Allocates projects for the Department of Physiology, Development, and Neuroscience, University of Cambridge.

By Rudolf Cardinal ([email protected]), Department of Psychiatry.

GNU GPL v3; see https://www.gnu.org/licenses/gpl-3.0.en.html.

• Create and activate a Python 3 virtual environment.

• Install direct from github:

  pip install git+https://github.com/RudolfCardinal/pdn_project_allocation

You should now be able to run the program. Try:

pdn_project_allocation --help

To run some automated tests, change into a directory where you're happy to stash some output files and run

pdn_project_allocation_run_tests

This produces solutions to match the test data in the pdn_project_allocation/testdata directory.

• There are a number of projects p, and a number of students s.
• Every student needs exactly 1 project.
• Every project can take a certain (project-specific) number of students.
• Every supervisor may, optionally, set a limit on the number of projects they can run (from the ones they offered), and/or the total number of students they can take.
• Students rank projects from 1 (most preferred) onwards.
  • If they don't rank enough, they are treated as being indifferent between all other projects.
• Output must be consistent across runs, and consistent against re-ordering of students in the input data.
• Supervisors can also express preferences (on a per-project basis; the students they prefer for one project might not be the students they prefer for another). The overall balance between "student satisfaction" and "supervisor satisfaction" is set by a parameter.
• Some student/project combinations can be marked as ineligible (e.g. a student might not have the necessary background, no matter how much they want the project).

Absolute constraints

• The course administrator and the supervisors determine which projects are available, how many students each project can take, and, optionally, how many projects each supervisor can actually run simultaneously, and/or how many students each supervisor can take in total.
• The course administrator and the supervisors also determine student eligibility for each project. (Students might be ineligible because they're taking the wrong course modules, or because the supervisor requires that all students pre-discuss projects with them and these students haven't, etc.)

Preferences

• Students rank their projects: 1 (best), 2 (next best), and so on.
• Supervisors rank potential students: 1 (most preferred), 2 (next), and so on.
• The course administrator decides how much to weight student preferences (e.g. 70%) versus supervisor preferences (e.g. 30%).

More on preferences

• Preferences are expressed via dissatisfaction scores. Ranks are naturally dissatisfaction scores: if I rank something as #1, and something else as #2, I am happiest when I get my lowest score.
• If I am a student and there are 20 projects, I have an allowance of 1 + 2 + 3

  + ... + 20 = 210 dissatisfaction points. My course administrator might allow me to rank all 20 projects.

• Alternatively, I might choose to rank only 5 (or the course administrator might permit only this). In that case, I will have "used" up scores 1, 2, 3, 4, and 5 (totalling 15). I will then have 15 unranked projects, and 195 unallocated dissatisfaction points. In this situation, each of those 15 projects will be given a dissatisfaction score of 195/15 = 13.
• If the adminstrator permits, I might also express ties. For example, I might express "joint second and third" by giving preferences 1, 2.5, 2.5, 4, 5.
  • This is "fractional" ranking (https://en.wikipedia.org/wiki/Ranking). It's the default input format. It's the only format used in calculation, because alternatives are biased, discussed below. (And it's the only output format, because this makes output ranks consistent with other output scores, which include means/sums of ranks.)
  • You can choose to use "competition" ranking as an input format; the equivalent competition ranks for our example would be 1, 2, 2, 4, 5.
  • You can also choose "dense" ranking as an input format; that would be 1, 2, 2, 3, 4 in our example.
  • Note that both competition and dense rankings are biased, so must not be used for calculation. For example, if person A rates two projects top and then a third project (competition ranks 1, 1, 3, totalling 5; dense ranks 1, 1, 2, totalling 4), and person B ranks three projects in order (1, 2, 3, totalling 6), then there are more ways to make person A completely happy; the maximization of overall happiness favours person A. In fractional ranking (in which person A would be 1.5, 1.5, 3, totalling 6), every person has the same notional number of "dissatisfaction points" to allocate. Rankings are always converted immediately to fractional rankings internally.
• The program enforces the requirement that a student's scores for all the projects (those explicitly ranked and those ranked by default) must add up to the total dissatisfaction (210 in this example). It also enforces that students can only allocate "from the best upwards in rank" -- for example, if the student expresses 5 preferences, those scores must add up to 15 (you can't say "1, 2, 3, 4, 6").
• Supervisor preferences are handled in exactly the same way. Each project supervisor can rank all of the students, or rank some (being indifferent between the others), or not rank anyone (having no preference between any students). Their dissatisfaction scores are calculated in exactly the same way.

