Scheduling for Trinity School at Meadow View - … · Scheduling for Trinity School at Meadow View Faculty ... •Classes are split by gender and grade ... These algorithms require

George Mason University SEOR Department

Scheduling for Trinity School at Meadow View

Faculty Presentation

Fall 2016

Mary BarthelsonMarissa BrienzaZachary Nuzzo


Introduction

Trinity School at Meadow View is a local, private, co-ed junior-high and high school with 145

students and 21 teachers that provides a classical education for grades 7-12.

1


Trinity’s Education System

• Teachers are pre-assigned to specific classes

• Trinity has developed a locked set of classes for all grades

• Classes are split by gender and grade

• Ex: 7th Grade Boys

• Each grade/gender section takes every class together

• Ex: All 8th Grade Girls take Algebra together

2


Problem Description

Currently, Trinity assigns students and teachers to periods and classrooms by hand which is a time intensive process.

They are seeking a tool that given input parameters can generate feasible schedules for future school semesters.

3


Background

Many schools, like Trinity, are faced with large classroom assignment problems every semester. The majority of these institutions solve the educational timetabling problem by:

Completing a brand new classroom assignment every semester with a large room booking log taking about 2-3 weeks (Tovey)

OR

Leaving classroom assignments unchanged and making a minimum number of changes to satisfy faculty requests (Tovey)

4


Problem Scope

• There is no specific metric that the Client is seeking to optimize

• The Client desires a set of feasible schedules

• The tool will not be a web-enabled application

• The team will build a self-contained tool that is usable by an audience not necessarily in the math/science field

• A User Manual will be developed

5


Project Approach

Requirements Development

Literature Review & Data Collection

Tool & Solution Development

Test & Evaluation

Final Delivery

Time

6


Requirements

Functional RequirementsThe tool shall…

• output at least two feasible schedules.• produce output in a spreadsheet/table format.• allow the user to export and save feasible schedules.• prompt the user for the following input parameters:

- Classrooms: room size, course restrictions- Teachers: names, part/full-time status, hours of availability, subjects taught- Sections: curriculum, size

• save the input parameters on the first use.• allow the user to edit all input parameters directly in the tool through the use of

tables.• allow stored fields to be reset.• notify the user if there is an error or additional information is required with the

inputs.• notify the user if the solution is infeasible.

7


Requirements

Nonfunctional Requirements

The tool shall…

• be supported by Windows OS.

• be available via a CD/DVD or flash drive.

• be saved on a local Trinity School at Meadow View computer.

• run using no cost software and all modules will be downloadable.

• have a User Interface that allows the user to add/edit all input parameters, but will not allow interaction with the scheduling algorithm.

• provide a User Manual and troubleshooting best practices.

8


Literature Review

Integer Programming (IP)

This approach is one of the most popular in solving the class-scheduling or educational timetabling problem.

Thoughts:

• Typically expressed in terms of a decision variable Xijk which is equal to one if class i is assigned to room j during a period k

• Commonly used objective functions: A linear cost model that reflects the preferences of the teachers (Glassey) & A null objective function and a set of constraints to solve for feasibility (Tovey)

• No access to commercial-grade solvers could greatly effect run-time

9


Literature Review (cont.)

Heuristic Methods

These algorithms require an initial solution and use search strategies to find an optimal solution by iteratively searching through the solution space.

Thoughts:

• Difficult to construct an initial solution

• Hard to determine which algorithm is best for our problem

• Commonly used heuristics in educational timetabling: Simulated Annealing (Ramirez) and Genetic Algorithms (Aycan)

10


Literature Review (cont.)

Constraint Programming

This solution method focuses on assigning variable values based on a set of constraints (no objective function) to produce a feasible schedule.

Thoughts:

• Constraint Satisfaction exploits the structure of timetabling problems

• Partial Constraint Satisfaction is used to overcome over constrained problems

• Sometime used in combination with heuristics (Aycan)

11


Data Sources

Provided by the client:

• Teacher Course Assignments

• Room Capacities

• Course-Specific Rooms

• Spring Semester Course Requirements by Grade

• Fall Semester Course Requirements by Grade

• Number of Students by Grade and Gender

• Map of School

12


Solution Implementation

Selected Method: Integer Programming

User Interface: MS Access VBA

Integer Program: MPS File

Solver: NEOS Server w/ Gurobi

Output Interface: MS Excel VBA

User’s Manual: PDF

13


Solution Strategy

• Create a User Interface to add/edit input parameters

• Construct an Integer Program to find an optimal master schedule

• Integrate the UI and IP to make a cohesive, easy-to-use tool

• Design an Output Interface that produces a clear, concise schedule

• Develop a User Manual to assist trouble-shooting and share information on maintenance

14


Trinity School Scheduler User Interface

15





TSS - Entity Relationship Diagram

TeacherTeacherIDLast NameFirst Name

Full/Part Time Status

ClassroomRoomIDName

Room Size

PeriodPeriodID

PeriodName

SectionSectionID

NameSemester

Teacher_SectionTeacherIDSectionID

Room_SectionRoomID

SectionID

Room_Period_AvailabilityPeriodIDRoomID

Availability

Section_Gender_GradeSectionID

Grade_GenderIDGrade_GenderGradeGenderID

Name Size

Teacher_AvailabilityTeacherIDPeriodID

Availability

16


IP Formulation

Indexes:i= section (grade, gender, subject)i*= subset of sections assigned to a teacherj= classroomk= period

Data/Input:Ai*k= availability of a teacher for period k {0,1}Bi*= full-time status of a teacher {0,1}Dij= section i can be taught in room j {0,1}Ejk= room j is available during period k {0, 1}Fj= size of room j {integer}Gi= size of section I {integer}Hjm= # of hops from room j to room m {integer}

17


IP Formulation (cont.)

