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ZDENKA ZENZEROVIĆ, Ph. D. E-mail: z.zenzerovic@gmail.com

SINIŠA VILKE, Ph.D. E-mail: svilke@pfri.hr

NATAŠA ANTONINI, M.Sc. E-mail: natasaa@pfri.hr

University of Rijeka, Faculty of Maritime Studies Studentska 2, HR-51000 Rijeka, Republic of Croatia

COST MODEL IN FUNCTION OF OPTIMAL CAPACITY PLANNING OF PORT CONTAINER TERMINAL1

ABSTRACT The aim of this paper is to show how the application of the queuing theory, as one of the quantitative research methods, could plan an optimal capacity of a port container terminal, which is one of the conditions for efficient functioning of the terminal. The problem of determining optimal capacity of the container terminal is reduced to calculating an optimal number of berths, since the capacity of the berth defines the necessary capacity of the other subsystems, and thereby the capacity of the entire container terminal. For the actualization of the mentioned research aim a model of total costs is set, whereby a combination of the number of berths and cranes for which the total costs are minimal is determined. This model can be applied for taking decisions related to manageability of any container terminal as terminals in current or future business conditions. The presented model is tested on an example of the container terminal at the Port of Rijeka. Key words: cost model, queuing theory, optimal capacity planning, port container terminal, container terminal at the Port of Rijeka.

1. INTRODUCTION Since the eightieth's of the last century the world container traffic has shown a continuous

and intensive development generated by the growth of global transport and the advantages of containerization, such as: the safety of goods inside containers, easier and faster cargo handling, the reduction of the turn-around time of ships in port and the integration of different transport modes into a unique transport chain by „door to door“ service. The container transport is nowadays the most perspective way of transporting cargo for which the further growth being predicted within the world foreign trade exchange.

The optimal capacity planning is one of the conditions for efficient functioning of a port container terminal. However, since the fluctuations of container reload conditioned by the uneven arrival of ships to the terminal and uneven duration of manipulations with containers

1 This scientific paper is the research result on the project: “Quantitative Methods in the Function of the

Optimal Management of Marine Systems”, project number: 112-1121722-3308, which is financed by the Ministry of Science, Education and Sport of the Republic of Croatia.

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caused by various factors, it is not simple to determine and size in practice an optimal capacity of a port container terminal. From the port service customer point of view it would be ideal for the terminal to handle the reserve capacity that should be applied to container reload necessity in case of current maximum capacity achieving of the terminal. However, the mentioned additional capacity would reduce the usage degree of the terminal and increase a part of fixed charges in its business activity.

The capacity of the port container terminal [1, pp.168] represents the amount of throughput of certain berths of the terminal in the selected time unit. As the capacity of the berth determines the throughput of the entire system of the container terminal, and indirectly the adequate capacity of the other subsystems of the terminal, the problem of determining optimal capacity of the port container terminal is reduced to calculating an optimal number of berths.

One of the ways of determining the optimal capacity of a port container terminal is the application of quantitative methods, of which in this paper the queuing theory was implemented.

In respect to the research object, in this paper the following hypothesis has been set: for the optimal capacity planning of the port container terminal a model of total costs based on the queuing theory can be applied. By applying this model a combination of the number of berths and cranes at the berth with the minimum costs for the given traffic of the terminal is determined, whereby the optimal functioning of the terminal can be achieved.

