Anchorage-Ship-Berth-Yard Link Modeling in Container Port
Branislav DRAGOVIĆ1,2*, Nam Kyu PARK3 and Zoran RADMILOVIĆ4
1)
3)
4)
*
Maritime Faculty, University of Montenegro, Maritime Transport and Traffic Division
2)
Korea Maritime University, Department of Logistics System Engineering
Department of Distribution Management, Tongmyong University, 535, Yongdang-dong, Nam-gu,
Busan, 608-711, Korea, Tel.: 82-51-610-8481; Fax: 82-51-610-8499; e-mail: nkpark@tit.ac.kr
Faculty of Transport and Traffic Engineering,University of Beograd,Vojvode Stepe 305, 11000 Beograd
Tel./Fax: +381-11-309-1324; e-mail: z.radmilovic@sf.bg.ac.yu
Corresponding Author. Address: Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor,
Montenegro, Tel.: +381-82-303-188; Fax: +381-82-303-184, e-mail: branod@cg.ac.yu; bdragovic@cg.yu
Address in Korea: 1, Dongsam-dong, Youngdo-gu, Busan, 606-791, Korea; e-mail: bdragovic@hhu.ac.kr
Abstract
This paper gives a Anchorage-Ship-Berth-Yard (ASBY) link modeling methodology based on statistical analysis of
container ship traffic data obtained from the Pusan East Container Terminal (PECT). Implementation of the presented
procedure leads to the creation of a simulation algorithm that captures ASBCY link performance well. All the main
performances of the ASBY link are given. As a ASBY link at a container terminal is the large and complex system, a
performance model has to be developed. The basic approach used consists of two models. Results from both models are
compared with each other. The first is a simulation model adapted to the problem of analysing ship movements in port.
The second model applies the results of the queueing model to an analytically formulated average container ship cost
function in port. The aim of this function is to minimize average container ship costs in port, including the allocation
planning of berths/terminal and quay cranes/berth. The analytical model has fewer inputs and requires less computational
times, whilst the simulation model can handle more practical situations with more manipulated variables and less
constraints. Both the simulation and analytical models were applied to existing in Korea Container Terminals.
Keywords: Anchorage-ship-berth-yard link, Simulation and analytical modelling, Model validation, Average ship cost
Topic Area: A2 Maritime Transport and Ports
1. Introduction
As a anchorage-ship-berth-yard link at a container terminal is the large and complex system, a performance
model has to be developed. This can be an analytical model, which uses mathematical concepts and
mathematical notations to describe the processes at the anchorage-ship-berth-yard link. In contrast with it, a
simulation model is basically a computer program, which mimics the important aspects of the studied link.
Since the container port facilities are very expensive to run and purchase, it is very important to determine
whether the existing container terminal capacities are large and efficient enough to handle the changeable
container flows during the considered periods of time. Determining the effect of changes in throughput, as
well as the influence of various operational, technological and economic aspects on efficiency of container
port operations, have been widely analyzed by using port analytical and simulation models. The basic
approach used consists of two models. The first is a simulation model adapted to the problem of analysing
ship movements in port. The second model applies the results of the queueing model to an analytically
formulated average container ship cost function in port. The aim of this function is to minimize average
container ship costs in port, including the allocation planning of berths/terminal and quay cranes/berth.
Numerical results and computational experiments are reported to evaluate a study on the improvement of the
calculation system of optimal throughput per berth in Korea Container Terminal.
This paper describes the criteria for the combined evaluation of important parameters of the total port cost
function such as: ship service and waiting time, the number of berths and related average container ship cost
in port and the optimal combination of berths/terminal and quay cranes/berth. These parameters are the basis
of simulation and analytical approaches used in this paper. Furthermore, the advantage of simulation modeling
over analytical modeling of ship-berth link is that it allows a higher level of detail and avoids too many
simplifications. In analytical models, the level of detail is often restricted by the limited expressiveness of the
analytical method, while for simulation models it is only restricted by the available time and resources.
This paper is organized as follows. In Section 2 we review the related literature and discuss main topics and
problems analyzed in the literature. Section 3 deals with the models for the analysis of ship movement in a
port by using simulation and analytical approaches. Also, this section is concerned with the evaluation of
functional estimation models in container port. In Section 4, numerical results and computational experiments
are reported to evaluate the efficiency of the models for PECT. The results, analysis and conclusions given in
this section are intended to provide guidance on achieving time efficiency and accuracy in the modeling of
anchorage-ship-berth-yard link and calibration of ASBCY link simulation and analytical models for PECT.
Finally, in Section 5 we conclude by summarizing the results and contributions of this paper.
2. Literature review
Simulation models have been used extensively in the planning and analysis of ASBCY in port. Many
different simulation models regarding port operation, especially ASBCY link planning, have been developed
in papers which are explained below. These models are coded in different simulation languages. In addition,
Kozan (1997) gives a review on recent analytical and simulation models.
Tugcu (1983) used a port simulation model to aid investment planning for the Istanbul seaport. El Sheikh et
al. (1987) developed a simulation model to help the planning of future berth requirements of a third-world
port. Chung, Randhawa, and McDowell (1988) proposed a method that uses buffer space to reduce container
loading times and optimize equipment utilization, and a simulation was developed to justify their method.
Silberholz, Golden, and Baker (1991) employed simulation to study the impact of work crew schedules on
container port productivity. Hassan (1993) presented a simulation to be used as a decision support tool for
evaluating and improving port activities. Ballis and Abacoumkin (1996) developed a simulation with
animation to simulate the operational activities of a container terminal with straddle carriers. Ramani (1996)
developed a simulation to support the logistics planning of seaports. The simulation provided estimates of port
performance indicators such as berth occupancy, ship output and ship turnaround times for various strategies.
Merkuryev et al. (1998) used simulation to improve logistics processes at Riga Harbour Container Terminal.
Merkuryeva et al. (2000) considered simulation of containers processed at the Baltic Container Terminal in
