Yesterday I realized that all my classes this semester involve math to some degree. In Elements of Macroeconomics, we’re calculating various economic metrics and representing consumption and income using aggregate expenditure models. In Public Health Biostatistics, we’re applying basic statistical knowledge to medical problems, focusing on whether or not there is a significant difference between the medical expenditures of smokers and non-smokers. In Differential Equations, we’re currently characterizing equilibria, analyzing asymptotic behavior of functions, and using these in applications such as population growth and chemical reaction rates. And lastly, in Intro to Optimization, we’re covering the basics of linear programming to minimize certain objective functions under various constraints.
While I do enjoy getting my numbers fix in every single one of these classes, there’s something fantastic about my optimization class. Affectionately referred to as “opti”, the course’s main focus is on how to find the best solution to a given problem, theoretical or real-world. How do we minimize waste? Minimize cost? Maximize profit? Where is the largest function value within this feasible region? These are the types of questions we are currently discovering how to answer and back up with mathematical evidence. I’ve been finding myself extremely interested in what we’re learning, and actually having a little bit of fun finishing our assignments. I was joking with my friend about how this past week’s homework took an ungodly amount of effort to do but that “there was beauty in the struggle” (I swear – I said it ironically). Take a look at the last problem of the homework I finished yesterday. You don’t have the read the whole thing unless you’re actually interested; the point is just that it is ridiculously long:
(This problem comes from Winston and Venkataramanan, originally from Sullivan and Secrest.) Lizzie’s Dairy produces cream cheese and cottage cheese. Milk and cream are blended to produce these two products. Both high-fat and low-fat milk can be used to produce cream cheese and cottage cheese. High-fat milk is 60% fat; low-fat milk is 30% fat. The milk used to produce cream cheese must average at least 50% fat, and that for cottage cheese at least 35% fat. At least 40% (by weight) of the inputs to cream cheese and at least 20% (by weight) of the inputs to cottage cheese must be cream. Both cream cheese and cottage cheese are produced by putting milk and cream through the cheese machine. It costs $0.40 to process 1 lb of inputs into into a pound of cream cheese. It costs $0.40 to produce 1 lb of cottage cheese, but every pound of input for cottage cheese yields 0.9 lb of cottage cheese and 0.1 lb of waste. Cream is produced by evaporating high-fat and low-fat milk. It costs $0.40 to evaporate 1 lb of high-fat milk, and each pound of high-fat milk that is evaporated yields 0.6 lb of cream. It costs $0.40 to evaporate 1 lb of low-fat milk, and each pound of low-fat milk that is evaporated yields 0.3 lb of cream. Each day, up to 3000 lb of input may be sent through the cheese machine. Each day, at least 1000 lb of cream cheese and 1000 lb of cottage cheese must be produced. Up to 1500 lb of cream cheese and 2000 lb of cottage cheese can be sold each day. Cream cheese is sold for $1.50 per lb and cottage cheese for $1.20 per lb. High-fat milk is purchased for $0.80 per lb, and low-fat milk for $0.40 per lb. The evaporator can process at most 2000 lb of milk daily.
Formulate a linear program in canonical form that can be used to maximize Lizzie’s daily profit. In working on this problem, provide the matrix A and vectors b and c for the LP.
The problem actually took almost an entire page just to describe what was happening in this situation. Just trying to figure out what was even happening with this godforsaken cottage cheese was confusing at first, but I painstakingly began to translate these constraints — e.g. the milk used in production of cream cheese must average at least 50% fat is simply the constraint equation,
(.60x + .30y)/(x+y) ≥ .50
where x = weight of high-fat milk and y = weight of low-fat milk.
And as time went on I realized that I’d been actually enjoying creating this linear program to help out poor Lizzie and her low-efficiency dairy business. So there really is “beauty in the struggle” in my optimization class, but I’m still trying to figure out a less pretentious way of saying it.
In all seriousness though, this course reminded me why I wanted to major in applied mathematics and statistics in the first place. I like math because while it’s all based in theory, it is extremely useful in almost every facet of day-to-day life — and I find both the theory and the application of it to be pretty amazing. This dairy problem proves the latter, but take a look at this definition of an interior point of a set, S.
x is said to be an interior point of S if there exists a ? > 0 such that (x – ?, x + ?) ⊆ S
For all you anti-math kids out there, what this is essentially is saying is that if you have a certain space (a set), (like a circle, a rectangle, a strangely-shaped blob figure), called S, then a point, let’s call it x, is defined as an “interior point” of this set if you can draw a circle with radius ? around this point x and have the entire circle still be within the set S.
Thinking about it more deeply, consider a point (let’s call it a) that is an interior point really, really, really close to the “edge” (or boundary) of this set S. Like super microscopic close. Say that the distance between this point a and a point on the boundary (let’s call it b) is .00001. Wow, that’s a tiny distance, right? These two points are basically RIGHT next to each other. Guess what though, a is still an interior point, because you can draw a circle around it with radius .000001. And as you move this point a closer and closer to the boundary, you’ll still NEVER have a be a boundary point, no matter how close it gets to the boundary because even if the distance between a and b is .0000001, you can create a circle around a with radius .00000001 and so on and so forth. Basically, you can move a infinitely closer and closer to the boundary, and you can always create a circle that is still within the set S, no matter how close your point is to the boundary. There’s an infinite amount of interior points between .000000001 and .0000000001 — but also an infinite amount of interior points inside the entire set S — how INSANE is that to fathom? Two separate infinities with seemingly different sizes but that are yet… the same?
That was an extremely long aside, but the point is that math is, in my mind, seriously cool. (All my math teachers would be proud of me, I think). And I’m really glad I get to study it here at Hopkins. If you’re a student here, you know that the spectrum of things your friends are passionate about is endless. I say, take a second to really acknowledge and admire everything you’re learning — and then proceed to geek out about whatever it is you enjoy — French colonialism, social theory, biomaterials, Klimt, or linear optimization because frankly, you just should.