The function of the total costs was published by P. Schonfeld and S. Frank in 1984 [2, pp. 56-62] for a container port with one berth. Later, in 1995 Z. Zenzerović in her dissertation [3, pp. 224-244] expands the model on the port with several berths and through graphics and calculations determines the change of every type of costs depending on the number of berths and cranes. Č. Dundović and Z. Zenzerović publish in 2000 [4, pp. 217-221] the model of costs for the general cargo port. The previous model is in the book System Optimisation of Croatian Container Ports in 2001 [5, pp.73-98] applied on real data for container terminal at the Port of Rijeka. The extension of the previous researches [6, pp. 45-69] is related to the cost model with the formulas for some types of costs modified and applied to the container terminal at the Port of Rijeka. In order to apply the cost model that is based on the queuing theory it is necessary to define the port container terminal as a serving system with adequate parameters, which is shown in papers [1, pp. 169], [7], [8], [9], [10], and then to calculate the indicators of a port container terminal operation using queuing theory formulas from books [11, pp. 20-26], [12] or, which is more practical, to apply relevant software, i.e. WINQSB [13]. In this paper in the first place the impact analysis of container terminal capacity on the total ship stay at terminal is performed as one of the essential indicators for the efficient business activity of the terminal. Following, introducing the costs as optimization criterion, the cost model that can be applied in optimal capacity planning of the terminal, tested on an example of container terminal at the Port of Rijeka, is set.

2. IMPACT OF CONTAINER TERMINAL CAPACITY ON THE TOTAL SHIP STAY AT TERMINAL

The berth capacity is defined by the number of ships that can be serviced in an observed time unit at a determined berth. For terminals with more berths the total capacity is calculated with the formula S , (1)

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where: S – number of container berths μ – berth service rate (the number of ships that can be serviced in a time unit). As mentioned, the port container terminal capacity depends on the number of berths and the throughput of each berth, so it is useful to determine in which way and to what extent these parameters affect the total ship stay at the terminal, and thus the total costs. 2.1. IMPACT OF THE NUMBER OF BERTHS ON THE TOTAL SHIP STAY AT TERMINAL

Since the port container terminal is defined as a serving system, the impact of the number

of berths on the total ship stay at a terminal can be tested using the queuing theory [3, pp.141-147], [11, pp.42-46].

If the ship arrival rate λ and the serving time Wserv are constant, then the increase of the number of berths S affects the progressive decrease of the average time of ship's waiting WQ and the total time of ship stay at the terminal W, and the value for which the time of ship stay at the terminal will be reduced depends on the values of berth occupancy ρ and the number of berths S.

The difference )(1 SWSW can be calculated by the difference L S L SQ Q( ) ( ) 1 that is obtained by the following equation (the formula derivation is shown in detail by the dissertation [3, pp. 143-144])

2200

1

)1()(1)1()(

!)()1(

SSSPSP

SSLSL

S

QQ

SSSS

S

S

)(2)(!

...1 3

0)1()!1(

221

SSS

S

, (2)

where: S – number of berths ρ – berth occupancy, ρ = λ/μ W – time of ship stay at terminal LQ – number of ships in queue P0 – probability that there are no ships at the terminal. In Table 1. the values of the difference L S L SQ Q( ) ( ) 1 calculated according to the

number of berths (from 1 to 10) and the berth occupancy (from 0.1 to 0.9) are shown.

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Table 1. Values of the difference L S L SQ Q( ) ( ) 1 in relation to the number of berths and the berth occupancy

Based on the results from Table 1. it is concluded:

1. The values of the difference L S L SQ Q( ) ( ) 1 are negative, which means that with an increase in the number of the berths the number of ships in queue is reduced regardless of the increased number of berths and the value of particular berth occupancy.

2. The absolute values of the difference L S L SQ Q( ) ( ) 1 are reduced with in an increase in the number of berths, while with an increase of the berth occupancy the same are increased. That means at the terminal with an increase of ship arrivals rate λ, the berth occupancy ρ is also increased, and with an increase in the number of berths the number of ships in queue is progressively reduced. 3. The previous Table can be used for planning the number of berths at the container terminal since it shows the results useful for decision making regarding the number of berths (values on the left from stepwise lines). Since WQ = LQ / λ, and λ is constant, it follows that WQ changes analogously to LQ, so this Table can also be useful for the analysis of the waiting time of the ship. 2.2. IMPACT OF THE SERVICE RATE ON THE TOTAL SHIP STAY AT TERMINAL

In assumption that the number of berths S and the ship arrivals rate λ are constant, then any increase of the service rate μ affect the progressively reduction of the average duration of service time Wserv and the total time of ship stay at the terminal W.

