Crane sequencing and storage location assignment problem in container terminals: An integrated approach

Crane sequencing and storage location assignment problem in container terminals: An integrated approach

Stavroula Boulaniki (S2617757)

Supervisor: dr. E. Ursavas Co-assessor: dr. S.A. de Blok

Abstract

In this research two significant optimization problems of quay crane sequencing and yard storage space allocation in the context of container terminals are studied. Due to their substantial interrelations, the problems can be combined and solved as a single large scale optimization problem with significant theoretical and practical contributions. The proposed model aims in reserving a yard storage location for each container, while scheduling the QC sequences in such a way that the total travelling distance is minimized. Alongside with an exact mathematical model formulation, a solution approach based on branch and price is developed. The proposed solutions are tested and validated through numerical experiments based on realistic instances from literature. Computational results suggest that small scale problems can be solved to optimality within reasonable computational time for the compact formulation model, while near-optimal results can be obtained for large scale instances solved with branch and price.

Master’s Thesis, MSc Supply Chain Management, University of Groningen, February 2015

1 Table of Contents

1. Introduction ... 2

2. Problem Definition ... 4

3. Literature Review ... 7

4. Mathematical Model ... 9

5. Computational Experiments ... 13

5.1 Input Data ... 13

5.2 Model Output ... 14

5.2 Model Evaluation ... 18

6. Solution approach ... 19

6.1 Reformulation ... 19

6.2 The Branch and Price Algorithm ... 21

6.3 Numerical experiments ... 22

7. Conclusions and Future Research ... 23

References ... 25

Appendix ... 28

2 1. Introduction

While the international shipping industry is responsible for carrying 90% of the world trade, seaborne trade itself is estimated to grow three times more than the world GDP by 2030 (International Chamber of Shipping, 2014). Alongside with the ever increasing containerized cargo, ports are considered the most vital aspect of transport infrastructure, while they provide a primary economic multiplier for prosperity (Alderton, 2013). Therefore the pressure for efficient operations in ports and container terminals is growing.

Container terminals are the facilities were container vessels are loaded and discharged and cargo containers are temporarily stored for further transportation. Their management and operation involves decisions on a strategic, tactical and operational level with the main goal of minimizing a ship’s berthing time and resources needed to manage the operations (Vis and de Koster, 2003). Fundamental elements for a container terminal appear to be the efficient stacking and transit of containers to and from the vessel’s side, with high productivity and maximum container throughput as key success factors (Stahlbock and Voß, 2008). Due to the specialized equipment required to handle all the workload and its high cost, container terminals are considered highly capital intensive and therefore a majority of academic literature has focused on improving their utilization (Fu et al., 2014). Examples of this are the quay crane (QC) assignment and sequencing problem, which are part of the quayside operations. Quay cranes are considered as primary equipment in container movement, therefore their efficient employment is of high importance (Imai et al., 2008).

At the same time however, recent research states the bottleneck of the port operations is moving from the quay side to the yard side of the terminals (Zhen et al., 2013).

Standard operations at the yard involve storage and retrieval of containers (Carlo et al., 2014). Specific issues that need to be resolved are the dispatching of vehicles to containers, their exact routing at the terminal, the optimal storage location for each container, as well as the conduction of a traffic control system (Bish et al., 2001).

These decisions are interrelated with each other and can affect the general

performance of a terminal. More specifically, space allocation arrangements in the

yard have a direct impact on the workload of the cranes and travel distances of the

3 yard trucks, while affecting the efficiency of the QCs at the same time (Zhang et al., 2003). Due to the complexity and size of these operations, the most common practice of researchers is to treat these issues sequentially, breaking them down into smaller sub-problems that can easily be optimized. Although this can be very interesting from a theoretical perspective, solving the problems separately does not correspond to real- life conditions. On the other hand, building integrated models allows faster scheduling with possibility of scenario analysis, insights in the interactions of various components and general economic benefits deriving from better operational decisions, as has been seen for instance in the oil industry with the integrated asset modelling approach (Perez et al., 2012).

In this framework, the research objective of this paper is to solve the crane sequencing and storage location assignment problem concurrently. The problem will be formulated via a mathematical model, partially based on the recent work of Choo, Klabjan and Simchi-Levi (2010). Their paper addresses the issue of crane sequencing by additionally taking into account QC clearance and yard congestion constraints that pose significant modeling challenges. After forming the original problem, the authors reformulate it as a set covering problem and solve it to optimality by branch and price.

