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
Noon1:00 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 $-----