International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 6

Number 1

March 2012

1 INTRODUCTION

Ship weather routing is defined as an optimum track

of ship route with an optimum engine speed and

power for an ocean voyage based on en-route

weather forecasts and ship’s characteristics. Within

specified limits of weather and sea conditions, the

term optimum means a maximum of safety and crew

comfort, a minimum of fuel consumption and time

underway, or any desired combination of these fac-

tors. It can be clearly seen that, the accuracy of de-

termining the optimum route depends on three as-

pects.

− The accuracy of the prediction of the ship’s hy-

drodynamic behavior under different weather

conditions.

− The accuracy of the weather forecast.

− The capability and practicability of the optimiza-

tion algorithm.

The focus of this study is on research of the opti-

mization algorithm. Many optimization algorithms

have been developed for solving ship routing prob-

lems in which ship fuel consumption and/or passage

time are minimized. Most popular methods include

calculus of variations (Bijlsma S.J 1975), modified

isochrone method (Hagiwara H 1989, Hagiwara H &

Spaans JA 1987), two dimensional dynamic pro-

gramming (2DDP) method (De Wit C 1990, Calvert

S et al. 1991,) and isopone method (Klompstra MB

et al. 1992, Spaans JA 1995) .

The method of calculus of variation treats the

ship routing as a continuous optimization problem.

Inaccuracy in the solution may arise in the functions

where second order differentials are required. The

errors could be expanded to an unacceptable level at

the end of the calculation.

The modified isochrone method is a recursive al-

gorithm. The route with the minimum of passage

time is obtained by repeatedly computing isochrones

(or time fronts) which are defined as the outer

boundaries of the attainable region from the depar-

ture point after a certain time. This method offers a

route with minimum fuel consumption by keeping

the propeller revolution speed as a constant during

the voyage first, then applying the modified iso-

chrone method to determine the minimum time of

passage. By varying propeller rotation speed this

method is able to find the propeller rotation speed at

which the minimum time of passage satisfies with

the desired arrival time. This minimum time route

will be treated as the minimum fuel route. Thus, the

fuel consumption of this route itself is not mini-

mized.

The 2DDP method based on Bellman's principle

of optimality is similar to the modified isochrone

method. It uses a recursive equation to solve ship

Development of a 3D Dynamic Programming

Method for Weather Routing

S. Wei & P. Zhou

Department of Naval Architecture and Marine Engineering, University of Strathclyde,

Glasgow, UK

ABSTRACT: This paper presents a novel forward dynamic programming method for weather routing to min-

imize ship fuel consumption during a voyage. Compared with the traditional two dimensional dynamic pro-

gramming (2DDP) methods which only optimize the ship’s heading, while the engine power or propeller rota-

tion speed are set as a constant throughout the voyage, this new method considers both the ship power setting

and heading control. A float state technique is used to reduce the iteration on the process of optimization for

computing time saving. This new method could lead to a real global-optimal routing in a comparison with a

tradition weather routing method which results in a sub-optimal routing.

routing problems formulated as a discrete optimiza-

tion problem. The accuracy of the solution depends

on the fineness of the grid system used. Compared

with the modified isochrone method, the advantage

of the 2DDP method is that it allows the operators to

take into account of navigation boundaries by means

of an appropriate selection of the grid points. Both

the modified isochrone and 2DDP methods assume

that ship sails at a constant propeller rotation speed

or a constant engine power for the entire voyage.

The isopone method is an extension of the modi-

fied isochrone method. An isopone is the plane of

equal fuel consumption that defines the outer bound-

ary of the attainable region in a three-dimensional

space, i.e. position and time. This method enables

the operators to consider the variations of ship en-

gine power to optimize the route. SPOS, a weather

routing software, adopted the isopone method at the

beginning of the software development. Although

the isopone method is mathematically more elegant

and theoretically offers better results than that of the

modified isochrone method, finally SPOS had never-

theless abandoned the isopone method and applied

the modified isochrone method. The main reason for

this change was due to the fact that the isopone

method appeared to be more difficult to understand

by the operators onboard ships, whereas the modi-

fied isochrone method is straightforward and easy to

understand.

