European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
INCREASING ENVIRONMENTAL SUSTAINABILITY BY
OPTIMIZING VESSEL SPEED IN LINER SHIPPING
Nurşen Aydın, Brunel Business School, Brunel University London, nursen.aydin@brunel.ac.uk
Afshin Mansouri, Brunel Business School, Brunel University London, afshin.mansouri@brunel.ac.uk
Habin Lee, Brunel Business School, Brunel University London, habin.lee@brunel.ac.uk
Abstract
Due to the increasing concern on the environmental impact of the Maritime operations, shipping
companies have focused on regulating their fuel consumption. Sailing speed is directly related to the
fuel consumption and green house gas emissions. We consider the speed optimization problem in the
liner shipping. In this problem, vessel tracks a fixed route to transport cargo and the handling time at
ports follows a stochastic process. The objective is to find an optimal speed policy between ports so
that the total voyage cost is minimized. We approximate the port times and formulate the problem as
nonlinear programming model. The proposed model is tested against an existing shipping schedule.
Computational experiments reveal that the proposed method performs well under a variety of
conditions.
Keywords: Speed optimization, Fuel emission, Liner Shipping.
1
INTRODUCTION
In recent years, economic and environmental concerns have brought a new perspective to maritime
operations. According to the International Maritime Organization (IMO), the international shipping
industry is responsible for approximately 2.2% of global green house gas emissions in 2012 (Third
IMO GHG Study, 2014). CO2 emissions from maritime transport represent a significant part of total
global greenhouse gas emissions. It is known that CO2 emission is directly proportional to fuel
consumption which significantly depends on the speed of the vessels (Third IMO GHG Study, 2014).
Therefore, many operational strategies focus on reducing vessel speeds. Although sailing with the
slowest speed is favourable with respect to the fuel cost and CO2 emissions, it may not be always
feasible due to the time constraints imposed by the ports. Mansouri et al. (2015) provide an extensive
review on the current studies in environmental sustainability in Maritime shipping.
The recent studies in maritime literature have focused on environmental impacts of the ships.
Christiansen et al. (2013) examine the ship routing and scheduling problems in liner, tramp and
industrial shipping. In liner shipping, port rotation is fixed and vessels follow the planned schedule
with weekly frequency. On the other hand, vessels do not have to follow a fixed route in tramp and
industrial shipping, and the ship routing and scheduling decisions are similar. Although the overall
objective function is different for each shipping area, the common objective is to decrease the
operational costs. Ronen (2011) points out the importance of reducing vessel speed on operating cost.
He works on the speed optimization problem by considering the service frequency and the required
number of vessels. Fagerholt et al. (2010) and Hvattum et al. (2013) work on the speed decisions in
fixed shipping routes with port time windows. They assume that vessel always arrives within the time
window of each port. Fagerholt et al. (2010) discretize the arrival times and solve the problem by
using shortest path algorithm. Hvattum et al. (2013) develop an exact solution algorithm for the
deterministic problem. Wang and Meng (2012) work on the speed optimization problem with
transhipment and container routing. They formulate the problem as mixed-integer nonlinear model and
propose outer-approximation algorithm to obtain approximate solution. Norstad et al. (2011)
incorporate speed decision in the tramp ship routing and scheduling problem and propose a local
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
search method. They first develop a solution algorithm for speed optimization problem with fixed
route. Then, they utilize this algorithm to generate initial solution for the proposed local search
method. Zhang et al. (2014) extend the work of Fagerholt et al. (2010) and Norstad et al. (2011), and
study on the optimality properties. We refer reader to Psaraftis and Kontovas (2013) for a
comprehensive review of the studies concentrated on speed decisions.
In this paper, we study speed optimization problem in liner shipping. Our study is closely related to
the work of Fagerholt et al. (2010). We focus on minimizing fuel consumption by optimizing the
speed of the vessel along a fixed route. Different than the model of Fagerholt et al. (2010), we do not
restrict the vessel to arrive within the time window. In real-life applications, vessels may arrive outside
of the time window. Therefore, in our problem formulation, we consider early arrivals and delays
while constructing the optimal schedule. We also focus on minimizing the time spent at ports. To sum
up, our objective is to find an optimal speed policy for a vessel along a fixed route by considering time
window violation at each port.
