“Double-Line Ship Mooring” (DLSM) mode has been applied as an initiative operation mode for solving berth allocation problems (BAP) in certain giant container terminals in China. In this study, a continuous berth scheduling problem with the DLSM model is illustrated and solved with exact and heuristic methods with an objective to minimize the total operation cost, including both the additional transportation cost for vessels not located at their minimum-cost berthing position and the penalties for vessels not being able to leave as planned. First of all, this problem is formulated as a mixed-integer programming model and solved by the CPLEX solver for small-size instances. Afterwards, a particle swarm optimization (PSO) algorithm is developed to obtain good quality solutions within reasonable execution time for large-scale problems. Experimental results show that DLSM mode can not only greatly reduce the total operation cost but also significantly improve the efficiency of berth scheduling in comparison with the widely used single-line ship mooring (SLSM) mode. The comparison made between the results obtained by the proposed PSO algorithm and that obtained by the CPLEX solver for both small-size and large-scale instances are also quite encouraging. To sum up, this study can not only validate the effectiveness of DLSM mode for heavy-loaded ports but also provide a powerful decision support tool for the port operators to make good quality berth schedules with the DLSM mode.
With the deepening of economic globalization, the world container transportation volume has increased dramatically in recent years [
In general, berth schedules are determined by specifying berthing time and position for the coming vessels by taking into account various constraints, such as the berth capacity, announced arrival time and departure time of container ships, and certain specific berthing requirements. In order to avoid collisions between vessels, the single-line ship mooring (SLSM) mode, which specifies that “no more than one vessel can be allocated to the same berth position at the same time,” is normally applied in container terminals over the world, and this rule is regarded as a default in most of the studies on berth scheduling [
As one of the most important economic role in the world economy, China is continuously developing the economic innovation, resulting in huge container throughput of the international hubs in China. Yangshan deep-water port is located in Shanghai, a mega city in China. As the largest sea-island artificial deep-water port, Yangshan deep-water port is an important part of Shanghai International Shipping Center, and its annual throughput has increased constantly since it was built in the year of 2005.
Having been one of the world’s busiest container terminals, Yangshan deep-water port has applied a so-called “Double-Line Ship Mooring” (DLSM) mode to build berth schedules since the year of 2019. Different from the widely used SLSM mode, DLSM mode allows two container ships to be moored simultaneously at the same berth location, enabling more container ships to be moored at their ideal berth and cranes to operate two container ships at the same time, a reasonable way to improve the efficiency of berth operation. Despite that, the real efficiency of the port with DLSM mode depends on how sophisticated the operators are according to the investigation made at Yangshan port, because berth scheduling with DLSM is much more complex than with the SLSM system. In consequence, it is essential to develop an effective decision support system to help berth operators improve the quality and efficiency of daily schedules with the DLSM mode.
This study aims at minimizing the additional operation costs for the vessels not placed at their ideal berthing position and penalties for the ships not being able to leave before their preplanned departure time, for a continuous berth allocation problems (BAP) model with the DLSM mode. The main contributions of this study are as follows: (1) as the first study on berth scheduling with the DLSM mode, it validates the contribution of DLSM model in comparison with the widely used SLSM mode; (2) a mixed integer programming model is constructed for the continuous BAP with DLSM mode, enabling a benchmark for the future studies on such topic; (3) a PSO algorithm is developed to obtain good quality DLSM berthing schedules within reasonable execution time, offering the berth operators at busy ports a powerful decision support tool to improve their work efficiency.
