# Mathematical Modeling and Exact Optimizing of University Course Scheduling Considering Preferences of Professors

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## Abstract

**:**

## 1. Introduction

- Mathematical modeling and exact optimizing of the UCS problem, taking into account professors’ preferences and minimizing the number of empty classrooms.
- Determining suitable time intervals while ensuring a minimum gap between courses within each group.
- Implementation of an exact search method to generate the most favorable course schedule, optimizing both timing and location of courses.
- Successfully performing the proposed mathematical model and solution method for UCS in a real dataset from a college in China and comparing the obtained scheduling results with the current program in the same college.

## 2. Theoretical Foundations of Research and Related Literature

#### 2.1. University Course Scheduling Constraints

- Hard constraints (HCs) are those constraints that should be followed and must not be violated. HCs guarantee the feasibility of the solution and are usually related to educational or administrative rules that are considered according to the university’s needs, the faculty’s requests, and the educational system.
- Soft constraints (SCs) are those limitations that determine the level of efficiency and usefulness of the timetable, which are not required to be exactly applied, and they can be seen as options for building a high-quality timetable. SCs depend on the requests of the planners and can include the opinions of the university staff, professors, and students. It is clear that there may not be a feasible solution that satisfies all the soft constraints, so the optimization models seek to find a solution that minimizes the violation of the soft constraints.

#### 2.2. Related Works and Research Gaps

- In most papers, the number of hard constraints considered in planning the schedule of university courses is small. Many researchers have worked on the problem of scheduling university courses, and each of them considered constraints in the form of soft constraints. Moreover, the models usually do not take into account many of the constraints that exist in real-world UCS problems, and their results are far from the actual conditions.
- According to the comparison of the research literature with the existing conditions in real-world universities, it is possible to find some constraints that have not been considered in the previous studies. Therefore, there is a need to efficiently consider these constraints and incorporate them into the problem model.
- There are some challenging constraints in educational systems which are less mentioned in the literature. Moreover, most of the existing techniques focus only on satisfying the schedules of professors and courses, and less attention is paid to the preferences of the students.
- In the past, few articles have dealt with the timing of predetermined courses, presenting them at a specific time, placing optional courses in a group, and not interfering with them. According to the conditions of most universities, some courses are shared between different disciplines, and the time of presentation of these courses is communicated. According to the time set for specific courses, their timing with other courses should also be examined.
- In the case of the students’ constraints, more attention has been paid to the non-interference of the students’ programs. Accordingly, it is necessary to consider different levels of education, including holding prerequisite and post-requisite courses simultaneously.
- In the formulation of the constraints related to the professors, more attention has been paid to non-interference with the professors’ schedules and the planning of courses based on their attendance. To focus on the preferences of professors and reduce their fatigue, some attention should also be paid to this point.
- To reduce the interference of schedules between different groups of academic fields, the schedule of courses during the week should be compressed as much as possible. Therefore, more classrooms can be freed, and a certain number of classrooms are assigned to the specified group.

## 3. Research Method

#### 3.1. Defining the Problem of Scheduling Classes

#### 3.1.1. Objectives

- (1)
- Compression of the students’ schedule. One of the main objectives of the UCS problem is to minimize the distance between two consecutive classrooms and the minimum distance traveled by the students. In other words, the schedule of the students should be connected to each other as much as possible, which means that their schedules should have the least gap between courses.
- (2)
- Compression of the classroom schedule. This objective is used to utilize as few classrooms as possible to minimize interference with the rest of the educational groups.
- (3)
- Compression of the professors’ schedule. Another objective is to plan the courses according to the times that the professors have in mind to present the courses. Moreover, it tries to coincide as much as possible with the program of the professors in such a way that there is the least empty space possible in the schedule of each professor and a minimal distance traveled by the professors.
- (4)
- Maximize the number of courses to be presented. According to this objective, all the courses should be presented in the semester, so that all students can choose their desired units.

#### 3.1.2. Constraints

- Professors impose the following constraints on the program:
- Each professor has the ability to teach a specific set of courses.
- Each professor has specific time periods for giving courses.
- The maximum allowed teaching hours of the professors must be respected.
- The master programs should be as compact as possible.

- The courses must be presented in one semester, considering the following constraints:
- Each course should be presented to the students with unique entries.
- Courses presented in two sessions must be given as far as possible one day apart.
- For courses with more than two units, two sessions are held during the week.
- The timing for predetermined courses has to be considered.
- The number of class meetings should be held based on the relevant courses and their number of units.

- The constraints for the students are as follows:
- The program for incoming students of a particular year should be held in consecutive time periods as much as possible.
- The students’ program must not be spread throughout the week as much as possible.
- The senior students’ program should be scheduled as much as possible over two days.

