6. MODIFIED VERSION of Chapter 13, Problem 15 (Rather than answering parts a through d as shown in the textbook, answer the following for this problem: a. Model and solve the problem as described in the textbook to minimize the total staffing cost. What is the cost of this optimal solution? How many full-time employees are required in this solution? How many part-time employees are required? b. Add constraints to insure that at least one full-time employee is on duty at all times and that the staff for this workday includes at least 5 full-time employees. What is the cost of this optimal solution? How many full-time employees are required in this solution? How many part-time employees are required?

(Hints for Chapter 13, Problem 15: Your decision variables will be how many full-time employees to start work in each hour and how many part-time employees to start work in each hour. Your constraints will be based on the number of tellers required to work in each hour time slot. Consider the time slot of 10AM - 11AM. You require 4 bank tellers to be working during this time. The number of bank tellers working 10AM-11AM is calculated as: full-time employees who begin their day at 9AM + full-time employees who begin their day at 10AM + part-time employees who begin their day at 9AM + part-time employees who begin their day at 10AM. This total must be >= 4. You will need to create similar constraints for each hour time slot.)

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The Northshore Bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank including adequate customer service, employee breaks, and so on. On Fridays, the bank is open from 9:00 a.m. to 7:00 p.m. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here: Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning on the hour. Considering salary and fringe benefits, full-time employees cost the bank $15 per hour (\$105 a day), and part-time employees cost the bank $8 per hour ( $22 per day). a. Formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. (Hint: Let xi5 number of full-time employees coming on duty at the beginning of hour i and yi5 number of part-time employees coming on duty at the beginning of hour i.) b. Solve the LP Relaxation of your model in part a. c. Solve your model in part a for the optimal schedule of tellers. Comment on the solution. d. After reviewing the solution to part c, the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full-time employee is on duty at all times and that there is a staff of at least five full-time employees. Revise your model to incorporate these additional requirements, and solve for the optimal solution