Optimization Models for Generating Graduation Roadmaps

Optimization Models for Generating Graduation Roadmaps

A. Dechter and R. Dechter

“Four-Year Colleges” in Name Only…

College Graduation Rates Statistics:

4 Years 6 Years US Universities 37% 63% Cal State University 8% 40%

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 ExampleFour-Year Course Schedule

Department of Marketing, Marketing OptionCalifornia State University, Northridge

http://www.csun.edu/marketing

YEAR 1 YEAR 2 YEAR 3 YEAR 4Fall Spring Fall Spring Fall Spring Fall Spring

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 * GE * OPEN -

3 UNITS GE(UD) * GE(UD) *

OPEN -3 UNITS

Internship (Recommend

ed)

TITLE 5 TITLE 5 GE * GE * FIN 303 GE(UD) * OPEN -

2 UNITSOPEN -3 UNITS

GE * GE * GE * GE * SOM 306

MGT 360

OPEN -3 UNITS

Internship (Recommend

ed)

OPEN -3 UNITS

15 UNITS 15 UNITS 15 UNITS 15 UNITS 16 UNITS 15 UNITS 15 UNITS 15 UNITS

http://www.csun.edu/marketing

Degree Programs as Projects

Project Degree Program Activities Courses Precedence relationships Prerequisite requirements Resource limitations Study-load limits Goal: minimize completion time Goal: minimize time-to-degree

A Sample Degree Program

Course Units Prerequisites Requirements Prerequisite Courses – Select as needed

C1 4 None C2 3 None C3 2 C4 C4 3 None

Required Courses C5 3 C1 C6 4 C1 and either C2 or C3

- or - C7 4 C4 C8 3 C4

Electives – Select at least 6 units from the following C9 3 C5 and C10 C10 3 C6 C11 3 C7, C8 and C10 - or - C12 3 C13 C13 4 C8

Minimum number of units required for degree – 24

A Network Representation of the Degree Requirements

C9(3)

C6(4)

C5(3)

C13(4)

C12(3)

C11(3)

C10(3)

C4(3)

C3(2)

C2(3)

C1(4)

C8(3)

C7(4)

Electives (min 6 units)Required CoursesPrerequisites

Two Degree Plans for the Example Plan A

Or

C9(3)

C6(4)

C5(3)

C13(4)

C12(3)

C11(3)

C10(3)

C4(3)

C3(2)

C2(3)

C1(4)

C8(3)

C7(4)


Plan B

C9(3)

C6(4)

C5(3)

C13(4)

C12(3)

C11(3)

C10(3)

C4(3)

C3(2)

C2(3)

C1(4)

C8(3)

C7(4)


Total Units = 24Longest Path = 4 terms

Total Units = 27Longest Path = 3 terms

Minimum Length Schedules for the Two Plans

Term 1 Term 4Term 3Term 2

Plan A

Plan B

C4 (3)

C5 (3)C1 (4) C6 (4)

C12 (3)C13 (4)C8 (3)

C10 (3)

C5 (3)

C6 (4)

C13 (4)C8 (3)C4 (3)

C1 (4)

C2 (3)

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

1 if course is scheduled for term 1 13,1 8

0 otherwiseit

i tx i t

1 if course is selected for the program

1 130 otherwise i

iy i

8

1

1 13i itt

y x i

if course is scheduled for term , 1,2,...81 13

0 if course is not selected for the programi

t i t ts i

i

The Objective Function

• In Both IP and CP:

• In Integer Programming

• In Constraint Programming

Minimize Z

8

1

1 13itt

tx Z i

1 13is Z i

Required Courses Constraints (A)

• Requirement: Course C5 must be taken• IP model constraint

• CP model constraint

5 1y

5 0s

Required Courses Constraints (B)

• Requirement: Either C6 or C7 must be taken

• IP model constraint

• CP model constraint

6 7 1y y

6 70 or 0s s

Elective Courses Constraints

• Requirement: Select 6 units from courses C9 through C13; C11 and C12 may not both be counted.

• IP model constraints:1 if course is used to satisfy the elective requirement

{11,12}0 otherwisei

iy i

11 11

12 12

11 12

9 10 11 12 13

13 3 3 3 4 6

y yy yy yy y y y y

Elective Courses Constraints (cont.)

• CP model constraint:

9 10 11 12 133 0 3 0 max 3 0 ,3 0 4 0 6s s s s s

Prerequisite Constraints (A)

• Requirement: Course C1 is a prerequisite for course C5

• IP model constraints:


1 5

8 8

1 5 51 1

1 9 1t tt t

y y

tx t x y

5 1 50 0s s s

Prerequisite Constraints (B)

• Requirement: Either C2 or C3 satisfies the prerequisite requirement for course C6

• IP model constraints:

6,

1 if course satisfies the prerequisite requirement of C2{2,3}

0 otherwisej

jy J

6,2 2

6,3 3

6,2 6,3 6

8 8

2, 6, 6,21 1

8 8

3, 6, 6,31 1

1 9 1

1 9 1

t tt t

t tt t

y y

y y

y y y

tx t x y

tx t x y

Prerequisite Constraints (B, cont.)


6 2 6 3 60 0 or 0s s s s s

The Two Complete Models

8

1

8

1

5 8 6 7

11 11 12 12 11 12

9 10 11 12 13

8 8

1 1

Minimizesubject to

1 13

1 13

1; 1; 1; ; 1

3 3 3 3 3 6, s.t. is a unique prerequisite of

1 9 1

i itt

itt

j i

jt itt t

Z

y x i

tx Z i

y y y yy y y y y yy y y y yy y i j j i

tx t x y

6,2 2 6,3 3 6,2 6,3 6

8 8

2, 6, 6,21 1

8 8

3, 6, 6,31 1

13

1

13

1

, s.t. is a unique prerequisite of ; ;

1 9 1

1 9 1

24

6 1 8

0,1 1 13,1 8

0

i

t tt t

t tt t

i ii

i iti

it

i

i j j iy y y y y y y

tx t x y

tx t x y

u y

u x t

x i t

y

11 12 6,2 6,3

,1 1 13

, , , 0,1

i

y y y y

5 6 7 8

9 10 11 12 13

6 2 6 3 6

13

1

13

1

Minimizesubject to

1 13

0 and 0 or 0 and 0

3 0 3 0 max 3 0 ,3 0 3 0 6

0 0

, s.t. is a unique prerequisite of 0 0 or 0

0 24

i

i j i

i ii

i ii

Z

s Z i

s s s s

s s s s s

s s s

i j j is s s s s

u s

u s

6 1

0,1,2,...,8 1i

t t T

s i N

The IP Model The CP Model

An Optimal Solution to the Example

C4 (3)

C5 (3)C1 (4)

C6 (4)

C8 (3)

C3 (2) C10 (3) C9 (3)

Term 1 Term 4Term 3Term 2 Term 5

Next Steps

• “Real life” case studies• Computational analysis

Documents

Optimization Models for Generating Graduation Roadmaps