Windship Routeing

824825

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

WINDSHIP ROUTEING

by

Prof.ir. J.A. Spaans

Reportnr. 654-P-3

Journal of Wind Engineering and

Industrial Aerodynamics, 19(1985)

Elsevier Science Publishers B.V.

Delft University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2

2628 CD DELFT

The Netherlands

Phone 015 -786882

WINDSHIP ROUTEING

by

r.

J. A. SPAANS

Delft University of Technology

Dept. Marine Technology

Section Hydronautics

Del ft.

Ship routeing is a procedure where an optimum track is determined for a particular vessel on a particular run, based on expected weather, sea state and ocean currents.

Optimisation can be performed in terms of minimum passage time

minimum fuel consumption within a specified passage time minimum damage to ship and / or (deck) cargo by the sea maximum comfort to passengers

or a combination of the above criteria.

For most commercial purposes a combination of the criteria (i) and (iii) or (ii) and (iii) is preferable.

G. HISTORICAL REVIEW

For many centuries navigators have gathered climatological and hydrographic data to be used for later voyages.

The first published "Sailing Directions" now known, are the 13th century "Compasso de Navigare" for the Mediterranean and the 15th century"Al Muhet" bij Ibn Majid for the Northern Indian Ocean and adjacent waters. This famous Arabic navigator was Vasco da Gamas pilot on his first trip across the Indian Ocean in 1498.

Prince Henry The Navigator (1394-1460) promoted thescientific approach to navigation and gathered the best cartographers, astronomers andother scientists of that time

in his nautical centre in Sagres Portugal. Shipmasters collected information on their trips and had afterwards to report in Sagres wherethe information was gathered and processed into charts and pilot books.

In the 16th century the know-how of the Iber ans spread over Europe, mainly by the standard work

"Arte de Navigar" by Pedro de Medina Valladolid 1545.

This work was translated and published in Italy, France, Holland, England and Germany (1).

Matthew Maury (1806-1873) was the founder of Maritime meteorology and oceanography. He

published the first "Pilot Charts" in 1845, containing seasonal wind and current information. The average passage time from New York to California round the Horn for

instance was reduced from 183 days to 139 days by using Maury's season charts (2).

At the end of the 19th century and in the beginning of the 20th century the "Deutsch Seewarte" published detailed sailing directions (Segelhandbucher) for shipmasters of sailing vessels. Figure 1 gives the average timefronts (isochrones) for sailing

vessels outward bound from the Lizard as given in (3). The perfect voyage planning was certainly one of the major reasons why the Germans, like Ferdinand Laeisz flying P-line, managed to operate sailing vessel profitably until far into the 20th century, when other seafaring nations had already totally switched to engine propulsion. (Only the Finish sailing fleet of Gustaf Erikson from Mariehamn managed to operate square rigged sailing vessels until the second world war, with cheap second-hand vessels, boyscrews and relatively low wages). A historical lesson is to be learnt from Ferdinand Laeisz

The ship owner operating with optimal voyage planning will survive where others have to give up!"

In 1952 as first commercial shipping company the Amen can President Lines started te recommend routes to bypass the worst storms for the Line's ships operating in North Pacific. _{A meteorologist was enlisted to provide forecasts in more detail than}

government predictions (4). Simultaneously other shipping companies were independently tninking along the same lines. By the end of the 1950s two American consulting firms - Oceanroutes and Pacific Weather Analysis, which later became Weather Routing Incorporated - had been formed, later followed by Bendix Marine Science Services.

In 1960 the Royal Netherlands Meteorological Institute at De Bilt was the first office in Europe to provide routeing services for the North Atlantic, followed by Bracknell in the U.K. and the Marine Meteorological Office in Hamburg.

Figure 1.

Isochrones outward bound from the Lizard.

3. CLIMATOLOGICAL ROUTEING

Average values or frequency distributions of winds, sea state, ice limits and ocean currents can be found in climatological atlasses or special season or monthly charts. The mariner can find this information including recommended tracks in

- Ocean Passages for the World (B.A.) - Pilot Charts (USA)

- Sailing Directions and Pilot Books (B.A. and others)

- Climatological Atlasses published by Meteorological or Hydrographic

Offices

"Ocean Passages" gives recommended routes for sailing vessels, low powered vessels and powered vessels, see figure 2.

N

and recommended tracks, see figure 3. The B.A. Pilots give additional information.

Using climatological data for routeing is only justified in areas where the weather situation is stable like the trade wind and monsoon areas outside the cyclone season.

