I have a list of objects with $N$-dimensional criteria (actually hotels in my case). Now I'd like to optimize the order of the criteria by which I filter to spent the least time on some evaluation process per object. Each property $i$ takes a fixed time $t_i$ to evaluate per object, while it will reduce the total amount of considered objects by a factor $q_i$. Which order will give me the least total time of evaluation?
I got as far as saying I need to minimize
$((t_{N}q_{N-1}+t_{N-1})q_{N-2}+t_{N-2})q_{N-3}+t_{N-3}\dotsb\to\text{min}$
where the indices now represent a new fixed permutation! Which permutation would that be?
Any ideas?
EDIT: If it's any help one can of course also write
$\begin{pmatrix} q_1 & t_1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} q_2 & t_2 \\ 0 & 1 \end{pmatrix}\dotsb\begin{pmatrix} q_N & t_N \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0\\ 1 \end{pmatrix}\to\text{min}$
I'm hoping for an algorithm that assigns "efficiencies" to all criteria and states that I should order all steps by efficiency.