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 4 2 00 p m 3 00 p m 5 10 00 a m 11 00 a m 3 3 00 p m 4 00 p m 3 11 00 a m Noon 7 4 00 p m 5 00 p m 6 Noon 1 00 p m 11 5 00 p m 6 00 p m 4 1 00 p m 2 00 p m 7 6 00 p m 7 00 p m 7 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 Let x i number of full time employees coming on duty at the beginning of hour i and y j number of part time employees coming on duty at the beginning of hour j ) If your answer is zero enter 0 and if the constant is 1 it must be entered in the box Min x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 s t x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (9 00 a m 10 00 a m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (10 00 a m 11 00 a m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (11 00 a m Noon ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (Noon 1 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (1 00 p m 2 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (2 00 p m 3 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (3 00 p m 4 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (4 00 p m 5 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (5 00 p m 6 00 p m ) x 9 x 10 x 11 y 9 y 10 y 11 y 12 y 1 y 2 y 3 Time (6 00 p m 7 00 p m ) x i , y j 0 and integer for i 9, 10, 11 and j 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 If your answer is zero enter 0 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 If your answer is zero enter 0 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 (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 six 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 If your answer is zero enter 0 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 Total Cost $

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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.	4	2:00 p.m.-3:00 p.m.	5
10:00 a.m.-11:00 a.m.	3	3:00 p.m.-4:00 p.m.	3
11:00 a.m.-Noon	7	4:00 p.m.-5:00 p.m.	6
Noon-1:00 p.m.	11	5:00 p.m.-6:00 p.m.	4
1:00 p.m.-2:00 p.m.	7	6:00 p.m.-7:00 p.m.	7

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: Let x_i = number of full-time employees coming on duty at the beginning of hour i and y_j = number of part-time employees coming on duty at the beginning of hour j.)

If your answer is zero enter 0 and if the constant is "1" it must be entered in the box.

Min

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

s.t.

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x₉

x₁₀

x₁₁

y₉

y₁₀

y₁₁

y₁₂

y₁

y₂

y₃

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

x_i, y_j 0 and integer for i = 9, 10, 11 and j = 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. If your answer is zero enter 0.

x₉
x₁₀
x₁₁
y₉
y₁₀
y₁₁
y₁₂
y₁
y₂
y₃

Total Cost: $

(c)

Solve for the optimal schedule of tellers.

If your answer is zero enter 0.

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.

(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 six 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. If your answer is zero enter 0.

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.

Total Cost: $

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