# Assignment problem of Professors to courses

I have a problem in my textbook and already know the answers (the answers to every question are listed at the end of the text); however, I can't for the life of me figure out the steps in the process to solve this one. It looks simple enough (it's a simple assignment problem), but I'm stumped.

The quantitative methods department head at a major Midwestern university will be scheduling faculty to teach courses during the upcoming autumn term. Four core courses need to be covered. The four courses are at the UG, MBA, MS, and Ph.D. levels. Four professors will be assigned to the courses, with each professor receiving one of the courses. Student evaluations of the professors are available from previous terms. Based on a rating scale of 4 (excellent), 3 (very good), 2 (average), and 1 (poor), the average student evaluations for each professor are shown.

(see attached file)

Professor D does not have a Ph.D. and cannot be assigned to teach a Ph.D. level course.

QUESTION:

If the department head makes teaching assignments based on maximizing the student evaluation ratings over all four courses, what staffing assignments should be made?

ANSWER:

A to MS, B to Ph.D., C to MBA, D to undergrad (UG)

Maximum total rating = 13.3

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#### Solution Preview

Let us denote the professors as A-1, B-2, C-3, and D-4. Also for courses, UG-1, MBA-2, MS-3, and Ph.D-4. This enables us to simplify variable definitions and clarity.

The numbers are provided in the following tables

Course

1 2 3 4

Professor UG MBA MS Ph.D.

1 A 2.8 2.2 3.3 3.0

2 B 3.2 3.0 3.6 3.6

3 C 3.3 3.2 3.5 3.5

4 D 3.2 2.8 2.5 --

Now let us define variables. Let x11 be the variables to indicate the assignment of professor A to UG course. First number of after x denotes professor and second one denotes course. Similarly we define other variables x12, x13, x14, x21, x22, x23, x24, x31, x32, x33, x34, x412, x42, x43, and x44. These are defined in the similar fashion as x11. All of these variables are binary/bivalent/Boolean such that they can take value of either 1 or 0, value of 1 when a particular professor is assigned to a particular course otherwise 0.

We can represent this problem as below

In the above figure left side nodes indicate professors and right side nodes indicate courses. The link connecting each node to other nodes indicated assignment. The ratings for each combination of professor and course are the ...

#### Solution Summary

Step by step formulation to the problem is given along with optimal solution