0
$\begingroup$

So here is the problem:

A small machine tool manufacturing company entered into a contract to supply $80$ drilling machines at the end of the first month, $120$ at the end of the second month and $150$ at the end of the third month. The unit cost of manufacturing a drilling machine in any month is given by $50x + 0.2x^2$, where $x$ is the number of drilling machines manufactured that month. If the company manufactures more units than needed the first month, there is an inventory carrying cost of $8$ per unit for each unit carried into the next month. Assume that the company has enough resources to produce up to $200$ drilling machines per month and there is no initial inventory. Formulate a two-variable nonlinear program to find the number of drilling machines to be manufactured in each month to minimize the total cost.

This is confusing to me. I can only identify one variable that is already been told in the question which is the drilling machines manufactured, $x$. I could say that I have $x_1$ for the machines manufactured in month $1$ and $x_2$ for the machines manufactured in month $2$ but then I would need $x_3$ for month $3$ and the problem only states to formulate a two-variable program. Also to minimize the total cost the extra machines manufactured might aswell be $0$.

Any idea how to solve this? This shouldn't be something complicated, but I seem to be missing something.

$\endgroup$

2 Answers 2

Reset to default
1
$\begingroup$

Since you have a constraint on the total number of drills, you can write $x_3 = 350 - (x_1 + x_2)$.

Let me add something with a more general scope: as a rule of thumb, if you don't see how to write the problem in two variables, writing it out in $N$ variables first can be very useful.

$\endgroup$
1
$\begingroup$

Calling

$$ \cases{ f(x) = 0.2x^2+50 x\\ g(x) = 8x } $$

we have the proposal

$$ \min_{x_1,x_2,x_3}\{f(x_1)+f(x_2)+f(x_3)+g(x_1-80)+g(x_1+x_2-120)+g(x_1+x_2+x_3-150)\} \\ \text{s. t.} \cases{80\le x_1\le 200\\ 120\le x_2\le 200\\ 150\le x_3\le 200} $$

$\endgroup$
2
  • $\begingroup$ It is said to formulate a two-variable nonlinear program, your solution has 3 variables? $\endgroup$ Commented May 29, 2023 at 22:43
  • 1
    $\begingroup$ Reading the statement of the problem, with its ambiguities, this is the model that I was able to extract. $\endgroup$ Commented May 29, 2023 at 22:54

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.