This is the text for following linear problem:
In one factory there is a production machine which is available 170 hours a month. Using this machine it is possible to produce 50 pieces of product A per hour or 80 products B per hour. Product A brings the factory a profit of 3 dollars per piece, and product B a profit of 2 dollars per piece. It is not possible to sell more than 7000 pieces of product A per month or more than 10000 pieces of product B. How many products are both types needed by this factory per month to maximize total profits?
This is how I modeled this problem.
Let $x$ be number of produced products of product A, and $y$ the number of produced products of product B.
The goal of the problem is
\begin{align} maxZ=3x+2y \end{align}
With constraints
\begin{align} x \leq 7000\\ y \leq 10000 \\ x \geq 0, y\geq 0 \end{align}
My biggest problem is how to make the following sentence as an constraint
Using this machine it is possible to produce 50 pieces of product A per hour or 80 products B per hour.
Does anyone know to how to convert the sentence to a constraint?
Best Answer
So far so good. Next you convert the pieces per hour into hour per piece. If $x$ is the number produced units of product $A$, then you produce one piece in $\frac1{50}$ hours. And $x$ pieces in $\frac1{50}x$ hours. Similar for product $B$. That means that the constraint is $$\frac1{50}x+\frac1{80}y\leq 170 \ \color{grey}{\textrm{hours}}$$