Question

Anderson Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the Steakhouse operates with full time employees work 8 hrs/ day. The rest of employees are part time employees who are scheduled for 4 hrs shifts during peak meal times. On the Saturdays the Steakhouse is open from 11:00AM to10:00PM management want to develop a schedule for part time employees that will minimize labor cost and provide excellent customer service. The average wage rate for the part time employees is $7.60/ hr. The total number of full time and part-time employees needed varies with time of a day as shown.

Time	Total number of employees needed
11:00AM-Noon	9
Noon- 1:00PM	9
1:00PM-2:00PM	9
2:00PM-3:00PM	3
3:00PM-4:00PM	3
4:00PM-5:00PM	3
5:00PM-6:00PM	6
6:00PM-7:00PM	12
7:00PM-8:00PM	12
8:00PM-9:00PM	7
9:00PM-10:00PM	7

One full time employee comes on duty at 11:00 AM works 3 hrs.takes 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.

1. Write the complete linear programming model(objective function and constraints) and develop a minimum cost schedule for part time employees?

2. 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.)

3. The restaurant wants to open an extra hour from 10 PM-11 PM with six part time employees needed during this hour. How much does payroll cost change?

4. 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? (restaurant closing time is 10 PM)

5. The restaurant decides to give part-time employees a raise to $8.50 per hour. How much does the raise cost in part d?

Answer 1

a) Linear programming model is following:

Let

X1 = number of part-time employees coming to duty at 11 am

X2 = number of part-time employees coming to duty at 12 pm

X3 = number of part-time employees coming to duty at 1 pm

X4 = number of part-time employees coming to duty at 2 pm

X5 = number of part-time employees coming to duty at 3 pm

X6 = number of part-time employees coming to duty at 4 pm

X7 = number of part-time employees coming to duty at 5 pm

X8 = number of part-time employees coming to duty at 6 pm

Objective: Min 7.6*4*(X1+X2+X3+X4+X5+X6+X7+X8)

s.t.

X1 >= 9-1

X1+X2 >= 9-1

X1+X2+X3 >= 9-2

X1+X2+X3+X4 >= 3-1

X2+X3+X4+X5 >= 3-2

X3+X4+X5+X6 >= 3-1

X4+X5+X6+X7 >= 6-2

X5+X6+X7+X8 >= 12-2

X6+X7+X8 >= 12-2

X7+X8 >= 7-1

X8 >= 7-1

X1, X2, X3, X4, X5, X6, X7, X8 >= 0

b) Solution of the Linear program using LINGO is following:

Lingo 17.0 Solution Report Lingol File Edit Solver Window Help Lindo Model Lingol 3 回Solution Report . Lingo! Global optimal solution found Objective value: Infeasibilities: Total solver iterations: Elapsed runtime seconds: 577.6000 0.000000 Min 30.4X1+30.4X2+30.4X3+30.4X4+30.4X5+30.4X6+30.4X7+30.4X 0.14 Model Class: LP X1+X2 8 Total variables: Nonlinear variables: Integer variables: X1+X2 +X3 >= 7 Total constraints: Nonlinear constraints: X2+X3+X4+X5 1 Total nonzeros: Nonlinear nonzeros: 40 X3+X4+X5+X6 - 2 X5+X6+X7+X8 10 Variable Value 8.000000 1、000000 0.000000 0.000000 0.000000 ·000000 0.000000 6.000000 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X6+7+X8-10 X2 X3 X4 X5 x8 X7 x8 Row Slack or Surplus 577.6000 0.000000 1 . 000000 2.000000 7.000000 0.000000 2.000000 0.000000 Dual Price -1.000000 -30.40000 0.000000 0.000000 0.000000 -30.40000 0.000000 0.000000 7 For Help, press Fl Ln 1, Col 1 4:29 pm

Variable X1 X2 X3 value 8.000000 1.000000 0.000000 0.000000 0.000000 4.000000 0.000000 6.000000 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X5 X6 X7 X8 Dual Price -1 000000 -30.40000 0.000000 0.000000 0.000000 -30.40000 0.000000 0.000000 0.000000 -30.40000 0.000000 0.000000 Row Slack or Surplus 577.6000 0.000000 1.000000 2.000000 7.000000 0.000000 2.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 7 10 12

Total payroll for the part-time employees = $ 577.6

Surplus variables for shift 2, 3, 4, 6 are 1,2,7,2 respectively.

