a new method for robust route optimization in ensemble ...608598/fulltext01.pdf · a new method for...

A new method for robust route optimization in ensemble

weather forecasts

Lukas Skoglund Jakob Kuttenkeuler Anders Rosén

September 28, 2012

Abstract

This paper presents a new dynamic programming method for multi-objective route optimization ofships. The method, which is an extension of the known Dijkstra's algorithm, uses the concept of Paretoe�ciency to handle multi-objective optimization and can be used with both deterministic and ensembleweather forecasts. The advantage of the presented method in combination with deterministic weatherforecasts is demonstrated in comparison to Dijkstra's algorithm. The comparison between the methods(non surprisingly) shows that both �nd the same minimum time route, but only the method suggestedin this paper was able to �nd the true minimum fuel route, with about 15% saving. Evaluation ofthe presented method in combination with ensemble weather forecasts show that there is an advantagewhen the objective of the optimization is to minimize fuel consumption.

En ny metod för ruttoptimering med ensemble-väderprognoser

Lukas Skoglund Jakob Kuttenkeuler Anders Rosén

28 september 2012

Sammanfattning

Denna artikel presenterar en ny metod för optimering av fartygsrutter där �era olika målfunktio-ner betraktas. Metoden bygger på en dynamisk programerings algoritm som påminner om Dijkstrasgrafsöknings algoritm, men som använder sig av Pareto-optimalitet för att hantera optimeringen av�era målfunktioner. Metoden kan användas med både deterministiska och ensemble-väderprognoser.Fördelarna med att använda metoden tillsamans med deterministiska väderprognoser demonstreras ien jämförelse med en metod som baseras på Dijkstras algoritm. Jämförelsen visar, som väntat, attbåda metoderna hittar samma rutt för minimal restid, men bara den metod som presenteras här hittarden rutt som minimerar bränsleförbrukningen, med en besparing på ungefär 15%. Utvärderingen avmetoden tillsammans med ensemble-väderprognoser visar att det �nns en stor fördel jämfört med attanvända deterministiska väderprognoser om målet för optimeringen är minimerad bränsleförbrukning.

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A new method for robust route optimization in ensemble forecasts

1 Introduction

1.1 Route optimization

Route optimization (or weather routing) in the con-text of this paper is the practice of �nding the bestpossible route to sail a ship between two locationsgiven a weather forecast. The purpose of route op-timization is typically to ensure safe passage, but italso serves to decrease operational costs and helpvessels to stay on schedule. Thus, which route isthe best is often subjective and depends on howone weighs the di�erent objectives, which can in-clude, but is not limited to: the safety of crew,ship and cargo, the time of passage, the operationalcosts and the comfort of the crew and passengers.Many of these objectives are often competing, i.e.the improvement of one objective usually comesat the expense of another. Conventional gradient-based optimization methods do typically not workfor weather routing due to the non-convex natureof the problem and lack of continuous derivatives.Thus, at best a gradient based approach wouldlikely converge to a local minimum with far worseperformance than the global minimum.

1.2 Bene�ts of route optimization

Numerical route optimization is proven to be suc-cessful. In ocean yacht racing, one would not con-sider not using it to minimize sailing time and e.g.in [1] it was shown that using route optimizationthe encountered wave height was signi�cantly re-duced.

In [2] routes purposed by a route optimizationmethod were compare to great circle routes forthe same voyage in simulations over a large set ofweather data and it was shown that the routes pur-posed by the route optimization method performedbetter with regards to operational costs.

In a paper by Henrik Rinder of Seaware AB [3]a statistical analysis of the bene�ts of route opti-mization was presented and it was concluded thatthe reduction in fuel consumption, averaged over16 di�erent journeys spread out over the year, was5.2%. The analysis also showed that the biggest re-ductions were to be found during periods of roughweather where the reductions in fuel consumptioncould be as high as 18%.

1.3 Previous Research

Most existing route optimization methods use stan-dard deterministic forecasts. The development ofrouting methods that use ensemble forecasts (de-scribed in section 2.1) has though been ongoing forover a decade and several approaches have beenproposed. This section does not attempt to be acomplete bibliography of the research in to ensem-ble weather routing, but introduces some importantreferences.

