Optimization Models for Generating Graduation Roadmaps A. Dechter and R. Dechter

“Four-Year Colleges” in Name Only… College Graduation Rates Statistics:

Reasons for Poor Graduation Rates Students are not sufficiently prepared academically Students do not enroll full-time because they need to work Insufficient or inadequate academic advisement Not enough courses are offered so students cannot enroll in classes they need The requirements for completing a degree are complicated or unclear

An Example from the CSUN Catalog MATH 255A. CALCULUS I (3) Prerequisites: Passing score on or exemption from the Entry Level Mathematics Examination (ELM) or credit in MATH 093, and either a passing score on the Mathematics Placement Test (MPT) or completion of MATH 105, or both MATH 102 and 104, or articulated courses from another college equivalent to MATH 105, or both MATH 102 and 104, with grades of C or better.

CSU Taskforce Recommendation Develop 4-year, 5-year, and 6-year graduation roadmaps for all academic degree programs. These roadmaps should be term-by-term depictions of the courses in which students should enroll over the entirety of their academic careers (general education and major) and should address both day and evening programs when program size is sufficient to support both patterns. After the plans have been developed, they should be accessible to students at feeder community colleges and high schools.

Graduation Roadmap Example Four-Year Course Schedule Department of Marketing, Marketing Option California State University, Northridge YEAR 1YEAR 2YEAR 3YEAR 4 FallSpringFallSpringFallSpringFallSpring ENGLISH 155 BUS 105 ACCT 220 ACCT 230 BUS 302/L MKT 346 MKT elective MKT 449 MATH 102 SOM 120 OR MATH 140 ECON 161 BLAW 280 MKT 304 MKT 348 MKT elective BUS 497 COMP 100 ECON 160 GE * OPEN - 3 UNITS GE(UD) * OPEN - 3 UNITS Internship (Recommend ed) TITLE 5 GE * FIN 303 GE(UD) * OPEN - 2 UNITS OPEN - 3 UNITS GE * SOM 306 MGT 360 OPEN - 3 UNITS Internship (Recommend ed) OPEN - 3 UNITS 15 UNITS 16 UNITS15 UNITS

Degree Programs as Projects

A Sample Degree Program

A Network Representation of the Degree Requirements

Two Degree Plans for the Example Total Units = 24 Longest Path = 4 terms Total Units = 27 Longest Path = 3 terms

Minimum Length Schedules for the Two Plans

Degree Planning as Constrained Optimization Objective: Minimize time-to-degree (i.e., number of terms) Constraints: –Requirements for the degree –Prerequisite requirements –Study load limits –Minimum total unit requirement –(Course availability)

Modeling the Problem Integer Programming –Traditional –Standard solvers Constraint Programming –“Natural” –Flexible

Defining the Decision Variables Integer Programming Constraint Programming

The Objective Function In Both IP and CP: In Integer Programming In Constraint Programming

Required Courses Constraints (A) Requirement: Course C5 must be taken IP model constraint CP model constraint

Required Courses Constraints (B) Requirement: Either C6 or C7 must be taken IP model constraint CP model constraint

Elective Courses Constraints Requirement: Select 6 units from courses C9 through C13; C11 and C12 may not both be counted. IP model constraints:

Elective Courses Constraints (cont.) CP model constraint:

Prerequisite Constraints (A) Requirement: Course C1 is a prerequisite for course C5 IP model constraints: CP model constraint:

Prerequisite Constraints (B) Requirement: Either C2 or C3 satisfies the prerequisite requirement for course C6 IP model constraints:

Prerequisite Constraints (B, cont.) CP model constraint:

The Two Complete Models The IP ModelThe CP Model

An Optimal Solution to the Example

Next Steps “Real life” case studies Computational analysis