To address the high environmental risk related to the increased oil tanker traffic in the High North, the Norwegian Coastal Administration (NCA) manages one of its vessel traffic service (VTS) centers in the town of Vard ø, Norway. The fleet of tugboats, controlled by the VTS center operators, patrols the coastline to hook-up with any potential drifting oil tanker in the region of interest, before it runs ashore. Presently, the tugboats are controlled manually, which is not only challenging but less effective. In this paper, we develop two alternative binary integer programming models that give better tugboat policies in less computational time compared to previous work. Promising results with historical data illustrate great potential for optimal environmental risk reduction along the northern coast of Norway
Maritime transportation plays an essential role in the international trade as it provides a cost-effective means to transport large cargo volumes. It is, however, characterized by a high level of uncertainty, which creates various risks in terms of fatalities, environmental pollution, and loss of property. In particular, oil spills from oil tankers grounding accidents have a devastating effect on the marine ecosystem [
In pursuit of sustainable sea transportation in the High North, the Norwegian coastal administration (NCA) administers one of its vessel traffic service (VTS) centers in the town of Vardø, Norway. About 200 vessels are monitored daily by the VTS center of which five to six oil tankers receive special attention due to their size or risk of pollution. Through the automatic identification system (AIS), the VTS center obtains static information (cargo, identity, dimensions) and dynamic information (heading, position) from oil tankers moving in the region. Additionally, dynamic models of wind, ocean currents, wave heights and weather forecasts are used to predict potential drift trajectories and grounding locations of vessels. Moreover, the oil tankers are required by law to move along a predefined corridor approximately 50 km away from the coastline. Any oil tanker that losess its maneuverability through steering or propulsion failure is immediately assigned to the closest patrol tugboat for rescue operation before it runs ashore. The size of the zone of interest is about 1100 km of coastline, and the number of tankers entering the region makes it difficult to effectively move the tugboats at the right place in time.
Previous works [
The remainder of this paper is organized as follows. Section 2 formulates the tugboat positioning problem and presents the two linear integer models that minimize the environmental risk from oil tanker grounding accidents. We discuss the integration of a receding horizon control (RHC) scheme into the mathematical models in Section 3. In addition, we present the numerical results with realistic test instances as well as case studies with historical events in Section 4. Finally, conclusions and further research are provided in Section 5.
Following the formulation approach from [
For every oil tanker (vessels) v, in the set V , entering the region of interest, we consider independent potential drift trajectories at each time period in the defined time horizon. Let Ω ¯ represent the set of possible scenarios in the planning horizon. Obviously, ω ¯ ∈ Ω ¯ is a combination of drift trajectories (vessel scenarios), or normal routes in absence of an incident, followed by each vessel. That is, ω ¯ = ( ω 1 , ⋯ , ω v ) , where ω v ∈ Ω v denotes the vessel scenario for vessel v in a given time period and Ω v is the set of all possible scenarios for vessel v. In case of drift, a vessel will follow a path denoted by p = ( c 1 , c 2 , ⋯ , c T ) , c t ∈ C , which is a succession of cells followed by the drifting vessel. Although the model inputs are updated every time period as discussed in Section 3, uncertainty on drift trajectories is addressed by predicting more than one single potential path. That is, ω v = ( t , p i ) ∀ p i ∈ P ω v t , where P ω v t represents the set of all predicted paths for vessel scenario ω v at time of distress call t and we denote by N the cardinality of P ω v t . Thus, ω v = ( t , p i ) represents the potential scenario for vessel v, where t ∈ T ∪ { T + 1 } is the time the VTS center notices or is alerted to the distress of vessel v and p i are the predicted paths followed by the drifting vessel. In the absence of incident, t is set to T + 1 .
Let G be the set of tugboats run by the VTS center in the town of Vardø. At the beginning of the planning, each tugboat g ∈ G is positioned at an initial cell c 0 g ∈ C . The tugboats can only transit between neighborhood cells at each time period, which is determined by their maximal speeds in the planning horizon. Accordingly, let F ( c ) ⊂ C be the set of cells that are adjacent to c ∈ C . Thus, F ( c ) represents cells that are reachable from cell c within one time period.
The main objective is to determine the position of tugboats at each time period such that the expected environmental consequence of oil tanker grounding accidents is minimized. Thus, let K ω v denote the environmental consequence associated with oil tanker v if vessel scenario ω v occurs and no tugboat manage to rescue it before it runs ashore. In the next subsection, we present the risk
model R i s k ω v for any vessel scenario ω v that helps to derive the risk for all
scenarios ω ¯ , which represents the main function to be minimized in the two binary integer programming (BIP) models presented in Subsection 2.2.
