Epsilon Airlines services predominantly the eastern and southeastern United States. The vast majority of Epsilon’s customers make reservations through Epsilon’s website, but a small percentage of customers make reservations via phone. Epsilon employs call center personnel to handle these reservations, deal with website reservation system problems, and rebook flights for customers whose plans have changed or whose travel is disrupted.

Staffing the call center appropriately is a challenge for Epsilon’s management team. Having too many employees on hand is a waste of money, but having too few results in poor customer service and the potential loss of customers. Epsilon analysts have estimated the minimum number of call center employees needed by day of the week for the upcoming vacation season (June, July, and the first two weeks of August). These estimates are as follows:

- **Monday:** 90
- **Tuesday:** 45
- **Wednesday:** 60
- **Thursday:** 50
- **Friday:** 90
- **Saturday:** 70
- **Sunday:** 45

Call center employees work for five consecutive days and then have two consecutive days off. An employee may start work on any day of the week. Each call center employee receives the same salary. Assume that the schedule cycles and ignore start-up and stopping of the schedule.

Develop a model that will minimize the total number of call center employees needed to meet the minimum requirements.

Let \(X_i\) = the number of call center employees who start work on day \(i\) (where \(i = 1\) for Monday, \(i = 2\) for Tuesday, etc.).

Minimize:

\[ X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 \]

Subject to:

\[ X_1 + X_4 + X_5 + X_6 + X_7 \geq 90 \]

\[ X_1 + X_2 + X_5 + X_6 + X_7 \geq 45 \]

\[ X_1 + X_2 + X_3 + X_6 + X_7 \geq 60 \]

\[ X_1 + X_2 + X_3 + X_4 + X_7 \geq 50 \]

\[ X_1 + X_2 + X_3 + X_4 + X_5 \geq 90 \]

\[ X_2 + X_3 + X_4 + X_5 + X_6 \geq 70 \]

\[ X_3 + X_4 + X_5 + X_6 + X_7 \geq 45 \]

\[ X_1, X_2, X_3, X_4, X_5, X_6, X_7 \geq 0 \]

Find the optimal solution:

- \(X_1 = \)
- \(X_2 = \)
- \(X_3 = \)
- \(X_4 = \)
- \(X_5 = \)
- \(X_6 = \)
- \(X_7 = \)
- Total Number of Employees =

Give the number of call center employees that exceed the minimum required:

- Excess employees:
- **Monday:**
- **Tuesday:**
- **Wednesday:**
- **Thursday:**
- **Friday:**
- **Saturday:**
- **Sunday:**