611

1 INTRODUCTION

Since 1994 there has been a tendency in many countries

other than the Russian Federation to show an

increasing interest in usability the Northern Sea Route

as an alternative shipping lane to traditional ones lanes

connecting Europe with ports of the Far East via the

Suez or Panama Canals. A number of simulation

results of the navigation and economic performance of

ship voyages by the NSR were published [9,10,11,12,19,

20,23, 28, 29]. Mulherin et al. [11] took into

consideration statistical ice navigation conditions with

the use of a cumulative distribution function of ice

conditions. Kitagawa et al. [9] simulated average

environmental conditions used that not reflected short-

term environmental conditions changes. Smith and

Stephenson [23] and Nam et al. [12] used very averaged

data input organized in various ways. Wang et al. [28]

approached multicriteria route planning including cost

and risk. Zhang et al. [29] included meteorological risk,

fuel consumption and time of voyage through the

North Atlantic Ocean in the multicriteria algoritm

using Pareto criteria. Matsuzawa et al. [10] and

Pastusiak [19, 20] simulated ship’s passage using

already known ice conditions. All of them did not show

input and output of simulated data errors. In the

author's previous work [20], many examples of error

values influencing the results of planning a ship's

voyage in the regions of ice occurrence were presented.

However, the impact of individual input data errors on

the output error of the navigation and economic result

of the voyage was not determined. The nature of the

error and the probability for it was not specified

(68.3%, 95.0% or 99.7%). Where different error values

were given, it was not specified how the information

on single input error could be used to evaluate the

Modeling Navigation, Ecologic and Economic Results of

Planned Voyage of Ship on the Northern Sea Route

T. Pastusiak

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: Over the years, many publications have been published on planning ship’s passage through the NSR

and input and output data modeling. Results were not considered in terms of errors of modeled output data.

Basic research for the years 2008 - 2020 was carried out in the study on opening and closing dates of ice-free transit

corridor on the NSR and probable number of tariff zones where paid assistance of icebreakers would be required.

Mathematical relationships of average values and probable errors of navigational, ecological and economic ship

voyage performance were developed. For this purpose, partial derivatives of functions were used. The outcomes

of the study have shown that changes of total average cost of vessel’s transit voyage depends mostly on number

of tariff zones where assistance of icebreakers is required. But navigation with icebreakers assistance ensures

smaller standard deviation of total voyage time and thus more precise scheduling of ship's voyage timetable

through the NSR. The higher probability of meeting requirements set by decision-maker when planning a long-

term voyage of ship, the shorter time window for meeting these requirements. Outside the favorable time

window, the navigational, ecological and economic results of voyage deteriorate very quickly.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 16

Number 4

December 2022

DOI: 10.12716/1001.16.04.02

612

navigational or economic performance of a ship's

voyage through ice covered areas [20]. Pruyn [22]

analyzed conclusions coming from many publications

and stated that most of them expected smooth

navigation through the NSR without delays for

inspections at the initial ports, without waiting for the

convoy and without waiting for good environmental

conditions. This author considered the need of

icebreakers assistance as important cost building and

potential elongation of voyage time factor. Due to

above, in the present work possible use of icebreakers

assistance was taken as a main problem for analysis.

In order to make rational decisions, one needs not

only information about average values but also about

their error values. It was assumed that the assessment

of the navigational (time, date), ecological (used fuel

producing CO2 and Greenhouse Gas) and economic

performance (total cost of a voyage) should be based

on more balanced criteria than pessimism or optimism

of the decision maker. Modeling the ecological

performance can be used to search for the methods to

reduce CO2 and Greenhouse Gas production by

improving the planning of ship operations. This work

highlights possibility of using the mean error of the

mathematical function instead of several data input

errors to calculate the navigational, ecological and

economic performance of the “long-term”, it means at

least three month period planned voyage, together

with the determination of the uncertainty range well in

advance of beginning of the voyage (Estimated Time of

Departure ETD).

The aim of the study was to develop a method for

long-time planning navigation, ecological and

economic results of transit voyages and scheduling

voyages through the Northern Sea Route (NSR) where

ice may be encountered. Navigational results were

based on length of route, speed of ship on selected

sections of route, total voyage time, date of beginning

of voyage at the port of departure and date of end of

the voyage at the port of destination. Economic results

of the voyage included the costs of fuel, charter,

icebreaker assistance on selected sections of the route

and total cost of entire voyage. Ecological results were

presented by fossil fuel consumption. Final results of

simulation involved average values and standard

deviation. The study has shown that the number of

tariff zones requiring icebreaking assistance has the

greatest impact on performance of planned ship

voyage.

2 RESEARCH METHOD

The first general principle was adopted, according to

the ISM Code [21], that the model of navigational,

ecological and economic performance of long-term

voyage planning should be simple and transparent,

and the results ready for direct implementation in the

decision-making process. The International Maritime

Organization [8] recommends that a voyage plan

should include information on the accuracy and

quality of each of the used electronic navigational

charts, accuracy of ship position and reasonable

accuracy of the timetable of voyage plan. A voyage

plan should also determine impact of the above-

mentioned accuracy and quality of data on the safety

of navigation [8]. Following this recommendation as a

second general principle, the inaccuracies of data input

affecting the process of planning and scheduling, and

navigational, ecological and economic performance of

voyage on the NSR, were taken into consideration and

worked out in this study. Taking into account the

statistical mean values, errors and uncertainty ranges

should be helpful for the substantive assessment of the

profitability of the project concerning the activation of

seasonal navigation with ships without ice

reinforcements through the NSR (transit navigation) or

for destination navigation (to / from a port located

inside the NSR) as a link of intermodal transport

connecting the European and Far East ports with ports

located deep in Siberian rivers.

The input and output data should be measurable to

allow comparisons to be made. This is to enable

evaluation and objective decision to be made. These

data should therefore be presented using historical

information, i.e. statistical values - mean values and

theirs errors values. Data processing systems

transforming input data into output data should base

on mathematical formulas only. Previously mentioned

concept "long-term" should be understood as a time

advance of at least 3 months, when only historical, i.e.

statistical information, is available to voyage planners.

The aim of this research is to map the phenomena

occurring in real conditions and play an important role

in long-term planning of transit voyages of ships in the

regions of ice existence on the Northern Sea Route.

