Week 5: The simplex method

Profits and pollution

This week's question is what we as a local community can do to get a steel plant to pollute less. We accept that they want to maximize profits. We have leverage through taxes and levies. How should we deploy them to minimize pollution? Hopefully you noticed the words maximize and minimize! There should be a way to model this as an optimization problem. And there is, but it has three variables. That means we will have a feasible set defined by three axes, so we can't draw it on a sheet of paper which means we can't solve it graphically. What to do? Turns out we can represent a multidimensional linear programming model in a matrix, and find the optimal vertex by pivoting the matrix according to some well-defined rules.

The rules we will follow for our pivoting are defined by the simplex method. It's a fairly involved procedure, and as long as any we will encounter in this course, so if you can manage this you've seen the worst. Like last week we'll start with some simple problems to learn the technique, and then turn our attention back to the question we really care about, how to diminish pollution.

So, back to pivoting this week, but this time we'll have our computers do the pivoting for us using an online tool. (Excel can do this, but only through its Solver add-in which is not included in the educational edition of Office 365).

The end of the beginning

This week marks the end of the first segment of the course on linear models. Next week we will turn to nonlinear models and from there work our way through some topics from calculus that will enable us to model change and accumulation in systems.

Don't stay stuck!

We all get stuck working on problems, but my heart sinks when students tell me that they spent two hours (or more!) stuck on a problem. Unless they were steadily trying new things for those two hours, it was time wasted. Time spent stuck is not time spent learning.

I you get stuck on how to proceed on a problem, check your notes and the course resources, give it ten minutes, but then set it aside and move on to the next one. If you get stuck on that one too, it's time for a break. Switch tasks for a bit. If you don't get any new ideas ask for help! If it's an office hour day you can see me there, but you can email me anytime. Or post to the discussion forum. I check both fairly frequently, and will get back to you! 🤗 Hopefully with a way to get unstuck.

Reminder: Exercise Sets 4.1 and 4.2 are due by 4 pm today!

I am especially looking forward to seeing your course surveys! I'll prepare a summary of your feedback and share it with you later this week so we'll all have a sense of where we're at.

Office Hours: Get help, or just come by to say "Hi"

I'll have office hours again this week (and every week) on Monday, Wednesday and Friday from 10:30 am to 11:30 am in my Zoom "office". I hope you'll come by at some point with your questions or just to say hi.

That's all for now!

Stay well and stay in touch,
Tim