Transportation Research Part E - VLIZ · Container storage and transshipment marine terminals Etsuko Nishimuraa,*, Akio Imaia, Gerrit K. Janssensb, Stratos Papadimitriouc a Graduate

Transportation Research Part E 45 (2009) 771–786

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Container storage and transshipment marine terminals

Etsuko Nishimura a,*, Akio Imai a, Gerrit K. Janssens b, Stratos Papadimitriou c

a Graduate School of Maritime Sciences, Kobe University, Fukae, Higashinada, 658-0022 Kobe, Japanb Department of Applied Economics Sciences and Research Group Data Analysis and Modelling, Hasselt University, Campus Diepenbeek, Agoralaan,Gebouw D, BE-3590 Diepenbeek, Belgiumc Department of Maritime Studies, University of Piraeus, 80 Karaoli and Dimitriou Str., GR185 32 Piraeus, Greece

a r t i c l e i n f o

Article history:Received 9 November 2007Received in revised form 8 December 2008Accepted 24 March 2009

Keywords:Transshipment containersContainer storageMega-containershipsMathematical programming

1366-5545/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.tre.2009.03.003

* Corresponding author. Tel./fax: +81 78 431 625E-mail address: [email protected] (E.

a b s t r a c t

This study addresses the storage arrangement of transshipment containers on a containeryard, in order to carry out efficiently the ship handling operations at a terminal wheremega-containerships call. An optimization model is specified to investigate the flow of con-tainers from the mega-containership to feeder ships using intermediate storage at the yard.A heuristic based on the lagrangian relaxation is formulated. The quality of the heuristicapproach is tested in a number of experiments. In the experiments, various situationsare analyzed with respect to mega-containership arrival rates, some strategies for stackarrangements and terminal layouts.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The introduction of a container system shows several benefits on saving the handling trouble, reducing the damage po-tential for goods, and decreasing the handling costs. Such a system contributes to a positive development of the intermodalsystem of freight transportation which enables a simplified movement of goods in bulk from one transportation mode toanother. In those days when container systems have been introduced, the port facility and operation systems also had tochange for it.

Recently, due to the continuously increasing container trade, many terminals are currently operating at or close to capac-ity. In addition, the containership ‘‘Emma Maersk” from Maersk Line, over 10,000 TEU containership (a mega-containershipcalled hereafter) has finally been introduced. The emergence of the mega-containership is one of great interest for logisticscompanies, shippers, port authorities and those engaged in maritime planning. Considering the trend towards larger con-tainer ships, the need for efficient terminal operations is more important than ever. Mega-containerships call mega-ports.From there, feeder ships transport the containers to a port close to the consignee. Thus, at the terminal where mega-contai-nerships call, the handling container volume is huge. Therefore, an efficient terminal for very large ships is one that facilitatesa fast transshipment of containers to and from the ships.

The efficiency of a container terminal depends on the smooth and efficient handling of containers. There are three basictypes of container handling systems engaged in loading and discharging operations at a container terminal: chassis, straddle-carrier and transfer crane systems. The latter system is the most popular in major terminals due to the need for high con-tainer storage capacity in the yard. The transfer crane system requires several types of handling equipment such as quaycranes, yard gantry cranes and yard trailers. The bottleneck in the loading and discharging operations occurs at the quay

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772 E. Nishimura et al. / Transportation Research Part E 45 (2009) 771–786

crane operation. Quay cranes should not stop their operation by waiting for trailers to pick up containers from them or todeliver containers to them.

At a terminal where mega-containerships are calling, most cargoes to be handled are transshipments from port of originto port of destination via that terminal. Thus, associated handling operations are undertaken between mega-containershipsand feeder ships. In order to decrease the turnaround time of ships calling at this terminal, smooth handling of operations isof major importance.

This study is concerned with the container storage arrangement on the container yard for the transshipment containers,in order to carry out the container handling operations in the most efficient way.

This paper is organized as follows. In the next section the problem is defined. Section 3 reviews the related literature. InSections 4 and 5, the formulation for the problem and the solution procedure are described, respectively. In Section 6 numer-ical experiments are carried out and results are presented. The final section reports the paper’s findings and conclusions.

2. Problem definition

Liner networks can be classified by ship calling type, such as hub-and-spoke network or multi-port calling routes. Theterminal where the mega-containership calls is not only used by the mega-containership and feeder ships but also by shipsof other sizes. From our knowledge, the yard area where the containers stored is divided roughly according to the direction ofcontainers passing through that terminal such as inbound, outbound and transshipment. This paper deals with the trans-shipment containers related mega-containerships. It means that we deal with a part of containers which all ships call ata relevant terminal. As the storage area for these containers is divided according to the direction of container movementin the terminal, the interference with the other types of containers is less, so we do not have to deal with the ships of othersizes in this paper.

First the target area of this problem is described. Containers flow into a terminal in two ways: from the mega-container-ship to the feeder ships and from the feeder ships to the mega-containership, as shown in Fig. 1. Containers need to be storedin the yard for several reasons: time differences between ship arrivals, repositioning of containers in ship stowage and mod-ifying the work sequence of container handling.

In this study, to simplify the treatment of the problem, we consider the container flow in only one direction, from themega-containership to feeders. In other words, the containers are stored in the yard area after the mega-containership ar-rives. At a later stage, these containers are stowed into the relevant feeder ship. Note that, in general, the loading operationfor a feeder ship can start only after the unloading operation for that ship is completed. This is also assumed in this paper.

Therefore, when the containers for the relevant ship from the mega-containership have already been stored in the yard,but the unloading operation for a feeder ship has not completed yet, the start of loading operation for this ship has to bepostponed until the unloading operation is completed. When the unloading operation for a feeder ship is finished, butthe containers from the mega-containership have not been stored in the yard, the start of loading for this ship also has todelay starting until the containers are stored.

There are two types of transshipment containers as follows: (1) containers unloaded from the ship are stored at yardareas, and then those are loaded to the other ships, or (2) containers unloaded from the ship go to the other ships, and thoseare loaded to the other ships directly.

