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Associate Professor Branislav DRAGOVI Ć 1

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MODELING OF SHIP-BERTH-YARD LINK

PERFORMANCE AND THROUGHPUT OPTIMISATION

Associate Professor Branislav DRAGOVIĆ1

1)Maritime Faculty, University of Montenegro, Maritime Transport & Traffic Division

Dobrota 36, 85330 Kotor, Montenegro, [email protected], [email protected]/bdragovic

Professor Nam Kyu PARK2

2)Department of Distribution Management, Tongmyong University of Information

Technology, Korea

Professor Zoran RADMILOVIĆ3

3)Faculty of Traffic and Transport Engineering, University of Belgrade, Serbia

This paper presents a ship-berth-yard link modeling methodology based on statistical analysis of container ship traffic data obtained from the Pusan East Container Terminal (PECT). The efficiency of operations and processes on the ship-berth link has been analyzed through the basic operating parameters such as berth utilization, average number of ships in waiting line, average time that ship spend in waiting line, average service time of ship, average total time that ship spend in port, average quay crane (QC) productivity and average number of QCs per ship.

- The rest of this lecture is organized as follow: methodology based on statistical analysis of container ship traffic data obtained from the Pusan East Container Terminal (PECT). The efficiency of operations and processes on the ship-berth link has been analyzed through the basic operating parameters such as berth utilization, average number of ships in waiting line, average time that ship spend in waiting line, average service time of ship, average total time that ship spend in port, average quay crane (QC) productivity and average number of QCs per ship.
- In the next section we will provide an overview of the literature related to the
- port simulation and analytical models and especially ship-berth-yard link models.
- Following section presents brief description of ship-berth-yard link simulation and analytical modeling procedure, consisting of model structure, data collection and applied simulation algorithm – flowchart.
- This is followed by the next section which gives model validation and
- simulation analysis of ship-berth link at PECT.
- Following section presents two types of models that are developed on the basis of the queuing theory: mathematical model and simulation model. Results from both models are compared with each other. The mathematical 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.
- In conclusions we draw and incorporate suggestions to continue research on
- ship-berth-yard link performance

The basic approach used consists of two models. The first is a simulation model adapted to the problem of analyzing 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 for PECT.

As a 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 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.

Table 1: a simulation model adapted to the problem of analyzing 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 for PECT. Literature review of a container port and ship-berth link planning

There are few studies dealing with ship-berth link planning. Researches related to a container port and particularly ship-berth link planning, which use simulation, are summarized in Table 1.

Table 1. Continue a simulation model adapted to the problem of analyzing 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 for PECT.

PROBLEM STATEMENT a simulation model adapted to the problem of analyzing 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 for PECT.

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. Main problem in analytical modeling of container terminal relates to the fact that they lose in detail and flexibility, 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.

A simulation model of a container terminal is basically a computer program written in a general purpose language or in a special simulation-oriented language. The different types of simulation languages that have been used for modeling of the processes at the ship-berth link include MODSIM III, AweSim, Arena, Extend, Witness, GPSS/H. The simulation models are used to analyze queuing and bottleneck problems, container handling techniques, truck and vessel scheduling (departure and arrival rates), equipment utilization, and port throughput and operational efficiency (yard, gate and berth). So, a simulation implements the most important aspects of the processes at the container terminal, often in a simplified manner. However, the advantage of simulation modeling over analytical modeling of container terminal is that it allows for a greater level of detail and to avoid too many simplifications.

Analytical modeling of container terminal consists of setting up mathematical models and equations which describe certain stages in the functioning of the system. Specifically, the probabilistic models are, often, used to describe the evolution of these systems in the process of its modeling. This choice accounts for the fact that the events like ship arrivals, service time, waiting time, etc., at the container terminal are often unpredictable, and hence assumed to be random. The big advantage of analytical modeling is that it requires a thorough understanding of the system. The biggest disadvantage in analytical modeling of ship-berth link is that many related processes are too complex to be in reach of analytical methods. Therefore, a lot of simplifications and approximations have to be made during the modeling process, which lessens the accuracy of the results. However, often analytical models can give a rough feeling for the influence of certain factors on the performance measures at considered ship-berth link. A second disadvantage is that the analyst needs to know the necessary mathematics very well, including their respective abilities to model processes at ship-berth link.