Besides these methods, there are many other

methods that have been used for weather routing in

recently years, like iterative dynamic programming

algorithm (Kyriakos Avgouleas 2008), augmented

Lagrange multiplier (Masaru Tsujimoto, Katsuji Ta-

nizawa 2006), Dijkstra algorithm (Chinmaya Prasad

Padhy et al. 2008), genetic algorithm (Harries S,

Hinnenthal J 2004) and so on.

Weather routing was first developed for determin-

ing shipping courses during a voyage with a mini-

mum of passage time. However, nowadays shipping

companies began to show more interesting in reduc-

ing fuel consumption driven by the fuel oil prices,

environmental considerations and maintaining a cer-

tain time schedule which is specified in the charter-

ing contract of a merchant vessel. In this paper, a

new forward three dimensional dynamic program-

ming (3DDP) method is presented for minimizing

fuel consumption during a voyage. It is an extension

of the traditional 2DDP method, allowing change

heading and speed with both time and position, thus,

it is able to realize a real global optimum result.

Compared with the isopone method, the 3DDP

method is straightforward and easy for program-

ming.

2 PROBLEM STATEMENT

Ship engine power and shipping course directly de-

cide the shipping route in the ocean. Ship speed over

the ground depends on the engine power. There is a

one-to-one relationship between them. Thus, both of

them can be equally treated as the control variables

in a weather routing process. In this paper, ship

speed over the ground and shipping course measured

from the true north are chosen as the control varia-

bles. The control variables are denoted as a control

vector



( , )U Uu



, where u represents ship

speed over the ground and ψ is shipping course

measured from the true north. Ship position



is al-

so a vector, specified by longitude φ and latitude θ,

( , )XX

ϕθ



Ship position



and voyage time t determine the

ship trajectory. Using



to denote weather condi-

tions (speed and direction of wind, significant

height, direction and peak frequency of wave and

swell),



is a function of position



and time t,

( , )E EX t=



(1)

During a voyage, constraints



must be met. The

constraints include geographic constraints, control

constraints and safety constraints.

Thus, ship position



at time t can be described

by the function below:

( , , , )X f X UEC

′ ′′′



  

(2)

Where

, , , XUEC

′ ′′′



 

correspond to time t’, t –

t’ = ∆t, ∆t is a time step used in calculation.

Because

′



is a function of

′



and t’, so



can

also be described by:

( , , , )X fX UtC

′ ′′ ′



 

(3)

This function can be explained that while comply

with the constraints, the ship will arrive at the pre-

sent position



at the present time t from

′



under

control of

′



during ∆t time step.

Instantaneous fuel consumption rate q can be ob-

tained by:

( , , , )q q XUtC=





(4)

The total fuel consumption for a voyage can be

obtained by

( , , , )

end

C q X U t C dt=

∫





(5)

Where:

Initial conditions:

(, )

s ss

ϕθ



tt=

Final conditions:

(, )

end end end

ϕθ



end

tt=

The constraints



: Geographic constraints, con-

trol constraints, safety constraints.

3 DYNAMIC PROGRAMMING

3.1 Advantages of the 3DDP method

Dynamic programming is a method which can solve

complex problems by breaking them down into

many simpler sub-problems. A stage is defined as

the division of sequence of the sub-problems in the

optimization procedure. The procedure of this meth-

od is to solve the sub-problems stage by stage. The

variables used to define a stage must be parameters

which are monotonically increasing with the pro-

gress of problem solving going-on. There are two

choices of variable selection to specify a stage for

ship routing problems, i.e. time and a measure of the

progress of the vessel from departure (voyage pro-

gress). Each stage consists of many states which can

be defined as a specific measurable condition of the

ship operation, such as time and location. If time is

chosen as the stage variable, the state can be defined

by possible locations where the ship could pass. If

voyage progress is chosen as the stage variable,

states should be defined by time and possible posi-

tions away from the great circle.

The 2DDP method chose voyage progress as the

stage variable. Because this method assumes that

ships sail at a constant propeller rotation speed or a

constant engine power for the entire voyage, there is

a one-to-one relation between ship position and

time. Thus, time variable is not needed to specify

states in this method. Several authors have already

attempted to solve the weather routing problem by

using 3DDP treating both engine power and ship-

ping course as the control variables during a voyage.