A very simple but not very correct way of considering this problem would be to assume that the containers move from themega-containership to the feeder ships directly by simply transporting the containers through the yard area. In that case a

...Feeder ship arrival

1st 2nd 3rd

Fig. 1. Concept of container flow in the terminal.

E. Nishimura et al. / Transportation Research Part E 45 (2009) 771–786 773

constant time for handling operation between the mega-containership and feeder ships could be included in the objectivefunction. The total handling time would not depend on the way of container storage but only on the number of containershandled.

However, from our knowledge, in order to rearrange the working sequence to ship stowage, most of containers are storedat the yard area and not just passing through. Therefore, in this study, we do not consider direct loading of containers tofeeder ships.

In general, the container stack situation and the container storage volume of the yard area are changing slowly during therelevant planning horizon. Our aim of this study is to formulate this problem in a simple way. Then, we investigate the qual-ity of our approach in some cases with some simplifying constraints. In future research, the implementation will be testedand further developed for more realistic applications. Thus, we consider the static model in this paper.

Secondly, we describe the handling of containers. A yard has multiple container blocks, and one container block consistsof multiple containers. Containers for a particular ship are treated as one container group. For example, one container groupfor ‘‘ship A”, which consists of 100 containers, is stored in one container block in the yard. We do not assign a single containerto a single location in the container block. Generally, if we should treat the problem container by container, we should haveto deal with issues like:

– precedence constraints in the working sequence and stowage location for ship stowage, and also– re-handle works caused by the working sequence and storage allocation in the yard,

by considering weight class and destination port for each container. However, as mentioned above, there is no need todeal with these issues in this study.

Thirdly, we consider the quay crane allocation to each ship. To shorten the handling time until mega-containership’sdeparture, several quay cranes will be assigned to the mega-containership simultaneously. Also, the number of quay cranesassigned to one feeder ship depends on the handling container volume. If two or more quay cranes are assigned to one feedership, the yard area for containers handled to that ship is wider than that for containers handled to the other feeder ship as-signed to one quay crane. The former ship can depart from the terminal after all quay cranes for that ship complete the han-dling operation. And from our knowledge, among quay cranes of a relevant feeder ship, containers of a feeder ship are storedat the blocks which are divided by each quay crane assigned to a relevant feeder ship, in order for quay cranes, yard cranesand yard trailers not to have a mutual influence on each other. Thus, the formulation of this problem with multiple quaycranes will be more complicated than that of one quay crane. We need more constraints and the solution method for theproblem with multiple quay cranes which is different from that for one quay crane. Therefore, to simplify the problem,we assume that only one quay crane is assigned to a feeder ship.

Fourthly, in the case that multiple ships arrive at a relevant terminal almost simultaneously, and if the stay time of someship is different from that of others, they can share a single block among them. Thus, we assume that a part of the yard areain the target block is used for the containers handled of other ships in another planning period. If we assume that we con-sider this problem only with containers handled from the mega-containership during its stay, it is reasonable to assume that,containers which are loaded to one feeder ship are stored in a single yard block, the yard block is not shared by containers ofmultiple ships.

Additionally, the assignment to feeder ships and its working sequence for quay cranes assigned to the mega-container-ship are given in advance, because they are determined at the time when the containers are stowed in the mega-container-ship. Note that, the containers for feeder ships are treated by a particular quay crane. To reduce the complexity, multiplequay cranes do not handle the containers for a feeder ship. It is also assumed that the berthing position of feeder ships isknown in advance.

For a feeder ship which arrives earlier, the loading operation needs to be worked out as soon as possible. For a feeder shipwhich arrives later, we do not have to do so, but have to be in time. We have to assign the location as close as possible fromits berthing location.

The determinant, the objective function and constraints of our problem are as follows: the determinant is storage alloca-tion for transshipment containers in the yard, the objective function is total service times from the mega-containership to fee-der ships including the waiting time for feeders until start of the loading operation, constraints are

– every container group of a feeder ship is stored exactly once in the yard,– every container group for only one feeder ship is assigned to a container block in the yard, and– the loading operation for feeders cannot be started until the containers from the mega-containership are stored in the

yard.

3. Literature review

Storage space is a critical resource in container terminals. Kozan and Preston (1999) consider the determination ofoptimal storage strategies and container handling schedules by yard cranes. They propose a heuristic method by a genetic


algorithm (GA) and analyze the factors that influence container transfer efficiency through lower throughput time at a con-tainer terminal with different types of handling equipment, storage capacities and layouts.

Kim and Kim (1994) discuss how to allocate the yard space which is released by loading export containers to a ship. Theydevelop a quadratic programming model which minimizes the material handling cost of yard cranes under the constraints ofspace requirements and space balance equations. They determine only the amount of space allocation for a ship. Kim and Bae(1998) propose mathematical models for three sub-problems: the bay matching, the move planning, and the task sequenc-ing, in order to shorten the turnaround time of ships in container terminals. The task sequencing minimizes the completetime of the re-marshaling operation. Kim and Kim (1999) consider how to allocate the storage space for import containers,in order to minimize the expected total number of re-handles. They propose mathematical models and solution procedures,have analyzed the cases where the arrival pattern of containers is constant, cyclic and dynamic. Kim et al. (2000) propose asolution method for determining the storage location of export containers considering their weights. They consider the stackarrangement and the weight distribution of containers in the yard bay. They develop a dynamic programming model fordetermining the storage allocation, in order to minimize the number of relocation movements. Kim and Park (2003) discusshow to allocate the storage space for outbound containers. They formulate the model as a mixed-integer linear program inorder to allocate the space for utilizing space efficiently and make loading operations more efficient. Two heuristic algo-rithms are proposed to solve this problem.

