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, but the temp agency managing the part time staff will charge the steakhouse one extra dollar per hour for shifts starting after 3:00 PM. The total number of full-time and part-time employees needed varies with the time of day as shown.

Time Total Number of Employees Needed

11:00 a.m. - Noon 9

Noon - 1:00 p.m.    9

1:00 p.m. - 2:00 p.m. 10

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. 4

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. Develop a minimum-cost schedule for part-time employees. What is the total payroll for the part-time employees? If required, round your answer to the nearest dollar.

Total daily salary cost = $????

Answer 1

a. Let x 1 = number of part-time employees beginning at 11:00 a.m. x2 = number of part-time employees beginning at 12:00p.m. x3 = number of part-time employees beginning at 1:00 p.m. *4 = number of part-time employees beginning at 2:00 p.m. *5 = number of part-time employees beginning at 3:00 p.m. x6 = number of part-time employees beginning at 4:00 p.m. x 7 = number of part-time employees beginning at 5:00 p.m. x8 = number of part-time employees beginning at 6:00 p.m. Each part-time employee assigned to a four-hour shift will be paid $7.60 (4 hours) = $30.40. Min 30.4x] + 30.4x2 + 30.4x3 + 30.4x4 + 30.4x5 + 30.4x6 + 30.4x7 + 30.4x8 Part-Time Employees Needed s.t. IV + IV x1 + x2 + IV + + IV x3 x3 + + + + x4 X + 4 + x4 + I 15 xy + x5 + x5 + X6 X6 + X6 + X6 + I 11:00 a.m. 12:00 p.m. 1:00p.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. + + IV > 1 25 x8 ? 10 x8 > 10 x8 2 6 xg > 6 x7 27 + x2 + x7 + IV j20 j = 1,2,...8

x20 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 x = 0). x=0, x=0, x4 = 0. x3 = 2). Xx=O. x = 4, and x3 = 6. In other words, the optimal schedule calls for 8 starting at 11 a.m., 2 starting at 3p.m., 4 starting at 5p.m., and 6 starting at 6 p.m.

(6) Substitute the optimal solution into the objective function to find the total payroll. 30.4x +30.4x2 + 30.4x3 +30.4x4 +30.4x3 +30.4x5 +30.4x, +30.4xg = 30.4 (8) +30.4(0)+30.4(0)+30.4(0)+30.4(2)+30.4(0)+30.4(4)+30.4(6) = 5608 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.