With their method they manage to incorporate several operational constraints in one model, which performs better solution quality and less computational effort than other existing models, thus leading to more efficient applications in an industrial setting and significant cost reductions. In the same way, it is possible to improve this further by considering additional factors. In this case the scope of the research will be yard storage location decisions. The methodology will include the use and extension of previous research on this field, in order to formulate related constraints and come up with an enhanced model which represents more comprehensive conditions.

Consequently, the main theoretical contribution of this paper will be to provide a

solution tool which integrates the complex issue of quay crane sequencing with yard

block storage location decisions and can be used for any general job-shop problem

with parallel machines. This augments the scheduling theory and combinatorial

optimization, thus improves operations research in general. Therefore the research

question arises as follows: ‘’How can the crane sequence and yard storage location

assignment problem be solved simultaneously?’ Addressing these decisions at the

same time can mitigate any negative effects resulting from sequential planning, as

4 well as reduce the total time needed to organize the entire discharging operations. As a result, increased efficiency in practical applications can be expected. The final model will be evaluated with data from existing literature and real life cases. Based on numerical experiments of the solution algorithm, key findings and recommendations for practical use will be offered.

The rest of this paper will be organized as follows. First the integrated problem of crane sequencing and yard space allocation will be presented, followed by a literature review on the topic. Subsequently, the mathematical model alongside with some initial results and an evaluation of the model will be presented. The next section will illustrate the solution algorithm applied to improve the results of the model, while insights into the outcomes and limitations of the research will follow. Finally, suggestions for future research will be provided.

2. Problem Definition

Decisions on the quay crane scheduling and yard storage location are typically made independently. However, the total handling time of each loading or discharging operation consists of both its processing time and the transportation time from/to the yard. Separate scheduling of these decisions ignores the interrelations between them and their influence in the total berthing time of a vessel, leading to plans of poor overall quality (Bierwirth and Meisel, 2010).

A comprehensive explanation of the terminal operations and the related processes will

hereafter be presented. To begin with, arriving vessels at container terminals are

assigned to a berth. After mooring, the containers must be unloaded with the use of

quay cranes according to a discharging plan. Transfer vehicles are then transporting

them to the yard where they will be stored temporarily until they need to be

transported further by truck or rail. A schematic view of the functional areas of a

container terminal and the various equipment used for its operations are presented in

fig. 1 (Meisel, 2009).

5

Transport operations between the quayside and yard side constitute the most important aspect of a container terminal. Considering the quay side, the focus here is the QC scheduling problem, which will specify the sequence of ship bays that each QC will serve and their time planning. With respect to a given number of QCs available to serve a vessel, the aim is to define a QC schedule that minimizes the ship’s total berthing time. This becomes challenging for various reasons; first of all, information about the specific cargo of each ship, the layout and list of containers that need to be discharged reach the terminal operators a few hours before a ship arrives at the terminal (Lee et al., 2007). Therefore the planning can only be determined at the time. Different constraints need concurrently to be satisfied; such as that QCs cannot cross each other (since they roll on the same rail tracks) or that a safety distance must be maintained between them (Chen et al., 2011).

At this planning stage, the operator has also to decide about the allocation of incoming containers to the storage yard, with the aim of minimizing the traffic congestion (Lee et al., 2006). In essence, the containers must be stored in such a way that the handling time needed to store and retrieve them when needed is minimal. This becomes more important as in reality most of the times it is not known when or in what order the containers will be called and the storage area has considerable constraints (Kozan and Preston, 2006).

A typical yard configuration usually consists of blocks (Chen and Lu, 2012). Their

number can vary according to the size of the yard, but generally each block is formed

by 20-30 sub-blocks. Each of them has six rows one next to each other, where 4-5

6 containers can be stacked one over another (tiers). Finally, blocks are divided into 20’

sections called slots and a typical block is forty slots long (Petering, 2009). Two main decisions need to be taken regarding the storage of containers: first the specific block assignment and subsequently the specific location on the chosen sub-block. The final schedule aims on reducing the travelling distance of the yard cranes, while balancing the workload of each block.