The rest of the paper is organized as follows. In Section 2, we formulate the speed optimization
problem in liner shipping route. We discuss our solution approach in the same section. In Section 3,
we present computational experiments on a real liner shipping route. Section 4 provides the
concluding remarks and future research directions.
2
MODEL FORMULATION
We consider a liner shipping company which provides shipping services over a number of ports
denoted by set N = {0,1, … , n}. Port 0 shows the stating node of the network. A vessel can visit port i
within its time window. If it arrives earlier than the available slot, it has to wait until port service is
open. If it arrives later than the time slot, it will lose a fraction of its overall service level. Service time
of a vessel in each port is a stochastic variable. We assume that service time follows uniform
distribution. We have to decide the speed between ports in order to minimize total fuel consumption
and maximize service level. The general parameters used in the problem formulation are summarized
in Table 1.
N
𝑆𝑖
[𝛼𝑖 , 𝛽𝑖 ]
𝑣𝑖
𝑑𝑖
𝑡𝑖𝑎
𝑡𝑖𝑑
𝜑
𝜃
𝑓𝑝
𝑓𝑠
Table 1.
Total number of ports
Random service time in port i such that 𝑆𝑖 ∈ 𝑈[𝑙𝑖 , 𝑢𝑖 ] and 𝑆0 = 0
earliest and latest planned arrival times at port i,
Average speed between ports (𝑖 − 1) and i, and it is limited by [𝑣𝑚𝑖𝑛 , 𝑣𝑚𝑎𝑥 ]
distance between port (𝑖 − 1)-th and i-th port-of-call
Arrival time of vessel at port i, (𝑡0𝑎 = 0)
Departure time of vessel at port i, (𝑡0𝑑 = 0)
Fuel consumption per hour during waiting and service at each port
Penalty for delay per hour
Price of fuel per ton consumed at ports
Price of fuel per ton consumed during sailing
The general parameters used for the problem formulation
In the literature, quadratic function of sailing speed is generally used to compute fuel consumption of a
vessel. The vessel can sail between the upper and lower speed limits and we assume that the speed of a
vessel is constant in each leg, i.e. between two consecutive ports. We use the empirical formula of
Fagerholt et al. (2010) to calculate fuel consumption rate per nautical mile at sailing speed 𝑣𝑖 . The fuel
consumption function is as follows:
g(𝑣𝑖 ) = 0.0036𝑣𝑖2 − 0.1015𝑣𝑖 + 0.8848
(1)
Then, the fuel consumption between ports (𝑖 − 1) and 𝑖 is given by 𝑑𝑖 g(𝑣𝑖 ). We assume that the
vessel consumes a fixed amount of fuel per hour during waiting and service time at each port. Given
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
the prices of fuel consumed during sailing and berthing, the total fuel consumption cost is computed
as,
𝑛
∑
𝑖=1
(𝑓𝑠 𝑑𝑖 g(𝑣𝑖 ) + 𝑓𝑝 𝜑(𝑡𝑖𝑑 − 𝑡𝑖𝑎 ))
(2)
Arriving later than the given time window will result in missing the available time slot. Finding a new
slot for berthing would be difficult and costly. To maximize the service level and avoid delays, we
penalize the vessel for each hour of being late. Then, the overall cost function is given by
𝑛
𝑛
∑ (𝑓𝑠 𝑑𝑖 g(𝑣𝑖 ) + 𝑓𝑝 𝜑(𝑡𝑖𝑑 − 𝑡𝑖𝑎 )) + ∑ 𝜃[𝑡𝑖𝑎 − 𝛽𝑖 ]+
𝑖=1
[𝑡𝑖𝑎
]+
(3)
𝑖=1
𝑚𝑎𝑥{𝑡𝑖𝑎
where
− 𝛽𝑖 =
− 𝛽𝑖 , 0}. Given the speed decision 𝑣𝑖 and service time 𝑆𝑖 at port i, the
states of the system at the following ports are defined by the following system dynamics equations;
𝑑
𝑡𝑖𝑎 = 𝑡𝑖−1
+
𝑑𝑖⁄
𝑣𝑖
𝑡𝑖𝑑 = 𝑚𝑎𝑥{𝑡𝑖𝑎 , 𝛼𝑖 } + 𝑆𝑖 , 𝑖 = 1, … , 𝑛
The problem is to find optimal sailing speeds between ports in a fixed route. We first work on the
deterministic formulation. This model assumes that all random quantities take on their expected
values. Therefore, it is computationally efficient and popular in practice. The optimization model has
the form:
minimize ∑𝑛𝑖=1 (𝑓𝑠 𝑑𝑖 g(𝑣𝑖 ) + 𝑓𝑝 𝜑(𝑡𝑖𝑑 − 𝑡𝑖𝑎 )) + ∑𝑛𝑖=1 𝜃[𝑡𝑖𝑎 − 𝛽𝑖 ]+
(4)
𝑑
subject to 𝑡𝑖𝑎 = 𝑡𝑖−1
+ 𝑑𝑖 ⁄𝑣𝑖 ,
𝑖 = 1, … , 𝑛
(5)
𝑡𝑖𝑑 = 𝑡𝑖 + 𝛦[𝑆𝑖 ],
𝑖 = 1, … , 𝑛
(6)
𝑡𝑖 ≥ 𝛼𝑖 ,
𝑖 = 1, … , 𝑛
(7)
𝑡𝑖𝑎 ,
𝑖 = 1, … , 𝑛
(8)
𝑖 = 1, . . . , 𝑛
(9)
𝑡𝑖 ≥
𝑣𝑚𝑖𝑛 ≤ 𝑣𝑖 ≤ 𝑣𝑚𝑎𝑥 ,
where 𝑡0𝑎 = 𝑡0𝑑 = 0. Constraints (5) and (6) correspond to the system dynamics equations.