Zhang et al. (2003) address the storage space allocation problem in the yard. They solve the problem using a rolling-hori-zon approach. This problem is decomposed into two levels: placing the total number of containers in each block at the timeperiod, and determining the number of containers associated with each vessel in order to minimize the total distance totransport the containers between yard blocks and the ship berthing location. They deal with four types of containers as fol-lows: inbound containers on vessels and already stored in yard blocks, outbound containers grounding in yard blocks andalready stored in yard blocks. However Bazzazi et al. (2009) consider only inbound containers at the first level of the problemthat Zhang et al. (2003) dealt with, they determine the number of inbound containers of each vessel stored in each block, inorder to minimize the imbalance among workloads allocated to blocks.

Lee et al. (2005) consider the container-to-yard location assignment problem, as given the concept to reserve dedicatedclusters for each vessel. They determine the number of containers discharged during each period, in order to minimize thetotal number of yard cranes required to handle the workloads. Lim and Xu (2006) propose a meta-heuristic procedure,named the critical-shaking neighborhood search, in order to minimize the yard space needed, as given for yard spaces torequests at each time.

Summarizing the related works, in most cases only export containers arriving from the hinterland are considered withouttransshipment containers. It is important that the arrival time of containers from the hinterland are estimated in advance,but this is not an easy task. However we deal with only transshipment containers, and therefore we assume that the arrivalpattern of these containers is given in advance as these containers come from ships calling to the terminal.

4. Problem formulation

In this section, a mixed integer programming formulation of the container storage allocation problem (CSAP) and thelagrangian relaxation of this formulation are presented.

4.1. Formulation of the container storage allocation problem ðCSAPÞ

In formulating the CSAP, a binary variable xjl is defined to specify whether the container group for ship j is stored at theyard location l. We consider the CSAP formulation, which is modified from the formulation of the ship-to-berth allocationproblem (BAP) in Imai et al. (2001). This formulation is further called [P-CSA]. Afterwards, we deal with berths, ships andorder of service sequence in the BAP as quay cranes assigned to the mega-containership, feeder ships, an order of workingsequence for quay cranes assigned to the mega-containership.

½P� CSA� MinimizeX

i2QC

X

j2V

X

l2YP

fðTi � Kj þ 2ÞCMYBðiÞjl þ CYF

BðjÞjl þ AM � FjgQ ijxjl þX

i2QC

X

j2V

X

l2YP

Q ijyjl ð1Þ

Subject toX

j2V

xjl ¼ 1 8 l 2 YP; ð2ÞX

l2YP

xjl ¼ 1 8 j 2 V ; ð3Þ

X

j02V

X

l02YP

1Kj � Kj0

maxfKj � Kj0 ;0gCMY

BðiÞj0 l0Qij0xj0 l0

þ Q ijfyjl � ðFj � AM � CMYBðiÞjlÞxjlgP 0 8 j 2 V ; l 2 YP; i 2 QC; ð4Þ

xjl 2 f0; 1g 8 j 2 V ; l 2 YP; ð5Þyjl P 0 8 j 2 V ; l 2 YP; ð6Þ


where i (=1, . . . , NQ) e QC, set of quay cranes; j (=1, . . . , NV) e V, set of feeder ships; l (=1, . . . , NP) e YP, set of container blocksin the yard; Ti, number of ships handled by quay crane i assigned to the mega-containership; Kj, order of unloading sequenceat quay cranes assigned to the mega-containership for handling of feeder ship j’s container group; Qij, 1 if quay crane i han-dles the container group of feeder ship j, 0 otherwise; AM, arrival time of the mega-containership; Fj, completion time of fee-der ship j‘s containers for unloading; CMY

BðiÞjl, handling time spent by feeder ship j ‘s container group movement from location ofquay crane i assigned to the mega-containership B(i) to yard block l; CYF

BðjÞjl, handling time spent by feeder ship j ‘s containergroup movement from yard block l to ship j ‘s berthing position B(j); xjl, decision variable; 1 if the container group of feedership j is stored in yard block l, 0 otherwise; yjl, stay time of feeder ship j ‘s container group in the yard block l until beginningof loading operation to a relevant ship.

The objective function (1) minimizes the sum of handling time for the container movements from the mega-container-ship to yard blocks, and of handling time for the container movements from yard blocks to the feeder ships’ berthing loca-tion, plus the waiting times for the feeder ships. Constraint set (2) ensures that the container group for each feeder ship mustbe stored at a yard block. Constraint set (3) enforces that each yard block services up to the container group for a feeder ship.Constraints (4) assume that the loading operation for a feeder ship must start after its arrival and its unloading operation.

In the objective function, a handling time CMYBðiÞjl is weighted by (Ti � Kj + 2). (Ti � Kj + 1)CMY

BðiÞjl, this results from the obser-vation that the handling time CMY

BðiÞjl of a specific ship j serviced by quay crane i assigned to the mega-containership contrib-utes to the waiting time of the feeder ships, which are to be serviced by the same quay crane after it. In other words, thewaiting time of a particular ship is represented by the cumulative handling time of its predecessors.

CB(1)1lYF

CB(2)2lYF

C*jl MY

CB(j)jl YF

CB(3)3lYF

F1

Mega-containership AM

handling time spent by feeder ship j 's container group movement from location of the quay crane assigned to the mega-containership (*) to yard block l

CB(4)4lYF

=AM-F1+C*1lMY

=AM-F2+C*1lMY+C*2l

MY

=AM-F3+C*1lMY+C*2l

MY+C*3lMY

=AM-F4+C*1lMY+C*2l

MY+C*3lMY+C*4l

MY

handling time spent by feeder ship j 's container group movement from yard block l to ship j 's berthing position B(j)

F2 F3 F4 Time

Waiting time for Ship-3C*3l

MY


MY


MY


MY

(a) Case 1: Short interval of feeders’ completion time for unloading

CB(1)1lYF

C*2lMY

CB(2)2lYF

C*3lMY

CB(3)3lYF

C*4lMY

CB(4)4lYF

y3l

=AM-F1+C*1lMY

=AM-F2+C*1lMY+C*2l

MY=F3- (AM+C*1l

MY+C*2lMY+C*3l

MY)

= F4- (AM+C*1lMY+C*2l

MY+C*3lMY+C*4l

MY)

TimeF1 F2 F3 F4

Mega-containership AM

y4l


MY

Waiting time for Ship-2

(b) Case 2: Long interval of feeders’ completion time for unloading

Fig. 2. Objective function (a) case 1: short interval of feeders’ completion time for unloading (b) case 2: long interval of feeders’ completion time forunloading.


