Hinge Doorknob Pull Switch plate Window lock Steel(gm) 0.5 1.2 0.8 0.2 1.3 Zinc(gm) 1.6 1.8 0.5 2.3 1.1 Lacquer(ml) 0.06 0.13 0.04 0.27 0.09 Labor(hr) 0.023 0.017 0.008 0.004 0.065 Energy(cal) 4.1 5.7 3.8 4.9 5.5

The table above gives the amount of each resource needed to make one unit of product.

Problem: Given that this month's supplies of 200 kg of steel, 450 kg of zinc, 30 liters of lacquer, 6300 hr of labor, and 1500 k cal of energy must be used in their entirety, how many units of each product can be produced? Why is this not feasible? What can be changed in the supplies of resources to make your answer feasible?

I created the following augmented matrix:

0.5 1.2 0.8 0.2 1.3 200000 1.6 1.8 0.5 2.3 1.1 450000 0.06 0.13 0.04 0.27 0.09 30000 0.023 0.017 0.008 0.004 0.065 6300 4.1 5.7 3.8 4.9 5.5 1500000

I then found the row reduced echelon form gettingx=

1.8065*10 ^{5}-51191.94 1.61083*10 ^{5}64212.96 22612.05

Clearly the issue is the negative number of doorknobs; however, I haven't a clue on how to fix the problem. I know you can increase the amount of Steel and decrease the amount of Energy to roughly 223kg and 1300k cal respectively. This would yield positive values for each product. I only found this out by tinkering around with some test values though which leads me to believe there are countless answers like this. Is there a more precise way to figure this out? How would you go about answering this problem?

The next part of the problem is:

The profit per unit for each product is given by:

Hinge Doorknob Pull Switch plate Window lock $0.58 $1.12 $0.33 $0.76 $1.25

Next year the price of zinc and cost of energy will be increasing. Therefore, management has decided to reduce the supply of one of these two resources. the reduction of which one will have the best effect of total profit?

I know that whichever one cause profit to drop the least will be the best to reduce; however, I haven't a clue on how to figure out the answer. I am sure I can fool around with the numbers and see what happens but I know there is a more precise way to go about this problem.

Any help that can be given would be appreciated. I should let it be known that I have access to Maple and used it to get to the point I am at so far.