Constraints (7) and (8) ensure that the vessel starts service after it arrives and the time window starts.
Constraints (9) guarantee that the speed of the vessel is within the lower and upper limits. This model
provides a scheduling and speed policy to minimize total fuel consumption. It does not capture the
temporal dynamics of the problem due to the approximation of stochastic variables. However, we can
improve its performance by refining the speed decision variable during the practical application.
Specifically, given the state variable 𝑡𝑖𝑎 at port 𝑖 (after vessel arrives to port 𝑖), we update the
constraints (5) and (8) with realized 𝑡𝑖𝑎 and solve the deterministic problem accordingly.
3
PRELIMINARY RESULTS
In this section, we present our preliminary results. We conduct numerical experiments to evaluate the
effects of speed decisions on the total fuel consumption. We solve the model (5)-(10) with a non-linear
programming solver in MATLAB using the interior point algorithm.
We conduct experiments by using data from a real liner shipping route of a major European shipping
company. This data includes the distances between ports, vessel arrival and berthing times, average
port service time and the average vessel speed between ports. Table 2 presents an existing schedule of
this shipping company. Lacking the actual service time distribution, we assume that service time at
port 𝑖 ∈ 𝑁 follows uniform distribution with mean 𝜆𝑖 . For a fair comparison, we set 𝜆𝑖 to the service
times given in Table 2. Since we use the fuel consumption function given in Fagerholt et al. (2010),
we assume that sailing speed ranges from 𝑣𝑚𝑖𝑛 = 14 knots to 𝑣𝑚𝑎𝑥 = 20 knots. Our cost function
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
includes three main parts; fuel consumption cost, port time cost (waiting and service) and delay
penalty. The first two cost types are directly related to the fuel cost. Vessels consume two types of
fuels during route trip. According to the data obtained from the shipping company, the fuel consumed
at ports is more expensive than the one consumed during sailing. In our experiments, we set the unit
price of fuel consumed in port twice the price of the one used during sailing. We fix the fuel cost to
$250/ton and $500/ton during sailing and berthing, respectively. The delay penalty, on the other hand,
is related to the service level and it is difficult to estimate. Another important parameter affects the
speed policy is time window. In real time applications, the schedule of the vessel is planned with
respect to the available time slots of the ports. According to the data of the shipping company, each
port specifies an available time slot (𝑡𝑖∗ ) and provides a time window around that slot. We define 𝜔 to
compute the time window in our numerical example. In other words, the earliest and latest possible
service times of port i are given by 𝛼𝑖 = 𝑡𝑖∗ and 𝛽𝑖 = 𝑡𝑖∗ + 𝜔.
Although the route schedule (arrival times and vessel speed) is determined before the vessel departs
from the first port, it can be changed during the journey. Weather conditions, disruptions and
congestion at ports are the main reasons of the deviation from the planned schedule. According to the
existing schedule in Table 2, the vessel generally arrived much earlier than the berthing time. This
behaviour can be attributed to the impact of uncertainty in ports and weather conditions. Being earlier
than the planned schedule resulted in long waiting times at ports. As it is seen in Table 2, the vessel
waited more than two days in port P.6 and it sailed at 15.2 knots on average between P.5 and P.6
instead of sailing at the minimum speed. Since sailing with high speed and arriving much earlier than
the berthing time only increases the costs, we conjecture that there is a miscommunication between the
vessel and the port during this journey.