Therefore, there is a need to design selected fragments

of the studied reality. It was assumed that selected

navigational and economic results of ship's voyage are

significant elements of reality. Navigational results are

based on length of route, speed of ship on selected

sections of route, total voyage time, date of beginning

of voyage (Estimated Time of Departure ETD) and date

of end of the voyage (Estimated Time of Arrival ETA)

at nodal points, i.e. the port of departure and port of

destination. It has been assumed that such ports will be

places where safety clearances of ships are carried out

before commence of voyage on the NSR. It will be the

port of Murmansk roads on the western side of the

NSR and the port of Provideniya roads on the eastern

side. The study considered the case of a ship passage

from west to east. Economic results of the voyage

include the costs of fuel, charter, icebreaker assistance

on selected sections of the route, if ice conditions on

route require it, and total cost of entire voyage. Due to

the involvement of ship-operators in counteracting

climate change, attention was paid to fuel consumption

as a factor generating greenhouse gases. Ships were

assumed to be assisted only by nuclear powered

icebreakers. Nuclear energy generation has near to

zero green-house gas emissions in the energy

generation phase [2, 3]. Under the EU Taxonomy

Regulation, certain nuclear activities that fulfil nuclear

and environmental safety requirements can be added

to those already covered by the first Delegated Act as

transitional activities to contribute to the transition to

climate neutrality [2, 3]. From this point of view,

nuclear propulsion assumed is an ecological

propulsion. In such a situation, possible assistance of

icebreakers during ship's voyage was not added to

environmental costs of ship's voyage. Other

components of planning were considered irrelevant to

study of reality and were omitted in the study. For this

reason, possibility of ship awaiting possible

613

improvement in ice navigation conditions was omitted

in considerations.

The study used ice concentration maps in simplified

Marginal Ice Zone scale [13]. The work concerns

navigation on ships without ice reinforcements, which

should avoid areas of ice occurrence. It will be

sufficient to know edges of the lowest ice

concentration. These maps contain this information.

They are published within 24 hours. This is relatively

high frequency of publication compared to other

published ice maps. It ensures minimization of

influence of ice drift on location of its edges used in the

research.

The work was based on historical, it means

statistical data. All statistical results are affected by

errors. Therefore, there was the need to determine the

values of possible deviations from the mean values.

This concerned input data, output data and ranges of

uncertainty. To define them, mathematical formulas

reflecting the processes taking place and errors of these

formulas were obtained by means of derivatives of

functions.

3 MODELING THE RESULTS OF

NAVIGATIONAL, ECOLOGICAL AND

ECONOMIC OF A SHIP VOYAGE

The first navigation information is date and time of

beginning of voyage (ETD). Date of opening transit

corridor through the whole NSR in the most difficult to

overcome chocking zone TCZ for defined ice class of

ships determines date of departure of a ship from the

port of departure T0, i.e. the date of the beginning of

voyage (ETD). The chocking zone is area on the NSR

which opens the latest and closes the first. Date T0 can

be determined with Equation 1. Variables included in

the formula are independent. Statistical values will be

required for distance from departure port to the edge

of the first tariff zone at the west side of the NSR

DOUT1 and for distance from the edge of first tariff

zone at the west side of the NSR to the mean location

of the chocking zone DIND1 and the ship speed in

these sections. It was assumed that a ship commence its

voyage when ice conditions between port of departure

and chocking zone allow for independent navigation

of this ship.

24 24

OUT IND

FA IND

= − −



(1)

The second navigation information is total voyage

time between the port of origin and port of destination

TTTL (Equation 2). To make calculations, it will be

necessary to determine statistical values for the length

route segments of DOUT1 on the western side of the

NSR, DOUT2 on the eastern side of the NSR and the

lengths of route segments in individual tariff zones.

The latter may apply to independent navigation of a

ship on the section of DIND length or navigation with

the icebreakers assistance along the section of DIB

length, which is payable according to the NSR

Administration tariff [4]. Specified ship speeds VFA for

navigation outside ice areas, VIND for independent

navigation and VIB for navigation assisted by the

icebreakers correspond to ice navigation conditions.

The maximum number of tariff zones is 7 [5]. What is

needed is knowledge of the number of tariff zones

requiring the assistance of icebreakers because of the

ice conditions occurring within the Julian day of the

year.

( )

1 1 1 2

24 24 24 24 24 24

IND

OUT IND OUT OUT

END CZ

FA IND FA IB IND FA

n k D

D D D D

V V V V V V



−





= − − + + + +



     



(2)

After adding the time of ETA (date) of the

beginning of the voyage T0 and the total time of the

voyage TTTL, the time (date) of the end of voyage

TEND (ETA) at the destination port is obtained

(Equation 3).

( )

1 1 1 2

24 24 24 24 24 24

IND

OUT IND OUT OUT

END CZ

FA IND FA IB IND FA

n k D

D D D D

V V V V V V



−





= − − + + + +



     



(3)

The fuel consumption was adopted as ecological

result of the planned voyage. In addition to the

previously specified factors, it is necessary to know

ship’s daily fuel consumption under way related to

areas where ice not exists CFA, in areas where ice exists

but the ship proceed without assistance of icebreakers

CIND or the ship proceed with assistance of

icebreakers CIB (Equation 4). After adding up the costs

of charter, fuel used and paid icebreakers services, the

total cost CTTL of the ship's voyage through the NSR

was obtained (Equation 5).

( )

24 24 24 24

IND

OUT OUT

FA IND IB FA

FA IB IND FA

nkD

FC C C C C

V V V V

−



=  +  +  + 

   

(4)

TTL TTL CH F T

C T C FC C C R GT=  +  +  

(5)

Cost of assistance of icebreakers CT in Equation 5 is

published by the NSR Administration [5] for ships of

particular ranges of Gross Register Tonnage (GRT). It

depends on number of the tariff zones where a ship

require assistance of icebreakers due to ice navigation

conditions existing there. For the ships in the range

from 40,000 till 100,000 GRT without ice strengthenings

(steel hull, no ice class) particular tariff points of CT for

tariff zones 1-6 are lying on the line presented by

Equation 6 (coefficient of determination R2 = 1). It is

representing tariff in Russian Rubles per unit of the

ship's Gross Tonnage.

89.369 357.47

Ck=  +

(6)

For defined rate of Russian Rubel R (1 RUB = 0.0136

USD at 01.09.2020 [24]) and PANAMAX class ship

tonnage 38,500 GT the Equation 5 can be transformed

into Equation 7.

( )

89.369 357.47

TTL TTL CH F

C T C FC C GT R k=  +  +    +

(7)

A significant source of uncertainty for all decision

factors is variability of beginning date of entire transit

corridor to be opened for ice-free navigation TCZ. Then

the number of tariff zones requiring assistance of

icebreakers is equal to 0 (k=0). The geographic location

of the chocking zone is also variable [16, 20]. This is

where the transit corridor opens at the latest and closes

at the earliest [18]. These variations can be determined

by the standard deviation σ for 1M (i.e. 1 · σ), 2M (i.e. 2

· σ), or 3M (i.e. 3 · σ). The current IMO requirements for

614

defining the value of navigational errors at sea apply to

2M (double standard deviation σ).

