Chicago is known for its unique take on the hot dog $($as seen in Figure $1),$ and the family$-$owned

regional chain Frankly Delicious has recently found its niche in the market. While it prides itself

on quality and fresh ingredients, the company has determined that transporting products from its

headquarters in downtown Chicago to its many restaurants in the northeastern region of Illinois

on a daily basis is becoming too costly$.$ As a result, the company has made the decision to open

regional warehouses to house products that it can then distribute to its retail locations. Each

warehouse will be based in one of the $20$ counties in which Frankly Delicious has restaurants and

will provide products for the operations in that county and all adjacent counties that are directly

accessible via a major roadway. It is possible for a county to be served by two or more warehouses,

in which case they will share operational responsibility.

Frankly Delicious$$s management has chosen to partner with you to determine where to place its

warehouses, with the goal of minimizing the total amount it spends on the monthly operations of

the warehouses. Expenditures are dependent on the county in which the warehouse is located and

are based on land usage and$/$or rent, the number of stores the warehouse would potentially serve,

salaries for warehouse employees, and other factors. A map of the region in which Frankly Delicious

operates and a table of counties, which includes information on county adjacency and associated

warehouse costs, are provided on the next page as Figure $2$ and Table $1,$ respectively. $($Note that

some counties border each other but are not directly accessible via a major roadway. The table

indicates counties as adjacent if and only if they are accessible as such.$)$ In summary, you must

develop a linear programming model for this optimization problem, perform an appropriate analysis

using Excel, recommend the counties in which Frankly Delicious should place its warehouses, and

present a formal business report discussing your findings and conclusions.