In the end, depending on the specific assigned storage location, the distance between each container bay and the yard block changes. This has an impact on the total travelling time of containers, therefore it can alter the best combination of which quay crane should serve each bay. A good example of where this can be of significant value is that of indented berths. In contrast to conventional terminals, at indented berths vessels can be handled simultaneously from both sides (Imai et al., 2007). For example (fig. 2), assuming that the designated storage location for a set of containers is S1, then assigning this task to quay crane Q6 is more beneficial in terms of transportation time compared to assigning it to Q1. The proposed model in this paper will hence be choosing a block storage location in the yard for each bay of containers, while deciding upon the best QC positioning and handling of tasks simultaneously.

Fig. 2. Indented berth with stacks (source: Vis & van Anholt, 2010)

7 3. Literature Review

Scientific research has focused extensively in port container terminal operations. A thorough review of the literature regarding the various operations and decision and planning problems involved with them have been presented by Vis and de Koster (2003), Steenken et al. (2004), Stahlbock and Voß (2008), Carlo et al. (2014) and SteadieSeifi et al. (2014).

Regarding the optimization of QC scheduling, most studies consider it independently from other resources in the terminal. The main focus is on specifying the job sequence for each quay crane which minimizes a ship’s turn time or equivalently the QC total makespan. At the same time, yard management literature revolves around yard cranes and block resources allocation. Yard crane deployment papers aim at optimizing the yard crane assignment and movement among the yard blocks, so as vessel and truck operations are not delayed. Yard storage allocation studies deal mostly with the problems of efficient stacking of containers, reshuffling and traffic congestion.

In the last decade several researchers have drawn their attention to examining two or

more operations of the terminal together. More specifically, a few attempts have been

identified to jointly examine the QC scheduling and the yard subsystem by seeking

integrated approaches to the problems. Chen et al. (2007) presented an articulated

hybrid flow shop scheduling problem (HFSS) for QCs, yard cranes (YCs) and yard

trucks (YTs) with presedence and blocking constraints, which was solved with the use

of a tabu search algorithm. The initial results were promising but needed further

improvement. Zeng and Yang (2009) developed a simulation optimization method for

scheduling loading operations for QCs, YCs and YTs, formulated as an HFSS

problem. Their method yielded improved results in comparison with single

optimization algorithms or simulation, but suffered from long computation time. Chen

et al. (2013) formulated the same problem as a constraint programming model and

developed a three-stage algorithm to solve it. Their approach proved effective for

finding high quality solutions for large problems, but with the assumption of a

predetermined QC-assignment to containers. Cao et al. (2010) integrated the QC and

YT scheduling problem for inbound containers and developed a genetic algorithm for

its solution, while Jinxin et al. (2010) did the same by formulating the problem as a

mixed integer programming model and solved it with the use of a ruled based

8 heuristic and a genetic algorithm. However, both considered only one QC. Lau and Zhao (2008) combined the schedules of different handling equipment with a mixed integer programming model which they solved by using a multi-layer genetic algorithm. They concluded that in order to solve real-world large size problems a planning strategy that takes into account smaller number of containers needs to be developed. As can be seen, because of the required complexity in modeling, all above papers somehow simplify the conditions of the problems, for instance either by considering only one QC or ingnoring existing operational constraints, such as presedence relationships between tasks, quay crane interference or safety margins.

Kaveshgar and Huynh (2014) appear to have taken these into account and derived an integrated schedule of QCs and yard trucks which was superior to the sequential planning one. Although there were still limitations related to computational times and problem sizes, their results confirmed the asserted benefits of integrated models.

In all above beneficial research, the yard sub-system is only considered in regard to its handling equipment and the container storage location is either taken as a parameter or ignored at all. In that sense, very few articles concern the equipment planning and storage assignment problem respectively. Bish (2003) considered the loading/unloading of containers and the storing at the terminal yard. The problem attempted to specify the storage location for each unloaded container, while dispatching vehicles to them and scheduling the operations of the cranes. A set of potential storage locations was assumed to be known though. Kozan and Preston (2006) modeled the seaport system with the goal of establishing the optimal storage strategy and container-handling schedule. They presented an iterative search algorithm that integrates a container transfer model with a container location model, which enabled them to specify both optimal locations and related handling schedule.

