A new numerical model for the on-demand computation of optimal ship routes based on sea-state forecasts has been developed. The model, named VISIR (discoVerIng Safe and effIcient Routes) is designed to support decision-makers when planning a marine voyage.

The first version of the system, VISIR-I, considers medium and small motor vessels with lengths of up to a few tens of metres and a displacement hull. The model is comprised of three components: a route optimization algorithm, a mechanical model of the ship, and a processor of the environmental fields. The optimization algorithm is based on a graph-search method with time-dependent edge weights. The algorithm is also able to compute a voluntary ship speed reduction. The ship model accounts for calm water and added wave resistance by making use of just the principal particulars of the vessel as input parameters. It also checks the optimal route for parametric roll, pure loss of stability, and surfriding/broaching-to hazard conditions. The processor of the environmental fields employs significant wave height, wave spectrum peak period, and wave direction forecast fields as input. The topological issues of coastal navigation (islands, peninsulas, narrow passages) are addressed.

Examples of VISIR-I routes in the Mediterranean Sea are provided. The optimal route may be longer in terms of miles sailed and yet it is faster and safer than the geodetic route between the same departure and arrival locations. Time savings up to 2.7 % and route lengthening up to 3.2 % are found for the case studies analysed. However, there is no upper bound for the magnitude of the changes of such route metrics, which especially in case of extreme sea states can be much greater. Route diversions result from the safety constraints and the fact that the algorithm takes into account the full temporal evolution and spatial variability of the environmental fields.

The operational availability of high spatial and temporal resolution
forecasts, for weather, sea state, and oceanographic variables paves the
way to a realm of downstream services, which are increasingly closer to
end-user needs

VISIR [vi'zi:r]

The aim of this paper is to lay a sound scientific foundation of VISIR-I, including all its main components: the optimization algorithm, the ship model, and the processor of the environmental fields.

After reviewing the literature in Sect.

The main mathematical schemes available in the literature to solve ship routing problems are reviewed in the following.

Initially devised as a manual tool for navigators, the isochrone method is
based on the idea of building an envelope of positions attainable by a vessel
at a given time lag after departure. This envelope is called an
“isochrone”. In the work by

The variational approach involves searching for trajectories making an
objective functional stationary, such as total time of navigation or
operational cost, given a set of constraints. The search is achieved by
varying the parameters controlling the trajectory. This approach is
equivalent to solving the Euler–Lagrange equation. In

The time-dependent problem instead can be addressed through the technique of
optimal control

The work by

Monte Carlo methods discard exact solutions in favour of faster solutions.
Also, they provide a viable technique for fulfilling multiple and competing
objectives. A class of Monte Carlo methods makes use of genetic algorithms.
They start with guessed routes (“chromosomes”) whose subparts (“genes”)
cross each other and mutate in a random way, in order to find a new route
(“offspring”) that better fits the objective function of the actual
problem. The use of Monte Carlo methods in the context of multi-objective
optimization is reviewed in

In discrete methods, the spatial domain is represented by some kind of grid
(regular or not) and the optimization is based on recursive schemes. A key
concept is the so-called principle of optimality: given a point on the
optimal trajectory, the remaining trajectory is optimal for the minimization
problem initiated at that point

There are several recurrent shortcomings in the ship routing literature: the
limited capability to deal with complex topological conditions, such as in
the coastal environment

All these issues need to be addressed simultaneously by a model aimed at feeding an operational system that also works in coastal waters, for a wide class of vessels and environmental conditions, taking into account navigation safety according to the latest international standards. In VISIR-I all the above-mentioned shortcomings are overcome. The method is based on an exact graph search algorithm, modified in order to manage time-dependent environmental fields and voluntary vessel speed reduction. It is validated against analytical results. In addition, the graph grid is designed to deal with the topological requirements of coastal navigation. VISIR-I also includes a dedicated motorboat model, and safety constraints for vessel intact stability are considered.

