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Slow Steaming in Container Shipping

Jasper Meyer

Student at the

University of Hamburg and

Hamburg University of Applied

Sciences, Germany

jasper_meyer@gmx.net

Robert Stahlbock

Institute of Information Systems

University of Hamburg

and Lecturer at FOM University

of Applied Sciences,

Essen/Hamburg, Germany

stahlbock@econ.uni-

hamburg.de

Stefan Voß

Institute of Information Systems

University of Hamburg

stefan.voss@uni-hamburg.de

Abstract

International supply chains heavily rely on

maritime shipping. Since the beginning of the latest

economical crisis, the containership fleet is slowing

down. This paper gives a short overview of the slow

steaming history as well as the widely assumed

coherence between a ship’s speed and its fuel

consumption. Calculating fuel consumption as a

function of speed provides decision support regarding

the decision to which extent slowing down should be

performed. It can be assumed that, compared to sailing

at full speed, a speed reduction has a positive

economic and also environmental impact. This paper is

focused on the economic aspects. We show the

considerable cost saving potential of a lower ship

speed as a result of the decreasing fuel consumption.

In combination with other variables of a container

vessels’ profit function, this may lead to the profit

optimizing speed of a container carrier.

1. Introduction

In the last decades, container shipping companies

were trying to deliver their goods as quickly and

reliably as possible. Even the ever-increasing fuel

prices could not stop this trend. The resulting costs

could be compensated by the growing revenues

resulting from the worldwide increasing demand of

transport capacity due to globalization. However,

based on the impacts of the economic crisis on the

global trade market in the last years, activities on the

transport market as well as revenues dropped severely.

Not only the demand of transport capacity was

shrinking in an unexpected way, but additionally the

supply was growing extremely fast. This vicious cycle

seems typical for the container shipping industry. In an

economic boom, shipping companies order large

capacities (a large number of ships and/or ships with a

large capacity), which are delivered later, possibly in a

recession phase. In combination with the trend of

growing ship size and the decreasing demand as a

result of a recession, this cycle leads to a large

mismatch between supply and demand of transport

capacity. As a result, freight rates decrease. One

strategy to cut down operational costs is to moor some

vessels with minimal crew for a longer time until new

cargo has to be loaded. Indeed, an increase of the

number of laid-up vessels could be observed as a result

of the global crisis.

An additional strategy for shipping companies is to

slow down vessels compared to sailing at full speed.

The basic idea of this slow steaming is not new as it is

well known, that the fuel consumption of large cargo

vessels is rising exponentially with a vessel’s velocity.

Due to this fact, ships were operated with a lower

speed in former times as well. But compared to today,

it was never applied to such a large part of the

worldwide fleet because of the exceptional

circumstances in the latest crisis. However, even

nowadays, as the crisis in the transport sector is nearly

over, slow steaming remains a common operating

mode for container ships. Due to the lack of interest in

former times, important parts of the theoretical

background of slow steaming are unknown or not

reflected in some parts of the literature.

In this paper we provide decision support regarding

the question to which extent slow steaming is

profitable and how profit optimizing vessel speeds can

be calculated. After a literature review we discuss

various effects of slow steaming in Section 3.

Calculations are shown in Section 4 and Section 5

concludes the paper.

2. Literature review

The calculation of optimal speed for freight vessels

and related performance indicators such as freight rates

were analyzed a few decades ago, e.g., in [9, 10]. In

[27], an analysis of the effect of oil price on the

optimal vessel speed is presented. The calculations for

2012 45th Hawaii International Conference on System Sciences

978-0-7695-4525-7/12 $26.00 © 2012 IEEE

DOI 10.1109/HICSS.2012.529

1306

optimal speed are different in these publications but the

main principles of the relationship among impact

factors and speed seem to be correct. However, the

research was based upon the common but old-

fashioned ‘admiralty formula’ which assumes that the

daily fuel consumption is rising by the power of three

with regard to the speed. This admiralty formula stems

from times when ships were operated by coal. In

particular today, this formula is not appropriate as a

basis for reliable calculations of fuel consumption

under real world conditions.

