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35
УДК 656.615
THE RELATIONSHIP BETWEEN THE LOADING LEVEL AND CAPACITY OF
THE SEA PORT TERMINAL
O.A. Malaksiano, Ph.D.
M.O. Malaksiano, Ph.D.
Odessa national maritime university, Odessa, Ukraine
МалакЫано O.A., МалакЫано M.O. Стввгдношення завантаження i пропускиo'i спроможноат термшалу морсъкого порту.
Дослщжуеться взаемозв'язок м1ж р1внем завантаження термшалу морського порту i показниками об-слуговування суден. Запропоновано математичну модель, яка заснована на застосуванш Teopi'i масового об-слуговування. Отримаш результата можуть бути вико-ристаш для знаходження оптимального р1вня завантаження термшалу.
Ключовг слова: показники обробки суден, р1вень завантаження комплексу
Малаксиано А.А., Малаксиано Н.А., Соотношение загрузки и пропускной способности терминала морского порта.
Исследуется взаимосвязь между уровнем загрузки терминала морского порта и показателями обслуживания судов. Предложена математическая модель, основанная на применении теории массового обслуживания. Полученные результаты могут быть использованы для нахождения оптимального уровня загрузки терминала.
Ключевые слова: показатели обработки судов, уровень загрузки терминала
Malaksiano О.A., Malaksiano М. О. The relationship between the loading level and capacity of the sea port terminal.
The relationship between the vessels processing indicators and the loading level of the terminal is investigated. The mathematical model based on the application of queueing theory was proposed. The obtained results may be used for the estimation of the optimal terminal loading level.
Keywords: vessels processing indicators, loading level of the terminal
The necessity to determine the optimal loading level of a sea port and its terminals arises when the optimal structure of the park of cargo handling equipment is being investigated [1]. Consider the terminal which consists of interchangeable berths equipped with facilities of the same type. So any unused berth is available for the cargo operations when ship arrives.
The capacity of the sea cargo front (SCF) of terminal is one of the most important characteristic which determines capacity of the terminal in whole. By capacity of SCF we mean the maximal amount of cargo that can be processed over a given time period (usually a year) with the fullest use of the equipment and calendar operational time. In accordance with [2], daily capacity of SCF which consists of berths can be determined as
П = E П, n,
i= 1
Q
'c, tef
Qci/Ni Ptl, auxi where n denotes daily capacity of SCF of i -th berth, Qci denotes the vessel capacity, tef is the
average time period when berth is busy during a day, Nl is the number of technological lines on the SCF; is the performance of a technological line, rauxi is the duration of auxiliary operations performed with the ship at the berth, including the waiting time.
The loading level of SCF Q should not exceed its capacity, otherwise it will be impossible for the terminal to cope with the cargo traffic. The situation, when the loading level matches SCF capacity is acceptable. In this case the roads will be empty only if vessels will arrive strictly in time with intervals which equals their processing time. However, the irregularities in time of ships arriving and their processing time which, in fact, cannot be eliminated, cause a large number of vessels in the roads when loading level of SCF approaches its capacity, which in turn cause significant financial losses for port clients. That is why the loading level should be restricted by the appropriate value which is noticeably less then capacity of SCF. In accordance with specifications [2], the suggested reasonable loading level of SCF should be found as
Q = k.n,
where kz is the employment ratio, which characterizes the rate of employment of berths. Despite the practical importance, however, no significant results have been proposed to estimate the value of kz so far. The suggested value of employment ratio should be chosen from 0,6 to 0,7 for universal terminals [2], from 0,5 to 0,6 for specialized bulk or timber terminals, and from 0,4 to 0,5 for container, ro-ro or oil terminals. This specifications does not take into account such important factors as vessels traffic intensity and cargo handling operation technology. Very often prescribed in this way employment ratio can considerably vary from its optimal. For more precise estimations of employment ratio different authors propose evaluations based on empirical exploration of relation between vessels processing indicators and SCF loading level under different circumstances. This approach has obvious disadvantages which considerably complicate its implementation in practice.
