1
Master Thesis
Crane schedule using double-cycling strategy:
modelling, heuristic and evaluation
Msc, Supply Chain Management
University of Groningen, Faculty of Economics and Business
2 Abstract
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Table of Content
1. Introduction ... 4 2. Problem description ... 5 3. Literature Review ... 7 4. Model Chapter ... 9 5. Heuristic approach ... 11 5.1 Johnson’s rule ... 11 5.2 Model experiments ... 135.3 Sequencing in each hatch ... 14
5. 4 Hatch sequencing ... 17
5.4.1 Gap Reduction ... 18
6. Evaluation ... 20
7. Conclusion and further research ... 22
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1. Introduction
In recent years, the volume of goods moved by containers via the container terminal has increased continuously. Driven by the increasing trend of containers transhipment through the terminal, much attention is paid to how to improve the operational efficiency in the container terminal. One of the most important performance indicators in terms of the terminal efficiency is the stay time of a vessel. To minimize the vessel stay time, an optimal schedule for Quay Crane (QC) should be made to reduce the number of cycles of a QC to complete the (un)loading activities. Quay Cranes are the most expensive and important piece of handling equipment to unload and load containers (Goodchild, 2005), and because of this, the efficiency of a container terminal can be best improved by reducing the total number of cycles of a QC. Several studies pay attention to optimize the process of QC activities. Shen, J. et al. (2006) developed a genetic algorithm for optimal container pick-up operation scheduling in order to minimize the total operation cost. Moccia, L. et al. (2006) developed a model to determine the sequence of unloading and loading movements for a QC assigned to a vessel to minimize the vessel service time. Bierwirth and Meisel (2009) studied more practical QC scheduling considering the crane interference constraints. Furthermore, they used a branch and bound algorithm to produce better solutions. Recently, Legato, P. et al. (2012) developed an enriched model for the Quay Crane Scheduling Problem (QCSP) that incorporates the processing time of a task. However, these studies about QCSP did not consider the double cycling strategy which is a method to reduce the number of cycles of QC by allowing simultaneous unloading and loading activities.
The double cycling strategy has not been clearly researched in previous studies. Because of this fact, most of container terminals use the single cycling strategy that QCs normally handle loading activities after all the unloading tasks are completed. However, one breakthrough being studied in recent literature about the containers (un)loading is the double cycling strategy. Goodchild and Daganzo (2007) studied crane double cycling in container ports and they proved that double cycling method does have some benefits based on the result that double cycling can reduce the operation time, improve vessel and crane productivity. The reason why double cycling strategy has these advantages is that by using this strategy, containers are loaded and unloaded in the same crane cycle. By doing so, some empty crane moves can be converted into productive ones. Besides, Goodchild and Daganzo (2005) developed a mathematical model to address a QC scheduling problem with regard to double cycling strategy.
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This paper extends the study of Goodchild (2005) about the QC scheduling by considering the constraint imposed by the deck factor. The objective of this study is to find an optimal unloading and loading sequence by a QC using double cycling strategy. The problem is solved with the aim of minimizing the total number of cycles by a QC. The QC scheduling problem considering the deck constraint with dual cycling is reformulated by a mixed integer programming model. Additionally, a heuristic approach based on Johnson’s rule is used to address the QC scheduling issue. Regarding the heuristic approach, the problem is decomposed into two parts: sequencing in each hatch and hatches sequencing. This study compares the result by the programming model with the result by the heuristic approach. Based on the experiment results, conclusion will be further offered.
The remainder of the paper is organized as follows. Section2 will introduce how do (un)loading operations work in the container terminal, double cycling process and the role of deck. The relevant literature review will be presented in section3. The mathematical model with parameters, variables and constraints will be introduced in section 4. Section 5 will present the heuristic approach based on Johnson’s rule. Evaluation part based on the experiment results will be demonstrated in section 6. The conclusion will be presented in section 7.
2. Problem description
This paper addresses the scheduling of containers unloading and loading activities by a QC using double-cycling strategy. How do the (un)loading operations work in the terminal can be seen from figure1 (a).
