What is the 'best' order for a list of items to minimising a cost function that is dependent on their order

Question

I'm trying to plan a work out with sit-ups/press ups etc. Each exercise will fatigue a muscle group by 'x' amount. After fatiguing a muscle, that muscle will recover (for simplicity linearly) at a rate of 'r' amount per second.

I want to order a set of exercises to minimise the maximum fatigue for any muscle at any point.

This feels like it may be similar to a standard problem which has been solved. Could you please point in this known problem?

Answer 1

If we're talking about 14 exercises, the best option would probably be to use the branch and bound method. Let's go through all the permutations recursively. If our current maximum is bigger than the upper bound estimate we had before, we should not continue this permutation, because it is guaranteed to be worse. If we calculated one full permutation, we should update the upper bound. The total number of states is
14! = 8e10, but a good number of them would be cut off.

It also helps to have a decent greedy solution to use as the initial estimate.

I'm also pretty sure that no polynomial solution exists because the function we're trying to optimize is too complicated.