In Table 2 we give a brief qualitative comparison of the simulation and analytical techniques for performance evaluation of ship-berth link.

Table 2. Qualitative comparison of the simulation and analytical techniques

for performance evaluation in port

SIMULATION MODELING OF SHIP-BERTH LINK PERFORMANCE analytical techniques

Generally speaking, we can realize the simulation

modeling by using GPSL or GPPL.

Ship-berth link is complex due to different interarrival times of ships, different dimensions of ships, multiple quays and berths, different capabilities of QC 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 in ship service presented in Figure 1.

Figure 1. Port operation with ship movement in port and

process flow diagram of the terminal transport operations

Figure 2. times of ships, different dimensions of ships, multiple quays and berths, different capabilities of QC 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 in ship service presented in Figure 1.

Flowchart for a ship arrival/departure

LOGIC OF ALGORITHM FOR SIMULATION MODEL times of ships, different dimensions of ships, multiple quays and berths, different capabilities of QC 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 in ship service presented in Figure 1.

Second come

Berths are not available! Wait in queue!

First class prioritiy

Compare priorities

Higher

Berth 4 available!!!

Berths are not available! Wait in queue!

First come

Second class prioritiy

Cranes are available!!!

Service completed

Service completed

There is no crane available!

Wait for crane!

Service completed

Berth 1

Berth 2

Berth 3

Berth 4

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 ship-berth link processes as accurate as possible the following phases need to be included into simulation model (Dragović et al. 2005a,b; 2006a,b):

Model structure: Ship-berth-yard 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 (Figure 1).

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: 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). 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 algorithm shown in Figure 2.

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 1 - 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 3.15 minutes from apron to stacking area including unloading/loading time by transfer crane. The distance between apron and stacking area is assumed to be 700 meters.

ANALYTICAL MODELING OF SHIP-BERTH LINK PERFORMANCE 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 1 - 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 3.15 minutes from apron to stacking area including unloading/loading time by transfer crane. The distance between apron and stacking area is assumed to be 700 meters.

Queueing theory (QT) models for analyzing movements of ships in port is proposed and shown in Fig.1.

In the analysis of various aspects of average time that ships spend in port, tws, including ns, nb, , , ncand , (e.g., Plumlee (1966), Nicolaou (1967, 1969), Wanhill (1974), Noritake (1985), Noritake and Kimura (1983,1990), Shabauek and Yeung (2001), Taniguchi et al. (1999)) defined tws as the sum of the average waiting time and average service time.

The average service time,

where

includes ships loading/unloading time in hours per containership, tc, expressed as

(1)

(2)

It follows that

(3)

Further, it can be shown that 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 1 - 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 3.15 minutes from apron to stacking area including unloading/loading time by transfer crane. The distance between apron and stacking area is assumed to be 700 meters.

(4)

where

(5)

for the (M/M/nb) model.

For minimizing tws, the Eq. (5) can be transformed in the form

(6)

Also, the Eq. (3) becomes

(7)

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 adopted relation to average port waiting time of ships (Noritake (1985), Noritake and Kimura (1983,1990), 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

(8)

the coefficient of variation of ships service time distribution and

k = the number of phases of an Erlang distribution

(9)

The Eqs. (8) related to ((M/Ek/nb)I) and (9) related to ((M/Ek/nb)II) for the (M/Ek/nb) model present average time that ships spent in port as a function of .

BERTH UTILIZATION FACTOR steady-state condition which were obtained by Morse (1958) refer to the (

(12)

SHIP TRAFFIC INTENSITY

Further, as a port operation parameter, i.e. berth occupancy index, can be defined in the following manner (Nicolaou (1967 and 1969); Noritake (1983)).

