Brewer Services contracts with outsourcing partners to handle various customer service functions. The customer service department is open Monday through Friday from 8 a.m. to 5 p.m. Calls vary over the course of a typical day. Based on a study of call volumes provided by one of the firm's partners, the minimum number of staff needed each hour of the day are:

Mr. Brewer wants to hire some permanent employees and staff the remaining requirements using part-time employees who work four-hour shifts (four consecutive hours starting as early as 8 a.m. or as late as 1 p.m.). Develop and solve an integer optimization model to answer the following questions.

a. Suppose Mr. Brewer uses five permanent employees. What is the minimum number of part-time employees he will need for each four-hour shift to ensure meeting the staffing requirements?

b. What if he uses 11 permanent employees?

c. Investigate other possible numbers of permanent employees between 5 and 15, compare the differences, and make a recommendation.