Very often we want to make decisions over a time horizon when decisions made for one time period may impact decisions for other periods.
A good example are production problems where there is the option to hold inventory from one period to the next. The Pastesian and the Frisbee Square use cases illustrate this situation very well.
One thing that become apparent as soon as we start to solve these problems is that there is an underlying network where nodes are period in time and arcs are links between there period--it's typically helpful to picture that network when solving the problem. As a result, flow balance constraints always show up in the formulation of these problems.
Transportation problems, including routing, will often have a time component too. For example, when there are time windows to pick up or delivery. In this case, the underlying network spans in space and time.
Mip Solar manufactures solar panels. One of the production stages is to heat the panel so that different layers of the material melt together. This heating process occurs inside the so-called laminators, and we assume it lasts 10 seconds.
Panels move from the previous production stage to laminators over conveyors. Each conveyor operates independently, meaning that a conveyor can move a panel back and forth at any time. Additionally, a panel can stay on a conveyor for as much time as needed, and it takes 1 second for a panel to move from one conveyor to another.
The planers at Mip Solar want to maximize the throughput of solar panels.
OBS: This is a simplified version of a problem that Mip Wise solved for one of its clients. We presented it here just to illustrate a relevant real-world application of the topic being studied in this module. But due to contractual reasons, we won't solve this problem.
Let's revisit the MipEx use case.
This time, we assume that each commodity has an arrival time, the hour of the day that the commodity is ready to leave its origin station, and a due time, the hour of the day that the commodity must be at its destination station. All this information is provided in the Commodities table.
The transit distance, transit time, transportation cost, and the maximum number of trucks that can travel on each arc in a day are given in the Transportation Lanes table.
The goal is to deliver all commodities on time at minimum cost.
It's important to note that the cost of a truck is the same whether the truck is full of partially full. On the other hand, commodities can stay at a site for as long as needed to leverage loads consolidations.
In the next section, we will study scheduling problems.