The total time of a ship stay at the terminal is calculated with the formula [11, pp. 20-26]

1

!!21

!1

112

2

1

SSSSSW

SSS

(3)

The difference of time of a ship stay at the terminal when the berth occupancy changes to

the value of h ( 0h ) is )()( WhWWh . The change value of the total time of ships stay at the terminal is calculated by the

following equations (detailed formula derivation is shown by the dissertation [3, pp. 138-140]):

- if the change value of berth occupancy ρ is given

5

2hWhWh (4)

,)!1()(

2)(1)()!1(

11)(1

20

20

SSP

SS

SP

SW

SS

(5)

- if the change value of service rate μ is given

1

1

1

1

21

hhW

hhWh , (6)

where: P0 – probability that there are no ships at the terminal h – change of berth occupancy ρ h1 – change of service rate μ . In order to simplify the determining of the difference value hW the values of the numeric

expression X W ( ) according to the number of berths (from 1 to 10) and the berth occupancy (from 0.1 to 0.9) are calculated and shown in Table 2.

Table 2. Values X W ( ) in relation to the number of berths and the berth

occupancy

From Table 2. it is concluded: 1. The values W ( ) are positive for any value of ρ, S and λ. However, the difference

value hW will be positive or negative depending on the fact if h presents the increase or decrease of the berth occupancy.

2. The values W ( ) show different tendencies regarding the changes of the number of berths S and the berth occupancy ρ, namely: with an increase in the number of berths for determined ρ they are reduced, and with an increase of ρ for determined S they are increased.

3. The shown Table can also be used for planning the number of berths at the container terminal: by an increase of service rate, beside unchanged number of berths, the total time of ship stay at terminal will be progressively reduced, and thereby also the total costs of ship stay at terminal.

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2.3. INTERDEPENDENCY OF PORT CONTAINER TERMINAL INDICATORS

Taking a decision regarding the optimal capacity of the container terminal one of significant indicators is the time of ship stay at the terminal. Based on the impact of the number of berths and service rate on the total ship stay at the terminal it is possible to compare the variants in relation to the volume of traffic and the number of cranes per berth.

For example, the total traffic of the container terminal at the Port of Rijeka in the year 2010 was 262 ships at I. berth with two ship gangs and 23 ships at II. berth with one ship gang; therefore, with 2 berths and 3 cranes (see section 3.2. of this paper). The reduction of the number of cranes at I. berth would affect the change of serving parameters: the arrival rate λ = 0.7176 ships/day remains equal, but the service rate is reduced to μ = 1.1959 ships/day

why the berth occupancy is increased to ρ = 0.60. Using the equation (6) and the values from Table 2. of this paper, it follows that the

expected time of ship stay would arise by 5197,007176,0/12535,0 WWh , respectively for more than 35 hours, which would negatively affect the total costs value.

However, the change of crane number does not affect only the change of time of ship stay at the terminal but also other indicators related to the operating of the terminal, such as the change of the number of berths, which is confirmed by the example from table 3.

Table 3. The functioning indicators of the container terminal according to the number of

berths (λ = 0.7176 ships/day, μ = 2.0712 ships/day, ρ = 0.3465)

S ρ/S P0 Pw LQ L WQ W

1 0.3465 65.3534 34.6466 0.1837 0.5301 0.2560 0.7388

2 0.1732 70.4691 5.1157 0.0107 0.3572 0.0149 0.4977

3 0.1155 70.7053 0.5541 0.0007 0.3472 0.0010 0.4838 Note: S - number of berths; P0, Pw – probabilities (in %), LQ i L – related to ships, WQ, W – time

(days)

From Table 3. it is concluded that the change of the number of berths (by constant λ and μ) affects the increase, respectively the decrease, of the indicators values of container terminal. Namely, by an increase of the number of berths, the following indicators are reduced: the average number of ships in queue as well as at the terminal, the average time of ship's waiting, the time of ship stay at the terminal and the probability of waiting; but the vacancy of berth is increased.

