FCC Company has $M$ production sources $($e$.$g$.,$ production plants$)$ for satisfying the demand of $N$ different products in each of $T$ time periods. One unit of product $)=(1,2,$dots,$N$ can be produced at a cost $c_{p,i,t}$ dollars by production source $)=(1,2,$dots,$M$ in period $)=(1,2,$dots,$T.$ Production source i has a scrap rate $r_i$ and its production capacity for product $p$ in period $t$ is $Ca{p}_{p,i,t}$ units. When production source $i$ is used in period $t,$ then a fixed $cost{A}_{i,t},$ which is independent from the product and the number of units produced by this source, incurs. All products produced by the production sources are distributed to $L$ warehouses located at different cities. $s_{i,w,p,t}$ is the shipment cost of one unit of product $p$ from production source $i$ to warehouse $)=(1,2,$dots,$L$ in period $t.D_{w,p,t}$ is the demand $($in units$)$ given by warehouse $w$ for product $p$ in period $t.$ The initial inventory level of each product is zero, and product $p$ may be stored from period $t$ to period $t+1$ at a cost $h_{p,t}$ dollars per unit. The storage space requirement $($in volume$)$ of product $p$ is $v_p{m}^3,$ and the storage capacity $($in volume$)$ of warehouse $w$ in any month is $M{S}_w{m}^3.$ Shortages of product $p$ in period $t$ are backordered at a cost of ${}_{p,t}$ dollars per unit; but shortages in the last period are not allowed.

Formulate the problem as a mixed$-$integer linear programming $($MILP$)$ model. That is$,$ define the decision variables explicitly, write the mathematical expressions for the objective function and constraints in CLOSED FORM, and explain them clearly.