Figure 1(a) top view of the vessel
Steenken, D. et al. (2004) described the terminal operation as containers flow system with two interfaces: quayside with ship unloading and loading and landside with container unloading and loading on/off trucks and trains. The figure1 (a) illustrates the quayside. QC resides where the vessel moors and it is mounted on a railway. The
vessel QC container
Quay
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schedule is about assigning the QC to each unloading and loading task with a certain sequence. The QC scheduling problem in this paper aims at finding a schedule for a QC with respect to a give objective function. In most studies, the objective is the minimization of the vessel stay time. While in this paper, the minimization of the total number of cycles a QC requires to complete the tasks is pursued. The side view of a vessel can be seen from figure1 (b). We can see from the figure1 (b) that the deck divides the containers into two parts above the deck and below the deck.
Figure1 (b)side view of a vessel
Deck & Hatch
In this study, the deck factor is considered when developing the model. The deck and hatch can be seen from the figure2. The deck which separates the stacks of containers into two parts calls for a precedence constraint. Containers above the deck should be unloaded first and then container below the deck can be unloaded. Similarly, in terms of the loading activity, containers below the deck should be loaded first and then containers above the deck can be loaded. In this paper, three hatches and fourteen stacks will be used as modelling objective to get the solution with a sequence of unloading and loading tasks. In terms of the heuristic approach, the problem will be decomposed into two sub-processes. The first step is sequencing the stacks in each hatch and the second process is sequencing the three hatches.
figure2 deck and hatch Double cycling
deck
hatch1 hatch2 hatch3
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The figure3 shows the comparison between single-cycle and double-cycling. For the single-cycling which can be seen from figure3 (a), a crane is assigned to load containers and then the crane returns without containers. So, this leads to an empty movement because the QC idles when it returns. On the other hand, in terms of the double-cycling that can be seen from the figure3 (b), a crane is assigned to load containers and the crane returns with the unloaded containers. This dramatically reduces the number of cycles of a QC to complete the unloading and loading tasks because it converts an empty movement to an unloading movement. The double cycling allows a QC to unload containers in the same cycle as a loading operation, thus doubling the number of QC activities in one cycle. As a result, the total number of cycles of a QC required to complete (un)loading tasks can be reduced. Similarly, the double cycling also allows a QC to load containers in the same cycle as an unloading operation. Because of this advantage, more and more container terminals begin to realize the benefits from the double cycling. Therefore, it is crucial to study the double cycling strategy in this paper.
(a) (b) figure3 single cycling vs double cycling (Haipeng and Kim, 2009)
3. Literature Review
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Moccia, L. et al. (2006) studied quay crane scheduling problem with the aim of determining a sequence of unloading and loading operation. The objective function is the minimization of the vessel completion time. The problem is formulated as a vehicle routing problem with side constraints. The results are derived through the branch-and-cut algorithm. In real setting, the collision of quay cranes should be taken into account. Bierwirth and Meisel (2009) added the interference constraints into the quay crane scheduling problem model in order to ensure the safety of quay cranes movement. Some behavioural issues are also considered in the QCSP. For example, the experience of crane drivers is incorporated into the model by Legato, P. et al. (2012). They converted the driver experience to the crane-individual processing time and the objective function is the minimization of the whole schedule.
In terms of double cycling operation, Goodchild and Daganzo (2005) considered the dual cycle in their studies and formulate the problem as a scheduling problem with the aim of minimizing the job completion time. Furthermore, they proposed a greedy algorithm to obtain the optimal solution. In another double cycling study by Goodchild and Daganzo (2005), they compared the performance of two algorithms used to determine a sequence of (un) loading tasks with a QC. They used greedy strategy and proximal strategy and the results show that the both strategies show a significant benefit, which means that the double cycling does have an advantage over single cycling. Rather than minimize the vessel completion time, Zhang and Kim (2009) formulated the quay crane scheduling problem with double cycling strategy as a mixed integer programming model in order to minimize the total number of required cycles of QC to complete the unloading and loading activities. Goodchild and Daganzo (2007) also focused on minimizing the required number of cycles of a QC to complete the (un)loading tasks. Besides, they focused on a long term impact of double cycling on quay crane operation. They compared the expected number of cycles using single cycling with the expected number of cycles using double cycling. The results show that double cycling can reduce the unnecessary crane moves, thus leading to fewer cycles to complete the tasks.
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4. Model Chapter
This chapter will introduce a mathematical model based on Goodchild (2005) which is discussed in the literature review section. The model assumption, parameters, variables and constraints are illustrated as below.