Further to the previous conclusions it follows the question how to define the number of berths as well as the capacity of other elements included in the serving process at the container terminal, since it is not possible to eliminate both the ship's and berth's waiting.

For taking such a decision it is necessary to introduce a certain criterion, for example, the percentage of berth usage, the duration of ship's waiting in queue or the number of ships in the queue, thereby the indicator representing the criterion that is considered most important for the effective functioning of container terminal is chosen.

The efficiency is increased either by increasing the number of berths or reducing the average serving time; in the first case the vacancy of berths will “increase”, and in the second case it is possible the effect on the quality of service and reduce the number of ship arrivals.

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Therefore the efficiency of the container terminal can be best determined by using the value indicators, in other words by applying the model of costs. 3. MODEL OF TOTAL COSTS OF PORT CONTAINER TERMINAL In order to determine the optimal solution an adequate optimization criterion is chosen. In this paper the criterion is expressed in terms of value, i.e. by means of costs calculated for the defined time unit. The shown model shall be applied to actual data of container terminal at Port of Rijeka. 3.1. DEFINING THE MODEL OF TOTAL COSTS OF PORT CONTAINER

TERMINAL

The cost model of port container terminal is presented with the function of total port costs (C) which includes the following costs: berths costs (Cb), costs of container cranes (Cd ), costs of transport-handling equipment (Cpp ), crane operator costs (Cld ), operator costs (Clp ), stacking area costs (Cwh ), costs of ship stay in the port (CW ) and cargo costs (CQ ).

Therefore, the function of total costs of the container terminal system has the form:

QWwhlpldppdb CCCCCCCCC , (7)

where C is the symbol for total costs of the port container terminal expressed in money units in the observed time unit, for example in €/hr.

The amount of a single type of cost is calculated with the matching formulas (8) – (25):

bb cSC (8) dd cdSC (9) pppp cpC (10)

ldldld ctdC (11)

lplp cnC (12)

whwhwh cakC (13)

WW cWC (14)

QQ cQWC (15)

where: S – number of berths in port container terminal, cb – cost per berth (€/hr); if cb is not given, it can be calculated taking into consideration

the capital recovery factor and the costs of berth maintenance:

24365

1

11

10

bM

bNi

bNiiBbc . (16)

B0 – initial price of berth (€), Nb – economical life cycle of berth (years), i – interest rate, Mb – annual maintenance cost per berth (€).

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d – number of container cranes per berth, cd – costs per crane (€/hr) calculated on the basis of the initial price of crane, the

economical life cycle of crane and the annual maintenance cost per crane

24365

111

)1(0

dN

N

d Mi

iiDcd

d

. (17)

D0 – initial price of crane (€), Nd – economical life cycle of crane (years), Md – annual maintenance cost per crane (€).

p – number of transport-handling equipment used in operations per one ship, cpp – unit cost per transport-handling equipment (€/hr).

24365

1

1)1(

10

ppN

N

pp Mi

iiPc

pp

pp

. (18)

P0 – initial price of transport-handling equipment (€), Npp – economical life cycle of transport-handling equipment (years), Mpp – annual maintenance cost per equipment (€).

λ – intensity of ship arrivals (ship /hr), d – number of crane operators tld – operating time for crane operators, or the duration of ship reload (hr/crane/ship)

calculated with formula: f

ld dyxt /)( . (19)

However, if tld is less than the minimal operating time in each shift, for example 7 hours, then the port pays minimal hours, and not the real value tld. That means that tld = max (tld, tmin), where tmin stands for the minimal duration of working shift.