Riga as a basic simulation research, and then its complementing by a metamodelling study is discussed.
Nevins et al. (1998) simulated the operations of a seaport, and provided detailed statistics on seaport
throughput and resource utilization. Gambardella, Rizzoli, and Zaffalon (1998) developed models of an
intermodal container terminal to aid container allocation in the terminal yard, resource allocation and
operations scheduling. Thiers and Janssens (1998) used a port simulation model to investigate the hindrance
of a river quay. Bruzzone and Signorile (1998) employed genetic algorithms and simulation to make strategic
decisions about resource allocation and terminal organization. Yun and Choi (1999) proposed a container
terminal simulation model using an object-oriented approach. Gambradela et al. (2001) presented a solution to
the problems of resource allocation and scheduling of loading and unloading operations in a container
terminal. Legato and Mazza (2001) focused on the berth and allocation of berths to arriving ships with
queueing network based on the model which is simulated by Visual SLAM software in various scenarios.
Their model was tested with data from Gioia Tauro container terminal. Key issues of the application of
modeling and simulation for the management of the Malaysian Kelang container terminal are discussed in
paper by Tahar and Hussain (2000). Nam et al. (2002) examined the optimal size of the Gamman Container
Terminal in Pusan, in terms of berths and quay cranes using the simulation analyses which were performed in
four scenarios, representing different operational patterns. Shabayek and Yeung developed simulation model
employing the Witness program to analyze the Hong Kong’s Kwai Chung container terminal performance. It
is shown to provide good results in predicting the actual operation system of the terminal. Kia et al. (2002)
investigated the role of computer simulation in evaluating the container terminal performance in relation to its
handling techniques and their impact on the capacity of terminal. Pachakis and Kiremidjian (2003) presented a
ship traffic modeling methodology based on statistical analysis of container ship traffic and cargo data
obtained from a port in the United States. Sgouridies et al. (2003) focused on the simulated handling of
incoming containers. Results on the service level, i.e., service times, utilization factor, and queues, are
generated for analysis. Demirici (2003) developed simulation model to analyze port operations and was run
especially for investment planning. This paper discussed the simulation model results of Trabzon port. Bielli
et al. (2005) proposed simulation model which can improve ports efficiency and they gave the architecture
components that are implemented with Java. Simulator calibration and validation were also presented in the
paper at the Casablanca container terminal. van Renzburg et al. (2005) described a computer simulation model
of ocean container carrier operations. Their simulation is called SimSea. Ali Alattar et al. (2006) simulated
different condition to find out the queue of containers at the port and also analyses the effect of increase in the
facilities at the port to reduce this queue. Dragović et al. (2005a) gave the simulation model results for ship
berth link of the Pusan East Container Terminal (PECT). They developed simulation model which can be used
by the port management to improve different operations included in the process of ship service at the shipberth link. Dragović et al. (2005b) developed simulation models of ship-berth link with priority service in
container port. The ship berth-link performance for five alternative strategies was evaluated, and system
behavior observed. The results revealed that simulation modeling is a very effective method to examine the
impact of introducing priority, for certain class of ships, on the ship-berth link performance at PECT. In order
to determine the performance evaluation of ship-berth-CY link in port Dragović et al. (2006a,b,c) proposed
two models based on simulation and queueing theory, respectively. Numerical results and computational
experiments were reported to evaluate the efficiency of the models for PECT.
Numerous studies have been conducted regarding the improvement of the efficiency of ship operations or
berth and QC scheduling and planning problem in port. Plumlee (1966) has discussed the problem of optimum
number of berths such as that the annual cost of the time that ships spend waiting for a berth plus annual cost
of vacant berths and related facilities that stand idle waiting for a berth are on minimum. Nicolaou (1967,
1969) has defined the port performance measures as “degree of occupancy” and “degree of congestion”,
“percentage occupancy” and “percentage congestion”. Noritake and Kimura (1983) extended the application
of queueing models to the ship-berth link in relation with the researches of Plumlee (1966) and Nicolaou
(1967). In addition, they defined the approaches for determination of optimal number of berths and optimal
berth capacity. Noritake (1985) has considered the port congestion in relation with the cost and the optimal
berth utilization. Schonfeld and Sharafeldien (1985) were the first who discussed the problem of optimal berth
and crane combinations in containerports. In the same context, Huang, Chishaki and Li (1995) and Huang et
al. (1997) assumed a similar form of equations to measure the transfer time a ship spends in a port.
Furthermore, Chu and Huang (2002) discussed the degrees of interferences for multiple cranes that work
simultaneously in the port of Kaohsiung. They found that a degree of mutual crane interference among
terminals depends on operational modes applied in the container yard. Haralambides et al. (2002) presented a
general overview of dedicated container terminals (DCTs) and stresses, trough the use of a generalized port
cost function, that one of the main factors that could explain this development is the increasing gap between
the objectives of ports and those of shipping lines. The main implications of a DCT, from a port viewpoint,
are analysed next through the employment of a simple queueing model. Yamada et al. (2003) presented a
mathematical model with the queuing theory for determining optimal container handling systems so that the
total cost incurred in a container terminal is minimised. A simulation model was also developed to investigate
the performance of the mathematical model.
3. ASBY link modeling
The crucial terminal management problem is optimizing the balance between the shipowners who request
quick service of their ships and economical use of allocated resources. Since both container ships and
container port facilities are very expensive, it is desirable to utilize them as intensively as possible. The main
problem in analytical modeling of container terminal relates to the fact that they lose in detail and flexibility,
and so they simplify the real situation. On the other hand, simulation modeling is better than analytical one in
representing random and complex environment of container terminal.
3.1. Simulation modeling
Most container terminal systems are sufficiently complex to warrant simulation analysis to determine