All these features are described in detail in what follows.

In this section we present the method employed by VISIR-I for solving the
route optimization problem. First, the problem is formally stated
(Sect.

The mathematical problem addressed and solved in an operational way by VISIR-I can be stated as follows.

A ship route is sought departing from

Ship speed

If set

Speed

The problem of finding the least-time route in any meteo-marine conditions is
thus equivalent to the minimization of

If the time dependence in refractive index

The first component of VISIR-I presented here is the shortest-path algorithm. The term “shortest path” is used both in the literature and hereafter with a more general sense than a direct reference to the geometrical distance. Indeed, “shortest” may refer to the spatial or temporal distance, as well as the cost or any other figure of merit of the optimal path.

Let us consider a directed graph

Graph spatial grid. Outgoing edges from the central node are
displayed as arrows pointing to the respective tail
node. Just the six edges relative to the first quadrant are shown (24-connectivity). The value of the angle

In VISIR-I, the resulting graph is first screened for nodes and edges on the
landmass. An edge is considered to be on the landmass if at least one of its
nodes is on the landmass or if both nodes are in the sea but the edge linking
them intersects the coastline. In such a case, the edge is removed from

Given that environmental conditions change over a timescale comparable with
or shorter than the vessel route duration, edge weights cannot be considered
as constants. Thus, in order to solve Eqs. (

With reference to the nomenclature in Table

Edge weight

Graph notation and relevant graph quantities used in this paper.

Parameters of the graph for the Mediterranean Sea after the removal of nodes and edges on the landmass (GSHHG coastline
used). In the actual route computations, just a subdomain of the whole basin is selected. Due to border effects, the
connectivity ratio

Thus, in VISIR-I, edge weights

There are various methods for computing shortest paths on a graph. For an
overview, see

A key concept in graph methods is the node label, which can be either
temporary or permanent. The permanent label

Depending on the way node labels are updated, graph algorithms may be classified into label setting or label correcting algorithms. A label setting single-source single-destination algorithm with fixed departure time is used here.

The fact that in VISIR-I destination node is assigned (through

In general, the fact that a graph is time-dependent implies that the shortest
path can have special features. In fact, under specific circumstances, the
strategy of traversing an edge as soon as possible does not always lead to
the shortest path. Also, the shortest path may not be simple (there may be
loops) or even not concatenated (Bellman's optimality not fulfilled). This
has consequences on the class of algorithm to be applied.

Condition Eq. (

Examples of time-dependent edge delays

Before the algorithm is run, edge delays

As seen above, VISIR-I's strategy regarding navigational safety is to remove
unsafe edge delays from the graph by setting their edge weight to

Engine throttle levels employed in VISIR-I (

VISIR-I defines, for a vessel with maximum engine power

The second component of VISIR-I is a ship model describing vessel interaction
with the environment (specified by the forecast fields of
Sect.

The following presentation comprises of a balance equation for the propulsion
system in Sect.

Motorboats are the focus of VISIR-I route optimization.

For these vessels, propulsion is provided by a thermal engine burning fuel
and delivering a torque to the shaft line and, when present, to a gearbox
(Fig.

Main vessel dimensions and seaway nomenclature. The red part of the
hull is normally underwater. The angle of wave encounter

A full modelling of this energy conversion mechanism is a highly complex task
involving, just to mention a few, the efficiency of each of these conversion
steps, the effect of hull-generated wake on propeller efficiency and
corresponding thrust deduction, and the load conditions of the engine

For the purposes of VISIR-I, it was deemed sufficient to derive the vessel
response function from a balance of thrust and resistance at the propeller.
That is, given the brake power

In this paper we restrict our attention to displacement vessels. Indeed high-speed planing hulls are characterized by a different dynamic behaviour and
deserve a more sophisticated treatment

When underway, a displacement vessel is subject to various forces hindering
its motion. A possible decomposition of the resulting force is to distinguish
calm water resistance

As outlined in

For specifying the drag coefficient

Parameters of the ship model. The numerical factor in the formula
for

However, it is our aim that VISIR-I runs without specifying too many vessels
parameters. Thus,

In addition to calm water resistance, sea waves are an additional source of
ship energy loss

Database of vessel propulsion parameters and principal particulars
used in this work. See Fig.