In all the above relationships (Equations 1-7) there

are numerous components of uncertainty. The

question is, on the basis of which factor the person in

charge of a long-term voyage planning (3-12 months in

advance) should make his decisions. There is a

problem of multi-criteria decision making. Using

different single input and output data for a decision

system leads to the calculation of different results, of

different nature and thus difficult to compare.

However, it is possible to reduce multi-criteria. This

can be done using the method of determining the mean

errors of functions defined by the formulas (Equation

1-7). Then only the basic navigational, ecological and

economic criteria for making decisions will be

maintained. To do this, one performs the

differentiation, i.e. calculation of derivatives [25] of

equations 1-5 and 7 with respect to the components of

equations with errors [6, 26]. Thus, the formulas

(Equation 8-12) are obtained, which define the mean

values of errors of functions for the mean values

(Equations 1-5 and 7). In order to use the formulas 1-

12, it is necessary to know the mean values and the

standard deviation of all variables included in the

formulas above. Based on the mean values and their

errors (standard deviation), ranges of decisive outputs

can be determined.

' 2 2 2 2

24 24

OUT FA IND IND

OUT IND

D V D V

FA IND

T m m m m



−−

=   +  +  + 







(8)

( )

2 2 2 2 2

24 24 24

24 24

OUT FA IB IB IND

IND OUT

OUT

D V D V D

FA IB IND

FA IB

TTL

IND

k n k

m m m m m

V V V

n k D

   



   

−

−

−

 +  +  +  

   



   

  



   



   

=



− − 



 + 











2 2 2

24 24

OUT IND

IB IND



−

 − 











(9)

1 1 1

2 2 2 2 2

' 2 2

1 1 1

24 24 24

24 24

OUT FA IND IND OUT

FA IB

OUT IND

D V D V D

FA IND FA

FA IND

OUT

END V D

m m m m m

V V V

T m m



   

−−

 +  +  +  



   



  



   





−

=   







( )

2 2 2

24 24

24 24 24

IB IND IND

OUT FA

IND

V D V

IND

IB IND

OUT IND

D V k

FA IB IND

n k D

m m m

V V V

+ + +



− − 



−

−

+   











−

+  +  − 



  





(10)

( )

2 2 2

2 2 2 2 2

24 24 24

24 24

OUT FA FA IND IB

OUT FA OUT

FA IB IB IB

D V C D V

FA FA IB

FA IB

IND

IB IND

D C D

C k C

m m m m m

V V V

n k C

   

     

−

  

 +  +  +  

   

     

   

  



     

   

−





=  +  +











−

( ) ( )

2 2 2

24 24 24

IND IND IND

OUT FA FA

IND IND IND

D V C

IND

OUT FA OUT IND

FA IB IB

D V C

FA FA IB

n k D C n k D

m m m

D C D D C

C D C

m m m

V V V



   

− −   − 

  + 



   





   





   

−



+   +  −



   



  



  







IND











(10)

( )

' 2 2 2 2 2

89.369

TTL

TTL CH T F FC k

C C m C m GT R m=  +  +   

(11)

4 INPUT VARIABLES FOR THE MODELS

Long-term planning in advance from 3 to 12 months

and more is necessarily based on historical or statistical

data. It has been noted that main source of uncertainty

at long-term voyage planning stage is date of opening

transit corridor for navigation permissible for the

specific ice class of ship, voyage time through the NSR

and in particular additional costs of icebreaker

assistance services. These costs of icebreakers services

are rising discretely according to the number of tariff

zones where ice conditions require a ship to use

icebreakers assistance.

Responsible persons in the offices must take into

account a number of factors when making long-term

decisions related to planning of cargo transport by sea

through the NSR. Earlier in the work, they were

divided into navigational, ecological and economic

factors. Navigational factors are length of route, date of

beginning and end of voyage, time of voyage, ice class

of ship and speed of ship under certain ice conditions.

Ecological factors are type of fuel and fuel

consumption along the entire route. The economic

factors are cost of fuel consumed, charter cost for entire

voyage and cost of tariffs for required icebreakers

assistance. They can be determined using

mathematical relationships. To use the formulas for

average values (Equations 1-7) and to determine the

uncertainty ranges (Equations 8-12), the standard

deviation value of all variables included in the above

formulas should be determined. Some of them require

more extensive research. They are presented below.

4.1 Dates of beginning and end of independent navigation

and with icebreaker support

Main criterion for navigation in areas where ice exists

is routing through the lightest ice conditions. These

lightest ice conditions on the NSR exist when an

uninterrupted transit corridor is created along the

entire NSR completely free of ice. For the long-term

planning of a ship's voyage, it is important to know the

opening and closing dates of such corridor. For this

purpose, an analysis of the NIC MIZ ice maps was

performed [13] for the period from 2008 to 2020. The

number of tariff zones in which ship has to use services

of icebreakers increases the cost of voyage by leaps and

bounds, significantly affecting the economic result of

the planned voyage of the ship. For this reason, it was

decided to use same ice maps to determine number of

tariff zones with ice before opening and after closure of

ice-free transit corridor.

Earlier in the work has been noted that main

uncertainties in long-term voyage planning are date

when the NSR ice-free transit zone opens and closes,

distance from port of departure to the chocking zone,

which usually opens the latest, and the distance from

port of departure on voyage to the chocking zone. As

the research concerns a voyage from west to east,

Murmansk was chosen as port of departure. Changes

over time in number of tariff zones requiring (or not)

use of paid icebreaker services for western and eastern

parts of the NSR and changes over time for opening

and closing of ice-free transit corridor are shown in

Figures 1 and 2.

The graphs show that results for the opening of

transit corridor (beginning of voyage) are consistent on

both sides - west and east (Figure1). When planning the

date of arrival of the ship at the zone blocking transit

corridor for the longest time ("chocking zone" on the

NSR) on basis of trend line for one tariff zone (red dash

line), there is a high probability of not incurring

charges for two or more tariff zones (blue, green or

magenta lines) requiring assistance of icebreakers.

When planning ship's arrival date to "chocking zone"

on the NSR on basis of trend line for two tariff zones

(blue dash line), there is very high probability of not

615

incurring charges for three or more tariff zones

requiring icebreaker assistance (green or magenta

lines).

Planning ship's passage at chocking place on the

NSR based on trend line for no tariff zones requiring

icebreaker services, it is highly likely to incur charges

for one tariff zone (red line) and a little likely to incur

charges for two tariff zones (blue line).

Figure 1. Number of tariff zones to be paid from given day

foreward (and after that day) when opening transit corridor:

(a) in western part of the NSR; (b) in eastern part of the NSR.