In [2] Saetra provided important informationabout the relationship between ensemble spreadand routing performance, con�rming that the ap-plication of ensemble forecasts to weather rout-ing has merit. Ho�schildt [4] evaluated severalapproaches to route optimization using ensembleweather forecasts, however none of those performedbetter than methods based on the deterministicforecast.

Allsopp, Phillpot and Mason [5, 6] presenteda method based on a dynamic programming ap-proach to solve a minimum time routing problemunder consideration of uncertainties. The methodis similar to the Bellman method [4] but expandsthe state space to include the weather scenario.The weather scenarios are part of a branching treeof scenarios with speci�ed probabilities associatedwith each branch. The method is implemented fora yacht racing problem where the only objective istime, but the method could be expanded to morecomplicated problems. Treby [7] used a slightly dif-ferent dynamic program to solve the same problem.The big di�erence is in the way the recursion in thedynamic program works.

Harries et al. [8] introduced a novel approachto route optimization that uses a genetic algorithmto generate Pareto optimal solutions to a multi-objective routing problem. The concept of Paretooptimality is explained in depth in part 2.1, butessentially a set of Pareto optimal solutions are so-lutions that are all optimal for di�erent objectivefunctions. In addition to varying the route, the ve-locity pro�le along the route was also varied, mak-ing the problem more complex but also more real-istic. Hinnenthal [9], together with Saetra [10] andClauss [11], later expanded this method to makeuse of ensemble forecasts.

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1.4 The contribution of this paper

This paper presents a method for numericalweather routing in ensemble forecasts which isbased on a dynamic programming approach. Themethod is similar to Dijkstra's algorithm [12] and�nds a set of Pareto optimal solutions to the rout-ing problem, like the method presented by Harrieset al. [8]. The algorithm used by the method isnot novel in itself, but its tailoring and applica-tion to the weather routing application is. The ba-sics of the algorithm is described in [13] and [14].The algorithm can solve multi-objective, time de-pendent routing problems, something Dijksta's al-gorithm can not (see section 3.3), and is thereforof interest for route optimization. The Algorithmis also well suited for use with ensemble forecastsand can incorporate the concept of robustness in-troduced by Hinnenthal and Saetra in [10], but,so far, only in a limited way since no modeling ofsafety is used for the testing presented in this pa-per. Application of the method is demonstratedfor a certain routing case and the potential of themethod is discussed.

2 Basics of route optimization

This section introduces some of the basic con-cepts of weather forecasting and route optimizationaswell as the concept of robustness, which is key tothe way the method presented here handles routeoptimization with ensemble weather forecasts.

2.1 Weather Forecasting

The standard weather forecast, here referred toas a deterministic forecast, is typically computedby �rst analyzing the current state of the weatherby collecting observational data from weather sta-tions and satellites and then calculating the stateof the atmosphere, the analysis. The analysis thenserves as the initial condition from which a nu-merical model of the atmosphere is integrated intime. The progression in time of the determinis-tic forecast is thus naturally strongly dependent onthe initial condition (the analysis). Since in-datato the analysis contains inaccuracies, imperfectionsand maybe errors it is interesting to study the im-pact of such perturbations in initial condition on

the progression of the forecast. Thus, as an alterna-tive, or a compliment, to the deterministic forecastone can produce what is called an ensemble fore-cast which is a large set of deterministic forecaststhat are generated from slightly perturbed initialconditions, resulting in several forecasts that thusevolve di�erently over time.All of the forecasts in the ensemble are typically

considered equally likely to be accurate. The indi-vidual forecasts of the ensemble forecast are calledensemble members. The spread of the ensemblemembers is called the ensemble spread and can beused to asses the accuracy of the deterministic fore-cast. A large spread at a given lead time indicateshigh uncertainty and vice versa. The ensemble fore-cast is a relatively new approach to forecasting, �rstbeing put to use in the 1990ies, and therefore thepotential applications and bene�ts to weather rout-ing has not yet been fully explored.

2.2 Performance model

Regardless of which method one uses for route op-timization a model for the ship performancel isneeded. Typically , a performance model calcu-lates the velocity and fuel consumption rate frominformation about the ship, the engine setting, anda weather forecast. Alternatively one can choose avelocity and the performance model will calculatethe engine setting and the fuel consumption rate.