Aligne, F. et al. (1998) chose time as the stage vari-

able and used the forward algorithm; Henry Chen

(1978) and Simon Calvert (1990) employed the voy-

age progress as the stage variable and used the

backward algorithm. The method presented in this

paper employs the voyage progress as the stage vari-

able together with the use of the forward algorithm.

The advantages of using the forward algorithm

can be stated as the following: When optimizing a

route, the initial departure time is fixed, the arrival

time can be treated as a flexible parameter, allowing

a set of route with a minimum fuel consumption to

be obtained corresponding to different specified ar-

rival times in one calculation.

To compare using voyage time as stage variables,

the advantages of using voyage progress as stage

variables are:, ship headings are pre-defined by voy-

age progress on grid points, so that ship speed over

the ground becomes the only explicitly defined con-

trol variable to be optimized during the routing op-

timization process. This method doesn’t need a finer

grid system. It can save much more computing time

than the methods which choose voyage time as a

stage variable.

3.2 Grid design

Since the great circle is an optimum route under

calm water conditions from the departure to the des-

tination, it is chosen as a reference for the construc-

tion of the grid system used in the route optimiza-

tion.

As describe above, states are of three dimensions,

i.e. time and geographic location with a unit spacing

ΔY located on a stage, perpendicularly away from

the great circles. The farthest states on a stage from

the great circle are the possible locations the ship

may pass to avoid a bad weather or certain sea

Figure 1. Projections of space grid system on a longitude × lat-

itude plane.

conditions. Unlike the tradition dynamic program-

ming, the variable of voyage time t of states are de-

termined as the optimization procedure is progress-

ing. Grids should be deleted when a shipping route

crosses islands/rocks.

Fig. 1 shows an example of stage projections on a

longitude × latitude plane where 16 stages have been

allocated (1, 2, 3…16) from the departure to the des-

tination of a shipping route. The distance between

two stages can be equally spaced ∆X. The total

number of stages is determined according to the total

distance of the route and the availability of compu-

ting capacity.

3.3 Estimate of fuel consumption between two

stages

Fuel consumption is a function of ship hydrodynam-

ics. The hydrodynamics of ship are modeled and

simulated in the process of fuel consumption estima-

tion. Thus, the accuracy of the modeling of the ship

hydrodynamics is critical in the accuracy of fuel

consumption estimation. As a consequence of added

-10

-20 0 20 40 60 80 100

Longitude

Latitude

Grids

Great Circle

stage 1

∆X

∆Y

resistance due to wind and waves as well as the in-

creased hull roughness over time, ship speed is often

smaller than the designed speed so called involun-

tary speed reduction. Besides that, a voluntary speed

reduction is needed to insure ship safety to minimize

or avoid slamming, deck wetness, propeller racing,

parametric rolling, motion sickness, engine over-

loading and so on. All these factors need to be con-

sidered in routing optimization as the constraints.

Since the focus of this paper is to discuss the optimi-

zation algorithm of fuel consumption during a ship

voyage, how to accurately predict ship hydrodynam-

ic at the sea is not discussed in depth or further.

However, the procedure of prediction of fuel con-

sumption between two stages is presented here. This

procedure can be treated as a sub-problem of a dy-

namic programming problem. The optimized fuel

consumption during an entire voyage is obtained by

adding up of all individual fuel consumption be-

tween two stages along a route with the newly de-

veloped 3DDP method.

As a ship voyage follows a predefined grid system,

her heading is fixed between two stages. The ship

speed over the ground is the only control variable

which directly determines the fuel consumption be-

tween any two stages during a course of shipping.

Fig. 2 shows the procedure determining fuel con-

sumption between two stages. In detail, it is as the

following:

Step 1: Calculation of ship resistance. Ship re-

sistance is calculated based on the ship speed

over the ground, draft, trim and the weather con-

dition. Ship resistance can be divided into three

main components: a). the calm water resistance;

b). the added resistance due to wave; c). the add-

ed resistance due to wind. Ship sea trial data,

model test data and numerical simulation results

are used to estimate these resistances.

Step 2: Estimation of engine power. The engine

power is calculated to overcome the above calcu-

lated resistances based on the propeller character-

istics.

