841
1 INTRODUCTION
Every port consulting and design organization has its
own carefully selected toolkit of port project and
design techniques and know-how to use them
efficiently. In the same time, all the designers face the
very same problem: at early stages of the project it is
not wise to use sophisticated and advanced tools that
give accurate results, but simple tools permit to gain
only rough estimations [1-3]. The methodological
problem behind this contradiction is in the nature of
the data used by these approaches. Simple and easy
formula computations deal with the deterministic
data, while the reality demands to take into account
the fluctuations caused by the stochastic character of
all main variables [4-6]. This paper offers an extension
of the classical deterministic approach that will help
to cover the gap between analytical and simulation
approaches. Moreover, the results could turn to be
Analytical Assessment of Stochastic Spread of Demand
for the Port Storage Capacity
A.L. Kuznetsov
Admiral Makarov State University of Maritime and Inland Shipping, Saint-Petersburg, Russia
H. Oja & A.D. Semenov
Konecranes Finland Corp., Hyvinkää, Finland
ABSTRACT: At design stages of any sea port development projects one of the key tasks is to estimate the
amount of cargo volume to be stored on the port warehouse. The shortage of the warehouse facilities would
disrupt port operations and affect the port marketing position, while the surplus capacity would raise the self-
cost of the services rendered by the port. Many port developing projects and long years of operational practice
have resulted into certain commonly accepted mathematical techniques that enable to assess all main
parameters sufficiently accurately. With ever-growing completion between the ports worldwide, to find a
delicate balance between the cost and quality becomes a core task behind nearly every aspect of port design
activity. The tools that have been used for centuries in port design and development started to lose their
adequacy in modern economic and logistic environment. As the response for this challenge the port designers
more and more move to simulation models. In the same time, an adequate simulation models need not only
accurate and reliable data, but also requests quite long time. Moreover, the models of the kind usually are
created ad hoc, reflecting particular features of the primal object under development and forfeiting the
generality and universality of analytical models. At beginning stages of port developing one need to have
simple and easy tools for the preliminary accession of project parameters, since usually there are several
variants and the full-scaled simulation of them is excluded. Still, these tools should be more enhanced
sophisticated than common analytical formulae. The main drawback of the formula calculation (streaming
computing by the current IT terminology) is they principally deal with deterministic values, while the real
worlds is inhibited with the stochastic ones. The study represented here is an attempt to narrow this gap. The
area selected to demonstrate the approach is the port warehouse size, regardless of the cargo type handled. In
the same time, this technique can be spread on many other port project parameters needed to be assessed.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 4
December 2020
DOI: 10.12716/1001.14.04.07
842
useful in proving the adequacy of any simulation
models, which itself is a very big problem.
2 METHODS AND MATERIALS
Let’s consider the calculation of the average amount
of cargo that dwells in the port
E
, when we know
the average duration of the cargo party formation
(accumulat ion on the warehouse)
f
T
, mean
interval of party arrivals (ship calls interval)
int
T
and
mean ship party’s volume (call size)
V
. Under the
assumption of triangular form of the party
accumulation (i.e. with the constant rate of cargo
arrival/departure on the terminal), within the one
party’s dwell time on the terminal
2
f
dw
T
T =
there
will be
1
2
f
dw
int int
T
T
TT
=
cargo parties of the volume
V
entering the warehouse, so the average total
amount is
1
2
f
int
T
ЕV
T
=
.
All of variables in this formula are stochastic
values, thus the resulting volume of the cargo storage
is the stochastic one, too. What judgments on the
character of the end value can we made?
For the sake of convenience let us make the
substitution of the working variables, particularly
VX=
,
f
TY=
, Z =
1
int
T
. In this notation the
formula will take the form of
1
E XYZ
2
=
. Let us
assume that we know the values of two main numeric
characteristics of the values
X, Y, Z
, specifically
the mathematical expectations
x
M X m=
,
y
Ym=
,
z
M Z m=
and dispersions
x
D X D=
,
y
D Y D=
,
z
D Z D=
.
Let us compute the characteristics of the target
computation value of the amount of storage, or
E
M E m=
and
E
D E d=
.
The evaluation of the mean arithmetic values
causes no difficulties since by the theorem on the
computation of independent stochastic values we
directly have
11
XYZ XYZ
22
E
M E m M M
= = =
.
In other words, the mathematical expectations of
cargo store amount is
1
2
=
x y z
M E m m m
The computation of the dispersion is a bit more
laborious. Actually, let us denote
XYZ A=
. Since
for any non-stochastic value
c
we have
2
D cA D Ac=
, then
11
D A D A
24
=
.