2.3 Resolution

Another aspect of route optimization that appearsregardless of the method used is resolution. Typ-ically, the continuous problem of �nding the bestroute and velocity is simpli�ed in to a discrete prob-lem and as with most discretized problems there isa trade of between high resolution, high accuracy,and computation time that has to be considered.For route optimization one is also limited by theresolution of the weather forecast. Forecast reso-lution usually guides the decision as to what res-olution should be used for the routing, althoughinterpolation in time and space is common.

2.4 Robustness

The concept of robustness, as introduced by Hin-nenthal and Saetra in [10], is used in this study

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to evaluate the feasibility of a route. If a routeis evaluated for performance and safety in severalensemble members and is found to be feasible, i.e.possible to sail without violating any constraints,in some members and infeasible in others we cal-culate the robustness of the route as the numberof members in which the route is feasible dividedby the total number of members. Hence a robust-ness close to one indicates a very safe route and arobustness close to zero indicates an unsafe route.

3 Purposed method

For brevity the method presented in this paper willbe referred to as POWER (Pareto Optimal WEtherRouting) and a reference method based on Dijk-stra's algorithm will be referred to as DWR (Dijk-stra's Weather Routing).

3.1 Pareto optimality

The concept of Pareto optimality originates fromthe �eld of economics and game theory. It is namedafter Vilfredo Pareto and will here be explainedshortly.

Let f̄(x̄) = (f1(x̄), f2(x̄), . . . fn(x̄)) be the func-tion to optimize, for example by minimization ofall of the functions min

x̄fi(x̄),∀i. However it is not

clear what constitutes an optimal solution since,presumably, not all functions fi(x̄) will have aglobal minimum for the same value of x̄. One com-monly used approach is to create a weighted sumof the di�erent functions fi. This gives us one op-timal solution, but the problem of specifying theweights on which the solution strongly depends re-mains. An alternative approach is to generate a setof optimal solutions, the Pareto optimal solutions.

To de�ne Pareto optimality the concept calleddominance is introduced. Let x̄′ and x̄′′ be two dif-ferent candidate solutions to the optimization prob-lem. Then:

If fi(x̄′) ≤ fi(x̄

′′) ∀iand fi(x̄

′) < fi(x̄′′) for at least one i

then x̄′ dominates x̄′′

Now, a solution is called Pareto optimal if no otherfeasible solution dominates it. Of course the sign

of the inequality in the de�nition of dominance de-pends on whether maximization or minimization ofthe objective fi(x̄) is done. The set of all Pareto op-timal solutions is called the Pareto frontier. All so-lutions on the Pareto frontier are optimal for someset of weights in the weighted sum approach. Theplot in �gure 2 titled 'Labels of A', shows an exam-ple of a two-dimensional Pareto frontier where theobjectives are time of travel and fuel consumption.

3.2 The Basic Algorithm

Here the algorithm used by POWER will be ex-plained shortly and some notes related to its appli-cation to weather routing will be discussed. Sincethe algorithm is a graph search algorithm some ex-planations of concepts related to graph are neces-sary. The following de�nitions are fairly generalbut are here explaind in relation to routing. Someof the de�nitions below are illustrated in �gure 1.

• De�nition: A vertex is, in the context of rout-ing, a location (longitude and latitude). Theyare usualy spread out around the great circleroute to cover an area of ocean that is consid-ered to be of interest, not to far from the greatcircle route.

• De�nition: An edge is a path between two ver-tices that does not pass through any other ver-tices. An edge is directed if it is only possibleto travel along it in one direction.

• De�nition: A graph is a collection of verticesand edges.

• De�nition: Two vertices are said to be neigh-bours if there is an edge connecting them. Fordirected edges one vertex may have another asits neighbour when the reverse is not true. Forexample: if there is a directed edge from vertexA to vertex B and no edge from B to A, then Bis a neighbour of A, but A is not a neighbourof B. This is the case in �gure 1.

• De�nition: The edgecost is the cost of travel-ing along an edge, e.g. time of travel or fuelconsumption.

• De�nition: If a graph with directed edges isacyclic then: If there is a way of traveling fromvertex A to vertex B there can not exist a wayof traveling from vertex B to vertex A.

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Figure 1: The �gure shows a schematic graph withdirected edges. Travel is only alowed in the direc-tion of the arrows. The vertices are illustrated bycicles and are label 'A' through 'E'. The edges arereferred to by which vertices they connect, e.g. ABand AE. There is a cycle present in the graph madeup of the edges BE and EB. For the algorithm pre-sented in this paper that is not allowed and eitherthe edge BE or the edge EB would have to be re-moved.