The purpose of the article
The aim of the article is to investigate the relationship between the vessels processing indicators and the loading level of the terminal by means of the appropriate mathematical model based on the application of queueing theory. The main material
Consider a terminal which consists of berths. Despite the attempts to draw up and follow schedules, very often the vessels arrival dates and processing times are subject to various arbitrary factors. Denote Py(t) the probability that the number of busy berths in the moment of time t equals y. It is obvious, that
m
V Py= 1. If the vessel arrives and several berths are
y=0
not busy, we will assume that the vessel will occupy the berth with the highest capacity. Also assume that every berth can receive only one vessel at the same time and all available resources are to be shared equally between vessels which occupy the same straight-line area of the terminal. For some reasons it might be interesting to consider another rules of resources allocations and different queue regulations. Although the following model is also applicable for various cases, they are not considered here.
The ability of stationary machines to be displaced from one berth to another depends first of all on berthage configuration of the terminal. Consider the simplest case, when terminal has broken-line shaped berthage, which does not permit stationary machines to be displaced to another berths, and berths have different capacities. Let tj be the average vessel
processing time for the j -th berth,
Qj24
Then if y of berths are occupied then vessels processing intensity equals
v i
j ll
If vessel processing times are the same for all
berths ti = t1 = const, then /uy =y— . If terminal has
ti
a straight-line shaped berthage, then stationary equipment can be displaced by means of crane tracks from a vacant berth to adjacent occupied one. In this case the intensity of cargo handling operation when y of berths are occupied equals
y^—, if yK < N,
My = i
y-
t( K ) 1
if yK > N,
t (N/ y)'
where K is the upper bound of mooring machines concentration on a vessel [3], N is the general number of machines at the terminal, t(n) is the mean processing time when vessel is processed by n machines.
Now consider the influence of vessels traffic intensity on the mean vessels processing time, waiting time in the roads and the other indicators, provided all other conditions and parameters of the terminal stay unchanged. It is obvious that the simple birth and death scheme is not applicable for this case. That is why for studying this problem we will proceed from the general model of Markovian chain. Assume that the time between vessels arrival and vessels processing time are random variables with the Poisson distribution. Consider a terminal which consists of three berths and restricted road with upper bound of vessels. Under the circumstances, the terminal can be modeled as a stochastic system each state of which can be defined by the set of four variables (i, j, k, r), where i, j and k are equal to 1 if respectively the first, second and third berths are busy, and equal to 0 if appropriate berths are vacant. Value of the last variable r indicates the number of ships in the roads, r = 0,...,n . For convenience denote: C = (0,0,0,0), C2 = (1,0,0,0), C3 = (0,1,0,0), C4 = (0,0,1,0), C5 = (1,1,0,0), C6 = (1,0,1,0), C7 = (0,1,1,0), C8 = (1,1,1,0), C9 = (1,1,1,0), C10 = (1,1,1,1), C„ = (1,1,1,2), C8+n = (1,1,1, n)
For the referred above model, the intensity of vessels processing on the i -th berth in the moment of time t depends on state of other berths at the moment. Let /uiy be the mean intensity of vessels processing on the i -th berth, when terminal is in the y-th state, and 2 is the mean intensity of vessels
for all i = 1,3,
t, =-
J W/
+ Tauxj ■ J = 1 m ■.
arriving. Assume that
y = 9, n . Denote random events which consist in finding the terminal
Mi8 = Miy /j. = ¡u18 + m28 +M38 • Consider
EKOHOMIKA: pecwii uacy
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ECONOMICS: time realities
in the state Ci, i = 1,...,8 + n at the fixed moment of time t. Denote pi (t) the probability to find the terminal in the state Ci at the moment of time t. If all probabilities pi (t) are known, one can easily obtain all required vessels processing indicators. In order to find these probabilities using well-known method [4] we will reduce the problem to study of appropriate differential equation. Fix an arbitrary moment of time and small time interval At. The probability that at the moment of time t + At the terminal will be in the state C1 equals
p1(t + At) = p1(t)-(1 -A-At) + p2(t)-u12 ■At +
- (1)
+ p3 (t) ■ u23 - At + p4 (t) ■ u34 ■ At + o(At)
The first summand in the right part of (1) expresses the probability of situation when there were no vessel at the time at the terminal and none has arrived during the interval after. The sum of the rest of summands express the probability that there were one vessel at the moment of time at the terminal and by the moment of time it will be processed. After the passage to the limit when from the equation (1) follows
p^ =-p(t) ■ A+P2 (t) ■ UU2 + P3(t) - U23 + P4© - U34
In much the same way one can obtain equations for the rest of states of the terminal. As the result we obtain the following system of differential equations.
