Linear Programming: Optimizing with Inequalities [Art of Problem Solving Intro to Algebra 9.40]

If you're a student who likes to ask "when am I ever gonna use this in real life?", then linear programming is the topic for you. Businesses actually do this kind of process every single day.

Linear programming, also sometimes called linear optimization, is all about maximizing or minimizing some quantity given certain constraints, limitations, or restrictions. Today, I want to take a look at the process by working through an example from the Art of Problem Solving Introduction to Algebra textbook. The problem is 9.40.

There are three main questions you want to ask yourself as you model these scenarios:
1) What are the constraints
2) What do we NOT know?
3) What is the objective?

Unlike a system of equations, for linear programming, you don't really want to solve it algebraically. Instead, you want to set up your linear inequalities, then graph those inequalities. The region of possible solutions will have some number of "corners", and it is those points that you check in your objective equation.

#linearprogramming #linearoptimization #artofproblemsolving

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[misugijun]

Thanks for this topic. it would be nice to see analytical solution of the scenarios too.

[yoseftreitman]

For part c) tell me if this approach makes sense from a teaching perspective. Consider the line connecting the "pure video games" corner at (0, 9) with the "some of each" corner at (40, 5). This line has slope -1/10. Thus, to increase the number of video games by 1, you have to reduce the number of movies by 10. This will be a good move if the number of video game friends gained exceeds the 10 movie friends lost.

My thought is that this nicely sets students up for the simplex method.

[yoseftreitman]

If memory serves, things get really screwy if one of the corners has non-integer coordinates.

[ryansunil]

Woudnt it make sense to make the weight 10 aswell?

[unitlvt]

You should have at least million subs unfortunately you dont 😢