As the “first service station” for ships in the whole port logistics system, the tugboat operation system is one of the most important systems in port logistics. This paper formulated the tugboat scheduling problem as a multiprocessor task scheduling problem (MTSP) after analyzing the characteristics of tugboat operation. The model considers factors of multianchorage bases, different operation modes, and three stages of operations (berthing/shifting-berth/unberthing). The objective is to minimize the total operation times for all tugboats in a port. A hybrid simulated annealing-based ant colony algorithm is proposed to solve the addressed problem. By the numerical experiments without the shifting-berth operation, the effectiveness was verified, and the fact that more effective sailing may be possible if tugboats return to the anchorage base timely was pointed out; by the experiments with the shifting-berth operation, one can see that the objective is most sensitive to the proportion of the shifting-berth operation, influenced slightly by the tugboat deployment scheme, and not sensitive to the handling operation times.
Container terminal is an important part in international logistics and plays a significant role in world trade. Recently, more and more people become to recognize the importance of global logistic business via container terminals. As the throughput of containers in container terminal increases and competition between ports becomes fierce, how to improve the efficiency in container terminal has become an important and immediate challenge for port managers. One of the most important performance measures in container terminals is to schedule all kinds of equipment at an optimum level and to reduce the turnaround time of vessels. Tugboat is one such kind of vital equipments in container terminal.
The performance of the tugboat operation scheduling has a direct influence on time when a ship can start its handling operation and when a ship can leave the port. Scheduling on tugboats with good performance may lower the turnaround of ships in a port. Thus the tugboat scheduling problem is an important one to be solved in the field of the port logistics.
When ships arrive at a port, if their target berths are not available immediately, they cannot enter into the berths directly and have to wait in the anchorage ground. Then they have to be tugged by certain amount of tugboats according to some rules. Moreover, the moving between two berths and the department of vessels also need to be tugged by tugboats. To improve the ship operation efficiency, tugboats should be scheduled at an optimum level.
According to the analysis mentioned above, the three types of service that a tugboat can provide are (a) tugging coming ships to the berth (viz., berthing); (b) tugging ships from one berth to another (namely, shifting-berth); (c) tugging ships leaving the berth (viz., unberthing). Not every ship will experience all the three types of services. That is, the shifting berth operation is not necessary, while the berthing and unberthing operations are necessary for all ships.
A typical tugboat operation process is illustrated in Figure
Illustration of a typical tugboat operation process.
Practically, tugboat scheduling managers allocate suitable tugboats to ships according to their length. Each ship can have one or more tugboats serving for it simultaneously by the scheduling rules.
The main idea of the scheduling rules is that big ships should be served by big tugboats (as with the horsepower), and small ships should be served by small tugboats; if more than one tugboat with the same horsepower are available, the allocation among the available tugboats is made by some heuristic rules.
For example, there are six types of tugboats in a port according to the horsepower unit, such as 1200PS, 2600PS, 3200PS, 3400PS, 4000PS, and 5000PS. The scheduling rules for allocating are as follows:
And the heuristics rules concluded from real-world practice include TSD rule: choosing the tugboat with the shortest distance from the scheduled ship to serve for it; FAT rule: choosing the tugboat which is the first available one for the scheduled ship; UWAT rule: from the perspective of balancing all tugboats’ working amount, choosing the tugboat with the minimum working amount up till now to serve for the scheduled ship.
According to the hybrid flow shop theory, the tugboat scheduling can be considered as a multiprocessor task scheduling problem (MTSP) with 3 stages. In the scheduling system, tugboats are taken as movable “machines,” and ships have to experience the berthing, shift-berth (if there exists this operation), and unberthing operations operated by tugboats sequentially.
On the other hand, compared with a typical MTSP, the tugboat scheduling problem has its own characteristics. Firstly, the exact same tugboat can provide all the three types of service (berthing, shifting berth, and unberthing), which means that the machine set for all the three stages is the same. This is different from a typical MTSP in which the available machine set in each stage is not the same. Besides, not all ships have to experience the shift-berth operation, which makes the problem different from a typical MTSP with the characteristics that all jobs have to experience all the stages.
Anyway, the tugboat scheduling problem is a kind of unconventional scheduling problem, an NP-hard problem which cannot be solved by exact methods. Some scholars have begun to make research on the topic.