A risk is a combination of the probability of an event and its consequence. In drift grounding accidents, the risk model for each potential vessel scenario ω v
is the product of the probability of failure R ω v , the probability of grounding given that it is adrift, 1 − Q ω v and the environmental consequence, K ω v :
R i s k ω v = R ω v ( 1 − Q ω v ) K ω v (1)
An oil tanker might start drifting at any time period with a certain probability,
R ω v , that depends on the internal factors from the oil tanker itself as well as wind and current forces, and wave heights.
In Equation (1), Q ω v represents the probability of successful hook-up of the
drifting vessel with the nearest tugboat. The VTS center detects every drifting oil tanker and informs the nearest tugboat. Practically, the tugboat response time is determined by three main factors: 1) preparation time (reaction time and mobilization time), 2) sailing time and 3) connection or towing time. [
the vessel is adrift, denoted by Q ω v , mainly depends on t l . Accordingly, let Q g c ω v denote the probability of successful hook-up by tugboat g with drifting
vessel v, given tugboat g is in cell c at time of distress call t and vessel v follows scenario ω v = ( t , p i ) . This probability depends on the position of the nearest tugboat at time of distress call, currents, wind, waves, distance of the vessel to shore and property of the drifting vessel such as type, draft, size and loading
condition. All these dependencies are captured in λ g c ω v , which is the predicted time left once tugboat g, in cell c at time of distress call t, reaches the drifting vessel in scenario ω v = ( t , p i ) . As in [
operational speed of the nearest tugboat and its location relative to the drifting vessel’s trajectory, and set
Q g c ω v = β ω v exp ( δ ω v ( λ g c ω v − t min ) ) 1 + exp ( δ ω v ( λ g c ω v − t min ) ) . (2)
The parameter t min represents the minimal remaining drift time required to attempt a hook-up. If λ g c ω v is less than t min , Q g c ω v is set to 0. In addition, β ω v ∈ [ 0,1 ] and δ ω v ≥ 0 represent the influence of weather conditions [
The environmental consequence of a drift grounding accident depends on the expected oil spill size (S) and the impact (I) of one tonne of oil on the environment
[
include both bunker and cargo spill. The spill size depends on the vessel type, size, loading condition and on whether the ship is single or double hulled. It is
found by combining the probability of an oil spill τ ω v , given that the vessel run aground with the expected oil outflow in the event of oil spill, O ω v , in scenario ω v : S ω v = τ ω v O ω v . Moreover, O ω v = α ω v γ ω v Dwt , where α ω v is the expected outflow rate given as a percentage of the tank content volume and γ ω v is the
volume of cargo and bunker oil as a percentage of vessel dead-weight tonnage Dwt.
The oil spill impact per tonne depends on the type of oil spilled and the vulnerability of the affected area. This is modeled as environmental sensitivity index,
E ω v and oil type significance index, L ω v ( I ω v = E ω v L ω v ). The value E ω v depends on oil type and incorporates the vulnerability and ecological significance of the geographical area. In addition, L ω v describes the significance of the oil
type spilled. In case of drift for a given vessel scenario ω v = ( t , p i ) , the impact of an oil spill will depend on the distance to shore, the weathering processes, the chemical composition of the oil, and the drift trajectory, which depends on the local wind and current condition.
We present two different binary integer models that minimize the expected environmental consequences from oil tanker grounding accidents. The first model, BIP-1, allocates the potential drifting vessels to the nearest tugboat, while the second model, BIP-2, focuses on the number of vessels that could not be rescued within a predefined threshold.
We denote by z g c ω v a binary variable taking the value 1 if tugboat g is in cell c
and is the nearest tugboat at time of distress call t of vessel scenario ω v = ( t , p i ) , and 0 otherwise. In addition, we assume that the probability of vessel scenarios ω v is mutually independent. This assumption may not always be reasonable, however, we justify it by the fact that vessels in distress are usually spatially separated with few common environmental factors [
a scenario ω ¯ is given by R ω ¯ = ∏ ω v ∈ ω ¯ R ω v . In addition, we define x g c t as a
binary variable taking the value 1 if tugboat g is in cell c at time t, and 0 otherwise. The environmental risk function to be minimized is then written as followed:
f ( z ¯ ) = ∑ ω ¯ = ( ω 1 , ⋯ , ω v ) ∈ Ω ¯ R ω ¯ ∑ ω v ∈ ω ¯ ∑ g ∈ G ∑ c ∈ C ( 1 − Q g c ω v ) K ω v z g c ω v , (3)
where z ¯ denotes the vector with components z g c ω v . The binary integer programming model below is developed to optimally minimize the objective function f ( z ¯ ) subject to some constraints.
