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= = ℜ
∫
, (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(ℜ) = length depending on route ℜ
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:
( )
min
T
A P t dt A∆= −
∫
(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».
389
Figure 2. Roll amplitude assessment membership functions
The corresponding set of rules is given in table 3.
Table. 3. Risk evaluation rules set.
№
Roll amplitude,
ϕ
Probability
coefficient,
K
SGR
Conclusion
Risk level,
R
1 IF Non significant AND Low Non significant
2 IF Non significant AND Moderate Low
3 IF Not dangerous AND Low Non significant
4 IF Not dangerous AND Moderate Low
5 IF Significant AND Low Allowable
6 IF Significant AND Moderate Allowable
7 IF Dangerous OR High Not allowable
Thus, the risk level for each route leg can be as-
sessed on the basis of weather prognosis data and
measured or predicted ship motion parameters. Such
prediction can be made by ship dynamic model ei-
ther linear or non-linear which satisfies accuracy and
computational costs criteria. To meet these require-
ments the combination of linear and non-linear ship
motion models were used for calculations in (Pip-
chenko 2009).
Figure 3. Function surface R(
ϕ
1/3
, K
SGR
).
5 ENGINE LOADS ESTIMATION
To estimate engine power required to keep preset
safe speed the functional relationship between speed,
power and additional resistance in waves shall be
determined.
Ship speed with regard to environmental disturb-
ances, basing on equality condition of propeller
thrust to water resistance in calm water can be found
as follows:
( )
W eW
U fT R= −
; (17)
where Т
е
= propeller thrust in calm water; R
W
= av-
erage additional resistance due to wind and waves,
calculated in this work using methods of Boese
(1970) and Isherwood (1973).
Engine load, required to keep specified speed un-
dergoing the wind and waves influence can be de-
termined as:
( ) ( )
( )
2
1 23
,
wW
W
T fU R U
cU c U c R U
= +
=⋅ +⋅++
(18)
,
w
w
TU
P
η
⋅
=
(19)
where Р
w
= engine power; с = approximation coeffi-
cients, determined from experimental data.
Additional resistance in constant weather condi-
tions can be represented as function of ship speed.
Therefore if required speed cannot be reached due to
lack of engine power and wave impacts, maximum
possible speed can be found applying next recursive
procedure:
E(0) = U’(0), U’(0) = U
max
;
WHILE
E
ε
>
,
0
ε
→
( )
2
1 23wW
T cU cU c R U
′′′ ′
=⋅ +⋅ +−
;
{ }
{ }
24
max 1 3
max 0, min ,
ww
cT cT
U U ce ce
′′ ′′
⋅⋅
′′ ′ ′
= ⋅ +⋅
;
;EUU
′′ ′
= −
UU
′′ ′
=
.
END OF CYCLE
Where U’ = calm water speed; U’’ = predicted max-
imum speed in waves, defined as inverse function of
T
w
; c’ = approximation coefficients, determined
from experimental data.
6 ROUTE OPTIMIZATION ALGORITHM
The route optimization is performed by following
algorithm.
390
− Ship motion parameters in specified load condi-
tion are calculated for defined range of speeds
and courses in wave frequency domain. The re-
sult of such calculation is a group of four-
dimensional arrays X = f(U,
µ
,
ω
), where X –
specified motion parameter.
− Initial transoceanic route is given as great circle
line, on which the optimal engine load and corre-
sponding minimal work А
min
needed to perform
the voyage in calm water are estimated.
− Weather prognosis for the voyage is given as
multidimensional array with discrecity 1-2° ϕ х
λ.
− After indexing of cells containing weather data,
correspondence between route legs and chart grid
shall be defined.
− On each route leg (1:N): ship motion parameters
for specified wind and wave conditions are recal-
culated using spectral analysis techniques; risk
level and corresponding safe speed are deter-
mined.
− If the safe speed on any route leg is less then
specified minimum threshold, algorithm switches
to route variation stage, if no – engine power in-
puts and additional work are calculated.
