I am trying to model $M$ sellers offering $N$ items at different prices. Each seller charges a different amount of shipping depending on how many items are purchased from them. For example, seller $s_1$ charges:
- \$0 for 0 items
- \$2 for 1 item
- \$3 for 2-5 items
- etc.
The other sellers will use the same quantity breakpoints, but could charge different amounts. Each item needs to be purchased once and only once, and I want to minimize the total cost of the items plus shipping. I'm just stuck on how to model the shipping, or even the correct terms to search for online.
The shipping cost for a given vendor is a step function of the volume. You can search for "step function" and "integer program". Perhaps one of my old blog posts might help: "Step Functions and Mixed Functions". It's a bit more general than what you have (yours is a pure step function, not a mixture of step and linear parts).