While the speed of a vessel may be optimized,

especially in the liner and container shipping business

various side constraints may come into play. Among

others, this concerns the interplay between different

vessels of a fleet operating to achieve some common

goals. In [24], fuel costs are modeled as a nonlinear

function of a vessel’s speed. The problem of vessels’

allocation to routes is combined with the problem of

speed selection in an optimization model. Based upon

[15, 25], an integer programming model for mini-

mizing operating and lay-up costs for a fleet of liner

ships operating on various routes is presented in [26].

Basic fuel consumption characteristics of vessels are

used as model input. However, environmental aspects

were not in the focus at that time. In [17], the optimal

vessel speed considering costs and environmental

aspects by lowered fuel consumption is briefly

analyzed and discussed.

Independent from the container shipping industry,

[3] provides a simple and yet effective spreadsheet

based approach for saving considerable amounts of

fuel for US navy ships without the need of new

equipment or ship modifications based upon analysis

of fuel curves that show the fuel consumption as a

function of power plant mode and speed, based upon

ship engineering publications. It is assumed that the

ship can operate with one or more of its propulsion

plants idled to save fuel. According to [2], this

estimation of fuel consumption is one part of the

logistics planning factor ’demand,‘ which is used for

optimizing the US navy’s supply by planning the

worldwide fleet of transport ships.

In [23], the effect of high fuel prices on the service

(e.g., the schedule, speed of vessels, number of vessels

serving a loop) of container companies providing liner

services on the Europe-Far East trade is analyzed. In

[6] a profit function is developed reflecting container-

ship and route characteristics. Two scenarios are

considered (no extra ships and extra ships for

maintaining a given cargo flow) as well as the

interrelation of costs, fuel prices, speed, fuel use, and

carbon dioxide (CO

2

) emissions. However, one basic

assumption is that the per-trip fuel consumption of the

main engine is basically given by the cubic law with

respect to the ratio of operational and design speed.

According to [7], the relationship between speed and

fuel consumption depends on an engine’s type and its

load. In particular with loads below 25% maximum

continuous rating, common rules of thumb fail. The

study reports potential emission reductions in the order

of 30% without the need of specific slow steaming

equipment. Recent calculations and detailed analysis of

economical and technical aspects in [11, 12] indicate

that the fuel savings potential by speed reduction is

considerably higher than claimed in numerous previous

publications.

According to our observation as well as [22] in

most formulations of maritime transportation

problems, time and cost of sailing are not varied

regarding speed. The latter paper builds upon [8] and

provides an extended formulation by introducing

variables for the sailing speed for each ship and sailing

leg, as well as an adjusted cost function and constraints

to incorporate speed as decision variables. For advising

solution methods such as multi-start local search based

methods the authors advise discretized arrival times. In

[4], it is shown for container shipping that slow

steaming has reduced emissions by around 11% over

the years 2008-2010 without the adoption of new

technology. Furthermore, a bunker break-even price

with the slow steaming strategy and the resulting

emission reduction being sustainable in the long run is

calculated. For the main container trades it is found

that considerable reductions can only be sustained with

a high bunker price of at least $350–$400. Therefore,

’market-based solutions‘ (e.g., tax levies and/or cap-

and-trade systems) are recommended in order to

sustain bunker prices.

Operational decisions aiming at fuel and emission

savings, such as slow steaming, in combination with

strategic decisions (e.g., fleet, alliances) are useful for

vessels that are already built and in operation.

However, there are ways of influencing a vessel’s

economic performance during the early designing and

construction (or modification) phase of a ship by

making decisions on, e.g., a ship’s shape, engine,

propulsion, fuel, etc. (see, e.g., [14, 29, 32]).

3. Effects of slow steaming

In practice as well as many publications simplified

formulas are used to describe the costs in relation to

velocity of vessels. To better understand related

approximations and to be able to better judge on

specific calculations we provide some physical

background. This might seem superfluous at first sight.

That is, one might argue that decision support is

possible without this due to available systems and

prototypes (see, e.g., the contributions in [2, 3] as well

1307

as [21]). However, based on current practices and

references especially in the container shipping industry

including maritime economics one needs to convince

that previous approaches are somewhat too simplified

to be used as entry points for building decision support

systems. Moreover, it seems necessary to consider the

option to provide entry points into necessary

extensions in problem settings. One among several

examples refers to situations, when optimal speed or

changes in speed influence the number of employed

vessels necessary to keep frequencies of sail.

