Quay Crane Scheduling: An exploration into double-cycling and ship stability

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Quay Crane Scheduling: An exploration into

double-cycling and ship stability

Michael Koçak s1740911

University of Groningen

MSc Supply Chain Management

Abstract

Increasing use of container transportation is putting pressure on harbors to find ways to operate more efficiently. The most important piece of equipment involved with (un)loading activities of containers at a port is a Quay Crane. Several models and solution algorithms have been developed to minimize the service time of a vessel by the QCs. The use of double-cycling has been shown to be beneficial for lowering the service time. Current literature on the double-cycling strategy does not consider the stability issue of the ship and considers operations for a single QC. Ship stability puts restrictions on the (un)loading operations of the vessel; balancing issues have to be neutralized by costly and time consuming measures; furthermore it limits the possible positions of containers on the vessel. We developed a stability model and solve it to optimality. We also propose a model incorporating the stability issue and the double-cycling strategy.

July 30, 2014

Supervisors:

dr. E. Ursavas

dr. X. Zhu

Content word count: 5885

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2

1. Introduction

Over the past two decades the use of container transportation has been steadily increasing. The most expensive and important piece of equipment to unload and load containers from and to a vessel at a terminal in a port is the Quay Crane (QC). The productivity of a port can be measured by the time it takes for the QC’s to unload and load a vessel before it is ready to leave for its next destination (Moccia et al., 2005). The optimization of the process has received a lot of attention in recent literature and several scheduling models and algorithms have been proposed to find a satisfying schedule within a reasonable time frame (Kim and Park, 2004) (Moccia et al., 2005) (Goodchild and Daganzo, 2005) (Ambrosino et al., 2006) (Zhu and Lim, 2006) (Bierwirth and Meisel, 2009) (Choo et al., 2010) (Chen et al., 2011)

The Quay Crane Scheduling problem (QCSP) is often preceded by the Quay Crane allocation problem (QCAP). The QCAP concerns the amount of QC’s to be assigned to a certain vessel. These two are often done sequentially (Meisel and Bierwirth, 2013). Meisel and Bierwirth (2013) propose a framework for integrating the QCAP and the QCSP, but in many cases the amount QC’s to be assigned is determined by the importance of the load, the size of the ship and/or by financial motives regarding the cargo and/or the shipping company involved (Akio et al., 2003) (Boile et al., 2006)

Some interesting development being addressed in recent literature with regard to the unloading and loading of the vessel are the so-called double-cycling strategies, where the ship is being loaded and unloaded

at the same time by different or even the same QC’s. Research by Goodchild and Daganzo (2006 and 2007) with regard to double-cycling shows its importance and benefits for the productivity of the ports. Some models have been developed regarding the double-cycling (Goodchild and Daganzo, 2005) (Zhang and Kim, 2009) which consider container scheduling under the assumption of service by only one QC.

These models do not take into account the stability of the ship nor do any of the models in the cited literature with regard to the QSCP. The stability of the ship is however one of the most important issues in practice and several instances can be found on any news platform of ships toppling over, because of the weight being improperly balanced. Ship stability during (un)loading can be safeguarded by pumping water in ballast tanks; this can be a costly and time consuming procedure. The stability issue imposes an important constraint on the QCSP. Thinking logically we can deduct that double-cycling could actually help the balancing issue in a way that specific cargo can be loaded and unloaded in such a sequence that balancing issues incurred without simultaneous unloading and loading can be offset.

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4 instances using MIP (Mixed Integer

Programming) and an exploration into a larger model also incorporating the double-cycling strategy.

2. Problem Description

For the problem we discuss in this paper we first consider an implementation of the double-cycling strategy. We assume each container on the vessel has a known location, known weight and known destination.

