International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 3
September 2011
385
1 INTRODUCTION
Ship routing is one of traditional navigational tasks
directly related to her safe and efficient exploitation.
Rising fuel prices and overcapacity spurring owners
to implement on more of their ships the slow steam-
ing policy that obviously makes ocean passage stage
of the voyage longer and consequentially increases
risks connected with ship operation in heavy weather
conditions. Hence, the problem of optimal compro-
mise between safety and economy become even
more crucial. At the present time series of routing
methods exist, such as isochrones (James 1959,
Bijlsma 2004, Szlapczynska, Smierzchalski 2007),
graph (Vagushchenko 2004, Padhy et al. 2008), ex-
pert (Oses, Castells 2008) and intelligence methods
(Nechayev et al. 2009). All of them allow to perform
route optimization by number of preset criteria.
However the main problem, connected to optimiza-
tion process, that’s still remaining, is to obtain the
objective function based on formalized relationship
between ship motion parameters, power inputs,
needed for environmental disturbances compensa-
tion, and route economical efficiency. One of the
possible solutions of this problem is given below.
2 OPTIMIZATION TASK & OBJECTIVE
FUNCTION
The solution of the above mentioned problem is
based on the next hypothesis (Pipchenko 2009): op-
timal route in prescribed weather conditions is such
combination of route legs and corresponding engine
loads, on which expended ship power inputs are
closest to minimum and predicted voyage time does
not exceed scheduled one, with regard to the safety
limits.
To assess the economical efficiency of the route,
one can divide overall ship costs in two categories:
minimal-unavoidable costs, needed for the voyage in
ideal conditions that can be expressed by minimal-
unavoidable work А
min
, and additional work А.
Therefore, total work, performed during the voyage
can be given as:
min
AA A= +∆
, (1)
or as voyage time integral of variable engine power:
( )
T
A P t dt=
, (2)
where Р = main engine power.
Minimal-unavoidable work can be defined from
the condition of minimum work performed during
specified time with constant engine power on the
shortest distance between ports:
( )
( )
min
, with S min
v
const v
T
A P dt S= =
, (3)
On the Method of Ship’s Transoceanic Route
Planning
O. D. Pipchenko
Odessa National Maritime Academy, Odessa, Ukraine
ABSTRACT: In this article control of ship on a transoceanic route is represented as multicriteria optimization
problem. Optimal route can be found by minimizing the objective function expressed as ship integral work for
a voyage, taking into account ship’s schedule, weather conditions, engine loads and risks connected with ship
dynamics in waves. The risk level is represented as non-linear function with heterogeneous input variables
which estimated by means of multi-input fuzzy inference system on the basis of pre-calculated or measured
ship motion parameters. As the result of this research the optimal transoceanic route planning algorithm is ob-
tained.
386
where S
v
= the shortest route length; T
v
= scheduled
voyage time; S() = length depending on route
configuration.
Thus, the main voyage optimality criterion, with-
out risks consideration, is the minimum of addition-
ally performed work, appeared due to weather, time
and distance limitations. This work can be given as
voyage length integral of additional resistance R
W
arisen due to environmental disturbances:
W
S
A R ds∆=
(4)
From equations (2), (3) the additional work can
be obtained as:
(5)
Therefore, the objective function representing the
specified route optimality can be expressed as:
( )
min
,
v
TT
Z P t dt Z A
=
(6)
For the full-valued solution of the problem, it is
also necessary to take into account corresponding
limitations. For this purpose the risk assessment
concept was used and next was formulated: the op-
timal route is found if the total work for the voyage
is closest to minimal, voyage time does not exceed
the scheduled one, and the risk level on each route
leg is less then specified limit. Thus, the objective
function will be given as:
( )
( )
( )
( )
( )
max
R,
min ,
R,
p
safe
TT
W safe
Ut
Z P dt
RU t




