A study on multi-ASC scheduling method of automated container terminals based on graph theory

Accepted Manuscript A study on multi-ASC scheduling method of automated container terminals based on graph theory Houjun Lu, Sai Wang PII: DOI: Reference:

S0360-8352(19)30067-1 https://doi.org/10.1016/j.cie.2019.01.050 CAIE 5676

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

27 February 2018 13 December 2018 25 January 2019

Please cite this article as: Lu, H., Wang, S., A study on multi-ASC scheduling method of automated container terminals based on graph theory, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie. 2019.01.050

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Title name: A study on multi-ASC scheduling method of automated container terminals based on graph theory.

Author details: The first author & corresponding author Houjun Lu, a Dr. from the College of Logistics Engineering of Shanghai Maritime University who’s field of research is terminal operation system development and port equipment scheduling. E-mail: [email protected]. The second author Sai Wang, a software development engineer from the Department of Information Technology of SAIC Motor Commercial Vehicle Technical Center who’s major research field is Manufacturing Execution System development. E-mail: [email protected]

Postal address: College of Logistics Engineering, Shanghai Maritime University, 1550 Haigang Avenue, Shanghai 201306, PR China.

ABSTRACT Compared with traditional terminals, two automated stacking cranes (ASCs) are configured for each container block of automated container terminal (ACT), which interact with automated guided vehicles (AGVs) and container trucks at the two ends of a container block individually. To increase the capacity, container yards with multiple rows of blocks perpendicular to the terminal’s shoreline are considered. To utilize the yard spaces, twin ASCs are set to share transport tracks installed at the two sides of a block, while interferences between the ASCs causes the routing and sequencing operations. In order to control the scheduling of twin ASCs, the interference model is established by analyzing the time overlap between tasks. Considering the influence of AGV transportation time, the model are then established to sequence the container handling tasks under the minimization of waiting time and makespan. A particle swarm optimization algorithm (PSO) based on graph theory model is design to solve the problem. Numerical experiments show that the algorithm is more competitive than traditional algorithm. Based on the model and experimental result, the practical significance of applying the algorithm to the actual situation is discussed. Keywords: Automated container terminal, Automated stacking crane, Automated guided vehicle, Particle swarm optimization algorithm, graph theory.

Funding project of the paper: The academic project of Shanghai Maritime University (YXR2017059)

1

ABSTRACT Compared with traditional terminals, two automated stacking cranes (ASCs) are configured for each container block of automated container terminal (ACT), which interact with automated guided vehicles (AGVs) and container trucks at the two ends of a container block individually. To increase the capacity, container yards with multiple rows of blocks perpendicular to the terminal’s shoreline are considered. To utilize the yard spaces, twin ASCs are set to share transport tracks installed at the two sides of a block, while interferences between the ASCs causes the routing and sequencing operations. In order to control the scheduling of twin ASCs, the interference model is established by analyzing the time overlap between tasks. Considering the influence of AGV transportation time, the model are then established to sequence the container handling tasks under the minimization of waiting time and makespan. A particle swarm optimization algorithm (PSO) based on graph theory model is design to solve the problem. Numerical experiments show that the algorithm is more competitive than traditional algorithm. Based on the model and experimental result, the practical significance of applying the algorithm to the actual situation is discussed. Keywords: Automated container terminal, Automated stacking crane, Automated guided vehicle, Particle swarm optimization algorithm, graph theory.

1. Introduction With the increase of import and export container number, lack of port efficiency has aroused widespread concern in the industry (Huth, T., Mattfeld, D.C., 2009), how to use the terminal resources effectively, improve port efficiency, reduce the energy consumption of the equipment scheduling has become a close concern of port management, 70% of the freight volume in the trade is transported by container ship (Wang, S., Liu, Z., Bell, M.G.H., 2015). In 2000, the container ship capacity reached 8 million Twenty-foot Equivalent Units(TEUs), in 2013, the quantity has reached 18 million TEUs(Rodrigue, J.P., Comtois, C., Slack, B., 2013). For large ports, the traditional operation mode is hard to meet the needs at this stage (Cabral, A.M.R., de Sousa Ramos, F., 2014). At the same time, with the maturity of automation equipment technology and the updating and iteration of the port integrated management system, the development model of the traditional container terminal is turning to automation. The concept of ACT was first introduced in the middle of 80s. Currently, the world's most automated ports are Holland Rotterdam port, Hamburg CTA, Qingdao port, Xiamen port, Shanghai Yangshan Port and so on(Rui Yang, Qing Li, 2017). In order to solve the shortcomings of traditional terminal operation ability, the devices of advanced automated terminal operation equipment are configured, such as overhead bridge crane (OBC), ASC, AGV, I/O Point support area(Vis, I.F.A., Carlo, H.J., 2010) . In the automated terminal, container is operated by OBC from unloaded container ships and placed on the surface of AGV firstly, AGV is assigned the path by the vehicle control system (VCS), and the terminal operating system (TOS) distributes the target container position(He Gang, 2017), the target container is transported to I/O Point, and the target container is operated by ASC in the block and transported to the target bay. The advantages of the automated terminal lie in the following two points. (1) In the automated block, as shown in Fig.1, each side is provided with AGV exchange area, the target container can be loaded and unloaded through the exchange area, the yard truck and AGV are prohibited from entering the block to achieve the ASC automation collaborative operation. (2) Automated terminal adopts the process mode of operational combination of OBC, AGV and ASC to reduce the operation time.