Ying and Lin [
As we can see from the previous research, scholars have begun to use many approaches to solve the tugboat scheduling problem, including the genetic algorithm, ant colony optimization, and particle swarm optimization. However, most literature only considers the situation of single operation stage and single anchorage base and neglects the influence of the tugboats’ and ships’ location information on the problem difficulty. That makes the model formulated far from reality. Thus, this paper will make research on tugboat scheduling problem considering multi-anchorage bases, different operation modes, and three operation stages.
The rest of paper is organized as follows. Section
The following assumptions are introduced for the formulation of the problem. The planning horizon is one day. Three operation stages (i.e., berthing, shifting-berth, and unberthing) are taken into consideration, but not all ships have to experience the shifting-berth operation. For ship which does not have to experience the second operation, assume there is a virtual shifting-berth operation and the operation time for that is zero. The ready times for all the tugboats are 0, and all the tugboats are at the anchorage bases at time 0; all the ships to be served have arrived at the anchorage ground at time 0. There are three types of locations in a port: berths for ships to load/unload cargoes, meeting locations where ships meet tugboats at the entrance of port, and the anchorage bases. Two operation modes (restricted cross-operation mode RCOM and unrestricted cross operation mode UCOM) may be adopted to schedule the tugboats in a port. All the ships enjoy the same precedence. The scheduling rules for allocating tugboats to ships are what we mentioned in Section The sailing speeds of all tugboats whenever sailing are the same. The tugboats may return to the anchorage base during the planning horizon according to the scheduling plans.
In assumption (e), the RCOM means that all anchorage bases have their fixed service area in the port, which means that each tugboat in every base can only operate in its corresponding service area, while the UCOM means that all tugboats can operate in the whole area of a port.
Before the tugboat scheduling model is formulated, a concept named scheduling round should be introduced first.
In practice, a scheduling round is used to define the duration from the time when a tugboat leaves for its target place from the anchorage base to the time when it returns to the base after finishing a certain amount of tasks (may be one task, maybe more than one). As Figure
Illustration of the tugboat scheduling rounds.
: Stage index,
In this paper, the objective is to minimize the total operation times of tugboats, which can be equal to the total duration for all the scheduling rounds of all tugboats. Thus we have to derive the calculation method for scheduling rounds.
From the definition of the scheduling round, the relation between the decision variable (
Equation (
Define the set of tasks right before which tugboat
Equation (
As it has been discussed before, the total operation times of tugboats are equal to the total duration for all the scheduling rounds of all tugboats. Thus the objective function can be expressed as follows:
The constraints in the proposed model include the following equations:
Constraint (
Ant colony metaheuristic is a concurrent algorithm in which a colony of artificial ants cooperates to find optimized solutions of a given problem (see Boveiri [
The inspiring natural process of ACS is the foraging behavior of ants. A colony of ants can identify the shortest pathway from a food source to their anthill without using visual cues; they communicate through an aromatic substance, called pheromone. While walking, ants secrete pheromone on the ground and follow, in probability, the pheromone previously laid by other ants. Ants are more likely to follow pathways marked by a larger accumulation of pheromone from other ants that have previously walked that route. Since ant searching a food source by shorter pathways will come back to the anthill sooner than ants traveling via longer pathways, the shorter pathways will have a higher traffic density than those of the longer ones. Hence, the pheromone accumulation will build up more rapidly on shorter pathways than on longer ones. Consequently, the fast accumulation of pheromone on the shorter pathways will cause ants to quickly choose the shortest routes. The described foraging behavior of ants can be used to solve scheduling problems by simulation: the objective value (e.g., flow time) corresponds to the quality of the food source (e.g., distance), artificial ants searching for the solution space simulate real ants searching for their environment, and an adaptive memory corresponds to the pheromone trail. In addition, the artificial ants are equipped with a local heuristic function to guide their search through the set of feasible solutions [
The main procedure of the ant colony algorithm is as follows. Generate ant (or ants). Loop for each ant (until complete scheduling of tasks). Select the next task with respect to pheromone variables of ready tasks. Deposit pheromone on visited states. Daemon activities. Evaporate pheromone.
The flowchart of ant colony algorithm is illustrated as Figure
Flowchart of ant colony optimization.