3.1. Positive effects

For companies in the shipping sector, the main

reason to implement slow steaming was to reduce the

consumption of petroleum products in the combustion

of the main engine. These products are fuel, but also

lubricating oil, which is combusted in large two-stroke

engines. This paper starts by analyzing the effects on

the fuel consumption, where a large rise in price was

noticeable. Compared to the nineties of the last

century, the average price of heavy fuel oil increased

until the period of 2007/2008 by more than 800% [5].

Thus, there was a high pressure to cut down costs in

this sector.

Claims in articles or publications regarding the

potential of fuel saving by slow steaming are often not

replicable as they do not explain required details of the

ascertainments. But as shown below, the physical

principles of the fuel consumption are too complex,

results are ambiguous, and conclusions are disputable,

making simple and generalized explanations virtually

impossible.

The physical formula for the force ܨ

ோ

needed to

move a ship through a flow depends on the velocity

difference between the ship and the surrounding

medium and consists of three single forces [31]:

• the wave resistance R

W

, which is a result of the

energy needed for the wave field around the

ship’s hull,

• the turbulent flow resistance R

T

, resulting from

occurring vortexes due to collapsing flow around

the hull,

• the laminar flow resistance R

L

, which is the

frictional resistance between a ship’s hull and the

medium.

In combination with the related velocity

dependencies, the needed force can be described by the

following function (1), with parameters ܽ

ௐ

ǡܽ

்

, and

ܽ

reflecting the wave resistance, the turbulent

flow resistance, and the laminar flow resistance:

ܨ

ோ

ൌܴ

ௐ

ܴ

்

ܴ

ൌ

ሾ

ܽ

ௐ

ܽ

்

ሿ

ȉݒ

ଶ

ܽ

ȉݒ (1)

To travel a distance D with a constant ship velocity

ݒൌܦȀݐ against this force, the work ܹ

ோ

ൌܨ

ோ

ȉܦ is

required. In the time t, the power ܲ

ோ

ൌܹ

ோ

ݐ

Τ

ൌܨ

ோ

ȉݒ

must be reached. Inserting into formula (1) leads to a

ship’s power requirement depending on the velocity:

ܲ

ோ

ൌ

ሾ

ܽ

ௐ

ܽ

்

ሿ

ȉݒ

ଷ

ܽ

ȉݒ

ଶ

(2)

The power requirements of a 8,500 TEU container

vessel as a function of the velocity is depicted in Fig. 1.

This function is valid for the parts below and above the

waterline. But the coefficients ܽ

ௐ

, ܽ

்

and ܽ

do not

only depend on the flowing medium and the relative

velocity between ship and medium, but they also

depend on many other factors such as the scale of the

hull, fouling, or the varnish condition. Furthermore,

these conditions can change over time and in

dependence of the speed as well, e.g., fouling

decreases with increasing speed. So the theoretical

dependence of the required power on the ship’s speed

can hardly be represented more precisely. Even more

factors have to be considered to calculate the fuel

consumption. In particular the levels of efficiency of

the engine, driveshaft and propeller have a

considerable impact. The speed dependence of these

levels of efficiency exacerbates the calculation, too.

Therefore, it is an option to determine the coherence of

the fuel consumption empirically. According to the

approach in [11], the fuel consumption ܨܥ

per

nautical mile (nm) can approximately be represented

by:

ܨܥ

ൌܨܥ

ܿ

ி

ȉݒ

(3)

Figure 1. Power requirements (8,500 TEU

container vessel) as a function of the velocity;

Data source: [13]

With ܿ

ி

as fuel consumption parameter and based

upon the engines minimum consumption ܨܥ

to

drive at all, this approach is assuming a fuel

consumption exponentially rising with the speed. There

are different statements in scientific papers about the

speed dependence. The most common assumption

1308

about this is the admiralty formula. But because of the

shown complexity, a ship’s fuel consumption must be

appraised individually instead of applying generalized

simplified values, e.g., based upon the admiralty rule,

which may result in misleading calculations.