0=-P1 ■ A+P2 ■ U12 + P3- U23 + P4- U34 0=P1 ■A-P2 ■ (A+U12) + P5 ■ U25 + P6 ■ uu 0 = -P3 ■(A + U23) + P5 ■ U15 + P7 ■ U37 0 = -P4 ■ (A+U34) + P6 ■ U16 + P7 ■ U27 0 = -P5-(A+UU5 +U25) +(P2 + P3)-A + P8- U38 0 = -P6-(A + UU6 +U36) + P4 -A + P8-U28 0 = -P7 - (A+U27 +U37) + P8- UU8 0 = -P8 ■ (A+u) + (P5 + P6 + P7) ■ A+P9 ■ u
(3)
0=Pi-1- A-Pi •(A+U) + Pi+1-u
0 = P7+n-A-Pt+n- u
(i = 9,...8+n)
Now when solution of (3) is obtained, the vessels processing indicators can be easily calculated.
In accordance with the established order, the capacity of a terminal is calculated over a long period of time, usually longer than year allowing for meteorological factors and necessary equipment service. In accordance with specifications [5], we will distinguish the following types of repair works: maintenance, scheduled repair (routine repair, capital repair) and emergency repair. Maintenance is usually carried out during the reception and delivery of shifts, or in the intervals of cargo operations. Rarely for this purpose, machines can be removed from service, but usually no more than for one or two shifts a month. So maintenances do not affect noticeably the capacity of the terminal. Frequency and duration of the scheduled repairs are governed by existing specifications [5], and their implementation is controlled by appropriate state supervisory authorities. The duration of one routine repair and capital repair equals approximately to one and three months respectively. The frequencies of such repair works are determined by operating time. For example the scheduled repairs of portal cranes at the average are carried out once in a two o three years. The capital repair is carried out once for every five routine repairs. During its life cycle portal crane for example, usually undergoes one or two capital repairs. The economically substantiated terms of repairs and retirement for terminal port equipment when forecast level of employment is subject to change can be determined by [6]. The scheduled repairs take a lot of time and that is why they should be taken into account when capacity of the terminal is estimated. Denote T the forecasting time-frame. Let TL be the total time when L machines are under repair. It is obvious that
N
The solution of system (2) provided normalization V TL = T, where N is general number of machines
dfl(t)
= -P1 (t>■ A+ P2 (t) ■ Wu + P3(t) ■ u23 + P4(t) ■ u>4
dt
dPPj-L = P1(t> A-P2(t) ■ (A+M1i)+P5(t) ■ №25 + P6(t> u dt
dP7t) = -P3(t) ■ (A+ub)+P5 (t) ■ u + P7 (t) ■ u dt
dP4(t)_
dt dPs(t)
" = P4 (t) ■(A + u34 ) + P6(t)-u6 + P7 (t) ■ Wn
— = P5 (t) ■ (A+ u + u25)+(P2 (t)+P3W) ■ A+ P8(t) ■ u
dt
dp6(tL = -P6(t)-(A+u6 + u36) + P4 (t) ■ A + P8 (t) ■ u28 dt
dp;(L = P7(t) ■ (A+M27 + u37)+P8(t> u dt
dP8(t)_
(2)
dt dPi (t)
= -P8 (t) ■ (A+ u)+(P5 (t)+P6 (t)+P7 (t)) ■ A+P9 (t) ■ u
dPj+n (t) .
= PM(t)-A-Pi (t)-(A+u)+Pi+1(t)-u (i = 9,...8+n)
P7+n (t)-A-P8+n (t)-u
condition V P;- = 1 and appropriate initial conditions
i=1
gives a full information about the queuing system. By means of passage to the limit when t ^ & system of differential equations (2) transforms into the following system of linear equations (3), where Pi = lim p; (t) expresses the limiting probabilities
when the terminal runs in the stationary mode.
L=0
at the terminal. The values of L and TL can be easily found from the preventive overhaul schedule. Besides scheduled repairs, machines can be put out of action for long terms because of emergency repairs. The emergency breakdown of equipment is a random event and the duration of its repair is also a random variable. That is why the number of machines AL being at the same time under emergency repair is a random variable. The mean number of up state machines at the terminal NL should be found
allowing for number and characteristics of equipment at the terminal and number of available service stations [7]. And the capacity of the area of a sea cargo front n(Nl ) is based on the number of the good state machines. The mean capacity of the whole sea cargo front on the forecasting time-frame T, allowing for the interruptions caused by meteorological reasons and repairs can be estimated by the formula
nr = S T L kmL n N )
L=0
where kmL is the coefficient of outage caused by meteorological conditions [2].