When carried out on board, climatological routeing is usually done without rigorous computer programs. The shipmaster assisted by his navigating officer compares loxodromic and great circle routes, takes advantage of currents, avoids areas with a high storm frequency especially when couterwinds are expected, and avoids areas with low visibility; the ice limits and the regulations of the load-line convention are taken into account when necessary. Often the recommended route Or near-by one is taken as the best as these routes are the result of age-long experience of mariners. However, different types of vessels in operation will require different routes. For instance a fully loaded tanker is less subject to damage by heavy seas than a vessel with cargo on deck and consequently the latter will be routed on a track with a lower storm frequency.

Navigators are taught at nautical colleges how to tackle these problems. Case-studies of different types of vessels and different routes are carried out. In the following section it is shown that better results can be reached with weather routeing.

4. WEATHER ROUTEING IN GENERAL

4.1 Graphical Method

In it's infancy in the 1950's the tactics of weather routeing were only aimed at bypassing storm centres of tropical storms to reduce the damage to ship and cargo.

In 1957 R. W. James (5) introduced a graphic method to determine a "least-time-track° for an ocean passage. In many routeing offices this procedure is still practised. The ship's speed as function of the significant wave height for four different conditions of wave directions was used by James, as shown in figure 4.

Figure 3. Pilot Chart North-Atlantic.

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---71--SHIP'S SPEED

SIGNIFICAI4T WAVE HEIGHT

Figure 4. Ship's speed as function of seastate.

At the Royal Netherlands Meteorological Institute (KNMI) empirical data are used to plot tne ship's speed in a polar diagram for different Beaufort wind forces and courses relative to wind and sea, see figure 5.

The influence of the significant period of swell and sea on the ship's speed is hardly taken into account at Routeing Offices until now, although the fuel consumption in tons per mile is significantly higher in a specific interval of wave periodo for a particular vessel. Figure 6 shows this for a 23000 tons container vessel according to Journee (11).

Before the computer took over, wave charts were hand-drawn using the information from the weather charts. The diagram of figure 7 is used by KNMI for this purpose, giving the significant wave height and period as function of wind speed, wind duration and wind fetch (= distance to windward where the wind is building up the waves).

FOLI,,NOW,WES BEAMWAVES

HEADWAVES

---0.3

0.0

Figure 6.

180°

5 10 15 20

,

significant wave period in seconds

90°

I

speed 19 knots

23000 TDW containervessel

sea/wind:

5

a, /1

m/sec

--2, rn/ 10 m/Lec slack

30

20 11. 10 8 6 5 1. 2 0,8 0.7 0.6 0.5

04

0,3 0,2

01

0.1 5

05

1 2 3 6 9 12 18 24

36 48

Figure 7. Significant wave height and period as function of wind speed, wind duration and wind fetch.

Nowadays, the enormous amounts of meteo data are processed by computers to produce the actual and predicted weather and wave charts. European Meteorological offices use a b-day weather prediction for the North Atlantic mainly based on the data from the "European Centre for Medium Range Weather Forecasts" in Reading U.K. The wave cnarts are however manually corrected, on the basis of the wave information from weather reports from individual ships, which reports are transmitted from selected ships with 0600 hours GMT interval.

For the prediction of the track to be followed by a depression the 500 nib charts are used. Tse direction of a line of equal altitude of the 500 mb pressure level

is roughly followed by the centre of a depression. In figure 8 the actual and predicted 500 mb charts, the weather chart and the wave chart for the actual situation on 18 February and for the situations 24 h, 48 h, and 72 h thereafter are shown.

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charts.

coincide with the centre of lowest pressure, but lies considerably south-easterly to southerly off this centre. The charts in figure 8 from the Netherlands Meteorological office show how a particular vessel was routed outside the area of highest waves.

The graphic method of determining the least time track to the destination is shown in figure 9. From the point of departure the distance covered in 12 hours is plotted

in a number of directions. The envelope of these vectors is the 12 h isochrone. The distance covered in 12 hours is determined, taking into account the speedloss in waves, see figure 4 and 5, and the wave chart as given in figure 8. From a number of points of the 12 h isochrone a "local" 24 h isochrone in determined; the envelope of these local isochrones is the overall 24 h isochrone. As far as the weather prediction extends this procedure is continued as shown in figure 9.

('-'01t0)

Figure 9. Least-time-track construction.

