QE: Here’s a simplified version of real problems that occur in the electric power industry. That reminds me, I really encourage you to consider looking for careers in that industry--it’s pretty important!
Suppose that at Generator A, it costs $20+$1*x dollars per megawatt to produce x megawatts this hour. At Generator B, it costs $30+$1.1*x dollars per megawatt to produce x megawatts this hour. Dollar amounts in this problem are actually somewhat realistic.
i) You need a total of 10 megawatts this hour. What’s the best way use some of A and some of B to get 10 megawatts total? It’s a good idea to do it both “by hand” and by graphing your overall objective function in Desmos and spotting the best point.
ii) Did you notice that Generator B costs more than A no matter what? Did Generator A therefore produce all the power, and B none of the power, in the optimal solution? What do you think?
QF: Here’s a simplified version of real problems that occur in various manufacturing industries.
Suppose that at Factory A, it costs sqrt(2/x) per car to make x cars this week, and at factory B it costs sqrt( (2.1/x)+(400/x^2)) per car to make x cars this week. Dollar amounts in this problem are not meant to be realistic, but the general trends are realistic.
i) You need a total of 2500 cars this week. What’s the best way to achieve that? Here, I’d give up on the by-hand approach and just find the best solution graphically.
ii) Factory B has a larger first coefficient than Factory A (2.1 vs 2.0); did this prevent Factory B from getting any production in the optimal solution?
iii) How is this problem different than the electric-power problem above?
iv) How are the undersea-power-cable, the dog-water-fetch, the subway-vs-drive, the electric-power, and the car-manufacturing problems similar in terms of modeling and #variables? Are any other problems in this homework similar to those?
QG: Chapter 3.1 warned us that the optimal solution isn’t always where the derivative=0 or is undefined; sometimes it’s at an endpoint of the interval. What problem(s) in this homework (if any) had the optimal solution at an endpoint instead of at a critical point of the objective function?
1 answer