The terminology used to identify the location of the container on the vessel can be seen from figure 1. The location is denoted as Row,Bay,Tier; where 1,1,1 indicates the container located at the left side of the ship, closest to the harbor and at the bottom of the stack (see the red circle in figure 1). Figure 2 is a legend showing the destination of each container from the example provided by figures 3, 4 and 5. Figures 3, 4 and 5 show respectively a top side view of the vessel, a harbor side view of the vessel and the containers to be loaded at the current harbor; all containers are shown with corresponding characteristics indicating: location, weight and destination.

Row

Bay

Ti

er

Figure 1: Terminology used for container location

Destination Current Harbor Destination Next Harbor Destination: Harbor after Next Destination: Beyond

Harbor after Next

Figure 2: Destination of container

2,1,3 MW 2,3,2 LW 2,2,3 LW 2,4,2 LW 2,5,2 LW 2,6,3 LW 2,7,3 MW 3,1,3 LW 3,3,2 LW 3,2,3 MW 3,4,2 LW 3,5,2 LW 3,6,3 LW 3,7,3 MW 4,1,3 LW 4,3,2 MW 4,2,3 LW 4,4,2 LW 4,5,2 LW 4,6,3 MW 4,7,3 MW 1,1,3 MW 1,3,2 MW 1,2,3 MW 1,4,2 MW 1,5,2 LW 1,6,3 LW 1,7,3 MW

Figure 3: Top side view of loaded ship 1,1,2 HW 1,3,2 MW 1,2,2 HW 1,4,2 MW 1,5,2 LW 1,6,2 LW 1,7,2 MW 1,1,3 MW 1,2,3 MW 1,6,3 LW 1,7,3 MW 1,1,1 HW 1,3,1 MW 1,2,1 MW 1,4,1 HW 1,5,1 HW 1,6,1 HW 1,7,1 MW

Figure 4: Harbor side view of loaded ship

LW HW HW MW MW MW LW LW MW HW

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5 In the above schematic consideration of

the unloading/loading problem 3 different weight categories for the containers are assumed. Namely: Light Weight (LW), Medium Weight (MW) and Heavy Weight (HW) containers. As can be noted no heavy weight containers are situated at the top regardless of destination, even though not all containers can be seen from the top view and the harbor side view alone. Kang and Kim (2002) addressed the fact that heavier containers should be placed at lower positions to keep the ship from capsizing and concluded that this may result in conflict with minimizing the make span of unloading/loading operations when heavier containers have nearer destinations.

For the balancing issue we have to consider the center of gravity of the ship. We limit ourselves to balancing issues of the ship regarding: inclination towards starboard or port and inclination towards the front end or the back end of the ship. We assume the balancing issues with regard to the height of the center of gravity can be adequately tackled by placing heavier containers at lower positions in the ship.

Figure 6: Top side view of loaded ship. Horizontal and vertical axis added, separating the containers with distinct impact on the balance of the ship.

Our consideration is illustrated by figure 6. If the assumption is made the ship is balanced before the ship arrives at the port for unloading/loading of containers, then any container movement in section β1 influencing the weight more than a defined level of acceptance should be offset by a container movement in section β2. The same holds for ρ1 and ρ2. The containers which are on the axis are not considered since they do not influence the balance, because we assume the weight within the container is equally distributed. Tasks situated further away from the center of gravity have a relatively larger impact on the stability and this should be accounted for by assigning a permutation to the weight values accordingly.

It is not necessary to portray all side views nor is it necessary to show the surrounded containers on the vessel. The problem is clear: all red containers (both visible and non-visible) have to be unloaded at the current harbor. Meanwhile the 10 containers indicated have to be loaded, each having their own destination and weight constraint. As can be seen from the schematic view at the very least five containers have to be moved before four of the red containers (1,1,2; 1,2,2; 1,6,1 and 1,7,2) can be reached.

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6 Figure 7: Single-Cycling vs. Double-cycling (Goodchild

and Daganzo, 2005)

The specific attributes of the containers regarding the weight and destination of the load will impose restrictions on the optimal schedule and determine the sequence in which each consecutive unloading and loading activity can be carried out.

