Problem $3$ Uncapacitated Facility Location Problem $(15$ points$)$

Given a set of potential depots $N=\{1,$dots,$n\}$ and a set $M=\{1,$dots,$m\}$ of clients, suppose there is a fixed cost $f_j$ associated with the opening of depot $j,$ and a transportation cost $c_{ij}$ if all of client $i\text{'}$s order is delivered from depot $j.$ The problem is to decide which depots to open and which depot serves each client so as to minimize the sum of the fixed and transportation costs.