Considering, e.g., [3], it seems necessary to use

detailed fuel burn rate tables for different ship types

based upon empirical observations in the container

shipping industry. The following calculations are based

on the consumption of an 8,000 TEU container vessel

[16] for showing a real-world example of a ship’s fuel

consumption (see Fig. 2).

Based on a least squares approximation and

formula (3), the following function values are

obtained: FC

min

= 90, c

F

= 0.00012, and ݊ൌͶǤͶ. The

fuel consumption per mile of the regarded ship rises in

dependence of the speed to the power of 4.4.

Additionally, it must be pointed out, that the

consumption per mile is one power less than the daily

consumption. According to [11], this is a characteristic

value for large cargo ships. This shows that for

container ships the potential for fuel savings is

considerably higher than assumed and claimed in

numerous previous publications.

Figure 2. Fuel consumption as a function of

vessel speed (8,000 TEU container vessel);

Data source: [16]

For example, with data provided in Table 1, a

container vessel on a trip from Europe to Far East is

expected to save approximately 2,550 tons of fuel,

resulting in financial savings of 1,785,000 $.

Table 1: Data used for exemplary calculation

of fuel savings by slow steaming

Parameter Value Unit

Distance 24,000 nm

Fuel Price 700 $/t

Speed v

1

= 25

v

2

= 20 (slow steaming)

kn

[nm/h]

Data sources: [5, 16, 33]

As mentioned above, parts of the lubricating oil are

combusted inside the engines as well. These

consumptions are also a considerable cost factor with

price increases similar to the fuel prices. The

lubricating oil consumption depends on the speed

dependent power ܲ

ா

generated by the engine and its

performed work ܹ

ா

ሺ

ݒ

ሻ

ൌܲ

ா

ሺ

ݒ

ሻ

ȉݐ, respectively. As

in formula (3), a certain minimal consumption ܮܥ

is

assumed, resulting in lubricating oil consumption per

nautical mile as shown in formula (4) with ܿ

as

lubricating oil consumption parameter:

ܮܥ

ൌܮܥ

ܿ

ȉܹ

ா

(4)

By assuming a linear coherence between fuel

consumption and lubricating oil consumption, it is

possible to assess the dimension of the cost saving

potential of formula (4) even without a specific power

demand curve. Based on the values in Table 2, the cost

savings for the above shown example trip from Europe

to Far East are 63,000 $ for lubricating oil.

Table 2: Data used for exemplary calculation

of lubricating oil savings by slow steaming

Parameter Value Unit

Specific fuel

oil consumption

175 g/kWh

Specific

lubricating oil

consumption

0,8 g/kWh

Lubricating

oil price

4,950 $/t

Speed v

1

= 25

v

2

= 20 (slow steaming)

kn

[nm/h]

Another positive effect resulting from a decreased

fuel consumption is the reduction of some emissions.

While nitric oxides and soot emissions may rise under

certain circumstances, the amount of CO

2

and sulfur

oxide (SO

X

) is decreasing severely, which is in

particular a benefit because there is some pressure on

the ship owners to reduce these emissions. Since the

International Maritime Organization (IMO) is

exacerbating its regulations on the SO

X

emission it has

also announced regulations on CO

2

emissions for the

near future (see, e.g., [30]).

3.2. Negative effects

Obviously, a ship can move less cargo in a fixed

time, when it is operated with a lower speed. This

coherence is represented in the maximum transport

performance ܨ

௦

, with ܿܽ

as the actual usable cargo

space (effective capacity) which is less than the

nominal cargo space due to weight limitations [28],

1309

and ݂

்

as the maximum number of round trips during

the operating time period ܶ

ை

:

ܨ

௦

ൌܿܽ

ȉ݂

்

ൌܿܽ

ȉܶ

ை

ܶ

்

Τ

(5)

The required time of a tour ܶ

்

is the sum of the

times spent at sea (shipping) ܶ

ௌ

and in harbors

(waiting) ܶ

ு

as shown in formula (6), with ݐ

ுǡ

as time

spent in a specific harbor of segment i within the tour,

ܦ

as distance of that segment, and ݒ

as speed on that

trip:

ܶ

்

ൌܶ

ு

ܶ

ௌ

ൌσݐ

ுǡ

σܦ

ݒ

Τ

(6)