2

OBC

I/O Point

AGV block ASC

BAY 01 03 05

07 09 11 13 15 17

19 21

23 25 27 29 31

yard crane

Fig.1. An aerial view of an ACT

Container terminal operation is an important link between loading and unloading operation. As shown in Fig.1, it is ASC operation of automated terminal. A block of automated container terminal is equipped with twin ASCs which are respectively used to the sending and receiving container in sea side and land side (Park et al., 2010). Because ASC can not be shared between blocks, the whole automated terminal efficiency is affected by ASC sequencing task. The ASC scheduling problem belongs to NP-hard problem, how to carry out ASC scheduling is of great importance to improve the port production efficiency and reduce the stay time of ships.

2. Related studies 2.1. Single ASC operations optimization For the single ASC scheduling study, Chunqian Zhang et al.(2002) adopted the integer programming model to optimize the dynamic task arrangement and path planning of single ASC by improving Lagrange heuristic algorithm. Linn R. (2003) studied the yard crane scheduling between blocks, which only considered the scheduling time, and the priority problem of the target box was ignored. Kim K.H. (2003) established a mixed integer programming model to solve the optimal path of single ASC. Kim K.H. et al.(2005) studied the problem of sequence in optimization of single ASC scheduling. Y.G. Chung et al.(1988) was the first one who first studied the optimization problem of ASC path selection in order to improve the efficiency of yard working efficiency and reduce the operation distance of ASC. 2.2. Twin ASC operating in a block 3

For the twin ASC scheduling study, based on buffer capacity constraint and time window constraint, Kim et al. (2009) established mixed integer programming model to solve the priority loading and unloading sequence of target container, and heuristic algorithm is designed to verify relevant examples, the result shows that the sequence of target container is arranged in order of priority and loading and unloading, and the delay time of the target container is reduced. Based on twin ASC scheduling problem, B. Skinner et al. (2013) design two-stage genetic algorithm to solve the optimization sequence of container loading and unloading. The results show that the sequence of loading and unloading target container is optimized and the cost of yard machinery and equipment is reduced. Yan Wei et al. (2008) proposed a container yard crane scheduling strategy based on the optimal priority search algorithm, and used the algorithm to simulate the case. C. Zhang (2000) studied the path optimization problem of yard crane, and the optimal model is solved by Lagrangian relaxation method with minimum total operation time. Dirk Briskorn and Panagiotis Angeloudis (2016) proposed a dynamic programming model for the twin ASC operation strategy, which effectively solved the problem of twin ASC cooperative operation. U. Speer et al.(2011) proposed two yard crane cooperative operation mode for the problem of interference between ASCs. 2.3. Research on twin ASC scheduling based on mixed storage mode It can be seen from the above literature that the research on ASC scheduling of container yard is based on traditional storage mode, and the research on ASC scheduling of hybrid stack mode is less. L.M.Gambardella et al.(2011) proposed the optimization model of container storage in mixed storage mode. Chunqian Zhang and Liu Liyin (2003) proposed an hybrid stack mode based on integer programming model, which distributed the loading and unloading target boxes to each block. But didn’t take into account the priority of the target container. Wang Bin (2005) studied the optimal distribution method of the import and export random container in a dynamic rolling period considering the randomness of the target container. Kang Haigui et al.(2009) introduced integer programming method based on the rolling plan. Zheng Hongxing and Yu Kai (2015) built a ASC scheduling mode with the target of the minimum cost of waiting time and ASC total travel, and designed the corresponding genetic algorithm considering the arrival time of yard truck. But it didn’t consider the uncertainty of the arrival time of yard truck. Liang Chengji et al.(2016) established the integer programming model considering the safety distance and equipment interference between multi-yard cranes, and the model was optimized by CPLEX. H.J. Carlo et al.(2015) designs a scheduling model based on ASC priority rule in mixed mode, applies branch and bound algorithm to solve the model, and quantifies different ASC scheduling priorities through simulation. M. Sha et al.(2017) established integer programming model to minimize the ASC energy cost based on mixed storage mode and simulation using field data, the result shows that ASC operation time was reduced, energy consumption of related equipment was saved, but this method can only solve the ASC local scheduling optimization problem, it is failed to optimize the sequence of target container scheduling. 2.4. Contributions In order to improve the utilization rate of automated yard, considering the mixed stack mode, we propose a twin ASC scheduling model, which is characterized as follows. (1) We have interviewed the schedulers from the main container terminals in China on the scheduling methods used in their information systems. Batch-based task assignment and scheduling solutions are widely used by them. Solutions based on rolling-horizon algorithms are usually developed to cope with the uncertain and large-scale operations optimization problems at container terminals. (2) The remarshaling, rehandling or repositioning operations within a block are usually not mixed with release and retrieval operations in a batch because their combinations make the problems even complex. Prior rehandling and prior repositioning can be used as effective means of cooperating the ASCs.

3. Twin ASC scheduling model 4

3.1. Problem description Considering that there are sequencing tasks on both sides of the sea and the land in the ASC scheduling problem, we uses the relay operation mode for ASC scheduling. The relay operation indicates that ASC1 runs to the designated bay and unloads the target container. ASC2 loads and unloads the target container and runs to the target bay for unloading. This operation can save the running time of ASC and optimize the AGV waiting time. The twin ASC scheduling problem should not only consider the assignment of loading and unloading tasks to two ASCs to arrange the optimal scheduling strategy, but also the mutual interference between ASCs. The interference situations include ASC collision, traversal, and execution of the same loading and unloading tasks. In the actual operation of the automated terminal, the AGV waiting time is also the key to affect the twin ASC scheduling problem. Therefore, the AGV waiting time is included in the influencing factors. we assume that the horizontal transport equipment does not wait indefinitely while waiting for the loading and unloading task. Under the premise that the container position to be loaded and unloaded, the container pick-up time, and the AGV arrival time are known, the container loading and unloading tasks are completed one by one to minimize the ASC working time and AGV waiting time.