When planning date for the ship to pass through

zone that blocks transit corridor at end of summer

navigation season (Figure 2) on basis of trend line for

three tariff zones (blue dash line), there is a high

probability not to pay for four or more tariff zones

requiring icebreaker assistance (green or magenta

lines). Planning ship's passage at chocking zone on the

NSR based on trend line for no tariff zones requiring

icebreaker services, it is highly likely that ship will

incur charges for one or two tariff zones (red and / or

blue lines).

Figure 2. Number of tariff zones to be paid from specified day

backward (and before that day): (a) during transit corridor

closing in western part of the NSR; (b) during transit corridor

closing in eastern part of the NSR.

Graphs in Figures 1 and 2 do not illustrate complex

phenomena occurring. Information obtained on their

basis does not constitute enough synthetic assessment

of situation for purposes of making decisions. In that

case, in next step, above relationships were analyzed.

Time span of the navigation season (number of days)

and the number of tariffs to be paid for required

icebreakers assistance with theirs probability were

found. The graph of average dates of changes in

number of tariff zones requiring icebreaker assistance

along the entire NSR was obtained, resulting from

2008-2020 with probability of occurence 68.3% (Figure

3). The x axis on this and next Figures is related to day

of the year when a ship arrives at the chocking zone.

Standard deviation of date when change number of

tariffs occurred is ± 11.9 days for range from 0 to 7 tariff

zones. For the range from 0 to 5 tariff zones requiring

icebreaker assistance, standard deviation for

determining number of zones is ± 11.2 days. Average

day of opened ice-free corridor is 264 (Julian day) with

standard deviation ±7.4 days. Edges of zero number of

tariffs are average dates of ice-free time window for the

whole NSR. The averaged data set approximated with

polynominal function of 5th degree (Equation 13).

Coefficient of determination R2 is 0.9870. The only 4

tariff zones existing simultaneously are expected

maximal value of tariffs to be analyzed. It is because

the associated 30 days before opening of ice-free

corridor and 30 days after closing ice-free corridor ice

conditions existing on the NSR are too severe for ships

without ice strengthenings – both for independent

navigation and with assistance of icebreakers.

As a result of use of statistical relationships, it is

possible to predict required day of arrival to the

chocking zone on the NSR well in advance, more than

3 months before the commence of voyage occurs. That

is, when reliable forecasts based on current

hydrological and meteorological information cannot be

expected yet.

Figure 3. Number of expected tariff zones to be paid for

icebreakers services on the whole NSR.

9 5 6 4 3

2.60985 10 3.03634 10 0.00137753

0.304823 32.9934 1408.78

CZ CZ CZ

CZ CZ

k T T T

−−

= −   +   −  +

+  −  +

(13)

It was assumed that the formula is valid for the

number of tariff zones from null to 4. If the ice

conditions on the NSR do not allow the ship to

independently navigate more than four areas of tariff

zones, it is assumed that ice conditions existing before

and after number of days related to designed number

of four tariffs are inaccessible for ships without ice

strengthenings. Standard deviation of expected

number of tariffs to be paid for required icebreakers

assistance mk is represented by Equation 14.

( )

9 4 6 3 2

k TCZ

13.04925 10 12.14536 10 0.00413259

0.609646 32.9934

CZ CZ CZ

T T T

−−



−   +   − 

=  



+  −



(14)

4.2 Time required to arrive from port of departure to the

chocking zone

If the ship arrives too early at the chocking zone, it will

have to wait for ice conditions to improve and / or use

expensive icebreakers services in one or more tariff

zones. In this study, it was assumed that ship will not

wait for improvement of hydrological conditions, but

will use paid icebreakers assistance services in as many

tariff zones as ice situation requires. Both waiting for

ice conditions to improve and use of icebreakers are

causing additional voyage costs.

It is therefore important to determine statistical

location of chocking zone and hence length of route

from origin port to chocking zone. In order to plan

616

ship's voyage from west to east through the NSR,

distances of center chocking zone from starting port (in

our case, Murmansk) were determined for navigation

seasons from 2008 to 2020. Found average value

increased by standard deviation σ equal 2,221 nm,

average value equal 1,709 nm, average value reduced

by standard deviation σ equal 1196 nm, standard

deviation σ equal 512 nm and also 1st quartile equal

1,384 nm, Median 1,547 nm and 3rd quartile equal 2,056

nm (Figure 4). Figure 4 shows that the vast majority of

the chocking zones of summer navigation period in the

years 2008-2020 were located close to Vilkitsky Strait

(Mys Chelyuskin) at Kara Sea and Laptev Sea. The only

clear anomaly was in 2012. Then chocking zone was

located at Wrangel Island on New Siberian Sea. Based

on known length of route from port of departure to

chocking zone and speed of a ship, it is possible to

calculate time needed to cover this part of route and

date of commencement of voyage at the port of

departure. It should be noted so appointed distances in

between port of Murmansk roads and the Chocking

Zone they do not apply to the route of the ship sailing

straight ahead, but to the route that resulted from the

circumstances of the ice navigation conditions. For this

reason, the geographic location of the middle of the

Chocking Zone (ice barrier) is better reflected by

average geographic longitude of locations of these

barriers along most frequently available route through

the NSR. This gives in approximation average equal

120°E, standard deviation ±27.6°, median 109°E, 1st

quartile 100°E and 3rd quartile 151°E.

Figure 4. Distance from the port of departure (Murmansk) to

chocking zone.

4.3 Speed of the ship and daily fuel consumption

In order to determine statistical values of speed for

ships without ice strengthening passing through the

NSR during independent voyage and with assistance

of icebreaker an analysis was carried out based on

navigation permits for the NSR issued by the NSR

Administration [14] and daily reports produced by the

above-mentioned ships during their voyage in 2016

[15]. In order to determine speed of the ship and fuel

consumption outside areas of ice occurrence and fuel

consumption in areas of ice occurrence, relationships

developed in author's earlier work were used [17, 21].

Statistical results are summarized in Table 1.

Table 1. Speed and fuel consumption of PANAMAX type

ship. Compiled by author based on [14, 15, 16, 20].

________________________________________________

Component Waters Waters Waters

outside inside inside

the NSR the NSR; the NSR;

independent voyage with

voyage assistance of

icebreakers

________________________________________________

Speed [kn] 13.70 10.70 9.14

St. dev. Of 0.33 2.23 1.86

speed [kn]

Daily fuel 31.7 34.2 39.3

consumption

[mt/day]

St. dev. of 1.68 1.81 2.08

consumption

[mt/day]

________________________________________________

4.4 Route lengths in particular tariff zones and outside

them for corridor opening towards east

In order to determine statistical lengths of the voyage

routes through particular sections of the route and

tariff zones, course of these routes was examined as if

ship was proceeding just behind front of ice-free zone

during opening of transit corridor through the NSR.