Decision Variables:Xijk= section i is assigned to room j during period k {0,1}

Objective Function:Minimize NullMaximize Student DistanceMaximize Sum of a Random Subset of Xijk

18



Teacher Constraints:

𝑖𝑗𝑋𝑖𝑗𝑘 ≤ 𝐴𝑘 ∀ 𝑘, 𝑖∗

𝑗𝑘𝑋𝑖𝑗𝑘 ≤ 4𝐵𝑖 + 2 1 − 𝐵𝑖 ∀𝑖*

𝑋𝑖𝑗1 + 𝑋𝑖𝑗2 + 𝑋𝑖𝑗3 + 𝑋𝑖𝑗4 ≤ 3

𝑋𝑖𝑗2 + 𝑋𝑖𝑗3 + 𝑋𝑖𝑗4 + 𝑋𝑖𝑗5 ≤ 3 ∀𝑖∗

𝑋𝑖𝑗3 + 𝑋𝑖𝑗4 + 𝑋𝑖𝑗5 + 𝑋𝑖𝑗6 ≤ 3

𝑗𝑋𝑖𝑗𝑘 ≤ 1 ∀𝑘, 𝑖∗

Class scheduled only when teacher is available

Full-time teachers can only teach 4 sectionsPart-time teachers can only teach 2 sections

Teachers shouldn’t teach more than three consecutive classes

Only one of the teacher’s sections can be scheduled a period

[1]

[2]

[3]

[4]

19



Classroom Constraints:

𝑖𝑋𝑖𝑗𝑘𝐺𝑖 ≤ 𝐹𝑗 ∀𝑗, 𝑘

𝑋𝑖𝑗𝑘 ≤ 𝐷𝑖𝑗 ∀ 𝑖, 𝑗, 𝑘

𝑖𝑋𝑖𝑗𝑘 ≤ 𝐸𝑗𝑘 ∀𝑗, 𝑘

Classroom should be able to hold the class (size)

Classes that need content-specific rooms are satisfied

Only rooms that are available should be scheduled

[5]

[6]

[7]

20



Scheduling Constraints:

𝑗𝑘𝑋𝑖𝑗𝑘 = 1 ∀𝑖

𝑖𝑗𝑋𝑖𝑗𝑘 ≤ 1 ∀𝑘, 𝑠𝑢𝑏𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖

𝑖𝑋𝑖𝑗𝑘 = 1 ∀𝑗, 𝑘

Each section should be scheduled once

Each grade/gender subsection should have one required class taught a period

Only one class per room every period

[8]

[9]

[10]

21



Cutting Plains Constraints:

𝑖 𝑥𝑖𝑗𝑘 = 12 ∀𝑘 [11]

𝑖𝑗𝑘 𝑥𝑖𝑗𝑘 = 72 [12]

Only 12 classes taught per period

Total of 72 classes scheduled

22



& Semester Specific Constraints

Humane Letters taught in consecutive periods

9th and 10th Grade (by gender) Music taught together

11th Grade Language taught together

12th Grade Drama & Art taught together

23


The NEOS Server

24


The NEOS Server


IP Specifics

25

Number of Variables 2454 (binary)

Number of Constraints 796

Run Time to Create MPS File 2 minutes 35 seconds

Run Time of Gurobi 2.02 seconds

Number of Simplex Iterations 5252


Output UI

26


Master Schedule

27





Test & Evaluation

The project team manually test the Trinity Scheduling IP by making sure:

• all classes were scheduled

• all teachers’ scheduling constraints were met

• all part-time teachers are assigned to a maximum of two classes

• all full-time teachers are assigned to a maximum of four classes

• teachers have no more than one class a period

• teachers are at most scheduled for a maximum of three consecutive classes

• all rooms hold at most one class per period

• all specific semester constraints were met

28


Success CriteriaThe product was delivered on time and met or exceeded all of the system requirements.

29

Success Measure Met Failed

Nonfunctional Requirements

2.1.1 The tool shall be supported by Windows OS.

2.1.2 The tool shall be available via a CD/DVD or flash drive

2.1.3 The tool shall be saved on a local Trinity School at Meadow View

computer.

2.1.4 The tool shall run using no cost software and all modules will be

downloadable.

2.1.5 The tool shall have a User Interface that allows the user to add/edit all

input parameters, but will not allow interaction with the scheduling algorithm

2.1.6 The tool shall have an Output Interface that allows the user to create a

master schedule

2.1.7 The tool shall include a User Manual and troubleshooting best practices.

Functional Requirements

2.2.1 The tool shall output at least two feasible schedules.

2.2.2 The tool shall produce output in a spreadsheet/table format.

2.2.3 The tool shall allow the user to export and save feasible schedules.

2.2.4 The tool shall prompt the user for the following input parameters:

classrooms: room size, course restrictions; teachers: names, part/full-time status,

hours of availability, subjects taught; sections: curriculum, size

2.2.5 The tool shall save the input parameters on the first use

2.2.6 The tool shall allow the user to edit all input parameters directly in the tool

through the use of tables.

2.2.7 The tool shall allow stored fields to be reset.

2.2.8 The tool shall notify the user if there is an error or additional information is

required with the inputs.

2.2.9 The tool shall notify the user if the solution is infeasible and a modification

to inputs is needed.

Satisfied Client

Delivered Product On Time


Future Recommendations

• Extend the formulation to assign teachers to specific classes

• Allow flexibility in student curriculum

• Develop a more generalized scheduling formulation

• Continue to explore heuristic and constraint satisfaction techniques

30


Questions?

31

Documents

Scheduling for Trinity School at Meadow View - … · Scheduling for Trinity School at Meadow View Faculty ... •Classes are split by gender and grade ... These algorithms require