Western Family Steakhouse offers a variety of low-cost meals and quick service.

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.

Time Total 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 AM works 3 hrs, take one hr. off, and returns for another 5 hrs. The other full-time employee comes to work at 1:00 PM and works the same 3 hours-on, 1-hour-off, 5-hours-on pattern. A. Write the complete linear programming model (objective function and constraints) and develop a minimum cost schedule for part time employees? B. What is 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 hrs shifts.) C. Assume that part-time employees can be assigned either 3hr or 4 hr shifts. Write the complete linear programming model and develop a minimum cost schedule for the part time employees. How many part time shifts are needed and what is the cost saving compared to the previous schedule? D. The restaurant decides to give part-time employees a raise to $8.50 per hour. How much does the raise cost? Use the results of ranges of optimality and feasibility for the Model in part c answer the following questions: E. Explain the dual values for constraints 1, 2 and 3.(-15.2, -7.6 and 0) F. How come there are no 4-hr part timers scheduled until 6 PM? What does it take for 4-hr part timers to be scheduled at 1PM and 4 PM?

Explanation / Answer

A) Objective Function is Minimize the cost=x+y=$7.60 per hour

Subjct to constraints:

11:00 w1 +12w2=8

12x1+1:00x2 = 8

1:00 y1 +2:00y2= 8

2:00 z1 +3:00 z3=2

3:00 a1+ 4:00 a2= 2

4:00 b1+ 5:00 b2= 2

5:00 c1+6:00 c2= 4

6:00 d1+7:00 d2=10

7:00 e1+8:00 e2= 10

8:00 f1+9:00 f2= 6

9:00 g1+10:00 g2=. 6

Cost Schedule

12 to 3=$7.60 * 8 employees=$60.8

3to7=$7.60 * 2employees=$15.20

5to9=$7.60*$4employees=$30.40

6to10=$7.60*6 employees=$45.60

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B)$60.8 + $15.20+ $30.40+$45.60=$152.00

4 part time shifts are needed if 4 hours are used.

if 3 hour used

11 to 2=8 employees

2to5=2employees

5to8=4employees

6to10=6employees

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c)Objective Function is Minimize the cost=x+y=$7.60 per hour

Subject to Constraints

11x1+ 2x2=8

2y1+5y2=2

5z1+8z2=4

6a1+10a2=6

Cost Schedule

11 to 2=8 employees*$7.60=$60.80

2to5=2employees*$7.60=$15.20

5to8=4employees*$7.60=$30.40

6to10=6employees*$7.60=$45.60

=$60.80+ $15.20+$30.40+$45.60=$152

4 shifts are needed,

Same Cost, no more changes

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d)

Cost Schedule

12 to 3= $8.50 * 8 employees=$68

3to7= $8.50 * 2employees=$17

5to9= $8.50*$4employees=$34

6to10= $8.50*6 employees=$51

=$68 +$17+$34+$51=$170

old one is $152, new one is$170

=$170 -$152=$18

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