Further on, by the definition of dispersion we
have
( )
2
D XYZ D A
A
M A m
= = −
. Since
the values
X, Y, Z
are independent,
A x y z
m m m m=
. Respectfully
( )
2
D XYZ
x y z
M XYZ m m m
= − =
2 2 2 2 2 2
2 XYZ
x y z x y z
M X Y Z m m m M m m m
= − +
With independent
X, Y, Z
the values
2 2 2
, , X Y Z
also are independent, so
2 2 2 2 2 2
M X Y Z M X M Y M Z
=
and
XYZ
x y z
M m m m=
and further
2 2 2 2 2 2
D XYZ
x y z
M X M Y M Z m m m
=−
.
In the same time,
2
MX
is the second initial
moment of the stochastic value
X
, so it could be
expressed through the dispersion as
22
x
M X D X m
=+
. Similarly,
22
y
M Y D Y m
=+
and
22
z
Z D Z m
=+
.
If inserted in the received formulae, these
expressions will give
2 2 2 2 2 2
D XYZ
x y z
M X M Y M Z m m m
= − =
( )
( )
2 2 2 2 2 2
( )
x y z x y z
D X m D Y m D Z m m m m+ + + − =
2 2 2 2 2
2 2 2 2
x y z x y z x y z x y z x y z
x y z x y z
D D D m D D D m D D D m m m D
m D m D m m
+ + + + +
++
Eventually, the seeking dispersion of the total
amount of cargo stored at a terminal could be
expressed as
843
2 2 2
2 2 2 2 2 2
(
)
1
4
= + + +
+ + +
E x y z x y z x y z x y z
x y z x y z x y z
D D D D m D D D m D D D m
m m D m D m D m m
The standard deviation of this value is
E
=
E
D
.
3 RESULTS AND DISCUSSION
We could expect by the central limit theorem (stating
that the sum of many weakly interdependent values
has a distribution close to the normal one) that the
stochastic value of total amount of cargo stored at a
terminal has the Gaussian distribution [7]. Since we
managed to assess the values of its mathematical
expectation and dispersion (standard deviation), we
could construct the correspondent cumulative
distribution function, as Fig. 1 shows.
Figure 1. Gaussian cumulative distribution function
By the definition, the cumulative distribution
function is the probability of the event
eE
,
where
e
is a current variable. In the context of our
study, we could interpret this function as the
probability that the warehouse of the size
e
would
be sufficient to contain the required amount of cargo
[8-9]. If the size is equal to the mathematical
expectations, in 50% it will be enough and in 50%
there will be the shortage of store facilities.
Knowing the properties of Gaussian distribution,
we could expect also, that the interval
2 , 2
E E E E
mm
− +
holds around 95% of all
values, thus the warehouse with the size
2
EE
m
+
will be insufficient only in 2,5%, as Fig. 2 shows.
Figure 2. Assessment of the required warehouse size
In order to prove the correctness of the results, the
simulation experiments were conducted. Fig. 3 shows
the results of an experiment of this simulation, with
vessels arrival intervals, party sizes and dwell times
distributed by Gaussian distribution.
Figure 3. Simulation of the warehouse dynamics
The values of the referenced parameters are
represented by the tab. 1.
Table 1. The referenced parameters of simulation
Description Notation Units
M
x
σ
x
Cargo party V [units] 2400 240
Formation time T [hours] 120 12
Arrival interva t [hours] 12 2
The results of the statistical processing (i.e. the
experimental distribution of values shown by Fig. 3,
are displayed on Fig. 4 a) and b).
Figure 4. Distribution of the simulated values
Density of distribution b) Cumulative distribution function
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The calculation by the technique described in this
study gives the characteristics of the final distribution
shown by the Tab. 2.
Table 2. Calculated parameters of simulation
Description Notation Units Value
Math.expectation of E
M
E
[units] 12014
Dispersion of E D[XYZ]
[units
2
] 9012302
Standard deviation σ [units] 3002
Spread 2*σ [units] 6004
Upper limit
M
E
+2*σ
[units] 18018
The calculated mathematical expectation and
dispersion enables to build the model cumulative
distribution function (Fig. 5) which practically
coincides with the one produced by the processing of
the simulation data shown by Fig. 4.
Figure 5. Reconstructed cumulative distribution function of
the warehouse size
The proximity of these functions was confirmed
by statistically valid amount of experiments with
different distributions (some of them even non-
Gaussian), thus enabling to state that this simple
technique is adequate. The designer would only make
several reasonable assumptions over the distribution
of the input values and immediately could see the
spread of the output values. Maybe not for 100%
reliable, this method could find a proper slot in the
port designer’s toolkit.
4 CONCLUSIONS
1 The analytical (formula) calculations cannot give
the perception of the values’ spread over mean
values.
2 The full-scaled simulation could bring the desired
results but at the cost of developing laborious
procedures not justified on the beginning stages of
the port projects.
3 The method proposed in this study enables to
receive a reasonable estimation of the stochastic
values by rather small extension of common
formula deterministic technique.
4 The method does not take into account any
specific properties of the cargo and thus could be
used for all types of port warehouses.
5 The approach described in the paper using the
example of the warehouse could be extended to
cover the assessments of other technological
parameters treated as stochastic values.
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