3. Theoretical background

Few papers have yet considered the double-cycling strategy even though its benefits have been shown (Goodchild and Daganzo, 2006). Correspondingly few harbors yet implement the double-cycling strategy and rather unload and load the vessel in consecutive order. At some ports in Asia, Europe and at San Pedro and Tacoma small scale trials of double-cycling have been executed (Goodchild and Daganzo, 2005).

Bierwirth and Meisel (2010) provide an overview of recent literature revolving around the QCSP. Most models proposed throughout recent literature revolve around minimizing the total (un)loading time called the total make span. Several detailed QSCP models have been developed considering different constraints such as the interference effect (Bierwirth and Meisel, 2009) to try and achieve reliable stay times for the ship at the harbor.

Furthermore Vis & Koster (2003) emphasize the importance of a quick

unloading/loading process of the vessel at the port to allow for efficient use of large container vessels. Concerning the unloading operations at the port they note that the crane driver is almost free to determine the order with which to unload the containers. This results in largely differing unloading times and probable efficiency losses. The loading process on the other hand, allows for very little freedom. The location to which each container has to be loaded on the vessel is subject to many constraints. Furthermore they conclude that existing models of simple cases should be extended to more realistic situations. The latter will ensure the applicability of the models.

They also note that the number of import containers is often not known very far in advance and that finding an optimal solution within reasonable solving time can be unrealistic, because the stowage plans is made across a number of harbors. This may call for more information transparency and closer cooperation between different harbors and shipping companies.

Stahlbock (2007) also highlights the increasing necessity of optimizing processes related to container transportation. Where Vis & Koster (2003) discussed an increased ship size of up to 8000 TEU (a standard size of a container unit), Stahlbock (2007) mentions vessels capable of handling up to 12,000 TEU’s. In the review paper of Stahlbock (2007) several authors are mentioned which all consider the importance of the stability issue with regard to the loading (and unloading) of the ship (Ambrosino et al., 2006) (Imai et al., 2006) (Dekker et al., 2006) (Álvarez, 2006).

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7 speed container carriers are getting more

popular for covering shorter distances. For the latter stability issues during unloading and loading activities are even more important, since they are more prone to stability issues. Besides the restriction posed by the stability issue on the sequence of the (un)loading activities, it also puts an important restriction on the possible container locations on board the vessel. ‘’Stability problems not only can damage containers, but also can incur human causalities’’, as stated by Lee (2012) when discussing the unloading and loading activities.

Literature regarding the QCSP that deals with several constraints at the same time is scarce (Bortfeldt and Wäscher, 2012) and so is the case for literature considering the double-cycling strategy. Bierwirth and Meisel (2010) identified several instances where developing a complex model considering many different constraints turned out to be profitable with regard to reducing vessel handling time at the port, however they also observed the opposite situation were simpler models are better suited, because of the limited advantages offered by a more complex model. Their research showed there is no superior model available yet that outperforms any other with regard to the QSCP across arbitrary setups.

In this paper we will propose a model following the double-cycling strategy whilst taking the balancing issue of the ship into account. The models discussed in the introduction which follow the double-cycling strategy (Goodchild and Daganzo, 2005) (Zhang and Kim, 2009) consider the time element with regard to the make span as the number of cycles rather than including

time as a distinct variable of varying operations. To determine ship balance at specific intervals it is necessary to incorporate the time element in the QCSP. Hereafter the mathematical model is introduced; it is then decomposed into two parts and partially reformulated and solved to optimality.

4. Mathematical Model

4.1. Assumptions and constraining factors

The model is partly based upon the Kim and Park (2004) formulation, considering the revisions by Moccia et al. (2005). The following assumptions are made:

Ship is balanced before operations begin.

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Container weight is known. (b) Container destination is known. (c) Container location on the vessel is

known.

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QC travel time is known and constant. (e)

Time of a loading/unloading task includes moving the crane over the designated container, picking the container and putting the container in its destination. As a consequence a penalty is assigned to the operation sequence loading-loading and unloading-unloading, because of the empty travel time of the QC.

