A library to assist registrars in partitioning students into groups for socially-distanced learning.
I am a computer science teacher at a private highschool in Northern California. We are being asked to prepare our students for a "partiioned" schedule when we return to school in the fall.
We intend to assign to each student one of two possible labels or colors ("blue and gold" for reference here) - and on each given week, only the students from one group are allowed to visit school in person. We're looking for an assignment of these colors to each student in a way that minimizes an objective function - in our school's case, we care about minimizing two things: the color-variance inside a class, and the gender-variance within these color groups.
Suppose you had a class of 15 students, and you randomly assign labels. You might end up with six students in one color, and nine in another. This is less optimal than a class with eight students in one color and seven in another. Because we're striving for socially distanced learning, it is our goal to ensure that no class accidentally has a large amount of students from one label present on any given day. That is, we're trying to minimize the "imbalance" in a classroom across different labels - this is minimized perfectly when the counts of students for each label is equal, or when the sample variance of the label counts is zero.
Additionally, we'd like to ensure that our classes are as gender balanced as possible within these blue and gold labels.
Finally, we'd like to constraints students who have siblings at the school such that all students in one family share the same color, to minimize logistics overhead for the family.
Instead of relying on random labelings to get good partitions, we can use some extra information: a mapping from students-to-classes, as well as a sibling collection.
We model this problem as a mixed-integer quadratic program.
The objective function here is a weighted combination of two terms: a "color-variance" term and a "gender-variance" term.
By default, the weight on the color-variance term is 1.0 and the gender-variance term is 0.5.
The color-variance term is summed across all classes. For each class, we compute the "optimal split" as having half the students in the class be blue and half be gold.
The gender-variance term is summsed across all class/color pairs. For each class/color pair we compute the "optimal split" as half the students in the blue section be male and half the students in the blue section be female - same for gold.
Specifically, our variance terms look like the following:
Color-variance: (optimal_people_in_gold - num_gold)^2
Gender-variance: (optimal_females_in_gold_split - num_gold_females)^2 + (optimal_females_in_blue_split - num_blue_females)^2
(note: the color-variance term is invariant on which color you choose, and the gender-variance term is invariant on which gender you choose - these were chosen arbitrarily).
The constraints in our problem are "sibling constraints" - we only require that siblings share the same color.
We encode this as a special "free variable" set for each family, F_i. For each student S_n in that family / set of siblings, we constrain
S_n = F_i
For instance, in a family with three siblings, we have three constraints:
S_1 = F_1
S_2 = F_1
S_3 = F_1
In practice, most students do not have siblings, and of the ones who do, their families are small (not more than three or four members) so this approach does not add a lot of constraints or variables.
We require all variables to be boolean-valued.
party depends on a commercial solver known as Gurobi. Obtaining an academic license is fairly straight forward for qualifying teachers or educators.
party also depends on cvxpy, a nice interface to various LP/QP solvers.
To install our dependencies, download gurobi's installer and set your license up following their instructions (the grbkey thing...) Then start a new virtual environment:
python3 -m venv ./env
Activate the virtual environment:
. env/bin/activate
Install our first dependency with pip:
pip install cvxpy
Install our second dependency manually, with setup.py file. On my machine, the install directory for Gurobi was /Library/gurobi/mac64.
You'll want to navigate there (still inside of the virtualenv!) and run sudo python setup.py install.
Test things out: open a python shell and check that import cvxpy and import gurobipy both work.
Call party.py with the three input files (students.csv, siblings.csv, schedules.csv)
TODO(mrdmnd): make this more flexible - for now this is completely useless to other educators because our data format is weird.