Fig. 2 shows the time chart of containers serviced by a quay crane, which is assigned to the mega-containership.Regarding the work, originated by the mega-containership, to the yard blocks, the container group for each feeder ship isstored in the yard block assigned in order of the quay crane’s pre-determined working sequence. Thus, the container group,which is first in line in the working sequence of a quay crane, starts being unloaded immediately after the mega-container-ship’s arrival. After, the next working order at the same quay crane begins being unloaded. For example, the handling timeCMY�1l for feeder ship-1 in Fig. 2 is a part of waiting time for feeder-2, feeder-3 and feeder-4, respectively. This is expressed by

(4 � 1 + 1)CMY�1l . Additionally, 1� CMY

�1l and 1� CYF�1l refers to the handling time spent by feeder ship-1’s containers movement

from the mega-containership to yard block l, and from yard block l to feeder-1 in the objective function. Since AM � F1 is anegative value if F1 < AM for ship-1, and AM � F4 is a positive value if F4 > AM for ship-4, the objective function value can beobtained.

Next, the loading operation to feeder ships begins after its own unloading operation, the container group waits until theunloading operation completes. This waiting time is not waiting time for the feeders, as it concerns the time that the con-tainer group is stored in the yard. In the model this time is expressed as the variable yjl.

When the container group for a feeder ship completes unloading, but the container group for this ship is not yet fullystored in the yard block, the feeder ship has to wait until the storage operation is completed. The waiting time for this feederis included in the objective function.

Therefore, as shown in Fig. 2, when the waiting time and the container group’s storage time for each feeder ship are cal-culated, the objective function value can be obtained. In case 1, the interval of feeder ships arrival is relatively shorter, and so,the interval of completion time for its unloading is also shorter. In case 2, the interval of feeders’ arrival is relatively longer,and so, the interval of completion time for its unloading is also longer. In case 2 some containers face some waiting time inthe yard, so the loading of containers to ships 3 and 4 can start immediately after their unloading operation.

The yjl variables technically have integer values to indicate the precedence relationship between completing containersfor feeder ship j stored at the yard block l and starting containers loaded for a relevant ship.

In constraints (4), ship j refers to a ship of which the containers are unloaded by a quay crane. The termP

j02V

Pl02YP

1Kij�Kij0

maxfKij � Kij0 ;0gCMY

BðiÞj0 l0Q ij0xj0 l0 refers to the completion time of the ship loading immediately before ship

j in the service sequence. The term (Fj � AM)Qijxjl refers to the completion time of the loading operation for ship j.

Thus, ifP

j02V

Pl02YP

1Kij�Kij0


BðiÞj0 l0Q ij0xj0 l0 þ CMYBðiÞjlQ ijxjl þ Qijyjl > ðFj � AMÞQ ijxjl, the interpretation should

be as follows: after it completes unloading feeder j’s containers from the mega-containership, those containers are storedin yard block l for yjl time units, and then they are loaded to feeder j.

IfP

j02V

Pl02YP

1Kij�Kij0


BðiÞj0 l0Q ij0xj0 l0 þ CMYBðiÞjlQ ijxjl þ Qijyjl ¼ ðFj � AMÞQ ijxjl, the interpretation should be as

follows: since it has already completed unloading feeder j’s containers from the mega-containership, immediately after itcompletes feeder j’s containers unloading, the containers from the mega-containership are loaded to feeder j.

4.2. Lagrangian relaxation of CSAP

The formulation of the CSAP is a mixed integer programming problem formulation which is not known to be solvable inpolynomially-bounded time as well as the ship-to-berth allocation problem in Imai et al. (2001). In this section, we develop aheuristic procedure by the subgradient optimization procedure based on the following lagrangian relaxation of the originalproblem [P-CSA]:

½P-LR� MinimizeX

i2QC

X

j2V

X

l2YP


BðjÞjl þ AM � FjgQijxjl þX

i2QC

X

j2V

X

l2YP

Q ijyjl

�X

i2QC

X

j2V

X

l2YP

kijl

X

j02V

X

l02YP

1Kj � Kj0

maxfKj � Kj0 ; 0gCMY

BðiÞj0 l0Q ij0xj0 l0

�X

i2QC

X

j2V

X

l2YP

kijlQ ijfyjl � ðFj � AM � CMYBðiÞjlÞxjlg ð7Þ

Subject toX

j2V

xjl ¼ 1 8 l 2 YP; ð2Þ

X

l2YP

xjl ¼ 1 8 j 2 V ; ð3Þ

xjl 2 f0; 1g 8 j 2 V ; l 2 YP; ð5Þ

yjl P 0 8 j 2 V ; l 2 YP; ð6Þ

where kijl is the lagrangian multiplier for quay crane i assigned to the mega-containership, feeder ship j, and yard block l andhas a non-negative value.


Because the yjl ‘s do not appear in any constraint, they are redundant. By this, this formulation can be rewritten as follows:

½P1� MinimizeX

i2QC

X

j2V

X

l2YP


BðjÞjl þ AM � FjgQ ijxjl �X

i2QC

X

j2V

X

l2YP

kijl

X

j02V

X

l02YP

1Kj � Kj0

�maxfKj � Kj0 ;0gCMYBðiÞj0 l0Q ij0xj0 l0 þ

X

i2QC

X

j2V

X

l2YP

kijlðFj � AM � CMYBðiÞjlÞQijxjl ð8Þ

Subject toX

j2V


l2YP

xjl ¼ 1 8 j 2 V ; ð3Þ

xjl 2 f0; 1g 8 j 2 V ; l 2 YP: ð5Þ

Problem [P1] is further reformulated by introducing the representative cost Ejl in the objective function.