Symbol d in formula (11) refers to the number of crane operators, and in formula (19) to the number of cranes. The difference between the number of cranes and crane operators occurs because of the fact that one crane is operated by 2 crane operators.

x – number of reloaded containers per ship, y – duration of one life cycle of crane (hr/TEU), f – interference coefficient (0<f<1).

cld – unit cost for crane operator (€/hr), n – number of operators participated in transport-handling operations per ship, clp – unit cost for operator (€/hr). kwh – required stacking area capacity

xkwh (20)

a – stacking surface per container (m2/TEU), cwh – cost per unit of stacking surface (€/TEU/hr) calculated with formula

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24365

11)1(

)1(0

whN

N

hwh Mi

iiWcwh

wh

(21)

Wh0 – initial price of stacking area (€), Nwh – economical life cycle of stacking area (years), Mwh – annual maintenance cost of stacking area (€).

W – average time of ship stay at container terminal, i.e. the ship's time spent in a queue and the ship's servicing time on the berth (hr/ship),

cW – cost per unit of ship stay at container terminal (€/h) calculated on the basis of the initial ship value, the economical life cycle of a ship and the annual costs of ship maintenance according to formula

24365

11)1(

10

WN

N

W Mi

iiWcW

W

(22)

W0 – initial ship value (€), NW – economical life cycle of a ship (years), MW – annual maintenance cost of a ship (€).

Value W is one of the indicators of the port functioning obtained by applying the queuing

theory, and the manner of calculating depends on the type of problem of the queue considering the elements that determine the type of problem of the queue: distribution of ship arrival, distribution of serving time, serving discipline and number of berths. The types of queues M/M/l/∞ and M/M/S/∞ are used for this paper since they are the most common cases of serving processes in port container terminals.

The average time of ship stay at container terminal with more berths is calculated with

formula /1QWW (23)

WQ – waiting time for available berth

11

02

1

)/1(!!)()!1(

SSnSSW

SS

n

nS

Q , (24)

where ρ = λ / μ , and μ is the service rate (ships/hr) calculated according to formula:

)/(1 mld tt (25)

tld – duration of ship reload (hr/crane/ship). tm – duration of docking/leaving port.

Q – average amount of cargo, i.e. average number of containers (TEU/ship), cQ – cost per container (€/hr).

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Because of practical reasons, for costs calculation the applying of relevant software is recommended. 3.2. CASE STUDY - CONTAINER TERMINAL AT THE PORT OF RIJEKA

The container terminal of the Port of Rijeka has two berths: I. berth, the Kostrensko quay – south and II. berth, the Kostrensko quay – west. The southern berth is 295 m long and the sea depth 12 m at the quay, while the western berth is 164 m long with the sea depth of 11 m at the quay.

On the western berth the older port container crane “Metalna” is located. This crane was moved from the south quay in 2009 in order to increase the operating capacity of the terminal. Since the limited crane outreach and the possibility for lower containers lifting, “Metalna” is used only for the handling of smaller container ships, when the south berth is not available. There are two port container cranes of the manufacturer Samsung on the southern berth, whose technical-technological features ensure the existing needs of the container terminal.

The actual total surface area of the container terminal in the Port of Rijeka is around 140 000 m2 while the stacking surface is 56 100 m2. Immediately, the horizontal transport-handling machinery that includes reachstackers and tugmasters with trailers and semi-trailers is used for ship and stacking operations at the terminal.

The parameters values of the model of total costs of the container terminal in the Port of Rijeka are the following: Number of berths S = 2

Initial price of berth - I. berth B0 = 2 040 301 € - II. berth B0 = 1 050 107 €

Interest rate i = 5

Economical life cycle of berth Nb = 50 years

Annual maintenance cost per berth - I. berth Mb = 204 030.10 € - II. berth Mb = 105 010.70 €

Number of cranes - I. berth d = 1 - II. berth d = 2

Initial price of crane - “Metalna” crane D0 = 1 736 689 € - “Samsung” crane D0 = 5 382 517 €

Economical life cycle of crane Nd = 10 years

Annual maintenance cost per crane - “Metalna” crane Md = 173 668.90 € - “Samsung” crane Md = 538 251.70 €