The reminder of this paper is structured as follows. Section
According to the literature, berth allocation problem (BAP) is regarded as the most important issue faced by the management of container terminals. As the quality of berth schedule has a great influence on the improvement of port operation efficiency, a lot of researchers have been studying BAPs, and numerous results were published [
With regard to the methodologies applied to BAPs, it can be observed in the literature that the BAPs are normally formulated as Mixed Integer Programming (MIP) models, which can be solved with commercial programming solvers for small-size problems [
Particle swarm optimization (PSO) is an evolutionary computation technique firstly developed by Eberhart and Kennedy [
In general, the berth operators are in charge of arranging each vessel arriving at the port to a suitable berth position according to the availability of the wharf resources and respecting the preplanned arrival and departure of the vessel. According to the practices observed at the targeted container terminal, a set of assumptions are defined as follows: Each vessel arrives at the port on the preplanned arrival time If a vessel cannot leave the port before the preplanned departure time, the port has to incur a penalty The coordinate corresponding to the leftmost end of the vessel is used to represent its berthing position, using the leftmost boundary of the wharf as the coordinate reference point Each vessel has a predefined minimum-cost berthing position, which is determined according to the goods that will be loaded/unloaded at the port, and if a vessel cannot be berthed at its ideal berthing position, additional operation cost will occur DLSM mode is applied, i.e., at most, two vessels can be simultaneously moored at the same berth position When two vessels are moored in double-line, the length of the inner-side ship cannot be longer than the vessel moored outside When two vessels are moored in double-line, the inner-side one must be berthed earlier and leave later than the outside one
For a better understanding, two types of notations are applied in this study: (1) Latin letters are used to denote parameters; (2) Greek letters are applied to represent decision variables.
As shown in Figure
Timeline corresponding to the berth scheduling of vessel
It is worth noting that (1) a gap between the preplanned arrival time
When the DLSM mode is applied at the port and two vessels will be scheduled to be moored in double-line at the same berthing position, it is necessary to determine the relative berthing line positions between these two vessels. Through the interview with the berth operators at the targeted port, a general rule is applied to ensure the berthing safety: Vessel
Figure
Time-side-wharf schematic diagram of a DLSM example.
For a better understanding of this example, the corresponding side-wharf section and time-wharf one are detailed in Figures
Side-wharf section of the DLSM example.
Time-wharf section of the DLSM example.
As for vessel
As mentioned in Section Minimization of the additional operation cost for vessels not located at its minimum-cost berthing position Considering that the ports normally predefine the minimum-cost berthing position for each vessel according the cargos that will be unloaded/loaded to maximize the operation efficiency, it is obvious that the larger is the deviation between the berthing position and the predefined minimum-cost position of a vessel, the more is operation cost at the wharf; in this study, this part of the objective function is formulated as Minimization of the penalty cost for vessels not leaving before their preplanned departure time Let To sum up, the objective function can be formulated as follows:
Considering that formula (
Let
Objective function (
Constraint (
Constraints (
In PSO algorithms, the particle swarm concept originated as a simulation of a simplified social system by introducing a number of simple entities—the particles—in the search space, where each particle represents a solution approach corresponding to a given position and velocity, which can be used to evaluate the objective function at its current location.
The movement of each particle is guided by their position, according to their own best position and a swarm’s best position, which represents the quality of searching, and the velocity decides the direction in which the particle would move in the next generation. These particles search for optimal solutions through updating generations. Formulas (
Considering that the classical PSO algorithm mentioned above may lead the particles to grow unlimitedly, which influences the particles’ convergence to the optimal solution, Shi and Eberhart [
The PSO algorithm proposed in this study is based on the updating mechanism proposed by Shi and Eberhart (1998).
Assuming that
Sort those generated random numbers in descending order and then allocate the corresponding vessels to berth positions one after another, i.e., the greater the random number
For a better understanding, here illustrated in Figure
Schema of the encoding process.