(env) mredmond@jontron68419 party $ python miqp.py data/students.csv data/siblings.csv data/schedules.csv
Using license file /Users/mredmond/gurobi.lic
Academic license - for non-commercial use only
Parameter OutputFlag unchanged
Value: 1 Min: 0 Max: 1 Default: 1
Changed value of parameter QCPDual to 1
Prev: 0 Min: 0 Max: 1 Default: 0
Changed value of parameter Threads to 4
Prev: 0 Min: 0 Max: 1024 Default: 0
Changed value of parameter TimeLimit to 60.0
Prev: inf Min: 0.0 Max: inf Default: inf
Gurobi Optimizer version 9.0.2 build v9.0.2rc0 (mac64)
Optimize a model with 1015 rows, 1518 columns and 13808 nonzeros
Model fingerprint: 0x32912ddd
Model has 855 quadratic objective terms
Variable types: 855 continuous, 663 integer (663 binary)
Coefficient statistics:
Matrix range [5e-01, 1e+00]
Objective range [0e+00, 0e+00]
QObjective range [1e+00, 2e+00]
Bounds range [1e+00, 1e+00]
RHS range [5e-01, 4e+01]
Found heuristic solution: objective 19940.125000
Presolve removed 172 rows and 172 columns
Presolve time: 0.04s
Presolved: 843 rows, 1346 columns, 9305 nonzeros
Presolved model has 843 quadratic objective terms
Variable types: 693 continuous, 653 integer (503 binary)
Root relaxation: objective 3.679375e+02, 7312 iterations, 2.47 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
H 0 0 17151.125000 1.50000 100% - 2s
0 0 367.93750 0 503 17151.1250 367.93750 97.9% - 2s
H 0 0 894.8750000 367.93750 58.9% - 3s
H 0 0 737.8750000 367.93750 50.1% - 5s
0 0 368.20342 0 503 737.87500 368.20342 50.1% - 7s
0 2 368.20342 0 503 737.87500 368.20342 50.1% - 7s
3 8 368.20342 2 638 737.87500 368.20342 50.1% 10.3 10s
126 135 371.39120 18 578 737.87500 368.20342 50.1% 21.1 15s
274 291 376.67650 38 549 737.87500 368.20342 50.1% 24.5 20s
H 396 416 724.1250000 368.20342 49.2% 24.4 23s
510 552 388.54260 70 482 724.12500 368.20342 49.2% 24.3 25s
725 773 400.60557 106 440 724.12500 368.20342 49.2% 22.6 30s
H 736 773 720.3750000 368.20342 48.9% 22.5 30s
H 811 842 675.8750000 368.20342 45.5% 21.9 34s
841 887 407.15140 125 408 675.87500 368.20342 45.5% 21.8 35s
H 843 887 597.6250000 368.20342 38.4% 21.8 35s
H 867 887 575.1250000 368.20342 36.0% 21.6 35s
H 908 911 505.3750000 368.20342 27.1% 21.4 39s
910 1061 409.87962 133 408 505.37500 368.20342 27.1% 21.4 40s
1801 1886 458.15290 340 211 505.37500 368.20342 27.1% 16.3 45s
2181 2233 382.31801 49 544 505.37500 368.22090 27.1% 16.3 50s
2609 2575 413.02943 131 503 505.37500 368.22090 27.1% 16.3 57s
Explored 2610 nodes (56307 simplex iterations) in 60.04 seconds
Thread count was 4 (of 4 available processors)
Solution count 10: 505.375 575.125 597.625 ... 19940.1
Time limit reached
Best objective 5.053750000000e+02, best bound 3.682208975725e+02, gap 27.1391%
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M|5 |5 |
F|6 |6 |
Intro to Computer Science
_|Blue |Gold |
M|5 |6 |
F|2 |3 |
Foundations 2 Spanish
_|Blue |Gold |
M|5 |5 |
F|5 |6 |
English 2
_|Blue |Gold |
M|4 |3 |
F|5 |6 |
Accelerated Chemistry (H)
_|Blue |Gold |
M|5 |3 |
F|3 |4 |
Foundations 2 Spanish
_|Blue |Gold |
M|3 |3 |
F|4 |3 |
Conceptual Chemistry
_|Blue |Gold |
M|3 |4 |
F|4 |2 |
Algebra 2 W/ Trig.
_|Blue |Gold |
M|2 |3 |
F|5 |5 |
US History
_|Blue |Gold |
M|6 |4 |
F|2 |3 |
Algebra 2 W/ Trig.
_|Blue |Gold |
M|3 |2 |
F|3 |4 |
AP Computer Science A
_|Blue |Gold |
M|4 |5 |
F|3 |3 |
US History
_|Blue |Gold |
M|4 |4 |
F|3 |2 |
Algebra 2 W/ Trig.