Variability of length of ship's routes defined in this way

by the NSR is intended to illustrate impact of

variability of ice conditions occurring in the

subsequent navigation seasons on length of the routes,

and thus uncertainty ranges. Statistical results are

summarized in Table 2. Lengths of the particular route

segments were very similar except tariff zones number

1, 6 and 7. Summed lengths of the route segments up

based on average value and median value were very

similar.

Table 2. Statistical route lengths and their standard deviation. Compiled by the author based on ice maps of Marginal Ice

Zone in ESRI Shape format [13]. Provided courtesy of the U.S . National Ice Center.

___________________________________________________________________________________________________

Tariff zone OUT 1 1 2 3 4 5 6 7 OUT 2 Averaged length Length of the

West East of tariff zone whole route

___________________________________________________________________________________________________

Average length [NM} 744 204 466 295 267 391 461 381 161 352.3 3371

St. dev. [nm] 4.9 95.6 125.3 41.6 37.2 40.9 109.1 142.2 1.3 84.6 66.5

Median length [nm] 741.1 162.1 441.7 288.7 262.4 398.9 431.5 340.3 161.0 340.3 3328

___________________________________________________________________________________________________

617

4.5 Ship’s operating costs

Cost of IFO 380 fuel in Rotterdam as of 1st September

2020 of USD 275.5 per metric ton based on current

prices given by Ship&Bunker [27]. Cost of chartering

PANAMAX-type ships assumed to be USD 14,250 for

waters in the Atlantic Ocean outside the Arctic on the

basis of the current prices given by Hellenic Shipping

News [7]. For delivery of ship in waters of the Arctic

Ocean, it was increased by a factor of 2.0 based on

separate studies [19]. Therefore, cost of charter in

Arctic waters assumed to be USD 28,500 per day and

pro rata for use of ship in the NSR region.

5 DISCUSSION OF RESULTS

On basis of formulas presented in chapter 2 and 3.1 as

well as average values and standard deviations of

input data presented in chapter 3, values of

navigational, ecological and economic results of the

PANAMAX type ship's voyage in period from 215 to

300 Julian days of the year were calculated (Figures 5-

10). It was assumed that this is a maximal time window

in which it is possible to consider transit navigation of

ships with steel hulls without ice reinforcements on the

NSR. Average values (black lines) were enlarged and

reduced by one standard deviation ± 1M representing

probability of 68.3% and by two standard deviations ±

2M representing probability of 95% (Figures 5, 6, 7, 9

and 10). Value of 95% probability meets the IMO

maritime navigation recommendations.

Total average time of transit voyage depends on

severity of ice conditions existing during whole

voyage. This severity is represented by number of

traffic zones where icebreakers assistance is required.

The highest severity (number of tariff zones requiring

assistance of icebreakers) is at the beginning and at the

end of the navigation season, and reduces till null at the

mean day of ice-free transit corridor opened on the

NSR.

Standard deviation of the total time of voyage

changes during the navigation season and ranges from

18.2% at the beginning and end of the season to 24.7%

of average value during ice-free window (for 68.3%

probability). For a probability of 95% (2M) deviation

values of total time of voyage are twice as high, from

36.4% at the beginning and end of the season to 49.4%

of average value during ice-free window (Figure 5).

Reason of it is greater standard deviation of ship’s

speed during independent navigation and increasing

share of independent navigation from the beginning

and end of the navigation season towards average day

of ice-free navigation period. Navigation with

icebreakers assistance ensures a much smaller

standard deviation of total voyage time and thus more

precise scheduling of ship's voyage timetable through

the NSR.

Total average fuel consumption depends on

severity of ice conditions existing during the whole

voyage. The highest average fuel consumption is at the

beginning and at the end of navigation season, and

decreases till minimal at the mean day of ice-free

transit corridor opened on the NSR. This is due to

higher daily fuel consumption during more severe ice

conditions (icebreakers assistance) and lower fuel

consumption during lighter ice conditions

(independent voyage).

Standard deviation of total average fuel

consumption ranges from 19.6% of average value on

the beginning and end of the navigation season up to

25.5% during ice-free window (for 68.3% probability).

Deviation values for probability of 95% (2M) are twice

as high, from 39.2% of average value at the beginning

and end of the navigation season to 51.0% during ice-

free window (Figure 6). Navigation with icebreakers

assistance is associated with a higher standard

deviation of fuel consumption and thus less precise

determination of deviations in voyage costs at the

beginning and at the end of the navigation season.

Figure 5. Total average time of transit voyage and voyage

time estimation error.

Figure 6. Total average fuel consumption during transit

voyage and fuel consumption estimation error.

Changes of total average cost of ship’s transit

voyage depend mostly on number of the tariff zones

where assistance of icebreakers is required due to

severity of ice conditions (Figure 7). Costs of fuel

consumed and time of voyage (charter of ship) are

increasing from external days of summer navigation

season towards mean day o ice-free transit corridor but

are relatively flat in relation to costs of assistance of

icebreakers (Figure 8). Changes of charter cost are

around 0.9-3.0% of total costs, changes of fuel

consumption costs are around 0.6 – 2.0% of total costs.

But changes of cost of required assistance of

icebreakers are much higher and are around 32.9 –

42.3% of total costs. To maintain relative stable total

costs of voyage it is recommended absolute avoidance

618

of voyage during period of time when severity of ice

conditions require assistance of icebreakers.

Figure 7. Total average cost of ship’s transit voyage and total

cost estimation error.

Figure 8. Total average cost of ship’s transit voyage and its

components.

In the next step, the average fuel consumption per

day was calculated for the whole transit voyage for

each day the ship was to pass the chocking zone. They

were obtained by dividing total fuel consumption by

number of voyage days (Figure 9). Values of average

fuel consumption increased and reduced by one

standard deviation (1M) and two standard deviations

(2M) were included on the diagram. The lowest fuel

consumption is related to ice-free transit corridor

existing along the whole NSR. Fuel consumption

increasing with severity of ice conditions towards the

beginning and end of the summer navigation season.

This is due to daily fuel consumption increasing with

the worsening of ice conditions represented by number

of tariff zones required assistance of icebreakers (Table

1). Particular lines for fuel consumption with theirs

standard deviations are sensitive for standard

deviation of daily fuel consumption that is smallest at

the mean day of opened ice-free transit corridor and

the highest at the beginning and at the end of

navigation season (number of tariff zones where was

required assistance of icebreakers).

Average fuel consumption per nautical mile passed

was calculated for the whole transit voyage for each

day the ship was to pass the chocking zone. They were

obtained by dividing the total fuel consumption by the

average length of the route for the years 2008 – 2020

(Figure 10). Values of average fuel consumption per

nautical mile increased and reduced by one standard

deviation (1M) and two standard deviations (2M) were

included on the diagram. The lowest fuel consumption

is related also to ice-free transit corridor existing along

the whole NSR. Fuel consumption per nautical mile

increasing with severity of ice conditions towards the

beginning and end of the summer navigation season.

