I am wondering what examples you like to give when introducing linear programming, where the examples are not clearly better suited as integer linear programs. I would like a few examples where we can discuss the motivating story, then formalize and write it down as an LP.
- I can motivate production problems as LPs, when I am careful to phrase everything in terms of rates: rate of resources consumed per day, rate of products produced per day, maximizing rate of profit per day. This way, fractional solutions make sense. Same for their duals (min-cost procurement).
- I will use zero-sum games as a more intricate example.
- I think scheduling and assignment problems are generally better as ILPs.
- Flows, paths, and matchings deserve their own separate discussion in my opinion, due to the Integrality Theorem.
It seems like most practically motivating applications are scheduling and assignment problems that really should be ILPs. What are your favorite motivating LP applications? I also appreciate thoughts about how to navigate this question pedagogically, for example if your approach is to just motivate ILPs and treat LPs as a means to an end.