If you are trying to express that "the student absolutely cannot do this project", see eligibility above.

If you're the course administrator, consider letting students and supervisors express as many preferences as they want. It won't cause any harm and may sometimes help, if competition is fierce for projects.

Optimization

• Within hard constraints (every student needs a project; maximum number of students per project; eligibility)...
• ... the program maximizes total satisfaction (minimizes total dissatisfaction).
  • Specifically, every student-project pairing is associated with dissatisfaction from the student, and dissatisfaction from the supervisor, as described above. These are weighted (e.g. 70% student, 30% supervisor, as above). The total weighted dissatisfaction score is minimized.
• Optimization is achieved via the Python-MIP package (https://python-mip.readthedocs.io/), which solves so-called mixed integer linear programming problems. This impressive software suite finds optimal solutions efficiently.

Fairness

• Algorithmic assignment is fair compared to human assignment, in that it prevents people "cherry-picking" during manual allocation. It's also fair in that it maximizes an objective measure of "happiness" (even though that won't exactly reflect real-world happiness).
• It is almost guaranteed, as a reflection of human nature, that students and supervisors who didn't get what they wanted will complain about the results (or the method). Anticipate this by getting everyone to agree to the procedure in advance. Ensure that supervisors are clear about any absolute eligibility criteria, convey these to the administrator along with their preferences, and agree to accept the result.
• If you run the program several times with the same input, you will get the same answers. (It would be unfair otherwise: there would be a temptation to keep "flipping the coin" until the operator gets the answer they want.) The program achieves this by shuffling its inputs in a "deterministic random" way (via a random number generator seed).
• The code is open-source and free for all to use or inspect.

Advanced options

• The course administrator may choose to say that students can only be allocated to projects that they've explicitly ranked. (For example, if a student chose 5 most-preferred projects, only those projects can be allocated to that student.) However, this may cause the algorithm to fail: there may be no such solution. (It is also open to "gaming" if a student is allowed to enter only one preference!) If it fails, the program will say so.
• By default, a dissatisfaction score of 2 is "twice as bad" as a score of 1 (dissatisfaction is linear). Optionally, the course administrator may set this to be non-linear by raising dissatisfaction scores to a power (exponent). For example, an exponent of 2 would map dissatisfaction scores of {1, 2, 3, ...} to {1, 4, 9, ...} for the optimization step.

Happiest on average, or stable?

• The basic solution is not always stable (the technical meaning of stability is given below). A supposedly optimal "stable" algorithm did not provide projects for all the students with our Sep 2020 real-world data (see below), so that wasn't much use. In response to this, firstly the software will report and explain any instability. Secondly, moreover, a new (?) algorithm is developed to produce stable solutions despite non-strict preferences, and this is now an option. It's possible to say "stable if you can, optimal if you can't". There can still be a tradeoff between average "happiness" and stability, which is up to the course administrator to decide on. (Instability may produce more complaints!)

This is an "assignment problem" or "maximum weighted matching" problem (see https://en.wikipedia.org/wiki/Assignment_problem).

It is different from the "stable marriage problem" (see https://en.wikipedia.org/wiki/Stable_marriage_problem), used for hospital/resident matching in the US via the Gale-Shapley algorithm and derivatives (https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm; https://www.nrmp.org/nobel-prize/). The stable marriage problem aims to pair couples (person A and B in each couple) such that there is no pairing A1-B1 where A1 prefers another (B2) over their allocated B1, and B2 also prefers A1 to their own allocated partner. That would be unstable, because A1 and B2 could run away together.