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Time of loading/unloading task is deterministic and constant. ‘The transfer to vehicles has a low variance when many moves are considered’, as stated by Moccia et al. (2005).

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8 All loading tasks have predetermined

bay locations. This will allow calculating the weight value on forehand incorporating the distance of each task from the center of gravity. Task further away from the center of gravity have a relatively larger impact on the weight value of the respective sections indicated in figure 6.

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With predetermined bay locations the model does not take into account the destination of specific containers, nor does it ensure heavier containers are placed at the top. This has to be done manually by assigning the load container to specific bays in a logical manner.

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The schedule is subject to the following limitations:

Each QC can only operate 1 task at a time.

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QCs are on the same track and they cannot cross each other.

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There exist some precedence relationships between certain tasks.

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Some tasks cannot be performed at the same time.

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The stability of the ship must be preserved throughout the entire make span and the ship balance may not shift towards one side more than a predetermined value.

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A penalty should be assigned to the operation sequence loading-loading and unloading-unloading, because of the empty travel time of the QC.

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4.2. Descriptives

The following notations are used for a mathematical formulation: Indices: Task (0...1……n……m…Z) QC number (1…K) Moment (0…T) Input Parameters:

Processing time of task i Bay of task i

Earliest time QC k is available Maximum total stability value of β1 Minimum total stability value of β1 Maximum total stability value of ρ1 Minimum total stability value of ρ1 Maximum stability value of β1 Minimum stability value of β1 Maximum stability value of ρ1 Minimum stability value of ρ1

Travel time between task i and j (only applies for task with different bay number, otherwise travel time is incorporated in processing time under assumption (f)

t Travel time for moving alongside a single bay

Sufficiently large constant Weight factor for make span

Weight factor for total completion time

Set of indices:

Set of all tasks 1…z Set of all load tasks 1…n Set of all unload tasks n…z

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9 Set of pairs of tasks that cannot be

performed simultaneously

Set of pairs of tasks between which there is a precedence relationship Set of all QCs 1…K

μ Set of all moments within the make span 0…T

Set containing the weight impact of task i (0…Z) on section β1; task 0 and Z are empty with weight value 0 Set containing the weight impact of

task i (0…Z) on section ρ1; task 0 and Z are empty with weight value 0

Decision Variables

1, if QC k starts task j immediately after having completed task i; 0, otherwise

1, if QC k starts task j at moment m within the make span; 0, otherwise. Completion time of QC k

Completion time of task I

1, if task j starts later than the completion time of task i; 0, otherwise

W Time at which all tasks are completed

Stability value of β1, 0 = 100% stability at moment 0; bounded by ∑ at moment T

Stability value of ρ1, 0 = 100% stability at moment 0; bounded by ∑ at moment T

Moment within the make span 0…T

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10 1 for , (16) , for , (17) , for , (18) ∑ ∑ w for , (19) ( ) for , , , (20) βm + ∑ ( ) for , , (21) βm + ∑ ( ) for , , (22) ρm + ∑ ( ) for , , (23) ρm + ∑ ( ) for , , (24) ( ) for , , (25)

For the objective function (1) α1 is considered to be strictly larger than α2; minimizing the make span is the main objective, because this determines the service time of the vessel. Using this formulation the objective function will ensure that the solution with the shortest

total completion time will be selected amongst all solutions with the minimal make span. This will make sure that more QCs may become available to other ships as soon as possible (Kim and Park, 2004). Constraint (2) defines the make span. Constraint (3) and (4) respectively selects the first and the last task for each QC.

Constraint (5) ensures that each task is started exactly once by a single QC. Constraint (6) ensures that tasks are performed in a ‘well-defined sequence’, as stated by Kim and Park (2004). The constraint makes sure that every task j has a predecessor as well as a successor; therefore we had to define the first and the last task of any QC to make the problem finite.