Cao et al. (2008) integrated the transfer vehicle planning with the storage space assignment on a block level to minimize the completion time of the unloading operations. But in their model, the QC schedule was assumed to be known. Storage space allocation and dispatching strategies for straddle carriers were examined by Soriguera et al. (2007) in a simulation performed by solving a mixed-integer program.

Froyland et al. (2008) studied the problem between the transfer and storage yard

operations, by decomposing it into smaller sub-problems with a small planning

interval. Their approach yielded effective but not optimal results. Finally, Wong and

9 Kozan (2010) examined the relationship between the QCs, yard machines and storage locations at the yard. The objective was to improve the operation efficiency of the QCs and to develop an analytical tool for yard operation planning, but problems were encountered with realistically sized cases in terms of computational time.

Although these articles deal with integrated solutions for the yard storage assignment problem, the focus is on some type of yard handling equipment and not the QCs. The difference of this work lies exactly on the synthesis of a decision system that combines the scheduling of the quay side equipment (QCs) with the simultaneous reservation of the best storage allocation at the yard. Such an integrated framework is expected to moderate any negative effects resulting from sequential planning and yield results which ensure sufficient resource utilization and service quality at low cost. Recently this has started to been achieved by similar approaches in other combined operations of the port, for example that of berth allocation and quay crane assignment (e.g. Imai et al., 2008; Giallombardo et al., 2010; Meisel and Bierwirth, 2013) and berth allocation and yard planning (e.g. Zhen et al., 2011, Zhen, 2012, Hendriks et al., 2013; Robenek et al., 2014 ;).

4. Mathematical Model

The quay crane scheduling problem in this paper is considered for a single ship. This means that no other vessels berth during the maximum time needed for all crane operations to be completed. Time is split in discrete periods and each interval equals the time needed for a QC to handle a standard 20-foot container. The QCs can only move and be assigned to a bay at these intervals. Although analogous, only unloading operations will be examined. The work of Choo et al. (2010) is modified as the basis of this model, while afterwards a solution algorithm will be presented based on its compact formulation described below. Before the complete description, several assumptions must be made:

 The QC gantrying time is insignificant in comparison with the container

handling time, so it can be excluded from the calculation of a ship’s

makespan.

10

 The number of available QCs per vessel is fixed and does not change before the handling of all workload

 All QCs are the same and have similar work rates

 Each yard storage block has the same capacity in terms of number of containers that it can store

 The difference in distance between bays and sub-blocks within the same block is minimal and therefore it will not be taken into account in estimating the distance

The following decriptives will be used for the mathematical formulation:

Indices

j Bay number, in increasing order of their location on the vessel and from left to right

k QC number, in increasing order of their relative position and from left to right b Yard storage location number, in increasing order of their location in the yard and from left to right

t Time period index, denoting the interval of time from t-1 to t.

Parameters

C Number of allocated QCs H Number of bays in the vessel

𝑓

Number of containers to be discharged in bay j

𝑑

Distance between block b and bay j if served by QC k B Number of blocks in the yard

T Total number of time periods in the planning horizon

𝑤

Yard activity threshold, in terms of number of QCs allowed to work on containers headed for the same storage location b at any time

r QC clearance value, in terms of the number of bays

𝑍

Yard block capacity, in number of containers (twenty foot equivalent units)

11 The problem is to determine the QC sequence and block storage location that minimizes the total travelling distance of QCs and therefore the discharging handling time of all containers, while respecting QC and yard congestion related constraints.

The model will use the following decision variables:

Decision Variables

𝑥

(𝑡) 1 if QC k b is positioned at bay j at time period t and 0 otherwise

𝛿

(𝑡) 1 if QC k is handling a container at bay j allocated to storage block b at time period t and 0 otherwise

Both 𝑥 and 𝛿 variable are used to differentiate operating QCs from QCs that are just positioned at a bay but their status is idle.