The decoding process, which is applied in this study to construct the berth schedule corresponding to a given solution obtained by the proposed PSO algorithm, consists of three steps as follows: Step 1: initialization of the berthing schedule The initial berth schedule can be generated by arranging each vessel one after another in the order defined by the solution to its minimum-cost berthing position. It is worth mentioning that although placing vessels to their pre-defined minimum-cost berthing positions can avoid additional operation costs, it is hardly possible for berth operators to arrange all the vessels to their minimum-cost berthing positions without overlapping any of them at a busy port. In consequence, there is a good chance that the berth schedule obtained at this step is infeasible due to the overlaps, and therefore, actions have to be taken to detect and resolve possible overlaps. Step 2: detection of overlaps Considering that the overlap between two vessels takes place if and only if both berthing periods and spaces of these two vessels are partly overlapped; the overlap between two vessels Step 3: overlaps resolving Once overlaps are detected, the current berth schedule is not yet feasible, and thus actions must be taken to remove those overlaps. The procedure resolving overlaps between two vessels Step 3.1: removing overlap detected between two vessels In this study, the overlap detected between two vessels is eliminated by fixing one vessel and moving the other one towards all possible directions until no overlap is observed between them. Here shown in Figure Upon further analysis of the four movements mentioned above, it can be observed that only movement (iii) can result in a feasible solution because (1) movement (i) is not available because there is not enough space on the left (dashed rectangle exceeds the left boundary of the wharf); (2) movement (ii) introduces an overlap between vessel Step 3.2: improving the feasibility of berthing schedule by taking into account the nearby vessels having overlaps with certain moved vessel Since it is possible to introduce new overlaps between the vessel being moved and some of the nearby ships, the relationship of all vessels that may have overlaps with the newly placed vessel must be considered to avoid introducing new overlaps. For a better understanding, let us continue the illustration with the example mentioned in step 3.1. Since moving vessel Step 3.3: accepting the best feasible berthing schedule Compare all of the possible feasible berthing schedules generated by the adjustments described in steps 3.1 and 3.2 and accept the best one, i.e., the feasible berthing schedule with the smallest objective value as the one that corresponds to the given solution obtained by the proposed PSO algorithm.
An example of three vessels obtained at step 1 with overlap detected.
Illustration of possible movements made to remove the overlap between two vessels in time-wharf section.
Possible movements of vessel
The general procedure of the proposed PSO algorithm is as follows. Step 1: set up the parameters of the PSO algorithm, such as the number of particles and the value of inertia weight coefficient. Step 2: initialize the position and velocity in allowable ranges for each particle and set iteration Step 3: calculate the fitness value, which is equal to the objective value of the proposed model, for each particle. Step 4: set the local-best value and global-best value for each particle, where the former equals the particle’s current position and the latter the position of the best particle. Step 5: update the velocity and the position for each particle. Step 6: update the fitness value for each particle. Step 7: compare the current fitness value of each particle with the local-best one. If the current fitness value of a particle is better, update the local-best position of this particle; otherwise, it remains unchanged. Step 8: find out the particle with the best fitness function from the current swam. If the current best fitness value is better than that of the recorded global-best one, replace the global-best position with the position of the current best particle; otherwise, the global-best one remains unchanged. Step 9: if the number of iteration
In this study, instances of different scales are randomly generated with the method introduced by Park and Kim [
The cost coefficients
Parameters used in the experiments.
Parameter | Distribution type | Range |
---|---|---|
Uniform distribution | ||
Uniform distribution | ||
Uniform distribution | ||
Uniform distribution |
The numerical experiments are programmed in C# (VS2017) on a PC with 2.3 GHz Intel Core i5 CPU and 4 GB RAM, and CPLEX 12.5 is applied as the programming solver for small-size instances. Both programming solver and the proposed PSO algorithm are set to terminate within 3 hours (10,800 s).
First of all, experiments are conducted to compare between two different mooring modes, i.e., DLSM mode and SLSM mode by considering both objective values and execution time for small-size instances, i.e., the instances with up to 25 vessels.
As shown in Table
Comparison between DLSM and SLSM modes for small-scale instances.
