# Linear programming model

Based on the notation in the previous subsection, the objective function, i.e. the total cost, can be expressed as follows:

subject to

44 *Freight logistics and shipment routing*

Constraint (3.6) represents that the fulf illed demands are no more than trade demands. Equations (3.7) and (3.8) define the number of containers (including transhipment) for od-pair that are loaded and unload on the ith port call on route *r,* respectively. Equations (3.9) and (3.10) indicate that the fulfilled demand of the od-pair must be loaded onto a vessel at the origin port о and unloaded from a vessel at destination port d. Equation (3.11) represents that if some of the fulfilled demands of od-pair are unloaded at a transhipment port, then these containers must be loaded to another vessel at the same transhipment port. Equations (3.12)—(3.13) define the total number of containers (including transhipment) that are loaded and unloaded at port *p.*

The right-hand side (RHS) of (3.14) represents the transhipment of od-pair demand to be unloaded at port call *ri* via service route r; the left-hand side (LHS) of (3.14) represents the transhipment of od-pair demand from service route *r* to be transferred to other route *s* at port call *ri.* Similarly, the RHS of (3.15) represents the transhipment of od-pair demand to be loaded onto service route s at port call *sj;* and the LHS of (3.15) represents the transhipment of od-pair demand from other service routes to be transferred to route *s* at port call *sj.* Equation (3.16) defines the total transhipment volume at port p,

which can also be equivalently defined as *y' _{p} =* yj, —

**X**

*У*

*pd*

*У*

*p*

**^**

*У*

*op*•

*dG**P **OG **P*

Equation (3.17) defines the number of containers for od-pair that is carried on board of vessel on leg i (from the ith port call to the <+lth port call) on route *r;* where *[rl, rm]* represents the set of port calls starting from *rl* to *rm,* which is a subset of *I _{r}.* Constraint (3.18) represents the service route carrying capacity constraints on each leg for each route during the planning horizon. Constraints (3.19) and (3.20) represent the non-negative of the relevant decision and intermediate variables.

The above model is a linear programming problem. From the decision variables *{yod.ri.rj}* and the intermediate variables *{y' _{0(l rj s}j},* the path of any satisfied demand of

*afd*-pair can be determined explicitly. In addition, the transit time at sea and the transferring time at transhipment ports are also explicitly modelled. The results are useful for key stakeholders in the container transport industry. For example, for shippers, the inventory-in-transit holding costs can be considered when making the shipment routing decisions. For ocean carriers, the freight rates can be specified for each port-pair covered by individual service route. For terminal operators, the unit loading, unloading, and transhipment costs can be specified separately to evaluate their impact on the pattern of container flows.

The top-down approach takes the global optimisation perspective by ignoring the identities of service providers and shippers. The results would provide the most efficient container movements in the world as if all stakeholders are coordinating their behaviours. However, a global shipping network involves a large number of different players and many of them are competitors especially when they offer similar services. Therefore, the result may not reflect the industrial practice. Nevertheless, the model may be used for service network design for a single ocean carrier if its customer demand can be reasonably forecasted.