Here, the following differentiation is necessary: If

the freight performance of a ship is lower than the

demand of transport performanceܨ

, a lower speed

does not result in a loss of revenues. Contrary, slow

steaming could reduce the mismatch between supply

and demand by absorbing a large amount of the global

container ship fleet’s capacity. So with ܨ

ǡ

as demand

of transport performance on a specific trip i, the actual

transport performance on that trip is defined as

ܨ

ǡ

ൌ൫ܨ

ௌ

Ǣܨ

ǡ

൯. Thus, the freight incomeܫ

ௌ

for a

tour is the sum of the income per trip, which depends

on ܨ

ǡ

and the trip specific freight rates

ிோǡ

:

ܫ

ௌ

ൌσܫ

ிோǡ

ൌσ

ிோǡ

ȉܨ

ǡ

ൌσ

ிோǡ

ȉ൫ܨ

ௌ

Ǣܨ

ǡ

൯

(7)

Hence, in case of demand exceeding the maximum

transport performance, a lower speed results in a

proportional loss of income for the shipping company.

Another negative factor of the extended traveling time

affects shippers and their customers since the longer a

trip takes the longer the cargo is bound to the sea. This

means additional capital costs for shippers and for their

customers (see, e.g., [12] for a simple calculation, or

[1] for considering an internal rate of return for

calculating opportunity costs). From this point of view,

faster operated ships are more attractive to both of

them. This has to be regarded as a competitive

disadvantage of slow steaming. However, this aspect is

not in the focus of the following calculation. A brief

discussion of the effectiveness and costs of slow

steaming for reducing emissions is, e.g., presented in

[6].

4. Calculation of profit optimizing speed

For calculating the profit maximizing vessel speed,

a profit function is required (the calculation is based

upon [11, 12]). Profit is the difference of revenue and

costs. The revenue is the above mentioned freight

income. The total operating cost of a vessel ܥ

comprises the following three costs:

• consumption costs ܥ

, as the sum of discussed

fuel consumption costs ܥ

ி

and lubricating oil

consumption costs ܥ

,

• harbor costs (e.g., fees) ܥ

ு

,

• usage costs ܥ

, e.g., labor costs, capital

consumption, maintenance, insurance.

Usage costs can be considered as more or less fixed

with respect to the vessel’s speed. If the vessel is

chartered, the value should be adjusted by taking the

contract’s details into account (e.g., by deducting costs

for lubricating oil). For the sake of simplicity, we

assume fixed ܥ

in the subsequent calculation. Harbor

costs do not depend on a vessel’s speed. Therefore, one

can simplify the calculation by considering average

harbor costs. With ܰ

ு

being the number of harbors on

the round trip and

ு

as the average harbor price, ܥ

ு

can be calculated as follows:

ܥ

ு

ൌ݂

்

ȉܰ

ு

ȉ

ு

ൌ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉܰ

ு

ȉ

ு

(8)

Consumption costs for shipping are the largest and

most important part of the total operating costs, with

fuel costs being the largest part of the consumption

costs. Total fuel costs are the sum of costs for each

segment of a tour, resulting from the fuel consumption

per segment and fuel costs for that segment. Thus, fuel

costs ܥ

ி

can be calculated as follows:

ܥ

ி

ൌσ

ிǡ

ȉܨܥ

ȉܦ

ൌσ

ிǡ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

ȉܦ

(9)

Costs for lubricating oil can be derived from the

above mentioned power requirements. Since this part

of shipping costs is by far the less significant part

compared to the fuel costs, we simplify the calculation

by incorporating them with a specific percentage of the

fuel costs. Herewith, we assume a proportional inter-

dependence of power and fuel consumption (i.e., a

constant specific fuel oil consumption independent

from engine load). This simplification from real world

seems appropriate for our purpose, in particular taking

modern electronic motor management into account.