This severity is represented by quantity of tariff zones

where assistance of icebreakers was required.

Figure 9. Average fuel consumption per day and estimation

error.

Figure 10. Average fuel consumption per nautical mile

passed and estimation error.

Following formula for the expected number of

tariffs to be paid for required icebreakers assistance

(Equation 13) and formula for standard deviation of

these expected number of tariffs (Equation 14) found

probable considerable increase or reduce of number of

tariff zones (severity of ice conditions) as far as time

span from mean day of opened ice-free transit corridor

(Figure 11). As negative number of tariff zones

requiring assistance of icebreakers is impossible, the

limit border for minimal number of that tariff zones

should be applied. It is for number of zones requiring

assistance of icebreaker k equal null (presented with

violet dash line). Following this observation

recalculations were performed for formulas 2, 4 and 7.

As a result, total average time of voyage, total average

fuel consumption, total average costs of voyage,

average cost of ship charter, average cost of fuel

619

consumed, average cost of icebreaker assistance, fuel

consumption per day and fuel consumption per

nautical mile graphs including theirs standard

deviations were obtained (Figures 12 – 19).

Lines of average value on graphs that are taking

into consideration deviations of number of tariff zones

where is required assistance of icebreakers are the

same as in the graphs taking into account the standard

deviation. However, the lines including probable

number of tariff zones k increased or decreased by

standard deviation of k with 1M and 2M narrow time

window of the lowest values. Average values provide

the lowest value time window from 240 to 280 Julian

day of the year. Average values of number of tariff

zones k increased and decreased by one standard

deviation mk (1M) provide the lowest value time

window from 250 to 270 Julian day of the year. Average

values k increased and decreased by double standard

deviation mk (2M) provide the lowest value time

window from 260 to 265 Julian day of the year. From

the above it should be concluded that in order to ensure

the lowest navigational, ecological and economic

results of the ship's transit voyage through the NSR for

average value the ship should pass the chocking zone

between 240 and 280 Julian day of the year. For k

increased by single standard deviation mk with

probability of 68.5% should pass between 250 and 270

Julian day. For k increased by double standard

deviation mk, ship should pass the chocking zone

between 260 and 265 Julian day. The higher the

probability of meeting the requirements set by

decision-maker when planning a long-term voyage of

ship, the shorter time window for meeting these

requirements. Outside the favorable time window, the

navigational, ecological and economic results of

voyage deteriorate very quickly. This is in line with

outcomes of Pruyn [22] work.

Figure 11. Number of tariff zones k with assistance of

icebreakers and its deviation for possible changes

Figure 12. Total average time of voyage and its deviation for

possible changes of number of tariff zones required

icebreaker assistance.

Figure 13. Total average fuel consumption and its deviation

for possible changes of number of tariff zones required

icebreaker assistance.

Figure 14. Total average costs of voyage and its deviation for

possible changes of number of tariff zones required

icebreaker assistance.

620

Figure 15. Average cost of ship charter and its deviation for

possible changes of number of tariff zones required

icebreaker assistance.

Figure 16. Average cost of fuel consumed during voyage and

its deviation for possible changes of number of tariff zones

required icebreaker assistance.

Figure 17. Average cost of icebreaker assistance and its

deviation for possible changes of number of tariff zones

required icebreaker assistance.

Figure 18. Fuel consumption per day and its deviation for

possible changes of number of tariff zones required

icebreaker assistance.

Figure 19. Fuel consumption per nautical mile and its

deviation for possible changes of number of tariff zones

required icebreaker assistance

6 VERIFICATION OF RESULTS

The formulas presented in the previous chapters were

created on the basis of statistical (historical) data for the

period of time from 2008 till 2020. These results of the

statistical research require verification. Here was

decided to use sample of real passage of the ship

through the NSR for the year out of the statistical

period of time used for creation of the formulas. For

creation above mentioned equations the PANAMAX

ship was used with the following parameters:

DWT=73,140 t, GT 38,489 tons, L=225m, B=32m. In this

case decided to use same size ship for verification.

Decided to use voyage data also of the bulk carrier

Golden Enterprise that made her voyage in early stage

of summer season. She departed from Murmansk on

08.08.2021 at 09.8 UTC and guided to Kara Gate. The

NSR entered on 10.08.2021 at 2300 UTC and left the

NSR at the Bering Strait on 23.08.2021 at 2300 UTC. Her

parameters were as follows: DWT 79,471 tons, GT

43,498 tons, L= 229m, B=32m. Her voyage data were

available on the interactive map on the CHNL

Information Office website [1]. The routes outside the

NSR were not exactly same like these used for

construction of Equations 1-14. In this case the

distances DOUT1 and DOUT2 from the Equations were

used. The corresponding speed was received from

averaged speed before entering the NSR and after

leaving the NSR. It gave VOUT1= 11.0 knots and

VOUT2=12.3 knots. Times of passage through these

621

sections of route were determined with corresponding

distances and speeds.

The analyzed ship passed average location of

historical ice barrier (chocking zone) on 228 day of the

year on 22:55 GMT. According to statistical data

(Figures 5-19) it was few days before opening of the

NSR for ice-free navigation. Total time of voyage

received from formulas was average 13.5 days in the

range 10.8 to 16.2 days taking into consideration 1M of

standard deviation and from 8.1 to 18.9 days taking

into consideration 2M of standard deviation.

Recalculated time of voyage duration (taking into

consideration the equivalent ports departure and

arrival) of analyzed ship in 2021 navigation season was

16.9 days. Result was inside of 2M range, very close to

1M edge. Then prediction of total time of voyage was

correct. Number of expected tariff zones requiring

assistance of icebreaker assistance received basis

formulas was average 2 with possible range from 1 to

3 tariff zones. The analyzed ship in 2021 navigation

season encountered 2 ice barriers. One tariff zone at

Severnaya Zemlya archipelago and one at East part of

Est Siberian Sea. Then prediction of quantity of tariff

zones requiring assistance of icebreaker using

presented formulas was correct. Following above

verification of presented in this work formulas

received from historical data assumed are satisfactory

for long-term forecasting.

