1. Upper Division Mathematics Course Schedulings
Chalmer Tomlinson
Tanawat Trakoolthai
Brian Tran
Mohsen Gol shiri Esfahani
March 11, 2015
Abstract
There are currently 59 upper division mathematics courses being oered for the
2015-2016 school year at Cal State Super Happy Fun Time. Tragically however, there
are only 18 faculty members available to teach these courses; and each with their
own individual preferences and constraints. Here at Make Everyone Happy Inc. we
have developed a exible program to handle the complex constraints placed upon
the scheduling of these courses and optimally solve the assignment problem. We had
tried many dierent Linear Programming techniques before nally concluding that our
unique algorithm generates the maximum amount of happy points. In this report, we
examine the results and methods.
1
2. 1 Setting up the Model
From the onset, we knew wanted to constrain the variables to be binary. To do this, we
created a set of dummy variables for each professor to represent the second class that they
would teach and a set for courses that were nonexistent. We then converted our table of
color-coded preferences into numerical happiness values using excel. This allowed us to
model our objective function as a maximizing happiness function. Thus, our objective
function looks like,
Max z =
n
i=1
m
j=1
cij · xij
where
xij = Professor i assigned to class j
cij = Happiness value for professor i assigned to class j
We then created the standard sparse matrix in order to model the constraints for a typical
assignment problem. For the No Two of the Same Courses' constraint, we grouped the
courses, which repeated in the quarter, by the professor and their dummy and set the bounds
to 1. Since the variables are binary, this ensured that only one class from each repeating
group would be taught from that professor. And as for the No Two Dierent Courses
constraint, we rst decided to consider the professors who were only okay with teaching
two dierent courses as not okay. We reasoned that if a feasible solution could be obtained
without requiring these professors to teach two dierent courses, and if every basic variables'
coecient was at least 3, then we would have arrived at an optimal solution.
Next, we created a set of auxiliary variables for every professor not willing to teach two
dierent classes. We grouped the courses that did not repeat into one constraint and set it to
be less than or equal to one of the auxiliary variables y. Then, for the repeating courses, we
grouped the professor and their dummy's courses into individual constraints and set them
to be less than or equal to M · y for each individual set of repeating courses. Then we set
the sum of the y variables to one in order to restrict the class assignments. And nally, after
running our algorithm for one quarter, we would place restrictions on the following quarter
by xing the professors who need to teach their sequenced courses. The results were then
calculated in MATLAB and tabulated as seen in Table 1.
2
3. Note: There exists multiple optimal solutions. For our initial results, professor 15 was not
assigned any courses for the whole year. We then split one of the courses that was not in a
sequence and assigned it to professor 15 by hand. This kept our happiness level consistent
and did not conict with any constraints. It should also be noted that our algorithm has
found a feasible solution in each case with the maximum amount of happiness points every
time. This tells us that the model worked and we have found an optimal solution for every
quarter.
2 External Report
Assigning professors sections to teach based on their personal preferences may seem like a
simple task on paper; but like many things in life, it can be made much more complicated
(and more fun) by mathematical modelling. A typical scheduling problem was given to our
company Make Everyone Happy, Inc. in which we were tasked to assign 18 mathematics
professors a variety of courses spanning across three quarters. The restrictions placed upon
us were as follows:
1. Every course has exactly one professor
2. Every professor has a maximum of two courses
3. Any professor teaching a part of a sequence must teach the second part in the following
quarter
4. A subset of professors cannot teach two of the same classes
5. A subset of professors cannot teach two dierent classes
Using these restrictions, we were able to rst model the problem into a pseudocode and then
rewrite it into MATLAB in order to solve the assignment. Based on our ndings and the
computations of our algorithm, we have determined an optimal solution to the scheduling
problem. The results are presented in Table 1. The table represents our recommendation
for assigning classes and making everyone as happy as possible. There are alternate optimal
solutions that may also be explored but it was unnecessary for our report. Thus this stands
as our best recommendation for the scheduling coordinator at Cal State Super Happy Fun
Time. The method we used to create our model is a blend of a traditional Assignment
Problem techniques, Binary Integer Programming, Auxiliary Variable techniques, and some
Good `ol Fashioned Linear Programming techniques. A printout of the coding used for our
3
4. algorithm is included in this report for the readers' pleasure.