With a given percentageܽ

Ψ

, the costs for lubricating

oil ܥ

can be calculated as in (10), with

ǡ

as the trip

specific price for lubricating oil:

ܥ

ൌσ

ǡ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

ȉܦ

ȉܽ

Ψ

(10)

For an operating time period ܶ

ை

, the resulting

consumption costs ܥ

are calculated as :

ܥ

ൌ݂

்

ȉ

ሺ

ܥ

ி

ܥ

ሻ

ൌ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉ

ሺ

ܥ

ி

ܥ

ሻ

(11)

The sum of the three cost components results in the

total operating costs ܥ

of a vessel:

1310

ܥ

ൌܥ

ܥ

ு

ܥ

ൌܥ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉܰ

ு

ȉ

ு

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉ

൫

σ

ிǡ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

ȉܦ

൯

ሺ

ͳܽ

Ψ

ሻ

(12)

This formula allows for deriving the cost

optimizing speed. This knowledge about the

relationship of speed and costs is an important

instrument in fleet planning allowing for even higher

profit than in case of operating with profit maximizing

speed. However, subsequently the paper is focused on

the profit optimizing speed. Hence, the profit is

calculated as difference between revenue and costs.

The revenue or income function is given by

formula (7). With the maximum transport performance

exceeding demand, the profit optimizing speed equals

the optimal speed with regard to costs. Therefore, it is

now assumed that the vessel’s capacity is completely

utilized. In this case, the function for the income

generated by a utilized vessel is given by formula (13):

ܫ

ൌσ

ிோǡ

ȉܨ

ௌ

ൌ

σ

ிோǡ

ȉܿܽ

ȉ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

(13)

The profit function ܲ

as difference between

income and costs is:

ܲ

ൌܫ

െܥ

ൌܫ

െܥ

െܥ

ு

െܥ

ൌ

ிோǡ

ȉܿܽ

ȉ

ܶ

ை

σ

ݐ

ுǡ

σ

ܦ

ݒ

Τ

െܥ

െ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉܰ

ு

ȉ

ு

െ

்

ೀ

σ

௧

ಹǡ

ା

σ

௩

Τ

ȉ

൫

σ

ிǡ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

ȉܦ

൯

ሺ

ͳܽ

Ψ

ሻ

(14)

This function allows for calculating the profit

optimizing speed for each segment of a tour. This

approach is simplified by making some assumptions

close to reality in order to calculate values without

requiring computer based approximation. First of all,

consumption functions can be simplified by assuming

that ݒ

, the speed for a segment i, can be expressed as

deviation from an average speed ݒҧ resulting inݒ

ൌ

ݒҧേοݒ

. Since the fuel consumption increases

disproportionately high to the increase in speed, the

positive deviations are always higher than the negative

ones. Thus, the fuel consumption is always higher with

various speeds in various segments compared to

shipping with constant speed throughout the entire trip

having the same total travel time. Furthermore, the

required multiple acceleration for shipping with

different speeds on a segment results in additional fuel

consumption. This leads to the basic rule that a

minimum of fuel consumption can be achieved by

shipping with a constant speed on each segment.

Secondly, it is assumed that the shipping time clearly

exceeds the wait time at harbors (ܦݒ

Τ

بܰ

ு

ȉݐ

ு

՜ͳ

ሺ

ܰ

ܪ

ȉݐ

ܪ

ܦݒ

Τ

ሻ

̱

ݒܦ

ΤΤ

), which is in addition

taken as an average value for further simplification.

Lastly, constant freight rates ሺ

ிோ

ൌσ

ிோǡ

ܰ

ு

ሻ

Τ

and

constant prices for fuel and lubricating oil are assumed.

The simplifications and resulting changes of the profit

function are listed in Table 3.

Table 3: Simplifications for profit calculation

Simplification

Calculation

without

simplification

Calculation

with simpli-

fication

Constant speed

σܦ

ݒ

Τ

ܦݒ

Τ

Shipping time wait

time at harbors

(average)

ܦݒ

Τ

ܰ

ு

ȉݐ

ு

ܦݒ

Τ

Constant freight rate

ிோǡ

ிோ

Constant prices for

fuel and lubricating oil

ிǡ

ǡ

ி

By considering these assumptions, formula (14) for

calculating the profit can be simplified to:

ܲ

ൌ ܶ

ை

ȉ

ቀ

ಷೃ

ȉ

ȉݒȂܥ

Ȃ

ே

ಹ

ȉ

ಹ

ȉݒെ

ሺ

ȉܽ

Ψ

ி

ሻ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

ȉݒ

ቁ

(15)

