This paper introduces a novel quay crane design, double girder bridge crane (DGBC). DGBC is capable of handling containers of two adjacent bays simultaneously, avoiding crane collisions, saving travelling and reposition cost, and eventually improving terminal efficiency. This problem is formulated as a resourceconstrained project scheduling with objective to minimize the maximum completion time. A twostage heuristic algorithm is proposed in which an operating sequences on each bay is obtained by double cycling, and the integrated timetable for both bays is constructed by solving resource conflicts using the proposed minimum cost strategy. We examine effectiveness and performance of applying DGBC with double cycling. A case study is presented to illustrate how DGBC works with the twostage method. Three extreme cases with respective conflict types are investigated to develop the performance bounds of DGBC with double cycling. The results show that DGBC can significantly improve terminal productivity, and outperforms single girder crane in both makespan and the lift operation percentage. The highest DGBC efficiency does not require maximum double cycles in two bay schedules; rather the integrated timetable for two bays is the main contribution to the DGBC performance as it yields better cooperation between two spreaders and the driver.
The rapid growth in global trade has led to remarkably higher shipping volumes and increased vessel carrying capacity. Technological innovations and highefficient scheduling strategies are required to meet the demand of increasing throughput in container terminals, especially in managing larger capacity vessels while reducing operating cost and maintaining service reliability. It is important to ensure their operating efficiency by incorporating new technologies and operating strategies when developing infrastructures.
Limited by the current technologies of transportation, the previous work has been mainly focused on the operating strategies for the existing equipment, that is, the traditional single girder quay crane (SG). The crane productivity is greatly improved by those researches; for example,
Having this assumption in mind, we propose a new cranebased design method in this paper, double girder bridge crane (DGBC). The availability of double girders would considerably increase the crane’s handling capacity while reducing its travelling cost, because this method enables DGBC to serve two adjacent bays at the same time with only one driver. Two girders share the infrastructure, DGBC is therefore operated closer to the economic purpose of the terminals, and its benefits can be obtained with limited investments, such as equipping SG with double girders.
DGBC can be installed in a terminal as shown in Figure
Comparing DGBC with SG.
The main contribution of this work is that a DGBCbased scheduling design in shoreside is proposed, which identifies its benefits including the capabilities to serve two bays simultaneously and save crane travel and reposition cost. Based on double cycling, a twostage heuristic algorithm is developed to demonstrate how DGBC is implemented. It is found from the comparison that DGBC outperforms SG, and double cycling plays less important effect on DGBC than SG.
This paper is organized as follows. Section
The quay crane is a key bottleneck for overall terminal efficiency [
A drawback of the traditional problems mentioned above is that there are many empty movements existing in crane operating cycles due to the use of single cycling, which would significantly affect the crane serving efficiency. However, it is reported that double cycling can reduce empty movements and improve the utilization of quay cranes [
Many works existed in the literature focusing on the different mathematical formulas for the quay crane scheduling problem. A rich model given by Legato et al. [
In order to demonstrate the DGBCbased scheduling problem, we first introduce the framework of DGBC and then give the problem description and settings, as well, and the application of double cycling to DGBC is discussed.
DGBC is a quay crane equipped with twin container spreaders on double girders. Each girder is positioned on one bay with the spreader handling the containers in this area, while another serves the adjacent bay. They are able to work on adjacent bays simultaneously. Both handling concepts are supported by common frame, cable, and power drives. Although this design increases the energy requirements compared with two single girder cranes, the savings on mechanical consumption and maintenance cost are worth more consideration. Furthermore, only one driver is required for two spreaders’ operations.
The mechanical structure of DGBC is depicted in Figure
Top view of DGBC.
The problem is described by a directed graph
Take a simple case as an example where the side view of the stowage plan for bays
Stowage plan for two bays.
Graph
There are three resources: one driver
Double cycling is considered in the operating strategy to improve processing efficiency, in which DGBC performs the unloading when the spreader carries an import container from the vessel to the shore and then conducts the loading when the spreader moves from the shore to the vessel with an export container. Then the empty movement in single cycling is replaced by the full movement resulting from double cycling, and the number of operating cycles is reduced. However, the processing time of a double cycle is longer than that of a single cycle as the spreader has to move slower while carrying a full container.
A series of movements must be executed between the adjacent lift activities in the same bay, because the spreader must move to the assigned location before raising or lowering a container. In other words, each
Four modes of movement combination.

