# Solutions to ECR from the modelling technique perspective

From the modelling technique perspective, the ECR modelling studies may be categorised into two research streams. The first stream adopts the network flow models and often applies mathematical programming to produce a set of arc-based matrices. The element in each matrix is a numerical value representing the quantity of empty containers to be moved on an arc (i.e. from one node to another node) in the network.

The second stream aims to develop effective state-feedback control policies that often uses inventory control, dynamic programming, and simulation- based optimisation methods. The solutions of these empty container repositioning policies are similar to those in traditional inventory control in manufacturing systems, and they normally consist of a number of decisionmaking rules associated with system dynamic states such as inventory levels of empty containers. By applying the rules at a decision epoch, the number of empty containers that need to be repositioned out or into a node can be determined dynamically. Several inventory-based control policies have been proposed in the literature, e.g. the double-threshold policy, the dynamic port-to-port balancing policy, the coordinated (s, S) repositioning policy, and the single-level threshold policy (Song and Dong 2015).

The contrast between network flow model and inventory-control model for empty container repositioning problems may be understood from the relationship between material requirements planning (MRP) and just-in-time (JIT) control. The former (network flow model) is good at planning, whereas the latter (inventory-control model) is good at control. The former is suitable for centralised management requiring extensive and accurate data, whereas the latter is suitable for decentralised management requiring modest data and being more robust to uncertainty.

In the remainder of this chapter, we present a few specific mathematical models for empty container repositioning problems, which represent the above modelling techniques. To simplify the exposition, we make the following common assumptions:

Assumption 4.1: containers are measured in TEUs. One FEU is treated as two TEUs.

Assumption 4.2: the shipping services are of weekly frequency.

Assumption 4.3: the vessels deployed in the same service route are of similar sizes.

Since empty container repositioning is driven by the trade imbalance, it is desirable to model both laden container movements and empty container repositioning simultaneously.