In VISIR-I, following the cited literature, a reduced non-dimensional
resistance

Sustained Froude number

Empirical methods are often used for deriving

Ship and environmental parameters (

Substituting Eqs. (

Sustained speed

Our results also prove that, by varying engine throttle, sustained speed does
not vary by the same factor at all

Furthermore, by comparing performances of vessel V1 (ferryboat) and V2
(fishing vessel), it can be seen that the former sustains a larger fraction
of its top Froude number at any given significant wave height. This different
dynamic behaviour is mainly related to the maximum engine brake power

Engine throttle needed for sustaining a given

The comparison between V1 and V2 also shows that the two vessels behave
quite differently in extreme seas, whereby vessel V1 (the ferryboat) is able to
reach more than 30 % while V2 (the fishing vessel) reaches less than
20 % of its top

Resistances are evaluated from the sustained speed

While calm water resistance

List of main approximations done in VISIR-I.

Resistance experienced by the vessel at constant power setting

Wave added resistance

In comparison to V2, vessel V1 exhibits larger resistances. However, for both
vessel classes, the

The ship model described so far needs to be complemented by navigational constraints in order to reduce dangerous or unpleasant movements for the ship itself, the crew and cargo.

Such situations cannot simply be ruled out by designing a vessel in
accordance with the Intact Stability (IS) Code,

VISIR-I checks for three modes of stability failure: parametric roll, pure
loss of stability, and surfriding/broaching-to. The theoretical hints below
are mainly based on

A realistic assessment of stability failure would require a detailed
knowledge of ship hull geometry. In the current version of VISIR-I, however,
just principal particulars of the vessel (length, beam, draught) are
employed. In addition, even vessel-internal motions and mass displacements,
such as the positioning of catch within a fishing vessel

In the following sections, we use the deep water approximation of the
wave dispersion relation in order to gain a rapid estimation of the threshold
conditions. We can thus estimate the wavelength

Parameters of the environmental fields.

When a ship is sailing in waves, the extent of the submerged part of the hull
changes in time. For most hull shapes, this also involves a change in the
waterplane area. This in turn influences the curve for the righting lever
(GZ), which is fundamental to ship stability. Indeed, if wavelength

The mathematical formulation of parametric roll is based on the solution of
Mathieu's equations and the computation of Ince–Strutt's diagram. It shows
that parametric roll occurs when encounter wave period

In VISIR-I the encounter period

Following

Formula Eq. (

This mode of stability failure is triggered by a similar condition to the parametric roll. However, it does not involve any resonance mechanism and thus may be activated by a single wave. In fact, if the crest of a large wave is near the mid-ship section, stability may be significantly decreased. If this condition lasts long enough (such as during following waves and a ship speed close to wave celerity), the ship may develop a large heel angle, or even capsize.

According to

Using also Eqs. (

Surfriding is the condition where the wave profile does not vary relative to
the ship. That is, the ship moves with a speed equal to wave celerity:

The simplest modelling of this mode of stability failure starts with the
computation of the force of the wave-induced surge which is able to balance
the difference between total resistance and thrust provided by the ship.
A critical point may then be reached, where surging is no longer possible and
the ship is captured by the surfriding mode

In

In VISIR-I, the following surfriding hazard criteria reported in

Of note is that all VISIR-I safety constraints described above,
Eqs. (

We distinguish the environmental fields between static (bathymetry and
coastline) and dynamic fields (waves, winds, currents). In VISIR-I,
bathymetry and coastline are employed to ensure that navigation occurs in not
too shallow waters and far from obstructions. Of the dynamic fields, just
wave forecast fields are used, as explained in
Sect.