Lift 
Movements  Lift 


U 

U 

L 

L 

U 

L 

L 

U 
Unloading
In this section, assumptions and major notations are given, and the DGBC scheduling problem model is presented.
In order to model the DGBC problem, we make the following assumptions. All the containers can be implemented by DGBC. The processes within shoreside are simplified by ignoring
The major notations used in the remainder of this paper are summarized as follows.
DGBC cannot finish the project until all the activities on two bays have been completed, so the makespan is the maximum completion time of all activities. The optimization objective is to minimize the makespan, whose function can be expressed as
All activities and resources are available from the start of the project. Consider
One lift handles one container at a time:
The spreader is blocked after its movement if the driver is still performing another spreader. Consider
Related to constraint (
Special movements are defined for the dummy nodes 0 and
The proposed model cannot obtain the optimum in acceptable time by the existing mathematical programming solver. Naturally, DGBC can be implemented by a twostage framework, where the first stage problem schedules the containers loading and unloading in each bay, and the second stage handles the coordination between the two spreaders’ operation under the driver constraint; thus a twostage heuristic algorithm is proposed to solve the problem. The proposed approach decomposes the DGBC problem into two stages, which can be solved in sequence.
First of all, each spreader is regarded as an independent crane to handle each bay. It is a traditional quay crane scheduling problem for single bay. The target of this problem is to find the best (un)loading sequence for each bay with minimum operating cycles. To achieve better crane processing efficiency, double cycling is introduced, which permits a quay crane to perform the unloading and loading in one operating cycle. The objective function of this problem is to minimize cycles required for loading and unloading, and the constraints are corresponding to constraints (
The double cycling procedure is constructed according to our previous work [
Lifts require the driver’s participation in controlling the spreaders to pick up and drop off containers. However, there is only one driver in charge of two spreaders for DGBC. As a result, there would be resource conflicts between two bays, in which one spreader cannot perform lifting directly after moving and has to wait for the driver to be released from the previous lifting with the other spreader. Based on the single bay scheduling results from Stage 1, a compact timetable for two bays is obtained in which the resource (driver) conflicts are solved by minimum cost strategy.
Although the objective of single bay scheduling is to obtain as many double cycles as possible, there would still exist single cycles. To help distinguish different conflicts, SU is defined hereafter as the interval during which only unloading operations exist, SL is the interval only involving loading operations, and DUAL is the double cycling part. In an operation cycle Gantt chart (OCGC); see Figures
Operation cycle Gantt chart (a)–(d).
Single cycling for
Double cycling for
Single cycling for
Double cycling for
The overlap between two lifts on different bays is defined as a conflict. There are three types of conflict between single cycles and double cycles; SS, DD, and SD. As listed in Table
Conflicts type.
Conflict  Single cycle  Double cycle 