For calculating the profit optimizing speed ݒ

,

the derivative of function (15) with respect to ݒ is set

to zero resulting in:

ௗ

ೇ

ௗ௩

ൌܶ

ை

ȉ

൬

ಷೃ

ȉ

Ȃ

ே

ಹ

ȉ

ಹ

െ

ሺ

ȉܽ

Ψ

ி

ሻ

ȉ

ሺ

ܨܥ

ܿ

ி

ȉݒ

ሻ

൰

ൌͲ (16)

Solving (16) for ݒ results in the profit optimizing

speed ݒ

as follows:

With given data, this formula allows for calculating

opt

for a trip of any vessel. This is exemplified by a

calculation for a round trip from Europe to Far East.

Table 4 shows data required for the calculation of

(17), resulting in ݒ

ൌʹͲǤͲͻ. Taking this profit-

optimal speed ݒ

ൌʹͲǤͲͻ and formula (15) for

profit calculation into account, the maximal profit for

this example can be calculated with ܲ

ǡ

ൌ

25.1 million $, while shipping with design speed

instead of profit-optimal speed results in a profit of

17.4 million $ only. The optimized speed results in a

profit increase of 7.7 million $ or 44% compared to the

design (maximum) speed.

ݒ

ൌ

ۉ

ۈ

ۇ

ிோ

ȉܿܽ

െܰ

ு

ȉ

ு

െ

ሺ

ȉܽ

Ψ

ி

ሻ

ȉܦȉܨܥ

ሺ

ȉܽ

Ψ

ி

ሻ

ȉܦȉ

ሺ

݊ͳ

ሻ

ȉܿ

ி

ی

ۋ

ۊ

ଵ

(17)

1311

Table 4: Data for calculating profit optimizing

speed for an exemplary trip Europe – Far East

Influencing factor Symbol Value

Effective capacity

(with ρ = 0.87;

[28])

ܿܽ

ͺǡͲͲͲȉͲǤͺ

ൎǡͲͲͲ

Trip length

ܦ

ʹȉʹͶǡͲͲͲ

ൌͶͺǡͲͲͲ

Operation time

ܶ

ை

ͳ՜͵Ͳ՜ͺǡͶͲ

(5 days for maintenance)

Number of

harbors

ܰ

ு

2

Speed exponent n 4.4

Consumption

parameter fuel 1

ܿ

ி

0.00012

Consumption

parameter fuel 2

ܨܥ

90

Lubricating oil

consumption [%]

ܽ

Ψ

0.005

Fuel price

ி

ͲͲ̈́ ݐ

Τ

ൌͲǤ̈́Ȁ

Lubricating oil

price

ͶǡͻͷͲ̈́

Τ

ൌͶǤͻͷ̈́Ȁ

Harbor price

ு

42,000 $

Freight rate

ிோ

ʹȉͳǡͳͲͲ̈́

Τ

ൌ

ʹǡʹͲͲ̈́Ȁ

*

Usage costs

(without

lubricating oil)

ܥ

͵ͲǡͲͲͲ̈́

Τ

ൌͳǡʹͷͲ̈́Ȁ

*

Note: The freight rate is assumed to be equal for both

directions for the sake of simplicity [36]; see, e.g., [12] for a

calculation with different rates

Revenue, costs, and profit as functions of the speed

are depicted in Fig. 3, demonstrating that the cost-

optimal speed is only affected by the relation of

shipping costs and freight rates. However, this

quantitative, cost-oriented horizon should be

broadened by taking also qualitative factors such as

image improvement (environmental friendly shipping)

or customer satisfaction into account. These factors

should be observed during real world operation in

order to be able to react as quickly as possible.

Furthermore, it can be seen that the profit

optimizing speed is usually higher than the cost

optimizing speed. The profit optimizing speed

decreases inversely proportionally with the 3

rd

root of

fuel price slightly faster than the cost optimizing speed.

Contrary to the cost optimizing speed, the profit

optimizing speed is independent from usage costs. In

addition, with increasing number of harbor stops or

harbor time, the profit optimizing speed is only

moderately decreasing. As far as the freight rates

increase proportionally to the travel distance, the profit

optimizing speed does not change significantly.