Number of reachstackers per ship - 1 ship gang p = 1 - 2 ship gangs p = 2

Number of tugmasters per ship - 1 ship gang p = 3 - 2 ship gangs p = 6 Number of trailers/semi trailers per ship - 1 ship gang p = 3

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- 2 ship gangs p = 6

Initial price of transport-handling equipment - reachstacker P0 = 338 750 € - tugmaster P0 = 140 000 € - trailer P0 = 35 000 €

Economical life cycle of transport-handling equipment Npp = 7 years

Annual maintenance cost of transport-handling equipment - reachstacker Mpp = 33 875 € - tugmaster Mpp = 14 000 € - trailer Mpp = 3 500 €

Arrival rate of ships - 1 ship gang λ = 0.0026 ship/hr - 2 ship gangs λ = 0.0299 ship/hr

Number of crane operators - 1 ship gang d = 2 - 2 ship gangs d = 4

Number of reloaded container per ship - 1 ship gang x = 116 TEU/ship - 2 ship gangs x = 454 TEU/ship

Duration of one life cycle of crane y = 0.042 hr/TEU

Interference coefficient - 1 ship gang f = 1.00 - 2 ship gangs f = 0.85

Number of operators - 1 ship gang n = 15 - 2 ship gangs n = 30

Unit cost for crane operator cld = 3.87 €/hr

Unit cost for operator clp = 3.87 €/hr

Number of reloaded containers per ship (average) x = 427 TEU/ship

Stacking surface per container a = 4.35 m2/TEU

Initial price of stacking area Wh0 = 10 387 403 €

Economical life cycle of stacking area Nwh = 100 years

Annual maintenance cost of stacking area Mwh = 311 622 €

Duration of ship docking/leaving port tm = 1 hr

Initial ship value W0 = 28 706 760 €

Economical life cycle of ship NW = 20 years

Cost of annual ship maintenance MW = 574 135.20 €

Average amount of cargo per ship - 1 ship gang Q = 116 TEU/ship - 2 ship gangs Q = 454 TEU/ship

Unit cost per container cQ = 2.7 €/hr .

Applying the model shown in section 3.1. and the previously mentioned actual data of

container terminal at the Port of Rijeka, two variants are compared: A – 2 berths, 3 cranes; two on I. and one on II. berth. B – 2 berths, 2 cranes; one on I. and one on II. berth.

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The single costs are included in Table 4. For the details regarding single costs calculations see the paper [6] mentioned in the literature. Table 4. Total costs of container terminal at the Port of Rijeka (€/h)

Variant (€/h) Type of cost

A B Cb 54,60 54,60 Cd 327,54 186,52 Cpp 80,70 53,80

Cld + Clp 179,15 120,61 Cwh 5 664,00 5 664,00 CW 179,24 430,29 CQ 654,71 1 591,50

Total 7 139,94 8 101,32 With the application of variant B, that takes into account the momentary stacking area

capacity of the terminal and achieved traffic in 2010, but with the usage of one crane per berth, the costs of the cranes, transport-handling equipment and human resource are reduced, as expected. However, there is a significant increase of ship costs that finally leads to increasing total costs for nearly one thousand €/h in relation to variant A.

So, in conditions of the existing traffic and terminal capacity, the optimal solution is variant A with two berths and three cranes.

Beside the application for the existing traffic, the shown model can be applied for future expected traffic, which is shown in paper [6], and thus be the basis for taking adequate business decisions. 4. CONCLUSION

For optimal functioning of the container terminal it is especially important to define the capacity of the terminal that affects the possibility of achieving the production plan, and the realisation plan for port services.

The problem of determining the optimal capacity of the port container terminal is reduced to calculating the optimal number of berths, since the capacity of berths determines the necessary capacity of the other subsystems of the port container terminal, and the throughput of the entire container terminal.

One of the ways of determining the optimal capacity of port container terminal is the application of quantitative methods, namely, queuing theory. This paper shows the cost model that determines the combination of number of berths and cranes per berth with the least costs for the given traffic of the terminal.