"Maximum satisfaction" problems aren't always stable, and vice versa (see e.g. Irving et al. 1987, https://doi.org/10.1145/28869.28871, and examples at https://en.wikipedia.org/wiki/Stable_marriage_problem#Different_stable_matchings).

A supposedly optimal stable algorithm for student-project allocation is that by Abraham, Irving & Manlove (2007, https://doi.org/10.1016/j.jda.2006.03.006), or "AIM2007". The "two algorithms" of the title are the one that is student-optimal, and the one that is supervisor-optimal. These algorithms are implemented in the Python matching package (https://matching.readthedocs.io/). In theory, this also brings extra sophistication, such as the ability to set supervisor capacity as well as project capacity. However, that implementation can fail completely (e.g. test example 4 in the testdata directory), by failing to allocate some students to any project. The example has no specific supervisor preferences, ten projects each with capacity for one student, and preferences like this:

P1  P2  P3  P4  P5  P6  P7  P8  P9  P10

S1 1 2 3 S2 1 2 3 S3 1 2 3 S4 1 2 3 S5 1 2 3 S6 1 2 3 S7 2 3 1 S8 3 1 2 S9 1 2 3 S10 1 2 3

The AIM2007 algorithm gave:

Preferences (re-sorted):

For student S1, setting preferences: [P1, P2, P3]
For student S2, setting preferences: [P1, P2, P3]
For student S3, setting preferences: [P4, P5, P6]
For student S4, setting preferences: [P4, P5, P6]
For student S5, setting preferences: [P7, P8, P9]
For student S6, setting preferences: [P7, P8, P9]
For student S7, setting preferences: [P10, P1, P2]
For student S8, setting preferences: [P9, P10, P1]
For student S9, setting preferences: [P8, P9, P10]
For student S10, setting preferences: [P5, P6, P7]
For supervisor Supervisor of P1, setting preferences: [S2, S8, S1, S7]
For supervisor Supervisor of P2, setting preferences: [S2, S1, S7]
For supervisor Supervisor of P3, setting preferences: [S2, S1]
For supervisor Supervisor of P4, setting preferences: [S3, S4]
For supervisor Supervisor of P5, setting preferences: [S3, S4, S10]
For supervisor Supervisor of P6, setting preferences: [S3, S4, S10]
For supervisor Supervisor of P7, setting preferences: [S5, S6, S10]
For supervisor Supervisor of P8, setting preferences: [S5, S6, S9]
For supervisor Supervisor of P9, setting preferences: [S8, S5, S6, S9]
For supervisor Supervisor of P10, setting preferences: [S8, S9, S7]

Result:

st  pr  student's rank
S1  P2  2
S2  P1  1
S3  P4  1
S4  P5  2
S5  P7  1
S6  P8  2
S7  --  --  [projects P1, P2, P10 already taken; P3 free but student didn't want it]
S8  P9  1
S9  P10 3
S10 P6  2

The AIM2007 algorithm requires each supervisor to rank all those students that have ranked at least one of their projects (https://matching.readthedocs.io/en/latest/discussion/student_allocation/index.html#key-definitions). In the absence of a real ranking, we have to give an arbitrary order. Nonetheless, in this example, an order was given, across all students who picked that project, and the algorithm (or this implementation) failed.

In contrast, dissatisfaction minimization solves this happily, e.g. with

st  pr  student's rank
S1  P1  1
S2  P3  3
S3  P4  1
S4  P5  2
S5  P9  3
S6  P7  1
S7  P2  3
S8  P10 2
S9  P8  1
S10 P6  2

... which is also stable, as it happens.

Likewise, with real data (Sep 2020), large numbers of students were unallocated by this method.

So: a potential extension for future years is to extend supervisor rankings and retry an algorithm such as AIM2007, but it can't (apparently) cope with the current situation.

Another possibility is that the algorithm would have worked if students ranked more projects. However, that would seem unsatisfactory in the sense that it would necessarily involve more dissatisfaction to bring stability.