A

Along with the coastline database, bathymetry is needed for computing
a land–sea mask for safe navigation. The first step is to select edges

In other words, just a strictly positive under keel clearance UKC

Bathymetry is needed also for a more accurate estimation of wavelength

can be rewritten as follows:

As seen from Eq. (

The coastline database is used in VISIR-I for a preliminary removal of graph
edges on the landmass (Sect.

To this end, the NOAA Global Self-consistent, Hierarchical, High-resolution
Geography Database
(GSHHG

A joint depth-coast land–sea mask is obtained by multiplying the mask
defined by Eq. (

Due to the quite different spatial resolution of the coastline and the
environmental fields, a regridding procedure is employed for reconstructing
the coastal fields.

Fields are extrapolated inshore by replacing missing values of sea fields
with the average of the first neighbouring grid points,
Fig.

The fields are bi-linearly interpolated to the target grid. In VISIR-I
this is the bathymetry grid. Thus, spatial resolution of wave fields is
enhanced from the original

Sea-over-land extrapolation.

The dynamic environmental fields are used in VISIR-I for the computation of
sustained ship speeds and safety constraints. In the present version, just
the effect of waves is considered, which is deemed to be the most relevant
for medium- and small-size vessels. The effect of wind and sea currents is
planned for future development. In fact,

wind drag may be significant for vessels with a large freeboard and/or
superstructure area

sea current drift is relevant especially in proximity to strong ocean
currents

wave effects include both drift and involuntary speed reduction.
The drift is due to nonlinear mass transport in waves

Flow chart of the computer code of VISIR-I model. Functioning mode 1 is run just once for preparing graph nodes and edges; mode 2 is the operational one, using sea nodes and edges computed from mode 1.

The current version of VISIR-I employs wave forecast fields from an
operational implementation of the Wave Watch III (WW3) model

1/8

Here we present the main steps in the computational implementation of VISIR-I
into a computer code. The code itself and a data sample can be obtained
following the instructions provided in Sect.

The flow chart in Fig.

Mode 1 is needed to produce the database of nodes and edges neither lying on
the landmass nor crossing it; see Table

Mode 2 is the functioning mode for the operational use of VISIR-I. First of
all, the ship model is evaluated. Equation (

Such weights, like those in Eq. (

Cycloidal benchmark: vessel speed

In VISIR-I, for long routes, the computing time is dominated by the
preparation of the edge weights and the shortest-path computation. The
computing time

An exact validation of the optimization algorithm of VISIR-I and the
forthcoming post-processing phase is possible in the case of time-invariant
fields. However, algorithmic complexity and pseudocode do not substantially
differ for the case of time-invariant and time-dependent fields, as pointed
out in Sect.

We exploit the cycloidal curve, being the solution to problem
Eqs. (

Figure

VISIR model performance metrics. The coefficients are identified by
least-square fits of Eq. (

Case study #1. Geodetic (black markers) and optimal (red markers)
route from Trapani (Italy) to Tunis (Tunisia) for vessel V1 of
Table

Case study #1. Information along geodetic (black) and optimal (red)
route of Fig.

Case study #2. Geodetic (black markers) and optimal (red markers)
route from Crete to Rhodes (Greece) for vessel V2 of Table

Cycloidal benchmark: length and duration of the three routes shown
in Fig.

Case study #2. Information along geodetic (black) and optimal (red)
route of Fig.

Summary metrics for the case study routes displayed in
Figs.

Note that the cycloidal profile is compatible with Snell's law of refraction,
as the route is refracted in order to reach the optically more transparent
(higher speed) region the soonest. Instead, the rhumb line connecting
departure and arrival points does not sufficiently exploit such a high-speed
region and lags behind by more than 18 %; see Table

In VISIR-I, the choice of the vessel parameters (Table

In this case study, vessel V1 of Table

In Fig.

