Applied Simplex Method For Deciding Future Coursework

June 28, 2015

I am pursuing a Masters of Computer Science. Why? Because learning is fun, I desire self-improvement, and because I can. I hope that the gains acquired from the degree support my goal of servitude. This is not a pursuit rooted in career-advancement. My motivation for learning drives my proposed model's objective, generally dubbed Z, which describes the target value I want to maximize at school. Before we can break down what Z is and the components involved, allow me to set the context.

The trouble with beginning my masters program is that there are many course options. Within the program, there are various tracks (or specializations) that a student may take. Each student must select a track. Each track has its own subsets of course categories, generally comprised of inter-related courses. Additionally, each category has minimum credit hour requirements for the courses in the set.

To clarify, suppose the program offers courses c1, c2, ... , c100. track1 may require that you take at least one course from (c1, c3 ) and at least one course from (c2, c10). track2 may require 2 courses from (c2, c3, c30), and so on. There may be one course per category, or there may be 100 courses per category. At my current school of enrollment, I have observed as many as four of these categories falling underneath a single track. I have even observed track catagories define rules that consume each other!

Consider a hypothetical track3. track3 may represent a CS degree with a speciality in robotics. Suppose track3 has the following category set:

  • category_1: take one course from (c1, c2, c3)
  • category_2: take one course from (c2, c4, c5)
  • category_3: take at least 3 courses from category_1 & category_2. In this case, category_3 is a composite category of category_1 & category_2.

Add more courses and more category conditions, and you may begin to see that even within a track, you can receive a very different education experience.

I want to take courses that I am most interested in. The objective variable, Z will represent fun in school. Using a Binary Integer Program (BIP), I suggest a model to assert that I get the maximum fun out of my selected courses. I achieve this by modeling real school-imposed constraints in a basic mathmatical form, as required by the BIP. The BIP consumes the over-arching master's program constraints, the track & category constraints, and a fun rating (or desireability rating) for each course in order to find an optimal solution using the simplex method.

Cool. Let's try it. Set the objective function: (Maximze) Z = c1*C1_DESIRABLITY + c2*C2_DESIRABILITY ... cN*CN_DESIREABILITY

  • Z ~= integer representation of net-fun,
  • cX ~= boolean indicator if I take the course named cX, and
  • CX ~= numeric "fun" vaule of taking course cX

Build a set of constraints to model:

  • take at least one track in the program
    • track1 + track2 ... + trackN >= 1
  • take the required number of graduation credits
    • c1*C1_CREDIT_HRS + c2*C2_CREDIT_HRS ... + cN*CN_CREDIT_HRS = 30
  • satisfy taking the # of courses required by each track category
    • c1 + c2 + c3 - 2*track1 >= 0, meaning at least 2 of c1, c2, or c3 must be taken if track 1 is selected
  • take sufficient foundational courses to sasify overall program requirements
    • c1 + c9 + c10 >= 2, where c1, c9, and c10 are the foundational courses in the program

I now must generate a course-wise score system to feed the model. I'll keep it simple and score all courses in 'fun'-ness, rated 0-5 (where 5 is good, 0 is bad). Certain contributing factors are particulary difficult to model. For instance, I omit scheduling, course availability, and professor influence when I generated my personal desirability ranking. These factors could all be modeled, but I leave them out for simplicity. Accepting PRs :) Note that there are some interesting things you can do with these factors. Applying sufficiently large or low (negative, even) ranking points can force the algorithm to put you in a course, or help you dodge a course entirely if at all possible. There are limits on its ability to do this, but know that it is a feature of this model.

Enter these into the Simplex algorithm, and allow it to hunt for a potential solution! The actual model is very small, and can be viewed here. After all was said and done, my model predicted that my interests fall most inline with the "Machine Learning" track, and assigned me the associated courses to satisfy that path!

('Status:', 'Optimal') // a maximized solution was found!
('track:_machine_learning', '=', 1.0) // 1 of 5 tracks was picked
('course:_Computational_Complexity_Theory_6520', '=', 1.0)
('course:_Data_and_Visual_Analytics_6242', '=', 1.0)
('course:_Distributed_Computing_7210', '=', 1.0)
('course:_Internetworking_Architectures_and_Protocols_7260', '=', 1.0)
('course:_Introduction_to_Operating_Systems_8803_002', '=', 1.0)
('course:_Machine_Learning_7641', '=', 1.0)
('course:_Pattern_Recognition_7616', '=', 1.0)
('course:_Software_Analysis_and_Testing_6340', '=', 1.0)
('course:_Software_Architecture_and_Design_6310', '=', 1.0)
('course:_Special_Topics:_Big_Data_for_Health_Informatics_8803_BDHI', '=', 1.0)

I tend to stick to javascript whenever feasible, however, Linear Programming packages in the JS community lack Integer & Binary variable support out-of-the box. Thus, I have implemented my solution in Python using the rad++ Pulp package.