7 SUMMARY AND CONCLUSIONS

Analysis of the resulting graphs leads to several

conclusions. The maximal time window in which it is

possible to consider transit navigation of ships with

steel hulls without ice reinforcements on the NSR is

from 215 to 300 Julian days of the year. Total average

time of transit voyage and total average fuel

consumption are directly proportional to severity of ice

conditions encountered during whole voyage. The

highest severity (number of tariff zones requiring

assistance of icebreakers) is at the beginning and at the

end of the navigation season, and reduces till minimal

at the mean day of ice-free transit corridor opened on

the NSR. A higher standard deviation of fuel

consumption during navigation with icebreakers

assistance makes less precise determination of voyage

costs at the beginning and at the end of navigation

season. Total average cost of ship’s transit voyage

depends mostly on number of tariff zones where

assistance of icebreakers is required due to severity of

ice conditions. Costs of fuel consumed and time of

voyage are increasing from external days of summer

navigation season towards mean day o ice-free transit

corridor but are relatively flat in relation to costs of

assistance of icebreakers. There is recommended

avoidance of voyage when severity of ice conditions

require assistance of icebreakers in order to maintain

relatively low and stable total costs of voyage. Average

fuel consumption per day and average fuel

consumption per nautical mile are directly

proportional to severity of ice conditions. The lowest

fuel consumptions are related to ice-free transit

corridor existing along the whole NSR. The fuel

consumption of icebreaker, when its assistance was

required was not taken into consideration.

A ship should pass the chocking zone on the NSR

in between 240 and 280 Julian day of the year to ensure

the lowest navigational, ecological and economic costs

of transit voyage for average value of expected number

of tariff zones to be paid for required icebreakers

assistance. A ship should pass the chocking zone

between 250 and 270 Julian day or between 260 and 265

Julian day to include single standard deviation (68.5%)

or double standard deviation (95.0%) of expected

number of tariff zones with assistance of icebreakers

respectively. The higher probability of meeting

requirements set by decision-maker when planning a

long-term voyage of a ship, the shorter time window

for meeting these requirements. Outside the favorable

time window, the navigational, ecological and

economic results of voyage deteriorate very quickly

and it is in line with outcomes of Pruyn [22] work.

Verification of presented in this work formulas

received from historical data found consistent with real

ship voyage data on early stage of summer navigation

season in 2021 year and assumed are satisfactory for

long-term forecasting.

The presented statistical method of calculating the

results of navigation, ecological and economic voyages

of ship that is taking into consideration ice conditions

on the basis of historical input data allows forecasting

the results of planned long-term transit voyage of the

ship through the regions of ice occurrence (3 months in

advance and more) on the Northern Sea Route. The

method can be applied to other areas where ice is

present and in areas where there is no ice. It is then

required to use appropriate historical data and develop

new input data. The use of formulas for average values

of voyage results and theirs standard deviations

significantly simplifies construction of the decision

model and thus facilitates long-term decision making

in case of presence of many input data. Planning

information containing average values and standard

deviations, and thus average values increased and

decreased by single and double standard deviation

meet the requirements of the International Maritime

Organization. Above all, it facilitate the understanding

of the uncertainty ranges. Standard deviations for

possible changes of the number of tariff zones where is

required icebreaker assistance that are included in the

diagrams indicate the time window when the best

navigational, ecological and economical results of the

ship transit voyage should be obtained. Therefore, the

method described in this work can be used in decision

support systems.

ACRONYMS

CFA – daily fuel consumption where Full Ahead speed is

applied [metric tons/day],

CIND – daily fuel consumption where independent voyage

speed is applied [metric tons/day],

CIB – daily fuel consumption where icebreaker assistance

speed is applied [metric tons/day],

CT – cost of required assistance of icebreakers related to

Gross Tonnage of the ship in between 40,000 and 100,000

GRT [RUB] for tariff zones 1-6,

CTTL – total cost of the voyage [USD],

DIB – mean length of the route segments on the tariff zones

for the time period 2008-2020 [nm]; DIB = DIND,

DIND – mean length of the route segments on the tariff zones

for the time period 2008-2020 [nm]; DIND = DIB,

622

DIND1 – mean distance from the edge of first tariff zone at the

west side of the NSR to the mean location of the chocking

zone [nm],

DOUT – mean length of the route segments on the tariff zones

for the time period 2008-2020 [nm]; DIB = DIND,

DOUT1 – mean length of the route segments from the

departure port to the edge of the first tariff zone at the west

side of the NSR for the time period 2008-2020 [nm],

DOUT2 – mean length of the route segments from the edge of

the last tariff zone at the east side of the

FC – fuel consumption for the whole voyage [metric tons],

GT – Gross Register Tonnage of the ship (const.)[register

tons],

k – expected number of tariffs to be paid for required

icebreakers assistance (rounded down to positive integer)

during transit voyage of the ship through the NSR,

mk = k’ – standard deviation of expected number of tariffs to

be paid for required icebreakers assistance,

mTTTL = T’TTL – mean error of total time of the voyage [day],

mVIB – standard deviation of speed of the ship on the NSR

where assistance of icebreakers is required [kn],

mVIND – standard deviation of speed of the ship on the NSR

where areas covered by ice are expected [kn],

mVIND1 – standard deviation of speed of the ship from the

edge of the first tariff zone at the west side of the NSR till

the mean location of the chocking zone [kn],

mVFA – standard deviation of sea speed of the ship (Full

Ahead) outside of the NSR [kn],

n – maximal number of the tariff zones on the NSR (n=7),

R – rate of exchange USD to RUB (const.),

T0 – time of beginning of the ship’s voyage at the port of

departure [Julian day],

TCZ – time of arrival ship at the chocking zone on the NSR

[Julian days of the year].,

mTCZ – standard deviation of time related to defined number

of tariffs to be paid (required icebreakers assistance) [Julian

days of the year],

TEND – time of arrival of the ship’s at the port of destination

[Julian day],

TTTL – total time of the voyage [day],

T’0 – standard deviation of the time of beginning of the ship

voyage at the port of departure [Julian day],

T’END – mean error of expected time of end of the voyage

[Julian day],

mDOUT1 – standard deviation of the mean length of the route

segments from departure port to the edge of the first tariff

zone at the west side of the NSR for the time period 2008-

2020 [nm],

VFA – sea speed of the ship (Full Ahead) outside of the NSR

[kn],

VIB – speed of the ship on the NSR where assistance of

icebreakers is required [kn],

VIND – speed of the ship on the NSR where areas covered by

ice are expected [kn].

REFERENCES

1. CHNL Information Office, Transit voyages on the NSR in

2021, https://arctic-lio.com/transit-voyages-on-the-nsr-in-

2021-the-results-as-of-the-current-date/, accessed

06.10.2022.