Single cycle  SS  SD 
Double cycle  SD  DD 
To remove those three kinds of conflicts, the minimum cost strategy which aims to minimize the increment in the makespan of two bays is developed. In order to settle a conflict, we have two options; that is, either the overlapped lift on
SS: take the SS conflict between lifting containers 1 and 12 in Section
DD: DD can also be tackled by the minimum cost strategy, for example, containers 8 and 16 in Figure
SD: SD cannot be resolved by one delaying operator. Due to the different movements required in single/double cycles, SD conflicts can be distinguished by the various overlaps as depicted in Figure
Time Gantt chart (a)–(d).
Single cycling for
Double cycling for
Single cycling for
Double cycling for
Timetable for two bays.
Nine types of SD conflicts.
A twostage heuristic algorithm is developed in order to solve the DGBC problem. Firstly, double cycling is used to achieve a better spreader processing schedule for each bay. The critical thing is that the two spreaders cannot be treated as two independent cranes as there is only one driver available. Then, we present a heuristic for two bays to pursue an integrated timetable, in which three types of conflicts are settled by the minimum cost strategy. The twostage heuristic algorithm is described as in Algorithm
(1) Scheduling
(2) Compute the timetable and makespan
(3) While (there is any activity in
(3.1) Pick the earliest activities
(3.2) If there is no conflict between activities
(3.2.1) If
(3.2.1.1) Add activities (
(3.2.1.2) Remove activities
(3.2.2) Else //
(3.2.2.1) Add activities (
(3.2.2.2) Remove activities
(3.3) Else //there exist a conflict between activities
(3.3.1) Calculate the blocking time
(3.3.2) If
(3.3.2.1) Add activity
(3.3.2.2) Remove activity
(3.3.2.3) Right shift activity
(3.3.3) Else
(3.3.3.1) Add activity
(3.3.3.2) Remove activity
(3.3.3.3) Right shift activity
(4)
(5) Return
Suppose there are at most
In this section, we provide a case study to illustrate how DGBC works and the performance of the given twostage heuristic algorithm. Assume the stowage plan is given as the example in Figure
In the first stage, the operating schedule for each bay is obtained independently and depicted by OCGC in Figure
The solutions obtained by single cycling and double cycling are provided, respectively, to compare their performance. All the cycles in Figures
However, OCGC cannot describe the exact processing time as movements are not displayed. In this paper, each bay’s schedule is represented in time Gantt chart (TGC) (see Figure
Figure
In contrast, all four modes appear in Figure
In the second stage, the timetable for two bays will be constructed under the driver constraint, since each lift is set to be as early as possible in the obtained single bay schedules (see Figures
This section examines the DGBC performance by testing the problem in three extreme cases. The performance of DGBC improvement can be bounded by those three extreme cases; each of them is compared with the SG problem. The effectiveness of double cycling applied on DGBC is also discussed.
In DGBC evaluation, assume there are
The parameters of processing time.

U  L 








60  60 

40  40  20  80  80 
In the following discussion, SGSS/SGSD denotes SG using single cycling/double cycling. The objective of this problem is to minimize the makespan (measured in seconds) of the crane serving two bays. Besides, there are two measures corresponding to makespan. One is makespan percentage (MP) which is the current makespan as a percentage of the SGSS makespan. MP can be calculated as
There are three extreme cases with respect to three types of conflicts, respectively; they are considered independently to examine the performance of DGBC.
Suppose there is only one type of conflicts in Stage 2; the DGBC problem is regarded as an extreme case. There are three extreme cases with SS, DD, and SD conflicts, respectively.
DGSS is an extreme case which has only SS conflicts in the timetable. If single cycling is applied, the DGBC problem is a DGSS case.
DGDD is the one with only DD conflicts. DGDD is the most effective case with maximum double cycles.
DGSD is more complicated than the above two cases, because the timetable has various overlaps between the double and single cycles. One SD conflict can be classified into 9 types, as shown in Figure
For each extreme case, the blocking time can be determined by solving SS, DD, and SD conflicts separately, as showed in Table
Blocking time of DGSS, DGDD, and DGSD.
Crane  DGBC  