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11 same time; then QC q which is on the right

of QC k, cannot move to task i if QC k is doing task j and . This is referred to as the interference constraint, because it stops the QCs, which are on the same rail, from crossing each other. Constraint (13) and (14) together determine that if task i and j are situated in the same bay, QC k cannot start operations on task j if another QC did not have time yet to move out of the designated bay area after completing task i.

Constraint (15) determines that if QC k performs the final task j, then the completion time QC is at least the completion task j + the travel time between j and the final position T of QC k. Constraint (16) in turn determines that if task j is the first task of QC k, then it cannot be completed before the earliest available time of QC k + its travel time to task j from its initial state + the processing time of task j. As such it limits the earliest starting time of QC k. Constraint (17) defines the variables and to be binary. Constraint (18) defines the variables

and to be non-negative.

Constraint (19) ensures that at any moment in time only one task can be started simultaneously by a specific QC k. Constraint (20) makes sure that if a precedence relationship is set by the variable that is respected by the values of the variable and . Constraint (19) and constraint (20) together define a certain moment within the make span.

Constraint (21) ensures that the stability value β of section β1 cannot go beyond a maximum permitted value and thus stops the ship from tilting towards β1. Constraint (22) ensures that the stability value β of section β1 cannot go below a minimum allowed

value and thus stops the ship from tilting towards β2. Constraint (23) and constraint (24) follow the same logic for section ρ1.

Constraint (25) assigns a penalty of 80% to the processing time of task j directly following task i if the task sequence of i-j is loading-loading or unloading-unloading. This is because the crane now travels empty towards the bay side for completing the task instead of travelling loaded. Under assumption (f) moving the crane over the designated container, picking the container and putting the container in its destination are all included in the processing time. The penalty considered is not 100% since even in the sequence unloading and loading-unloading there is some small empty crane travelling time not considered.

4.4. Model evaluation

We were unable to solve the model under 4.3 to optimality, it would require dynamic modeling techniques to keep track of changing values during the make span, for example β (m) and ρ(m). Furthermore the original model of Kim and Park (2004) as revised by Moccia et al. (2005) already required high computational times even for smaller instances. Hence we propose to decompose the model into two parts and in this paper we will solve the stability part of the model. We re-formulate the model into a static model, which will allow us to solve the model to optimality using mixed integer programming.

5. Stability Model

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12 the stability part of the model. The problem

is re-formulated as follows. Minimize: (26) for (27) ∑ ∑ for , (28) ∑ for , w (29) ∑ ∑ for (30) ∑ for , (31) ∑ ∑ for (32) ∑ ∑ for (33) for , for , (34) = ∑ ∑ for (35) ∑ ∑ (36) (37) for (38) for (39) = ∑ ∑ for (40) ∑ ∑ (41) (42) for (43) for (44) , for , , (45) for (46)

In this part of the model we no longer consider minimizing the completion time of individual QCs and hence the objective function (26) is reduced to minimize the completion time of all tasks W.

Constraint (27) defines the make span, W is set to be strictly larger than the completion time of any task j.

Constraint (28) makes sure that each task i is started exactly once by a single QC. Constraint (29) ensures that any QC can carry at most 1 task at any given time.

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13 SO from carrying out a task until moment 1

within in the make span.

Constraint (34) is an interference constraint ensuring that any QC k situated to the right of QC v, cannot perform a task j situated in the same bay or to the left of task i, when QC v is carrying out task i. This constraint will stop the QCs from colliding with one another.

Constraint (35) determines how the weight value of section β1 is influenced at moment m by the tasks assigned at that moment. Constraint (36) and constraint (37) respectively determine that the sum of all tasks may not exceed a certain maximum and minimum value; and . These constraints can be used to determine if the assignment of tasks to bay numbers yields a feasible schedule guaranteeing ship stability on departure. If these constraints are unfeasible the containers have to be distributed differently across the bays of the ship or other tactics have to be deployed to ensure the stability of the ship.