2. European Commission, EU Taxonomy: Commission

presents Complementary Climate Delegated Act to

accelerate decarbonisation. European Commission –

Press release, Brussels, 2 February 2022, p 3

3. European Commission, Regulations, Commission

delegated regulation (EU) 2022/1214 of 9 March 2022,

amending Delegated Regulation (EU) 2021/2139 as

regards economic activities in certain energy sectors and

Delegated Regulation (EU) 2021/2178 as regards specific

public disclosures for those economic activities, Official

Journal of the European Union, L 188/1, 15.07.2022

4. Federal Tariff Service, Order of March 4, 2014 N 45-t / 1, On

approval of tariffs for icebreakers guiding of ships by

FGUP ATOMFLOT in the water area of the Northern Sea

Route (in Russian: Федеральная Служба по Тарифам,

Приказ от 4 марта 2014 г. N 45-т/1 Об утверждении

тарифов на ледокольную проводку судов, оказаемую

ФГУП "АТОМФЛОТ" в акватории Северного

Морского Пути), p 6, accessed 07.10.2020)

5. Federal Tariff Service, Order of March 4, 2014 N 45-t / 2, On

approval of rules application of tariffs for icebreakers

guiding of ships in the waters of the Northern Sea Route

(in Russian: Федеральная Служба по Тарифам, Приказ

от 4 марта 2014 г. N 46-т/2 Об утверждении правил

применения тарифов на ледокольную проводку судов

в Северного Морского Пути), p 2, accessed 07.10.2020

6. Górski, S. Ocena dokładności w nawigacji morskiej,

Wydawnictwo Morskie: Gdańsk, Poland, 1977; p 130

7. Hellenic Shipping News,

https://www.hellenicshippingnews.com/weekly-dry-

time-charter-estimates-september-02-2020, accessed

10.02.2021

8. IMO Guidelines For Voyage Planning, RESOLUTION

A.893(21) adopted on 25 November 1999, Annex 25, p 15

9. Kitagawa, H. (ed); Izumiyama, K.; Kamesaki, K.;

Yamaguchi, H.; Ono, N. The Northern Sea Route. The

shortest sea route linking East Asia and Europe, Ship and

Ocean, Tokyo, Japan, 2001; p 238

10. Matsuzawa, T.; Sogihara, N.; Tsujimoto, M.; Uto, S. NSR

transit simulations by the vessel performance simulator

“VESTA”. Part 1, Speed reduction and fuel oil

consumption in the summer transit along NSR. In:

Proceedings of the 23rd International Conference on Port

and Ocean Engineering under Arctic Conditions,

Trondheim, 2015; p 12, ISSN: 0376-6756

11. Mulherin, N.D.; Eppler, D.T.; Proshutinsky, T.O.;

Proshutinsky, A.Yu.;, Farmer, L.D.; Smith, O.P.

Development and results of a Northern Sea Route Transit

Model. CRREL Report 96–5, US Army Corps of

Engineers, 1996; pp 104

12. Nam J-H, Park I, Lee HJ, Kwon MO, Choi K, Seo Y-K

(2013) Simulation of optimal arctic routes using a

numerical sea ice model based on an ice-coupled ocean

circulation method. Int J Naval Archit Ocean Eng, 2013, 5,

p 210–226, http://dx.doi.org/10.2478/IJNAOE-2013-0128

13. NATICE, U.S. National Ice Center,

https://usicecenter.gov/Products/ArcticData. Accessed 20

Jan 2021

14. Northern Sea Route Administration,

http://nsra.ru/ru/grafik_dvijeniya_po_smp, Accessed

20.10.2020

15. Northern Sea Route Administration,

http://nsra.ru/ru/rassmotrenie_zayavleniy/razresheniya.

html?year=2016, Accessed 20.10.2020

16. Pastusiak, T. Ice conditions affecting passage of Polish

vessels convoy though the NSR in 1956. Long-term ice

forecasts and passage strategies. TransNav, the

International Journal on Marine Navigation and Safety of

Sea Transportation 2018, Vol. 12, No. 1, p 101-106,

doi:10.12716/1001.12.01.11

17. Pastusiak, T. Influence of hydro-meteorological

conditions on operational parameters of PANAMAX size

vessels [in Polish] (Wpływ warunków

hydrometeorologicznych na parametry eksploatacyjne

statków typu PANAMAX). Prace Wydziału

Nawigacyjnego Akademii Morskiej w Gdyni, 2016, 31, p

139-151, DOI: 10.12716/1002.31.14

18. Pastusiak, T. Scheduling Transit Voyages of Vessels of

Various Ice Classes Across the Northern Sea Route,

Annual of Navigation 26(1), 2019, p 114-126, doi:

10.1515/aon-2019-0012

19. Pastusiak, T. The Northern Sea Route as a Shipping Lane.

Expectations and Reality. Springer International

Publishing Switzerland, 2016; p 219, ISBN 978-3-319-

41834-6, DOI10.1007/978-3-319-41834-6

623

20. Pastusiak, T. Voyages on the Northern Sea Route,

Springer Nature Switzerland, 2020; p 279, ISBN 978-3-

030-25490-2, DOI10.1007/978-3-030-25490-2

21. Polish Register of Shipping, The International

Management Code for the Safe Operation of Ships and for

Pollution Prevention (ISM Code), as amended and the

amended Guidelines for the Implementation of the ISM

Code [in Polish] (Międzynarodowy kodeks zarządzania

bezpieczną eksploatacją statków i zapobieganiem

zanieczyszczaniu (Kodeks ISM)) z poprawkami oraz

znowelizowane wytyczne do wdrażania przez

administracje Kodeksu ISM). Polish Register of Shipping:

Gdańsk, Poland, 2001; p 40

22. Pruyn, J. F.J. Will the Northern Sea Route ever be a viable

alternative?, Maritime Policy & Management 2016, 43, 6,

pp 661-675, DOI: 10.1080/03088839.2015.1131864

23. Smith, L.C.; Stephenson, S.S. New Trans-Arctic shipping

routes navigable by midcentury. PNAS 2013, 26,

110(13):E1191–E1195, doi: 10.1073/pnas.1214212110

24. Stooq, https://stooq.pl/, accessed 15.02.2021

25. Symbolab, https://www.symbolab.com/solver/partial-

derivative-calculator, accessed 22.01.2021

26. Urbański, J.; Kopacz, Z.; Posiła, J. Ocena dokładności

pozycji okrętu, WSMW: Gdynia, 1976; p 259

27. Ship&Bunker,

https://shipandbunker.com/prices/emea/nwe/nl-rtm-

rotterdam#IFO380, accessed 10.02.2021

28. Wang Y, Zhang R., Qian Longxia, An Improved A*

Algorithm Based on Hesitant Fuzzy Set Theory for Multi-

Criteria Arctic Route Planning, MDPI, Symmetry, 2018, p

29. Zhao W., Wang Y., Zhang Z., Wang H., Multicriteria Ship

Route Planning Method Based on Improved Particle

Swarm Optimization–Genetic Algorithm. J. Mar. Sci. Eng.

2021, 9, 357, p 21. https://doi.org/10.3390/jmse9040357