Having trouble with Part C of this question. Unable to figure out the Min and developing a minimum cost schedule for the part-time employees. With this, I also need help to figure out the inequality for 7 pm, 8 pm, and 9 pm. I also need help with the optimal solution of Part C. The other parts are all correct.
A linear programming computer package is needed.
Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.
| Time |
Number of Employees Needed |
| 11:00 a.m.–Noon |
9 |
| Noon–1:00 p.m. |
9 |
| 1:00 p.m.–2:00 p.m. |
9 |
| 2:00 p.m.–3:00 p.m. |
3 |
| 3:00 p.m.–4:00 p.m. |
3 |
| 4:00 p.m.–5:00 p.m. |
3 |
| 5:00 p.m.–6:00 p.m. |
6 |
| 6:00 p.m.–7:00 p.m. |
12 |
| 7:00 p.m.–8:00 p.m. |
12 |
| 8:00 p.m.–9:00 p.m. |
7 |
| 9:00 p.m.–10:00 p.m. |
7 |
One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.
(a)
Develop a minimum-cost schedule for part-time employees. (Let
xi
= the number of part-time employees who start work beginning at hour i where
i = 1 = 11:00 a.m.,
i = 2 = Noon,
etc).Min
30.4×1?+30.4×2?+30.4×3?+30.4×4?+30.4×5?+30.4×6?+30.4×7?+30.4×8?
s.t.11:00 a.m.
x1?=8
Noon
x1?+x2?=8
1:00 p.m.
x1?+x2?+x3?=7
2:00 p.m.
x1?+x2?+x3?+x4?=1
3:00 p.m.
x2?+x3?+x4?+x5?=2
4:00 p.m.
x3?+x4?+x5?+x6?=1
5:00 p.m.
x4?+x5?+x6?+x7?=5
6:00 p.m.
x5?+x6?+x7?+x8?=10
7:00 p.m.
x6?+x7?+x8?=10
8:00 p.m.
x7?+x8?=6
9:00 p.m.
x8?=6
x1, x2, x3, x4, x5, x6, x7, x8 = 0
Find the optimal solution.
(x1, x2, x3, x4, x5, x6, x7, x8) =
(8, 0, 0, 0, 2, 0, 4, 6)
(b)
What is the total payroll (in $) for the part-time employees?
$
608
How many part-time shifts are needed?
20 shifts
Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.
There are 7 surplus employees that are scheduled from 2-3 p.m. and 4 surplus employees scheduled from 8-9 p.m. Based on this it would be desirable to rotate employees off sooner making this more beneficial.
(c)
Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let
xi
be the same variables from part a. Let
yi
= the number of part-time employees who start work beginning at hour i where
i = 1 = 11:00 a.m.,
i = 2 = Noon, etc).
Min : _________________________
s.t.11:00 a.m.
x1?+y1?=8
Noon
x1?+x2?+y1?+y2?=8
1:00 p.m.
x1?+x2?+x3?+y1?+y2?+y3?=7
2:00 p.m.
x1?+x2?+x3?+x4?+y2?+y3?+y4?=1
3:00 p.m.
x2?+x3?+x4?+x5?+y3?+y4?+y5?=2
4:00 p.m.
x3?+x4?+x5?+x6?+y4?+y5?+y6?=1
5:00 p.m.
x4?+x5?+x6?+x7?+y5?+y6?+y7?=5
6:00 p.m.
x5?+x6?+x7?+x8?+y6?+y7?+y8?=10
7:00 p.m.
_________
8:00 p.m.
_________
9:00 p.m.
_________
x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9 = 0
Find the optimal solution.
(x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9) =
(_______________________________________________)
How many part-time shifts are needed, and what is the cost savings compared to the previous schedule?
part-time shifts needed
20
cost savings
$106.40
