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 $p(p=1,2,$dots,$N)$ can be

produced at a cost $c_{p,i,t}$ dollars by production source $i(i=1,2,$dots,$M)$ in period $t(t=1,2,$dots,$T).$

Production source i has a scrap rate $r_i$ and its production capacity for product $p$ in period $t$ is

Cap ${?}_{p,i,t}$ 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 $w(w=1,2,$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.