NTS is a Swiss intermodal operator located in Lausanne. When a customer has to transport goods between origin and destination, NTS supplies him with one or more empty containers into which the goods can be loaded. Upon arrival at the destination, the goods are unloaded and the empty containers must be transported to the new customer pick-up point. Therefore, NTS management needs to reallocate empty containers periodically (every week). What the company focuses on is that empty container transportation activities are very expensive (the cost is almost 35% of total operational costs).

In May 2023, some 20 empty containers must be reallocated between terminals in Amsterdam, Berlin, Munich, Paris, Milan, Barcelona and Madrid. The number of empty containers available or requested at various terminals is reported, along with transportation costs (in euros/container) shown in Figure 1.

Figure 1. Cross flow of goods distribution

Determine how to solve the problem with the aim of minimizing transportation costs

Note: Node quantity is "+" (Supply) and quantity is "" (demand)

Arrows depict the flow of materials and costs.