Case  DGSS  DGDD  DGSD  
SD1  SD2  SD3  SD4  SD5  SD6  SD7  SD8  SD9  
Block(s)  60  140  20  40  100  80  60  40  60  40  20 
SS conflict occurs between two single cycles, which have the same timetable. Then delaying one lift to remove the first SS conflict in one DGSS part can solve all the remaining SS conflicts consequently, and the total blocking time is 60 s. Similarly, all the DD conflicts in one DGDD part can be removed by delaying one lift with the blocking time 140 s.
However, each SD conflict in a DGSD part must be addressed by the minimum cost strategy individually, and the successive SDs are transferred to another type while dealing with the current SD. All SD conflicts are rescheduled into the feasible timetables with the different blocking time depicted by the grid boxes in Figure
Through the twostage method, all conflicts are examined and addressed sequentially, and the DGBC problem can be separated into several parts each of which is of one extreme case. Generally, the adjacent two bays cannot be served at the same time because SGs have to keep safe distance with others; however, in order to conduct the comparison between DGBC and SG in the same scenario, one SG is assigned to serve two bays sequentially. In this section, both DGBC and SG are evaluated on
As shown in Figure
Comparison between SGSS and DGSS.
To make it clear, DGSS is divided into two scenarios:
DGDD can also be separated into two scenarios:
Comparison between SGSS and DGDD.
However, DGDD does not always lead to a better result than DGSS. For example, the outcomes with DGDD and DGSS are overlapped. In other words, more double cycles cannot always result in the better DGBC makespan. The results in Figure
Different from the SS/DD conflicts which can be solved by one time block in one DGSS/DGDD part, each SD conflict in a DGSD part must be handled individually. Therefore, the performance of each DGSD case with different SD conflict is compared with both SGSS and SGSD; see Figure
Comparison between SGSS, SGSD, and DGSD.
We assume that there are three lifts in this comparison which simplifies the conflict detection and makes evaluation tractable. For SGSS, the three lifts are supposed to be processed sequentially in single cycles. Therefore, the makespan is 540 s; the lifting time is 180 s; then the LP is 33.33%.
In contrast, SG performs the three operations sequentially in one single cycle and one double cycle, denoted by SGSD. The makespan is computed as 480 s, which is 88.89% of that in SGSS. With the same lifting time 180 s, the LP is improved to 37.5% higher than SGSS.
As shown in Figure
Although the bounds of the DGBC performance is not quantified, the time complexity of the heuristic is closely related to the conflicts type, number and position. However, the makespan of the DGBC problem can be bounded by comparing three extreme cases against SGs.
In Stage 2 of the proposed algorithm, the timetable of the DGBC problem is splitted into several parts; each of them has one type of conflicts. For example, in Figure
The comparisons show that DGBC outperforms SG on the three measures. MP of DGBC can be improved to 41.67% of DGDD at the best case. The best LP can be obtained for the DGDD as 80% while the lowest one (33.33%) may come from the DGSS with
The effect of double cycling on reducing the empty movement and increasing the crane processing efficiency is well established for SG. As shown in Figure
According to the full movement proportion of the total moving time, double cycling and single cycling are compared in Figure
Full movement percentage of double cycling.
However, the better makespan does not necessarily include the most double cycles, because the makespan domains of DGSS and DGDD are overlapped. Even in the case of the same number of double cycles, the efficiency of the double cycling is varied from the position of double cycles, such as SD1 and SD2 in Figure
Why does double cycling make less effective impact on the DGBC scheduling problem as it does for SG? Because the DGBC problem is sensitive to instances, and the conflicts are propagable. Double cycling only works in Stage 1 to get a good operating sequence for each single bay. An integrated timetable should be constructed for two bays in Stage 2, which has a larger effect on the makespan; a more compact timetable will enable the driver to handle the two spreaders more simultaneously and cooperatively. Double cycling reduces the number of operating cycle for each bay individually; new conflicts would exist for later containers. Therefore, the impact of double cycling is less significant on DGBC.
This paper describes how to implement DGBC and examines its performance. In addition to the reduction in cranes collisions, the crane travelling, and reposition cost, the crane serving efficiency can be improved significantly by DGBC with its capability to serve two adjacent bays simultaneously.
Based on the proposed twostage heuristic algorithm, the makespan of the DGBC problem is bounded by three extreme cases (DGSS, DGDD, and DGSD), and the best makespan takes 41.67% of that for SGSS. LP is improved from 33.33% (SGSS) to 66.67% (DGSS), even 80% (DGDD). As a result, the driver can perform the lifts more efficiently and productively.
In conjunction with double cycling, the makespan of the DGBC problem can be further improved with the full movement percentages increased to 66.67%, 80%, and 88.89% for DGSS, DGSD, and DGDD, respectively, all of which are better than or equal to SGSS with 66.67%. Therefore, the effectiveness of double cycling on DGBC is verified throughout the evaluation.
On a more ambitious scale, DGBC can be implemented in the railmounted container terminals. The horizontal movement of the driver will be taken into account in future work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Grant 61272377) and the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110027).