From the destination backwards a line is drawn perpendicularly to all isochrones. This line is taken as the "least time track". Theoretically this is not correct; in each point of the least time track the projection of the distance travelled per unit of time on the perpendicular of the isochrone should be maximal, as shown

in section 4.3. In practice, however, this manual method has proved to be very useful Areas with high waves - especially head seas - are avoided because of the inherent speed loss. The consequent advantage is, that less damage to ship and (deck) cargo will be experienced, so safety and economy benefit simultaneously.

1)1'

\\\,_

of the track followed in regard to the great circle track and the loxodromic track are given and also the time loss of the track followed in regard to the "evaluation track" is provided.

In figure 10 the evaluation chart is shown for a particular case. The track actually followed shows to be very irregular. The actual least-time track is not shown in the figure, but lies somewhat more to the south. The time profit in this case of a 14 day voyage was about 2 days compared to the great circle track and 3 days to the loxodromic track.

In (9) tse average results of the Dutch Meteorological office of 1971-1981 are published. The average time profit is different for 15 several geographical routes and varies between 1 hour and 10 hours.

For people who are suspicious of routeing offices publishing their own results, reference (8) is recommended which states that the results published by KNMI are even somewhat conservative.

In (4) a Shell spokesman confirmo an average time profit of 4 hours on an Atlantic crossing, which equals about 10 tons of fuel or about $5000.- per transit.

The price of routeing advice for the North Atlantic varies; in the Netherlands the charge is about $200.- plus telegram expenses. In view of the increased fuel costs, more ship operators are using the services of routeing offices.

In reference (10) an interesting case study is mentioned. Oceanroutes Inc. routeing office simulated the passage of the North Pacific from San Francisco to Nojima Saki for the month of December for several years, where 8 different tracks were compared, see figure 11 where the results for 1979 are shown. For each day departure from San Francisco the enroute time is plotted vertically for each of the 8 tracks. Differences of about two and a half days passage time exist on the 10th December and 21st December, but the choice of the route is reversed in both cases: This experiment clearly puts a query against routeing by Pilot Charts based solely on climatologi cal considerations.

Apart from time / fuel savings, damage to ship and cargo is diminished by avoiding nig'n (head) seas. In 1976 the Naval Weather Service (USA) contracted for a study

covering s years of Atlantic experience in heavy weather and its damage effects. Figure 12 summarizes the results, An average ship will experience $1000.- per day

damage to ship and cargo when proceeding in a significant wave height of 6 m, whereas tnis amounts to $10,000.- per day for waves of 8 m height; a dramatic increase:

An experiment conducted by Oceanroutes Inc. and mentioned in reference (10), was the analysis of tne transmitted weather reports of January 1977 of 660 ships enroute in a trans-Pacific voyage of which 39% were routed by a routeing office. The figure 13 shows the result, which is at first glance not very dramatic. However the sting

11.0 11,0 O. o

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51/33/ TO 140.1 YIA SAA O.J co-rrn . 2o o

Figure IIA.

Routeing results December 1979.

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Figure 11B. SELECED ROUTES

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unrouted ships encounter significant wave heights of more than 6.5 m than routed ships.

The same graph is made for ships in head seas in figure 14 where the distinction

between routed and unrouted ships is even clearer.

On an average the sea height was

reduced by 14 per cent.

damage

per day

$ 100,000

$ 10,000 $ 1,000 (1976) $ 10

Figure 12.

h

i-77..E7-77

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90

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, reported significant waveheight (all

_directions)

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, significant wave height (head seas)

Figure 14.

Another noteworthy result is mentioned in /10/. A Japanese automobile shipper found that after using routeing recommendations for trans-Pacific voyages from a routeing office, the average damage per automobile shipped, droppeddramatically from $2.95 to $0.85 which is a considerable profit in view of the 350,000 vehicles exported annually froni Japan to the USA.

The second major input for the computer program is the ship's speed in sea conditions which can generally be described for a (wind assisted) motor vessel with a given draught, trim and hall condition as

V = T

(n, y, Hs, Ts,

os,

Hsw, Tsw, Os, Vw, Ow) + (1) where

propeller revolutions per minute vessel heading

H, = significant wave height of sea

Ts significant pericd of sea

es = average course of sea waves

Hsw, Tsw, Osw = parameters for swell

v

wind speed

Ow wind direction

= current vector

= _{speed over the ground}

-17 = speed through the water

When meteorological and oceanographical data are known as function of the position and n is constant, formula (1) can be reduced to

-17 = (t,

X, 44

where 4, = latitude and X = longitude.

