Integer linear programs

Imagine you have a bunch of apples and bananas, and you need to figure out how to pack them into boxes. You have a limited number of boxes, and each box can only hold a certain number of apples and bananas.

Now, let's pretend that the number of apples and bananas you have is represented by x and y, respectively. The number of boxes you have is represented by z.

To make things even simpler, let's say that each box can only hold one apple or banana. So if a box has an apple in it, it can't have a banana, and vice versa.

An integer linear program is a fancy way of saying that you're trying to find the best way to pack those apples and bananas into boxes, while following certain rules.

Here's the catch: you don't just want to pack them any old way. You want to pack them in a way that maximizes or minimizes something. Maybe you want to minimize the number of boxes you use, or maybe you want to maximize the number of apples you can fit.

So you need to come up with a set of rules that will guide you as you pack those fruits into boxes. These rules are called constraints.

For example, let's say you have 6 apples and 5 bananas, and you have 8 boxes. You want to pack them in a way that maximizes the number of apples you can fit.

Your constraints might look like this:
- Each box can hold either one apple or one banana, but not both.
- The total number of apples packed cannot be greater than 6.
- The total number of bananas packed cannot be greater than 5.
- You have to use all 8 boxes.

Now you can start packing. You might put an apple in the first box, a banana in the second, an apple in the third, and so on.

Eventually, you'll reach a point where you can't fit any more apples into the boxes without breaking your constraints. Maybe you've packed 4 apples so far, and you're out of boxes that can hold apples.

At that point, you know you've found the best solution you can, given your constraints. That's your integer linear program!