The mathematical formulation for the model is as follows:

Minimize ∑

∑

∑

∑

𝛿

(𝑡) ∗ 𝑑

(1)

Subject to: ∑

𝑥

(𝑡)

𝑗=1

=1 ∀ k=1,..,C, t=1,..,T ; (2)

∑

𝑥

(𝑡) ≤ 1

𝑘=1

∀ j=1,..,H, t=1,..,T ; (3)

1 − 𝑥

(𝑡) ≥ ∑

𝑥

(𝑡) ∀ j=2,..,H, k=1,..,C,

m=1,..,C t=1,..,T ; (4)

12 1 − 𝑥

(𝑡) ≥ ∑

𝑥

(𝑡) ∀ j=1,..,H-1, k=1,..,C,

m=1,..,C t=1,..,T ; (5)

𝑥

(𝑡)≤ ∑

𝑥

(𝑡) ∀ j=1,..,H-1, k=1,..,C-1, t=1,..,T ; (6)

𝑥

(𝑡)≤ ∑

𝑥

(𝑡) ∀ j=2,..,H, k=2,..,C, t=1,..,T ; (7)

∑

∑

𝛿

(𝑡) = 𝑓

∀ j=1,..,H, b=1,.B, ; (8)

𝛿

(𝑡) ≤ 𝑥

(𝑡) ∀ j=1,..,H, b=1,.B, k=1,..,C, t=1,..,T ; (9)

∑

∑

𝛿

(𝑡) ≤ 1 ∀ k=1,..,C, t=1,..,T (10)

∑

∑

𝛿

(𝑡) ≤ 𝑤

∀ b=1,.B, t=1,..,T ; (11)

∑

𝛿

(𝑡) ≤ 𝑍

(12)

𝑥, 𝛿 binary. (13)

13 The objective function evaluates the sum of the total travelling distance of the QCs between bays and the reserved storage location blocks. Constraint (2) states that QCs cannot move away from the ship at any time and (3) does not allow any of them to be allocated to more than one bay at any time. Constraints (4) and (5) apply the QC clearance conditions, i.e. for safety reasons a QC must be positioned r minimum bays away from any neighbouring QCs on the left and right respectively. Constraints (6) and (7) enforce the QC ordering provision, where ‘’higher-numbered’’ or ‘’lower- numbered’’ QCs must be positioned to the right and left of any QC respectively.

Constraint (8) declares that all requested container jobs will be completed within the planning horizon, while constraint (9) makes sure that a QC will be positioned at a bay if it is working there. Constraint (10) ensures that a QC cannot discharge more than one container from a bay at any time and (11) that the total amount of QC work performed on containers headed for a particular storage location does not exceed the yard activity threshold. Constraint (12) determines the maximum number of containers that can be stored to each yard block, so that the assigned schedule will not exceed their capacity. Finally, constraint (13) defines all variables as binary.

5. Computational Experiments 5.1 Input Data

We modeled the problem with the use of FICO Xpress optimizer solver (www.fico.com) and the Xpress-IVE version 1.24.00/Mosel 3.4.2. In order to solve and evaluate the results we initially made use of the data sets provided by Choo et al.

(2010), augmented by yard template parameters (i.e. number of blocks and distances

between bays and blocks). At first we only used 2 yard blocks as possible destination

for the containers, so as to avoid overcomplicating the problem with more variables

and input data before we could have an indication of its actual performance and also

to facilitate better comparisons with the original model of Choo (2010). The first set

included 12 small problem instances, while the second investigated realistic sizes. The

input parametes differed in terms of bays, number of QCs and clearance values, while

the number of containers was randomly distributed among the ship’s bays. Table 1

presents the parameters used for the initial tests. The respective distance estimations

14 were built upon yard configurations used in Zhen et al. (2011), which can be found in the Appendix. The last two columns list the number of rows and variables of the model, which as anticipated are a lot more than the model of Choo et al. (2010).

Table 1. Initial Problem Instances

Instance Code H C R T B Constraints Variables

SS1-1

10 2 1 61 2 10136 3660

SS1-2

10 3 1 61 2 18676 5490

SS1-3

10 4 1 61 2 29412 7320

SS1-4

10 5 1 61 2 42344 9150

SS1-5

10 2 3 61 2 10136 3660

SS2-1

20 3 3 106 2 66376 19080

SS2-2

20 4 4 106 2 105172 25440

SS2-3

20 3 5 106 2 66376 19080

SS3-1

30 3 8 143 2 135308 38610

SS4-1

40 2 8 198 2 133888 47520

SS4-2

40 3 8 198 2 250708 71280

SS5-1

50 3 8 249 2 344964 112050

SSP-1

25 2 8 519 2 218524 77850

SSP-2

30 3 8 839 2 793724 226530

SSP-3

40 4 8 1385 2 out of

memory

out of

memory

5.2 Model Output

The experiments were conducted on an Intel® Core™ Duo 3.00 GHz and 4 GB RAM

personal computer run on MS Windows 7 Enterprise 64-bit operating system. Table 2

summarizes the results and additionally presents the relevant runtime of the single

ship’s compact formulation model of Choo (2010) solved by CPLEX.

