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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.
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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 ܽ
ǡܽ
, and
ܽ
reflecting the wave resistance, the turbulent
flow resistance, and the laminar flow resistance:
ܨ
ൌܴ
൅ܴ
൅ܴ
ܽ
൅ܽ
ȉݒ
൅ܽ
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:
ܲ
ܽ
൅ܽ
ȉݒ
൅ܽ
ȉݒ
(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 ܽ
, ܽ
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 ܹ
ݒ
ൌܲ
ݒ
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
, 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
ൌܿܽ݌
௘௙௙
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 performanceܨ
, 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௜
. Thus, the freight incomeܫ
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௜
ȉ൫ܨ
Ǣܨ
஽ǡ௜

(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
ಹǡ೔
σ
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
ܨܥ

൅ܿ
ி
ȉݒ
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 percentageܽ
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
ܨܥ

൅ܿ
ி
ȉݒ
ȉܦ
ȉܽ
Ψ
(10)
For an operating time period ܶ
, the resulting
consumption costs ܥ
are calculated as :
ܥ
ൌ݂
ȉ
ܥ
ி
൅ܥ
ൌ
σ
ಹǡ೔
σ
Τ
ȉ
ܥ
ி
൅ܥ
(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
ಹǡ೔
σ
Τ
ȉܰ
ȉ݌
σ
ಹǡ೔
σ
Τ
ȉ
൫
σ
݌
ிǡ
ȉ
ܨܥ

൅ܿ
ி
ȉݒ
ȉܦ
ͳ൅ܽ
Ψ
 (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
ಹǡ೔
σ
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
ಹǡ೔
σ
Τ
ȉܰ
ȉ݌
σ
ಹǡ೔
σ
Τ
ȉ
൫
σ
݌
ிǡ
ȉ
ܨܥ

൅ܿ
ி
ȉݒ
ȉܦ
ͳ൅ܽ
Ψ
 (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 inݒ
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
ܰ
ܪ
ȉݐ
ܪ
൅ܦݒ
Τ
̱
ݒܦ
ΤΤ
), which is in addition
taken as an average value for further simplification.
Lastly, constant freight rates ሺ݌
ிோ
ൌσ݌
ிோǡ௜
ܰ
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
ಷೃ
ȉ௖௔௣
೐೑೑
ȉݒȂܥ
Ȃ
ȉ௣
ȉݒ

݌
ȉܽ
Ψ
൅݌
ி
ȉ
ܨܥ

൅ܿ
ி
ȉݒ
ȉݒ
(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
ಷೃ
ȉ௖௔௣
೐೑೑
Ȃ
ȉ௣
݌
ȉܽ
Ψ
൅݌
ி
ȉ

ܨܥ

൅ܿ
ி
ȉݒ
ൌ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݌
݌
ȉܽ
Ψ
൅݌
ி
ȉܦȉܨܥ

݌
ȉܽ
Ψ
൅݌
ி
ȉܦȉ
݊൅ͳ
ȉܿ
ி
ی
ۋ
ۊ
(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])
ܿܽ݌
௘௙௙
ͺǡͲͲͲȉͲǤͺ͹
͹ǡͲͲͲ
Trip length
ܦ
ʹȉʹͶǡͲͲͲ
ͶͺǡͲͲͲ
Operation time
ܶ
ͳ՜͵͸Ͳ՜ͺǡ͸ͶͲ
(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
ൌͲǤ͹̈́Ȁ
Lubricating oil
price
݌
ͶǡͻͷͲ̈́
Τ
ൌͶǤͻͷ̈́Ȁ
Harbor price
݌
42,000 $
Freight rate
݌
ிோ
ʹȉͳǡͳͲͲ̈́ 
Τ
ʹǡʹͲͲ̈́Ȁ
*
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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1314
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Success in the shipping industry depends to large extent on the quality of the main asset: the ship. The owner or operator will ask herself: do I have the right ship for a certain job, and what performance can I achieve with a given ship. The relationship between the specifications of the ship, which are fixed in the design stage, and the economic performance of the ship has received some attention, but most of the previous work lacks in the extent to which economic performance is measured or expressed. This paper describes an attempt to operationalize the concept of design for service in the maritime industry. The paper presents a lengthy review of previous work, which shows that some attention was devoted in the past to the relationship between economic and technical aspects of shipping. However, the ‘economics' usually turned out to be simple cost calculations, instead of clear insight in costs and benefits of certain design decisions. The main variable for the shipowner to buy a ship, or to operate a ship on a certain route is earnings potential. The relation between technical specifications and earnings potential is fairly direct: desired earnings potential influences the design specifications, and the specification of the finished ship determine the earnings potential. The analysis in this paper shows that shipowners also consider cargo carrying capacity, speed and versatility, but no other, more detailed, design factors. Subsequently, we present the design for service framework for the shipping industry. This framework points the attention to a thorough service requirements analysis that drives the design stage. Finally, some preliminary work is presented on empirical studies that are currently developed in the Netherlands.
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We use linear programming (LP) for solving the problem of the optimal deployment of an existing fleet of multipurpose or fully containerized ships, among a given set of routes, including information for lay-up time, if any, and type and number of extra ships to charter, based on a detailed and realistic model for the calculation of the operating costs of all the ship types in every route and on a suitable LP formulation developed in earlier work of the authors. The optimization model is also applicable to the problem of finding the best fleet compostion and deployment, in a given set of trade routes, which may be the case when a shipping company is considering new or modified services, or a renewal of the existing fleet. In addition, two promising mixed linear-integer programming formulations are suggested.
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The tradeoff between fuel savings through slow steaming on the one hand, and the loss of revenues due to the resulting voyage extension on the other hand is analyzed, and three models for the explicit determination of the optimal speed of a ship are presented. Each model is applicable under different schedule of revenues, and the optmal speed is a solution to a cubic equation over the feasible range of cruising speeds.
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Fuel consumption and emissions on a shipping route are typically a cubic function of speed. Given a shipping route consisting of a sequence of ports with a time window for the start of service, substantial savings can be achieved by optimizing the speed of each leg. This problem is cast as a non-linear continuous program, which can be solved by a non-linear programming solver. We propose an alternative solution methodology, in which the arrival times are discretized and the problem is solved as a shortest path problem on a directed acyclic graph. Extensive computational results confirm the superiority of the shortest path approach and the potential for fuel savings on shipping routes.
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The background and the literature in liner fleet scheduling is reviewed and the objectives and assumptions of our approach are explained. We develop a detailed and realistic model for the estimation of the operating costs of liner ships on various routes, and present a linear programming formulation for the liner fleet deployment problem. Independent approaches for fixing both the service frequencies in the different routes and the speeds of the ships, are presented.
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Extending and improving an earlier work of the second author, an Integer Programming (IP) model is developed to minimize the operating and lay-up costs for a fleet of liner ships operating on various routes. The IP model determines the optimal deployment of an existing fleet, given route, service, charter, and compatibility constraints. Two examples are worked with extensive actual data provided by Flota Mercante Grancolombiana (FMG). The optimal deployment is solved for their existing ship and service requirements and results and conclusions are given.