=


+∆



P
P
, (7)
where U
safe
= maximum safe speed, at which the
specified hazardous occurrence risk R is below the
critical limit; P
max
= maximum engine power; Р =
engine power needed to keep defined calm water
speed;
Р(R
W
) = additional power needed to com-
pensate the resistance due to environmental disturb-
ances R
W
; R (0,1) = risk level on the specified
route leg.
3 RISK EVALUATION
3.1 Problem definition
According to the route optimality definition, given
above, the risk level conducted with ship activity in
prescribed weather conditions shall be determined
for each route leg. Therefore, we define the leg as
the part of the route on which ship control regime
(speed and heading) and weather conditions remain
constant. As opposed to classical definition two or
more different route legs may be situated on one line
between the waypoints, depending on weather grid
density.
Mathematically the risk level can be defined as
product of likelihood of hazardous occurrence and
its consequence. In our case we define likelihood as
probability of reaching defined dynamical motion
parameters that may lead to the series of negative
consequences, conducted with ship’s operation in
storm.
Assessing the risks of ship operation in heavy
weather conditions one can define the situations
connected with damages to hull structure, ship’s sys-
tems and machinery and the situations arising due to
violations of cargo handling technology.
For instance, the achievement of defined high
amplitudes of roll may lead to the series of situations
with different levels of consequences, such as shift-
ing or loss of cargo, flooding of ship’s compart-
ments, capsizing. Therefore, next risk levels can be
highlighted: insignificant, low, practically allowable
and not allowable. The risk management should
cover such measures which allow to vary the proba-
bility of definite event or to reduce the degree of its
consequence. When solving the problem of safe ship
control regime selection in heavy seas we assume
the degree of consequence as constant. From the
other hand by altering ship control settings operator
can affect the probability of reaching such ship mo-
tion parameters that lay beyond the limits of practi-
cally allowable risk. In this case the risk level can be
given as
( )
12
R , ,...,
n
fpp p=
, (8)
where p
1
, p
2
,…,p
n
= probabilities of reaching the
ship motion parameters, that may lead to definite
hazardous occurrence.
3.2 Seaworthiness criteria
To perform the risk assessment and to find a safe
control regime in given weather conditions it’s nec-
essary to define appropriate criteria, thereupon fol-
lowing factors should be taken into account:
frequency and force of slamming;
frequency of green water;
motion amplitudes;
hull stresses;
propeller racing;
accelerations in various ship points;
forced and controlled speed redaction.
387
Table 1.
General operability limiting criteria for ships.
Criterion
Cruikshank &
Landsberg (USA)
Tasaki et al.
(Japan)
NORDFORSK, 87
(Europe)
NATO STANAG
4154 (USA)
RMS of vertical accelerations on
forward perpendicular
0.25 g 0.8 g / p = 10
-3
0.275g (L
pp
< 100 m)
0.05g (L
pp
> 300 m)
-
RMS of vertical accelerations on the
bridge
0.2 g - 0.15g 0.2g
RMS of transverse accelerations on
the bridge
- 0.6 g / p = 10
-3
0.12g 0.1g
RMS of roll motions 15° 25°/ p = 10
-3
6° 4°
RMS of pitch motions - - - 1.5°
Probability of slamming 0.06 0.01
0.03 (L
pp
< 100 m)
0.01 (L
pp
> 300 m)
-
Probability of deck wetness 0.07 0.01 0.05 -
Probability of propeller racing 0.25 0.1 - -
*The significant motion amplitudes (Х
1/3
) can be obtained by doubling the corresponding RMS (root mean square value).
Table 2. Management level navigators inquiry results.
Roll motion
amplitude, °
Slamming, intensity
per hour
Deck wetness,
intensity per hour
Speed reduc-
tion, %
Deviation from
course, °
Small < 7 < 5 < 5 < 13 < 20
Not dangerous
< 14
< 11
< 10
< 24
< 38
Substantial
< 23
< 19
< 20
< 46
> 40
Dangerous
> 26
> 23
> 23
> 58
-
*The average values of inquiry data are given.
** Example: slamming probability with period of pitching 5 sec and intensity 20 times/hour: 0.028.
The comparative table of general operability lim-
iting criteria for wide variety of ships in waves com-
bined from data of Lipis (1982) & Stevens (2002) is
given in table 1. However criteria of NORDFORSK
and NATO STANAG appear to be too strict, and in
series cases, when ship proceeds through a heavy
storm, the motion parameters may exceed these cri-
teria.
According to inquiry of management level navi-
gators (captains and chief mates) passing the Ship
Handling course in Training & Certifying Centre of
Seafarers of Odessa National Maritime Academy
(TCCS ONMA) empirical values of ship operability
criteria were obtained (table 2).
Usage of last gives possibility to perform more
detailed, supported by personal seagoing experience
of navigators, assessment of ship state in waves.
It should be noted that risk assessment by only
threshold values, defined for the series of criteria is
ineffective. Therefore, we suggest to apply not two-
valued state assessment function, but numerical or
linguistic function, defined in range between two ex-
treme values: «0» - «1», «best» - «not allowable»
(minimal maximal risk level).
4 FUZZY LOGIC ASSESSMENT
4.1 Assessment algorithm
To implement above mentioned suggestion seawor-
thiness assessment system consisting of two fuzzy
inference subsystems (FIS) was built (fig. 1) on the
basis of more complex model given in (Pipchenko,
Zhukov 2010).
Figure 1. Multicriteria seaworthiness assessment system
x
1
…x
n
= motion parameters, S
1
…S
n
= corresponding rates, R
= risk level.
Following algorithm was adopted in the system to
define the generalized risk level from several motion
parameters. Ship motion parameters, taken as the
system input, pass the FIS structure of the 1
st
level.
As the result series of rates on each criterion in form
of numerical or linguistic variables (for instance,
slamming impact: “small”, “substantial” or “danger-
ous”) received on its output.
In course of definition system’s membership
functions (MF) it is suggested to form boundary
388
conditions on the basis of existing international op-
erability criteria, and MF’s intermediate values by
approximation of preliminary transformed expert in-
quiry data.
After that obtained rates pass the FIS of the 2nd
level, on the output of which the general assessment
on the set of conditions is obtained in the form of
risk level. For defuzzification Mamdani algorithm
was used in both subsystems.
4.2 Membership functions evaluation
Let’s describe the FIS membership functions (MF)
definition process on example of roll amplitude.
Maximum allowable roll amplitude can be deter-
mined from condition:
{ }
limit
1/3
min , , ,
shift flood capsize operator
ϕ ϕϕ ϕ ϕ
=
, (9)
where
ϕ
shift
= cargo critical angle;
ϕ
flood
= flooding
angle;
ϕ
capsize
= capsize angle;
ϕ
operator
= operator de-
fined maximum roll amplitude. For general case the
maximum angle of 30° was chosen.
For each linguistic term a numerical interval, on
which a membership function is defined, can be
found from condition:
{ }
( )
{ }
**
0,max , 0,1, 2..., max
TT
N
ϕ ϕϕ ϕ
=
, (10)
where
*
T
ϕ
= values declared by respondents as limits
for specified terms. For roll amplitude these terms
are: “Non Significant” NS, “Not Dangerous”
ND, “Significant” S, “Dangerous” – D.
The principal variable on which the computation
of experimental membership function made in the
work is relative term repetition frequency
max
T TT
ν νν
=
,
ν
T
= quantity of respondents, de-
clared specific value (i.e. roll amplitude is “non sig-
nificant”, if ϕ < 5°),
max
T
ν
= maximum number of
value repetitions for specified term.
Basing on relative term repetition frequency ex-
perimental data for membership functions
*
T
µ
ob-
tained in the way given below.
For “Non Significant” amplitude term
*
NS
µ
:
( ) ( ) ( )
( )
( ) ( ) ( )
( )
*
*
1 , for max
/ 2,for max
NS NS NS
NS NS NS
µ ϕ ν ϕ ϕϕ ν
µ ϕ ν ϕ ϕϕ ν
=−<
=