15 Table 2. Initial results

Instance Code

Total

Distance (m)

Mosel Runtime (secs)

CPLEX Runtime (Choo )

SS1-1

3050 0,3 0,87

SS1-2

3050 0,6 0,98

SS1-3

3050 1,4 1,1

SS1-4

3050 1,7 1,02

SS1-5

3050 0,4 1,13

SS2-1

5300 4,4 0,84

SS2-2

5300 14,3 11,58

SS2-3

5300 4,6 9,67

SS3-1

7150 15,5 6,54

SS4-1

10100 9,6 25,89

SS4-2

10100 64,8 368,7

SS5-1

12450 127,1 -

SSP-1

25950 17,1 808,09

SSP-2

41950 199,4 -

SSP-3

unknown - -

The output of the model for the first instance is illustrated as an example in the tables below:

Table 3. Results for SS1-1 / crane 1

k (crane) 1

t (period) 1 2 3 4 5 6 7 8 9 10

j (bay) to b (block) 1

1

1 1

2

1

3

1

4

5

6

7

16

8

9

10

j (bay) to b (block) 2

1

2

3

4

5

6

1

7

8

9

10

Table 4. Results for SS1-1 / crane 2

k (crane) 2

t (period) 1 2 3 4 5 6 7 8 9 10

j (bay) to b (block) 1

1

2

3

4

1

5

1

6

7

8

9

10

j (bay) to b (block) 2

1

2

3

4

17

5

6

7

1

8

1

9

1

10

1

As explained before, variable 𝛿

(𝑡) takes the value of 1 for every handling of container by a crane. These values have been highlighted with blue on the tables above. At the same time, the green boxes give the position of the cranes at every time period when they are inactive. The QC movements from bay to bay have been summarized in Table 5.

Table 5. Summarized QC schedule of SS1-1

crane/period T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 crane 1

bay 1 bay 4 bay 1 bay 3 bay 3 bay 3 bay 6 bay 2 bay 1 bay 2

bay 10 bay 9 bay 9 bay 8 bay 7 bay 9 bay 8 bay 4 bay 5 bay 4

It needs to be noted that based on this outcome no containers are handled on time period T2. This is a result of the programming of the model, which tries to distribute the workload across all time intervals and therefore produces wastage in terms of chosen bay-to-blocks destinations per schedule. This goes in line with the findings of Choo et al. (2010) and is not considered problematic, since a simple manual inspection can detect and move it, in this case a time interval behind. This, for example, will result into handling the jobs in 9 time periods instead of 10 and even result to a slightly less QC travelling time.

Some more tests were run subsequently based on varying number of 1-8 yard block

locations, which are considered realistic for a single ship. Some exhibitive examples

are presented in Table 6.

18 Table 6. Problem Instances with varying number of blocks

Instance Code H C R T B Total Distance (m) Mosel Runtime

SSB-1

10 2 1 61 8 3050 1,1

SSB-2

20 2 3 106 4 5300 5,3

SSB-3

20 3 3 106 8 5300 5,9

SSB-4

25 2 8 519 8 24950 57,4

It can be seen that the optimal value of the QC travelling distance remains the same for all small instances. This is not surprising as the total number of containers in these sets never exceedes the typical capacity of a block. This practically means that although there are more yard block locations available for storage, the same two are chosen in this yard configuration because of their proximity to the vessel’s bays. For the realistic data set we observe that the optimal value is smaller, which can be justified by the clearance and yard constraints. As it could also be seen on the presented example of SS1-1, when there are only two blocks available it might be necessary for the QCs to perform additional movements or wait for each other on an idle position in order to not violate the constraints, while when there are more blocks available for storage this can be avoided.