- The classrooms should be consistent with the following constraints:
- Classrooms should be selected based on their capacity (number of persons).
- Classrooms should be selected based on the required facilities of the course.
- The time of the classes that are determined outside the faculty should be included in the schedule.
- The schedule of the classrooms does not interfere with each other.

#### 3.1.3. Inputs

- Information about professors:
- Number of professors;
- The name of the courses that each professor will present;
- The attendance times of the professors to present the relevant courses.

- Information about classrooms:
- Number of classrooms;
- Classroom capacities;
- Classroom facilities.

- Information about study groups:
- Number of working days per week;
- Number of sessions per day;
- Information about reasonable times for the formation of courses.

#### 3.1.4. Outputs

#### 3.2. Mathematical Model

## 4. Numerical Results

#### 4.1. Number of Study Units

#### 4.2. Maximum Allowed Teaching Hours

#### 4.3. Preparation Times of the Professors for Teaching

#### 4.4. Course and Classroom Matching

#### 4.5. Determining the Time of the Courses

#### 4.6. Classroom Access Time

#### 4.7. Determining the Groups

#### 4.8. Execution Time Analysis

#### 4.9. Checking the Validity of the Solutions

- Speed of obtaining solutions. One of the significant advantages of the proposed model is its computation time. According to the considered solutions, this model is solved in a short and reasonable time.
- The possibility of analyzing the solutions. In the cases when the program is performed manually, by making a small change in the conditions, such as a change in the schedule of the professors, the number of courses, or a change in the classrooms, it is necessary to revise the program again and thus, sometimes one is forced to re-prepare the weekly schedule of the courses, which requires a long time to complete. However, using the proposed model is easy and quick, and it can check different results together and then choose the best one.
- Solution accuracy and error reduction. Considering that the designed mathematical model reaches an optimal solution and this means that all constraints are satisfied, if the data are entered correctly, the errors that may occur in manual programming will not occur.
- Proper allocation of classrooms, courses, and time. Comparing the proposed model and the manual model, it can be seen that fewer classrooms have been allocated, and even some classrooms have not been used. For example, in class 10 and class 11, the courses are not offered during the week in these two classes, and the classes are free. Courses are assigned based on the capacity and equipment of the classrooms. The schedule of classrooms is compressed as much as possible during the week. Moreover, the days of the week decreased from 5 working days to 4 working days.
- The quality of the obtained schedule. In addition to taking into account the conditions of the faculty of engineering, the designed model tries to reduce the time gaps between the professors, not provide same-group courses at the same time for students, compress the sessions of students, especially senior students, and limit the teaching time of the professors to 8 h. If the inputs of the model are entered carefully, the output solution will be very suitable. Therefore, it will lead to the maximum satisfaction of students and professors.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Reference | Year | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | F15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lü and Hao [1] | 2010 | * | * | * | * | |||||||||||