½P2� MinimizeX

j2V

X

l2YP

Ejlxjl ð9Þ

Subject toX

j2V


l2YP

xjl ¼ 1 8 j 2 V ; ð3Þ

xjl 2 f0; 1g 8 j 2 V ; l 2 YP: ð5Þ

By relaxing constraint set (4), formulation [P-CSA] becomes a two-dimensional assignment problem and is therefore easy tosolve.

5. Solution procedure

5.1. Subgradient method

The quality of the feasible solution obtained through the above procedure strongly depends on the ability to determinegood lagrangian multipliers kijl. In [P-LR], for each combination of (i, j, l), it is assumed that the container group for each shipcan be allocated to yard blocks while minimizing total handling time from the mega-containership to feeder ships and totalwaiting time for feeder ships. This may lead to an infeasible solution of the original problem [P-CSA] as the container groupsfor some feeder ships may be loaded before these ships have completed their unloading operation.

Good multipliers are also important as the quality of the lower bound, (i.e., the objective function value of [P-LR]), is afunction of those multipliers. The best lower bound corresponding to the optimal multiplier vector k� is determined as

ZLRðk�Þ ¼MaxkðZLRðkÞÞ; ð10Þ

where ZLR(k) is the value of the lagrangian function with a multiplier set (vector) k. To find the set of the lagrangian multi-pliers, we have used the subgradient optimization procedure (Fisher, 1981). This approach has been widely utilized. Forexample, Imai et al. (2001) have proposed the heuristics to solvable effectively the ship-to-berth assignment problem,and Imai et al. (2007) have also developed the algorithm to solve the vehicle routing problem with full container load. Itis sufficient to note that as a termination criterion the maximum number of iterations has been fixed at 200. The procedurealso terminates if the gap between the feasible solution value and the lagrangian bound becomes less than 1. Given integerobjective function coefficients, this condition is sufficient to detect an optimal solution. The process, discussed in the nextsection, is performed at each iteration of the heuristic to determine a feasible solution to [P-CSA]. At the time of termination,the subgradient optimization procedure reports the best feasible solution and the best lower bound generated in all theiterations.

During each iteration of the procedure, [P-LR] is solved to obtain a lower bound for [P-CSA]. Note that the objective func-tion value of [P-LR] is equal to the objective function value of [P2] plus the value related to yjl. As the objective function of[P-LR] is a lower bound, yjl values are fixed to zero.

The subgradient procedure is detailed as follows.In the subgradient optimization procedure, given a set of starting multipliers k0, a sequence of multipliers is generated

using the following expression:

kkþ1 ¼ kk þ tkðAxk � bÞ; ð11Þ

where xk is an optimal solution to the lagrangian problem [P-LR] (kk) and tk is a positive scalar step size and Ax 6 b is the setof constraints being relaxed (i.e., constraint set (4)). The following step size has been used frequently in the past:

tk ¼ dkðZBestF � ZP2ðkkÞÞ=kAxk � bk; ð12Þ


where ZBestF is the best known feasible solution value, ||Axk � b|| is the euclidean norm of Axk � b, and dk is a scalar satisfyingthe relation 0 < dk 6 2. This scalar is set to 2 at the start of the procedure and is halved whenever the bound does not improvein 20 consecutive iterations.

The procedure can be written in a formal way as follows:

Step 1: Maxiter = 200, d = 2, ZBestF = +1, k = 1, kk = 1, ZBestLB = 0, k ¼ k� ¼ f0g.Step 2: Solve the problem [P2], and calculate the objective function of [P-LR]. Let ZLR be the solution value of [P-LR]. If

ZLR > ZBestLB, let ZBestLB = ZLR, kk = 1, k ¼ k�, otherwise kk = kk + 1.Step 3: Perform the obtaining feasible solution process. Let ZFeas be the objective function value of the feasible solution. If

ZFeas < ZBestF, let ZBestF = ZFeas. If ZBestF � ZBestLB < 1, STOP.Step 4: Let k = k +1. If k > Maxiter, STOP; otherwise continue.Step 5: If kk > 20, let kk = 1, k ¼ k�, d = d/2, otherwise calculate step size tk and update multipliers kijl.Step 6: If kijl < 0, then set kijl ¼ 0. Go to Step 2.

5.2. Determination of a feasible solution

We propose the following process to find the feasible solution. The indices i and j indicate the quay crane number and theship number, respectively.

Step 1: Let i = 1, j = 1, ZFeas = 0.Step 2: If j > the number of ships, Let i = i + 1, j = 1 and FIRST = 0.If i > the number of quay cranes assigned to the mega-containership, STOP.Step 3: If Qij = 1 (i.e., ship j is serviced by quay crane i), go to step 4. Otherwise go to step 8.Step 4: Get CTMY; the completion time of ship j ’s containers unloading from quay crane i to yard location l as following: IfFIRST = 0, CTMY ¼ Fj þ CMY

BðiÞjl. Otherwise, CTMY ¼ CTMY þ CMYBðiÞjl. And let FIRST = 1.

Step 5: Get BTYFj ; the beginning time of ship j ’s containers loading from yard block l to feeder ship j ’s berthing position as

following: If Fj < CTMY ;BTMYj ¼ CTMY and yjl = 0. Otherwise, BTMY

j ¼ Fj and yjl = Fj � CTMY.Step 6: Get CTYF

j ; the completion time of ship j ’s containers loading from yard block l to feeder ship j ’s berthing position asfollowing:

CTYF ¼ BTFY þ CMY :
j j BðjÞjl
Step 7: Calculate the objective function of a feasible solution,

Z ¼ Z þ CMY þ CTYF � F :
Feas Feas j BðiÞjl j
Step 8: Let j = j + 1 and go to step 2.

6. Computational experiments

6.1. Experimental design

The solution procedure is coded in the ‘‘C” language. Problems used in the experiments are generated randomly, butsystematically.