Table 1: Optimal Solution for the 2015-2016 School Year
Fall 2015 Professor Winter 2016 Professor Spring 2016 Professor
MATH 113 Gonzalez MATH 113 Gonzalez MATH 136 Liu
MATH 136 Lewis MATH 136 Vinh MATH 149 Gonzalez
MATH 149 Gutierrez MATH 162 Chen MATH 151 Ayala
MATH 149 Gutierrez MATH 162 O'neill MATH 163 Chen
MATH 162 O'neill MATH 162 Tanaka MATH 163 O'Neill
MATH 162 Gupta MATH 163 Gupta MATH 163 Tanaka
MATH 180 Liu MATH 163 O'neill MATH 203 Lewis
MATH 202 Liu MATH 202 Liu MATH 203 Liu
MATH 202 Lewis MATH 202 Lewis MATH 212 Gutierrez
MATH 202 Martinez MATH 203 Martinez MATH 238 Martinez
MATH 303 Gupta MATH 203 Liu MATH 247 Lee
MATH 336 Harris MATH 203 Lewis MATH 261 Gonzalez
MATH 336 Harris MATH 211 Gutierrez MATH 336 Harris
MATH 368 Lee MATH 260 Gonzalez MATH 336 Rivas
MATH 368 Lee MATH 304 Gupta MATH 408 Luong
MATH 391 O'neill MATH 336 Harris MATH 576 Smith
MATH 425 Luong MATH 336 Harris
MATH 460 Ayala MATH 369 Lee
MATH 516 Richards MATH 369 Lee
MATH 576 Gonzalez MATH 407 Luong
MATH 448 Ayala
MATH 516 Richards
MATH 576 Smith
4
5. 3 Internal Report
The approach we took to this problem had not changed since we had started the project.
We decided early on to use binary variables and dummies of the professors/classes, since it
seemed the most clear and easy way to model the scheduling problem. The techniques for
typical Assignment Problems were useful in setting up a few of the constraints as a sparse
matrix. However, our rst diculties were in modelling the No Two Dierent Classes
constraints for certain professors. Because a few of the professors were only okay with
teaching two dierent courses, it was dicult for us to weight their preferences into the
constraints. We tried to scale their variables, but none of those techniques worked. We
eventually opted on considering their preferences as Never.
We were originally going to compare the dierences between Never and Great, but we
later reasoned out that if we can get a feasible solution where every basic variables' happiness
coecient was greater than or equal to 3 then the solution is optimal. Hence there would
be no need to recalculate the scheduling.
In order to model these complex constraints. we eventually settled on adding in auxiliary
variables. This, in turn, complicated our coding immensely in MATLAB. However, with
perseverance and hard work (and one talented programmer), we were able to write our
model as a exible algorithm to solve the problem.
The next biggest hurdle we faced was coding the restriction such that if a professor
taught the rst part of a sequence, then they have to teach the second part of the sequence.
A complex algorithm had to be written in order to force a professor (or their dummy) to
teach any section of the second part of the sequence.
As we were working through each quarter, it became clear that our algorithm would
consider the basic variables in the order in which they appeared. This would result in an
optimal solution where the bottom of the list of professors were given a lower priority than
the rest. For example, in our tables, professor 15 would not be assigned any real classes for
the whole year. This was circumvented by manually assigning a class to professor 15 in the
Spring Quarter. We searched through the list of preferred classes for professor 15 and picked
a class where the happiness value was at least 3. Then we searched for the professor who
was currently teaching that class and switched it to professor 15; so long as it didn't conict
with any other restrictions. It is interesting to consider what would happen if we randomize
the order of professors every quarter and/or if we revised our code to address this slight bias:
5
6. And if we would arrive at a dierent optimal solution that generated the same happiness
level with dierent basic variables.
4 Appendix
Figure 1: Excel Spreadsheet for the Constraints
In Figure 1, the cells with zeroes and ones represent the subset of professors with tighter
restrictions than the rest.
