Consider a supply chain network with three potential sites for warehouses and eight retailer regions. The fixed costs $(initial$ capital investment$)of$ locating warehouses $at$ the three sites are given below:

$*$ Site $1$: $$100,000$

$*$ Site $2$: $$80,000$

$*$ Site $3$: $$110,000$

The capacities $of$ the three sites are $100,000,80,000$ and $125,000,$ respectively. The retailer demands are $20,000$ for the first four retailers and $25,000$ for the remaining. The unit operating cost $is$ not considered. The unit transportation costs $($$$)$ are given $in$ the following table.

Formulate the optimization problem as a linear integer program to determine the optimal location and distribution plan that will minimize the total cost $($initial investment$+$ transportation costs$).$ Assume dedicated warehouses, that is$,$ each retailer has to be supplied by exactly one warehouse. You must define your variables clearly, write out the constraints, explaining briefly the significance of each, and write the objective function.