Date: Aug 7, 2013 5:06 PM
Author: Ben
Subject: Matrix Optimization

I have a problem dealing with optimization of student preferences, I can find the answer by trial and error, but the exercise is to do so systematically using matrices.  The problem asks for the maximum total utility for student assignments on sports classes given the students preferences.  There are 6 students (Al, Bob, Chad, Dave, Erin, and Fitz) and 3 sports classes (Rugby, Soccer, and Track) and each student will participate in one sport.  Rugby has 1 spot available, Soccer has 2 spots available, and Track has 3 spots available.  The students have unique utility based on their ranking of the sports (3 utility for a 1st choice, 2 utility for a 2nd choice, and 1 utility for a 3rd choice) given below:

________ A B C D E F
R(1 spots) 3 3 2 1 1 2
S(2 spots) 2 1 3 3 2 1
T(3 spots) 1 2 1 2 3 3

Just by trial and error, I can see that the maximum total utility is 17 when Rugby is Al, Soccer is Chad and Dave, and Track is Bob, Erin, and Fitz, but my question is how to optimize the matrix to give that solution?