Every year, our final year students are offered the chance to do a research project. These are credit baring and take the place of another taught course. Allocating students to project is not simple task as many students might want to work on the same project for example. In this blog post I’ll describe the mathematical model used to quantify how good an allocation is and then show how we used Python (specifically the pulp library) to get the best possible allocation.

In our School the process of allocating project starts with all members of staff writing project descriptions which are then circulated to students. Students are then asked to rank their top 4 projects.

## Problem inputs

Assuming we have $$I$$ students and $$J$$ projects, this can be translated mathematically to a preference matrix: $$A\in\mathbb{Z}^{I\times J}$$ where $$A_{ij}$$ is the rank of project $$j$$ assigned by student $$i$$. If student $$i$$ did not pick project $$j$$ then $$A_{ij}=0$$.

So for example the following matrix for $$I=4$$ and $$J=8$$ implies that the second student’s 3rd choice project would be project 7:

Because of the combinatorial nature of things: not all students can be guaranteed a project, however:

1. Some students must be given a project: in our case this corresponds to our Masters students.
2. When all else is equal the tie breaker between two students will be their year average from the previous year.

These two things can be captured mathematically by two vectors: $$\alpha, \omega \in\mathbb{R} ^ I$$, where $$\alpha_i$$ is the average mark of student $$i$$ and:

We will return to $$\alpha, \omega$$ shortly but first let us consider how we will represent a given allocation. This will be denoted by a matrix $$x\in\{0, 1\}^{I\times J}$$ where:

So for example the following represents an allocation where all students would get their first choice:

An important consideration when allocating students is not just the student preference but also the workload implications on staff who supervise them:

1. Projects can only be allocated a number of times (in our School this number is 2). Theoretically this could differ on a project by project basis and will be represented by $$\chi\in\mathbb{Z}^J$$.
2. Supervisors (who can offer multiple projects) can only take on certain number of students. This is usually 2 but in some cases for members of staff with high workloads this is 1.
3. Supervisors (who can offer multiple projects) can only take on a certain number of “credits”: for example Masters projects correspond to twice as many credits (40) as Bachelors projects (20). In general supervisors are limited to 40 credits (so 2 Bachelors projects or 1 Masters project).

Assuming there are $$K$$ supervisors these upper bounds will be denoted by:

We capture the projects supervised by each member of staff using $$S\in\mathbb{Z}^{K\times N}$$ where $$S_{kj}$$ is the number of credits of project $$j$$ if it is offered by supervisor $$k$$ (0 if not).

Now there are a number of constraints on what is to be considered a valid allocation:

## Problem constraints

### Student can only be allocated to projects they have chosen:

(Thus if $$A_{ij}=0$$ $$x_{ij}=0$$.)

### Every supervisor can only supervise a given number of projects

For the number of students each supervisor can take on we have the constraint:

For the number of credits each supervisor can take on we have the constraints:

## Evaluating a solution

Our goal is to give as many students as possible as high a ranked project as possible. If all things are equal we will use their marks in previous years to give them a better chance. Note that due to the combinatorial nature of this problem, that does not mean an individual student with a mark higher than an other will get priority over that student. Because of the feasability constraints and the numerous interactions this will not necessarily occur.

The objective function used is (in a maximisation framework):

thus $$\alpha_{i}/A_{ij}$$ is a scaling assigned to project $$j$$ when picked by student $$i$$:

• If the project is highly ranked it (so less preferable) it contributes less;
• If the student has a high mark it contributes more.

## Finding the best solution

An important part of the objective function $$c$$ is that it is linear in $$x$$ (there are no terms like $$x_{13} ^ 2$$). This implies that we can use a neat mathematical tool call Integer Linear Programming to find a solution to the problem of optimising $$c(x)$$ over the constraints listed above.

I’ve written about this sort of thing a few times now, if you’re interested in seeing other examples take a look at:

In this particular case, once the various inputs have been read in (in our case we did this by reading in an excel spreadsheet) the following Python function is all we need:

def create_prob(A=A, alpha=alpha, S=S, chi=chi, beta=beta, kappa=kappa, omega=omega):

number_of_students, number_of_projects = A.shape
prob = pulp.LpProblem("Project matching", pulp.LpMaximize)
x = pulp.LpVariable.dicts("x", itertools.product(range(number_of_students), range(number_of_projects)),
cat=pulp.LpBinary)  # Variables

objective_function = 0
for student in range(number_of_students):
for project in range(number_of_projects):
if A[(student, project)] > 0:
objective_function += x[(student, project)] * (alpha[student]) / ((A[(student, project)]))
prob += objective_function

# The upper bound
for student, project in x:
prob += x[student, project] <= float(A[student, project])

# At most chi_j student for every project
for project in range(number_of_projects):
prob += sum(x[(student, project)] for student in range(number_of_students)) <= chi[project]

# At most 1 project per student
for student in range(number_of_students):
prob += sum(x[(student, project)] for project in range(number_of_projects)) <= 1

# At least omega_i project per student
for student in range(number_of_students):
prob += sum(x[(student, project)] for project in range(number_of_projects)) >= omega[student]

# Number of projects supervised
for k, supervisor in enumerate(S):
prob += sum(supervisor[project] / (max(1, supervisor[project])) * sum(x[(student, project)]
for student in range(number_of_students))
for project in range(number_of_projects)) <= beta[k]

# Number of credits supervised
for k, supervisor in enumerate(S):
prob += sum(supervisor[project] * sum(x[(student, project)]
for student in range(number_of_students))
for project in range(number_of_projects)) <= kappa[k]

return prob, x

The following then creates the problem and solves it:

>>> prob, x = create_prob(A=A, alpha=alpha, S=S, beta=beta, kappa=kappa, omega=omega)
>>> prob.solve()
1

The return of 1 simply indicates that the solver arrived at the solution successfully.

### Reflections

Note that all of this can be implemented and done directly in Python which is Open source. The one library that is needed is called pulp which can be installed straightforwardly using pip install pulp.

The solution approach here is exact: it uses integer linear programming, theoretically, as the problem becomes more complex it could take more computational time to find a solution. In these cases heuristic algorithms such as simulated annealing or genetic algorithms can be used. However in practice, this should scale readily for the “usual” class size.

One of the great uses of this type of approach is that it is first of all transparent (if students ask why they have been allocated a certain project we can say why). Second of all, it takes no time to create a new solution given a set of inputs. Whilst it took a bit of time this Summer to write down the mathematical model and the Python code (most of the difficulties with the code was actually reading in the various inputs), next year this should take no time at all.

Finally, the mathematical model is generic enough to be usable by others however if there are certain constraints that are not captured these should be (famous last words) straightforward to add.