3.1.2 AGV Arrival Plan

In this paper, the set of waiting time is divided by the given AGV arrival time schedule. Considering the uncertainty of the AGV arrival time in each stage, the first arrival time in each stage is the initial time of the stage, and the end time of the stage is ASC. The moment when a move is performed and returned to the sea side bracket area. The factors to be considered in the classification of the AGV waiting time set are: the AGV arrival time, the time when the ASC completes the task, and the AGV marking the need to wait in each stage. (1) AGV arrival time The start time of each task phase is arranged according to the AGV arrival time, and the classification of the AGV waiting time set can be performed by determining the start time of each phase. (2) ASC's time period for completing the task The time period for the ASC to complete the task is mainly to record the time that the ASC completes a move under the current task, which can determine the number of AGVs in the waiting state. (3) AGV waiting time collection During the time period of each task determined in (2), when the AGV arrives, if the capacity of the sea side bracket area is overloaded, the currently arrived AGV is marked as into

, and according to the task set

incorporates

; if the capacity of the sea side bracket area is still available, the AGV does not enter the task set.

The AGV arrival time and the waiting time set in the task time period. The horizontal axis represents the time period (task phase) in which each ASC runs a move, and the vertical axis represents the arrival time of the rth AGV. For example, during the time period of ={

, the ASC is in the time period of running a move, and the set

} indicates that it is waiting for unloading container. The AGV is not recorded in the set if it does not

need to wait. Through this dynamic programming idea, all the AGVs in the state to be processed are classified into various stages to determine whether relay operations are required at each stage. At the same time, avoiding the peak loading and unloading period, the seaside ASC is under load due to the excessive number of AGVs to be loaded and unloaded and the wrong selection of the relay position, is depicted in Fig.2.

5

Fig.2. AGV arrival time and waiting time Gantt chart

3.1.3 ASC Scheduling Plan

Within the automated block, the ASC performs tasks according to a pre-arranged scheduling plan, and all scheduling commands are expected to execute from the required start to the final delivery time, but the tasks do not have to be performed as planned. Different from traditional yard crane scheduling, since the twin ASC moves only in the block, it is impossible to move across the block, which limits the flexibility of equipment scheduling in the yard. Operation tasks may create overlapping areas (collision constraints) especially during the high peak loading and unloading period of the yard, which leads to uncertainty of mission planning time. How to effectively dispatch ASC without affecting the efficiency of the operation becomes a key issue for the yard manager to make dynamic decisions is depicted in Fig.3.

Fig.3. diagram of ASC scheduling time

In the terminal, when the AGV transportation target container reaches the block, a current command list is generated for the specific ASC in the block, and the ASC performs the task according to the command list. Based on this case, we arrange the running tasks through the command list. In principle, each ASC can perform each task, but this paper avoids the task of the seaside ASC to perform the target bay on the landside by setting the penalty coefficient, and vice versa, is depicted in Fig.4. On the ASC task selection module, since the sea side bracket is an important parameter in this process, it needs to be considered in the scheduling task. If the sea side bracket area has a high occupancy rate, it should be arranged to schedule an operation task according to each target container priority. The evaluation function is used to ensure that the potential commands selected by each ASC have evaluation functions, and the evaluation values assigned to the ASC allocation commands should be divided. (1) The time ASC runs to each target container bay ( speed,

, L indicates the distance,

Indicates ASC running

indicates ASC travel to target bay time).

(2) Relaxation time of program command start time ( the scheduled completion time for this task phase,

, and

indicates relaxation time,

indicates

respectively represent the time when the ASC actually 6

completed the task and the planned completion time in the previous task phase, and the μ is a 0-1 variable indicating whether the seaside bracket area has a target container). (3) ASC work area penalty factor (ASC will get a penalty factor

in the task overlap area).

(4) Priority command (if the task object selects the target container according to the AGV task schedule, it will get a negative penalty coefficient

).

Fig.4. terminal operation command diagram

3.2 Mathematical Model The operation unit is divided into various loading and unloading links of the automated yard. Therefore, the working unit in this paper includes the following. (1)The AGV waiting time set in each loading and unloading task is taken as its working unit. (2) the AGV waiting time in the AGV waiting time set cannot exceed the maximum waiting time. If it exceeds, the penalty value is added for the AGV waiting time. (3) the seaside task and the landside task are each a working unit, and the seaside task priority is set to be greater than the landside task. According to some operations involved in the loading and unloading process of the block, this paper makes the following assumptions. Assumption 1: The AGV arrival time series has been obtained before the start of the ASC loading and unloading task. Assumption 2: The capacity of the landside truck transfer point is sufficient when the landside truck is taken into the port. Assumption 3: The target container involved in the business process are all the same container type. Assumption 4: The ASC operating speed is set to a uniform motion to ignore the acceleration and deceleration processes.

