Problem $1$

There are $m$ potential sites for building new facilities that ship a certain amounts of a single product to $n$ warehouses. If you construct a new facility on site jin$[m],$ there will be a yearly leasing cost $f_j$ and this new facility can only produce up to $M_j$ amounts of the product. It costs $c_{ij}$ to ship one unit of the product from the constructed facility jin$[m],$ to a warehouse iin$[n].$ For each warehouse iin$[n],$ there is demand $d_i$ of the product. Formulate a mathematical model that aims to find where to build the new facilities to satisfy the supply and demand constraints while minimizing the total costs $($including shipping and building costs$).$