systems performance. The GPSS/H simulation language, specifically designed for the simulation of
manufacturing and queueing systems, has been used in this paper (Schriber 1991).
In order to present the ASBCY link processes as accurate as possible the following phases need to be
included into simulation model (Dragović et al. 2005a,b and 2006a,b,c):
 Model structure: ASBCY link is complex due to different interarrival times of ships, different dimensions
of ships, multiple quays and berths, different capabilities of QCs and so on. The modeling of these systems
must be divided into several segments, each of which has its own specific input parameters. These segments
are closely connected with the stages of ship service (Figures 1, 2 and Figures 4 and 5).
 Data collection: All input values of parameters within each segment are based on data collected in the
context of this research. The main input data consists of ship interarrival times, lifts per ship, number of
allocated QCs per ship call, and QC productivity. Existing input data are subsequently aggregated and
analyzed so that an accurate simulation algorithm is created in order to evaluate ship-berth-yard link
parameters.
 Inter-arrival times of ships: The inter-arrival time distribution is a basic input parameter that has to be
assumed or inferred from observed data. The most commonly assumed distributions in literature are the
exponential distribution (Demirci 2003; Pachakis and Kiremidjian 2003; Dragović et al. 2006a,b); the
negative exponential distribution (Shabayek and Yeung 2002) or the Weibull distribution (Tahar and
Hussain 2000; Dragović et al. 2005a,b).
 Loading and unloading stage: Accurate representation of number of lifts per ship call is one of the basic
tasks of ship-berth link modeling procedure. It means that, in accordance with the division of ships in
different classes, the distribution corresponding to those classes has to be determined.
 Number of QCs per ship: The data available on the use of QCs in ship-berth link operations have to be
considered too, as this is another significant issue in the service of ships. This is especially important as total
ship service time depends not only on the number of lifts but also on the number of QCs allocated per ship.
Different rules and relationships can be used in order to determinate adequate number of QCs per ship. On
the other hand, in simulation models, it is enough to determine the probability distribution of various
numbers of QCs assigned per ship.
 Flowchart: Upon arrival, a ship needs to be assigned a berth along the quay. The objective of berth
allocation is to assign the ship to an optimum position, while minimizing costs, such as berth resources
(Frankel 1987). After the input parameter is read, simulation starts by generating ship arrivals according to
the stipulated distribution. Next, the ship size is determined from an empirical distribution. Then, the
priority of the ship is assigned depending on its size. The ship size is important for making the ship service
priority strategies. For the assumed number of lifts per ship to be processed, the number of QCs to be
requested is chosen from empirical distribution. If there is no ship in the queue, the available berths are
allocated to each arriving ship. In other cases ships are put in queue. The first come first served principle is
employed for the ships without priority and ships from the same class with priority. After berthing, a ship is
assigned the requested number of QCs. In case all QCs are busy, the ship is put in queue for QCs. Finally,
after completion of the loading and unloading process, the ship leaves the port. This procedure is presented
in the algorithms shown in Figures 2 and 3.
In order to calculate the ship-berth-yard performance, it is essential to have a through understanding of the
most important elements in a port system including ship berthing/unberthing, crane allocation per ship, yard
tractor allocation to a container and crane allocation in stacking area. As described in Figure 2 - process flow
diagram of the terminal transport operations, the scope of simulation, strategy and initial value and
performance measure will have to be defined. In addition, the operational aspect such as machine failures
having a direct impact on ship, crane and vehicle will have to be considered. To move containers from apron
to stacking area, four tractors are provided for each container crane. It takes average 7.5 minutes from apron to
stacking area including unloading/loading time by transfer crane. The average distance between apron and
stacking area is assumed to be 850 meters.
Fig. 1. Import/Export containers take a number of short trips in traveling from containerships to their eventual
recipients/loading, with specialized equipment handling each trip (see Figures 4 and 5)
Fig. 2. Port operation with ship movement in port and
process flow diagram of the terminal transport operations
Fig. 4. Flow of inbound/outbound containers
(1-Container ship; 2-QC; 3-Straddle carrier; 4-CY; 5-RTG;
6-Land carriers; 7-Terminal/land truck
Fig. 3. Flowchart for
a ship arrival/departure
Fig. 5. Flow of inbound/outbound containers
(1-Container ship; 2-QC; 3/8 Terminal/land truck; 4-Straddle
carrier; 5-RTG; 6-CY; 7- Land carriers
3.2. Analytical modeling
Queueing theory (QT) models for analyzing movements of ships in port is proposed and shown in Fig. 2,
which indicates that each symbol has the following meaning:  – average ship arrival rate in ships/hour;  –
average ship service rate in ships/hour; n b – number of berths per terminal; nc – number of quay cranes (QCs)
per berths; ns – number of ships present in port; ns – average number of ships present in port with n b berths in
the period T; ns w – average number of ships waiting for berths with n b berths in the period T; ns b – average
number of ships served at n b berths in the period T; t w – average waiting time in hours/ship; t s – average
service time in hours/ship; t ws – average time that ships spend in port in hours/ship; t du – maneuvering time
(berthing/unberthing) = const. in hours; n con – number of containers loading/unloading per containership;
rcon – QC move time in hours/container; t c – ships’ loading/unloading time in hours/containership; k c – QC
interference exponent;  – ship traffic intensity and  – berth utilization factor.
In the analysis of various aspects of average time that ships spend in port, tws, including ns, nb, , , nc and ,
(e.g., Plumlee (1966); Nicolaou (1967, 1969); Noritake (1985); Noritake and Kimura (1983, 1990); Taniguchi
et al. (1999) and Yamada et al. (2003)) defined t ws as the sum of the average waiting time and average
service time, i.e. t ws  t w  t s . These studies have made it clear that the types (M/M/nb) and (M/Ek/nb) of
queuing theory models are most practical to explain ship movements in port.
The average service time, t s , ts  1 /  , where   tc  tdu  , includes ships loading/unloading time t c ,
in hours per container ship, expressed as
1
tc  (ncon  rcon ) / nc  c
k
(1)
where
kc  ln nconrcon   ln tc  / ln nc 
(2)
It follows that
1 / kc
1 / kc
n r 
nc   con con 
 tc 