Formulation
m i n f ( z ¯ ) (4)
s.t.
∑ c ∈ F t g ( c ′ ) x g c t − 1 ≥ x g c ′ t ∀ g ∈ G , ∀ c ′ ∈ C , ∀ t ∈ T \ { 0 } (5)
∑ c ∈ C x g c t = 1 ∀ g ∈ G , ∀ t ∈ T (6)
x g , c 0 g , 0 = 1 ∀ g ∈ G (7)
z g c ω v ≤ x g c t ∀ g ∈ G , ∀ c ∈ C , ω v = ( p , t ) ∈ Ω v , v ∈ V (8)
∑ g ∈ G , c ∈ C z g c ω v = 1 ∀ ω v ∈ ω ¯ , v ∈ V (9)
x g c t , z g c ω v ∈ { 0 , 1 } ∀ g ∈ G , ∀ c ∈ C , ∀ t ∈ T , ∀ ω v ∈ Ω v , v ∈ V (10)
Constraints (5) ensure tugboats move only between neighborhood cells. In addition, constraints (6) make sure tugboats are located in only one cell at each time period. The initial positions of tugboats are given in constraints (7), such that cell c 0 g is the position of tugboat g at the beginning of the time horizon. Constraints (8) and (9) allocate nearest tugboats to vessel scenarios and ensure that each vessel scenario ω v is allocated to only one tugboat.
The main objective of this model is to minimize the expected environmental consequence associated with the potential drifting vessel scenarios that could not be rescued within a predefined threshold ρ . From the previous approach,
Q g c ω v = β ω v e x p ( δ ω v ( λ g c ω v − t min ) ) 1 + e x p ( δ ω v ( λ g c ω v − t min ) ) , (11)
where λ g c ω v represents the estimated drift time left once the vessel is reached by the tugboat. Thus, we define H g c ω v as a binary parameter taking the value 1 if tugboat g is at cell c at time of distress call t and is not able to hook-up with vessel v under scenario ω v = ( t , p i ) , within a predefined threshold time ρ and 0 otherwise. Additionally, let y ω v be a variable that takes the value 1 if no tugboat
is able to hook-up with vessel v, doing scenario ω v , within a predefined threshold ρ and 0 otherwise. The expected environmental consequence to be minimized in then written as follow.
g ( x ¯ ) = ∑ ω ¯ = ( ω 1 , ⋯ , ω V ) ∈ Ω ¯ R ω ¯ ∑ ω v ∈ Ω v , v ∈ V K ω v y ω v (12)
where x ¯ denotes a vector with components x g c t .
In addition to g ( x ¯ ) , we define u ( x ¯ ) as a function that gives incentive to tugboats to optimally position themselves once the threshold is reached:
u ( x ¯ ) = ∑ ω ¯ = ( ω 1 , ⋯ , ω V ) ∈ Ω ¯ ∑ ω v ∈ Ω v , v ∈ V ∑ c ∈ C ∑ g ∈ G K ω v x g c t λ g c ω v . (13)
The BIP-2 model that minimizes the objective functions g ( x ¯ ) and u ( x ¯ ) is presented below.
Formulation
m i n g ( x ¯ ) + u ( x ¯ ) (14)
s.t.
constraints (5)-(7) in BIP-1 and
∑ c ∈ C ∑ g ∈ G H g c ω v x g c t ≤ y ω v + c a r d ( G ) − 1 ∀ ω v ∈ Ω ¯ (15)
x g c t , y ω v ∈ { 0,1 } ∀ g ∈ G , ∀ c ∈ C , ∀ t ∈ T , ∀ ω v ∈ Ω v , v ∈ V (16)
Constraints (15) capture the vessels that could not be reached within the predefined threshold and the other constraints are the same as in BIP-1.
In this section, we integrate the RHC algorithm with the BIP models to account for uncertainty in weather conditions and dynamic changes of the input parameters. A RHC is a class of algorithms that make use of explicit process models to predict future response of a system, with optimizations as intermediate steps. The main idea is to dynamically run the BIP model in real time, while implementing only the first time period over the whole planing horizon (see [
We present the numerical settings in this section and discuss the quality and performance of the BIP models, compared with previous work, run with realistic test cases. In addition, the promising results with a historical event highlight the important features derived from the BIP models as a decision support tool for the NCA managers.
