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The bank is working to develop an efficient work schedule for full-time and part-time tellers. The...

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. 7
10:00 a.m.-11:00 a.m. 5 3:00 p.m.-4:00 p.m. 6
11:00 a.m.-Noon 7 4:00 p.m.-5:00 p.m. 8
Noon-1:00 p.m. 11 5:00 p.m.-6:00 p.m. 7
1:00 p.m.-2:00 p.m. 9 6:00 p.m.-7:00 p.m. 5

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 xi = number of full-time employees coming on duty at the beginning of hour i and yj = 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 - Select your answer -≤≥<>=Item 21 Time (9:00 a.m.-10:00 a.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 33 Time (10:00 a.m.-11:00 a.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 45 Time (11:00 a.m.-Noon.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 57 Time (Noon.-1:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 69 Time (1:00 p.m.-2:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 81 Time (2:00 p.m.-3:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 93 Time (3:00 p.m.-4:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 105 Time (4:00 p.m.-5:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 117 Time (5:00 p.m.-6:00 p.m.)
x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 - Select your answer -≤≥<>=Item 129 Time (6:00 p.m.-7:00 p.m.)
xi, yj ≥ 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.
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 four 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: $
business   Operations-Management   
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Homework Answers

Answer #1

(a)

Integer programming model is following:

Min 105x9 + 105x10 + 105x11 + 32y9 + 32y10 + 32y11 + 32y12 + 32y1 + 32y2 + 32y3
s.t. 1x9 + 0x10 + 0x11 + 1y9 + 0y10 + 0y11 + 0y12 + 0y1 + 0y2 + 0y3 >= 6 Time (9:00 a.m.-10:00 a.m.)
1x9 + 1x10 + 0x11 + 1y9 + 1y10 + 0y11 + 0y12 + 0y1 + 0y2 + 0y3 >= 5 Time (10:00 a.m.-11:00 a.m.)
1x9 + 1x10 + 1x11 + 1y9 + 1y10 + 1y11 + 0y12 + 0y1 + 0y2 + 0y3 >= 7 Time (11:00 a.m.-Noon.)
1x9 + 1x10 + 1x11 + 1y9 + 1y10 + 1y11 + 1y12 + 0y1 + 0y2 + 0y3 >= 11 Time (Noon.-1:00 p.m.)
0x9 + 1x10 + 1x11 + 0y9 + 1y10 + 1y11 + 1y12 + 1y1 + 0y2 + 0y3 >= 9 Time (1:00 p.m.-2:00 p.m.)
1x9 + 0x10 + 1x11 + 0y9 + 0y10 + 1y11 + 1y12 + 1y1 + 1y2 + 0y3 >= 7 Time (2:00 p.m.-3:00 p.m.)
1x9 + 1x10 + 0x11 + 0y9 + 0y10 + 0y11 + 1y12 + 1y1 + 1y2 + 1y3 >= 6 Time (3:00 p.m.-4:00 p.m.)
1x9 + 1x10 + 1x11 + 0y9 + 0y10 + 0y11 + 0y12 + 1y1 + 1y2 + 1y3 >= 8 Time (4:00 p.m.-5:00 p.m.)
0x9 + 1x10 + 1x11 + 0y9 + 0y10 + 0y11 + 0y12 + 0y1 + 1y2 + 1y3 >= 7 Time (5:00 p.m.-6:00 p.m.)
0x9 + 0x10 + 1x11 + 0y9 + 0y10 + 0y11 + 0y12 + 0y1 + 0y2 + 1y3 >= 5 Time (6:00 p.m.-7:00 p.m.)

(b)

Create spreadsheet model as follows:

EXCEL FORMULAS:

L3 =SUMPRODUCT(B3:K3,$B$19:$K$19) copy to L6:L15

Enter Solver Parameters as follows:

Click Solve to generate the solution:

Solution:

x9 0
x10 0
x11 0
y9 6
y10 4
y11 4
y12 0
y1 1
y2 2
y3 5

(c)

Time No. of Full-time No. of Part-time
employees employees
9:00 a.m.-10:00 a.m. 0 6
10:00 a.m.-11:00 a.m. 0 10
11:00 a.m.-Noon 0 14
Noon-1:00 p.m. 0 14
1:00 p.m.-2:00 p.m. 0 9
2:00 p.m.-3:00 p.m. 0 7
3:00 p.m.-4:00 p.m. 0 8
4:00 p.m.-5:00 p.m. 0 8
5:00 p.m.-6:00 p.m. 0 7
6:00 p.m.-7:00 p.m. 0 5

(d)

The additional requirement are formulated in the model and Solver Parameters as shown below:

ADDITIONAL EXCEL FORMULAS IN COLUMN O:

O6 =SUMPRODUCT(B6:D6,$B$19:$D$19) copy to O6:O15


Click Solve to generate the solution:

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