For a given situation -1% (y) is given by a speed polar as shown in figure 15. The speed

polar in figure 15 at right is concave in a given sector due to an obligatory speed reduction when proceeding in waves entering from abaft the beam. By 'symmetric cruising' a higher average speed can be attained. In a study by J.M.J. Journee of Delft University of Technology /11/, sponsored by the Ministry of Economic Affairs as part of a general

'energy saving' project, mathematical models are given for the ship's speed and the fuel consumption taking into account the relevant factors.

Figure 15. Speed polars.

To develop a computer algorithm, an optimisation criterion has to be defined. In most programo the transit time is minimized /12/, /13/, /14/ by computing

o (o,

s, t) for the voyage, keeping n constant. Only in high seas n will be reduced to avoid damage.

Basically two methods are in use

(i) the method based on the 'maximum principle' of Pontryagin /15/. A state vector x (t)

E e

and a control vector u (t) E Rm are defined. The differential equation for x (t) is given by

(t) = f (x (t), u (t), t)

X (0) = xo (3)

x (T) = xi

find x (t) and u (t) to minimize the object functional

J

= f

L (x, u, t) dt (4)

0

An adjunctive vector p is introduced and when L (.) and f (.) are twice continuously differentiable for x, u and t, the problem is solved by solving the simultaneous equations

= f (x, u, t)

pt = Lx - pt fx (5)

Hu . I

In the case of minimum time routeing

J:

_f

1 dt o L = 1 H = -1 + pt f = f (x, h, t) (speed polar) pt _pt p f, = x (0) = xo x (T) = H (x1, T) =

As Jx = p, and the isochrones as mentioned in section 4.1 are contours of equal J, the vector p is the gradient vector of J and is perpendicular on a contour line.

According to Pontryagin H = -1 + pt f = -1 + pt has to be maximized so the projection of the speed vector k on p has to be maximized, as used

in section 4.1.

In figure 16 the timefronts for an Atlantic transit computed by the algorithm of Bijlsma (13) are shown.

When the rate of decrease of fuel is described by the equation

= f (t. 0, v.

0

(8)

functions q (t) and v (t) have to be determined minimizing

J

= f

f (t, 0, v, *) dt

O

In (13) this is pointed out; it is a subject of further research at KNMI. (19)

Figure 16.

Computer produced by means of an incremental plotter using wave inforination over the

period 17 January-23 January 1970, fictitious ship's data and a 12-hour time step. The

Figure 17. Forward dynamic programming.

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70

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50

010

30

d. 10

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40

Figure 18.

APRIL

NEST PASSAGE

MIN SHIP SPEED

0 iilS

CO cl SO

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In the ocean area under consideration a series of gridpoints is appointed equally distributed "along track" and "cross track" along the great circle route, see figure 17. From the starting point the enroute-time to the gridpoints on the first cross-track line is calculated with the expected weather and ship's parameters.

From each gridpoint on the first line this same procedure is carried out giving different times of arri val at the second cross-track points for different routes. The earliest times of arrival and respective routes are stored, others are skipped. This procedure is continued until the destination is reached. The route belonging to the earliest arrival time is the 'least-time-track'. In figure 18 routes determined by this method by Mayes /16/ are shown. The method was used by Mayes

in simulation programs of sailing vessel routeings.

The second method is simpler, but a disadvantage is that interpolation in time for the metes data is necessary to compute the passage times between the gridpoints, as

meteorological data are available for the standard 12 h GMT intervals.

In /11/ a system is described to present to the shipmaster of a motor vessel continuous information about the performance of the ship, to face him directly with his fuel management. Average fuel consumption per hour, per mile and per kWh together with speed, power and rpm are printed after each selected period, for instance 20 minutes. Also information is given about the effect on the ship's performance by the degree of fouling of the ship's hull. An algorithm is developed which simulates the ship's performance depending on the ship's heading, the actual weather and the weather fore-casts. This algorithm is used to determine the optimum speed to arrive just in time at

the destination or to calculate the expected time of arrival at a fixed engine setting, rpm, etc. The total device including the computer was called 'energy clock and is

commercially for sale as a PERSUS (Performance Surveillance System). The device is shown in figure 19.

DATEI printer

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COIVAND CODES

voy.A.Gr,

1

list Positi=s

2

enter new positions

3

delete positions

4

insert positions

5

change data

6

calculate E.T.A

7

calculate R.P.M.