To reveal the effects of insufficient information sharing, we test our proposed model with two
scenarios. In the first scenario, we set the starting time of the time window (𝑡𝑖∗ ) to the berthing times
given in Table 2. The arrival and berthing times of the existing schedule are given by the following
vectors.
𝒕𝒆 = {0, 7, 66, 105, 239, 385.5, 456, 574, 661, 735, 813.5, 1062.5},
𝒕∗ = {0, 8, 88, 114, 256, 392, 511.5, 574, 700, 762, 817, 1062.5}.
Port
Service
Distance
Arrival
Berthing
Departure
P.0
-
-
-
12/04/14 22:00
-
16.6
P.1
116
12/05/14 05:00
12/05/14 06:00
12/06/14 12:15
30
15.4
P.2
409
12/07/14 16:00
12/08/14 14:00
12/09/14 06:00
15
14.8
P.3
15
12/09/14 07:00
12/09/14 16:00
12/10/14 14:30
22
15.0
P.4
1546
12/14/14 22:00
12/15/14 14:00
12/16/14 07:30
18
15.1
P.5
1681
12/20/14 23:30
12/21/14 06:00
12/22/14 10:30
27
15.2
P.6
537
12/23/14 22:00
12/26/14 05:30
12/28/14 10:20
48
14.9
P.7
142
12/28/14 20:00
12/28/14 20:00
12/30/14 17:45
29
15.1
P.8
617
01/01/15 11:00
01/03/15 02:00
01/04/15 03:30
21
15.1
P.9
143
01/04/15 13:00
01/05/15 16:00
01/07/15 07:30
35
15.0
P.10
180
01/07/15 19:30
01/07/15 23:00
01/08/15 20:30
19
14.2
P.11
3200
01/18/15 04:30
01/18/15 04:30
01/18/15 13:30
8
-
Table 2.
Time (h)
Avg. Speed
An existing schedule of the shipping company
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
We compare the performance of the proposed model against the existing schedule of the shipping
company. In this experiment, we set the width of the time window to 2 hours (𝛼𝑖 = 𝑡𝑖∗ and 𝛽𝑖 = 𝛼𝑖 +
2, 𝑖 ∈ 𝑁). According to the data of the shipping company, a vessel consumes approximately 0.1ton
fuel per hour. Therefore, we set the waiting costs to $50/hour. Since delays may result in poor service,
we set the delay penalty $1000/hour to prevent lateness. Table 3 and 4 summarize our findings. First
row of Table 3 presents the berting times at all ports. The next two rows present the arrival times of
the shipping company (𝒕𝒆 ) and the optimal arrival time of the optimization model (𝑡𝑖𝑎 ). The remaining
rows show the resulting waiting times of these schedules. The first observation we have is that the
waiting time in P.6 is extremely high although it is lower than the one resulted from existing schedule.
Early arrivals like this case occur due to the given time windows. The distance between some
consecutive ports can be short. If the consecutive time windows are not tight, the vessel always arrives
early to the next port even if it sails at minimum speed. In this scenario, time windows are loose.
Therefore, the optimum speed of deterministic model is always around 14.0 knots. Although the vessel
sails with its minimum speed, it arrives ports 2, 6, 8 and 9 much earlier than the available berthing
time. When we compare it with the resulting waiting times of existing schedule, we observe that
deterministic policy performs better. Table 4 presents the fuel consumption costs during waiting and
sailing. When we look into the total costs, we obtain that the percentage gap for waiting cost is
20.74%. This shows that, deterministic policy provides a more fuel efficient schedule.
Ports
P.6
P.7
P.1
P.2
P.3
P.4
P.5
8
88
114
256
392
511.5
𝒕𝒆
7
66
105
239
385.5
t ai
8.3
70.3
105.2
247.4
1
22
9
0.3
17.7
8.8
𝒕
∗
Waiting time
(𝒕𝒆 )
Waiting time
(𝒕𝒂𝒊 )
Table 3.