Constraint (38) and constraint (39) ensure the stability of the ship during loading operations by not allowing the loading and unloading operations to exceed a certain maximum and minimum value for any moment within the make span.

Constraint (40) – Constraint (44) are the same as constraint (35) – Constraint (39) except they ensure stability for the section ρ1.

Constraint (45) defines the variable to be binary and constraint (46) defines the completion time of all tasks to be nonnegative.

6. Results

6.1 Model input

For the model input we follow the example provided under section 2. We assume the following the weights: LW = 1, MW = 3, HW = 6. Furthermore we assume that for each container distance from the center the weight value should be multiplied by the distance. We disregard half bay distances. For example the task situated in slot 2,7,3 has a medium weight value. Its distinct weight impact on β1 will be 1*3*3 = 9.

- 1; because it is an unloading task situated in section β2 and thus it has a positive impact on the weight value of section β1 when it is removed. - 3; because it is a MW container and - 3; because it is 3 container distances

vertically from the center of gravity. Its distinct weight impact on ρ1 will be -1*3*1= -3.

- -1; because it is a loading task situated in section ρ1 and when it is removed it will thus have a negative impact on the weight value of ρ1 - 3; because it is a MW container and - 1; because it is 1 container distance

horizontally from the center of gravity.

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14 Task(i) G(i) H(i)

1 -3 2 2 -3 -1 3 0 2 4 1 1 5 1 -1 6 9 -3 7 -9 -6 8 -18 -12 9 -6 -6 10 -12 -12 11 2 -2 12 2 -2 13 12 -12 14 9 -6 15 9 -6 16 3 6 17 -3 6 18 0 -2 19 -1 -2 20 3 6 21 6 12 22 6 12 23 -1 -1 24 -3 6 25 0 6 26 0 -1 27 0 6 28 -2 -1 29 -6 -6 30 -6 12

Table 1: Weight values G(i) and H(i)

The process time of any task is constant and deterministic and we assume it is equal to 1. As mentioned under section 2, some tasks have to be removed to allow the QC to reach specific unloading tasks. We do not consider reshuffling and assume the containers moved become part of the set for loading operations. This yields several precedence

relationships and we need to include these relationships in our model. The precedence relationships following from our problem definition under section 2 can be found in table 2. set predecessor(j,i) /7.8, 9.10, 11.12, 12.13, 14.15, 8.16, 10.17, 13.18, 18.19, 15.20/

Table 2: Precedence relationships

Finally each task is situated in a specific bay ( ). To assign the loading tasks to specific bays one could use a linear programming (LP) model ensuring ship stability at moment of departure, in this paper we assigned the tasks manually. The bay in which each task is located can be found in table 3. Task(i) 1, 7, 8 1 2, 9, 10 2 16, 20, 21, 22 3 3, 18, 25, 26, 27 4 4, 5,17 ,19 ,23 ,24 , 30 5 11, 12, 13, 28, 29 6 6, 14, 15 7 Table 3: Location of task(i) according to bay number 6.2 Model output

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15 θ(m) β(m) θ(m) β(m) 1 4 2 -6 3 6 4 -5 5 -5 6 -3 7 -6 8 -1 9 6 10 6 11 -6 Table 4: Weight values for β(m)

θ(m) ρ(m) θ(m) ρ(m)

1 3 2 -1

4 -3 5 -1

6 -3 8 3

9 -1 10 -2

Table 5: Weight values for ρ(m)

Table 6 shows the completion time of each task i, note that the completion time of task SI is equal to the make span. Task 7 and task 20 are the last two ‘real’ tasks performed by QC 2 and QC 3 respectively.

