A TWO-STAGE BINARY PROGRAMMING APPROACH TO SOLVE CLASS TIMETABLING PROBLEM

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International Journal of Technology Management and Information System (IJTMIS) eISSN: 2710-6268 [Vol. 2 No. 2 June 2020]

Journal website: http://myjms.mohe.gov.my/index.php/ijtmis

A TWO-STAGE BINARY PROGRAMMING APPROACH TO SOLVE CLASS TIMETABLING PROBLEM

Zuraida Alwadood ^1*, Norlenda Mohd. Noor² and Nur Shahidah Zainuddin³

1 2 3 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam, MALAYSIA

*Corresponding author: [email protected]

Article Information:

Article history:

Received date : 14 February 2020 Revised date : 25 April 2020 Accepted date : 7 May 2020 Published date : 22 June 2020

To cite this document:

Alwadood, Z., Mohd. Noor, N., &

Zainuddin, N. (2020). A TWO-STAGE BINARY PROGRAMMING

APPROACH TO SOLVE CLASS TIMETABLING PROBLEM.

International Journal Of Technology Management And Information System, 2(2), 14-26.

Abstract: This study is concerned with the class scheduling in a department of a public university in Malaysia. This study used mathematical programming model as a method to solve the timetabling problem. The objective of this study is to solve the class timetabling problem and find the optimal solution that will minimize the teaching cost, considering there are two grades of lecturers with different pay schemes. The mathematical model is solved in two stages. The first stage involves allocating lecturer teaching load to the list of courses which consists of various groups. The second stage assigns the allocated load of lecturers to the time block in the timetable. The model was solved in a short computing time using MATLAB software. This study introduces novel constraints that successfully produced optimal solution, besides minimizing the teaching cost. This study helps to determine the optimal timetable for the lecturers in the department based on the modified model, at the same time satisfies the restrictions posed by the faculty policies.

Keywords: mathematical programming model, class timetable, scheduling.

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1. Introduction

Timetabling problems can be categorized under assignment problems which involved assigning a set of activities to a set of resources and time periods, which are subject to a set of predetermined restrictions or constraints. In education sector, timetabling problems normally deal with the construction of weekly class timetables of schools and tertiary level institutions. Other areas such as in transportation, a large number of timetabling problems involves the scheduling of buses, trains and aircrafts.

The university timetabling can be defined as the process of allocating the courses to a specific time slots within weekdays or the working days to a specific instructor or lecturer. The timetabling process is a complex task due to the big number of student enrolled, high number of courses offered, increased number of instructors and time constraint (Deris, 2000). Timetabling is a common problem in institutions like schools or universities in order to get the best classroom schedule. The process of organizing the classes or lecturers’ timetable in the past usually took a long time since it was done manually. Even now, certain institutions are still using the manual approach in creating the timetable in universities. By preparing the timetable manually, one has to consider many factors such as lecturers load, courses taught, pre assigned classes and many other factors. In addition, the different loads for each lecturer are to be considered by the person preparing the timetable. The manual timetabling process takes a lot of time in order to check all possibilities in order to get the best timetable. This may result the schedule to be less optimal and it needs to go through some additional steps before it is optimized. To undergo the same tedious process every semester to suit the current semester requirements is very time consuming and less effective.

The objectives of this study are to solve the class timetabling problem of a department in a local university using a binary programming model and find the optimal solution with minimum cost and to determine the most optimum timetable for lecturers based on the model developed. This study is intended to help universities in creating lecturers’ timetable and ensuring the most efficient timetabling process. Besides this, this study ensures that there is no clash on the lecturers’

timetable. The academic staff could save a lot of time creating the timetable every semester while taking into accounts all restrictions in terms of the teaching load, courses offered, time blocks, among others.

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2. Literature Review

Solving timetabling problem for a university is a very challenging and time consuming task due to diverse demands, staff preferences and availability, as well as the limitation of time slots. The context and complexity of the assignment problems are dependent on the relevant university systems. There are various mathematical model approaches have been proposed for the problem and solved by various researchers.

