The Northside 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 as follows. Time No. of Tellers 9:00 a.m. – 10:00 a.m. 6 10:00 a.m. – 11:00 a.m. 4 11:00 a.m. – Noon 8 Noon – 1:00 p.m. 10 1:00 p.m. – 2:00 p.m. 9 2:00 p.m. – 3:00 p.m. 6 3:00 p.m. – 4:00 p.m. 4 4:00 p.m. – 5:00 p.m. 7 5:00 p.m. – 6:00 p.m. 6 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 $16 per hour ($112 a day), and part-time employees cost the bank $9 per hour ($36 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 (in $ per day). (Let xi = number of full-time employees coming on duty at the beginning of hour i and yi = number of part-time employees coming on duty at the beginning of hour i where i = 9, 10, 11, 12, 1, 2, 3, 4, 5, 6. Assume that each employee must be able to finish their complete shift before the bank closes.) Min s.t.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. (b) Solve the LP relaxation of your model in part (a). Solving the LP relaxation tells us to use a total of full-time employees and part-time employees at a cost of $ per day. (c) Solve your model in part (a) for the optimal schedule of tellers. Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day. Comment on the solution. This solution may not be ideal for the Northside Bank since the cost per der day is over $1,000.This solution seems very cost effective for the Northside Bank since the cost per day is below $500. This solution may not be ideal for the Northside Bank since it only makes use of part-time employees.This solution may not be ideal for the Northside Bank since it only makes use of full-time employees.This solution seems reasonable since it requires the Northside Bank to use a near even mix of part-time and full-time employees. (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 on Fridays. Revise your model to incorporate these additional requirements. In addition to the constraints from part (a), what additional constraints should be added to the integer programming model? Full-Time during 9:00 a.m. – 10:00 a.m. Full-Time during 10:00 a.m. – 11:00 a.m. Full-Time during 11:00 a.m. – Noon Full-Time during Noon – 1:00 p.m. Full-Time during 1:00 p.m. – 2:00 p.m. Full-Time during 2:00 p.m. – 3:00 p.m. Full-Time during 3:00 p.m. – 4:00 p.m. Full-Time during 4:00 p.m. – 5:00 p.m. Full-Time during 5:00 p.m. – 6:00 p.m. Full-Time during 6:00 p.m. – 7:00 p.m. Total Full-Time Employees Solve for the optimal solution. Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day.
The Northside 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 as follows. Time No. of Tellers 9:00 a.m. – 10:00 a.m. 6 10:00 a.m. – 11:00 a.m. 4 11:00 a.m. – Noon 8 Noon – 1:00 p.m. 10 1:00 p.m. – 2:00 p.m. 9 2:00 p.m. – 3:00 p.m. 6 3:00 p.m. – 4:00 p.m. 4 4:00 p.m. – 5:00 p.m. 7 5:00 p.m. – 6:00 p.m. 6 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 $16 per hour ($112 a day), and part-time employees cost the bank $9 per hour ($36 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 (in $ per day). (Let xi = number of full-time employees coming on duty at the beginning of hour i and yi = number of part-time employees coming on duty at the beginning of hour i where i = 9, 10, 11, 12, 1, 2, 3, 4, 5, 6. Assume that each employee must be able to finish their complete shift before the bank closes.) Min s.t.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. (b) Solve the LP relaxation of your model in part (a). Solving the LP relaxation tells us to use a total of full-time employees and part-time employees at a cost of $ per day. (c) Solve your model in part (a) for the optimal schedule of tellers. Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day. Comment on the solution. This solution may not be ideal for the Northside Bank since the cost per der day is over $1,000.This solution seems very cost effective for the Northside Bank since the cost per day is below $500. This solution may not be ideal for the Northside Bank since it only makes use of part-time employees.This solution may not be ideal for the Northside Bank since it only makes use of full-time employees.This solution seems reasonable since it requires the Northside Bank to use a near even mix of part-time and full-time employees. (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 on Fridays. Revise your model to incorporate these additional requirements. In addition to the constraints from part (a), what additional constraints should be added to the integer programming model? Full-Time during 9:00 a.m. – 10:00 a.m. Full-Time during 10:00 a.m. – 11:00 a.m. Full-Time during 11:00 a.m. – Noon Full-Time during Noon – 1:00 p.m. Full-Time during 1:00 p.m. – 2:00 p.m. Full-Time during 2:00 p.m. – 3:00 p.m. Full-Time during 3:00 p.m. – 4:00 p.m. Full-Time during 4:00 p.m. – 5:00 p.m. Full-Time during 5:00 p.m. – 6:00 p.m. Full-Time during 6:00 p.m. – 7:00 p.m. Total Full-Time Employees Solve for the optimal solution. Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter12: Queueing Models
Section: Chapter Questions
Problem 59P
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The Northside 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 as follows.
| Time | No. of Tellers |
|---|---|
| 9:00 a.m. – 10:00 a.m. | 6 |
| 10:00 a.m. – 11:00 a.m. | 4 |
| 11:00 a.m. – Noon | 8 |
| Noon – 1:00 p.m. | 10 |
| 1:00 p.m. – 2:00 p.m. | 9 |
| 2:00 p.m. – 3:00 p.m. | 6 |
| 3:00 p.m. – 4:00 p.m. | 4 |
| 4:00 p.m. – 5:00 p.m. | 7 |
| 5:00 p.m. – 6:00 p.m. | 6 |
| 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 $16 per hour ($112 a day), and part-time employees cost the bank $9 per hour ($36 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 (in $ per day). (Let
xi
= number of full-time employees coming on duty at the beginning of hour i and
yi
= number of part-time employees coming on duty at the beginning of hour i where
i = 9, 10, 11, 12, 1, 2, 3, 4, 5, 6.
Assume that each employee must be able to finish their complete shift before the bank closes.)Min
s.t.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.
(b)
Solve the LP relaxation of your model in part (a).
Solving the LP relaxation tells us to use a total of full-time employees and part-time employees at a cost of $ per day.
(c)
Solve your model in part (a) for the optimal schedule of tellers.
Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day.
Comment on the solution.
This solution may not be ideal for the Northside Bank since the cost per der day is over $1,000.This solution seems very cost effective for the Northside Bank since the cost per day is below $500. This solution may not be ideal for the Northside Bank since it only makes use of part-time employees.This solution may not be ideal for the Northside Bank since it only makes use of full-time employees.This solution seems reasonable since it requires the Northside Bank to use a near even mix of part-time and full-time employees.
(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 on Fridays. Revise your model to incorporate these additional requirements.
In addition to the constraints from part (a), what additional constraints should be added to the integer programming model?
Full-Time during 9:00 a.m. – 10:00 a.m.
Full-Time during 10:00 a.m. – 11:00 a.m.
Full-Time during 11:00 a.m. – Noon
Full-Time during Noon – 1:00 p.m.
Full-Time during 1:00 p.m. – 2:00 p.m.
Full-Time during 2:00 p.m. – 3:00 p.m.
Full-Time during 3:00 p.m. – 4:00 p.m.
Full-Time during 4:00 p.m. – 5:00 p.m.
Full-Time during 5:00 p.m. – 6:00 p.m.
Full-Time during 6:00 p.m. – 7:00 p.m.
Total Full-Time Employees
Solve for the optimal solution.
Solving for the optimal schedule tells us to use a total of full-time employees and part-time employees at a cost of $ per day.
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