A firm produces three different types of finished material in which to construct shoes, canvas and rubber. Each of these materials requires the following amounts of production (in hours) to produce $\bbox[yellow]{100 }\,$square yards in three different parts of the factory.
Material $ \,\,\,\,\,\,\,\ A \,\,\,\,\,\,\,\,\,\,\ B \,\,\,\,\,\,\,\,\,\,\ C $
Leather $ \,\,\,\,\,\,\,\ 7.5 \,\,\,\,\,\,\,\,\,\,\ 8 \,\,\,\,\,\,\,\,\,\,\ 5.5 $
Canvas $ \,\,\,\,\,\,\,\ 3 \,\,\,\,\,\,\,\,\,\,\ 1.5 \,\,\,\,\,\,\,\,\,\,\ 5 $
Rubber $\,\,\,\,\,\,\,\ 4.5 \,\,\,\,\,\,\,\,\,\,\ 5 \,\,\,\,\,\,\,\,\,\,\ 3 $
There are 5000, 6000, and 7000 hours of production time available in each of the three different parts of the factory ( A, B and C) respectively. The profit realized in each of the sale of one square yard of leather, on one square yard of canvas, and on one square yard of rubber is $\$15$, $\$7$, and $\$3$. The goal is to determine a schedule that will maximize profit.
But I find it difficult to form equations related to this question.
If I take, [??? not sure]
L- total leather yards produced (or Time taken to produce 100 yards?)
C - total canvas yards produced
R - total rubber yards produced
L<100
C<100
R<100
Is this a good start? Or Am I wrong? Please help!
So I am assuming Leather, Canvas and Rubber have to go through all parts (A,B and C) of the factory before they can be sold for their respective profit:
The first step is to initialize variables that encode the number of leather, canvas and rubber (L,C,R).
Without the time constraints, you would simply produce as much as possible to maximize your profit, thus your goal function would be
$max \quad 15L + 7C + 3R$
To satisfy the constraints, for each part of the factory, you need to multiply the amount of stuff you produce (L,C and R) by the amount of time it takes to produce them (in this specific part), and the sum of those has to be at most the total production time (of this part).
Try to work out the actual constraints yourself.
(Do NOT forget "trivial" constraints that restrict the amount of things to be $\geq 0$ for example)