8

exit module

Figure 19.

INTEL

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MODUI2 7 a 9

DAIT. TMZ _INTETvALTIXE 4 5 6

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From the foregoing chapter it became clear that for optimal routeing knowledge of the

speed polar is of prime importance.

For sailing vessels without motor assistance this

polar diagram is primarily dependent of the wind, but also the speedloss by the

seastate snould be involved.

Mostly the speed polars are given for a 'completely

developed sea', which for instance means that when sailing in the lee of land higher

speeds are attained.

In figure 20 the speed polars of a 17,000 dwt Dynaship are given.

This sailing vessel

has a maximum sail area of 103,000 ft2.

Further details of this vessel are given by

Wagner in /17/.

As a function of the true wind angle the speed through the water

is plotted for wind forces from 1 to 9 Beaufort; the wind speed in knots follows from

tse Beaufort scale value 13 from V

= 1.625 BA.

The minimum true wind anale when

sailing close hauled is 500.

By 'symmetric cruising

at Beaufort 5, the speed made

good in the direction of the wind is 5.5 km, not taking into account the timeloss

(speedloss) by the tacking manoeuvres.

When sailing with following winds, symmetric

cruising is also more advantageous, as can be seen from the speed polars.

Using the polar diagram of the Dynaship the least time track for a trans-Atlantic

passage was determined at our Department, making use of the actual weather charts

of tnis area during a week in November 1980.

From the weather charts as produced

by the Royal Netherlands Weather Institute, charts were derived with lines of equal

wind speed, see figure 21 and 22; the wind directions are given by vectors.

From

tnis information the speed polar in any point along the track is known at 12 hour

intervals.

With the same method as described in section 4.1, the subsequent time

fronts of 12 hour intervals are determined.

In figure 23 the time fronts are shown

with tne least time track and the great circle track.

Some results of tnis simulation

- great circle distance from Bishop Rock (49.30' N

006°30' W) to Cape Race

(4o°40' n 52040' W) is 1829 n.m; the great circle course of departure is 2820;

the vertex latitude is 50.40.b' N;

passage time along the least tire track is 10 periods of 12 hours, see figure 23;

departure course is 3100;

distance along the least time track is 2034 n.m.;

- highest latitude along the least tire track is

56' N;

- average speed along the least time track was 17 kn;

Figure 20. Dynaship performance.

Polar diagram.

TRUE WIND

ANGLE

242

co?.,

L!

\x/

I 2,

.P.t+S!

Figure 23. Least time track and great circle track of routed sailing

vessel.

Table 5.1.

Per od Numper

_{Travelled distance}

ai Normed distance

_ai/a

Least time

Great circle

Least Circle

Great circle

1 ₂₀₄ 168 1.0029 1.4933 2 192 90 0.9440 0.8000 3 174 75 _{0.8555 0.6667} 4 225 89 _{1,1062 0.7911} 5 _{204 120} 1.0029 1.0667 6 195 _{82 0.9587} 0.7289 7 _{213 58} 1.0472 0.5156 8 186 _{62 0.9145} 0.5511 9 246 180 1.2094 1.6000 10 195 114 0.9587 1.0133 11 ₆₀ 0.5333 Id - 138 - 1.2267 13

-110 _0.9778 14 ₈₅ - 0.7556 15 ₂₁₆ -1.9200 16 _{- 153} - 1.3600

E ai

_{2034 1800} 10 16 a _{203.4 112.5} 1 1 a

_20.58

_47.0

0.1012 0.4178

results will be less spectacular because one has to work with predicted weather charts

whereas in this simulation the successive actual

_{weather charts have been used.}

_The

relation between routeing results from predicted and from actual weather charts will be

a research project in our Department in Delft in the coming tinie.

This is a research project in our Department where we have just started on /18/ and it appears to us that large savings of fuel can be obtained when using optimization programs as pointed out in section 4.3. The problem is now to define one unique speed polar for a given wind speed and direction, as there is one extra parameter: the propulsion power. Letcher /19/ has published on this issue before. To simplify matters we assure the gross income per voyage a fixed amount. Further we assume that the daily expense for the vessel is a fixed amount plus an amount that increases with the percentage of propulsion power, see figure 24 the circles on the left.