3.2.1 Model Parameters and Decision Variables

Based on the ASC relay bay selection problem, the ASC loading and unloading task scheduling model is established. The relevant notation definitions are shown in Table 1. 7

Table 1 Notation definitions Variables

Definitions Decision variable, target bay selected by ASC a in i task

I

{1,2,…,|I|}, loading and unloading task set of imported containers, |I|>=1

A

{1,2}, ASC set

B

{1,2，…,|B|}, bay-sorted collection from the sea side to the road side {1,2,…,|

|}, AGV set in wait state in i task

Target loading and unloading bay of imported containers in i task Target loading and unloading bay of the landside ASC in the i task AGV maximum waiting time i stage running time during ASC scheduling V

Number of bay passed by ASC per unit time

L

Safety distance between ASCs

M

Large enough positive number i is equal to 1 when the ASC a is idle in the i task, otherwise equal to 0 In the i task, at time t, ASC a is in the bay j In the i task, the ASC is equal to 1 when the container is loaded and unloaded at the time t, otherwise it is 0 ASC a time to complete task i i task is completed equal to 1, otherwise 0 AGV

arrival time in i task

Waiting time of the rth AGV in the i task Loading and unloading time of ASC in i task Remaining capacity of I/O Point in i task The sth target container in the remaining capacity of the sea side bracket area in the i task I/O Point equals 1 when there is no remaining capacity in i task, otherwise 0 is equal to 1 when a=1, otherwise 0 Weights, In the i task, select 1 in the relay i task to select the relay bay, otherwise 0.

3.3 Objective Function 3.3.1 Minimize AGV Wait Time The minimum AGV waiting time is the sum of the differences between the time when all AGVs leave the bracket area and the time when they reach the sea side bracket area during the planned time, indicating the sum of the AGV waiting times during the entire planned time period. The minimum AGV waiting time is the sum of the differences between the time when all AGVs leave the bracket area and the time when they reach the sea side bracket area during the planned time, indicating the sum of the AGV waiting times during the entire planned time period. This phase consists of three parts, AGV arrives at the seaside bracket area. The time during which the seaside ASC completes the loading and unloading task. The seaside ASC needs to determine whether to carry out the relay operation during this task phase. The part of the objective function is below. 8

(1) Eq.(1) represents the sum of the AGV wait times for each task phase. The objective function considers whether the ASC selects the relay operation under each task phase. When the relay operation is performed in the i phase, the time of each task phase is determined according to the ASC reciprocating relay position and the time of loading and unloading the target container. When the relay operation is not performed in the i phase, the ASC round-trip target container bay and loading and unloading Time determines the time of each task phase. Determining the AGV waiting time in each phase by determining the difference between the i stage task completion time and the AGV arrival time in the i-phase, is depicted in Fig.5.

Fig.5. Task time of ASC

3.3.2 Minimize ASC Wait Time After ensuring that the AGV waiting time is minimized, considering the global optimal situation, the ASC operation in the block may cause a scheduling time delay after satisfying the minimum waiting time of the AGV waiting time. Therefore, this paper sets the following objective function with the goal of minimizing the ASC runtime. The part of the objective function is: (2) Eq.(2) indicates the ASC scheduling time. When the seaside ASC and the landside ASC are loaded and unloaded during each mission phase, the overlap of the work area may be encountered. (1) In the sea side loading and unloading task, the target container bay is larger than the target container bay in the land side loading and unloading task, and when the twin ASC are facing each other at the same time, the two ASCs may collide; (2) The twin ASC operating areas overlap when relaying, is depicted in Fig.6. In order to improve the efficiency of ASC scheduling, this paper sets priorities for the scheduling equipment operating conditions in different situations to ensure that the relay operation can proceed smoothly. Since the sea side work priority is greater than the land side work priority, the land side ASC may avoid the sea side ASC or may perform relay operations to affect the land side dispatching efficiency in each mission phase. Therefore, this paper uses the max function to find the longest running time in the dual ASC to constrain the landside ASC efficiency degradation due to the seaside ASC priority limit.

9

Fig.6. ASC operation overlay diagram

3.3.3 Total Objective Function

In the relay position selection problem, this paper considers that the target decision value is different because the decision makers have different preference settings for each target sub-function. Since each sub-objective function value is in the same order of magnitude, this paper sets the objective function weight to construct the total objective function, which can be obtained: (3) Eq.(3) represents the total objective function obtained by weight offset combination. The total objective function will change with different weight settings. Different objective function values can be obtained according to different sub-goals of the decision makers. The general focus is on improving the efficiency of sea survey work. This paper sets

.

3.4 The Constraint

3.4.1 AGV wait time constraint

When the AGV reaches the seaside e target bracket area, it is necessary to classify the AGV waiting time according to the running time of each task phase, so as to subsequently count the AGV waiting time in each stage. The number of AGV arrivals and waiting time statistics in this phase can be obtained by setting the running time of each phase. Using the dynamic programming idea to determine the task execution time in task i, the analysis is as follows. If in task I, it can be obtained from Table 2 that when the ASC status number is N= {1, 2},

,

the state transition equation is showed as Eq.(4), where represents the set of containers to be loaded and unloaded in the I/O Point in the i task. If , and the status number is N={3,4}||{5,6} in task i, two cases need to be discussed. When N= {3, 4}, to avoid ，

，

，

，

, the state transition equation is showed as Eq.(5). When N= {5, 6}, ，

. After obtaining the task time of each stage, the AGV waiting time

set can be planned by the traversal algorithm, and the AGV waiting time is determined according to the AGV arrival time and the remaining capacity constraint of the bracket. Eq.(6) indicates that the AGV arrival time in task i must be after the task i-1 completion time to ensure that the AGV start waiting time in each phase task is counted. Eq.(7) indicates that the AGV arrival time in task i must be before the task i completion time. In task i, the AGV count in the AGV wait time set is based on the 10

number of AGVs that can be reached before the task completion time and the remaining capacity of the bracket area is full. Specifically, when the AGV reaches the sea side bracket area, the target container cannot be unloaded to the bracket area, and only the empty space of the sea side bracket area can be reserved for the target container to be placed, is depicted in Fig.7. Eq.(8) indicates that the AGV arrival time must satisfy the AGV arrival time sequence set constraint. Specifically, the AGV arrival order is fixed, and there is no overlap in the arrival time of the AGV.