We consider two types of terminal layout shown in Fig. 3. The mega-containership is serviced at berth 2 in the linear type,and also is serviced at berths 2 and 3 in the indented type. We assume that the average intervals of feeders’ completion timefor unloading are 1 hour and 2 hours. If we consider the problem for a long time period, the arrival time of the mega-con-tainership may not be so important. The lagrangian relaxation is used to solve the problem. For this reason, the number ofships with relaxed constraints depends on the arrival time of the mega-containership AM, and it means that better solutionsare likely obtained, when AM gets closer to the arrival time of the last feeder ship. Assuming AM has a unique value in ourexperiments, we performed four computations with different values for AM. In our experiments AM was set to the first feedership’s completion time of unloading operation, 1/3 and 2/3 of time duration between the first and last feeder ships’ comple-tion time of unloading operation, and last feeder ships’ completion time of unloading operation, indexed by time 1, 2, 3 and 4as shown in Fig. 4.

In this problem, it is assumed that only one quay crane is assigned to each feeder ship. For this reason, between 50 and250 containers are stowed to each feeder ship. Although the maximum number of containers flow from the mega-container-ship to feeder ships might be up to 10,000, we assume that about half of 10,000 containers are dealt with. Thus the mega-containership calling once treats the containers for about thirty feeder ships.

Berth 1 Berth 2 Berth 3 Berth 4

(a) Linear type

Berth 1 Berth 4

Ber

th 3

Berth 2

(b) Indented type

Fig. 3. Terminal layouts (a) linear type (b) indented type.

Time 1 Time 2 Time 3 Time 4

4emiT3emiT2emiT1emiT

Average interval = 1 hour

Average interval = 2 hours

Mega-containership’sarrival AM

Mega-containership’sarrival AM

Feeder ships’ completiontime of unloading

Feeder ships’ completiontime of unloading

Fig. 4. Arrival time of the mega-containership.


With respect to the handling time for each feeder ship’s container from the mega-containership to the yard block, we as-sume that it takes 1.2 min to handle a container at the yard blocks closest to either the locations of the quay crane assignedto the mega-containership or the feeder ships’ berthing location. This time is obtained by the following formulation with thenumber of containers handled and the distances from yard blocks to the location where a relevant feeder ship is berthing:

CYFBðjÞjl ¼ NCj � 1:2 � ð1þ lnðDBðjÞl=DBðjÞMINÞÞ; ð13Þ

where NCj, the number of containers loaded into feeder ship j; DB(j)l, the distance from the yard block l to the location wherefeeder ship j is berthing B(j); DB(j)MIN, the shortest distance among the yard blocks l to the location where feeder ship j is ber-thing B(j).

Table 1Handling times for feeder ships (hours).

Avg. and Std. among feeder ships Shortest Longest Average

Avg. Std. Avg. Std. Avg. Std.

Mega-containership to yard blocks Linear type # 1 2.5 1.1 5.6 2.4 3.4 1.5# 2 2.5 1.1 5.7 2.4 3.4 1.5# 3 2.5 1.1 5.6 2.3 3.4 1.5# 4 2.5 1.1 5.7 2.3 3.5 1.5# 5 2.4 1.0 5.6 2.3 3.4 1.4

Indent type # 1 2.5 1.1 7.1 2.9 5.2 2.2# 2 2.5 1.1 7.2 2.9 5.3 2.2# 3 2.5 1.1 7.2 2.9 5.3 2.2# 4 2.5 1.1 7.2 2.9 5.3 2.2# 5 2.4 1.0 7.1 2.8 5.2 2.1

Yard blocks to feeders Linear type # 1 2.4 1.0 6.1 2.5 3.8 1.7# 2 2.5 1.1 6.2 2.6 3.9 1.7# 3 2.5 1.1 6.1 2.5 3.8 1.7# 4 2.5 1.1 6.3 2.5 3.9 1.7# 5 2.4 1.0 6.1 2.4 3.8 1.6

Indent type # 1 2.5 1.1 6.7 2.8 4.8 2.0# 2 2.5 1.1 6.7 2.7 4.8 1.9# 3 2.5 1.1 6.7 2.7 4.8 1.9# 4 2.5 1.1 6.8 2.7 4.9 1.9# 5 2.4 1.0 6.6 2.6 4.8 1.8


Also, it is assumed that the handling time for each feeder ship’s container from the yard block to the location where eachfeeder ship is berthing, can be formulated as follows:

CMYBðiÞjl ¼ NCj � 1:2 � ð1þ lnðDBðiÞl=DBðiÞMINÞÞ; ð14Þ

where DB(i)l, the distance from the location where quay crane i is assigned to the mega-containership B(i) to yard block l;DB(i)MIN, the shortest distance from the location where quay crane i is assigned to the mega-containership B(i) to yard blocks.

From the formulations (13) and (14), we can obtain the handling time for feeder ships as shown in Table 1. Five patternsof those times are used as the input data. The table shows the shortest, longest and average handling times between the loca-tion where a ship is berthing and the yard blocks where its containers are stored. The average value and standard deviationamong ships are shown. There is hardly any difference in shortest handling times between both layouts. However, the aver-age handling times in the indented type are longer than those in the linear type.

6.2. Computational results

The quality of solutions obtained by our approach, is expressed as the gap between the lower bound and the feasible solu-tion value, and the iteration number at which the process terminates in Table 2. Table 2 parts (a) and (b) show the comple-tion times of the feeder’s loading for 1-h and 2-h average intervals, respectively. The item ‘Pattern #’ refers to twenty fivecombinations by five patterns of completion time for feeder ships’ unloading and five patterns of handling time for feederships.