Model assumption
1. The time to unload or load one container is identical.
2. One container refers to a container group consisting of ten containers with the same destination.
Notations:
h=index of hatches (1….h)
i,j=index of stacks (1….i) (1….j) W=the number of cycles
Su= stacks with unloading operation
Sl= stacks with loading operation
Shu= stacks with unloading operation in the hold of hatch h Shl= stacks with loading operation in the hold of hatch h
Dhu= stacks with unloading operation on the deck of hatch h Dhl = stacks with loading operation on the deck of hatch h
pair(i,j) describes the precedence relationship with the unloading and loading operation
Parameters:
niu=the number of containers to unload from stack i
nil=the number of containers to load into stack i
Procedure Variable:
Ciu=completion time of unloading all containers from stack i Cil=completion time of loading all containers into stack i
Cih=completion time of unloading containers above the deck from stack i
Cip=completion time of loading containers under the deck into stack i Decision Variable:
Xijk: binary variable for permutation of unloading stacks (X ij
k=1 if unloading for stack j
is performed by quay crane k immediately after unloading for stack i, 0 otherwise) Yijk: binary variable for permutation of loading stacks (Y
ijk=1 if loading for stack j is
performed immediately by quay crane k after loading for stack i, 0 otherwise)
10 Minimize W Subject to constraints: W≥Ciu, ∀ i∈I (1) W≥Cil, ∀i∈I (2) Ciu≥niu+Cih , ∀ i∈ Shu (3) Cil≥n i l+ C iu , ∀i∈ I (4) Cil≥nil+Cip , ∀i∈ Dhl (5) X0jk j∈Su =1, ∀k∈K (6) Xijk j∈Su =1, ∀i∈ Su ∀k∈K (7) Xijk j∈Su =1, ∀j∈ Su ∀k∈K (8) XiTk j∈Su =1, ∀k∈K (9) Y0jk j∈Sl =1, ∀k∈K (10) Yijk j∈Sl =1, ∀i∈ Sl ∀k∈K (11) Yijk j∈Sl =1, ∀j∈ Sl ∀k∈K (12) YiTk j∈Sl =1, ∀k∈K (13) Cju-Ciu+M*(1-Xijk)≥ n j u (14) Cjl-C i l+M*(1-Y ijk)≥ njl (15) In the expressionX0jk, X iT
k , the subscript 0 and T represent the start and completion of
unloading tasks, and the same meaning for loading tasks. Constraints (1) to (2) describe the definition of total completion cycle of quay cranes. Constraint (3) ensures that the unloading activities under the deck cannot start until the unloading activities above the deck are completed. Constraint(4) ensures that the loading activities in the stack cannot start until the unloading activities of the stack are done. Constraint(5) ensures that the loading activities above the deck cannot begin until the loading activities under the deck are completed. Constraints (6) to (13) define the operation sequence of loading and unloading. Constraints (14) and (15) describes the definition of Xijk and Y
ijk, where M is a sufficiently large positive number.
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5. Heuristic approach
To prove that the model can give a best result with the (un)loading sequence so as to minimize the total cycles of a QC, a heuristic approach using Johnson’s rule Goodchild and Daganzo (2005) proved that it can provide the minimum result about the total cycles required by a QC to finish the jobs will be introduced. Then we compare the results by the model with the results by the heuristic approach. Further, based on the comparison, we can see whether the model can give a best solution. The quay crane scheduling problem is divided into two sub-processes: sequencing stacks in each hatch and sequencing hatches. The rule used to arrange the order of stacks in this paper is Johnson’s rule.
5.1 Johnson’s rule
According to the explanation by Wikipedia, Johnson’s rule is a method to schedule the jobs in two work centers. The goal is to find an optimal job sequence to reduce the total amount of time it takes to complete all the jobs. The assumption to use the Johnson’s rule is that the time for doing each job is fixed and that all jobs go through the second work center only after proceeding with the first work center.
In this study, the Johnson’s rule can be used to handle the unloading and loading issue because this problem is similar with the job scheduling in two work centers. Firstly, in our study, the two work centers can be converted into the unloading task and loading task. Also, the assumption of using Johnson’s rule is also met because the time to unload or load a container is identical, which is the model assumption. Additionally, loading activities cannot begin until the unloading activities are completed, which is similar with the second assumption to use Johnson’s rule. Therefore, Johnson’s rule can be used to sequence the stacks unloading and loading problem in this study. The procedure of the Johnson’s rule applying in this study to find the optimal loading and unloading sequence is described as below:
Step 1. List the number of unloading containers and the number of loading containers in two vertical columns.
Step 2. Check all the stacks and identify the stack with the smallest value of number of containers.