(11)
For “Not Dangerous” amplitude term
*
ND
µ
:
( ) (
) ( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
( )
*
*
*
/ 2,for max
1 , for
max max
/ 2,for max
ND NS NS
ND ND
NS ND
ND ND ND
µ ϕ ν ϕ ϕϕ ν
µϕ νϕ
ϕνϕϕν
µ ϕ ν ϕ ϕϕ ν
= <
=
≤<
=



(12)
For “Significant” amplitude term
*
S
µ
:
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
( )
*
*
*
/ 2,for max
1 ,for
max max
/ 2,for max
S ND ND
SS
ND S
SS S
µϕ ν ϕ ϕϕ ν
µϕ νϕ
ϕνϕϕν
µϕ νϕ ϕϕ ν
= <
=
≤<
=



(13)
From table 2 it can be seen that limit values for
terms NS, ND & S roll amplitudes were defined
from condition
ϕ
<
*
max
ϕ
. At the same time term
“Dangerous” amplitude was defined from condition
ϕ
>
*
max
ϕ
,
therefore:
( ) ( )
*
DD
µϕ νϕ
=
(14)
On the basis of experimental membership func-
tions values, following function can be approximat-
ed for application in fuzzy inference algorithm:
( )
( )
( )
2
max
2
/
2
max
max
,
1,
c
e
ϕϕ
σ
µϕ ϕ ϕ
µϕ ϕ ϕ
−−
= <
=
(15)
where σ, с = function parameters.
As result of approximation four MF’s were ob-
tained (fig. 2.).
4.3 Rules set definition
To make an inference or to get a determined ship
state assessment applying fuzzy logic it is necessary
to construct corresponding set of rules.
As input parameters roll amplitude and “maxi-
mum probability” coefficient were applied in sug-
gested system. “Maximum probability” coefficient
K
SGR
(0,1) can be determined as:
max max max
min 1, мах , ,
S GW
R
SGR
S GW R
pp
p
K
ppp


=





(16)
( )
0,1
SGR
K
,
where p
S
, p
GW
, p
R
= slamming, green water and pro-
peller racing probabilities, superscript max means
maximum allowable criterial value.
The output risk level R is divided in four linguis-
tic terms: «non significant», «low», «allowable» and
«not allowable».