Considering the motion of the wave height field as well, the optimal route
attempts to maximize the time spent in calmer seas, where, due to the smaller
added wave resistance (Eq.

Figure

Case study #3. Geodetic (black markers) and optimal (red markers)
route from Gibraltar (UK) to Ben Abdelmalek Ramdan (Algeria) for vessel V2 of
Table

Case study #3. Information along geodetic (black) and optimal (red)
route of Fig.

Comparison of the effect of different parametrizations of

In the second case study, a transfer of fishing vessel V2 of
Table

In Fig.

In the third case study, a voyage of fishing vessel V2 of
Table

Figure

In this paper, we have presented the scientific basis of VISIR-I, a ship routing system, as well as results of its computation of optimal routes in the Mediterranean Sea. The system is designed for flexible modelling of the vessel and its interaction with the environment. Time-dependent analysis and forecast fields from oceanographic models are employed in input.

The optimal routes computed by VISIR-I were shown to correctly avoid islands and waters shallower than ship draught. Vessel course is generally refracted towards regions of larger sustained speed, allowing in some cases to sail a longer path and reach the destination earlier than along the rhumb line. VISIR-I optimal routes are checked for vessel intact stability, in terms of compliance with IMO regulations and more advanced research results. In some cases, it is these safety criteria, and not the refraction, being responsible for route diversions. The algorithm is also able to compute voluntary speed reductions. The vessel parameters needed to run the model are limited to basic propulsion data and hull principal particulars, making the system accessible for on-demand computations even by non-professionals of navigation.

Several issues require further improvements. In relation to the
time-dependent algorithm, in the case of rapid changes in the analysis or
forecast fields, the optimality of the route retrieved by the model is no
longer guaranteed (see FIFO condition in
Sect.

For the environmental fields, the most urgent upgrade seems to be accounting
for wind, especially for larger vessels, and for currents, at least for
slower boats. Wind was recently added to a variant of VISIR-I for sailboat
routing

Concerning the ship model, a more advanced parametrization of both calm water
and wave added resistance (Sect.

A summary of the main approximations employed in VISIR-I is provided in
Table

The first operational implementation of the system took place in the
Mediterranean Sea

In the future, VISIR could be generalized to other optimization objectives,
such as bunker savings, by suitably modifying the refractive index in
Eq. (

In conclusion, we would like to stress the potentiality of VISIR to offer the scientific and technical communities an open platform whereby various ideas and methods for ship route optimization can be shared, tested, and compared to each other. In this respect, the fact that in VISIR-I – through this paper and related source code - the various system components (vessel model, shortest-path algorithm, and processing of the environmental fields) are openly documented and made publicly available should enable unprecedented developments in the efficiency and safety of navigation.

The VISIR-I code is made available under the GNU General Public
License (Version 3, 29 June 2007) at

The pseudocode for the time-dependent shortest-path algorithm employed in
VISIR-I (see Sect.

It is organized into three main parts: initialization of node labels and indices (rows 2–6); main iteration loop (rows 7–10 and 13–17); exit condition (rows 11–12).

The input arguments are the start and end nodes

In order to numerically evaluate the impact of a constant drag coefficient

The case

Summary metrics for the routes of all case studies in Sect. 3 and
different values of

First of all, we note that, at maximum engine throttle, the vessel speed
curve as a function of

Finally, we can visualize the effect of

The steps leading to the expressions
Eqs. (

We start from

The l.h.s. of that
equation is in fact non-dimensional, as confirmed also by evaluating it with
the parameter values in

Funding through TESSA (PON01_02823) and IONIO (subsidy contract no. I1.22.05) projects is gratefully acknowledged. Mannarini and Pinardi were partially funded by the AtlantOS project (EC H-2020 grant agreement no. 633211). Edited by: R. Marsh