The basic algorithm �nds the Pareto optimal set ofpaths between two vertices in a graph granted thatthe following conditions are ful�lled.

• Condition: The two vertices are connected, i.e.there is a way of traveling between the two.

• Condition: All edgecosts are positive. For thepurpose of weather routing, objectives such astime of travel and fuel consumption are alwayspositive.

• Condition: If any of the edgecosts are time de-pendent, minimization of the total time mustbe one objective. For the purpose of weatherrouting, objectives such as time of travel andfuel consumption are always time dependent.

• Condition: The edges of the graph are directedand the graph is acyclic. If there is a way oftraveling from vertex A to vertex B there cannot exist a way of traveling from vertex B tovertex A.

The algorithm resembles the classic Dijkstra'sshortest path algorithm, and other label setting al-gorithms by assigning and correcting labels for eachof the vertices. The di�erence is in that the pre-sented algorithm saves all the Pareto optimal labelsfor each vertex instead of just one label. This col-lection of labels is called a Pareto optimal set of

labels. A label is the set of values of the di�erentobjectives upon reaching the vertex, e.g. time ofarrival and fuel consumption. The label also con-tains information on which vertex preceded the cur-rent vertex so that one can reconstruct the entireroute from the labels of the goal vertex when thealgorithm has �nished. Assuming an appropriategraph covering the ocean between our point of de-parture and our destination, the algorithm can bedescribed as follows:

1. Initiation:

1.1. Set the Pareto optimal set of labels toempty for each vertex.

1.2. Designate the vertex corresponding to thepoint of departure to 'Current'.

1.3. A label corresponding to the starting con-ditions is added to the Pareto optimal setof labels of 'Current'.

2. Evaluate edges from 'Current' to neighbours.

2.1. For each neighbour of 'Current' and eachlabel in the Pareto optimal set of labelsof 'Current' create a 'candidate label' byevaluating the journey between the 'Cur-rent' vertex and the neighbour startingfrom the time speci�ed in the label witha ship performance model. Add this can-didate label to the Pareto optimal set oflabels of the neighbour if the candidate isnot dominated in that set. If the candi-date is added to the set remove all labelsin the set which are dominated by it, thusmaintaining a Pareto optimal set.

3. Select next vertex for evaluation.

3.1. Select any of the vertices in the graphwhich has had all edges leading to it eval-uated already.

3.2. Set this vertex to 'Current'

3.3. If 'Current' is the vertex corresponding tothe destination (the goal vertex) go to 4,else go to 2.

4. Generate the Pareto optimal solutions to therouting problem from the Pareto optimal la-bels of 'Current'.

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Steps 2 and 3 above form the recursive part of thealgorithm and are illustrated in 2 where the eval-uation of two edges leading to the same vertex areevaluated and a Pareto optimal set of labels for thatvertex is established.

To expand the search space and relate the prob-lem more closely to reality waiting at vertices forsome discrete preset times could be included, butthis would rarely be of any bene�t in practice.However a similar and much more useful expansionis to include variation in velocity. This is a sim-ple extension of the algorithm and involves addinganother loop to evaluate the travel from each labelusing the di�erent speeds. To expand the methodto use ensemble forecasts one only has to evaluateeach travel between vertices in each ensemble fore-cast member and then average the values of the ob-jectives from the evaluation in each forecast mem-ber. Both of these extensions are included in thefull description of the algorithm in Algorithm 1.

3.3 Theoretical considerations

Dijkstra's algorithm, and by extension our refer-ence method DWR, can fail to �nd the optimalpath in graphs with time dependent edge costs andfor weather routing edge costs are always time de-pendent. However it does not fail for minimumtime routing as the travel times over the edges ofthe graph satisfy the non passing property, alsoknown as �rst-in �rst-out, i.e. if two identical shipssail the same route with the same velocity pro�le(or engine load pro�le), the one that leaves �rst willalways arrive at the destination �rst. This can beseen in �gure 4 where DWR has found the min-imum time route but has failed to �nd the min-imum fuel route, found by POWER. This is be-cause fuel consumption does not satisfy the nonpassing property, i.e. for two identical ships sail-ing the same route the one leaving �rst does notnecessarily use less fuel. This is important sincethe complexity and computational requirements ofDWR are signi�cantly lower than for POWER, sofor minimum time routing a method based on Di-jkstra's algorithm, like DWR, is better. But forcommercial �eet operations minimizing the timeof passage without considering fuel consumption isusually pointless and a method like POWER willbe preferable.

