Question 1155591: A toymaker makes soldiers and guns from wood and plastic. The store has 60 units of wood and 44 units of plastic. Each soldier requires 10 units of wood and 4 units of plastic, whereas each gun requires 6 units of plastic and 4 units of wood. The store has to use a minimum of 22 units of plastic. The demand for soldiers is no more than 2. Each soldier is sold along with at least two guns. A soldier and a gun earn $30 and $20 in profit, respectively. Formulate a linear programming model for this problem with an appropriate objective function.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Let x = number of soldiers
Let y = number of guns
The constraints are....
(1) 10x+4y <= 60 (maximum units of wood)
(2) 22 <= 4x+6y <= 44 {minimum and maximum units of plastic)
(3) x <= 2 (maximum demand for soldiers)
(4) y >= 2x (at least 2 guns for each soldier)
and, of course, x>=0 and y>=0
Sketch a graph of the constraint boundary lines and the resulting feasibility region.
Contrary to what is usually taught, it is NOT necessary to evaluate the objective function at every corner of the feasibility region.
Instead, the corner at which the objective function is maximized can be determined by comparing the slopes of the constraint boundary lines and the objective function.
The objective function is P = 30x+20y; its slope is -3/2. The maximum value of the objective function will be obtained at the corner of the feasibility region where a line with slope -3/2 just touches the feasibility region.
If you have drawn your graph carefully and know the slopes of the constraint boundary lines, you will see that happens at the intersection of the constraint boundary lines x=2 and 4x+6y=44. The coordinates of that corner are (2,6).
The value of the objective function at that corner is 30(2)+20(6) = 60+120 = 180.
ANSWERS: The maximum profit is $180, when 2 soldiers and 6 guns are produced.
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