optimization method to solve a problem

Question

A small petroleum company owns two refineries. Refinery 1 costs \$20,000 per day to operate, and it can produce 400 barrels of high-grade oil, 300 barrels of medium-grade oil, and 200 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs \$25,000 per day to operate, and it can produce 300 barrels of high-grade oil, 400 barrels of medium-grade oil, and 500 barrels of low-grade oil each day. The company has orders totaling 25,000 barrels of high-grade oil, 27,000 barrels of medium-grade oil, and 30,000 barrels of low-grade oil. How many days should it run each refinery to minimize its costs and still refine enough oil to meet its orders?

Using the optimization method, how can we solve this?

Answer 1

Hint:

The following diagram depicts what is happening. Yet, please, calculate the exact solution yourself.

I hope this helps $\ddot\smile$

Answer 2

Let $x$ be the number of days refinery $1$ is run and $y$ the number of days refinery $2$ is run.

Your function to optimize is : $$f(x,y) = 20000x + 25000y$$ Your constraints are : $$ 400x + 300y \geq 25000 \\ 300x + 400y \geq 27000 \\ 200x + 500y \geq 30000 $$ (Note we cannot use strict equality because it leads to a non-solution system).

Since $f$ is increasing in both argument we need to consider the boundary on the region defined by the constraint and analyze $f$ on this boundary to find its minimum.