5.2 Model Evaluation

The model yields optimal results for all instances, except for SSP-3 (which is the

largest realistic set), due to insufficient memory. The runtime is slightly longer for

only a few of the small problem instances in comparison to that of Choo et al. (2010),

but that was expected due to the different nature of the model, objective function and

respective number of variables and constraints. Nevertheless, it seems to perform

better for larger instances, both in terms of computational time and the actual fact of

being able to solve two instances which their model did not achieve. It should

however be noted that these differences ought to be due to the use of a different

modeling software and also the different specifications of the computers in use.

19 Therefore it can be said that although the two models are not directly comparable, ours is valid and equally significant, while by definition it gives results with enhanced practical importance due to its provision of simultaneous planning.

As expected, the mathematical formulation of the integrated problem of QC scheduling and yard storage allocation problem is large and complex, while the use of a commercial solver leads to long computation times or even infeasibility due to the huge number of variables in realistic data sets and lack of memory. Therefore a different solution approach will be taken based on a branch and price framework to solve the integrated problem. Choo et al. (2010) used the branch and price algorithm to achieve decreased computational time in comparison to the direct approach of the commercial solver. Robenek et al. (2014) also used it successfully in their effort to optimize the integrated berth allocation and yard assignment problem in bulk ports. It will later be shown that the proposed method is more efficient for dealing with our integrated problem as well.

6. Solution approach

As discussed in §5.2, this part will deal with the reformulation of the compact mathematical model described in §4. The problem will be codified as a generalized set covering problem and solved by branch and price, a method chosen for its suitability for solving large-scale problems (Barnhart et al., 1998). B&P basically constitutes of LP relaxations solved by delayed column generation. What happens in the algorithm in principle is the decomposition of the problem into two parts, a master problem and a pricing problem.

6.1 Reformulation

Reformulating the compact model into column generation form produces the so called

master problem. Each column here will consist of H*B rows, one for each bay-to-

block (𝑗, 𝑏) coordinate. The reformulated equivalent of the model has its core in the

creation of an entire QC position to-(bay, block) plan per time period as a decision

20 variable. Each such feasible assignment has to respect all clearance requirements, ordering QCs and yard congestion constraints. Therefore the master problem will consist of a ‘0-1’ set-partitioning structure, where the QC position-to-(bay, block) is encoded as a C-tuple and represents a (bay, block) couple. For instance ((𝑗

𝑏

), (𝑗

𝑏

), . . , (𝑗

𝑏

) states that the first QC is positioned at bay 𝑗

and will handle a container that is directed to storage location 𝑏

(so our original variable will have the value of 𝛿

(𝑡) = 1). At the same time, the right-hand vector symbolizes the load profile of each coordinate, which has to be handled completely as instructed by constraint (8). An example of a column is illustrated in the figure below. The values of 𝛿

, 𝛿

𝛿

are 1, which together depict one assignment.

𝑦 =

[ 0 1

− 1 0

− 1 0]

𝑓 =

[ 2 5

− 7 3

− 1 2]

Fig.3. Example of a column for H=3, B=2, C=3, r=0 and #20 containers spread across the bays

In this framework we let P be the set of all QC feasible work assignments. For p ϵ P we have a binary variable 𝑦

, which is 1 if assignment p is selected in a given time and 0 otherwise. The formulation then will read:

Minimize ∑

𝑦

𝑑

(14)

subject to: ∑

𝑦

= 𝑓

, ∀𝑗 = 1, . . , 𝐻; (15)

𝑦 nonnegative integer (16)

21 where 𝑑

symbolizes the traveling distance of one assignment, calculated from the total distance of all bay-to-block pairs incurred in that assignment. 𝑓

remains the number of containers that need to be discharged from each bay.

The objective function (14) minimizes the total distance travelled from QC position- to-(bay, block) assignments selected to handle all the workload of the vessel.

Constraint (15) establishes the unloading requirements, so that for every bay j, all 𝑓

necessary assignments are selected. From this formation the corresponding 𝑥 values of the compact model will be derived from a given feasible solution 𝑦. The 𝛿 values will thereafter be obtained by promptly setting the 𝑥 values to zero, so as the work completion constraint (8) is satisfied.