Step 3. If this stack belongs to the unloading containers column, this stack is then placed in first position.
Step 4. If this stack belongs to the loading containers column, this stack is then placed in the last position.
Step 5. Delete the line with number of unloading containers and loading containers for this stack.
Step 6. Repeat the steps from 2 to 5, and finally the loading and unloading sequence can be found.
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number of unloading containers and loading containers for the same stack, place the one which belongs to unloading containers column first.
To clarify the procedure, a flowchart is made as below.
To test the usefulness of Johnson’s rule, we test it using the data from literature by Goodchild and Daganzo (2005). Furthermore, we compare the result by the literature with the result by using Johnson’s rule for the same data. The data are illustrated as below. The four stacks are labeled as A B C and D.
List the number of unloading and loading containers into two columns
Identify stack with the smallest value
belong to unloading column?
yes no
Place this stack in the first position
Place this stack in the last position
Cross off this stack
All tasks are scheduled?
no yes
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No. Unloading containers No. Loading containers
A 3 2
B 3 5
C 2 0
D 2 3
The sequence result given by the literature is {B,A,C,D}. While if we use the Johnson’s rule, we can get a different result with the sequence {D,B,A,C}. Then we look at the total cycles resulting from different sequence. The detailed (un)loading plans are illustrated in figure 4.
(a)
(b)
Figure 4 total cycle’s comparison between (a) literature result and by using (b) Johnson’s rule
Clearly seen from figure4, the sequence by the study of Goodchild and Daganzo (2005) leads to thirteen cycles to complete the tasks. However, by using the Johnson’s rule, a different sequence is generated and it leads to twelve cycles to complete the jobs. One cycle is reduced, and therefore this illustration implies that the heuristic approach using Johnson’s rule does give an optimal solution.
5.2 Model experiments
The dataset we use in this study is illustrated in table1. Three hatches and fourteen stacks are considered to be handled by a QC with the aim of finding the optimal (un)loading sequence in order to minimize the total number cycles of crane moves.
Table 1 dataset studied in this paper Hatch Stack Above or under
14 2 I A 6 3 3 J A 7 5 3 K U 1 5 3 L A 9 3 3 M U 5 7 3 N U 8 6
Figure 5 data visualization
Figure 5 visualized the dataset studied in this paper. Fourteen stacks are labeled as letter A to N and the deck makes the stacks of containers into two parts: above the deck and below the deck. The detailed information can be seen from table1. The solutions derived by modeling are illustrated as below. For hatch 1, the unloading sequence is {A,C,B,D,E} and the loading sequence is {D,E,A,C,B}. For hatch 2, the unloading sequence is {F,I,H,G} and the loading sequence is {H,G,F,I}. For hatch 3, the unloading sequence is {J,L,K,M,N} and the loading sequence is {K,M,N,J,L}. The results are logical because unloading containers below the deck should be done after unloading containers above the deck. Similarly, loading containers above the deck should be done after containers under the deck are loaded.
5.3 Sequencing in each hatch
Heuristic approach based on Johnson’s rule is presented in this section. Firstly, we sequence the unloading and loading tasks in each hatch. The data for hatch 1 are shown in table 2.
Table 2 data for hatch 1
stack No. containers of unloading No. containers of loading
A 4 3
B 2 1
A B C F I J L
D E G H K M N
deck
15
C 5 7
D 8 6
E 6 4
Johnson’s rule is used to arrange the operation sequence. As clearly seen from table two, for stacks above the deck, 1 is the smallest value and it belongs to the loading column. Hence, the second stack is placed the last position. Then the line of the second stack can be deleted. 3 is the next smallest value which belongs to loading column, so the stack A is placed in the second to last position. The remaining stack C is put in the first position. In terms of the stacks under the deck, 4 is the smallest value and it belongs to the loading column. So stack E is put in the last position and stack D is put before it. Finally, the operation sequence is identified as {C,A,B,D,E}. Figure 6 compares the total cycles resulting from the operation sequence by modelling with the total cycles resulting from the sequence using Johnson’s rule. Compared with the solution derived from modelling, the cycles needed are the same as 40 cycles.
figure 6 solution comparison between modelling and Johnson’s rule
Then we aim to sequence the (un)loading tasks for hatch 2. Data for hatch 2 are illustrated in table3.