(13)

Furthermore, there holds

(14)

Then, the average number of ships present in port with nb berths in the period T is expressed as

(15)

Also, average number of ships waiting for berths with nb berths in the period T is obtained as

(16)

It follows from (15) and (16) that average number of ships served at nb berths in the period T can be written in the form

(17)

In view of that steady-state condition which were obtained by Morse (1958) refer to the (

the Eq. (15) becomes

(18)

or

(19)

From Eqs. (13), (14) and (19) we have

(20)

AVERAGE TIME THAT SHIPS SPEND IN PORT

The substitution of Eq. (12) into Eq. (8) yields

(22)

for the (M/M/nb):(FCFS//)model, and hence by (4) we have

(23)

(24) steady-state condition which were obtained by Morse (1958) refer to the (

(25)

In order to write tws from (4) as a function of in the form

(27)

we substitute (20) into (8) to obtain

(30)

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 Eq. (30) into Eqs. (10) and (11), respectively:

(31)

(32)

- SHIP LOADING/UNLOADING OPERATIONS MODELING steady-state condition which were obtained by Morse (1958) refer to the (

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 follow group:

- berth cost in $ per hour,

- QCs cost in $ per hour,

- storage yards cost in $ per hour,

- transportation cost by yard transport

equipment between quayside and

storage yard in $ per hour

- labor cost for QC gangs in $ per hour,

- ships cost in port in $ per hour,

- containers cost and its contents
- in $ per hour

The total cost function, would be concerned with

the combined terminals and containerships cost as

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

(33)

where TC - total port system costs in $/hour.

By substituting the Eq. (9) into Eq. (33) yields

(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.

From the total port cost function per average arrival rate, we can obtain

(35)

Since = , we get

(36)

or because of by the Eq. 12), = nb, the Eq. (36) also has the form

(37)

Eqs. (35), (36) and (37) show the average container ship cost in $/ship, AC. In this study, the trade-off will be simulative and analytically resolved by minimizing the sum of the relevant cost components associated with the number of berths/terminal and QCs/berth, and 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 solving 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.

NUMERICAL EXAMPLES WITH we can obtain

EXPERIMENTAL STRATEGY

This section gives a ship-berth link modeling methodology based on statistical analysis of container ship traffic data obtained from the PECT. PECTis big container terminals with a capacity of 1,963,304 twenty foot equivalent units (TEU) in 2004. There are four berths with total quay length of 1,200 m and draft around 14-15 m. Ships of each class can be serviced at each berth.

Figure 3.

PECT layout

INPUT DATA we can obtain

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 six months period from September 6, 2004 to February 27, 2005, which involves 711 ship calls, see Table 3 (PECT Management reports). The ships were categorized into the following three classes according to the number of lifts made per each ship: first class consists of ships with less than 500 lifts made, second class of 501 – 1,000 lifts and third class are those with more than 1,000 containers loaded/unloaded per ship. Ship arrival probabilities are as follows: 28.1% for first class, 42.3% for second class and 29.6% for third class. The ship arrival rate is 0.175 ships/hour. The total throughput during the considered period was 979,655 TEU. Also, the berthing/unberthing time of ships is considered to be 1 hour.

Table : Input data - Ship characteristics we can obtain

Note: T1– Scheduled time; T2 – Time of arrival; Cs – Capacity of ships in TEU;

L – Ship length; nc – Number of QCs assigned per ship; PQC – Productivity of QCs

The assignment of QCs per ship was assumed random with probabilities equal to the percentages of number of QCs that was allocated for ship servicing. Therefore, the results of analysis of frequencies of QCs assignment per ship expressed in % are given in Table 4. For first class of ships (under 500 lifts), 15% of total ships is given 1 QC, 66.5% are given 2 QCs, 17% are given 3 QCs and 1.5% are given 4 QCs. For second class of ships, 2 QC are assigned for servicing in 23% cases, 3 QCs in 67.3%, 4 QCs in 9.4% and 5 QCs in 0.3% cases. The data for third class of ships are as follows: 2 QCs in 1.9% cases, 3 QCs in 47.7% cases, 4 QCs in 43.8% cases and 5, 6 and 7 QC are assigned in 4.3%, 1.9% and 0.4% cases, respectively. Furthermore, average QCs productivity given in lifts per hour is shown in Table 5.