_|Blue |Gold |
M|4 |6 |
F|6 |6 |
Introduction to Journalism
_|Blue |Gold |
M|4 |3 |
F|5 |6 |
English 2
_|Blue |Gold |
M|3 |4 |
F|4 |3 |
Conceptual Chemistry
_|Blue |Gold |
M|4 |4 |
F|4 |4 |
US History
_|Blue |Gold |
M|4 |4 |
F|2 |2 |
Intermediate Spanish
_|Blue |Gold |
M|5 |6 |
F|5 |5 |
Photography
_|Blue |Gold |
M|3 |3 |
F|4 |3 |
Analytic Geometry and Algebra
_|Blue |Gold |
M|5 |3 |
F|4 |3 |
Latin 2
_|Blue |Gold |
M|4 |5 |
F|2 |1 |
Yearbook: Publication Design I
_|Blue |Gold |
M|1 |0 |
F|6 |4 |
Latin 3
_|Blue |Gold |
M|1 |1 |
F|1 |1 |
Intermediate Spanish
_|Blue |Gold |
M|5 |5 |
F|6 |6 |
Foundations 2 Spanish
_|Blue |Gold |
M|2 |2 |
F|2 |2 |
Intermediate Spanish
_|Blue |Gold |
M|6 |7 |
F|5 |5 |
Analytic Geometry and Algebra
_|Blue |Gold |
M|4 |3 |
F|5 |5 |
Analytic Geometry and Algebra
_|Blue |Gold |
M|5 |5 |
F|4 |4 |
Photography
_|Blue |Gold |
M|2 |3 |
F|3 |2 |
Analytic Geometry and Algebra
_|Blue |Gold |
M|3 |4 |
F|4 |4 |
Chamber Orchestra
_|Blue |Gold |
M|1 |2 |
F|4 |2 |
Foundations 1 Mandarin
_|Blue |Gold |
M|0 |1 |
F|4 |3 |
English 2
_|Blue |Gold |
M|4 |3 |
F|4 |4 |
English 1
_|Blue |Gold |
M|6 |5 |
F|3 |3 |
Modern World History
_|Blue |Gold |
M|4 |4 |
F|2 |2 |
Physics 1
_|Blue |Gold |
M|4 |4 |
F|5 |6 |
Freshman Seminar A Fall
_|Blue |Gold |
M|21 |20 |
F|17 |17 |
How Music Works 2S
_|Blue |Gold |
M|5 |2 |
F|3 |6 |
English 1
_|Blue |Gold |
M|5 |6 |
F|4 |3 |
Modern World History
_|Blue |Gold |
M|5 |4 |
F|5 |5 |
Physics 1
_|Blue |Gold |
M|4 |5 |
F|4 |3 |
Select Mixed Chorus
_|Blue |Gold |
M|0 |1 |
F|3 |2 |
Analytic Geometry and Algebra (H)
_|Blue |Gold |
M|4 |6 |
F|5 |3 |
Freshman Seminar A Spring
_|Blue |Gold |
M|17 |18 |
F|20 |20 |
Women's Chorus 1S
_|Blue |Gold |
M|0 |1 |
F|6 |5 |
English 1
_|Blue |Gold |
M|4 |5 |
F|5 |4 |
Modern World History
_|Blue |Gold |
M|3 |3 |
F|6 |5 |
Physics 1
_|Blue |Gold |
M|5 |4 |
F|4 |5 |
English 1
_|Blue |Gold |
M|4 |3 |
F|4 |5 |
Modern World History
_|Blue |Gold |
M|3 |3 |
F|4 |6 |
Body Maintenance 2S
_|Blue |Gold |
M|3 |2 |
F|6 |7 |
Modern World History
_|Blue |Gold |
M|4 |6 |
F|5 |4 |
Physics 1
_|Blue |Gold |
M|3 |5 |
F|6 |4 |
Modern World History
_|Blue |Gold |
M|6 |5 |
F|3 |3 |
Analytic Geometry and Algebra (H)
_|Blue |Gold |
M|5 |7 |
F|5 |4 |
English 1
_|Blue |Gold |
M|5 |6 |
F|4 |4 |