First, regarding the gaps among the mega-containership arrivals, the more feeder ships complete unloading after themega-containership arrives, the larger gaps become and it is more difficult to obtain good solutions. At time-1 of themega-containership arrival (AM = 1), the gaps range from around 1.0% to over 100%. The range of gaps is wider than withother time epochs of the mega-containership arrival. At AM = 2 or AM = 3, the gaps drop under 1.0% for all cases. Also on aver-age, the earlier the mega-containership arrives, the larger the gap becomes. This can be explained because the relaxed con-straints with respect to the feeder ships are less satisfied at an early arrival of the mega-containership than those at a laterarrival. Note that, the optimal solution is obtained when AM = 4, because this time is the same as the last feeder ship’s arrivaland all relaxed constraints are satisfied for all ships.

Secondly, regarding the gaps between both types of layout, the values in the indented type are smaller than those in thelinear type. The number of quay cranes assigned to the mega-containership in the latter type is higher than in the formerone. This means that more containers loading and discharging operations take place during a specific planning period. How-ever, at AM = 3, more feeder ships have already arrived and are waiting for loading. In this case, the number of quay cranesassigned to the mega-containership in the indented type is higher than those in the linear type. The scale of the problem inthe indented type is larger than those in the linear type. Thus the gaps in the indented type are larger than in the linear type.

Thirdly, comparing the gaps between 1-h and 2-h average intervals of completion time for feeders’ unloading, the valuesin the 1-h interval are smaller than those in the 2-h interval. This can be explained because the terminal situation in theformer arrival rate is more congested than that in the latter during a specific planning period. Thus in a relatively congestedsituation, the gap becomes smaller.

Table 2Solution quality of our proposed approach.

Prob.# Handlingtime

Interval ofcompletiontime

Terminal Layout Linear type Indented type

Mega-containershiparrival AM

1 2 3 4 1 2 3 4

(a) 1 h average interval of completion time for feeder’s unloading1 1 1 GAP (%) 4.7 0.4 0.0 0.0 1.6 0.2 0.1 0.0

No. of iterations 200 200 1 1 200 200 200 12 1 2 GAP (%) 2.2 0.0 0.0 0.0 0.5 0.5 0.3 0.0
























No. of iterations 200 200 1 1 200 200 1 1Average GAP (%) 39.9 0.8 0.0 0.0 10.7 0.8 0.1 0.0

No. of iterations 200.0 200.0 24.9 1.0 200.0 200.0 128.6 1.0

(b) 2 h average interval of completion time for feeder’s unloading1 1 1 GAP (%) 180.6 9.6 0.1 0.0 99.6 4.3 0.3 0.0






No. of iterations 200 200 200 1 200 200 200 1(continued on next page)


Table 2 (continued)

Prob.# Handlingtime

Interval ofcompletiontime

Terminal Layout Linear type Indented type

Mega-containershiparrival AM

1 2 3 4 1 2 3 4

7 2 2 GAP (%) 373.6 111.5 0.6 0.0 258.6 52.6 1.1 0.0No. of iterations 200 200 200 1 200 200 200 1



















Average GAP (%) 269.6 51.3 0.3 0.0 163.4 23.6 0.4 0.0No. of iterations 200.0 200.0 200.0 1.0 200.0 200.0 200.0 1.0

GAP (%) = (Feasible solution value � lower bound) � 100/lower bound.No. of iterations means the number of iterations at which the process terminates.


To evaluate the feasible solution values obtained by our proposed approach, they are compared with values by two otherstrategies.

In a first strategy, we consider the waiting times for feeder ships until the start of the loading operation to feeder ships inthe objective function. In order to investigate the effect, we calculate the objective function using the solution in which weobtain the containers’ yard location from the minimization problem of handling times without the waiting time of feeders.This minimization problem is formulated as follows:

½P�MINH� MinimizeX

i2QC

X

j2V

X

l2YP

ðCMYBðiÞjl þ CYF

BðjÞjlÞQijxjl ð15Þ

Subject toX

j2V

xjl ¼ 1 8 l 2 YP ð2ÞX

l2YP

xjl ¼ 1 8 j 2 V ð3Þ

xjl 2 f0; 1g 8 j 2 V ; l 2 YP ð5Þ

Problem [P-MINH] is a two-dimensional assignment problem, for which the optimal solution can be obtained easily. Withthis solution, we calculate the waiting time and the objective function value using the algorithm to obtain a feasible solutionas described in Section 5.2.

(a) 1-hour average interval (b) 2-hours average interval

Fig. 5. Total service time including waiting time for feeder ships.


(a) 1-hour average interval (b) 2-hours average interval

Fig. 6. Total waiting time for feeder ships.



In a second strategy, we aim to compare with the objective function obtained by the minimization of the work of the quaycranes assigned to the mega-containership. The containers from the mega-containership are stored in yard blocks close tothe location where quay cranes are assigned to the mega-containership. Then, by using this result, similar to the first strat-egy, we calculate the waiting time and the objective function value using the algorithm obtaining the feasible solution asdescribed in Section 5.2.

Fig. 5 shows the sum of the handling times from the mega-containership to feeder ships plus the waiting times until thestart of the loading operation by our proposed approach, by the minimization of handling times (strategy 1), and by the min-imization of the work of the quay cranes assigned to the mega-containership (strategy 2). The results for 1-h and 2-h averageintervals of completion time for feeders’ unloading are shown in Fig. 5a and b, respectively. The item ‘Pattern #’ refers to thecompletion times for feeders’ unloading generated by random numbers. The values show the average of the results by eachseries of handling times. We also show the average among twenty five patterns by five handling times and five completiontimes in the lower line. In Fig. 6, the waiting times until the start of the loading operation are shown for the three strategies.

First, we compare the total service time from the mega-containership to feeders including waiting for feeders until begin-ning the loading operation (hereafter called the total service time) among the above-mentioned three strategies. In mostcases, those times by our proposed approach are shorter than those by the other strategies.

To look at the results in detail, in the linear type, at AM = 1, there is not so significant difference between the total servicetimes by our proposed approach and those by minimization of total handling time. At AM = 2, AM = 3 and AM = 4, the times byour proposed approach are shorter than those by the other strategies, and those by minimization of quay cranes’ works as-signed to the mega-containership are the longest. This may be explained because the earlier the mega-containership arrives,the less feeder ships arrive after the mega-containership’s arrival. In the case of less feeder ships waiting, the total servicetimes by minimization of total handling time are shorter than the other strategies.