4 Initial evaluation

POWER was implemented in matlab in combina-tion with a simplistic but adequate ship perfor-mance model to evaluate the time of travel andfuel consumption of the routes. One limiting factorin the calculations was the undesirable property ofthe ship performance model only allowing for set-ting of engine load values and not ship speed as onewould like. This prevented direct explicit controlof the ship speed and caused some problems whenPOWER is used together with ensemble weatherforecasts as the arrival times at a vertex has to beaveraged over the evaluations from all the ensembleforecast members which introduces an uncertaintyabout where the ship is in the state space (posi-tion and time). This uncertainty increases withthe length of the voyage. Therefore a comparisonof the performance of POWER using deterministicforecasts and POWER using ensemble forecasts ismade di�cult, however a comparison is still pre-sented here.Due to the di�culty of direct control of ship

speed a shorter journey through rough weather waschosen as the test case since it o�ers the best com-parison between deterministic and ensemble rout-ing. The test case is of a bulk carrier sailing in thenorth Atlantic during a period of severe weatherwith signi�cant wave heights above 10 meters atthe center of the a storm system passing throughthe area. The point of departure is located of thecoast of Norway and the point of arrival is locatednorth-west of Ireland. In �gure 3 the great circleroute between the point of departure and the pointof arrival is depicted.Evaluation 1: The performance of POWER, us-

ing a deterministic weather forecasts, was com-pared to DWR. Both methods were used to op-timize the voyage described in the test case. DWRwas tasked with �nding the minimum time routeand the minimum fuel route to be compared withthe solutions of POWER, a two dimensional Paretofrontier (time and fuel consumption).Evaluation 2: To compare the results from de-

terministic routing and ensemble routing usingPOWER the solutions from the deterministic rout-ing were reevaluated in the ensemble weather fore-casts and were then plotted alongside the solutionsfrom the ensemble routing. This was done due tounavailability of analyzed weather to compare the

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(a) Initial state (b) Evaluation of edge AC

(c) Evaluation of edge BC (d) Final state

Figure 2: An example illustration the algorithm described in 3.2. a) The edges leading to vertices Aand B have already been evaluated and a Pareto optimal set of labels (labels marked by (o)) exist forboth. b) The algorithm selects A as the next vertex to be evaluated and proceeds to evaluate the travelbetween A and C (indicated by the red edge) starting from each of the conditions in the labels of A. Thisgenerates a set of candidate labels (candidate labels marked by (x)) of C. All candidates are kept sincenone is dominated by any other (can be seen marked with an (o) in c) and the initial set of labels of Cwas empty. c) The algorithm proceeds to evaluate the edge between B and C and a new set of candidatelabels of C are found. This time one of the candidates are dominated by one of the existing labels ofC (dominated labels marked as red) and is therefor not added to the Pareto optimal set of labels of C.Also one of the existing labels of C is dominated by one of the candidate labels and is therefor removedfrom the Pareto optimal set of labels. d) Now remains only the �nal Pareto optimal set of labels of C,since all edges leading to C have been evaluated and hence no changes will be made. All edges leadingaway from C (not shown here) may now be evaluated, i.e. C may be set as 'Current' by the algorithm.

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Figure 3: 936 nautical miles long great circle routefor the test case.

solutions in. Since some solutions are not feasiblein some of the ensemble members, i.e. it is not pos-sible to sail the route under the weather forecastedby the ensemble member either because the riskof slamming was to high or because an arrival be-fore a preset end time was not possible, the resultswere compared at di�erent levels of robustness (see2.4). Since the ensemble routing already evaluatesthe solutions in all ensemble members the robust-ness of a route is used as an objective to create athree dimensional Pareto frontier (time, fuel con-sumption and robustness). The deterministic solu-tions reevaluated in the ensemble weather forecastdo not necessarily constitute a three dimensionalPareto frontier, but it is a close approximation andis suitable for comparison with the results from theensemble routing.

5 Results

The results from evaluation 1 are shown in �gure 4.It is seen that both POWER and DWR has foundthe same minimum time solution but only POWERhas found the true minimum fuel solution. Theminimum fuel route suggested by POWER con-sumes more than 15% less fuel than the minimumfuel route suggested by DWR.