6
7. 1
Table of Contents
........................................................................................................................................ 1
objective function ................................................................................................................ 1
----- constraints ------ equal to ------ ........................................................................................ 2
----- constraints ------ less than or equal to ------ ....................................................................... 2
classes = 23;
% indexes of professors (only first index) that wil not teach two of the
% same UD
no2same = [1,3,31,33];
% indexes of professors (only first index) that wil not teach two different
% UD in the same quarter
no2diff = [1, 3, 9, 11, 13, 21, 23, 29, 31, 33, 35];
% first index of repeated classes
sameclass = [3,6,8,10,16,18];
% different classes
diffclass = [1,2,13,14,15,20,21,22,23];
% index of class if 3 sections exist
threepeat = [3,10];
% professor i in the middle of a sequence j
prof_seq = [13, 4, 10, 9, 12, 8, 4];
seq = [6, 6, 10, 10, 10, 18, 15];
variables = 1296 + size(no2diff,2) * (1 + size(sameclass,2));
objective function
% initialize coefficient vector for objective function
f = zeros(variables,1);
% Import the data
[~, ~, raw] = xlsread('C:Usersb_ran_000DesktopSpreadsheet A (Autosaved).xlsm','
% Create output matrix
data = reshape([raw{:}],size(raw));
% Populate f
for i = 1:36
for j = 1:36
f((i - 1)*36 + j) = - data(i,j);
end
8. 2
end
% Clear temporary variables
clearvars data raw;
----- constraints ------ equal to ------
% initialize coefficinet matrix Aeq and constraint vector beq
Aeq = zeros(72,variables);
beq = zeros(72,1);
% constraint - one teacher can teach one class
for i = 1:36
for j = 1:36
Aeq(i,(i - 1) * 36 + j) = 1;
end
beq(i) = 1;
end
% constraint - one class can have only one teacher
for i = 1:36
for j = 1:36
Aeq(i + 36,(j - 1) * 36 + i) = 1;
end
beq(i + 36) = 1;
end
% constraints for auxiliary variables used below
for i = 1:size(no2diff,2)
for j = 1:(size(sameclass,2) + 1)
Aeq(i + 72, j + 1296 + (i-1) * (size(sameclass,2) + 1)) = 1;
end
beq(i + 72) = 1;
end
for i = 1:size(prof_seq,2)
Aeq( i + 72 + size(no2diff,2), seq(i) + (prof_seq(i) - 1) * 36) = 1;
Aeq( i + 72 + size(no2diff,2), seq(i) + 1 + (prof_seq(i) - 1) * 36) = 1;
Aeq( i + 72 + size(no2diff,2), seq(i) + prof_seq(i) * 36) = 1;
Aeq( i + 72 + size(no2diff,2), seq(i) + 1 + prof_seq(i) * 36) = 1;
if any(seq(i) == threepeat) == 1
Aeq( i + 72 + size(no2diff,2), seq(i) + 2 + (prof_seq(i) - 1) * 36) = 1
Aeq( i + 72 + size(no2diff,2), seq(i) + 2 + prof_seq(i) * 36) = 1;
end
beq( i + 72 + size(no2diff,2))=1;
if prof_seq(i) == 8
beq( i + 72 + size(no2diff,2))=2;
end
end
----- constraints ------ less than or equal to ------
% initialize coefficinet matrix A and constraint vector b
9. 3
A = zeros(72,variables);
b = zeros(72,1);
% constraint - no two of the same Upper Division in same quarter
% no repeats of class i for prof j
for i = 1:size(no2same,2)
for j = 1:size(sameclass,2)
A( size(sameclass,2) * (i-1) + j, sameclass(j) + (no2same(i) - 1) * 36) = 1
A( size(sameclass,2) * (i-1) + j, sameclass(j) + 1 + (no2same(i) - 1) * 36)
A( size(sameclass,2) * (i-1) + j, sameclass(j) + no2same(i) * 36) = 1;
A( size(sameclass,2) * (i-1) + j, sameclass(j) + 1 + no2same(i) * 36) = 1;
if any(sameclass(j) == threepeat) == 1
A( size(sameclass,2) * (i-1) + j, sameclass(j) + 2 + (no2same(i) - 1) *
A( size(sameclass,2) * (i-1) + j, sameclass(j) + 2 + no2same(i) * 36) =
end
b( size(sameclass,2) * (i-1) + j)=1;
end
end
% constraint - no two different Upper Division in same quarter
for i = 1:size(no2diff,2)
for j = 1:size(diffclass,2)
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1,
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1,
end
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1, 1 +
b( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1) = 0
for k = 1:size(sameclass,2)
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
if any(sameclass(k) == threepeat) == 1
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1)
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1)
end
A( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
b( size(no2same,2) * size(sameclass,2) + (i-1)*(size(sameclass,2) + 1) + 1
end
end
%%[x , fval] = linprog(f,A,b,Aeq,beq,lb,ub);
%%intcon = 1:1296;
%%x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub);
[x , fval] = bintprog(f, A, b, Aeq, beq);
optimal = vec2mat(x,36);
Maximum number of iterations for LP Relaxation exceeded; increase options.M