Step 3: If the engine power is more than MCR

(maximum continuous rate), the ship speed will

be reduced by Δu, and then go back to step 1.

Step 4: Calculation of probability of slamming,

deck wetness, and propeller racing. To ensure the

ship safety, if these constrain values exceed cer-

tain pre-set limits, the ship speed will be reduced

Δu, and then go back to step 1.

Step 5: Calculation of fuel consumption and ship

position for next time interval

∆

Step 6: Execute step 1 to step 5 repetitively in a

fixed time interval

∆

t between two stages until

the ship (simulation step) arrives at the next stage

or final destination.

The time interval

∆

t for calculation is normally

chosen at the frequency of the reception of weather

forecasting onboard which is usually every 6 hours.

3.4 Algorithm description

The backward recursive algorithm has been used in

most dynamic programming of weather routing.

However, the forward dynamic programming offers

more convenience in programming. The forward dy-

namic programming can be interpreted as that a path

is optimal if and only if, for any intermediate stages,

the choice of the foregoing path is optimum for this

stage. By using this principle, the weather routing

procedure can be broken down into a sequence of

simpler problem solving. Notations defined in the

programming are as follows.

− K: total number of stage.

− N (k): total number of state projection on the lati-

tude × longitude plane on stage k, where: k = 1, 2,

3… K. N (1) = 1, N (K) = 1.

Figure 2. Estimate of fuel consumption between two stages.

− P (i, k): state project position on stage k, where: i

= 1, 2, 3…, N (k). P (1, 1) is the departure posi-

tion; P (1, K) is the destination position.

− J: total number of time interval between states on

a geographical position.

−

t∆

: time interval between states on a stage.

− X (i, j, k): a state on stage k, where: i = 1, 2, 3…,

N (k), j = 0, 1, 2 …, J - 1, k = 1, 2, 3…, K. The

geographic position of the state X (i, j, k) is P (i,

k) on stage k, the time variable of the state is t

i ,j, k

≦ t

i, j, k

≦

, and

t∆

× j,

t∆

× (j

+ 1). The state X (i, j, k) at stage k is a floating

point between position (P (i, k),

) and (P (i, k),

− X (1, 0, 1): the initial state. Time variable t

1, 0, 1

the initial state is 0.

− X (1, j, K): the states on the final stage K, j = 0, 1,

2…, J - 1, the position of these states is P (1, K).

− F

opt

(X (i, j, k)): the minimum fuel consumption

from the initial state to the state X (i, j, k).

− u (m): ship speed over the ground varying be-

tween 5 to 30 knot, u (m) = 5 + 0.1 × (m -1),

where: m = 1, 2, 3…, M.

The recursion procedure of the forward dynamic

programming can be described as follows:

Step 1: Set stage variable k = 1.

Step 2: Iterate step 3 to 6 below for each attainable

state X* = X (i, j, k) on stage k, where: i = 1, 2,

3…, N (k), j = 0, 1, 2…, J. if a state X* = X (i, j,

k) is unattainable due to constraints, its fuel con-

sumption F

opt

(X (i, j, k)) is set to infinitive.

Step 3: Calculate ship heading H* from X* to the

next stage position P (i’, k + 1), where: i’ = 1, 2,

3…, N (k + 1). Iterate step 4 to 5 for each H*. If a

ship heading H* violates the heading constraints

or geographic feasibility, calculation for this

heading is given up and go to the next heading

calculation.

Step 4: Iterate steps 5 for each u* = u (m), where:

m = 1, 2, 3…, M. If certain ship speed u* violates

the control constraints or safety constraints, skip

out of this loop and go to the next loop.