The results for evaluation 2 can be seen in �g-ures 5, 6, 7 and 8. As stated before a comparison

Figure 4: Comparison of deterministic POWER(stars) and DWR (circles).

between the deterministic and ensemble routing re-sults are made di�cult due to the behavior of theship performance model. The Pareto frontier ofthe ensemble method is as good or better than thedeterministic method. The largest di�erence is inthe high robustness region, where the solutions arerequired to be feasible in all or almost all of the en-semble forecast members (�gures 5 and 6), wherethe ensemble routing method �nds a wider rangeof solutions with better fuel consumption. For thelow feasibility region (�gure 8) it is seen that the en-semble routing has converged in a set of solutionsthat are very competitive, they are however onlyfeasible in a few of the ensemble forecast members.

6 Discussion

6.1 Summary and conclusions

A new method for route optimization is presented.The method is shown applicable with both de-terministic and ensemble weather forecasts. Themethod is based on a dynamic programming al-gorithm and computes Pareto optimal solutions toa multi-objective routing problem. The purposedmethod has been tested in comparison to a simpleroute optimization method base on Dijkstra's algo-rithm. Both methods used a deterministic weather

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Figure 5: Comparison of deterministic (Red) andensemble (Blue) POWER routing solutions wheresolutions are feasible in all ensemble forecast mem-bers.

forecast and attempted to optimize the route be-tween two locations in the north Atlantic duringa period of severe weather. The reference methodbase on Dijkstra's algorithm was tasked with �nd-ing the minimum time route and the minimum fuelroute and the method presented in this paper calcu-lated solutions along the two dimensional, time andfuel consumption, Pareto frontier. The compari-son between the methods (non surprisingly) showsthat both were able to �nd the same minimum timeroute, but only the method suggested in this paperwas able to �nd the true minimum fuel route, us-ing about 15% less fuel. The purposed method wasalso tested against itself, one version using a deter-ministic forecast and the other using an ensembleforecast. This comparison showed that the versionusing the ensemble forecast was able to �nd solu-tions that require less fuel compared to solutionsfrom the version using the deterministic forecast atthe same level of robustness. Both of the test aboveshows that the purposed method is promising andthat further study of the method is warranted.

6.2 Future work

The next step in evaluating the performance of themethod presented here is to test it in a large setof weather forecasts and then reevaluate the sug-

Figure 6: Comparison of deterministic (Red) andensemble (Blue) POWER routing solutions wheresolutions are feasible in atleast 40 ensemble forecastmembers.

gested solutions in analyzed weather. Further, toproperly evaluate the potential advantages of us-ing ensemble forecasts for route optimization usingthis method, the evaluation should be designed sothat the optimization is updated whenever there isa new forecast available. This would properly sim-ulate the way route optimization is used in prac-tice and would allow for a quantitative analysis ofthe bene�ts of route optimization using the pur-posed method, using both deterministic and ensem-ble forecasts. Before performing any full scale eval-uation of the method some improvements shouldbe made.

• Updating the ship performance model or in-corporation an e�cient autopilot to allow forexplicit control of the vessels velocity.

• Incorporating additional safety modeling in tothe ship performance model and including newobjectives such as the comfort of the crew andpassengers.

• Reducing the computation time required bythe method by optimizing the code and, pos-sibly, including parallelization strategies.

• Investigating the e�ect of the spatial and tem-poral resolution of the method on the accuracy

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Figure 7: Comparison of deterministic (Red) andensemble (Blue) POWER routing solutions wheresolutions are feasible in atleast 20 ensemble forecastmembers.

of the solutions. This will be important sinceit greatly e�ects the computation time of themethod and an unnecessarily high resolutionwill make the method to slow for practical ap-plication.

• The structure of the graph used by the methodalso greatly e�ects the computation time andan investigation of possible improvements tothe generation of the graph is of great interest.

Figure 8: Comparison of deterministic (Red) andensemble (Blue) POWER routing solutions wheresolutions are feasible in atleast 1 ensemble forecastmember.

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Algorithm 1 Dynamic programming algorithm

• De�nitions

� getLabels(vertex) returns the current Pareto optimal set of labels of the vertex.

� getNeighbours(vertex) returns the neighbours of the vertex.

