2D grid placement problem
If I understand your problem correctly, it seems like you're looking to examine a furthest-point weighted Voronoi diagram under the Manhattan ($\ell_1$) distance. The transformation is as follows. 

For each point $p_i$ on the grid, define the distance function $$d_i(x) = \max(0, \|x - p_i\|_1 - (t_{n+1}-t_i))$$ Then $d_i(x)$ is the (truncated) distance of $x$ from the ball of radius $t_{n+1}-t_i$ around $p_i$ under the $\ell_1$ distance.  

You want to find 
$$ x^* = \arg \min \max_i \ d_i(x) $$
which can be obtained by computing the furthest-point weighted Voronoi diagram under the $\ell_1$ distance (the weighted part is because of the offset associated with the $t_i$). 

If you want code, then I'm not sure where you might find it, but this site has an amazing collection of Voronoi demos written in Java (including furthest point Manhattan distance) and you might be able to adapt it for for your needs.