Section Allocation

Suppose you have $n$ students in a class who need to be assigned to $m$ discussion sections. Each student needs to be assigned to exactly one section. Each discussion section should have between 6 and 10 students. Suppose an $n \times m$ preference matrix $P$ is given, where $P_{ij}$ gives student $i$'s ranking for section $j$ (1 would mean it is the student's top choice, 10,000 or a large number would mean the student can not attend that section).

The goal will be to get an allocation matrix $X$, where $X_{ij} = 1$ if student $i$ is assigned to section $j$ and $0$ otherwise.

In [11]:
using Convex, GLPKMathProgInterface
# data.jl has our preference matrix, P

X = Variable(size(P), :Bin)

# We want every student to be assigned to exactly one section. So, every row must have exactly one non-zero entry
# In other words, the sum of all the columns for every row is 1
# We also want each section to have between 6 and 10 students, so the sum of all the rows for every column should 
# be between these
constraints = [sum(X, 2) == 1, sum(X, 1) <= 10, sum(X, 1) >= 6]

# Our objective is simple sum(X .* P), which can be more efficiently represented as vec(X)' * vec(P)
# Since each entry of X is either 0 or 1, this is basically summing up the rankings of students that were assigned to them.
# If all students got their first choice, this value will be the number of students since the ranking of the first choice is 1.
p = minimize(vec(X)' * vec(P), constraints)

solve!(p, GLPKSolverMIP())