Table: No. of QCs assigned per ship in %

The inter-arrival time distribution of ships at the PECT was plotted in the Figure 3. It is found that even though the arrivals of the ships at the PECT, taking the whole period of six months and each class of ships, are scheduled and not random, the distribution of inter-arrival times fitted very well exponential distribution.

In order to obtain punctual data, we have done fitting of empirical distribution of service times of ships with appropriate theoretical distribution for each class of ships. It is observed that service time of first class of ships follows 7-stages Erlang distribution (Figure 4), while 12-stages Erlang distribution fits very well the service time of second class of ships (Figure 5). Finally, service time of third class of ships follows the 3-stages Erlang distribution (Figure 6). The distribution types of service time for each class of ships are given in Figures 4 - 6.

Goodness of fit was evaluated, for all tested data, by both chi-square and Kolmogorov-Smirnov tests at a 5 % significance level.

Figure 3. plotted in the Figure 3. It is found that even though the arrivals of the ships at the PECT, taking the whole period of six months and each class of ships, are scheduled and not random, the distribution of inter-arrival times fitted very well exponential distribution.Distribution of ships inter-arrival

times (IAT) at PECT

Figure 4. Service distribution of first class of ships (the 7-stage Erlang distribution)

Figure 5. Service distribution of second class

of ships(the 12-stage Erlang distribution)

Figure 6. Service distribution of third

class of ships(the 3-stage Erlang distribution)

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 containership, 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 6 (PECT Management reports, Korea Maritime Institute (1996)). The described and tested numerical experiments contain four segments in relation to the input variables.

Table 3. Input data – Terminal characteristics

nc - average number of QCs assigned per ship (Real data and Simulation resluts); cnb = 62 million $; i = .0663; ny - 40, cnbm = 6.2 million $ ; cnb= 1215 $; cnc = 38.8 $/QC hour; ttcon= 188 hours; aconcy= 63.9 m2/container; ccy = 0.000292 $/m2 hour; cl= 357 $/gang hour; cw =1.4 $/container hour.

VALIDATION-VERIFICATION 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 containership, 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 6 (PECT

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 40 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 2.573 and 64.34%, while the simulation output shows the value of 2.564 and 64.12%, respectively, see Table 7. Average service time shows very little difference between the simulation results and actual data, that is, 15.12 h and 15.20 h, respectively (Table 4). The simulation results of the number of serviced ships completely correspond with the real data (i.e. the simulation result of the total number of ships is 712.3 and the real data is 711; the first class of ships: 201.25 and 201; the second class: 301.75 and 301 and the third class: 209.3 and 210), see Table 5. All the above shows that simulation results are in agreement with real data.

VALIDATION-VERIFICATION 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 containership, 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 6 (PECT

Table: Number of ships serviced in simulation periodSeptember 6, 2004 to February 27, 2005

Analitical,

Table: Average service time of ships, traffic intensity and berths utilisation

RESULTS 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 containership, 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 6 (PECT

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 4 displays the results, the key measures are average traffic intensity, berth utilisation and average service time of ships (all classes, first class, second class and third class), while Table 6 shows average time that ships spend in queue (all classes, first class, second class and third class). In addition, Table 7 gives average time that ships spend in port (all classes, I class, II class and III class). According to this, judging from the computational results for some numerical examples of the models: (M/Ek/nb)I – using the average waiting time given by Eq. (10) (for brevity, the analytical Model I is denoted as AM I) and (M/Ek/nb)II – using average waiting time 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.

RESULTS 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 containership, 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 6 (PECT

Table 9. Average time that ships spend in queue

Table 10. Average time that ships spend in port

AVERAGE CONTAINER SHIP COST 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 containership, 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 6 (PECT

Figure 7. Average container ship costs for various traffic intensity ( =0.5-3.5) - Minimum AC per ship first class are $50,202 for SM; $50,754 for AM I; and $50,908 for AM II

Figure 8. Average container ship costs for various traffic intensity ( =0.5-3.5) - Minimum AC per ship second class are $96,008 for SM;

$96,405 for AM I; and $96,769 for AM II

Figure 9. Average container ship costs for various traffic intensity ( =0.5-3.5) - Minimum AC per ship third class are $149,621 for SM ;

