Question

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.

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.

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b. What is the total payroll for the part-time employees? How many part-time shifts are needed? Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.

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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. How many part-time shifts are needed, and what is the cost savings compared to the previous schedule?

Answer 1

(a) Define the decision variables. Let:

A linear program model is a mathematical model with a linear objective function, a set of linear constraints, and nonnegative variables. Each part-time employee assigned to a four-hour shift will be paid $7.60 per hour or $30.40 for the shift.

Use EXCEL to solve for the optimal solution and sensitivity report using the following steps.

1. Select the Data tab from the Ribbon

2. Select Solver from the Analysis Group

3. When the Solver Parameters dialog appears:

Enter the objective function

Select the To: Min option

Enter the decision variables into the By Changing Variable Cells box

Select Add

4. When the Add Constraint dialog box appears:

Enter the constraints in in the Cell Reference box

Select

Enter the right-hand side in the Constraint box

Click OK

5. When the Solver Parameters dialog box reapers:

Click the checkbox for Make Unconstrained Variables Non-negative

6. Select the Select a Solving Method drop-down button.

Select Simplex LP*

7. Click Solve

8. When the Solver Results dialog box appears:

Select Keep Solver Solution

Select Sensitivity in the Reports box

Click OK

The optimal solution is , , , , , , , and . In other words, the optimal schedule calls for 8 starting at 11 a.m., 2 starting at 3 p.m., 4 starting at 5 p.m., and 6 starting at 6 p.m.

(b) Substitute the optimal solution into the objective function to find the total payroll.

The total payroll for the part-time employees is $608 with a total number of 20 part time shifts needed. There are 7 surplus employees schedule from 2-3 p.m. and 4 from 8-9 p.m. This suggests that the desirability of rotating employees off sooner would be beneficial.

(c) Define the decision variables. Let:

A linear program model is a mathematical model with a linear objective function, a set of linear constraints, and nonnegative variables. Each part-time employee assigned to a three-hour shift will be paid $7.60 per hour or $22.80 for the shift.

Use EXCEL to solve for the optimal solution and sensitivity report using the following steps.

1. Select the Data tab from the Ribbon

2. Select Solver from the Analysis Group

3. When the Solver Parameters dialog appears:

Enter the objective function

Select the To: Min option

Enter the decision variables into the By Changing Variable Cells box

Select Add

4. When the Add Constraint dialog box appears:

Enter the constraints in in the Cell Reference box

Select

Enter the right-hand side in the Constraint box

Click OK

5. When the Solver Parameters dialog box reapers:

Click the checkbox for Make Unconstrained Variables Non-negative

6. Select the Select a Solving Method drop-down button.

Select Simplex LP*

7. Click Solve

8. When the Solver Results dialog box appears:

Select Keep Solver Solution

Select Sensitivity in the Reports box

Click OK

The optimal solution is , , , , , , , , , , , , , , , and . In other words, the optimal schedule calls for 6 starting at 6 p.m. working a 4-hour shift, 8 starting at 11 a.m. working a 3-hour shift, 1 starting at 1 p.m. working a 3-hour shift, 1 starting at 3 p.m. working a 3-hour shift, and 4 starting at 5 p.m. working a 3-hour shift.

Twenty part time shifts are required, but the daily cost is reduced to $501.60. The surplus has been reduce by a total of 14 hours.