Fig.7. AGV arrival time sorting chart

，

，

(4)

， (5) ， ，

(6)

，

(7) ，

(8)

3.4.2 Task Constraint

In the container loading and unloading process, the target container loading and unloading operations according to the mission plan need to be completed. Specifically, each container in the loading and unloading plan needs to be loaded and unloaded by the ASC to the target bay in the block. Eq.(9) ensures that the target container in the unloading operation of the sea side vessel can perform the loading and unloading operation to enter the bay of the target block. Eq.(10) indicates that when the loading and unloading task i is completed, the current task i is marked to indicate that it has been completed; if the loading and unloading task i is not completed, the right side of the inequality is infinite positive number, and no valid value is generated on the left side. ，

(9) ，

，

(10)

11

3.4.3 Seaside Bracket Capacity Constraint

When the AGV reaches the bracket area of the sea side blocks, it is necessary to judge the remaining capacity of the current bracket area. When the capacity reaches saturation, the AGV in the current loading and unloading task is counted in the waiting time set. When the AGV reaches the bracket area of the sea side box area, it is necessary to judge the remaining capacity of the current rack area. When the capacity reaches saturation, the AGV in the current loading and unloading task is counted in the waiting time set. Therefore, the AGV needs to wait until the remaining capacity of the bracket area is not equal to 0 to unload the target container to the bracket area. When there is still capacity remaining, the AGV in the current loading and unloading task is not counted in the waiting time set, and the AGV directly unloads the target container to the sea side bracket area to leave. Eq.(11) ensures that when the AGV is waiting in the sea side bracket area, the remaining capacity of the bracket area must be 0 at the current time. Eq.(12) constrains the set 0-1 variable. Specifically,

can be equal to 1 when the

remaining capacity of the sea side bracket area is zero. ，

，

(11)

，

(12)

3.4.4 ASC Scheduling Constraints

In the unloading business process, a reasonable arrangement of container position will establish a good foundation for the dispatching of containers, loading, unloading, and departure equipment. However, automated terminal equipment is required for the storage of target position. In the actual operation process, the priority of the AGV operation task is higher because the priority of the sea side operation is generally higher than that of the land side. In the case of a small amount of work in the block, it is possible to prevent the sea side ASC from moving to the land side area of the block or the land side ASC moving to the sea side area of the block. In the case of a large amount of work in the block, the twin ASC operation area may overlap and cause a collision, and the ASC loading and unloading target bay is too long. In order to avoid excessive overlap of the scheduling equipment working area, this paper sets the penalty coefficient to reduce the excessive overlap of the twin ASC working area, and takes into account the ASC operating rule constraints in the field. Eq.(13) ensures that ASCs do not collide with each other and there is a safety distance L between the twin ASC. Eq.(14) indicates that ASC can only perform loading and unloading tasks at most once at the same time. Eq.(15) indicates that ASC can only be in one bay position at a certain time to prevent ASC from moving across the bay. Eq.(16) ensures that the ASC is in the target bay corresponding to the task when loading and unloading. ，，

，

，

，

，

，

， ，

(13) (14)

， ，

(15)

，

(16)

，

(17) 12

，

(18)

Eq.(17) and Eq.(18) respectively indicate that the time at which ASC completes task i must be greater than or equal to the current task completion time. Eq.(17) can obtain

by transform. Specifically, the left side of the

inequality indicates the time when the seaside ASC completes the task i, and the right side of the inequality indicates the actual time consumed by the task phase when the relay operation is selected in the task i or the relay operation is not performed; the Eq.(18) is similar to Eq.(17).

3.4.5 ASC Relay Rule Constraints

Before the AGV reaches the seaside bracket area, the ASC receives a list of pending ASC commands assigned by the terminal operating system. Considering the excessive overload of the work area in the block, before judging whether the relay operation is performed at each stage, this paper assumes that when the ASC selects the target container for loading and unloading operations in the sea side bracket area, the current ASC is selected by setting the loading and unloading task evaluation method. After the target container is determined, whether the relay operation is performed is determined according to the target container bay, the current AGV waiting time state, and the land side ASC operating state. Eq.(19) indicates that when there are multiple target containers in the sea side bracket area waiting for the ASC to be loaded and unloaded to the target bay, the ASC needs to select the target container for loading and unloading according to the evaluation function . According to the value of

, the target container is selected for loading and unloading operation according to the order

from low to high. Eq.(20) indicates that the target bay of the landside ASC is changed to the relay bay when the relay operation is performed in each task. Eq.(21) indicates that the landside ASC performs the landside task only after the ith relay task is performed and the relay operation is not performed in the i+nth task. Specifically, the landside ASC does not always wait to perform the relay operation, and the land side container loading and unloading task must be performed. Eq.(22) limits the number of relays in the scheduled operation. Specifically, the number of relay operations will not be greater than the total number of loading and unloading tasks. Eq.(23) indicates that the relay operation must be performed when the AGV waiting time is not less than the maximum waiting time. Eq.(24) indicates that the relay position

is determined when the

relay operation is determined in the loading and unloading task i. (19) ，

(20) ，

，

， ，

(21) (22)

