# Network flow models for ECR in complex shipping networks

This section considers a shipping network consisting of multiple service routes. When multiple service routes are considered, it is possible and common that transhipments will occur. In other words, a container may be lifted off the vessel in one service route at a port and then lifted on a vessel in another service route at the same port. For example, large container ports such as Singapore, Hong Kong, and Rotterdam involve a large number of transhipments that exceed their 50% of the port throughput.

The following assumptions are made to facilitate the model formulation:

• (i) The weekly demands for any port-pair are constant.
• (ii) Shipment is splittable; i.e., it can be fulfilled partially in different routes.

With the above assumptions, a linear programming model can be formulated to deal with both laden container routing and empty container repositioning in a given shipping network. We present an origin—destination (O-D)-link- based model below. It should be noted that if assumption (ii) is removed, then the model will become a mixed-integer programming model instead of a linear programming model. We introduce the notation as follows (some notation is based on Lee and Song 2017):

Input parameters:

P: the set of ports in the shipping network;

R: the set of shipping routes in the shipping network;

Rp. the set of routes that call at port peP;

Nf: the number of port calls on the route reR;

I,: the set of port call indices on the route reR, i.e. I,:=

{1,2, ...,Nr};

p(ri): the port that corresponds to the ith port call on route r;

I, p. the set of port call indices corresponding to port p on the route

reR, i.e. I, p:= {iel, p{ri) = p};

od: an index to represent the O-D port-pair from port oeP to port

deP;

D0(j: the weekly demand from port oeP to port deP;

C„ : the unit cost of transporting laden containers on vessel on leg

i on route reR;

C^: the unit penalty cost for lost sales for the od demand;

Capr the vessel capacity on route reR;

Decision variables:

yL ri’ xod ri: t^e number of laden (empty) containers for od that are loaded on the ith port call on route r;

y'od ri’ xod ri: t^e number of laden (empty) containers for od that are unloaded on the /th port call on route r;

i'od ri’ xod ri' t^e number of laden (empty) containers for od that are carried on board on leg i (from the ith port call to the (i+l)th port call) on route r;

y0l/: the fulfilled demand for od port-pair.

xp: the number of empty containers to be repositioned out of port p.

xo(f. the number of empty containers to be repositioned from port о

to port d.

The objective function is the total cost consisting of the following: (i) the laden and empty container loading (lifting-on) costs; (ii) the laden and empty container unloading (lifting-off) costs; (iii) the laden and empty container transhipment costs; (iv) the lost-sale penalty costs; (v) the laden container transportation costs on vessel; and (vi) the empty container transportation costs on vessel.

The laden container routing and empty container repositioning problem can be formulated as a linear programming model:

subject to