The 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 Time No of Tellers Time No of Tellers 9 00 a m 10 00 a m 6 2 00 p m 3 00 p m 6 10 00 a m 11 00 a m 4 3 00 p m 4 00 p m 4 11 00 a m Noon 8 4 00 p m 5 00 p m 7 Noon 1 00 p m 10 5 00 p m 6 00 p m 6 1 00 p m 2 00 p m 9 6 00 p m 7 00 p m 6 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 ($32 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 Letx i number of full time employees coming on duty at the beginning of houriandy i number of part time employees coming on duty at the beginning of houri ) Min x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 x 9 y 9 Select your answer Item 13 Time (9 00 a m 10 00 a m ) x 9 x 10 y 9 y 10 Select your answer Item 19 Time (10 00 a m 11 00 a m ) x 9 x 10 x 11 y 9 y 10 y 11 Select your answer Item 27 Time (11 00 a m Noon ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 Select your answer Item 36 Time (Noon 1 00 p m ) x 10 x 11 y 10 y 11 y 12 y 1 Select your answer Item 44 Time (1 00 p m 2 00 p m ) x 9 x 11 y 11 y 12 y 1 y 2 Select your answer Item 52 Time (2 00 p m 3 00 p m ) x 9 x 10 y 12 y 1 y 2 y 3 Select your answer Item 60 Time (3 00 p m 4 00 p m ) x 9 x 10 x 11 y 1 y 2 y 3 Select your answer Item 68 Time (4 00 p m 5 00 p m ) x 10 x 11 y 2 y 3 Select your answer Item 74 Time (5 00 p m 6 00 p m ) x 11 y 3 Select your answer Item 78 Time (6 00 p m 7 00 p m ) x i ,y j 0 and integer fori 9, 10, 11 andj 9, 10, 11, 12, 1, 2, 3 (b) Solve the LP Relaxation of your model in part (a) If required, round your answers to the nearest whole number x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Total Cost $ (c) Solve for the optimal schedule of tellers Time No of Full time employees No of Part time employees 9 00 a m 10 00 a m 10 00 a m 11 00 a m 11 00 a m Noon Noon 1 00 p m 1 00 p m 2 00 p m 2 00 p m 3 00 p m 3 00 p m 4 00 p m 4 00 p m 5 00 p m 5 00 p m 6 00 p m 6 00 p m 7 00 p m Comment on the solution The input in the box below will not be graded, but may be reviewed and considered by your instructor (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 If required, round your answers to the nearest whole number The new optimal solution is as follows Time No of Full time employees No of Part time employees 9 00 a m 10 00 a m 10 00 a m 11 00 a m 11 00 a m Noon Noon 1 00 p m 1 00 p m 2 00 p m 2 00 p m 3 00 p m 3 00 p m 4 00 p m 4 00 p m 5 00 p m 5 00 p m 6 00 p m 6 00 p m 7 00 p m

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Question: The 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,

The 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.

Time	No. of Tellers	Time	No. of Tellers
9:00 a.m.-10:00 a.m.	6	2:00 p.m.-3:00 p.m.	6
10:00 a.m.-11:00 a.m.	4	3:00 p.m.-4:00 p.m.	4
11:00 a.m.-Noon	8	4:00 p.m.-5:00 p.m.	7
Noon-1:00 p.m.	10	5:00 p.m.-6:00 p.m.	6
1:00 p.m.-2:00 p.m.	9	6:00 p.m.-7:00 p.m.	6

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 ($32 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: Letx_i= number of full-time employees coming on duty at the beginning of houriandy_i= number of part-time employees coming on duty at the beginning of houri.)

Min

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

x₉

y₉

- Select your answer -<>=Item 13

Time (9:00 a.m.-10:00 a.m.)

x₉

x₁₀

y₉

y₁₀

- Select your answer -<>=Item 19

Time (10:00 a.m.-11:00 a.m.)

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

- Select your answer -<>=Item 27

Time (11:00 a.m.-Noon.)

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

- Select your answer -<>=Item 36

Time (Noon.-1:00 p.m.)

x₁₀

x₁₁

y₁₀

y₁₁

y₁₂

y₁

- Select your answer -<>=Item 44

Time (1:00 p.m.-2:00 p.m.)

x₉

x₁₁

y₁₁

y₁₂

y₁

y₂

- Select your answer -<>=Item 52

Time (2:00 p.m.-3:00 p.m.)

x₉

x₁₀

y₁₂

y₁

y₂

y₃

- Select your answer -<>=Item 60

Time (3:00 p.m.-4:00 p.m.)

x₉

x₁₀

x₁₁

y₁

y₂

y₃

- Select your answer -<>=Item 68

Time (4:00 p.m.-5:00 p.m.)

x₁₀

x₁₁

y₂

y₃

- Select your answer -<>=Item 74

Time (5:00 p.m.-6:00 p.m.)

x₁₁

y₃

- Select your answer -<>=Item 78

Time (6:00 p.m.-7:00 p.m.)

x_i,y_j 0 and integer fori= 9, 10, 11 andj= 9, 10, 11, 12, 1, 2, 3

(b)

Solve the LP Relaxation of your model in part (a).

If required, round your answers to the nearest whole number.

x₉
x₁₀
x₁₁
y₉
y₁₀
y₁₁
y₁₂
y₁
y₂
y₃

Total Cost: $

(c)

Solve for the optimal schedule of tellers.

Time	No. of Full-time employees	No. of Part-time employees
9:00 a.m.-10:00 a.m.
10:00 a.m.-11:00 a.m.
11:00 a.m.-Noon
Noon-1:00 p.m.
1:00 p.m.-2:00 p.m.
2:00 p.m.-3:00 p.m.
3:00 p.m.-4:00 p.m.
4:00 p.m.-5:00 p.m.
5:00 p.m.-6:00 p.m.
6:00 p.m.-7:00 p.m.

Comment on the solution.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

(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.

If required, round your answers to the nearest whole number.

The new optimal solution is as follows:

Time	No. of Full-time employees	No. of Part-time employees
9:00 a.m.-10:00 a.m.
10:00 a.m.-11:00 a.m.
11:00 a.m.-Noon
Noon-1:00 p.m.
1:00 p.m.-2:00 p.m.
2:00 p.m.-3:00 p.m.
3:00 p.m.-4:00 p.m.
4:00 p.m.-5:00 p.m.
5:00 p.m.-6:00 p.m.
6:00 p.m.-7:00 p.m.

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