，

，

(23) (24)

3.5 Graph theoretic model

13

As mentioned in the first section, due to the loading and unloading tasks are determined, the model can be changed by cooperative degree of ASC to minimize the operation time. This paper creatively adopts graph theoretic model proposed by Dirk Briskorn and Panagiotis Angeloudis (2016) to express multiple ASC scheduling problems and translates the problem of ASC scheduling into Job-Shop problem. Then the improved particle swarm algorithm is used to solve the problem. Under the condition of mission determination, each ASC operates in turn. The distribution of

and

in

container yard is shown in Fig.8. (1) In the time period of i,

is located in stowage location of

.

is f-rom

(

to

). (2) In the time period of i,

is located in

(3) In the time period of i,

is from

is located in

(4) In the time period of i,

is from

and

to

(max(

to

(

is located in

and

)

is from

to

).

). (

).

Fig.8. ASC distribution map

Based on graph theoretic model, this paper translates the problem of ASC scheduling into Job-shop problems, analyzes the minimal time of ASC operation and the interference avoidance between ASC. As shown in Fig.9, path rules are as follows. (1) The starting point of any path is from (0, 0) in the direction of slope 1. (2) When the horizontal or vertical direction of the moving point meets the bottom or the left-most of the obstacle,it connects the bottom or the left-most of the obstacle in a direction with a slope of 0 or infinity, such as the path of -

-

or

.

(3) When the moving point meets the margin of obstacle(such as to obstacle(such as

to

,

to

or

), it continue moves along the margin of

) and get away from obstacle.

(4) When the moving point hits the bottom or the left-most of the obstacle, it moves along this side and the movement refers to the rule(3). In Fig.9, the horizontal axis and vertical axis respectively stand for the operation time of ASC(s-1) and ASCs with per unit scale length 1min. The area of Obstacle represents the time zone of interference between ASC(s-1) and ASCs. Time point of (

) in the time zone is the infeasible solution of objective function, that is ASC(s-1) and ASCs can’t operate at the same

time. C shows that during the time period of . L means that ASC(s-1) is entering . R represents that ASC(s-1) comes out from

and

, ASC(s-1) and ASCs is respectively located in

and ASCs is located in the time zone ASCs is located in the time zone

. Similarly, B and T is similar to L and R. TL shows that ASC(s-1) moves in the direction of zone of ASCs away from

with the length of with the length of and it is the time

.Similarly, TR, BL, BR and TL are the same.

14

t

Obstacle

TL

t4-t3

k2

L

r1-r2+1

ASCs

r1-r2+1

k4

BL

T

TR

C

B

k1

R

BR

k5

k3 r1-r2+1

t2-t1

r1-r2+1

o

ASC (s-1)

t

Fig.9. Twin ASC obstacle diagram

15

Fig.9 shows the example of interference when ASC(s-1) and ASCs is in operate. The slope of o-

is 1, which represents

that there is no interference in the operation of ASC(s-1) and ASCs. When it reaches to , ASC(s-1) and ASC(s-1) are ready to .

arrive to the target towage location (1)

-

means ASC(s-1) choose avoiding operation and ASC(s-1) enter into target towage.

comes to the target towage location when ASC(s-1) finishes the task in

.

-

shows the time that ASCs

means the time zone that ASC(s-1) is away

from the last target towage and get close to current target towage. (2)

-

means ASCs choose avoiding operation and ASC(s-1) enter into target towage.

comes to the target towage when ASCs finish the task in

.

-

represents the time that ASCs

means the time zone that ASCs is away from the last

target towage location and get close to current target towage location. (3) If there is no interference in ASC operation, objective function can be formulated as follows. (25) (4) If there is interference in ASC operation, the number of obstacle in graph theoretic model is positively correlated with the frequency of interference. The target is to optimize cooperative degree and the time complexity of algorithm is O(

).

3.6 Modified particle swarm optimization(MPSO) This paper makes graph theoretic model by grid method which is shown in Fig.10. The graph theoretic model is decomposed into cells with binary information and records the ASC operation information. In graph theoretic model, path segment represents ASC operation time. If the slope of line segment is 0 or infinite, the ASC operation time is the segment length. If the slope of line segment is 1, the ASC operation time is the length that the line segment projects on the horizontal axis.

ASCS

t

ASC(S-1)

t

Fig.10. Graph theoretic model based on grid method

This paper translates the problem of ASC scheduling into Job-shop problems and the improved particle swarm algorithm is used to get the optimal path in graph theoretic model. The optimal path is the ASC optimal scheduling. Detailed flow of the algorithm is shown in Fig.11.

