I am not sure if I used the corret terminology. I think the problem is a multi-stage combinatorial optimization problem. The problem is like this:
There is a dynamic equation $$ S_{k+1} = f(S_k,\pi_k) $$ where $S_k$ is a multi-dimension vector, $f$ is a function, $\pi_k$ the permutation of the elements in set $A=\{a_1,a_2,\dots,a_n\}$. There are $m$ stages. Given $S_0$, select an order from $\pi_A$. It will generate the next state $S_1$. At the stage $m$, we have $S_m$. We have an objective function $g$, which is a function of the state $S$. We would like to optimize $g(S_m)$.
The problem is NP-hard. At each stage, there are $n!$ possible solutions, and the total solution is $n!^{m}$.
I would like to know what is the terminology for such problems. Are there any efficient algorithms? What are the accuracy and time complexity?