I have set of machines with varying productivity.
I want put the machines in different groups so that the groups have approximately equal productivity.
Lets say, we have $M$ machines.
and we want to divide them into $G$ groups of equal size.
Size of a group, $S=M/G$.
The productivity of machine $m$ is given by $P_m\ge 0$ (some machine may have zero productivity).
What is an easy LP formulation?
$\textbf{Tried...}$
Let $x_{g,m}$ be a binary indicator. If $x_{g,m}=1$, machine $m$ belongs to group $g$.
So, we have $$\sum_{m=1}^Mx_{g,m}=S, \forall g$$
The productivity of group $g$ is given by
$$T_g=\sum_{m=1}^MP_m*x_{g,m},\forall g$$
I prefer $$T_1\approx T_2 \approx T_3 \approx \cdots\approx T_G$$
What would be a good objective function?
Let $\phi$ is a variable.
$$\phi=\frac{T_a}{T_b}$$
maximize $\phi$?
I am looking for an efficient implementation...
$\bf{EDIT}$
All the solutions proposed are hard to solve. I am rather looking for some greedy heuristic approach to solve this problem.
This is what I have tried so far...
Choose $G$ machines (machines with the G largest productivity) as the group head.
Then I follow an iterative steps for the the remaining $M-G$ machines. For a given machine, it is attached to a group that has the smallest productivity.
Do you think it a good heuristic?
Any suggestion with better heuristic?