However, the solutions were generated by each ant in the basic ant colony algorithm by random, and those solutions may not be the optimal solutions or satisfactory solutions. That makes the updating of the pheromone be done by random too, which may cause a lot of time costs to get the optimal value, and that value may also be the local optima. To avoid that phenomenon, the diversity of the population should be considered.
By that thought, we introduce the simulated annealing into the ant colony algorithm, which can guarantee the quality of the search and avoid the phenomenon of the local optima. Thus, the simulated annealing-based ant colony algorithm is proposed.
According to the analysis above, we introduce a simulated annealing-based ant colony algorithm to solve the formulated tugboat scheduling problem. The basic procedure of the algorithm is as Figure
Flowchart of the proposed algorithm.
Generate the initial tugboat scheduling plans (individuals) which act as representing codes for the simulated annealing algorithm.
Generate new scheduling nodes used to apply for the ant colony algorithm.
Apply the ant colony algorithm for the scheduling process.
Compute the total operation time for all tugboats in the planning horizon as the key indicator for the system.
If the current temperature is less than the final temperature, then go to
Reduce the temperature according to the predetermined rule.
Let the individuals having better fitness be new parents.
Based on the new parents, perform a new neighborhood search to get the new individuals.
Output the best solution.
In the ant colony algorithm for solving the proposed problem, jobs are defined as ants and resources are defined as nodes. The main procedure of ant colony optimization has been discussed in Section
According to the algorithm, a certain amount of ants have to be generated. In order to make the schedules by which ants travel satisfy the requirements of the scheduling system, three arrays were set in the algorithm:
The selection of nodes during the algorithm is referenced by the roulette wheel. Thus the state transit rule can be concluded as.
In (
After all ants of a generation have traveled all the tasks, compute the total operation times
In (
The key operations in the simulated annealing include individual coding, initial individuals’ generation, and the neighborhood search scheme.
In this paper, the real integer method is adopted to code for an individual. As every ship may experience at most 3 stages of operation, we set the number of columns as three times of the number of ships. Assume that there are 4 ships to be served (ship 1, 2 do not have to shift a berth, while ship 3, 4 will experience a shifting-berth operation) and 3 available tugboats, then the coding expression of the individuals should be a
Illustration of coding for an individual.
The first row of the coding representation means the service order for ships, and the next two rows are the indexes of tugboats serving for ships in the first row. Note that each index appears three times in the first row: if it is the first time an index appears, it means that the ship is berthing; for the second time it appears, it may be a virtual or real shifting-berth operation; otherwise, the unberthing operation. The fourth and fifth rows are descriptive parts which tell us whether tugboat 1 and 2 return to the base after finishing the task.
As ship 1 and 2 do not have to shift a berth, the virtual shifting-berth operations are proposed to keep the total operations three times of the number of ships. That can be illustrated as the shadow parts with diagonal lines in Figure
According to that individual coding, the service order for ships in Figure
The procedure for generating the initial schedule can be described as Figure
As we can see from Figure
The generation procedure of the initial individuals.
The procedure for the neighborhood search scheme can be concluded as Figure
The neighborhood search scheme.
Given a solution
However, during the neighborhood search process, the temporary solution may be an infeasible solution. For example, the virtual shifting-berth operation (the shadow parts in Figure
The infeasible solution generated by the three-point interchange.
Thus it is necessary to modify the temporary solution. Steps for modifying the temporary solution are as follows.
Initialize
Judge if the second and third rows of the If both the values are zero, which means that the task in the search for two columns: one for the berthing operation for ship served in the if if If the two values are not both zero, then go to
Judge if if else, set
After being modified according to the steps introduced above, the temporary solution can be changed to a new solution by deciding whether tugboats should return to the base according to (
To implement a comparison of the findings from the proposed algorithm, some experimental data were randomly generated, details of which are as follows. Location data: the sailing times between each location (P1–P8, M1-M2, B1-B2) are as Table Ship data: styles of ships are generated to S1, S2, S3, S4, and S5 which take up about 10%, 20%, 40%, 20%, and 10% of the total ships, berthing/unberthing times, loading and unloading times of ships are normally distributed in Tugboat data: quantities of the six kinds of tugboats in the two anchorage bases are all one.
Sailing times between each location.