Step 5: Choose u* and H* as the control variables,

calculate the fuel consumption Δf

m. i’

and the voy-

age time Δt

m, i’

between state X* on stage k and

next state on stage k+1 with a geographic position

P (i’, k + 1) by using the method described in sec-

tion 3.3, t

m, i’

is denoted as the arrival time at posi-

tion P (i’, k + 1) from the initial state, t

m, i’

= t

i, j, k

+ Δt

m, i’

, the position X’= (P (i’, k+1), t

m, i’

) forms

a new possible state on stage k + 1, The fuel con-

sumption at X’ is f* is determined by f* = F

opt

(i, j, k)) + Δf

m,i’

Step 6: When the calculation of all possible states

X’ between time

′

and

′

at position P (i’, k +

1) on stage k + 1 is completed, the possible state

which has the minimum fuel consumption f*

min

chosen as the state X (i’, j’, k + 1) Thus, F

opt

(i’, j’, k + 1)) = f*

min

. The departure state X* on

stage k, the arrival state X (i’, j’, k + 1)) on stage

k+1 and the corresponding control variables be-

tween the two states are saved for tracing the op-

timum route by a backward procedure at the end

of the calculation. During the optimization pro-

cess states within a time interval are floating. The

benefit of using float states is that it eliminates

the calculation of the interpolation. Thus, it can

save computing time. When the weather in

t∆

time does not change much this method will not

influence the accuracy of the optimized result.

Step 7: Let k = k + 1, then go back to step 2 until k

= K.

Once the final state on stage K has been obtained, a

backward calculation procedure is used to identify

the optimized fuel consumption route with the speci-

fied arrival time and the corresponding control vari-

ables during the entire voyage.

4 CASE STUDY

This section presents two case studies with the use

of above described 3DDP method. As a simplifica-

tion, the weather conditions are set artificially. Alt-

hough the weather conditions used are not real and

certain conditions may never happen in the reality,

the use of artificial weather conditions will offer the

same effect as the real ones in illustrating the meth-

odology and advantages of the 3D dynamic pro-

gramming. Holtrop method, a regression analysis

method, is used to predict the total resistance in calm

water. The engine power is calculated by propeller

characteristics of the case ship.

Two different sets of weather conditions are used

in the case studies with the following common pa-

rameters:

− Case ship: a 54,000 DWT container ship.

− Departure from:

(0, 0)

X =



− Arrival at:

(90, 0)

end

X =



− Time interval between states on a stage:

t∆

= 1

hour.

− Time step for fuel consumption calculation be-

tween two stages

∆

t = 6 hours

− Ship speed: u = 5 to 30 knots.

− Ship speed change step: Δu = 0.1 knots.

− Total stage number: K = 16.

− Total number of stage projection on a stage: N (1)

= 1, N (K) = 1, N (k) = 17, where k = 2, 3, 4…K -

− Stage space: ΔX = 360 nautical miles.

− State space: ΔY = 75 nautical miles.

4.1 Case study 1

The geographic constraints and weather conditions

for case 1 study are shown in Fig 3. The geographic

constraints are set as a rectangular area which can be

islands, rocks or mine fields. The scope of the geo-

graphic constraints is longitude from 50 to 70 degree

and latitude from -1 to 7 degree. The envelop of the

bad weather is set as a rectangular area as well, posi-

tioned longitude from 50 to 70 degree, latitude from

- 1 to - 9 degree at time t = 0. The bad weather stays

at this initial area for 60 hours before moving to-

wards south with a speed of 3 knots. The ship is not

allowed to enter into the bad weather area for safety

consideration. Fig.4 shows the results of fuel con-

sumption vs. time obtained from the route optimiza-

tion for the case study 1. When the specified arrival

time is smaller than 233 hours, both of the 3DDP

and 2DDP methods can get the similar strategies

which choose the route closed to the dotted line in

Fig. 5 and a constant ship speed during the voyage.

That means, changing the ship heading can get much

benefit than changing the ship speed at this weather

condition. When the specified arrival time is more

than 272 hours, the time during the voyage is rela-

tively long, so the bad weather already pass away

before the ship arrive there, the 3DDP method also

get a similar strategy with the 2DDP method which

choose the route closed to the solid line in Fig. 5 and

a constant ship speed. When the specified arrival

time is between 233 hours and 272 hours, the 3DDP

method can get a better result than it calculated by

the 2DDP method.

Fig. 5 and Fig. 6 show the optimized route and

optimized ship speed obtained by using the 2DDP

and 3DDP methods under a specified arrival time

end

= 264 hours. The results have demonstrated that

fuel consumption obtained by 2DDP is 1014.54 tons

for the given voyage conditions and that is 969.25

tons if the ship follows the route resulted from

3DDP. As a result, route and operation profile opti-

mized by the 3DDP offers about 4.5% of fuel saving

compared that with the 2DDP. The reason for the

fuel saving is that the 3DDP method permits the ship

to change the heading and speed during the route. In

the first section of the route, the ship slows down to

let the bad weather pass-by first. Once the bad

weather has passed, the ship increases her speed to

ensure the desired arrival time is achieved.

