0006. It's the end of the semester and it's time to optimize. I have a sociology final exam and a public policy final exam on the same day and I can only allocate 16 hours total to exam preparation.
I estimate that if I do not study at all, I'll get C- in public policy and a C+ in sociology. But, experience has shown that studying pays off. In sociology, each hour of study raises my grade by 1/3 (0.33) of a letter grade (C- to C, for example). In public policy, the payoffs are not as quick: each 2 hours yields 1/3 of a letter grade (0.167 grade per hour).
Thus, 3 hours of sociology study would take me from C+ to B+

C 1 hour > B- 1 hour > B 1 hour > B+

Our plans are subject to a number of constraints:

  • There is a four hour public policy study session planned with the professor and it would be unwise and impolitic not to attend. Thus, PPOL hours has to be at least 4.
  • Since the top grade is A, sociology will top out at 5 study hours since these will take me from C+…B-…B…B+…A-…A and in public policy my max is correspondingly 14 with each 2 hours taking me one grade of the seven steps from C- to A.
  • Total study hours must be 16 or less.
  • Since I am a public policy major, I feel strongly that I have to study more for PPOL than for SOCIOLOGY

I translate these into five inequalities

  • (Only 16 hours ) PPOL = -SOC + 16
  • (More PPOL than SOCIOLOGY) PPOL ≥ SOC
  • (PPOL Study Group) PPOL ≥ 4
  • (SOCIOLOGY max) SOC ≤ 5
  • (PPOL max) PPOL ≤ 14

I want to know how to divide my study time so as to maximize my grades. Let's start with a typical Excel layout for a simple LP problem.

LP-setup-1.gif
LP-chart-01.gif

Identify the lines in this chart, indicate the feasible set, and find the optimal value of the variables and the resulting value of the objective function.