ncon rcon 


1

   t du 


1 / kc
 n r 
  con con 
 1  t du 
(3)
Further, it can be shown that
tws  tw  ts
(4)
where
tw 
tw 

  

nb
 nb 1 1   ns 1   nb n  
b


    
 n

n
!

n
!

n



 b
b 
 s 0 s  

2
nb  1!nb    
n
1
(5)
b
nb
 nb1 1    n s

nb 1

nb  1! nb          1 nb 1nb



n
!

n
!

n



b
b
 ns 0 s  

(6)
2
tw 
n
b
 nb 1 1    n s  n
nb 1
nb  1! nb            b nb    
 n s  0 ns !    
2
(7)
for the (M/M/nb) model.
For minimizing tws, the Eq. (5) can be transformed in the form
nb
tw 

   


ns
nb
 nb 1
1 
1 
2
nb  1!nb           nb 
 n s  0 ns !   
nb !    n 
b









(8)
Also, the Eq. (3) becomes
1/ kc
 n r 
nc   con con 
   tdu 
(9)
Furthermore, the above form of t w will be used for minimizing the total port cost per container ship.
On the other side, the difference equations in the steady-state condition which were obtained by Morse
(1958) refer to the (M/Ek/nb) model. But, there is no theoretical formula which concerns the average time that
ships spend in port. Only some approximation formulae exist, which relate the average waiting time of ships
in the (M/Ek/nb) model to that in the (M/M/nb) model. In this study, formulae due to Lee and Longton (1959)
and Cosmetatos (1975, 1976) have been adapted concerning the average port waiting time of ships (Noritake
1985, Noritake and Kimura 1983 and 1990, Radmilović 1992 and Taniguchi et al. 1999). Accordingly with it,
when the ships service time has an Erlang distribution with k phases, the following equations are obtained
tws  twVc  ts
where Vc 
(10)
11 
  1 - the coefficient of variation of ships service time distribution, and k = the number of
2k

phases of an Erlang distribution;
1/ 2
11   1

4  5nb   2 

  ts
t ws  t w     1  1    nb    nb  1 

2
k
k
32
n





b


(11)
The Eq. (10) related to ((M/Ek/nb)I) and the Eq. (11) related to ((M/Ek/nb)II) for the (M/Ek/nb) model, present
average time that ships spent in port as a function of .
The average time that a ship spends in a port, t ws , increases monotonically as the average arrival rate of
ships,  grows. The value of t ws grows indefinitely as  asymptotically approaches the total capacity of
berths, i.e., nb  .
3.2.1. Berth utilization factor
The berth utilization factor of a container port is determined by the number of berths available, the average
cargo handling rate of each berth, the number of QCs per berth and the optimal combination of
berths/terminal and QCs/berth. Namely, the level of berth utilization caused by the ships calling at a port
depends on the relative magnitudes of the average ship arrival rate, the average ship loading/unloading time
(or the average ship service rate), the number of berths at a terminal and the number of QCs per berth.
In practice, the terminal operator is not much interested in terminal congestion as it ensures high berth
utilization, unless congestion scares away potential carriers. Also, the carrier is only interested in the terminal
operator’s overall cost to the extent that it leads to higher fees being charged.
The interrelationship between berth utilization, congestion, the fixed costs of terminal facilities and the cost
of waiting containerships, permits planners to evaluate terminal and berth capacity in the best interests of the
terminal operators and carriers.
In queueing theory models, the utilization of berth (or the utilization factor) as the product of the average
arrival rate of ships and average service time plays a significant role. Furthermore, the berth utilization factor
for a multiple-berth terminal with n b berths is

t s

nb

.
nb 
(12)
3.2.2. Ship traffic intensity
We use the following symbols: Ncon - total number of container loaded onto and discharged from ships in
port during the period T (in containers); rc - daily rate (in containers/day);
T - time of port operation
considered (in days). Then   N con / T  / ncon  , where Ncon/T is the average number of containers handled in
port per day. Similarly, it is seen that   rc / ncon . Hence,  /   N con / rcT .
The traffic intensity as the product of the average arrival rate of ships and average service time play a
significant role in the queueing models. Accordingly, this parameter with the notation , is called the ship
traffic intensity and it is equal     ts   /  . Further,  as a port operation parameter, i.e. berth occupancy
index, nb nb , can be defined in the following manner (Nicolaou 1967 and 1969, Noritake 1983).
nb 1
  nb n  nb   nb  ns  Pns  ,
b
(13)
ns  0
where  nb - degree of occupancy of port with nb berths.
Furthermore, there holds
N con  rcT  rcTnb nb .
(14)
Then the average number of ships present in port with n b berths in the period T is expressed as
ns 

nb
 ns Pns   nb
ns  0
 Pn   n 
n s  n b 1
s
b
nb
.
(15)
Also, the average number of ships waiting for berths with n b berths in the period T is obtained as
ns w 

 n
n s  nb 1
s
 nb Pns  .
(16)
It follows from (15) and (16) that the average number of ships served at n b berths in the period T can be
written in the form
nsb  ns  ns w 

nb
 n Pn   n  Pn  .
n s 1
s
s
b
n s  nb 1
s
(17)

In view of that
 Pn   1 , the Eq. (17) becomes
s
ns  0
nsb 
nb
nb
n s 1
ns 0
 ns Pns   nb  nb  Pns 
(18)
or
nsb  nb 
nb 1
 n
ns 0
b
 ns Pns  .
(19)
From Eqs. (13), (14) and (19) we have
  ns 
b
N con 
 .
rcT

(20)
3.2.3. Average time that ships spend in port
The determination of a theoretical relationship between the usage of berths within a port and the number of
berths/terminal, as well as the number of QCs/berth will minimize the combined costs of using port facilities
and average ship cost in port (in $/ship). The average time that ships spend in port, t ws , is a factor in ship
costs and shipper costs. These cost functions may be enhanced by taking into account the ships’ earning
capacity as well as theirs direct costs in port.
This investigation shows that total port costs and average time that ships spend in port are relevant criteria
for the planning container port operations, including the optimal combination of berths/terminal and
QCs/berth, as well as the berth utilization.
The substitution of Eq. (12) into Eq. (8) gives
nb  nb
t w  t w   

ns
nb 1





n
n
nb  1! nb 1 nb   nb    b   nb   b n  nb  
 n s  0 ns ! 

nb  n
n
 n 1
nb  1!  n 1nb 2 1   2   nb 
b

  nb nb nb nb nb  1   
ns ! 
s
b
b
 ns 0

1

(21)
According to it, the average waiting time, tw   , is shown as follows:
t w   
 n nb n
b
b
2
ns

 nb 1
1    nb  1!1     nb 

 ns  0 ns !


  nb nb 1 nb 



(22)
for the (M/M/nb):(FCFS//) model, and hence by (4) we have
t ws    tw    t s .
(23)
When the ships service time has an Erlang distribution with k phases, the following equations are obtained
by the substitution of Eq. (12) into Eqs. (10) and (11), respectively:
t ws    tw    Vc  t s
(24)
11   1
4  5nb 1 / 2  2   t
t ws    t w       1  1  1   nb  1
 s
32 nb
  k
2k