Modern World History
_|Blue |Gold |
M|4 |5 |
F|5 |5 |
Beyond Words 2S
_|Blue |Gold |
M|5 |5 |
F|4 |3 |
English 1
_|Blue |Gold |
M|5 |4 |
F|5 |5 |
Physics 1
_|Blue |Gold |
M|4 |4 |
F|5 |5 |
Foundations 2 Mandarin
_|Blue |Gold |
M|5 |5 |
F|2 |1 |
Modern World History
_|Blue |Gold |
M|5 |5 |
F|5 |5 |
English 1
_|Blue |Gold |
M|4 |4 |
F|5 |5 |
Men's Chorus 2S
_|Blue |Gold |
M|4 |4 |
F|0 |0 |
Physics 1
_|Blue |Gold |
M|6 |4 |
F|4 |4 |
Foundations 2 Spanish
_|Blue |Gold |
M|4 |4 |
F|6 |5 |
Foundations 2 Spanish
_|Blue |Gold |
M|3 |2 |
F|0 |0 |
Latin 2
_|Blue |Gold |
M|3 |3 |
F|0 |0 |
English 1
_|Blue |Gold |
M|3 |3 |
F|4 |5 |
Physics 1
_|Blue |Gold |
M|4 |5 |
F|4 |5 |
Foundations 2 French
_|Blue |Gold |
M|2 |2 |
F|2 |3 |
Art of Visual Design 2S
_|Blue |Gold |
M|3 |7 |
F|4 |1 |
Integrated Geometry & Algebra
_|Blue |Gold |
M|3 |3 |
F|2 |3 |
Conceptual Physics
_|Blue |Gold |
M|3 |3 |
F|2 |3 |
Foundations 1 Spanish
_|Blue |Gold |
M|3 |2 |
F|2 |3 |
Modern World History
_|Blue |Gold |
M|4 |3 |
F|2 |3 |
Analytic Geometry and Algebra (H)
_|Blue |Gold |
M|4 |4 |
F|2 |3 |
Body Maintenance 1S
_|Blue |Gold |
M|3 |3 |
F|4 |5 |
Foundations 2 French
_|Blue |Gold |
M|1 |2 |
F|2 |2 |
English 1
_|Blue |Gold |
M|2 |2 |
F|3 |4 |
Art of Visual Design 1S
_|Blue |Gold |
M|4 |3 |
F|5 |5 |
Beyond Words 1S
_|Blue |Gold |
M|6 |7 |
F|1 |3 |
Integrated Geometry & Algebra
_|Blue |Gold |
M|4 |4 |
F|3 |3 |
Conceptual Physics
_|Blue |Gold |
M|4 |4 |
F|3 |3 |
How Music Works 1S
_|Blue |Gold |
M|4 |4 |
F|4 |3 |
Analytic Geometry and Algebra
_|Blue |Gold |
M|1 |1 |
F|5 |5 |
Push Play 2S
_|Blue |Gold |
M|0 |0 |
F|1 |1 |
Yearbook: Publication Design Leadership
_|Blue |Gold |
M|1 |0 |
F|0 |1 |
Push Play 1S
_|Blue |Gold |
M|1 |0 |
F|1 |1 |
Yearbook: Publication Design II
_|Blue |Gold |
M|0 |0 |
F|0 |1 |
Improv: Whose Line is It Anyway 2S
_|Blue |Gold |
M|1 |0 |
F|0 |0 |
Latin 1
_|Blue |Gold |
M|2 |0 |
F|0 |1 |
Drama
_|Blue |Gold |
M|0 |1 |
F|2 |2 |
Foundations 1 Spanish
_|Blue |Gold |
M|1 |2 |
F|0 |0 |
Overall (by assignments)
_|Blue |Gold |Sum
M|1038 |1031 |2069
F|1067 |1075 |2142
Sum|2105 |2106 |4211
Overall (by student count)
_|Blue |Gold |Sum
M|144 |144 |288
F|147 |149 |296
Sum|291 |293 |584
Status: user_limit
Objective Value: 505.3750000000003
Assignments:
(elided for privacy)