# Multiple objectives: cost, unreliability, and emissions

Three types of objectives are considered. The first objective measures the economic efficiency of the shipping service using the annual total vessel operating cost (*Jcosi)>* which includes two components: the annual FC cost of all ships in the service (assume 365 days in a year) and the annual charter hire for all ships in the service:

Note that *n _{v}* represents the number of ships deployed in the service route,

*L N*

*E*—■ *g* represents the expected average FC per ship per voyage, fc=i /=1

and 365/(T/24) represents the number of voyages per year. Hence, the first term on the right-hand-side of (7.17) represents the annual total FC cost incurred by all ships deployed in the service route. The second term on the right-hand-side of (7.17) represents the annual charter hire for all ships in the service. According to empirical observations, the ship’s FC per nautical mile *g(.)* is a quadratic convex function of the sailing speed (Fagerholt et al. 2010). Moreover, it is reasonable to assume that ^(s) is monotonic increasing and nonnegative in the range from the minimum vessel speed to the maximum speed.

The second objective measures the service effectiveness using the average schedule unreliability *(Jsu)>* which can be defined as the probability that the ship arrives later than the planned arrival time plus the arrival time window. This objective can be estimated by averaging the delay frequency over *L* consecutive voyages in the planning horizon.

where /{} is an indicator function. It takes the value 1 if the condition in {} is true, 0 otherwise. As the arrival time window increases, the schedule unreliability is decreasing. In other words, a larger arrival time window will be in favour of schedule reliability. However, this would imply a higher degree of uncertainty for terminal operators and cargo owners since the ships may arrive at a port over a wider range of time interval.

The third objective measures the carbon emissions from ships using the annual total ССЬ emissions from all the ships deployed in the service route *(Jcoz* )• Since emissions are proportional to the FC, it follows

The multi-objective optimisation problem is to Find the optimal decision variables *{n _{r},s,,,*T,} that minimise the three objective functions

*JCost, Jsu>*and

*Jcch*simultaneously.