where b0, b1,
and b2 are regression parameters. Psaraftis and
Kontovas4,5 have discussed at length the nature of the
variation of ship speed consumption (per unit time) with ship speed,
showing its complexity in view of its dependence on many factors, and
pointing out4 that “It is known from basic naval
architecture that fuel consumption depends non-linearly on both ship
speed and ship payload”, and concluding that11 “the
function f can be a complex function which may not even be defined in
complex form”. It is worth noting that assuming that this function is
convex, then the optimal solution is one where ship speed is
uniform19; however, the convexity assumption may not
hold in practice, as for example when different ship fuel consumption
rate – ship speed curves are used for different fuels, as is the case
in the case study which is presented in Section 5. In almost all work
reported on ship speed optimisation problem variants the cubic function
is employed to represent the variation of ship fuel consumption rate
(per unit time) with ship speed. However, there is one notable exception
which has not been used in the ship speed optimisation literature,
namely that of Brown et al.10, whereby, instead of
using ship speed as a decision variable, a ship speed level mix is
selected a priori and time spent in each ship speed level (which is
known) is defined as the decision variable. In the next two Sections,
the nonlinear programming and linear programming models, respectively,
are formulated for the ship speed optimisation problem variant for a
ship voyage between two ports, wherby it is required to minimise ship
fuel consumption and having a due arrival date at destination
port.