Another possibility is that this is just a known failure mode of AIM2007. For example, Olaosebikan & Manlove (2020, https://doi.org/10.1007/s10878-020-00632-x) note that "... exactly the same students are unassigned in all stable matchings", and their Algorithm 1 has a termination condition of "until every unassigned student has an empty preference list" (not that no students are unassigned!).

We can go one step further, and enforce stability via integer linear programming, as per Abeledo & Blum (1996, https://doi.org/10.1016/0024-3795(95)00052-6). However, the algorithm assumes strict ordering (e.g. that each student strictly ranks all projects, and each supervisor strictly ranks all students that apply to their projects).

Since we can't have any student unassigned, and we are now up to Aug 2020 in the research literature, I've done two things: (a) offered a stability constraint via a (new?) algorithm that does not require strict preferences, allowing "dissatisfaction minimization" within that constraint, or (b) the option to choose, or fall back to, overall dissatisfaction minimization (though that may offer some unstable solutions).

• 2019-10-31: started.
  • Representations.
  • Brute force method.
  • MIP (MILP) method: Mixed Integer Linear Programming Problems.
  • Output.
• 2019-11-01:
  • test framework
  • 1-based dissatisfaction score by default (= rank, probably more helpful given that is the input)
  • Failed to find a clear example where you'd be clearly better off with a worse mean and a better variance.
  • Experimented with power (exponent); not much gain and adds complexity.
• 2019-11-02:
  • Excel XLSX input/output, in addition to CSV.
• 2019-11-03:
  • Excel only (removed CSV).
  • Supervisors can express preferences too.
  • Removed brute force method; now impractical. (With 5 students and 5 projects, one student per project, and no supervisor preferences, the brute-force approach examines up to 120 combinations, which is fine. With 60 students and 60 projects, then it will examine up to 8320987112741389895059729406044653910769502602349791711277558941745407315941523456 = 8.3e81).
• 2020-09-11:
  • Save input data with output.
  • Change default weight to favour students (over supervisors).
• 2020-09-17:
  • Support eligibility.
  • Bugfix to data input checking.
• 2020-09-27, v1.1.0:
  • Option to exponentiate preferences.
  • Configure behaviour for missing eligibility values.
  • Allow projects that permit no students.
  • Show project popularity.
  • Handle Excel sheets that appear to have 1048576 rows (always).
  • Tested with real data.
  • Speed up spreadsheet reading; student CSV output (e.g. for Meld).
• 2020-09-28 to 2020-09-29, v1.1.1:
  • Shows median/min/max in summary statistics.
  • --seed option (for debugging ONLY; not fair for real use as it encourages fishing for the "right" result from the operator's perspective).
  • Improved README.
  • Options to use the AIM2007 algorithms, as above.
  • Options to enforce stability via the MIP approach, and to try that but fall back to the overall "least dissatisfaction" approach (now the default).
  • New algorithm to produce a stable solution (and within that, the best stable solution) even despite non-strict preference orderings.
• 2020-10-05, v1.2.0:
  • Renamed column Project_name to Project in the "Projects" spreadsheet.
  • New "Supervisors" spreadsheet.
  • Optional constraint: maximum number of students per supervisor.
  • Optional constraint: maximum number of projects per supervisor.
• 2021-10-03, v1.3.0:
  • Support multiple supervisors per project. (Per-supervisor constraints continue to work.)
  • More helpful error messages.
  • Deal with superfluous whitespace (e.g. around project/student names).
  • Bump mip from 1.5.3 to 1.13.0.
  • Bump cardinal_pythonlib from 1.0.96 to 1.1.7.
  • Bump openpyxl from 3.0.5 to 3.0.9.
  • Bump lxml from 4.4.1 to 4.6.3.
  • Bump matching from 1.3.2 to 1.4.
• 2021-10-03, v1.4.0:
  • Project short titles, used as column headings.
• 2022-22-09, v1.5.0:
  • Support competition and dense rankings as input formats (but retaining fractional rankings for calculation and output).