Consider an assignment problem with $m$ machines and $t$ tasks. If $m < t$ and we allow some tasks to be not assigned, I found that we can assume dummy machines which do all the tasks with 0 cost.
I am looking for intuition to assign zero costs and not infinite costs to all the dummy machines. I referred to the following document - assignment problem. Page 6 and onwards talk about the unbalanced assignment problem which I did not understand well.
Any help will be appreciated.
Since it is an assignment problem (meaning, in particular, that every dummy machine gets exactly one task), the objective coefficients of the assignments to dummy machines do not matter, as long as they are all equal. Suppose every possible assignment of a task to a machine is given cost $c\in\Re$. Note that I make no assumption whether $c$ is positive or negative. Every feasible solution will have a contribution of $c(t - m)$ to the objective function from the dummy machines, and that is a constant. So the feasible solution that minimizes overall cost including the dummy machines is the feasible solution that minimizes total cost of assignments to real machines (which is equivalent to setting the cost to zero). A cost of zero is just a convenient choice.
Setting a cost of $\infty$ might cause some numerical adventures with solver software, as might using the largest double precision number possible for the cost rate.