 |

7 × 6 × 3 (120) | 468 | 35 | 2 | 0.1087 |

In this paper, the proposed PSO algorithm is applied to solve OCSLAP. Computational results indicate that, in the small and medium scale, the target function value obtained by the proposed algorithm is much smaller than PSO and GA and its improvement efficiency is 38% and 58%, respectively. In addition, results also prove that the proposed algorithm could solve combinatorial problems well.

This study could be further extended from the following perspectives. First, a new intelligent algorithm that is tree-seed algorithm (TSA) could be considered to solve OCSLAP, because the TSA has two different search equations and, for each tree, more than one seed (candidate solution) is created and these equations and seed production mechanism could provide balanced local and global search capability [

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was supported by the 2017 Founded Major Project of NSFC (71731006) and Ministry of Education (MOE) of China Project of Key Research Institute of Humanities and Social Sciences at Universities (17JZD020).