$2.(25$ Points$)$ Consider the following capacitated facility location problem. There are N sites where we can open facilities to deliver J types of products to M customers. The fixed cost of opening a facility at site i is fi$.$ The available capacity for facility i$,$ if opened, is denoted by Ki$.$ All products use the same capacity, however, with different rates. Specifically, to produce one unit of product j$,$ the required unit of capacity is wj$.$ These rates are the same for all facilities. Other parameters cilj and dlj represent unit shipment cost for product j from facility i to customer l and demand for product j at customer l respectively.

a$)(13$ points$)$ Write down the mathematical programming model for the minimum cost facility location and allocation solution.

b$)(6$points$)$SupposethatafacilitycanproduceuptoP$($P