Question

You are required to schedule helpers so that you have sufficient helpers during the event and with the lowest cost. The event will start from 9am to 5pm. Based on your past experience, the number of helpers required in each hour is as below:

Time	Number of helpers
9am – 10am	3
10am – 11am	3
11am – 12pm	6
12pm – 1pm	8
1pm – 2pm	6
2pm – 3pm	4
3pm – 4pm	3
4pm – 5pm	3

Only 2 of your friends will be coming to help during the event. One of them comes at 9am, works for 3 hours, takes an hour off and returns for another 4 hours. Another friend comes at 12pm and works for 4 hours. Besides, part-time helpers are also required and they work for 3-hour shift beginning on the hour.

Assume each part-time helpers is paid $10 each hour, you are required to determine how many part-time helpers required so that the cost can be mininised. Based on your output, develop a schedule to show the allocation of part-time helpers.

This question is to answer using linear programming. Please solve for sample answer for me. Thanks.

Answer 1

Answer:

Integer linear program is following:

Let

X1 be the number of part-time helpers starting on at 9 am

X2 be the number of part-time helpers starting on at 10 am

X3 be the number of part-time helpers starting on at 11 am

X4 be the number of part-time helpers starting on at 12 pm

X5 be the number of part-time helpers starting on at 1 pm

X6 be the number of part-time helpers starting on at 2 pm

Minimize 3*10*(X1+X2+X3+X4+X5+X6)

s.t.

X1 >= 3-1

X1+X2 >= 3-1

X1+X2+X3 >= 6-1

X2+X3+X4 >= 8-1

X3+X4+X5 >= 6-1-1

X4+X5+X6 >= 4-1-1

X5+X6 >= 3-1-1

X6 >= 3-1

X1, X2, X3, X4, X5, X6 integers

Solution using LINGO is following:

Lingo 17.0- Solution Report Lingol File Edit Solver Window Help Lindo Model Lingo Solution Report-Lingo1 Global optimal solution found Objective value: Infeasibilities: Total solver iterations: Elapsed runtime seconds: 330.0000 Minimize 30X1+30X2+30X3+30X4+30X5+30X6 0.000000 0.92 LP S.t. xi>= 2 X1+X2 2 X1+X2+X3 >- 5 X2+X3+X4 >= 7 Model Class: Total variables: Nonlinear variables: Integer variables: Total constraints: Nonlinear constraints: X3+X4+X5 24 Total nonzeros: Nonlinear nonzeros: X4+X5+X6- 2 X5tX6-1 Variable X1 X2 X3 Value 2.000000 3.000000 4.000000 0.000000 0.000000 2.000000 Reduced Cost 0.000000 0.000000 0.000000 0.000000 30.00000 0.000000 X5 xe Row Slack or Surplus 330.0000 0.000000 3.000000 4.000000 0.000000 0.000000 0.000000 1.000000 0.000000 Dual Price -1.000000 -30.00000 0.000000 0.000000 -30.00000 0.000000 0.000000 0.000000 -30.00000 For Help, press F1 NUM Ln 1, Col 1

Optimal solution:

X1 = 2

X2 = 3

X3 = 4

X4 = 0

X5 = 0

X6 = 2

Total cost = $ 330

THANK YOU!