� getT ime(label) returns the value of the time objective from the label.

� edgeCost(vertex, vertex, time) returns the values of the di�erent objective functions evaluatedfor traveling between the two neighbouring vertices starting at the speci�ed time.

� paretoAdd(label, setOfLabels) updates the Pareto optimal set of labels setOfLabels with thenew label in the appropriate way to maintain a Pareto optimal set.

� Next() returns the next vertex to be evaluated. For a vertex to be a candidate for the nextvertices to be evaluated all of the vertices which have edges leading to the vertex must havebeen evaluated previously.

� V elocities is a set of allowed velocities

� Forecasts is the ensemble of forecasts.

� edgeCost(forecast) is now an array of edge cost values for each ensemble member.

� average(edgeCosts) computes the average values of the di�erent objectives in edgeCost.

• Initialization

� All sets of labels are empty except that of the starting vertex which has one label correspondingto the starting conditions.

� Current is set to the starting vertex.

• Algorithm

while Current 6= goal dolabelsOfCurrent← getLabels(Current)neighboursOfCurrent← getNeighbours(Current)for each neighbour in neighboursOfCurrent do

labelsOfNeighbour ← getLabels(neighbour)for each label in labelsOfCurrent do

for each velocity in V elocities do

for each forecast in Forecasts do

startT ime← getT ime(label)edgeCost(forecast)← edgeCost(Current, neigbour, startT ime, velocity, forecast)

end for

newLabel← label + average(edgeCost)paretoAdd(newLabel, labelsOfCurrent)

end for

end for

end for

Current← Next()end while

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7 References

References

[1] Mikael Sternson and Ulf Björkenstam. Infu-lence of weather routing on encountered waveheights. International Shipbuilding Progress,49, 2002.

[2] Øyvind Saetra. Ensemble shiprouting. Tech-nical Memorandum 435, European Center forMid-ranged Weather Forecasts, Research De-partment, 2004.

[3] Statistical bene�ts of route optimization.Technical report.

[4] M. Ho�schildt, J. Bidlot, B. Hansen, andP.A.E.M. Janssen. Potential bene�t of ensem-ble forecasts for ship routing. Technical Mem-orandum 287, ECMWF, 1999.

[5] Toby Allsopp. Stochastic weather routing forsailing vessels. A thesis submitted in partialful�lment of the requierments for the degree ofmaster of engineering, The University of Auck-land, 1998.

[6] Toby Allsopp, Andrew Mason, and AndyPhilpott. Optimising yacht routes under un-certainty. In the 2000 Fall National Conferenceof the Operations Resaerch Society of Japan,pages 176�183, 2000.

[7] Asher Treby. Optimal weather routing usingensemble weather forecasts. In Proceedings of

the 37th annual conference, Operational Re-

search Society of New Zealand, 2002.

[8] Stefan Harries, Justus Heinmann, and JörnHinnenthal. Pareto optimal routing of ships.In International Conference on Ship and Ship-

ping Research - NAV 2003, 2003.

[9] Jörn Hinnenthal. Robust Pareto-Optimum

Routing of Ships utilizing Deterministic and

Ensemble Weather Forecasts. PhD thesis,Technischen Universität Berlin, 2008.

[10] Jörn Hinnenthal and Øyvind Saetra. Ro-bust pareto-optimal routing of ships utiliz-ing ensemble weather forecasts. Maritime

Transportation and Exploitation of Ocean and

Coastal Resources, Volume 1: Vessels for Mar-

itime Transportation, Volume 2: Exploitation

of Ocean and Coastal Resources, 2005.

[11] Jörn Hinnenthal and Günther Clauss. Ro-bust pareto-optimum routing of ships utilis-ing deterministic and ensemble weather fore-casts. Ships and O�shore Structures, 5:2:105�114, 2010.

[12] E. W. Dijkstra. A note on two problems in con-nexion with graphs. Numerische Mathematik,1:269�271, 1959. 10.1007/BF01386390.

[13] João Carlos Namorado Climaco and ErnestoQueirós Vieira Martins. A bicriterion shortestpath algorithm. European Journal of Opera-

tional Research, 11(4):399 � 404, 1982.

[14] Chi Tung Tung and Kim Lin Chew. Amulticriteria pareto-optimal path algorithm.European Journal of Operational Research,62(2):203 � 209, 1992.

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