Figure 3: Income, Costs, and Profit as

functions of vessel speed

5. Conclusion

The main purpose of this paper was to provide an

overview over the main financial effects of slow

steaming in order to evaluate economic aspects of this

operating mode of vessels which is receiving

considerable interest in particular as a result of the last

economic crisis.

Looking at these issues can be done from various

sides. In different disciplines and for different purposes

the objective may be different, i.e., a shipping liner

may look at ‘slow steaming’ from a different

perspective than an operator in case where ships may

be looked at as single entities.

The most important positive impact for a shipping

company is the savings of fuel and, therefore, fuel

costs. However, analysis of literature and

communication with experts revealed that some

literature is based upon false assumptions regarding

physical aspects and volume of cost savings. Taking

main drivers of fuel consumption into account, it can

be concluded that the often applied cubic function,

based on the old admiralty formula, is not appropriate

for reflecting the increase of fuel consumption as a

result of increased speed. The gained insight was used

for calculating fuel consumption in an exemplary case

in order to demonstrate the potential of slow steaming.

In addition, the often ignored costs for lubricating oil

were incorporated. Based on a more detailed analysis,

an enormous potential of cost savings for shipping

companies became apparent and better documented.

Environmental aspects were mentioned but this

paper is not focused on them. Without any doubt,

environmental aspects demand significant attention in

future research, in particular considering IMO

1312

regulations and pressure to comply with governmental

rules striving for environmental friendly shipping.

The increased tie-up of shipping capacity as a result

of slow steaming was briefly discussed as well. In

times of significant overcapacity, this tie-up and the

resulting increase of freight rates is a positive effect on

the market. Contrary, in times of demand exceeding

supply, the additional removal of transport capacity by

slow steaming is disadvantageous for shipping

companies since they lose income. For customers,

longer trip duration is disadvantageous due to their

tied-up capital being shipped. This has to be considered

as a comparative disadvantage for shipping companies

in a highly competitive market.

For giving an advice from an economic point of

view, the composition of profit was analyzed. Slow

steaming affects costs as well as revenue. The deducted

profit function delivered the formula for the profit

optimizing speed. An exemplary calculation illustrated

the findings. The presented considerations can be

helpful for calculating an optimum speed. However,

real world operation is even more complex. As in

aviation, exogenous variables such as weather

conditions have significant impact on fuel

consumption. The current version of our paper, like

other sources in the maritime economics literature,

provides no consideration of the potentially significant

effects of such exogenous variables. Taking a ship’s

characteristics and (forecasts of) weather and sea

conditions into account is the focus of ‘ship weather

routing‘ approaches aiming at the calculation of a track

for ocean voyages resulting in, e.g., maximum safety

and crew comfort, minimum fuel consumption,

minimum time underway, or any desired combination

of these factors (see, e.g., basic work in [18, 19, 20]). If

one has to take into account weather effects this could

dramatically change the modeling emphasis from a

static planning perspective to a dynamic, online

optimization application. When approaching a decision

support system (DSS) for the container shipping

industry this is an issue of future research, especially

when combining this with fleet deployment issues. For

example, a DSS should reflect the main influencing

technical and economical factors, such as vessel

characteristics, freight rates, emissions, weather

conditions, trim, etc., and goals, such as cost or

emission minimization or profit maximization. The

DSS can result in better decisions on operating a ship

during a specific voyage (speed, route), in particular if

a sensitivity analysis is provided for a better estimate

of decision impacts in an environment with uncertain

events.

The most important question regarding slow

steaming aims at its sustainability. The demonstrated

calculations show that the optimal vessel speed mainly

depends on freight rates and fuel prices. Hence, a

decreased speed is reasonable in particular in times

with high fuel prices and low freight rates. Assuming,

that fuel prices will not significantly drop in the near

future, it can be concluded that from an economical

perspective slow steaming is a good if not the best

operating mode for container vessels. However, there

are technical issues. For example, the lifespan of an

engine is expected to decrease due to suboptimal

usage. Therefore, engine manufacturers offer, e.g.,

‘slow steaming kits‘ in order to overcome such

problems (see, e.g., [34, 35]). These preparations

require additional investments that should be

incorporated into calculations and cost-benefit

analyses, e.g., in a lifecycle costing approach.

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