− Optimization task is reached if the minimum ad-
ditional work in given weather conditions is
found, and the maximum risk level on the route is
less then specified threshold.
In isochrones method proposed by James (1959)
the engine power is considered as constant, where
speed is only changed due to wind and waves effect.
Thus it’s not applicable with the objective function
(7). From the other hand directed graph method
(Vagushchenko 2004) allows to control the ship by
both speed & course. But to get the accurate solution
the dense waypoint matrix shall be built that leads to
high computational costs. Therefore we suggest to
make generation of alternative routes by setting ad-
ditional waypoints – “poles”. In this case, pole it is
intermediate point inserted for avoidance of adverse
weather conditions. Positions of poles may be
changed either manually or by optimization algo-
rithm.
Poles shall be set as:
1 11
2 22
, 1,2,...,
... ... ...
m mm
Pole
Pole
mM
Pole
ϕλ
ϕλ
ϕλ
= =
(20)
Position of each pole shall satisfy following con-
ditions (fig. 4.):
1 Length of perpendicular, dropped to the or-
thodromy line between start and destination
points must not exceed specified threshold:
( )
margin
dm d
≤
(21)
( )
( )
cos
cos arctan
cot
arctan
cot
AP
A
AP
l
dm
l
Ψ−Ψ
=
(22)
2 Absolute difference between courses put from
pole to start and destination points must exceed
90°. It provides that pole stays in the space be-
tween start and destination points:
( )
90
o
P
mΨ≥
(23)
3 Distance from start point to each next pole shall
increase:
( ) ( )
1
AP AP
lmlm>−
(24)
Figure 4. Pole position in relation to the route.
Route legs are rebuilt depending on poles posi-
tions. Quantity of waypoints is determined propor-
tionally to the distances between poles. Route legs
before or after poles normally built as great circles.
If М≤4, optimization is carried out by Nelder-
Mead method. If М>4, optimization is carried out by
Genetic Algorithm method, because of Nelder-
Meads coefficient quantity limitations.
391
Figure 5. Example of imitated transatlantic route optimization.
7 EXAMPLE
As an implementation example of the given route
optimization method planning of imitated transatlan-
tic route for handymax container vessel (L = 200 m,
B = 30 m, GM = 1.0 m) is illustrated on figure 5.
The comparison of initial great circle and alternative
routes is given in table 4.
Table. 4. Initial and alternative routes comparison.
Parameter
Great circle
Alternative route
Min
Max
Min
Max
Distance, nm
2125
2212
Poles positions
42.0 N 45.0 W // 42.0 N 40.0 W //
43.0 N 30.0 W
Specified voyage
time, hours
106
Voyage speed,
knots
20
20.8
∆А, % from A
min
14.9
11.2
Risk level, %
7
88
6 56
Engine load, % 61
95
67 67
Rolling ampli-
tude, deg
0 18
1 22
Slamming inten-
sity, times/hour
0 3
0 0
Green water in-
tensity,
times/hour
0 103
0 0
As can be seen from this data, optimization leads
to quite good results both for safety and efficiency of
the route. Thereupon the time of the voyage remains
the same, with even less total engine loads (and ob-
viously less fuel consumption). From the other hand
on alternative route slamming and green water prob-
abilities reduced to a minimum. However the rolling
amplitude remains high on separate parts of alterna-
tive route, but it should be taken into account that
there are not many good choices to make as the imi-
tated weather conditions are almost everywhere ad-
verse.
8 CONCLUSION
To optimize the transoceanic route objective func-
tion which represents ship work expended for the
voyage was suggested. The work in that case repre-
sented as time integral of main engine power-inputs
needed to keep specified speed and to compensate
the additional resistance arisen due to environmental
disturbances with regard to ship’s safety.
To perform the safety assessment a fuzzy logic
system which represents relation between ship mo-
tion parameters and corresponding risks was devel-
oped. That allowed to perform the general risk level
value evaluation on the basis of multi input data.
Both these results give the opportunity to perform
effective transoceanic route planning on the basis of
formalized safety and efficiency assessment with re-
gard to specified weather conditions.
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