Now for illustration consider the particular case when the number of vessels in the roads is bounded by 12, terminal has a straight-line berthage and consists of three berths equipped by nine rigs with
Table 1. The relationship between the vessels processing indicators and the loading level of the terminal
loading level of the terminal Vessels arriving intensity, vessels per hour The probability to find no vessels at the terminal The probability to find one vessel at the terminal The probability to find two vessels at the terminal The probability to find three vessels at the terminal The probability to find at least one vessel in the roads The probability to find the road overflowed The mean number of vessels at the terminal without the road The mean cargo handling processing time, hours The mean waiting time in the roads, hours The mean overall waiting time (time in the roads + handling processing time), hours
0,05 0,002 0,878 0,114 0,007 0,114 0,000 0,000 0,130 65,126 0,012 65,139
0,11 0,004 0,770 0,200 0,026 0,200 0,000 0,000 0,262 65,467 0,097 65,564
0,16 0,006 0,675 0,263 0,051 0,263 0,002 0,000 0,396 65,976 0,326 66,302
0,22 0,008 0,591 0,307 0,080 0,307 0,005 0,000 0,534 66,619 0,774 67,393
0,27 0,010 0,515 0,335 0,109 0,335 0,011 0,000 0,675 67,371 1,523 68,894
0,33 0,012 0,448 0,349 0,136 0,349 0,022 0,000 0,820 68,210 2,674 70,884
0,38 0,014 0,388 0,353 0,161 0,353 0,038 0,000 0,971 69,122 4,348 73,470
0,44 0,016 0,334 0,347 0,180 0,347 0,061 0,001 1,125 70,093 6,711 76,804
0,49 0,018 0,285 0,333 0,195 0,333 0,092 0,003 1,285 71,112 9,981 81,093
0,54 0,020 0,241 0,313 0,203 0,313 0,132 0,005 1,449 72,170 14,462 86,632
0,60 0,022 0,201 0,287 0,205 0,287 0,183 0,009 1,617 73,256 20,559 93,815
0,65 0,024 0,165 0,257 0,201 0,257 0,246 0,016 1,790 74,359 28,785 103,144
0,71 0,026 0,133 0,224 0,189 0,224 0,320 0,024 1,964 75,464 39,701 115,164
0,76 0,028 0,104 0,189 0,172 0,189 0,404 0,034 2,138 76,547 53,768 130,315
0,82 0,030 0,079 0,153 0,150 0,153 0,496 0,044 2,308 77,578 71,069 148,647
0,87 0,032 0,057 0,119 0,124 0,119 0,591 0,054 2,465 78,518 91,011 169,529
0,93 0,034 0,040 0,089 0,098 0,089 0,682 0,062 2,604 79,334 112,211 191,545
0,98 0,036 0,027 0,063 0,074 0,063 0,764 0,065 2,719 80,001 132,780 212,781
performance of 50 tons per hour each. Assume the mean vessel capacity of 10000 tons. And put the upper bound of concentration equals three rigs per vessel. In this case the mean ships processing conditional intensities are
M12 = M23 = M34 = M15 = M25 = Mie = M36 = M27 = M37 = °,°153846 M18 = M28 = M38 = 0,0122448 The analysis of this example shows that the gradual increase in the loading level leads to the moderate increase in the mean vessels processing time from 65 up to 80 hours, while the mean waiting time in the roads at the same time undergoes the considerable growth from 1 up to 132 hours. The vessels processing indicators are almost independent from the loading level of the terminal if its value does not exceed 0,38, and are subjected to considerable change for the worse otherwise (table 1, picture 1).