In the indented type, the total service times by our approach are shorter than others in most cases. At AM = 1 and AM = 2,the times by the minimization of total handling time are shorter than or similar to those by minimization of quay cranes’work assigned to the mega-containership. At AM = 3 and AM = 4, the times by the minimization of total handling time arelonger than or similar to those by minimization of quay cranes’ works assigned to the mega-containership. The less feederships complete unloading operation before AM = 1 and AM = 2, the total waiting times for feeders are shorter as shown inFig. 6. Thus the minimization of total handling time has a large effect on decision making regarding container storageallocation.

Considering the waiting times as shown in Fig. 6, in all cases, the waiting times obtained by our proposed approach areshorter. Minimization of handling time leads to longer times. This may be explained because our proposed approach consid-ers the waiting time within the objective function.

Secondly, we compare the total service time from the mega-containership to feeder ships and the waiting time for feederships between both types of layout. In most cases, the total service times in the indented type are longer than those in thelinear type. At AM = 3 and AM = 4, the total service times in the linear type are similar to those in the indented type. Looking atthe total waiting times for feeder ships, those by our approach and the minimization of total handling times in the indentedtype are shorter than those in the linear type. This can be explained because more quay cranes are assigned to the mega-containership in the indented type.

Thirdly, we compare the total service time and the waiting time among the arrival times of the mega-containership. Thelater the mega-containership arrives, the more ships have to wait until its arrival time. For this reason, the total waitingtimes are longer, and also the total service times are longer.

Additionally, we compare the total service time and the waiting time between two average intervals of feeders’ comple-tion time for unloading. As shown in Fig. 4, the longer the time interval, the longer feeder ships wait until the mega-contai-nership arrives. By this, the total service times in the 2-h average interval are longer than those in the 1-h average interval. Inthis figure AM = 3 in the 1-h average interval case, and AM = 2 in the 2-h interval case are taken identical. The length of timeperiod from the first feeder ship arrival to the former case is similar to that from the first feeder ship arrival to the latter case.However, as shown in Fig. 4, the congestion on berths in the former cases are different from that in the latter cases. Withhigher congestion, more ships are waiting and the total waiting time is longer. Also the total waiting times and the total ser-vice times of the former are longer than those of the latter.

7. Conclusions

The port where the mega-containership make a call, plays an important role in the connection between a relevant shipand others. This paper addresses the storage planning problem of transshipment containers which are dealt with at themega-port, and suggests a solution procedure. The contribution of this paper to the literature is that, we consider the storageproblem at the terminal where the mega-containership calls under a container flow which is different from the previousresearch.

In order to have large effects on the introduction of the mega-containership, a relevant ship will be serviced on the high-est priority. However, if the operation for the relevant ship is maintained on schedule which is determined in advance,we can pay attention to the time of connection between the mega-containership and feeder ships. When the service timeincluding waiting time of feeders is kept as short as possible, it might be connected to succeeding ships smoothly. Thus


we consider the problem as the objective function including the waiting of feeders. The results, produced by this paper, un-der simplification of the real situation, show that the quality of solution by our proposed approach is very promising. Theproblem and the approach need, however, further study under more realistic constraints.

Acknowledgements

This research is partially supported by the MEXT under the Grant-in-Aid for Young Scientists-B Grant 17710136 and bythe JSPS under the Grant-in-Aid for Exploratory Research Grant 19656232.

References

Bazzazi, M., Safaei, N., Javadian, N., 2009. A genetic algorithm to solve the storage space allocation problem in a container terminal. Computers & IndustrialEngineering 36, 1711–1725.

Fisher, M.L., 1981. The lagrangian relaxation method for solving integer programming problems. Management Science 27, 1–18.Imai, A., Nishimura, E., Papadimitriou, S., 2001. The dynamic berth allocation for a container port. Transportation Research Part B 35, 401–417.Imai, A., Nishimura, E., Current, J., 2007. A lagrangian relaxation-based heuristic for the vehicle routing with full container load. European Journal of

Operational Research 176, 87–105.Kim, K.H., Bae, J.W., 1998. Re-marshaling export containers in port container terminals. Computers & Industrial Engineering 35, 655–658.Kim, K.H., Kim, D.Y., 1994. Group storage methods at container port terminals. In: The American Society of Mechanical Engineers, 75th Anniversary

Commemorative Volume, MH-vol. 2. The Material Handling Engineering Division, pp. 15–20.Kim, K.H., Kim, H.B., 1999. Segregating space allocation models for container inventories in port container terminals. International Journal of Production

Economics 59, 415–423.Kim, K.H., Park, K.T., 2003. A note on a dynamic space-allocation method for outbound containers. European Journal of Operational Research 148, 92–101.Kim, K.H., Park, Y.M., Ryu, K.R., 2000. Deriving decision rules to locate export containers in container yards. European Journal of Operational Research 124,

89–101.Kozan, E., Preston, P., 1999. Genetic algorithms to scheduling container transfers at multimodal terminals. International Transactions in Operational

Research 6, 311–329.Lee, L.H., Chew, E.P., Tan, K.C., Han, Y., 2005. An optimization model for container terminal stacking yard management. In: Proceedings of The First

International Conference on Transportation Logistics, Singapore, 27–29 July 2005 (available in CD-ROM).Lim, A., Xu, Z., 2006. A critical-shaking neighborhood search for the yard allocation problem. European Journal of Operational Research 174, 1247–1259.Zhang, C., Liu, J., Wan, Y.W., Murty, K.G., Linn, R.J., 2003. Storage space allocation in container terminals. Transportation Research Part B 37, 883–903.

Transportation Research Part E - VLIZ · Container storage and transshipment marine terminals Etsuko Nishimuraa,*, Akio Imaia, Gerrit K. Janssensb, Stratos Papadimitriouc a Graduate

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