(25)
The Eqs. (24) and (25) for the (M/Ek/nb):(FCFS//) model present the average time that ships spend in
port expressed as a function of  .
Since the average time that ships spend in port is a function of parameters, then we get


 
t ws   ns ,  , u nc  c , ncon , t c  t c .
k
(26)
In order to write t ws from (4) as a function of  in the form
tws    tw    ts ,
(27)
we substitute (20) into (8) to obtain
tw   
n 
n 1 n
2
 
b
nb  
nb  1!nb       

 ns  0 ns ! nb! nb    
b
s
n
(28)
b
or
n
tw   
b
nb
 nb 1 ns
nb 
nb  1! nb        

 ns  0 ns ! nb! nb    
Finally, the average waiting time t w   is given as follows
(29)
2
tw   
n
b
 nb 1 ns 
nb  1! nb         nb nb   
 ns  0 ns ! 
(30)
2
for the (M/M/nb):(FCFS//) model.
When the service time of ships obeys the Erlang distribution with k phases, the following equations are
obtained by substitution of the Eq. (30) into Eqs. (10) and (11), respectively:
t ws 
n
b


nb  1! nb          nb nb   
 ns 0 ns ! 
2
nb 1
ns
11  1
   1 
2k  
(31)
1


  
2 2


4

5
n
1
1
1




b
   1  1  1  nb  1
 1
t ws 


nb 1 ns
32


 2 k   k  nb 
 

nb  1! nb   2       nb nb     
n
!
n

0
 s s 
n
b
(32)
The Eqs. (31) ((M/Ek/nb)I) and (32) ((M/Ek/nb)II) for the (M/Ek/nb) model present the average time that ships
spend in port as a function of  .
3.3. Ship loading/unloading operations modeling
The main goal of our modeling of ship-berth link is performance evaluation of ship loading and unloading
operations in term of time and cost. The ship operations in ship-berth link are defined as loading or unloading.
Loading/unloading operations include lifting containers. The container handling operation begins with
preparing the ship for loading/unloading, and then QCs and gangs are assigned for unloading container.
The basic tasks in container port management are berth and QC allocation, container yard and storage
planning, dock labor, controlling container and cargo flows, and the logistics planning of container operations.
In this way the port management would be acting jointly in its own best interests and those of the shipping
lines using the container port. On the other hand, the objective cost function of the formulation is to minimize
total port costs, which includes service berth utilization, waiting time before berthing, length of queue, time
spent in the queue and ship turnaround time.
In general, this model integrates main actual operations of the container terminal by simplifying complex
activities, and these operations are defined according to ship class. In this section, various objects were
observed in the real terminal and model elements. Model elements of the container terminal can be separated
into following group: berth cost in $ per hour, c1  nb cnb ; QCs cost in $ per hour, c2  nb nc cnc ; storage yards
cost in $ per hour, c3  ncontt con aconcy ccy ; transportation cost by yard transport equipment between quayside
and storage yard in $ per hour, c4  nctc ncyc ct ; labor cost for QC gangs in $ per hour, c5  nctl cl ; ships
cost in port in $ per hour, c6  tws cs ; and containers cost and its contents in $ per hour, c7  t ws nrcon cw .
The total cost function, would be concerned with the combined terminals and containerships cost as
TC  i 1 ci .
7
It is necessary to know that only the total port cost function computes the number of berths/terminal and
QCs/berth that would satisfy the basic premise that the service port cost plus the cost of ships in port should
be at a minimum. This function was introduced by Schonfeld and Sharafeldien (1985). We point out that their
solutions may not be as good as ours because we have simulation approach to determine key parameters tw, t s,
, ,  and especially kc. Therefore, to find the optimal solution, their function can be obtained in the
following form


TC  f    nb cnb  nc cnc 


  ncontt con aconcy ccy  nctl cl  ncyc ct   t ws   cs  nrcon cw

(33)
where TC - total port system costs in $/hour; cnb - hourly berth cost in $, ( cnb1 - the initial berth cost, i interest
 
rate,
n y - economic lifetime in years,


cnb  cnb1 i1  i  y / 1  i  y  1  cnbm / 365  24 ;
n
n
cnb m - annual maintenance cost per berth),
cnc - crane cost in $/QC hour; tt co n - average yard container dwell
time, in hours; aconcy - number of m2 of storage yard per container; ccy - storage yard cost in $/m2 hour; ncyc hourly average number of cycle by yard transport equipment between quay side and container yard; ct -
transportation cost between quay side and container yard per cycle in $; tl - paid labor time in hour per gang
per ship, tl  max tc ; cl - labor cost in $/gang hour; cs - ship cost in port in $/ship hour; nrco n - average
payload in containers/ship; cw - average waiting cost of a container and its contents in $/container hour.
By substituting the Eq. (9) into Eq. (33) we obtain
1 / kc
 n r 
TC  f    nb cnb  nconttcon aconcy ccy   con con 
   tdu 
nb cnc  tl cl  ncyc ct   t ws   cs  nrcon cw





(34)
where tws   is defined by the Eq. (4) or the Eq. (31) or the Eq. (32) or it is a result of simulation modeling.
The total port cost function per average arrival rate, denoted as AC, is defined as
AC 
f  


f  

(35)
Since    , we can write
AC 
TC


f  

(36)
or because of by the Eq. (12),   nb  , the Eq. (36) also has the form
AC 
TC
f nb 

.
nb 
nb 
(37)
Eqs. (35), (36) and (37) give the average container ship cost AC, in $/ship. In this study, the trade-off will be
solved in simulative and analytical form by minimizing the sum of the relevant cost components associated
with the number of berths/terminal and QCs/berth and the average arrival rate. These three parameters are key
to the analysis of facility utilization and achieving major improvements in container port efficiency, increasing
terminal throughput, minimizing terminal traffic congestion and reducing re-handling time. A reduction in
operating cost can be achieved by jointly optimizing these parameters. In determining the berths/terminal and
QCs/berth, analysts and planners are concerned primarily with the average time that ships spend in port and
the average cost per ship serviced.
3.4. Algorithm for optimal solution
Suppose that N b is sufficiently large positive integer in the sense that nb  Nb . To obtain the optimal
solution  min for average cost AC, we consider the set of functions f nb    f   , nb  1,2,..., Nb , defined by
Eq. (20).

