$($Integer Model$)$ The Livewright Medical Supplies Company has a total of $12$ salespeople it wants to assign to three regionsthe South, the East, and the Midwest. A salesperson in the South earns $$600$ in profit per month for the company, a salesperson in the East earns $$540,$ and a salesperson in the Midwest earns $$375.$ The southern region can have a maximum assignment of five salespeople. The company has a total of $$750$ per day available for expenses for all $12$ salespeople. A sales person in the South has average expenses of $$80$ per day, a salesperson in the East has average expenses of $$70$ per day, and a salesperson in the Midwest has average daily expenses of $$50.$ 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.