16

Chromosome coding Initialization population

Increase the number of particles

Calculating the velocity and position of an individual

Population size reached

N

Calculation of individual fitness

Y

Meet the terminating conditions

Output Path processing

Calculation of pbest,gbest,sbest

Update gbest N

Y Fitness>pbest

Fitness>gbest Y

N Update pbest

Update sbest

Fig.11. MPSO flow chart

3.6.1 Initial population Particle in the initial group is the starting point of algorithm iteration. If the particle is not initialized, it will add the running time of the algorithm. This paper randomly chooses particles which meet the population size between origin and end point (the target point of the particle motion which is in top right) in graph theoretic model, in order that ASC(s-1) and ASCs don't interfere with each other in initial state. 3.6.2 Calculating position and speed Traditional particle swarm optimization algorithm is easier getting optimal solution to process function optimization in low-dimensional space. But PSO algorithm easily trapped into local optimal solution in solving complex multimodal problems. This paper adopts PPSO and performs local optimal search to the global optimal. The formula for speed and position is as follows. (26) In the formula above,

is the particle updated speed.

the particle previous speed. current position.

， ，

is the previous best position.

is inertia weight,

.

is population optimal location.

is particle acceleration constant, value range is [0,2].

， ，

is

is particle

is uniform random number

distributed between [0,1]. (27) 3.6.3 Calculating the fitness value This paper takes values of time sum of ASC operation and waiting for each other. In graph theoretic model, it is particles moving distance, distance function can be formulated as follows. 17

(28) is fitness value of particles.

is sequential pattern of particles.

3.6.4 Calculating pbest, gbest, sbest pbest is the fitness value of previous best position of particles. gbest is the fitness value of previous best position of population. sbest is the fitness value of current best position of population. 3.7 Path processing 3.7.1. Smooth path operation The path based on PPSO is connected by multiple lines with poor smoothness. So the path need smooth operation to optimize. The smoothness of the path is affected by the expected value of particles angle. If it is small, it may enter into Obstacle and lead to interference with ASC(s-1) and ASCs. If it is wide, it can’t take a shortcut and can’t move along the border which disturbs optimal order of ASC(s-1) and ASCs. This paper sets that the expected value of angle is 135 . If it is not 135 , it needs smooth operation. If it is 135 , the path remain unchanged, as shown in Fig.12.

Fig.12. Smooth path operation

In translation process of particle, the particle may move through obstacle, which represents that ASC(s-1) and ASCs during the time zone is in stowage location

and

, as shown in Fig.13. This paper changes the problem of searching

optimal solution to the path into the problem of the shortest distance from the starting point to the target point through the visual line by visibility graph. In visibility graph, there is no interconnection between lines. Whether there is a point of intersection between

or not is determined by vector equation. (In visual method, in order to ensure that there

is no intersection between connections, it is necessary to determine whether the intersection between intersected, and whether there is any intersection point between

and

and

is

can be determined by vector equation.) (29)

If

has a solution in [0, 1], it means there is an interconnection. The path is feasible, that is ASC(s-1) and ASCs,

which avoid the mutual interference.

Fig.13. Path translation operation

3.7.2. Set termination conditions

18

In this paper, the algorithm termination condition is the maximum number of iteration.

4 Numerical example We conduct simulation experiments to evaluate the performance of the formulated model and developed algorithms. The algorithms were coded in Matlab and the experiments were performed on the Intel(R)-64 Core(TM) 2 CPU, [email protected] GHz with 2 GB of memory. In order to facilitate data analysis, the model parameter settings are shown in table 2, the start time of task is set to the start time of the ASC scheduling operation. In the following, [A1], [A2], and [M] given below represent the improved particle swarm optimization algorithm, the particle swarm optimization algorithm, and the equation solution. Table 2 Parameters1 ASC number

Transportation speed

Loading speed

Initial position

1.5S/bay

200s/container

Bay00

1.5S/bay

200s/container

Bay50

4.1 Demonstration of the proposed programs and algorithms In order to analyze the data, the time of the ASC loading and unloading of each target container was set as int eger which is uniform discrete distribution in [1,5], The experimental results of [A1] are shown in Fig.14. when th e numder of iterations reaches 23, local optimal solution are discovered in algorithm execution. Then the target val ue reaches steady state and obtains the optimal solution 119min. In Graphical Model, the path of ASC is {(0,0),(6 0,60),(60,63),(60,66),(61,67),(107,113),(113,113)}. In Graphical Model, ASC moving path graph is shown as Fig.15. Contents in square brackets show the bay of interference between ASC1 and ASC2. For example, [bay11, bay13] indicates that ASC2 is in bay 11 and ASC1 is in bay 13, there is a cross between them, so the obstacle region is an infeasible solution area.

Fig.14. Evolution diagram of fitness of PPSO

In Fig.15, solid path is the global optimal solution path of ASC scheduling which represents that ASC 2 and ASC1 perform the tasks separately in 0-60min without of interference. At the time of 60min, ASC 2 gives way to ASC1 and ASC2 is located in bay 12.Then ASC1 enters into

bay 13. Within the time of 67-113min, ASC operate separately with no interference. The

overall operate time is 119min. Dotted line in the figure is local optimal solution, the total waiting time for AGV is 31min.