Task(i) D(i) Task(i) D(i) 1 3,00 2 10,00 3 5,00 4 5,00 5 6,00 6 7 7 12,00 8 7,00 9 8,00 10 6,00 11 9,00 12 2 13 4,00 14 11,00 15 10,00 16 2,00 17 11,00 18 11 19 3,00 20 12,00 21 6,00 22 7,00 23 2,00 24 9 25 10,00 26 9,00 27 8,00 28 3,00 29 5,00 30 4 SI 12

Table 6: Completion times

m 1 2 3 4 5 6 k.i 1.1 1 1.3 1 1.8 1 1.10 1 1.12 1 1.30 1 2.4 1 2.13 1 2.16 1 2.19 1 2.21 1 2.22 1 3.5 1 3.6 1 3.23 1 3.28 1 3.29 1 + 7 8 9 10 11 12 1.2 1 1.9 1 1.18 1 1.26 1 1.SI 1 2.7 1 2.17 1 2.24 1 2.25 1 3.11 1 3.14 1 3.15 1 3.20 1 3.27 1 + 0 1.SO 1

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16 Table 7 contains the optimal schedule for

satisfying all constraints. Loading tasks are marked blue and unloading tasks are marked orange. The table shows what QC k is carrying out which task i at which moment m. The make span can easily be derived from the table, because of the incorporated Sink. Although the last two ‘real’ tasks are scheduled at moment 11, they have a processing time of 1 and hence the solution provided by the model is 12.

6.3 Computational result

For this specific instance of 30 tasks, 3 QCs and 10 precedence relationships the model can be solved to optimality within 31.784 seconds using a computer with AMD Phenom II X 955 Processor and 8 GB RAM running on MS Windows 7 Enterprise 64-bit SP1.

7. Experiments

Using the same computer as under section 6.3 we conduct several experiments to test the model. To test the model we randomly generate weight values such that the sum for both G(i) and H(i) are equal to 0. The processing times of the jobs are deterministic and constant; equal to 1. The set of precedence constraints is the same for each instance. And the bay to which each is located is selected at random, but equal for instances with the same number of tasks.

Table 8 summarizes the results and it can be seen that for small to medium size instances the model yields the optimal schedule within reasonable time.

Furthermore when considering that each task i can be seen as a set of tasks then it is obvious the model could be used even for larger instances. #i #QCs ∑ ∑ W Time (s) 60 5 0/5/-5 0/3/-3 13 171 60 4 0/5/-5 0/3/-3 19 1000 60 4 18 8 40 5 0/5/-5 0/3/-3 11 31 40 5 11 7 40 4 0/5/-5 0/3/-3 11 56 20 4 0/5/-5 0/3/-3 8 5 20 3 0/5/-5 0/3/-3 8 31 20 2 0/8/-8 0/6/-6 15 378 Table 8: Experiments

We should however note the drastic difference in computational time between the instance with 40 tasks and the one with 60 tasks when both are assigned 4 QCs. Also interesting to note is that as the number of QCs decreases the computational time of finding the optimal solution increases. Furthermore for the two instances with all weight constraints disabled computational time becomes drastically lower, especially for cases with more tasks.

This all makes sense as the number of tasks and constraints are in direct relationship with the time it takes to solve the problem. It is why the problem formulation under section 4 is explicitly hard to solve unless it is decomposed.

8. Discussion & Future Research

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17 made it explicitly harder to solve the

problem to optimality unless it is decomposed. After decomposing the model we re-formulated the part of the model considering the ship stability to allow for it to be solved to optimality using MIP. We get reasonable solving times for instances up to 60 tasks with 4 QCs. By adding more QCs we can solve even larger problems to optimality within reasonable time. We have tried to link the time at which a certain task is handled together with the sequence in which tasks are handled. If this could be done then the double-cycling constraint could easily be added to the ship stability model. Several attempts have shown that this may require a dynamic model. Perhaps another series of constraints can be added to

the existing model ensure that double-cycles are preferred over single-cycles. Several models have been constructed in an attempt to do so, but this did not yield the desirable results and hence they have been omitted from this paper. Future research should also look into developing heuristics for scheduling container operations following the double-cycling strategy, whilst taking the stability issue into account.

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Quay Crane Scheduling: An exploration into double-cycling and ship stability