6.2 The Branch and Price Algorithm

Herewith we will present the most crucial components of the branch and price algorithm. To begin with, it should be noted that the LP relaxation of (14)-(16) is equally strong as that of (1)-(13) and it constitutes the so called restricted master problem (RMP). Subsequently follows the pricing problem, which is responsible for providing a column that prices out positively or proves that none exists. The dual variables 𝛾

and 𝜋

related to constraint (15) from the (RMP) are used to solve the pricing problem as follows:

Minimize 1 ^{− ∑}

^∑

^𝑑

∗ 𝛾

− ∑

𝜋

𝜗

(17)

subject to: ^∑

𝜗

= 𝐶 ∀ j=1,..,H ; (18)

𝐶(1 − 𝜗

) ≥ ∑

𝜗

∀ j=1,..,H-1 ; (19)

𝐶(1 − 𝜗

) ≥ ∑

𝜗

∀ j21,..,H; (20)

22 𝛾

≤ 𝜗

∀ j=1,..,H, b=1,..,B ; (21)

∑

𝛾

≤ 𝑤

∀ b=1,..,B ; (22)

𝜗, 𝛾 binary (23)

The decision variable 𝜗

is 1 if a QC is positioned at bay j and 0 otherwise, while 𝛾

is 1 if a bay-to-block assignment is selected and 0 otherwise (constraint 23). The objective function (17) embodies the reduced cost of a column and constraints (18)- (20) secure the QC positions according to clearance and ordering rules. Finally, constraints (21) and (22) declare the QC working positions and the respect of the activity threshold in the yard.

Based on the (RMP) and pricing problem, column generation and a branching procedure is performed right after. Hereby we will branch on the fractional variable 𝑦

, following the branching rules of rounding up and down to the closest integer value respectively. Our strategy is defined as depth-first search, since we select the node with the most components fixed, which is the deepest in the tree.

6.3 B&P Numerical Experiments

Numerical experiments anew produce the following results, presented together with the original values of the compact model:

Table 6. Branch and Price Results

23 Instance Code H C R T B Total Distance Mosel Runtime

SS1-2

10 3 1 61 2 3050 0,6

SS1-2(B&P)

10 3 1 61 2 3770 0,078

SS1-3

10 4 1 61 2 3050 1,4

SS1-3(B&P)

10 4 1 61 2 3210 0,078

SS1-5

10 2 3 61 2 3050 0,4

SS1-5(B&P)

10 2 3 61 2 3130 0,073

SS2-2

20 4 4 106 2 5300 14,3

SS2-2(B&P)

20 4 4 106 2 6020 0,124

SSP-1

25 2 8 519 2 25950 17,1

SSP-1(B&P)

25 3 8 519 2 30670 0,125

It can be seen that the new solution approach saves tremendous amount of computational effort. However, the total travelling distance of the QCs appears increased to some extent, therefore the results should be considered near-optimal.

The trade-off between the additional cost per QC travelling distance unit and planning and scheduling operations needs to be considered per case.

7. Conclusions and Future Research

In this paper we study the integrated problem of quay crane sequencing and yard

storage space allocation in the context of container terminals. So far very limited

research has attempted to deal with this decisional combination of port operations, at

least in terms of considering and scheduling simultaneously both aspects and not

simplifying or taking one of the two as parameter. The necessity of incorporating the

individual planning of these operations into one model has been highlighted

throughout this study. An exact mathematical model has been built partially based on

the work of Choo et al. (2010) and extended accordingly to reflect important QC

clearance requirements, yard congestion and block capacity constraints. The goal of

the model is to reserve a storage block location for each container, while sequencing

the QC jobs in such a way that their total travelling distance is minimized.

24 The mathematical formulation of the integrated problem of QC scheduling and storage block reservation is complex and extensive, but has been solved to optimality for small and relatively small realistic sizes. It has also been validated against the compact model of Choo et al. (2010) and it yields similar if not better results when examined under the same data sets. In order to improve the outcomes, we developed a solution approach based on a branch and price algorithm, which yielded results in considerably less computational time. This is considered highly significant in practical applications. However, a feasible level of suboptimality has been detected, which can constitute ground for future research.

Finally it should be noted that the simultaneous planning problem in this paper has

been considered for a single ship. Future research may extend this model to consider

multiple ships berthing in the planning horizon. Another possible extension may

additionally include the specific sub-block reservation in the yard, instead of only the

designated block destination.

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Appendix

Figure: Typical yard configuration (source: Zhen et al., 2011)