table 3 data for hatch2
stack No. containers of unloading No. containers of loading
F 5 5 G 9 2 A C B D E D E A C B unload load 25 40 Solution by modelling C A B C A B 25 40 unload load
Solution by Johnson’s rule
D E
16
H 2 6
I 6 3
Because unloading F and I should be completed before unloading H and G, we firstly use Johnson’s rule to sequence the stack F and I. Stack I has the smallest value with loading operation, so stack I is put last. Thus the sequence is {F,I}. Subsequently, we use Johnson’s rule to arrange the stack H and G. Stack G has the smallest value of 2 with loading column, hence G is put last and the sequence is {H,G}. Finally, the operation sequence in hatch 2 is identified as {F,I,H,G}, which requires 34 cycles in total. Figure 7 compares the sequence of unloading and loading derived from Johnson’s rule with the sequence of unloading and loading operation by modelling. As clearly seen from figure 7, the total number of cycles derived from heuristic approach is the same as solution obtained from modelling.
Figure7 solution comparison between modelling and Johnson’s rule
Finally, we schedule the (un)loading operation for hatch3 and the data for hatch3 are illustrated in table4.
table 4 data for hatch3
stack No. containers of unloading No. containers of loading
J 7 5 K 1 5 F I H G H G F I 22 34 unload load Solution by modelling F I H G H G F I 22 34 unload load
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L 9 3
M 5 7
N 8 6
Due to the fact that unloading J,L should be finished before unloading K,M and N, stack J and L are firstly considered to use Johnson’s rule. Stack L has the smallest value which belongs to loading column, so we put stack L after stack J. In terms of stack K,M and N, 1 is the smallest value and it belongs to unloading column, so stack K is placed in the first position. Then 5 is the smallest value and it belongs to unloading column, hence it is put in the second position. Finally, the operation sequence is identified as {J,L,K,M,N} requiring 44 cycles to complete the unloading and loading operation. Also, as can be seen from figure 8, a sequence generated by modelling also leads to 44 cycles to complete the jobs, which is the same as the result by Johnson’s rule.
figure 8 solution comparison between modelling and Johnson’s rule
5. 4 Hatch sequencing
After sequencing the (un)loading tasks in each hatch, we sequence the hatches based on a gap reduction way. The gap reduction will be introduced in the following section.
J L K M K M N J unload load 30 44 Solution by modelling J L M K N K N J L unload load 30 44
Solution by Johnson’s rule N
L
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5.4.1 Gap Reduction
The gap reduction method is proposed by Hasan et.al (2009) that the total make span can be reduced by swapping job activities with the aim of reducing the gap. Sometimes the schedule leaves some gaps between jobs, thus leading to unnecessary crane moves. While the gap can be reduced or even be removed by moving a job from the right side of the gap to the left side of the gap. In other words, if we swap the job sequence, the existing gap can be reduced, thus reducing the total number of required cycles. This gap reduction process helps to make a seamless schedule with the minimum number of cycles, because reduction of gaps may help to reduce the make span. A demonstration of improving solution because of gap reduction by swapping job sequence can be seen in figure9.
figure 9 a demonstration of gap reduction
As can be seen from the figure9, due to the working constraint, there is a gap existing between the first two loading operations. However, by exchanging the sequence in the way that {8,4,4} is put in front of {4,4,5}, this gap is removed and as a result, the make span is correspondingly reduced from 39 to 38. Therefore, to optimize the operation sequence to minimize the make span, efforts should be put on reducing the gap.
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figure 10 solution comparison
As illustrated in figure 10, the shadow areas between jobs refer to the gap that could be reduced. For a random sequence as {hatch3, hatch2, hatch1}, the number of gap is 29 which has a potential to be reduced. By placing hatch1 from the right side to the most left side as {hatch1, hatch3, hatch2}, the gaps are reduced from 29 to 22. Hence, the total number of cycles is reduced from 92 to 85. The reduction of gaps positively affect the reduction of the make span. Figure 10 best shows that the solution can be improved by placing hatch 1 in front of hatch 3 because in that way the unnecessary gaps can be reduced. We can see from figure 10 that the gap between H and G is removed by swapping hatch 1 and 3, which helps to reduce the total cycles. Furthermore, changing the sequence of hatch1 and hatch3, the tail of the schedule reduced, thus leading to a reduction on the total cycles.