Instances | SLSM | DLSM | Diff_Obj1 (%) | Diff_CPU1 (%) | ||
---|---|---|---|---|---|---|
OBJ1 | CPU1 (s) | OBJ2 | CPU2 (s) | |||
8-1 | 300 | 0.2 | 252 | 0.2 | −16.00 | 0.00 |
8-2 | 324 | 0.3 | 304 | 0.2 | −6.17 | −33.33 |
8-3 | 488 | 0.3 | 488 | 0.4 | 0.00 | 33.33 |
10-1 | 430 | 0.3 | 430 | 0.5 | 0.00 | 66.67 |
10-2 | 656 | 0.4 | 656 | 0.3 | 0.00 | −25.00 |
10-3 | 754 | 0.6 | 474 | 0.5 | −37.14 | −16.67 |
15-1 | 2642 | 4.4 | 2020 | 2.3 | −23.54 | −47.73 |
15-2 | 804 | 2.2 | 544 | 2.3 | −32.34 | 4.55 |
15-3 | 940 | 0.5 | 940 | 1.1 | 0.00 | 120.00 |
20-1 | 2816 | 733.2 | 1842 | 8.3 | −34.59 | −98.87 |
20-2 | 3574 | 205.2 | 2604 | 7.7 | −27.14 | −96.25 |
20-3 | 1932 | 116.4 | 1232 | 3.6 | −36.23 | −96.91 |
25-1 | 4532 | 3630 | 3384 | 668.7 | −25.33 | −81.58 |
25-2 | 6886 | 7245.2 | 4616 | 1510.1 | −32.97 | −79.16 |
25-3 | 4440 | 7280.1 | 2936 | 1421.0 | −33.87 | −80.48 |
Average | −20.35 | −28.76 |
With regard to the objective values, it can be observed that the DLSM mode obviously dominates the SLSM mode because the objective values of solutions with the DLSM mode (shown in column “OBJ2”) are at least as good as those with the SLSM mode (shown in column “OBJ1”). According to the difference rates shown in column “Diff_Obj1” (Diff_Obj1 = (OBJ2 − OBJ1)/OBJ1
To sum up, it can be concluded that DLSM mode can help the port operators in not only improving their work efficiency but also reducing overall operation costs.
It should also be mentioned that the optimal solutions cannot be obtained by CPLEX solver within 3 hours for the instances with more than 25 vessels for neither of these two modes. Therefore, we can conclude that CPLEX solver is only effective for solving small-scale problems regardless of whether DLSM is applied, and thus it is necessary to develop efficient heuristics to obtain good quality solution within reasonable execution time for large-scale instances so as to cope with the real requirements of the huge terminal containers such as Yangshan port.
As mentioned before, the CPLEX solver is just capable of solving BAP models for small-scale instances with both SLSM and DLSM modes, though much more vessels must be scheduled during even 120 hours. Thus, in this study, a PSO algorithm has been proposed to obtain good quality solutions within reasonable execution time for large-scale instances.
As shown in Table
Comparison between the performance of CPLEX and PSO algorithm for solving problems with the DLSM mode.
Instances | CPLEX | PSO | Diff_Obj2 (%) | ||
---|---|---|---|---|---|
OBJ2 | CPU2 (s) | OBJ3 | CPU3 (s) | ||
8-1 | 252 | 0.2 | 252 | 8.2 | 0.00 |
8-2 | 304 | 0.2 | 304 | 4.4 | 0.00 |
8-3 | 488 | 0.4 | 488 | 9.4 | 0.00 |
10-1 | 430 | 0.5 | 430 | 10.1 | 0.00 |
10-2 | 656 | 0.3 | 656 | 8.0 | 0.00 |
10-3 | 474 | 0.5 | 484 | 16.2 | 2.11 |
15-1 | 2020 | 2.3 | 2052 | 40.7 | 1.58 |
15-2 | 544 | 2.3 | 574 | 41.3 | 5.51 |
15-3 | 940 | 1.1 | 1066 | 23.6 | 13.40 |
20-1 | 1842 | 8.3 | 2102 | 73.7 | 14.12 |
20-2 | 2604 | 7.7 | 2974 | 73.9 | 14.21 |
20-3 | 1232 | 3.6 | 1232 | 73.6 | 0.00 |
25-1 | 3384 | 668.7 | 3904 | 103.3 | 15.37 |
25-2 | 4616 | 1510.1 | 4854 | 122.8 | 5.16 |
25-3 | 2936 | 1421.0 | 3022 | 111.3 | 2.93 |
30-1 | Cannot | >3 h | 7566 | 326.3 | — |
30-2 | obtain | >3 h | 6820 | 251.1 | — |
30-3 | the optimal | >3 h | 8248 | 211.4 | — |
35-1 | solution | 7898 | 272.2 | — | |
35-2 | 12006 | 273.1 | — | ||
35-3 | 12760 | 279.5 | — | ||
40-1 | 15800 | 354.1 | — | ||
40-2 | 15640 | 565.2 | — | ||
40-3 | 20558 | 474.2 | — | ||
45-1 | 26150 | 685.2 | — | ||
45-2 | 35374 | 744.6 | — | ||
45-3 | 31726 | 644.2 | — |
With regard to objective values, the proposed PSO can obtain optimal solutions for the instances with 8 vessels and most of the instances with 10 vessels and even one instance with 20 vessels; near-optimal solutions can be obtained for the rest of the instances with 10 vessels and most of the cases with 15 vessels and even most of the cases with 25 vessels with quite small difference rate, which can be illustrated in column “Diff_Obj2” (Diff_Obj2 = (OBJ3 − OBJ2)/OBJ2
Since it is observed in Table
As shown in Table
Comparison between optimal solutions with SLSM mode and solutions obtained by PSO with the DLSM mode.
