I need help with Linear Programming. I am very confused as to how to set this up and all the constraints. Any?

help would be great





Giapetto, Inc., sells wooden soldiers and toy trains. Each soldier produced requires 3 board feet of lumber and 3 hours of labor, while each train requires 5 board feet of lumber and 4.5 hours of labor. Giapetto’s lumberyard can supply up to 180,000 board feet of lumber per month at a cost of $1/board foot. His workers earn an hourly wage of $4, and at most 195,000 hours of labor are available each month. He sells soldiers for $80 each and the market will take up to 50,000 soldiers per month. He can sell as many trains as he can make at a price of $55 apiece, and he has decided that to keep potential customers happy, he must provide at least 15,000 trains per month. In addition to producing trains and soldiers himself, Giapetto can buy (from an outside supplier, Lioness) extra soldiers at $85 each and extra trains at $65 each. To keep his contract with Lioness, he must purchase at least 5000 items per month from them.





A. What is the maximum monthly profit t

I need help with Linear Programming. I am very confused as to how to set this up and all the constraints. Any?
As far as I've understood the condition, we have the following: let


s - number of soldiers produced, s' - bought from outside supplier, t - number of trains produced, t' - bought; s, s', t, t' ≥ 0, then


Profit = 65s + 32t - 5s' - 10t' /to be maximized/


1) 3s + 5t ≤ 180000 /lumber/;


2) 3s + 4.5t ≤ 195000 /labor/;


3) 0 ≤ s ≤ 50000 /upper bound for s/;


4) t ≥ 15000 /lower bound for t/;


5) s' + t' ≥ 5000 /according the contract/


The coefficients for the profit are 80-3-4*3 = 65, 55-5-4*4.5 = 32, 80-85 = -5 and 55-65 = -10. Solving the above with the simplex-method, my computer produced the following optimal solution:


maximum profit = 2 730 000,


s = 35 000, t = 15 000, s' = 5 000, t' = 0, so 3) is 15 000 short and 2) is 22 500 short. That doesn't match with the hint you've supplied: s = 50 000, the latter is impossible, because it leads to 180 000 - 3*50 000 ≥ 5t or t ≤ 30 000/5 = 6 000 and this contradicts to t ≥ 15 000.





Finally, if you change the right side in 5) to 8000, all other the same, the optimal values for s, t, t' are the same, the only changes are s' = 8 000 instead of 5 000, and the maximal profit value = 2 715 000.





Hope that helps!