Burke et al. [3] | 2010 | * | * | * | * | |||||||||||

Soza et al. [4] | 2011 | * | * | * | * | |||||||||||

Shiau [5] | 2011 | * | * | * | * | |||||||||||

Gunawan et al. [6] | 2012 | * | * | * | * | |||||||||||

Cacchiani et al. [7] | 2013 | * | * | * | * | * | ||||||||||

Basir et al. [8] | 2013 | * | * | * | ||||||||||||

Bolaji et al. [9] | 2014 | * | * | * | * | |||||||||||

Fong et al. [10] | 2014 | * | * | * | * | |||||||||||

Badoni and Mishra [11] | 2014 | * | * | * | * | |||||||||||

Al-Yakoob & Sherali [12] | 2015 | * | * | * | * | |||||||||||

Babaei et al. [13] | 2015 | * | * | * | * | |||||||||||

Méndez-Díaz et al. [14] | 2016 | * | * | * | ||||||||||||

Vermuyten et al. [15] | 2016 | * | * | * | * | |||||||||||

Soria-Alcaraz et al. [16] | 2016 | * | * | * | * | |||||||||||

Bellio et al. [17] | 2016 | * | * | * | * | |||||||||||

Cavdur et al. [18] | 2016 | * | * | * | ||||||||||||

Fonseca et al. [19] | 2016 | * | * | * | ||||||||||||

Borchani et al. [20] | 2017 | * | * | * | ||||||||||||

Song et al. [21] | 2017 | * | * | * | ||||||||||||

Bagger et al. [22] | 2018 | * | * | * | * | |||||||||||

Akkan et al. [23] | 2018 | * | * | * | ||||||||||||

Jamili et al. [24] | 2018 | * | * | * | * | * | * | |||||||||

Junn et al. [25] | 2019 | * | * | * | ||||||||||||

Müller et al. [26] | 2019 | * | * | * | ||||||||||||

Joolaei et al. [27] | 2020 | * | * | * | * | * | ||||||||||

Tavakoli et al. [28] | 2020 | * | * | * | * | |||||||||||

Kenekayoro [29] | 2020 | * | * | * | * | * | ||||||||||

Al-Khanak et al. [30] | 2021 | * | * | * | * | * | * | * | ||||||||

Guerriero et al. [31] | 2022 | * | * | * | ||||||||||||

Savio et al. [32] | 2022 | * | * | * | ||||||||||||

F1: No interference with optional courses | ||||||||||||||||

F2: Providing courses at a specified time | ||||||||||||||||

F3: Considering the time of predetermined courses | ||||||||||||||||

F4: Planning of professors’ courses based on attendance hours | ||||||||||||||||

F5: No interference in professors’ schedule | ||||||||||||||||

F6: Compression of professors’ program | ||||||||||||||||

F7: Considering the maximum hours allowed for teaching | ||||||||||||||||

F8: Considering the equipment needed for each course | ||||||||||||||||

F9: Non-interference in the location of classes | ||||||||||||||||

F10: The capacity of the classrooms | ||||||||||||||||

F11: Compressing the classroom schedule during the week | ||||||||||||||||

F12: No interference with the students’ schedule | ||||||||||||||||

F13: Maximum number of courses for students | ||||||||||||||||

F14: Considering different levels of education | ||||||||||||||||

F15: Compression of students’ program |

Sets/Indices | Definition |
---|---|

T | Set of time intervals of weekdays on which course planning is possible, enumerated by the index $t$. |

K | Set of classrooms that are available for the weekly course schedule, enumerated by the index $k$. |

R | The collection of professors who teach university courses, enumerated by the index $r$. |

C | Set of courses that are planned for the group of students, enumerated by index c. |

L | Set of study groups that become a special entry for students during a semester, enumerated by index $l$. |

Parameters | Definition |

${b}_{c}$ | The number of hours that the c-th course must be held per week (the number of units of course c). |

$m$ | The maximum time that can be scheduled for a professor on a day in hours. |

${a}_{rt}$ | Binary parameter, which takes the value one if the r-th professor is ready to present the course in the t-th period, and zero otherwise. |

${\beta}_{rc}$ | Binary parameter, which takes the value one if the r-th professor teaches the c-th course. |

${P}_{ck}$ | Binary parameter, which takes the value one if the c-th course can be held in the k-th class. |

${\gamma}_{ct}$ | Binary parameter, which takes the value one if the c-th course can be held at the t-th time. |

${f}_{kt}$ | Binary parameter, which takes the value one if the k-th class is available at the t-th time. |

${\lambda}_{cl}$ | Binary parameter, which takes the value one if the c-th course is in the l-th subject group. |

${h}_{dt}$ | Binary parameter, which takes the value one if the t-th time interval belongs to the d-th day (d = 1, 2, 3, 4, 5). |

${w}_{r}$ | The weight of the professor r, which is a numerical value in the range [0–1], is determined based on characteristics such as academic degree, experience, etc. |

${\delta}_{rdt}$ | The weight of the time period that expresses the preference of professor r over the interval t from day d, which is a numerical value in the range [0–1]. |

DecisionVariables | Definition |

${X}_{cktr}$ | Binary variable, which takes the value one if the c-th course is scheduled in the t-th 1 h interval ($t=1,2,\dots .,50$) in the k-th class and this course is presented by the professor $r$. |

${Y}_{ckt\prime r}$ | Binary variable, which takes the value one if the c-th course is scheduled in the t’-th 2 h interval ($t\prime =1,2,\dots .,25$) in the k-th class, and this course is presented by the professor $r$. |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | C3 R1 | C4 R2 | C5 R2 | C6 R2 | C7 R3 | C8 R3 | C9 R3 | - |

K2 | - | - | - | - | - | |||||

K3 | - | - | - | - | - |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 |
---|---|---|---|---|---|

K1 | - | - | - | - | - |

K2 | C4 R2 | C7 R3 | C1 R1 | C2 R1 | C3 R1 |

K3 | C8 R3 | - | C9 R3 | C5 R2 | C6 R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | C3 R1 | C4 R2 | C5 R2 | C6 R2 | C7 R3 | C8 R3 | C3 R1 | C4 R2 |

K2 | C4 | R2 | C7 | R3 | C1 | R1 | C2 | R1 | C3 | R1 |

K3 | C8 | R3 | - | C9 | R3 | C5 | R2 | C6 | R3 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Day 1 | K1 | C1 R1 | C2 R1 | C3 R1 | - | - | - | C8 R2 | C9 R2 | C10 R2 | |