In order to further prove that the gap reduction positively affects the total number of cycles and that the reduction of tail length leads to reduction of total number of cycles, the other possible sequences are illustrated as figure 11. As seen from the figure11, with sequence {hatch3, hatch1, hatch2}, the total gaps are 24 cycles and the length of tail is 10 cycles. For sequence hatch {2, 1, 3}, the total gaps are 28 cycles and the length of tail is 14 cycles, thus leading to more cycles with 90 cycles. With the sequence hatch {2, 3, 1}, the gaps are 29 cycles in total and the length of tail is 15 cycles. Besides, for the sequence {hatch1, hatch2, hatch3}, 27 cycles are identified as gaps and the length of tail is 14 cycles. Compared with figure10, it is clear that sequence {hatch 1, hatch3, hatch2} can leads to the minimum cycles because gaps in this schedule are the smallest with 22. Furthermore, the length of tail is also the smallest with 8 cycles compared with other sequences. Therefore, the reduction of gaps and length of tail can positively affect the total cycles.
unload load
92 A random sequence : {hatch3, hatch2, hatch1}
unload load
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sequence {hatch3, hatch1, hatch2}
sequence {hatch2, hatch1, hatch3}
sequence {hatch2, hatch3, hatch1}
sequence {hatch1, hatch2, hatch3}
figure 11 other possible hatch sequences
6. Evaluation
The quay crane scheduling problem is addressed by mathematical modeling and by a heuristic approach. The following figures and tables illustrate the result comparison of the case discussed above between modeling and proposed heuristic approach.
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The above diagrams compare the number of cycles and gaps derived from modelling and the proposed heuristic approach with each hatch. For hatch 1, the unloading and loading sequence from modelling are {A,C,B,D,E} and {D,E,A,C,B} respectively. With this operation sequence, the total number of cycles is 40 and there are 19 gaps. Based on the proposed heuristic approach, the number of cycles now is also 40 and the gaps are 19 too with the unloading sequence {C,A,B,D,E} and loading sequence{D,E,C,A,B}. Also, for hatch 2, the unloading and loading sequence obtained from modelling are {F,I,H,G} and {F,G,H,I}. Based on this sequence, 34 cycles are required to complete all jobs and there are 16 gaps existing in the job schedule. The proposed heuristic approach presents the same solution sequence and thus the number of cycles is also the same as 34. In hatch 3, the model gives an unloading and loading sequence solution with {J,L,K,M,N} and {K,M,N,J,L}. With this sequence, 44 cycles are necessary to finish all jobs and 18 gaps exist in the job schedule. The proposed heuristic approach gives a same solution with 44 cycles and 18 gaps. Therefore, it can be drawn that the model can provide a best solution for the (un)loading sequence with the aim of minimizing the total cycles of a QC to complete the jobs. Finally, by means of the gap-reduction approach, an optimal hatch sequence is found with 85 cycles in total.
Both the modeling and the heuristic approach give a solution for the quay crane scheduling problem. It can be seen that the mathematical model can give a best result because Goodchild and Daganzo (2005) stated that the heuristic approach based on Johnson’s rule can give the minimum cycles of QC to complete the jobs. Furthermore,
0 5 10 15 20 25 30 35 40 45 50
hatch1 hatch2 hatch3
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the results obtained from the modeling are the same as the results by the heuristic approach. Therefore, the model in this paper can provide a best result for (un)loading sequence so that the total cycles can be minimized. Additionally, the result gives schedulers in container terminal an insight on how to optimize the operation sequence. Firstly, the gaps existing in the schedule should be tried best to reduce or even remove, because by doing so, the total cycles can be reduced. In figure 9, the gap which is shadowed between the first two loading jobs is removed by changing the schedule sequence, thus reducing one cycle in total. Also, as can be seen from figure 10, the gap between the loading task H and G is removed by changing the schedule sequence. As a result, the total cycles reduce from 92 to 85. Secondly, by looking at the figure 10 and 11, the most important part influencing the total cycles is the tail part. Therefore, schedulers should place the tasks that require more cycles in the middle part as many as possible. By doing so, there will be tasks with fewer cycles required left in the tail part, thus leading to the minimum cycles.
7. Conclusion and further research
In this paper, with focus on minimizing the total number of QC cycles, we have proposed a mathematical model considering an additional deck factor based on an existing model and taking the double cycling strategy into account. Besides, we have also presented a heuristic approach which can give a best result with the number of cycles. Because the result by the model is the same as the result by proposed heuristic approach, the model proposed in this study can provide the best schedule for a QC to complete the (un)loading tasks under the double cycling operation.
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