Problem 9-09 (Algorithmic)

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 phones. Epsilon employs call center personnel to handle these reservations and to deal with website reservation system problems and for the rebooking of 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 very 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:

Day	Minimum Number of Employees Needed
Monday	50
Tuesday	90
Wednesday	45
Thursday	65
Friday	75
Saturday	65
Sunday	45

The 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	Xi = the number of call center employees who start work on day i
	(i = 1 = Monday, i = 2 = Tuesday...)

Min	X1 +	X2 +	X3 +	X4 +	X5 +	X6 +	X7
s.t.
	X1 +			X4+	X5+	X6+	X7	=	fill in the blank 2
	X1 +	X2+			X5+	X6+	X7	=	fill in the blank 4
	X1 +	X2+	X3+			X6+	X7	=	fill in the blank 6
	X1 +	X2+	X3+	X4+			X7	=	fill in the blank 8
	X1 +	X2+	X3+	X4+	X5			=	fill in the blank 10
		X2 +	X3+	X4+	X5+	X6		=	fill in the blank 12
			X3 +	X4+	X5+	X6+	X7	=	fill in the blank 14
	X1,	X2,	X3,	X4,	X5,	X6,	X7		0

Find the optimal solution.

X1	=	fill in the blank 15
X2	=	fill in the blank 16
X3	=	fill in the blank 17
X4	=	fill in the blank 18
X5	=	fill in the blank 19
X6	=	fill in the blank 20
X7	=	fill in the blank 21

Total Number of Employees = fill in the blank 22

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

Excess employees:

Monday	=	fill in the blank 23
Tuesday	=	fill in the blank 24
Wednesday	=	fill in the blank 25
Thursday	=	fill in the blank 26
Friday	=	fill in the blank 27
Saturday	=	fill in the blank 28
Sunday	=	fill in the blank 29