P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | M1 | M2 | B1 | B2 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
P1 | 0 | 18 | 14 | 20 | 32 | 34 | 35 | 30 | 19 | 29 | 15 | 32 |
P2 | 18 | 0 | 23 | 15 | 31 | 33 | 27 | 35 | 21 | 33 | 12 | 35 |
P3 | 14 | 23 | 0 | 12 | 39 | 34 | 30 | 32 | 15 | 38 | 16 | 31 |
P4 | 20 | 15 | 12 | 0 | 35 | 38 | 31 | 39 | 12 | 31 | 18 | 34 |
P5 | 32 | 31 | 39 | 35 | 0 | 18 | 12 | 19 | 31 | 12 | 29 | 11 |
P6 | 34 | 33 | 34 | 38 | 18 | 0 | 13 | 15 | 34 | 11 | 36 | 15 |
P7 | 35 | 27 | 30 | 31 | 12 | 13 | 0 | 12 | 29 | 18 | 25 | 19 |
P8 | 30 | 35 | 32 | 39 | 19 | 15 | 12 | 0 | 33 | 15 | 39 | 12 |
M1 | 19 | 21 | 15 | 12 | 31 | 34 | 29 | 33 | 0 | 30 | 15 | 25 |
M2 | 29 | 33 | 38 | 31 | 12 | 11 | 18 | 15 | 30 | 0 | 28 | 16 |
B1 | 15 | 12 | 16 | 18 | 29 | 36 | 25 | 39 | 15 | 28 | 0 | 26 |
B2 | 32 | 35 | 31 | 34 | 11 | 15 | 19 | 12 | 25 | 16 | 26 | 0 |
Suppose the basic data are as shown in Section
Results of the PHA versus existing scheduling rules.
Number of ships | RCOM | UCOM | ||||||
---|---|---|---|---|---|---|---|---|
PHA | TSD | FAT | UWAT | PHA | TSD | FAT | UWAT | |
10 | 3743 | 4222 | 4181 | 4280 | 3743 | 4200 | 4156 | 4245 |
15 | 4983 | 5551 | 5504 | 5609 | 4951 | 5485 | 5456 | 5575 |
20 | 6800 | 7436 | 7357 | 7483 | 6705 | 7353 | 7242 | 7451 |
25 | 8481 | 9132 | 9030 | 9220 | 8405 | 8900 | 8858 | 9100 |
30 | 10012 | 11185 | 10993 | 11530 | 9988 | 11031 | 10920 | 11235 |
As we can see from Table
Besides, we can see from Table
Results from the proposed algorithm under two different operation modes.
Number of ships | Total operation times of all tugboats | Average operation times of all tugboats | ||||
---|---|---|---|---|---|---|
RCOM | UCOM | GAP1 | RCOM | UCOM | GAP2 | |
10 | 3743 | 3743 | 0 | 312 | 312 | 0 |
15 | 4983 | 4951 | 32 | 415 | 413 | 3 |
20 | 6800 | 6705 | 95 | 567 | 559 | 8 |
25 | 8481 | 8405 | 76 | 707 | 700 | 6 |
30 | 10012 | 9988 | 24 | 834 | 832 | 2 |
After the basic analysis above, we compare the operation times on whether tugboats return to the anchorage base during the planning horizon, the results of which can be shown as Table
Comparisons between the total operation times on whether tugboats return to the base.
Number of ships | RCOM | UCOM | ||||
---|---|---|---|---|---|---|
Results if tugboats do not return to the base ( |
Results if tugboats return to the base ( |
GAP* | Results if tugboats do not return to the base ( |
Results if tugboats return to the base ( |
GAP* | |
10 | 3743 | 2675 | 39.93% | 3743 | 2675 | 39.93% |
15 | 4983 | 4005 | 24.42% | 4951 | 3952 | 25.28% |
20 | 6800 | 5285 | 28.67% | 6705 | 5206 | 28.79% |
25 | 8481 | 6575 | 28.99% | 8405 | 6503 | 29.25% |
30 | 10012 | 8007 | 25.04% | 9988 | 7951 | 25.62% |
Based on Table
In this section, sensitivity analysis of the three elements to the objective is to be made, and all the experiments done are under the UCOM mode and based on the assumption that tugboats can return to the anchorage base during the planning horizon.