Therefore, these many number of employees can be scheduled for 3 hrs shift.

c) The revised LP and its solution is following:

Lingo 170-Solution Report . Lingo! File Edit Solver Window Help Lindo Model-Lingol 3Solution Report-Lingol Global optimal solution found Objective value: Infeasibilities: Total solver iterations: Elapsed runtime seconds: 729.6000 Min 30.4X1+30.4X2+30.4X3+30.4X4 30.X5+30.4x6+30.4x7+30.4X830.4X9 0.000000 0.12 LP S.t Model Class: Total vriables: Nonlinear variables: Integer variables: X1+X2 -8 X1+X2 +X3 >= 7 X1+X2+X3+X4 2 Total constraintS: Nonlinear constraints: 13 X2+X3+X4+X5 1 Total nonzeros: Nonlinear nonzeros: >= 45 X3+X4+X5+X6 -2 X4+X5+X6+X74 XS+X6+X7+X810 Value 8.000000 0.000000 0.000000 0.000000 6.000000 4。000000 0.000000 0.000000 6.000000 Variable X1 X2 Reduced Cost 0.000000 30.40000 30.40000 30.40000 0.000000 0.000000 0.000000 0.000000 0.000000 10 XS X6 X7 X8 X9 Row Slack or Surplus 729.6000 0.000000 0.000000 1.000000 6.000000 5.000000 8.000000 Dual Price -1.000000 -30.40000 0.000000 0.000000 0.000000 0.000000 0.000000 For Help, press F1 NUM Ln 1, Col 1 4:32 pm

Variable X1 X2 X3 value 8.000000 0.000000 0.000000 0.000000 6 . 000000 4.000000 0.000000 0.000000 6 . 000000 Reduced Cost 0.000000 30.40000 30.40000 30.40000 0.000000 0.000000 0.000000 0.000000 0.000000 X5 X6 X7 X8 Dual Price -1 000000 -30.40000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -30.40000 0.000000 0.000000 0.000000 -30.40000 Row Slack or Surplus 729.6000 0.000000 0.000000 1.000000 6.000000 5.000000 8.000000 6 . 000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 7 10 12 13

New payroll cost = $ 729.6

Increase in payroll cost = 729.6-577.6 = $ 152

d) Complete Linear Program is following:

Let

Let

X1, X2, ... X8 = number of 4-hr shift part-time employees coming to duty at 11 am, 12 pm, ... 6 pm resp.

Y1, Y2, ... Y9 = number of 3-hr shift part-time employees coming to duty at 11 am, 12 pm, ... 7 pm resp.

Objective: Min 7.6*4*(X1+X2+X3+X4+X5+X6+X7+X8)+7.6*3*(Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y9)

s.t.

X1+Y1 >= 9-1

X1+X2+Y1+Y2 >= 9-1

X1+X2+X3+Y1+Y2+Y3 >= 9-2

X1+X2+X3+X4+Y2+Y3+Y4 >= 3-1

X2+X3+X4+X5+Y3+Y4+Y5 >= 3-2

X3+X4+X5+X6+Y4+Y5+Y6 >= 3-1

X4+X5+X6+X7+Y5+Y6+Y7 >= 6-2

X5+X6+X7+X8+Y6+Y7+Y8 >= 12-2

X6+X7+X8+Y7+Y8+Y9 >= 12-2

X7+X8+Y8+Y9 >= 7-1

X8+Y9 >= 7-1

Xi, Yi >= 0

Solution using LINGO is following:

Lingo 170-Solution Report . Lingo! File Edit Solver Window Help Lindo Model-Lingol 3Solution Report- Lingo1 Global optimal solution found Objective value: Infeasibilities: Total solver iterations: Elapsed runtime seconds: 494.0000 0.000000 +30.4X6+30.4X7+30.4X8+22.8Y1+22.8Y2+22.8Y3 +22. 8Y1 +22 . 8Y5+22. 8Y6+22 . 8Y7+22. 8Y8+22. 8Y9 0.13 Model Class: LP Total variables: Nonlinear variables: Integer variables: 17 X1+X2+71+72 >= 8 X1+X2+X3+71+72 13-7 X1+X2+X3+X4+Y2+Y3+Y42 X2+X3+X4+X5 Y34Y4 YS-1 Total constraints: Nonlinear constraints: 12 Total nonzeros: Nonlinear nonzeros: 76 Variable X1 X2 X3 X4 X5 X6 Value 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 6.000000 7.000000 0.000000 0.000000 1.000000 0.000000 0.000000 3.000000 0.000000 0.000000 Reduced Cost 0.000000 0.000000 7.600000 7·600000 0.000000 0.000000 7.600000 0.000000 0.000000 0.000000 7.600000 0.000000 7.600000 0.000000 0.000000 0.000000 7.600000 X7+X8+Y8+79-6 X8 Y1 Y2 Y3 Y4 Y5 Y6 רצ T8 Y9 For Help, press F1 NUM Ln 25, Col 9 4:39 pm

Variable X1 Value 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 6.000000 7.000000o 0.000000 0.000000 1.000000 0.000000 0.000000 3 . 000000 0.000000 0.000000 Reduced Cost 0.000000 0.000000 7.600000 7.600000 0.000000 0.000000 7.600000 0.000000 0.000000 0.000000 7.600000 0.000000 7.600000 0.000000 0.000000 0.000000 7.600000 X3 X4 X6 X7 X8 Yl Y3 Y4 9 Row Slack or Surplus 494.0000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Dual Price -1 000000 -7.600000 -15.20000 0.000000 -7.600000 -7.600000 -7.600000 0.000000 -15.20000 -7.600000 0.000000 -7.600000 10 12

Total payroll cost = $ 494

e) The LP model with the new cost and its solution is following:

Lingo 170-Solution Report . Lingo! File Edit Solver Window Help Lindo Model-Lingol 1G] _İEZ P Solution Report . Lingo! Global optimal solution found Objective value: Infeasibilities: Total solver iterations: 552.5000 Min 34X1+34X2+34X3+34X4+34XS +34X6+34X7+34X8+25.5Y1+25.5Y2+25.5Y3 +25.5Y4+25.5YS+25.5Y6+25.5Y7+25.SY8+25.5Y9 0.000000 0.13 LP Model class: Total variables: Nonlinear variables: Integer variables: 17 X1+X2+71+72 >= 8 X1+X2+X3+71+72 13-7 X1+X2+X3+X4+Y2+Y3+Y42 X2+X3+X4+X5 Y34Y4 YS-1 Total constraintS: Nonlinear constraints: 12 Total nonzeros: Nonlinear nonzeros: 76 Value 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 6.000000 7.000000 0.000000 0.000000 1.000000 0.000000 0.000000 3.000000 0.000000 0.000000 Variable X1 X2 X3 Reduced Cost 0.000000 0.000000 8.500000 .500000 0.000000 0.000000 8.500000 0.000000 0.000000 0.000000 3.500000 0.000000 8.500000 0.000000 0.000000 0.000000 8.500000 XS X6 X7 X8 X7+X8+Y8+79-6 Y2 Y3 Y4 YS Y6 רצ T8 Y9 For Help, press F1 NUM Ln 1, Col 1 4:42 pm

New payroll cost = $ 552.5

The raise in wages increase the cost by = 552.5-494 = $ 58.5