optimize labour and costs

[ski007]

I would appreciate it greatly if someone could provide me with a solution to the problem below:
If the contract runs late the business will be penalized $1000 for each late day. It is estimated that the contract needs about 2,500 man hours extra of unskilled labor to be delivered in four weeks. The ordinary working week is 37 hours and the pay rate for unskilled laborers is $20 per hour. Overtime for unskilled labor is $35 per hour and a worker cannot work more than 15 hours overtime per week. No more than 10 unskilled laborers can be assigned to this contract. There is an option of employing skilled workers at a rate of $40 per hour with no weekend work. These workers can work for more than the ordinary 37 hour week but must be paid $70 per hour overtime.
What is the most economical solution to this problem?

1. a) Where the contract finish on time
2. b) Where the total costs are minimized

I have only done some basic differential calculus. In my research this problem is a fit for solution by a mix of partial differentials and Lagrange multipliers. I tried to figure that out but got lost. I believe that the number of both types of employees, the number of weeks and the overtime should be variable, so six variables are needed. I used this approach in Excel solver using GRG Non-Linear method and found the solution as $68900 with 10 unskilled laborers working 15 hours overtime per week for 4 weeks and 10 skilled workers working for 5 hours overtime for 1 week, the total time equaled 2500 hours.
My main problem is how to I arrive at this solution on paper.I would really like assistance in first stating the problem properly as I think this is where I am making my first mistake and then an attempt at how to proceed to a solution.
My attempt at stating the problem is:

• X1 = total cost of unskilled labour for project duration
• X2 = total cost of skilled labour for project duration
• X3 = total hours for project completion
• Minimize: Z = x1 + x2
• Subject to:
• X1 = XaZa(37*20 + Ya (35) )
• X2 = XbZb(37*40 + Yb (70) )
• X3 = XaZa(37+ Ya) + XbZb(37 + Yb)
• X3 = 2500
• Xa = unskilled labourers ≤10, Xb = skilled labourers ≥ 0,
• Ya = unskilled overtime ≤ 15, Yb = skilled overtime ≥ 0,
• Za, Zb ≥ 0

 

[stapel]

I would appreciate it greatly if someone could provide me with a solution to the problem below:

If the contract runs late the business will be penalized $1000 for each late day.

It is estimated that the contract needs about 2,500 man hours extra of unskilled labor to be delivered in four weeks. The ordinary working week is 37 hours and the pay rate for unskilled laborers is $20 per hour. Overtime for unskilled labor is $35 per hour and a worker cannot work more than 15 hours overtime per week.

No more than 10 unskilled laborers can be assigned to this contract. There is an option of employing skilled workers at a rate of $40 per hour with no weekend work. These workers can work for more than the ordinary 37 hour week but must be paid $70 per hour overtime.
What is the most economical solution to this problem?

1. a) Where the contract finish on time
2. b) Where the total costs are minimized

I have only done some basic differential calculus. In my research this problem is a fit for solution by a mix of partial differentials and Lagrange multipliers. I tried to figure that out but got lost.

I believe that the number of both types of employees, the number of weeks and the overtime should be variable, so six variables are needed. I used this approach in Excel solver using GRG Non-Linear method and found the solution as $68900 with 10 unskilled laborers working 15 hours overtime per week for 4 weeks and 10 skilled workers working for 5 hours overtime for 1 week, the total time equaled 2500 hours.

My main problem is how to I arrive at this solution on paper.I would really like assistance in first stating the problem properly as I think this is where I am making my first mistake and then an attempt at how to proceed to a solution.
My attempt at stating the problem is:

• X₁ = total cost of unskilled labour for project duration
• X₂ = total cost of skilled labour for project duration
• X₃ = total hours for project completion

• Minimize: Z = X₁ + X²
• Subject to:
• X₁ = X_aZ_a(37*20 + Y_a (35) )
• X₂ = X_bZ_b(37*40 + Y_b (70) )
• X₃ = X_aZ_a(37+ Y_a) + X_bZ_b(37 + Y_b)
• X₃ = 2500

• X_a = unskilled labourers ≤10, X_b = skilled labourers ≥ 0,
• Y_a = unskilled overtime ≤ 15, Y_b = skilled overtime ≥ 0,
• Z_a, Z_b ≥ 0

You were given some suggestions (and were advised that, in the opinion of the resident experts, information was missing) at one of the other places you've posted this. What have you done with that information?