The cost model of the container terminal is presented with the function of total port costs that contains the following costs: berths costs, costs of container cranes, costs of transport-handling equipment, crane operator costs, operator costs, costs of stacking surface, costs of ship stay in the port and cargo costs. The single costs are calculated by appropriate formulas.

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The decision regarding the optimal solution, or the optimal capacity of the container terminal is established by comparison of different variants depending on traffic, number of berths and cranes per berth.

The application of the cost model enables to take the right business decisions for any terminal and for the terminals in present and future business conditions, which proves the hypothesis set in the introduction of this paper.

The model is tested on the real data of the container terminal at the Port of Rijeka. According to obtained results and in conditions of the existing traffic and terminal capacity, the optimal solution is the variant with two berths and three cranes. LITERATURE [1] Zenzerović, Z., Kvantitativne metode u funkciji optimalnog funkcioniranja sustava

kontejnerskog prijevoza morem, Pomorski zbornik, Društvo za proučavanje i unapređenje pomorstva RH, Pomorski fakultet u Rijeci, 43 (2005.), str. 165-191.

[2] Schonfeld, P., Frank; S., Optimizing the Use of a Containership Berth, Transportation Research Record, 984 (January 1984.), pp. 56-62.

[3] Zenzerović, Z., Optimizacijski modeli planiranja kapaciteta morskih luka, doktorska disertacija, Ekonomski fakultet Rijeka, Z. Zenzerović, 1995.

[4] Dundović, Č., Zenzerović, Z., An Optimal Capacity Planning Model for General Cargo Seaport, Traffic, Vol.12 (2000.), No. 5-6, pp. 207-215.

[5] Zenzerović, Z., Bešlić, S., Poletan, T., Model ukupnih troškova lučkoga kontejnerskog terminala, Optimizacija sustava hrvatskih kontejnerskih luka, poglavlje 4., Visoka pomorska škola u Rijeci, Fakultet za turistički i hotelski menadžment Opatija, 2001., str. 73-98.

[6] Zenzerović, Z., Vilke, S., Jurjević, M., Teorija redova čekanja u funkciji planiranja kapaciteta kontejnerskog terminala riječke luke, Pomorstvo, 25/1 (2011.), Pomorski fakultet u Rijeci, str. 45-69.

[7] Zenzerović, Z., Antonini, N., Vilke, S., Metodološki pristup istraživanju procesa opsluživanja – studija slučaja kontejnerskog teminala riječke luke, Pomorstvo, 26/1 (2012.), Pomorski fakultet u Rijeci, str. 371-393.

[8] Zenzerović, Z., Model određivanja optimalnog kapaciteta lučkoga kontejnerskog terminala, Optimizacija sustava hrvatskih kontejnerskih luka, poglavlje 3., Visoka pomorska škola u Rijeci, Fakultet za turistički i hotelski menadžment Opatija, 2001., str. 47-71.

[9] Zenzerović, Z.- Mrnjavac, E., Modelling of Port Container Terminal using the Queuing Theory, Trasporti Europei, Trieste, Anno VI, Numero 15 (2000.), ISSN 1129-5627, pp. 54-58.

[10] Zenzerović Z., Poletan, T., Utjecaj propusne moći pristana na efikasnost lučkog kontejnerskog terminala, Pomorski zbornik, Društvo za proučavanje i unapređenje pomorstva RH, knjiga 35 (1997.), str. 57-72.

[11] Zenzerović, Z., Teorija redova čekanja, Stohastički procesi II.dio, Rijeka, Pomorski fakultet u Rijeci, Sveučilište u Rijeci, 2003.

[12] Gross, D., Harris, C. M., Fundamentals of Queueing Theory, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., 1998.

[13] Chang, Y. L., WinQSB, Decission Support Software for MS/0M, John Wiley & Sons, Inc., 1998. http://winqsb.en.softonic.com/download (travanj, 2012.)