The region of interest covers about 1100 km of coastline and the corridor is on average 50 km away from the coast. We discretize the region by collecting the
| Step 1: |
|---|
| 1) Let t : = 1 ; x g , c 0 g , 0 : = initialvalue ∀ g ∈ G . |
| 2) Obtain the predicted drift trajectories and velocity of vessels and tugboats. |
| 3) Run the BIP model to obtain the optimal positions of patrol tugs. |
| 4) Implement only the first period of the BIP solution. |
| Step 2: |
| 1) Let t : = t + 1 ; x g , c 0 g ,0 : = x g c t − 1 ∀ g ∈ C . |
| 2) Update the predicted drift trajectories and velocity of vessels and tugboats. Additionally, update the current number of oil tankers moving along the zone of interest as well as the available number of tugboats. Update the probability of successful hook-up matrix. |
| 3) Run the BIP model to obtain the optimal tugboats policy. |
| 4) Implement the first period of the new BIP solution. |
| Step 3: Go back to Step 2 or stop if t = T + 1 . |
center position of each cell and transform them into Cartesian coordinates for input to the model. Once the optimal solution of the BIP model is obtained, the drift trajectories, oil tanker and tugboat positions are transformed back to geographical coordinates. The drift trajectories are obtained using the AIS and Norwegian Meteorological Institute (NMI) information with the algorithm presented in Section 3 (see [
Previous research, on the same region of interest, conducted by [
All computations are carried out on a personal computer with an Intel® Pentium® IV 3.0 CPU and 4.0 GB of RAM. The optimization software Gurobi 6.0.5 is used as a solver, with Python 2.7.3 and Pyomo 4.2, on Microsoft Windows 7.
This subsection discusses the numerical results for three different cases. For each case, the models are run for a total of 24 hours with 100 different instances and the environmental risk associated with each scenario is computed according to the tugboat positions from each model policy. A total number of 6 oil tankers, which correspond to the current average daily number, are used with random geographical positions, directions and speeds.
In this case, we use a large cell size of 5 by 5 km as in [
As presented in
| MIP-U | BIP-1 | BIP-2 | ||||
|---|---|---|---|---|---|---|
| Risk | Time (min) | Risk | Time (min) | Risk | Time (min) | |
| Avrg | 11.758 | 1.91 | 11.838 | 0.565 | 11.818 | 0.077 |
| Std.dev | 7.992 | 1.91 | 8.055 | 0.565 | 8.119 | 0.077 |
| Min | 1.835 | 5.589 | 1.835 | 4.535 | 1.835 | 1.378 |
| Max | 41.244 | 11.786 | 41.244 | 6.633 | 41.648 | 1.744 |
large cells size. Essentially, the models are run dynamically, where only the first step of one hour is implemented, as described in
In this subsection, we consider smaller cells size, of 2 by 2 km, compared to that of Case 1. We also use a single path as in Case 1 to predict the drift trajectory of each potential drifting vessel. The computational results for MIP-U and BIP-2 models are presented in
Noticeably, the computational time for both MIP-U and BIP-2 models have considerably increased compared to the values in Case 1. Indeed, smaller cells size increases the overall number of cells, which consequently expand the problem size. The average run-time for the BIP-2 model is equal to 5.5 minutes with a standard deviation of only 3.7 minutes. These values are significantly smaller than those of MIP-U model. The average performance of almost 40 minutes, with a maximum of 91.5 minutes, in the MIP-U model makes it impossible to be run dynamically and account for uncertainty with the algorithm in
This case study uses the same cells size of 2 by 2 km as in Case 2. The main difference, however, is the total number of scenarios. In this case, we use three drift trajectories to predict the path followed by each vessel scenario ω v . In addition, we consider up to two possible vessel scenarios ω v for each scenario ω ¯ . This gives more than 50,000 total number of scenarios.
For this case, none of the MIP-U and BIP-1 models are able to provide solutions in less than three hours. Thus, the numerical results presented in
| MIP-U | BIP-2 | |||
|---|---|---|---|---|
| Risk | Time (min) | Risk | Time (min) | |
| Avrg | 2.209 | 39.246 | 2.253 | 5.495 |
| Std.dev | 4.784 | 42.937 | 5.070 | 3.717 |
| Min | 4.034 | 5.589 | 3.577 | 1.378 |
| Max | 43.983 | 91.458 | 43.983 | 11.069 |
| Initial Risk | Optimal Risk | Time (min) | |
|---|---|---|---|
| Avrg | 31.172 | 3.769 | 12.116 |
| Std.dev | 10518 | 5.762 | 2.808 |
| Min | 8.637 | 0.349 | 5.589 |
| Max | 57.159 | 46.940 | 14.498 |
are those of the BIP-2 model only. The initial risk column in
In order to assess the effect of the threshold parameter on the solution values, we run the BIP-2 model with a real world instance. The test case consists of 9 vessels that moved along the coast over a time period of 15 hours. In addition, we discretize the region of interest into small cells with large scenarios number as in Case 3. The expected potential risk for each value of the threshold ρ ∈ { 2,5,8,11 } is presented in
As presented in
| Threshold | 2 | 5 | 8 | 11 |
|---|---|---|---|---|
| Avrg | 4.396 | 4.440 | 4.726 | 4.919 |
| Std.dev | 4.622 | 4.448 | 4.294 | 4.122 |
| Min | 1.036 | 1.245 | 1.928 | 3.105 |
| Max | 24.848 | 22.355 | 20.065 | 18.711 |
18.71 while the minimum risk value increase from 1.04 to 3.11 for a threshold value of 2 and 11, respectively. The threshold parameter gives more options with regards to tugboats policy and level of risk. That is, the managers at the VTS center will have to make a trade-off between having smaller standard deviation of the risk and avoiding the high risk of worst case scenarios at the expense of higher expected potential risk.