1
, and   tc  tdu  , it follows

that must be tdu    j . Obviously, the function f j ( ) is defined on the interval tdu , j  , and there holds
For a fixed nb , nb  j , 1  j  Nb , we proceed as follows. Since  
lim
 t du  0
f j     , f j  j    ,
Hence, the function f j   attains its minimum on tdu , j  at some point  j . Furthermore, by substituting
   j in Eq. (9) we obtain
1/ kc
 j
1 . nc  nc
0
 n r 
  con con 
   t 
du 
 j
,
(39)
whence

 
20.  j   tdu  nconrcon nc j 
kc

 j
Since we require that nc must be a positive integer, the corresponding value of  min
which minimizes f j  
is a one of the following values:

 
30.  'j   tdu  nconrcon nc j 
kc
, or 
''
j



  tdu  nconrcon nc j    1
kc
,
where nc j  is given by Eq. (39), and nc j   is the greatest positive integer that is not greater than nc j  .
 j
Furthermore,  min
is a value of x  tdu , j  which minimizes f j ( ) under condition that the corresponding
value nc given by Eq. (9) is a positive integer.
Observe that  'j'  tdu , i.e.  'j'  tdu , j  . On the other hand,  'j  tdu , j , i.e.  'j  j

  j . In this case we obtain
 tdu  ncon rcon nc j  
kc
 j
 min
 ''
 j if


 ' if
 j
 
 
 
 
f  'j  f  'j'
f   f 
'
j
(40)
''
j

50. In the case when  'j  tdu , j , i.e.  'j  j , or equivalently to  tdu  ncon rcon nc j  
 j
 min   .
'
j
Hence, the minimal average cost AC j for given nb  j is AC j 
 j
i 
kc
 .
 j
f  min