P.8
P.9
P.10
P.11
574
700
762
817
1062.5
456
574
661
735
813.5
1062.5
393.6
461.5
574.6
665.5
736.4
814.5
1064.5
17
6.5
55.5
-
39
27
3.5
-
8.6
-
49.5
-
34.5
25.6
2.5
-
The optimal arrival times for a given port time
Policies
Fuel Consumption
(Waiting)
Fuel Consumption
(Sailing)
Total Fuel
Consumption
Existing Schedule
$9,025.0
$369,003.35
$378,028.35
Optimization Model
$7,475.0
$363,592.61
$371,067.61
Potential Saving
20.74%
1.49%
1.88%
Table 4.The fuel consumption cost
In the second scenario, we set the starting time of the time window (𝑡𝑖∗ ) to the arrival times given in
Table 2 and we assume that handling service also starts at the arrival time. In other words, the existing
schedule does not incur any waiting cost. In this case, the arrival and berthing times of the existing
schedule are given by the following vectors.
𝒕𝒆 = 𝒕∗ = {0, 7, 66, 105, 239, 385.5, 456, 574, 661, 735, 813.5, 1062.5}
For a fair comparison with the previous scenario, we set the width of the time window to 𝜔 = 2. Table
5 presents the schedule for this scenario. When we look into the arrival times, we observe that the
optimal speed policy of optimization model always prevents early arrivals and delays. Moreover,
comparing optimal speed vaues in Table 5 with those in Table 2, we note that the sailing speeds of the
proposed model are always lower than the ones in the existing schedule. Table 6 presents the resulting
fuel consumption costs. In this case, the potential saving is around 0.85%. Since the starting time of
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
the service is earlier in this experiment, the optimum speed values are higher than 14.0 knots
(optimum speed policy of the previous scenario). As a result, total fuel consumption increases.
Port Index
P.6
P.7
P.1
P.2
P.3
P.4
P.5
𝒕∗
7
66
105
239
385.5
456
t ai
7.8
66.4
105.4
240.5
386.8
v (knot)
14.82
14.84
14.82
14.85
14.84
Table 5.
P.8
P.9
P.10
P.11
574
661
735
813.5
1062.5
458
575.7
663
736.9
815.5
1064.5
14.83
14.65
14.58
14.34
14.36
14.22
The optimal arrival times for scenario 2
Policies
Fuel Consumption
(Waiting)
Fuel Consumption
(Sailing)
Total Fuel
Consumption
Existing Schedule
-
$369,003.35
$369,003.35
Optimization Model
-
$365,924.34
$365,924.34
Potential Saving
-
0.85%
0.85%
Table 6.The fuel consumption cost for scenario 2
4
CONCLUSION AND FUTURE RESEARCH
In this paper, we addressed speed optimization problem in liner shipping with uncertain port times. By
considering the waiting and the delay costs in ports, we formulated the problem as a constrained
nonlinear model which can be solved by nonlinear programming solvers. We tested the performance
of our model by implementing on a real-life case from a liner shipping company. We observed that the
existing schedule results in long waiting times at ports. These long waiting times can be resulted from
insufficient information sharing between the vessel and the ports. Even if the time windows are not
tight, the vessel preferred to sail at higher speeds to avoid delays due to the uncertainties along the
journey. On the other hand, the optimal policy of the proposed model decreases the waiting cost by
around 20%. This result reveals the importance of communication during route trip. Many shipping
companies use decision support systems (DSS) to manage shipment operations. The proposed speed
optimization model can be integrated to DSS to decide and update sailing speed dynamically. For
instance, in case of a deviation from planned schedule, the sailing speed can be updated by using the
proposed model.
This is an ongoing research. An extension of this work could look into tackling the uncertainty in the
system. The proposed model assumes that the stochastic port times take on their expected values. This
may be unreasonable in situations where the service times at ports are highly variable. A promising
direction to pursue for future research would be to develop a dynamic model for speed decisions.
Dynamic model considers all possible combinations of service time and hence, it can cope with
uncertainty better than the deterministic models. As a future work, we would work on the dynamic
programming formulation.
Acknowledgement
This research was supported in part by EU FP7 project MINI-CHIP (Minimising Carbon Footprint in
Maritime Shipping) under grant number PIAP-GA-2013-611693.
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European, Mediterranean & Middle Eastern Conference on Information Systems 2015 (EMCIS2015)
June 1st – 2nd 2015, Athens, Greece
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