The histograms in
This case is based on real-world data collected from the AIS and the NMI. In addition, we use the basemap library in python to plot and draw the maps with drift trajectories, oil tanker and tugboat positions.
On the 21st of Mars 2014 at 11:10 pm, a vessel ran aground at N71˚01.06'N - 028˚27.46'E after 15 hours of drifting time. At the time of distress call, the nearest tugboat was located at N70˚40'N - 023˚40'E and unsuccessfully tried to reach the drifting vessel. The tugboat was located about 142.8 km away from the vessel at the time of grounding. We ran the BIP-2 model for 15 hours prior to the time of distress call for more than 50,000 scenarios. A total number of 7 vessels, including the one that ran ashore, sailed along the region during the considered planning horizon. Their corresponding directions, latitudes, longitudes and speeds over ground (SOG) at the beginning of the planning horizon are presented in
The results for the first and last time period are presented in
| Vessel 1 | Vessel 2 | Vessel 3 | Vessel 4 | Vessel 5 | Vessel 6 | Vessel 7 |
|---|---|---|---|---|---|---|
| North-west | North-west | West-north | West-north | North-west | West-north | North-west |
| N71˚1 | N71˚51 | N69˚28 | N71˚27 | N71˚25 | N71˚42 | N68˚55 |
| N028˚02 | N022˚56 | N013˚51 | N028˚29 | N026˚27 | N019˚48 | N012˚10 |
| 12.4 | 13.4 | 12.5 | 12.6 | 13.0 | 13.1 | 9.1 |
paths followed by the tugboats from the time of distress call to the time of grounding while the green solid lines linked with small squares represent the suggested movement of the tugboats by the BIP-2 model. The risk values associated with each vessel scenario is not presented in the figure because of the small visibility.
A zoomed-in view of the grounded location in
In this paper, we address the environmental risk related to oil tankers traffic along the northern Norwegian coast. We propose two alternative models that could be used as a decision support tool at the vessel traffic service center in the town of Vardø, for a better rescue operation of vessels in distress. These models are combined with a receding horizon control algorithm to account for uncertainty in weather conditions and to dynamically update the constantly changing input parameters. For a large cells size of 5 by 5 km and smaller scenarios number,
the BIP-1 and BIP-2 models outperform the MIP-U model from previous work. In addition, the BIP-A model is by far faster than the other models for large scenarios number and small cells size, which considerably adds complexity to the models. Moreover, the BIP-2 model gives flexibility to the operators at the VTS center by allowing different threshold levels. The results with a historical event indicate better decisions on tugboat patrol operations.
It is recommended that further research is done to determine the optimal fleet of tugboat required as well as extension of the BIP models to consider other search and rescue operations. Additionally, more research is needed to better assess the failure probabilities of vessels, oil spill rates, probability of oil spill given that an accident has occurred, and environmental consequence of the region of interest. Furthermore, the hook-up probability formula could be better estimated with empirical data including new features such as “ship arrestors” newly acquired by the NCA to reduce the speed of the drifting vessels.
This work was carried out as part of the project Dynamic Resource Allocation with Maritime Application (DRAMA), grant no. ES504913, with partial funding from the Norwegian research Council and NTNU—University of Science and Technology, Ålesund, Norway. We sincerely acknowledge Trond Ski and colleagues at the Norwegian Coastal Administration for their support and input during the work with this paper.
The authors declare no conflicts of interest regarding the publication of this paper.
Assimizele, B. and Bye, R.T. (2020) Minimizing the Environmental Risk from Oil Tanker Grounding Accidents in the High North. American Journal of Operations Research, 10, 83-100. https://doi.org/10.4236/ajor.2020.103005