K2 | C6 | R2 | C7 | R2 | - | - | - | ||||

Day 2 | K1 | C1 | R1 | C2 | R1 | C3 | R1 | C4 | R1 | C5 | R2 |

K2 | - | - | - | - | - |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Day 1 | K1 | C1 R1 | C2 R1 | C3 R1 | - | - | - | C1 R1 | C2 R1 | C3 R1 | |

K2 | - | - | C4 | R1 | - | - | |||||

Day 2 | K1 | C1 | R1 | C2 | R1 | C3 | R1 | C5 | R2 | - | |

K2 | C7 | R2 | - | - | - | - |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | ||||
---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | - | - | C6 R2 | C7 R3 | C8 R3 | C9 R3 | - |

K2 | C3 | R2 | C1 R1 | C2 | R1 | C4 | R2 | C5 | R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||
---|---|---|---|---|---|---|---|---|

K1 | - | C6 R3 | C5 R2 | C2 | R1 | C2 R1 | C1 R1 | C4 R2 |

K2 | C8 R3 | - | C7 R3 | C9 R3 | - | C3 | R2 | C1 R1 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | - | - | C6 R3 | C7 R3 | C8 R3 | C9 R3 | - | |

K2 | C3 | R2 | C1 | R1 | C2 | R1 | C4 | R2 | C5 | R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||
---|---|---|---|---|---|---|---|---|

K1 | C2 R1 | C1 R1 | C2 R1 | C3 R2 | C1 R1 | C8 R3 | C7 R3 | |

K2 | C6 R3 | - | - | C9 R3 | - | C5 R2 | C4 R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Day 1 | K1 | C1 R1 | - | C4 | R2 | - | C7 R3 | C8 R3 | C9 R3 | - | |

K2 | - | C12 | R1 | C1 | R1 | C2 | R1 | C3 | R2 | ||

Day2 | K1 | C11 R1 | - | C9 | R3 | C5 | R2 | C6 | R2 | C10 | R3 |

K2 | - | - | - | - | - |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Day 1 | K1 | C1 R1 | - | C4 | R2 | - | - | C10 R3 | C11 R1 | - | |

K2 | - | - | C1 | R1 | C2 | R1 | C3 | R2 | |||

Day2 | K1 | C7 R3 | C8 R3 | C5 | R2 | C6 | R2 | C9 | R3 | C10 | R3 |

K2 | - | - | - | - | - |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | - | - | C6 R3 | C7 R3 | C8 R3 | C9 R3 | - | |

K2 | C3 | R2 | C1 | R1 | C2 | R1 | C4 | R2 | C5 | R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | - | C2 R1 | C1 R1 | C7 R3 | C8 R3 | C1 | R1 | C5 | R2 | |

K2 | C4 | R2 | - | C3 | R2 | C6 R3 | C9 R3 | C2 | R1 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C1 R1 | C2 R1 | - | - | C6 R3 | C7 R3 | C8 R3 | C9 R3 | - | |

K2 | C3 | R2 | C1 | R1 | C2 | R1 | C4 | R2 | C5 | R2 |

Classroom/Time | 8–10 | 10–12 | 13–15 | 15–17 | 17–19 | |||||
---|---|---|---|---|---|---|---|---|---|---|

K1 | C3 | R2 | C7 R3 | C1 R1 | C8 R3 | C9 R3 | - | C6 R3 | - | |

K2 | C1 | R1 | C5 | R2 | C4 | R2 | C2 | -R1 | C2 | R1 |

No. Courses | No. Classrooms | No. Professors | Execution Time (Second) |
---|---|---|---|

9 | 3 | 3 | 38 |

10 | 2 | 2 | 40 |

9 | 2 | 3 | 37 |

12 | 2 | 3 | 48 |

10 | 3 | 5 | 63 |

15 | 3 | 5 | 148 |

20 | 3 | 5 | 412 |

20 | 5 | 7 | 637 |

25 | 7 | 10 | 1150 |

25 | 10 | 10 | 1533 |

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## Share and Cite

**MDPI and ACS Style**

Chen, M.; Werner, F.; Shokouhifar, M.
Mathematical Modeling and Exact Optimizing of University Course Scheduling Considering Preferences of Professors. *Axioms* **2023**, *12*, 498.
https://doi.org/10.3390/axioms12050498

**AMA Style**

Chen M, Werner F, Shokouhifar M.
Mathematical Modeling and Exact Optimizing of University Course Scheduling Considering Preferences of Professors. *Axioms*. 2023; 12(5):498.
https://doi.org/10.3390/axioms12050498

**Chicago/Turabian Style**

Chen, Mo, Frank Werner, and Mohammad Shokouhifar.
2023. "Mathematical Modeling and Exact Optimizing of University Course Scheduling Considering Preferences of Professors" *Axioms* 12, no. 5: 498.
https://doi.org/10.3390/axioms12050498