# Emission reduction through ship scheduling and planned speed optimisation

For any business organisation in logistics sectors, the key performance indicators (KPIs) include operational efficiency and service effectiveness. Operational efficiency is often achieved by cost minimisation, whereas service effectiveness is often achieved by delivery reliability. Clearly, emission reduction should be considered together with organisations’ business KPIs to survive and gain competitive advantages. This section addresses a joint tactical planning problem for the deployment of a number of vessels in a service route, the planned maximum sailing speed, and the service schedule of the shipping route to simultaneously optimise the shipping emissions, the expected cost, and the schedule reliability under uncertain port times. The problem is formulated into a stochastic multi-objective optimisation problem at the operational level. The relationships between the objectives and the decision variables are established analytically. A multi-objective generic algorithm (MOGA), which is based on non-dominated sorting and simulation, is presented to solve this problem. This section is mainly based on Song et al. (2015).

## Modelling a shipping service scheduling with uncertain port times

Consider a liner shipping route with weekly service frequency. As we discussed in Sections 5.6 and 5.7, the important tactical decisions include the ship deployment, the ship sailing speed, and the planned arrival times at each portcall. The ship sailing speed and the planned arrival times determine the timetable of the ship, i.e. the shipping service schedule. Because of the uncertainties at ports such as unavailability of resources and waiting time, the vessel may deviate from the planned schedule. As a result, the ship operator may adjust the actual sailing speed at each sea leg within a certain range to catch up the schedule. However, the ship operator may not be willing to speed up beyond the range because of the high-FC cost. For example, in the extra-slow-steaming practice, the vessel normally will not exceed 19 knots, whereas in the super-slow-steaming practice, the vessel normally will not exceed 16 knots. To capture such behaviour of the shipping lines, we introduce a decision variable, the planned maximum sailing speed, to allow the ship operator to adjust the actual sailing speed in the range between the minimum speed and the planned maximum sailing speed. It should be noted that the planned maximum sailing speed could be significantly smaller than the maximum speed.

Although the above decisions are largely at the tactical planning level, it is necessary to build an operational model so that the system dynamics and performance can be appropriately evaluated by explicitly simulating the operational behaviours such as the actual sailing speed in response to uncertainty realisation. The following notations are introduced first (Song et al. 2015).

*Input parameters*

N: the number of portcalls in a voyage (i.e. a round-trip).

L: the number of voyages that a ship sails in the planning horizon.

*dp* the distance from the ith portcall to the next portcall in nautical

miles.

*lj(uj):* the minimum (maximum) port times at the ith portcall.

£,■: the random variable representing the uncertain part of port

time at the ith portcall if the ship arrives at the portcall no later than the planned arrival time, s.t. 0 < < и, *—l _{r}*

*Tjj:* the random variable representing the uncertain part of port

time at the ith portcall if the ship arrives at the port later than the planned arrival time, s.t. 0 < 7], We assume that /),

has the first-order stochastic dominance over which represents the phenomenon that delayed ship has a higher probability to wait a longer time at port.

^{5}min(^{s}max)^{:} the minimum (maximum) ship’s sailing speed in knots at sea.

£(s): the ship’s FC in tonnes per nautical mile at sailing speed *s,*

which is assumed to be a monotonic increasing and quadratic convex function of the actual sailing speed. This is based on the empirical observations that the ship FC per unit of time has a cubic relationship with respect to the sailing speed.

*C(■.* the fuel price in US$ per tonne.

C„: the daily cost of a ship in US$, which excludes the FC cost.

This may be interpreted as the daily charter hire.

*W _{t}:* the ship’s arrival window time at the ith portcall.

*K _{c}:* the emission factor that converts the consumed fuel into the

emissions, which represents the kilograms of C0_{2} emitted per tonne of fuel burned by the engine.

*Decision variables*

*n„:* the number of ships to be deployed in the liner shipping route.

*s _{v}* the planned maximum sailing speed m knots, s.t. _<

_{min}

~~v~~

T,: the planned transit time in hours from the ith portcall to the

next portcall, i.e. the planned duration for the ship from arriving at the ith portcall to arriving at the (i + l)th portcall. The weekly service frequency requires thatTj + T_{2}+... + T_{jV} = 168w_{(},. Flere, 168d_{(}, represents the number of hours in a voyage.

*Intermediate variables and objective functions*

*T:* the journey time of a voyage (single round-trip) in hours. We

have T = 168-«„.

*tj f,:* the planned ship arrival time at the ith portcall in the *kth* voyage with tj i =0.

*t" _{k}:* the actual ship arrival time at the ith portcall in the

*kth*voyage.

*tf _{k}:* the actual ship departure time at the ith portcall in the

*kth*

voyage.

*JCosh* the annual total cost in US$ by all the ships deployed in the

shipping service route.

*Jsu ^{:}* the average schedule unreliability over all portcalls in all voyages in the planning horizon.

*Jco?'■* the annual total C0_{2} emission in tonnes from all the ships deployed in the service route.

354 *Sustainability in maritime transport *Assumption 7.1

Ships will not be served at a portcall before its planned arrival time, which reflects the phenomenon that the berthing plan and required resources at ports may not be available before the planned ship arrival time.

Assumption 7.2

If the ship arrives at the ith portcall no later than the planned arrival time window, the uncertain part of the port time is described by the random variable otherwise, the uncertain part of the port time is described by the random variable *TJj,* where *)],* and has the same range [0, и, — /,-], but *I],* has the first-order stochastic dominance over which represents the phenomenon that delayed ships are subject to a higher degree of uncertainty and may take longer service time atports. Mathematically, the first-order stochastic dominance relationship can be described as follows: Prob{7J, > *x}* > Prob {^, > x} for all x e [0,и, —/,•], and for some *x* e [0,и,- —/,■], Prob {/], > .v} > Prob{£, > .v}.

Assumption 7.3

After the ship departs from a port, the operator will determine the sailing speed from the current portcall to the next portcall in such a way that the ship would arrive at the next portcall at the planned arrival time as close as possible, subject to the constraint that the actual sailing speed is taking a value in the interval [s_{min},s„J. Here, *s _{v}* is a tactical decision variable representing the planned maximum sailing speed. This assumption reflects the phenomenon that ship operators are often willing to catch up the schedule by speeding up to a certain level (s„), but not beyond this level although the maximum speed (s,

_{mx}) could be greater than

*s*

_{t},.From Assumptions 7.2 and 7.3, the ship’s actual sailing speed in each sea leg in each voyage (denoted by s, * _{k}* for 1 <

*i*<

*N*and 1

*< k< L)*can be determined dynamically in response to the realised random variables over portcalls. The dynamic system is formulated as follows.

Equations (7.8)—(7.12) represent the regularity and continuity of the shipping service voyage by voyage. Equation (7.13) determines the ships’ sailing speed dynamically at each sea leg considering the ship speed constraints and Assumption 7.3. Equation (7.14) determines the ships’ actual departure time, where and *Tjjj,* should be regarded as the realised samples of the random variables and 7), respectively. Inequality (7.16) reflects the fact that if there is no uncertainty at the port, the ship should be able to reach the next portcall before the planned arrival time when it sails at the maximum sailing. From

*N*

- 1 V rf,
- (7.10) and (7.16), we have
*n*where_{v}>n_{v},*n„*= -/(/; +—'—) > i-^{e}- the - 168^ s
_{max}

L »=i

minimum number of ships required to be deployed in the service route.