Obviously, the optimal solution  min is equal to  min , 1  i  Nb , such that
ACi  min AC j 1  j  Nb .
Hence, nb  i is the optimal value of the number of berths.
Finally, the optimal value of the number of container cranes, nci  , is given by Eq. (39), i.e.
1/ kc
 n r 
nc   i  con con 
  min  tdu 
i 
if
.
  j , we assume
4. Numerical examples
This section gives a ASBY link modeling methodology based on statistical analysis of container ship traffic
data obtained from the Pusan East Container Terminal (PECT). PECT is big container terminals with a
capacity of 2,008,573 twenty foot equivalent units (TEU) in 2005. There are five berths with total quay length
of 1,500 m and draft around 15-16 m, Figs. 6 and 7 (PECT website). Ships of each class can be serviced at
each berth.
Fig. 6. PECT layout, 2005
Fig. 7. PECT layout, 2006
4.1. Input data
An important part of the model implementation is the correct choice of the values of the simulation
parameters. The input data for the both simulation and analytical models are based on the actual ship arrivals
at the PECT for the ten months period from January 1, 2005 to October 31, 2005 (Fig. 6) and January 1, 2006
to October 31, 2006 (Fig. 7), respectively (PECT website, PECT Management reports). This involved
approximately 1,225 ship calls in 2005 and 1,285 in 2006. The ship arrival rate was 0.168 ships/hour in 2005
and 0.176 in 2006. Total throughput during the considering period was 1,704,173 TEU in 2005 and 1,703,662
TEU in 2006. Also, the berthing/unberthing time of ships was assumed to be 1 hour.
The interarrival time distribution is plotted in the Figures 8 and 9. Interestingly, even though ship arrivals of
the ships are scheduled and not random, the distribution of interarrival times fitted very well the exponential
distribution.
Service times were calculated by using the Erlang distribution with different phases. To obtain accurate
data, we have first fitted the empirical distribution of service times of ships to the appropriate theoretical
distribution. It is observed that service time of ships in 2005 follows the 4-phase Erlang distribution (Figure
10), while the 5-phase Erlang distribution fits very well the service time of ships in 2006 (Figure 11).
Goodness-of-fit was evaluated, for all tested data, by both chi-square and Kolmogorov-Smirnov tests at a 5
% significance level.
We have carried out extensive numerical work for high/low values of the PECT model characteristics. Our
numerical experiments are based on different parameters of various PECT characteristics such as: number of
containers loading/unloading from container ship, the QC move time, hourly berth cost, average yard
container dwell time, transportation cost by yard transport equipment between quayside and storage yard,
number of m2 of storage yard per container, storage yard cost, paid labor time, labor cost, ship cost in port and
average payload of containers, presented in Table 1 (PECT Management reports, Korea Maritime Institute
1996). The described and tested numerical experiments contain four segments in relation to the input
variables.
Fig. 8. Distribution of ships inter-arrival times
(IAT) at PECT, 2005
Fig. 9. Distribution of ships inter-arrival times
(IAT) at PECT, 2006
Fig. 10. Service distribution of ships (the 4phase Erlang distribution), 2005
Fig. 11. Service distribution of ships (the 5phase Erlang distribution), 2006
Table 1
Input data – Terminal characteristics
Input data
Year
2005
2006
ncon
rcon
tl
cs
nrco n
nc*
kc
in no. of
containers
897
875
in hours per
container
0.042
0.042
in hours/
gang/ship
15.19
13.75
in $/ship
hour
1161
1161
in contain.
/ship
996
981
2.87
3.27
0.919
0.896
nc* - average number of QCs assigned per ship (Real data and Simulation results); cnb1 = 62 million $; i = .0663; n y - 40, cnb m = 6.2 million $;
cnb = 1215 $; cnc = 38.8 $/QC hour; tt co n = 188 hours; aconcy = 63.9 m2/container; ccy = 0.000292 $/m2 hour; ncyc = 8; ct = 5 $/cycle; cl = 357
$/gang hour;
cw =1.4 $/container hour.
4.2. Validation-verification
For purposes of validation of simulation model and verification of simulation computer program, the results
of simulation model were compared with the actual measurement. Four statistics were used as a comparison
between simulation output and real data: traffic intensity, berth utilization, average service time and average
number of serviced ships. The simulation model was run for 44 statistically independent replications. The
average results were recorded and used in comparisons. After analysis of the port data, it was determined that
traffic intensity and berth utilization are about 2.556 and 64.31%, while the simulation output shows the value
of 2.548 and 64.09% in 2005 and 2.609 and 52.29%, while the simulation output shows the value of 2.664 and
51.64% in 2006, respectively, see Table 2. Average service time shows very little difference between the
actual data and simulation results, that is, 15.35 h and 15.19 h in 2005 and 13.87 h and 13.75 in 2006,
respectively (Table 2). The simulation results of the number of serviced ships completely correspond with the
real data (i.e. the simulation results of the total number of ships are 1,227.02 in 2005 and 1,284.15 in 2006,
and the real data are 1,225 in 2005 and 1,285 in 2006). All the above shows that simulation results are in
agreement with real data.
Table 2
Average service time of ships, traffic intensity and berths utilisation
Results
Real data
Simulation
Analitical
Traffic intensity
2005
2006
2.556
2.609
2.548
2.664
2.811
2.887
Berth utilisation in %
2005
2006
64.31
52.29
64.09
51.64
71.98
59.83
Average service time of ships in hours
2005
2006
15.35
13.87
15.19
13.75
15.38
13.71
The attained agreement of the results obtained by using simulation model with corresponding values of real
parameters has also been used for validation and verification of applied analytical model. In accordance with
it, the correspondence between simulation and analytical results completely shows the validity to the applied
analytical model to be used for optimization of processes of servicing ships at PECT, i.e., at the considered
ASBL link, see Tables 2 and 3.
4.3. Results
The impact of different models is determined by comparing the key performance measures of simulation
and analytical approaches to those of the real data of PECT. Table 2 displays the results, the key measures are
average traffic intensity, berth utilisation and average service time of ships (in 2005 and 2006), while Table 3
shows average time that ships spends in queue (in 2005 and 2006). In addition, Table 3 gives average time
that ships spend in port. According to this, judging from the computational results for some numerical
examples of the models: (M/Ek/nb)I – using the average waiting time t w given by Eq. (10) (for brevity, the
analytical Model I is denoted as AM I) and (M/Ek/nb)II – using average waiting time t w given by Eq. (11) (for
brevity, the analytical Model II is denoted as AM II). It can be confirmed that the Eq. (10) is inclined to
estimate the values of average time that container ships spend in port, i.e. average waiting time of ships.
Table 3
Average time that ships spend in queue and average time that ships spend in port in hours
Simulation results
Analytical
(M/Ek/nb)I (AM I)
results
(M/Ek/nb)II (AM II)
Average time that ships
spend in queue in hours
2005
2006
2.429
1.156
3.191
1.854
2.632
1.295
Average time that ships spend in port
in hours
2005
2006
17.619
14.906
18.571
15.564
18.012
15.005
The average time that ships spend in port for SM is 14.906 h in 2006. This is about 15% less than SM,
17.619 h in 2005. The average time that ships spend in port is 15.005 h for AM II in 2006, about 16% less
than AM II, 18.571 h in 2005. Finally, the average time that ship spends in port for AM I is 15.564 h in 2006,
about 16% less than AM I, 18,571 h in 2005.
4.3.1. Average container ship cost
The results presented here support the argument that average cost per ship or container served, can be easily
obtained by the use of the average cost curves in function of berth utilisation, traffic intensity and QCs/berth.
The described and tested numerical experiments contain more segments in relation to the input variables. All
numerical results presented in Figs. 12 – 15 are obtained by using the input data from Table 1. Simulation
testing (Simulation model (SM)) was than carried out by using the GPSS/H. The solution procedure for AM I
and AM II models was programmed using the MATLAB program.
As expressed in Eqs. (35) and (37), AC can be considered as a function of  and  . Therefore, our
numerical results are given with respect of ,  and QCs. Recall that in our examples the obtained values of
average time that ships spend in port t ws , dominate in the sum defined by the right hand side of Eq. (34). This
shows that t ws plays the most important role for AC given by Eqs. (35) and (37). Figs. 12 – 14 present AC as a
function of  and  . The average costs, i.e. the optimization function of the handling processes at a container
terminal is examined by using SM, AM I and AM II models of the container ports. These models are used for
the operational analysis of the model to minimize the total container port cost, including the costs of dock
labor, transportation cost by yard transport equipment between quayside and storage yard, facilities and
equipment, ships, containers, and cargo.
Fig. 12 shows the optimization function as a function of the variable  while Fig. 13 presents AC as a
function of QCs. Fig. 12 presents how  reduces the average costs per ship for each model. In curve AM I,
the minimum cost per ship served decreases by about 8% in 2006 with respect to 2005. This decrease is about
7.7% in 2006 with respect to 2005 for curve SM from Fig. 12. Finally, in curve AM II from Fig. 12 in 2006,
