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Constrained optimisation to allocate modules at York

A week ago today I handed in my final-year project in Computer Science (the infamous dissertation that the fine people of Twitter are completely bored of hearing me talk about). Like most in the department, mine consisted of about 20,000 words over seventy pages and a ten minute presentation describing the work I’d been doing for the last year or so. According to the staff involved, a University software project implemented by a student was something that had never been tried before, and as I had such a fun time doing it I thought I’d write a bit here. The aim was to create software that allocates students to modules.

The project had two fairly distinct halves: a web interface to let students express their preferences for optional modules, and a bit of software to take that input and do the allocation. The end result had to satisfy constraints from the departments (e.g. a class can’t have more than x students), while also trying to keep students as happy as possible - it’s a constrained optimisation problem. The more I think about it, this project could not have been any more perfect for me; half web stuff, half maths. Lovely.

The web development side of things was nothing too outrageously challenging. A lot of research and testing with staff and students to make sure it was as easy to use as possible, though the end product is just a jQuery sortable implementation. Those of you who have used the YUSU site to vote during elections will recognise that this software goes for the same mental model. Students in the Departments of History and Archaeology noticed that too, which was wonderful. Here’s a bit of video of one of the earlier functional prototypes (on an iPad, because iPads make everything cooler) - please excuse the music:

The constrained optimisation and linear programming stuff was where it started to get really interesting from a computer science point of view. The first I’d heard of CO was in about May last year, so having an expert supervisor was more than a little helpful. We used a solver (in this case Gurobi) to save a whole load of work on the implementation, and I’m shocked by how powerful it is. It’s got wonderful interfaces for several languages and is completely free for academic use! Linear programming and all that related gunk is such a huge field that if you try hard enough I bet you’ll be able to think of somewhere you could use it.

Once it was all set up, it’s just a case of creating a ton of (binary) variables, defining the objective function (what the solver should try to achieve), loading in the constraints and hitting the big red button. This system used the following:

  • Binary variables: one for every possible allocation (an allocation is a student, module pair), set to 1 if the student is allocated the module or 0 otherwise.
  • Objective function: a linear function that can be maximised or minimised by the solver. I chose to include every binary variable with a coefficient indicating the “goodness” of that allocation, which is based on the rank the student gave.
  • Constraints: each class had a maximum and minimum size, and each student had to take the required number of classes.

We’re talking about painfully simple code to interact with Gurobi, too. Here’s a bit of it in Python:

from gurobipy import *

model = Model("modalloc") # new Gurobi model
gurobivar = model.addVar(vtype=GRB.BINARY, name=student+"_"+module) # new binary var
objfn = LinExpr() # objective function
objfn.addTerms(rank_coeff[ranks[(student, module)]], gurobimavs[student][module])
model.addConstr(num_students <= modules[module]['max'], "classmax") # constraint

(There’s a slightly more fully-featured gist available, if you’re that way inclined…)

I think what I love most about this software is that it’s a great example of a nice modular web project: none of the parts by themselves are horrendously complex, but they come together to make a system that actually solved a problem - code I wrote was used by 800 students in two departments, and the data it generated is now stored in the central student database. All my fingers are crossed that IT Services in York will be able to take the code on, spruce it up a little and offer it to many more departments next year. The report’s on GitHub, but private at the moment. I’ll make it public once it’s marked if I’m able to.

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