$149,329 for AM I ; and $149,536 for AM II

Figure 10. Average container ship costs for various traffic intensity ( =0.5-3.5) - Minimum AC per ship third class are $101,990 for SM ;

$101,323 for AM I ; and $101,536 for AM II

Figs. 11 intensity (– 16 show the optimization function AC of two variables nb (nb = 3 , 4, 5) and nc (nc = 1 , 2,…, 7) for constant value of . In Fig. 12 obtained results correspond to those from Fig. 7, in Fig. 14 to results from Fig. 8, and in Fig. 16 to those in Fig. 19. Still, even in Fig. 12, the study offers similar results, i.e. the minimum average cost per ship served are $50,202 in relation to $50,202 from Fig. 7 – curve AM I. 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.

Figure 11. Average container ship costs for various berths/terminal (nb= 3,4,5) and QCs/berth (nc = 1,2,…,7); Minimum AC per first class of ships is $52,160 for AM I ( = 2.85); nb = 4 and nc = 2.05

Figure 12. Average container ship costs for various berths/terminal (nb = 3,4,5) and QCs/berth (nc = 1,2,…,7); Minimum AC per first class of ships is $50,202 for AM I (= 2.95); nb = 4 and nc = 2.5

Figure 13. Average container ship costs for various berths/terminal (nb = 3,4,5) and QCs/berth (nc = 1,2,…,7); Minimum AC per second class of ships is $99,302 for AM II ( = 2.84); nb = 4 and nc = 2.85

Figure 14. Average container ship costs for various berths/terminal (nb = 3,4,5) and QCs/berth (nc = 1,2,…,7); Minimum AC per second class of ships is $96,008 for AM II ( = 2.93); nb = 4 and nc = 3

Berth throughput berths/terminal (nb = 3,4,5) and QCs/berth (n

To obtain a deeper understanding of terminal throughput, the PECT is compared by four parameters: operation efficiency and number of ships serviced in simulation period of PECT – Fig. 17 (RD – real data and SR – simulation result), ship operation efficiency in PECT – Fig. 18, operation efficiency and throughput of PECT – Fig. 19 and the TEU/hectare and TEU/berth meter of PECT – Fig. 20. Therefore, 500,000 TEU per berth is a high standard in PECT, which is achieved by top 50% of terminals operators. But 700,000 TEU per berth is standard is Chinese major ports, which have been achieved by top 50% of terminals operators. The ship operation efficiency has a significant relationship with throughput of berth. In 2010, the standard is expected to be higher, because the ship is bigger and advanced technology implemented in yard operations. Based on the performance achieved, and highly competitive environment in the Far East region, it is expected that PECT could achieved at least 700,000 TEU/berth as a new standard for major terminal operators in Asia.

Figure 17. Operation efficiency and number of ships serviced in simulation period of PECT

Figure 18. Ship operation efficiency in simulation period of PECT

in PECT

Figure 19. Operation efficiency and throughput in simulation period of PECT

of PECT

Figure 20. The TEU/hectare and in simulation period of PECT

TEU/berth meter of PECT

CONCLUSIONS in simulation period of PECT

Models described and developed in this paper, especially SM can be used: to estimate the improvements in performance of the ship-berth link operations when their handling capacities vary; for average cost analysis, as the simulation provides seven important parameters, i.e., average service time of ships in port, average arrival rates of ships, the number of QCs/berth, QC productivity, the berth throughput, 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).

From the features of average cost curves and global optimum solutions – various obtained delayed systems, the following facts are confirmed:

If the values of containers transferred per ship, the average QC move time, the number of berths/terminal, the number of Erlang phases of service time distribution and other numerical input values are given, then the optimum number of QCs/berth, the traffic intensity, the berth utilization, the average time that ships spend in port, the total port cost and average cost per ship or container served, can be easily obtained by the use of average cost curves and global optimum solution;

The results obtained here suggested that an increase in the number of QCs/berth could reduce the average cost per ship or container served.

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 the terminal efficiency and berth throughput.

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The purpose of this monograph is to present the achievements of the authors in the field of ports and container terminals modeling during the last few years. The material in the monograph is divided in two parts: ports modeling and container terminal modeling.

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