$(20$ pts$)$ Consider a scenario where C is the set of clients and S is the set of servers. Each client i in set C can be served by server j in set S with a service cost cij $.$ Because of the emerging issue of carbon emission and global warming at most k servers can stay functional at a time. Our objective is to find the optimal set of k servers so that each client can be served while minimizing the total service cost. Let OPT be the sum of the service costs over all clients for the optimal solution.

$1.$ Design an LP representing the problem. $[$clearly describe in language the variables, describe the constraints and objective function$]$

$2.$ Write the dual L