19

t (0,113)

(113,113)

(1)

(2) [bay11,bay12]

[bay13,bay13]

[bay11,bay13]

(3)

ASC1

[bay11,bay12] [bay13,bay13] [bay11,bay13]

[bay11,bay11]

[bay11,bay11]

(113,0)

(0,0)

t

ASC2

Fig.15. ASC moving path graph

4.2 Comparisons among [A1] [A2] and [M] To compare the optimality and computing times of the proposed methods ([A1], [A2] and [M]), container data volume to be loaded and unloaded is set to 10, 20, 30, … , 250. The results are presented in table 3, the start time of ASC scheduling is set to 0S in the experiment. The results are generalized bellow. First, the results are consistent when data size is 40, indicating that the results are almost the same when the data volume is small, and the performance of [A1] is the same as that of [A2]. Second, comparing to [A1] and [M], [A1] consumes much less time to obtain local optimal solutions. When data volume reaches 50, [A2] can bring better solution than [M], and the algorithm consumes less time, [A1] can consume less time than [A2] to obtain the equivalent optimal solution. Third, [A1] is fast and sometimes can obtain better solutions. Notably, [M] is truncated after four minutes. It is practical when the sequencing problem comes batch by batch. Fourth, [A1] is effective for solving small-sacle instance, and is competitive in solving medium and large scale instance. Therefore, the [A2] is used generate lower bounds for verifying the validity of the [A1], from these results, [A1] is fast and effective comparing to [A2]. Table 3 Computing times and results of solving the model and performing the algorithms NO

[M]

[A1]

[A2]

Time/s

Time/s

Time/s

10

2

71.32

6.12

0.01

71.32

6.12

0.05

71.32

6.12

20

100

124.68

14.92

0.03

124.68

14.92

1.05

124.68

14.92

30

180

184.51

38.97

0.03

184.51

38.97

5.40

184.51

38.97

40

180

238.22

59.15

0.07

238.22

59.15

8.37

238.22

59.15

50

200

334.73

128.18

0.07

294.72

80.31

14.02

313.55

98.11

60

200

401.72

201.32

0.13

350.52

113.90

20.73

359.72

125.19

70

200

496.68

271.68

0.23

412.52

128.22

28.43

416.68

141.41

80

200

592.45

329.05

0.38

468.87

143.27

40.21

479.45

150.03

90

200

717.22

368.20

0.61

533.75

159.72

61.90

552.60

162.32

100

200

845.52

411.75

0.91

599.93

178.32

89.20

783.67

354.45 20

110

200

1043.57

453.90

1.4

705.68

194.38

103.12

996.52

385.31

120

-

-

-

3.99

790.77

209.72

114.65

1092.60

418.20

130

-

-

-

5.43

901.50

234.25

156.10

1185.58

453.32

140

-

-

-

7.24

955.38

259.52

205.02

1316.78

503.18

150

-

-

-

9.57

1049.88

271.90

265.12

1402.68

564.32

160

-

-

-

12.61

1175.85

288.22

308.67

1606.20

628.68

170

-

-

-

14.51

1301.53

303.27

381.04

1755.82

679.05

180

-

-

-

16.65

1486.27

319.72

434.71

1941.20

756.53

190

-

-

-

20.47

1643.60

334.32

523.39

2091.65

784.75

200

-

-

-

31.14

1693.01

347.38

651.40

2183.68

811.52

210

-

-

-

46.25

1864.52

369.72

769.01

2385.05

864.20

220

-

-

-

58.46

2031.85

383.08

851.05

2634.75

904.40

230

-

-

-

69.12

2056.52

408.96

947.42

2878.22

1013.15

240

-

-

-

80.11

2217.72

420.83

1032.56

3148.20

1077.02

250

-

-

-

91.02

2290.12

435.20

1147.73

3332.52

1126.57

Based on the numerical results and analyses of the above experiments, the managerial implications and technological application possibilities are discussed below. (1) Based on the experimental results, for small-scale instances(10-50 with target container), the existing model solving and algorithm may find the optimal sequencing solution. The results of particle swarm optimization and improved algorithm are almost the same for medium-scale instances(69-90 with target container). When operation scale is large, the improved algorithm can be used to provide the solution. (2) Considering the engineering and purpose of implementing an ACT, the scheduling algorithms for key operations resources (ASC, AGV) are important to balance the stability and efficiency. Although some solvers are capable of finding optimal solutions, their performance are not stable and sensitive to the data. Therefore, simple and fast exact algorithms may be competitive comparing to mature solvers and stochastic search algorithms. (3) As mentioned in the introduction and problem analysis sections, the examined problem is simplified by considering the easy control of ASCs. In an efficient automation system, many complex conditions are isolated, the sequencing problem of ASCs must be coupled with the sequencing problems of quay cranes, ASCs and other resources. Therefore, based on the model, algorithms and solutions for localized problems, multi-scale systems should build to coordinate them.

5. Conclusion The ASC scheduling strategy of automated container terminal directly affects the waiting time of AGV, thus it affects the production efficiency of container terminal. Therefore, it is necessary to optimize the ASC scheduling method of the automated container terminal area. For dual ASC scheduling problem of automated container terminal based on hybrid stack mode, this paper established ASC scheduling model and translated the dual ASC scheduling problem into a Job shop problem in consideration of the target priority and waiting time constraints of container loading and unloading. ASC scheduling problem was represented by graphical model which was solved by PPSO. All feasible solution of ASC scheduling provides optimal scheduling plan for administrator. In the mixed pile mode, ASC scheduling problem didn’t consider coordination degree factor of ASC scheduling between the box areas, this paper assumed that the time that reached to target stowage location and the time of loading and unloading was known. Therefore, we can further study the uncertainty scheduling problem with actual situation.

References 21

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23

Graphical Abstract

24

Highlights of the paper 1. We aim to minimize ASC operation time and the waiting time of target container for loading and unloading. 2. The ASC scheduling problem is transformed into a Job-Shop problem by graph theoretic model. 3. The grid method is used to express the graph theoretic model, and the problem is solved by MPSO algorithm. 4. We set up smooth path operation and path translation operation, so that the model is solved efficiently and optimally. 5. We consider the priority problem of the target container and waiting time constraints of the container.

25