solution

Q5

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 x_i = 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. =_________

Noon =________

1:00 p.m. =________

2:00 p.m. =________

3:00 p.m. =________

4:00 p.m. =________

5:00 p.m. =_______

6:00 p.m. =_______

7:00 p.m. =_______

8:00 p.m. =_______

9:00 p.m. =_______

x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈ = 0

Find the optimal solution.

(x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈) = (__________________)

(b)

What is the total payroll (in $) for the part-time employees?

$ =_________

How many part-time shifts are needed?

shifts=___________

Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.

=____________________________________________________________________________________________________________________________

(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 x_i be the same variables from part a. Let y_i = 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. =_______

Noon =_________

1:00 p.m. =_________

2:00 p.m. =_________

3:00 p.m. =_________

4:00 p.m. =__________

5:00 p.m. =__________

6:00 p.m. =___________

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 =__________

cost savings = $__________