$5\mathrm{.}10($Adapted from Srinivasan, $2010)$ Consider a supply chain network with three potential sites for warehouses and eight

retailer regions. The fixed costs of locating warehouses at the three sites are given as follows:

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 transportation costs $($$$)$ are given in Table $5\mathrm{.}17$:

TABLE $5\mathrm{.}17$

Data for Exercise $5\mathrm{.}10$

a$.$ Formulate a mixed integer linear program to determine the optimal location and distribution plan that will minimize

the total cost. You must define your variables clearly, write out the constraints, explaining briefly the significance of

each and write the objective function. Assume that the retailers can receive supply from multiple sites. Solve using

any optimization software. Write down the optimal solution.

b$.$ Reformulate the optimization problem as a linear integer program, assuming dedicated warehouses, that is$,$ each

retailer has to be supplied by exactly one warehouse. Solve the integer programming model. What is the new optimal

solution?