the minimum cost per ship served decreases by about 7.7% in relation to 2005.
Figure 13 compares the average ship costs in 2005 and 2006 taken by SM, AM I and AM II models at a
PECT. They graphically show the sensitivity of the average ship costs to the various QCs per berth. Figure 13
presents how additional QCs reduce the average costs per ship for each model. In curve SM, the minimum
cost per ship served decreases by about 7.5% in 2006 with respect to 2005. This decrease is about 7.5% in
2006 with respect to 2005 for curve AM I from Figure 14. Finally, in curve AM II from Figure 13, the
minimum cost per ship served decreases by about 8% in relation to 2005.
Fig. 12. Average container ship costs for various berth
utilization (  = 0.1–0.9): 1) Minimum AC in 2005 are
$87,875 (SM,  =0.64); $87,917 (AM I,  =0.72) and
$87,932 (AM II,  =0.72); 2) Minimum AC in 2006
are $81,117 (SM,  =0.516); $81,232 (AM I,  =0.59)
and $81,241 (AM II,  =0.575)
Fig. 13. Average container ship costs for various
QCs/berth (QCs=1-8): 1) Minimum AC in 2005 are
$98,787 for SM (QCs=2.87), $98,876 for AM I
(QCs=3) and $98,651 for AM II (QCs=3.25); 2)
Minimum AC in 2006 are $91,415 for SM
(QCs=3.25), $91,498 for AM I (QCs=3.75) and
$90,852 for AM II (QCs=4.25)
Accordingly, it will be useful to show graphically the range of container capacity which can be optimally
handled with the specific number of berths, i.e. the optimal range of  . For the reason already stated in the
numerical experiments, the average container ship cost AC (in $/ship) has been adapted as a measure to
determine the  and the optimal number of cranes/berth nc ( nc  1,2,...,7 ) for the constant number of
berths/terminal in this study.
Figs. 14 compare the average ship costs in 2005 and 2006 taken SM, AM I and AM II models at a PECT.
These graphically show the sensitivity of the average ship costs to the various values  . Fig. 14 presents how
 reduces the average costs per ship for each model. In curve AM I, the minimum cost per ship served
decreases by about 4.8% (   2.664 ; nc*  3.27 ) in 2006 with respect to 2005 (i.e. $101.805 for   2.548
and nc*  2.87 ). However, the average cost per ship served decreases by about 5% in 2006 in relation to the
2005, see Fig. 14 – curves AM II. This decrease is about 5% in 2006 with respect to 2005 for curve SM from
Fig. 14.
Fig. 15 shows the optimization function AC of two variables nb (nb = 3 , 4, 5) and nc (nc = 1 , 2,…, 7) for
constant value of  . In Fig. 15 obtained results correspond to those from Fig. 14. Still, even in Fig. 15, the
study offers similar results, i.e. the minimum average cost per ship served are $96,383 in relation to $96,383
from Fig. 14 – curve AM II. These results will emphasize the effects of terminal and traffic intensity, average
time that ships spend in port, numbers of QCs/berth, QC productivity and numbers of berths/terminal. These
five parameters are keys to the analysis of the whole container port efficiency and achievement of economies
of scale. However, major improvements in port productivity, quality of service and costs reduction can be
achieved by joint optimizing these variables.
AC [$/ship] x 105
2
1.8
1.6
1.4
1.2
1
0.8
5
8
6
4
nb
Fig. 14. AC as a function of traffic intensity (  =0.5–3.5):
1) Minimum AC in 2005 are $101, 315 for SM; $101,805
for AM I; and $101,423 for AM II; 1) Minimum AC in
2006 are $96,289 for SM; $96,541 for AM I; and $96,383
for AM II;
4
2
3
0
nc
Fig. 15. AC for various berths/terminal and QCs/berth;
Minimum AC per ships in 2006 is $96,383 for AM II ( 
= 2.887);
nb  4 and nc*  3,25
The results presented here support the argument that the average cost per ship served could be decreased by
increasing number of QCs/ship and their productivity. At the same time, the objective is to minimize the
average time that ships spend in queue for the four berths at the PECT and hence the average time that ships
spend in port. Our results show that ships arrivals over time are needed as input data for the optimisation of
the problem. In addition to the arrival date and ships time in port, it also generates the number of lifts per ship
(i.e. number of containers to be served per ship). On the basis on a QC productivity, this number of lifts per
ship can easily be converted into the average service time of ships needed at the berth.
5. Conclusions
The ASBY link simulation and analytical models can be used as a decision support tool for analyzing and
evaluating ASBY link performance by the terminal management. This approach allows different models of
use of available facilities to be in for incorporated, and it facilitates the ship arrival patterns, lifts per ship call,
number of QCs assigned and QC productivity as actual data. In addition, it provides results of the possible
simulation and analytical approaches in comparison with the current situation, and the implication of the
results given through the input data and assumptions. These models and especially SM can be used: to
estimate the improvements in performance of the ASBY link operations when their handling capacities vary;
for average cost analysis, as the simulation provides six important parameters, i.e., average service time of
ships in port, average arrival rates of ships, the number of QCs/berth, QC productivity, the degree of
utilization and traffic intensity of container terminal, which are needed to establish average cost effective
system; and in the planning for future additional QCs/berth and berths/terminal that may be needed, through
the use of forecasted average interarrival time of ships (obviously, high average time that ships spend in queue
would indicate the need for additional QCs/berth and berths/terminal).
Container terminals in Busan Port, especially PECT, are trying to expand capacity and increase performance
at a maximum of investments. Often the container terminal operations are changing to meet increased
customer demands as well as to adapt to new technologies. Reasons for the decrease of the average cost per
ship served with the introduction of new container berth, QCs, container yard area and automated stacking
cranes (ASC) include that waiting time of ships and the average time that ships spend in port decrease with the
advanced handling systems improving the operations procedures.
As discussed above, this paper has grown from the observation that terminal-based improved handling
capabilities and development policies, analyzed in previous section, may be effective for reducing ship
turnaround times and average cost per ship serviced, especially when the number of lifts per ship is large.
Hence, in this paper we analyze terminal development policies, in which new QCs can be assigned per ships,
while focusing on easily implementable approaches. We develop simulation and analytical models, which
provides solutions to large-sized problems usually encountered in practice in reasonable computational times,
and analyze its effectiveness. In addition, computational analysis shows that these models are quite effective
in realistic settings, when the OCs are assigned closer related to the apron area, and when the number of
containers to be unloaded from/loaded onto each ship is in the hundreds. Thus, these models can be used to
obtain a good solution to the real problem.
The need for understanding the major determinants of terminal size and handling capacity are not hard to
establish. The amount of capital involved in the infrastructure development, handling equipment and
operational processes of container terminal is now being approached by the investment necessary in container
ship. With rapid change still occurring in sizes of container ships, incorrect investment decisions can be very
costly related to container terminal, and a proper understanding of the crucial terminal management problem,
which is optimizing the balance between the shipowners who request quick service of their ships and
economical use of allocated resources can reduce the investment risk considerably.
The results of this study imply that the economies of container ship operations are now, and are likely to be,
such that terminal operators must provide excellent service guaranteeing safety, on-time service, and
accuracy. To do that a hub ports facilitate adequate port facility, equipment, and handling system. In addition,
average cost per ship service and any costs involved in transshipment must be minimized. In case of hub port
calling rather direct calling, there is a possibility that most of transshipment cost will be paid by the shipping
lines. Accordingly, shipping lines operating very large container ships will seek ports with the lower
transshipment cost.
A simulation model employing the GPSS/H has been developed to ship-berth link performance evaluation
of PECT. It is shown to provide good results in predicting the ASBY link operations system of the PECT. The
attained agreement of the results obtained by using simulation model with real parameters has been also used
for validation and verification of applied analytical model. In accordance with that, the correspondence
between simulation and analytical results completely gives the validity to the applied analytical model to be
used for optimization of processes of servicing ships at PECT. Finally, these models also addresses issues
such as the performance criteria and the model parameters to propose an operational method that reduces
average cost per ship served and increases berth throughput.
Acknowledgement
This work is financially supported by the Ministry of Information and Communication Republic of Korea
(MIC) and the Institute for Information Technology Advancement (IITA) through the fostering project of the
University Information Technology Research Centre (ITRC).
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