EKOHOMIKA: peawii uacy
№2(12), 2014
ECONOMICS: time realities
Picture 1. The dependence of the mean vessels cargo handling processing time (a), the mean waiting time in the roads (b) and the mean overall waiting time (c) from the loading level of the terminal
This picture allows to analyze the relationship between the average vessels processing indicators and the loading level of the terminal. Although the average values of these indicators may be satisfactory, nevertheless it is possible that in some periods of time these indicators can vary significantly from their averages. In many situations these variations are highly undesirable. That is why in many cases it is important to plan the loading level of the terminal in accordance with its capacity so that the system would be able to return quickly to its normal state after the crisis situation has arisen. The considered above model allows to estimate the average time during
which the terminal would be able to return to its normal stationary mode after the crisis situation has occurred which has caused the accumulation of large number of vessels in the roads. The curves of the change in the probability to find no vessels at the terminal, the change in the probability to find one vessel at the terminal, the change in the probability to find two vessels at the terminal, and the change in the probability to find all berths busy, in the case when the loading level of the terminal equals 0,27 and the road is overflowed at the beginning of the period are depicted on the picture 2.
Picture 2. The change in the probability (a) - to find no vessels at the terminal, (b) - to find one vessel at the terminal, (c) - to find two vessels at the terminal, (d) - to find all berths busy, in the case when the loading level of the terminal equals 0,27 and the road is overflowed at the beginning of the period.
The changes in the probability that at least one of indicators can seem to be satisfactory, in view of the
the berths will be available if the loading level of the terminal equals 0,27 or equals 0,6 or equals 0,82, and the road is overflowed at the beginning of the period are depicted on the picture 3. The curves on picture 3 shows that although average vessels processing
fact that the system is unable to restore quickly after the crisis situations, in some cases it would be advisable to take steps to reduce the loading level of the terminal or increase the capacity of the terminal.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
500
1000
1500
2000
2500
3000
3500
4000
Picture 3. The change in the probability that at least one of the berths will be available if the loading level of the terminal (a) - equals 0,27, (b) - equals 0,6 or (c) - equals 0,82, and the road is
overflowed at the beginning of the period.
Conclusions
The obtained relations between the vessels processing indicators and the loading level of the terminal may be used for the investigations of the optimal loading level which brings maximal profits
for the "port-carrier-client" system. This method also can be used for the investigation of the reverse problem, that is the optimization of the various terminal characteristics in order to fit the given loading level.
References:
1. Malaksiano O.A. The mathematical model of development of the park of sea port stationary equipment [Text] / M.O. Malaksiano // Visnyk naukovyh prats ONMU. - Odesa: ONMU, 2003. -Vol. 12. - P. 147 - 155.
2. The specifications of technological design of the sea ports [Text] / VNTP 01-78 / Minmorflot. M. : CRIA "Morflot", 1980. - 122 p.
3. Malaksiano O.A. The mathematical model of the problem of the finding of the upper bound of concentration for the ships with the upper processing method [Text] / M.O. Malaksiano // Visnyk naukovyh prats ONMU. - Odesa : ONMU, 2002. - Vol. 8. - P. 176 - 184.
4. Kovalenko I.N. Introduction to Queueing [Text] / I.N. Kovalenko, B.V. Gnedenko -N.Y. Birkhauser Verlag AG, 1989. - 336 p.
5. The specifications of technical exploitation of cargo handling equipment of sea ports [Text] / KND 31.4.002-96. Part 1 - Odesa : Pivden NDIMF - 98 p.
ЕКОНОМ1КА: реалп часу
№2(12), 2014
ECONOMICS: time realities
6. Malaksiano M.O. On the optimal repairs and retirement terms planning for terminal port equipment when forecast level of employment is uncertain [Text] / M.O. Malaksiano // Economic cybernetics. - 2012. - № 4 - 6 (76 - 78). - P. 49 - 56.
7. Zubko N.F. The calculation of the mean overall time budget of the park of machines of the same type [Text] / N.F. Zubko // Visnyk naukovyh prats ONMU. - Odesa: ONMU, 2003. - Vol. 12. - P. 156 - 163.
Надано до редакци 01.02.2014
Малакаано Олександр Антоншович / Oleksandr A. Malaksiano
almala_nm@ukr. net
Малакаано Микола Олександрович / Mykola O. Malaksiano
malax@ukr.net
Посилання на статтю /Reference a Journal Article:
The relationship between the loading level and capacity of the sea port terminal [Електронний ресурс] / O.A. Malaksiano, M.O. Malaksiano //Економка: реали часу. Науковий журнал. — 2014. — № 2 (12). — С. 21-27. — Режим доступу до журн.: http://economics. opu. ua/files/archive/2014/n2. html