Instances | CPLEX-SLSM | PSO-DLSM | Diff_Obj3 (%) | ||
---|---|---|---|---|---|
OBJ1 | CPU1 (s) | OBJ3 | CPU3 (s) | ||
8-1 | 300 | 0.2 | 252 | 8.2 | −16.00 |
8-2 | 324 | 0.3 | 304 | 4.4 | −6.17 |
8-3 | 488 | 0.3 | 488 | 9.4 | 0.00 |
10-1 | 430 | 0.3 | 430 | 10.1 | 0.00 |
10-2 | 656 | 0.4 | 656 | 8.0 | 0.00 |
10-3 | 754 | 0.6 | 484 | 16.2 | −35.81 |
15-1 | 2642 | 4.4 | 2052 | 40.7 | −22.33 |
15-2 | 804 | 2.2 | 574 | 41.3 | −28.61 |
15-3 | 940 | 0.5 | 1066 | 23.6 | 13.40 |
20-1 | 2816 | 733.2 | 2102 | 73.7 | −25.36 |
20-2 | 3574 | 205.2 | 2974 | 73.9 | −16.79 |
20-3 | 1932 | 116.4 | 1232 | 73.6 | −36.23 |
25-1 | 4532 | 3630 | 3904 | 103.3 | −13.86 |
25-2 | 6886 | 7245.2 | 4854 | 122.8 | −29.51 |
25-3 | 4440 | 7280.1 | 3022 | 111.3 | −31.94 |
Average |
Considering that hundreds of vessels should be operated every day at huge container terminals, the proposed PSO will be much more practical than CPLEX for supporting the decision-making of the port operators to not only improve their working efficiency but also reduce operation costs related to berth scheduling operations.
The study aims at minimizing the total operation cost of the continuous berth scheduling problem by taking into account the Double-Line Shipping Mooring (DLSM) mode, where both the additional operation cost for vessels not moored at their minimum-cost berthing position and penalty cost related to vessels not being able to leave before its preplanned departure time are considered.
The problem is firstly formulated as a mixed integer programming model and solved by the CPLEX solver for small-scale instances. As for larger size instances that cannot be optimally solved by CPLEX solver, a PSO algorithm is proposed to obtain good quality solutions within reasonable execution time.
Numerical experiments are conducted to compare not only the efficiency between the traditional Single-Line Shipping Mooring (SLSM) mode and the innovative DLSM mode but also the performances between CPLEX solver and the proposed PSO algorithm. It can be concluded with the experimental results that (1) DLSM mode outperforms the SLSM mode in reducing not only total operation cost but also execution time. (2) The proposed PSO algorithm can generate optimal or near-optimal solution for small-scale instances. (3) The proposed PSO algorithm is much more efficient than the CPLEX solver for large-scale instances, which copes with the requirements of berthing management in Yangshan Deep-Water Port, one of the busiest container terminals in the world.
To sum up, as the first research dedicated to BAP with DLSM mode, this study can help not only in validating the advantages of DLSM mode but also offering an efficient decision support tool to berth operators in busy ports to improve their working efficiency.
Motivated by the results obtained in this study, it is interesting to keep improving the efficiency of the proposed algorithm and to apply such method in the targeted port.
All the experimental data can be generated with the rules described in the paper.
The authors declare that there are no conflicts of interest regarding the publication of this paper.