Linear optimization, coming up with the constraints

Question

A container manufacturer is considering the purchase of two different types of cardboard-folding machines: model A and model B. Model A can fold 30 boxes per minute and requires 1 attendant, whereas model B can fold 50 boxes per minute and requires 2 attendants. Suppose the manufacturer must fold at least 320 boxes per minute and cannot afford more than 12 employees for the folding operation. If a model A machine costs $15,000 and a model B machine costs $20,000, how many machines of each type should be bought to minimize the cost?

I know we must minimize the equation C= 15000A + 20000B but I'm having trouble coming up with the constraints. Any help is appreciated.

Answer 1

Start by defining appropriate variables: let $x_A$ and $x_B$ be the number of machines of type $A$ and $B$ to be bought, respectively.

Indeed, you need to minimize the function $$ 15000x_A+20 000 x_B $$ subject to the following constraints:

1. The manufacturer must fold at least $320$ boxes per minute: $$ 30x_A+50x_B \ge 320 $$
2. The manufacturer cannot afford more than $12$ employees: $$x_A+2x_B \le 12$$
3. Variables are non negative: $$ x_A,x_B \ge 0 $$

Once you have properly defined your variables, just go through each sentence of the exercise, and ask yourself how you can express these mathematically with your variables.

Always make sure your equations are homogeneous, i.e., each term is expressed in the same unit as the other ones. For example, the objective function is expressed in dollars, constraint $(1)$ is expressed in boxes per minute, and constraint $(2)$ is expressed in men.