An estimate is made of the total travelled distance for the passage and therefore the gross incore per travelled mile is known. When the speed polars for different

percentages of propulsion power are know, either from model studies or from measurements on board of the vessel, the gross incore per hour sailing is known, see the polar figures in figure 24 os the right. Subtracting the daily cost and propulsion cost polars from the gross incore polars will give the net incorepolars per hour sailing. The envelope curve of these net incore polars will result in figure 26 wnere tne optimal motor propulsion percentage is now uniquely defined asfunction of tne true wind angle and as a consequence there is a unique speed polar for a given wind situation, see figure 26.

With this speed polar the maximum net income route is determined with the method described in section 4; s multaneously the fuel expenses are accumulated, resulting

in

- predicted maximum income route;

- total fuel cost along the route.

The researcn program 'wind assisted vessels of the section hydronautics will cover the following subprograms in the coming tire

developing computer programs for 'maximum net income routeing'; developing 'performance programs' for wind assisted motorvessels;

investigate the sensitivity for 'assumed distance' in routeing results; simulation of ocean passages with 'real weather' (hindcasting) with different types of wind assisted motor vessels to find the best design;

investigate degradation in routeing results when using 'expected weather (forecasting)' instead of 'real weather (hindcasting)'.

co

motor

propulsion

power

(2 engines)

7

0°

1

net inca

100%

75 %

50%

25%

2 5 °A OVa

symmetric cruising

Figure 24. 'Income polars'.

0°

300

600

90°

120°

150° 180°

true wind angle

Figure 25. Propulsion power as function of true wind angle.

0 % 250/0 S 0 %75%

100%

ross income

per

our curves

i i

-

1

-;

symmetric

-1

_symm.

--cruising- - -r - - - - -

uising-engi nes [ I . I I II

symmetric cru

true windangte

fuelcost per hour

symmetric cruis

net income

per hour

speed

Figure 26. Polar diagrams for speed,

fuel cost and net income per hour,

as function of true wind angle for a given wind speed.

Haarlem, 1941.

/2/ American Practi cal Navigator, Originally by Nathaniel Bowditch,

411 Defense Mapping Agency, USA, 1977.

/3/ Segelhandbuch fi'm den Atlantischen Ozean, Deutsche Seewarte, L. Frederichsen

& Co., Hamburg, 1910.

/4/ Dodging the Weather, Quarterly Journal of the American Bureau of Shipping,

August 1982.

/5/ R. W. James, The Present State of Ships Routeing, Interocean '70 International

Conference.

/o/ G. C. Korevaar, Experiences and Results of the Ship Routing of the Royal Netherlands

Meteorological Institute, Scientific report 76-9, KNMI, De BiIt, 1976.

/7/ _{Results of Marine and Isthmian Lines Weather Routed Crossing 1961 - 1963,} Weather Routing Incorporated, New York.

/8/ _{Report on the Results of Weather Routing North Atlantic Traffic by the Royal}

Netherlands Meteorological Institute, Shell International Marine Ltd., MRP/22, July 1966.

/9/ D. Heyboer, KNMI Routering (Dutch), NTT-De Lee, 1981, No. 11.

/10/ W. G. Constantine, Weather Routing for Safety and Economy, Ship Operation and Safety Conference, Southampton, April 1981.

/11/ J. _{M. J. Journee, Fuel Saving by Surveillance and Simulation of Ship's Performance}

(in Dutch), NTT-De Zee, November 1982.

/12/ C. de Wit, Optimal Meteorological Ship Routing, Netherlands _{Scheepsstudiecentrum} TNO, report No. 142 S, August 1970.

/13/ S. J. Bijlsma, On Minimal-Tima Ship Routing, Royal Netherlands Meteorological Institute, report No. 94, 1975.

Maritime Research Center, Kings Point, New York, 1978.

/15/ L. S. Pontryagin, Mathematical Theory of Optimal Processes, Interscience Publishers Inc., New York, 1962.

/16/ James H. Mayes, Sailing Ship Weather Routing, Symposium on Wind Propulsion of Commercial Ships, London, 1980

/17/ B. Wagner, Calculation of the Speed of Travel of Sailing Vessels, Jahrbuch der Schiffbautechnischen Gesellschaft 61, Springer Verlag, 1967.

/18/ O. Steinert, Project Proposal 'Routing of Wind Assisted Vessels n Dutch) Dept. of Marine Technology, Section Hydronautics, August 1984.

/19/ J. S. Letcher, Optimal Performance of Ships under Combined Power and Sail, Journal of Ship Research, Vol. 26, No. 3, pp. 209-218, Sept. 1982.