A medical supply company has a total of $12$ salespeople available in any of its $3$ primary service

regions $-$ the East, the South, and the Midwest during the next month. A salesperson in the East earns

$$600$ in profit per month for the company, a salesperson in the South earns $$540,$ and a salesperson in

the Midwest earns $$375.$ The Eastern region requires at least $3$ salespeople in the upcoming month.

The company has a total of $$750$ per day available for expenses for the sales team. A salesperson in

the East has average expenses of $$80$ per day, a salesperson in the South has average expenses of

$$100$ per day, and a salesperson in the Midwest has average daily expenses of $$50.$ The minimum

number of salespeople assigned in the upcoming month is $10$ while the maximum assigned is $12.$ The

Southern region requires between $3$ and $6$ salespeople while the Midwest region requires $2$ or more.

The company wants to determine the number of salespeople to assign to each region to maximize

profit.

a$)$ Formulate an integer programming model for this problem.

b$)$ Solve this model by using the computer.

c$)$ What is the difference in profit for the model using integer constraints compared to the model

without the requirement of integers?