4.2 Case study 2

In case 2 study the geographic constraints is the

same as that of case 1. Whereas the bad weather area

at time t = 0 is longitude from 50 to 70 degree, lati-

tude from - 7 to - 15 degree. The bad weather moves

towards north with a speed of 3 knots. Fig. 7 shows

the geographic constraints and weather conditions.

Fig.8 shows results of fuel consumption vs. time ob-

tained from the route optimization for the case study

2. Because of the same reasons with the case 1,

when the specified arrival time is smaller than 270

hours or bigger than 303 hours, both the 3DDP and

2DDP methods can get the similar results; when the

specified arrival time is between 270 hours and 303

hours, the 3DDP method can get a better result than

it calculated by 2DDP method.

Fig. 9 and 10 present the route and ship speed op-

timized by the 2DDP and 3DDP methods under the

same arrival time t

end

= 278 hours. The fuel con-

sumption calculated by the 2DDP is 898 tons and

that is 852 tons from the 3DDP calculation. A 5.1%

of fuel saving has been achieved by using the 3DDP

compared with the 2DDP method. Unlike the case

study 1 where ship speed is reduced to wait for the

bad weather to pass during the first part of the route,

the ship speed is increased to pass the region before

the bad weather comes. Once the ship has passed the

region where the bad weather is going to pass, the

ship speed is slowed down and maintained the de-

sired arrival time.

5 CONCLUSION

A newly developed 3DDP for weather routing has

been presented. Case studies have shown that com-

pared with the use of traditional 2DDP method, fuel

saving can be achieved by using the newly devel-

oped 3DDP method in certain constraints and

weather conditions. Since the speed of the ship var-

ies according to the weather conditions and move-

ment, the newly developed 3DDP increases the safe-

ty of shipping.

The 3DDP method considers optimization of both

the ship speed and heading. Its operation and pro-

gramming are easier and straight forward.

In this paper, real weather forecast is not consid-

ered, but this 3DDP method can also give enlight-

enment for the weather routing problem. In the fu-

ture, this method will be used based on real weather

forecast and ship hydrodynamics.

Figure 3. Geographic constraints and weather conditions (case

1).

Figure 4. The fuel consumption vs. time curve (case 1).

-10

-8

-6

-4

-2

0 20 40 60 80 100

Longitude

Latitude

Grids

Geographic constraints

Bad weather

700

900

1100

1300

1500

1700

220 230 240 250 260 270 280 290 300

Time (hour)

Fuel (ton)

3DDP

2DDP

Figure 5. Optimized route (case 1).

Figure 6. Optimized speed (case 1).

Figure 7. Geographic constraints and weather conditions (case

2).

Figure 8. The fuel consumption vs. time curve (case 2).

Figure 9. Optimized route (case 2).

Figure 10. Optimized speed (case 2).

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Navigation 169: 95-106

-10

-8

-6

-4

-2

0 20 40 60 80 100

Longitude

Latitude

Grids Optimised route of 3DDP Optimised route of 2DDP

Geographic constraints

Bad weather

19.6

19.8

20.2

20.4

20.6

20.8

0 50 100 150 200 250 300

Time (Hour)

Speed(Knot)

3DDP

2DDP

-10

-8

-6

-4

-2

0 20 40 60 80 100

Longitude

Latitude

Grids

Geographic constraints

Bad weather

660

710

760

810

860

910

960

260 270 280 290 300 310

Time (hour)

Fuel (ton)

3DDP

2DDP

-10

-8

-6

-4

-2

0 20 40 60 80 100

Longitude

Latitude

Grids Optimised route of 3DDP Optimised route of 2DDP

Geographic constraints

Bad weather

18.8

19.2

19.4

19.6

19.8

20.2

20.4

0 50 100 150 200 250 300

Time (Hour)

Speed(Knot)

3DDP

2DDP