Assume that there are 0%, 5%, 10%, 15%, 20% of the total ships which have to experience the shifting-berth operation, the minimal total operation times of all tugboats when the number of ships is 10, 15, 20, 25, 30 are summarized in Table
As we can see from Table
Results with different proportion of the shifting berth operation.
Number of ships | Results with different proportion of the shifting-berth operation | ||||||||
---|---|---|---|---|---|---|---|---|---|
0%1 | 5%2 | GAP1a | 10%3 | GAP2b | 15%4 | GAP3c | 20%5 | GAP4d | |
10 | 2675 | 2845 | 6.36% | 3076 | 13.04% | 3224 | 20.52% | 3415 | 27.66% |
15 | 3952 | 4205 | 6.40% | 4518 | 12.53% | 4703 | 19.00% | 4961 | 25.53% |
20 | 5206 | 5485 | 5.36% | 5824 | 10.61% | 6121 | 17.58% | 6452 | 23.93% |
25 | 6503 | 6857 | 5.44% | 7331 | 11.29% | 7715 | 18.64% | 8035 | 23.56% |
30 | 7951 | 8381 | 5.41% | 9122 | 12.84% | 9423 | 18.51% | 9731 | 22.39% |
| |||||||||
Average | / | / | 5.79% | / | 12.06% | / | 18.85% | / | 24.61% |
Assume that the distribution characteristics of handling operation times of ships at berth are
As we can see from Table
Compared with the operation times of tugboats, the handling times are much larger. After completing a certain task, a tugboat can return to the base to have a rest and then sail to its next target location. With the increase of the handling operation times, the wait times in the base may also increase, which are not parts of the total operation times of tugboats. Thus, the objective does not reveal obvious reaction to the change of the handling times.
Results with different distribution characteristics of the handling operation times.
Number of ships | Results with different distribution characteristics of the handling operation times | ||||
---|---|---|---|---|---|
|
|
GAP1a |
|
GAP2b | |
10 | 2845 | 2815 | −1.05% | 2862 | 0.60% |
15 | 4205 | 4240 | 0.83% | 4200 | −0.12% |
20 | 5485 | 5522 | 0.67% | 5488 | 0.05% |
25 | 6857 | 6870 | 0.19% | 6840 | −0.25% |
30 | 8381 | 8387 | 0.07% | 8428 | 0.56% |
| |||||
Average | / | / | 0.14% | / | 0.17% |
We then assume different deployment schemes of the available tugboats in the port (i.e., Scheme 1: the number of all types are 1; Scheme 2: the number of type 6 are 2, others are 1; Scheme 3: the number of type 5 and 6 are 2, others are 1). The results solved by the PHA are summarized in Table
As Table
By the analysis, we can say that the objective is most sensitive to the proportion of the shifting-berth operation, influenced slightly by the tugboat deployment scheme, and not sensitive to the handling operation times.
Results with different tugboat deployment schemes.
Number of ships | Results with different tugboat deployment schemes | ||||
---|---|---|---|---|---|
Scheme 11 | Scheme 22 | GAP1a | Scheme 33 | GAP2b | |
10 | 2845 | 2833 | −0.42% | 2808 | −1.30% |
15 | 4205 | 4186 | −0.45% | 4159 | −1.09% |
20 | 5485 | 5421 | −1.17% | 5394 | −1.66% |
25 | 6857 | 6807 | −0.73% | 6785 | −1.05% |
30 | 8381 | 8325 | −0.67% | 8299 | −0.98% |
| |||||
Average | / | / | −0.69% | / | −1.22% |
This paper formulated the tugboat scheduling problem as a multiprocessor task scheduling problem (MTSP). The model considers factors of multi-anchorage bases, different operation modes, and three stages of operations (berthing/shifting-berth/unberthing). A hybrid simulated annealing-based ant colony algorithm is proposed to solve the addressed problem. By the numerical experiments without the shifting-berth operation, the effectiveness were verified, and the fact that more effective sailing may be possible if tugboats return to the anchorage base timely was pointed out; by the experiments with the shifting-berth operation, the paper proved that the objective is most sensitive to the proportion of the shifting-berth operation, influenced slightly by the tugboat deployment scheme, and not sensitive to the handling operation times.
Future work about the topic should be to extend the problem from the static situation to a dynamic one, although it may be much more difficult but more meaningful.