Jamili et al. (2018) presented a multi-objective 0-1 mathematical model to address a course timetabling problem. The purpose of the model is to allocate the courses to timeslots, so as to satisfy the restrictions in terms of availability times of instructors, the number of available classrooms in faculties, the eligibility of classes and timeslots for the courses, overlap prevention for teaching hours of each instructor, the maximum working times allocated to each instructor in day, overlap prevention for courses within course groups. In addition, it also attempted to increase satisfaction degree of instructors by maximizing their preferences to teach in their desired day and timeslot, as well as providing more times to do research. Another study of timetabling was also done by Daskalaki (2004) using 0-1 integer programming The objective function is to minimize the cost of assigning classes, considering the teaching staff preferences. Ma and Sen (2017) proposed an innovative two-steps approach to solve faculty timetabling problem using mathematical models. The resource allocation is successfully optimized while the faculty preferences are also satisfied. The problem is solved in a short running time of 30 minutes and it assists management team in term of avoiding manual and time consuming planning. The outcome has strictly reduced the conflict and, thereby, improves productivity and yields higher satisfaction among the faculty members. This approach also reduces the number of variables and errors in runtime; thus, it can be used for small and medium size organizations for their resource planning project.

The research done by Mabini (2015) applied zero-to-one integer programming in solving the class timetabling problem of a university. This study was divided into two phases where the second phase of the study used the result from the first stage of the study. In the first stage, the goal was mainly to assign lecturer or instructor to a course while the second stage was to schedule the results achieved in stage one to weekly time slots. The objective function of the research is to minimize the cost. Bucco et. al (2015) considered mixed integer programming and the objective of the study is to maximize the use of every classroom in the faculty. The objective of the research is to minimize the number of assigned meetings to the time block with the greatest number of meetings.

The university offers 1688 number of courses with total of 1083 number of lecturers and 258 classrooms. The meetings or the classes offered are between Monday to Saturday with a total of ten time blocks during day and night. The class was equally distributed to the entire timetable throughout the week with purpose of utilizing the use of the classrooms. The result achieved has reduced the classroom demand during the previously was the most assigned time block hence, the objective function is satisfied.

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Komijan and Koupaei (2015) determined the efficiency of a lecturer on teaching a course is given based on five criteria: teaching experience, number of related papers or books, number of related research plans, relationship of the course with the Ph.D. dissertation and willingness to teach. The objective of the study is to maximize the education quality by ensuring optimum assignment of courses to each of the lecturer. The mathematical model was successfully solved using GAMS software and assigned each course to the most eligible lecturer. In addition to this, Vermuyten et al. (2016) presented a two-stage integer programming approach for a university course timetable that aims at minimizing the resulting student flows in the buildings.

A model proposed by Ojha (2013) using a mixed integer programming to solve the scheduling problem of a dancing studio. The dancing studio previously used the same schedule for years which was prepared manually by the studio clerk. However, by using the same schedule it may not be the best method as there are changes in the variable. The objective function of the study is to minimize the number of classes taught per week and to minimize the number of times the dance and instructor need to come to the studio per week. The study considered four different situations where the researcher uses different number of students to examine the optimality percent of each situation. After four trials the model becomes near optimal after including the preferences constraints. The result achieved was much close to the optimal solution but the schedule still needed some alteration for it to be optimal.

Most educational institutions must schedule a set of examinations at the end of each session or year (Carter, 1986). Other than class scheduling, exam-room assignment problems have also been extensively investigated in literature. Among these researches which were using mathematical optimization approaches are Ergul and Ozturk (2017), Abayomi-Alli et al. (2019), Bania and Duarah (2018), Mohamed (2018) and Kahar and Kendall (2014).

3. The Modelling of Data

The data was collected from a department coordinator from a local university in Malaysia. The data collected includes the lecturers’ position, the list of courses taught by the department members, the list of lecturers, the maximum and the minimum teaching hours allowed for each grade of lecturer, the total hours each course must be taught each week and the faculty policies regarding the timetable preparation.

The data provided by the coordinator is for the March 2018 academic session. There are 20 lecturers with two different grades, 14 courses with different number of groups for the semester.

The faculty has limited the number of class meetings to two hours in each meeting which makes a total of 18 time blocks available for the weekly timetable from Monday to Friday. Since each meeting is limited to two hours, lecturers will be assigned a total of two meetings for a group of course in each week.

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Among the input data is the number of lecturers that will be assigned to the schedule. The total number of lecturers in the department is 20 including one contract or part time lecturer teaching a bachelor degree courses. Another input data is the courses taught by the lecturers in the department.

There are total of 14 courses with various numbers of group and each group must undergo 4-hour lesson every week. There is a difference in the loading hours for the lecturers depending on the lecturer’s grades. The lecturer with higher position (Grade 1) will be assigned less loading hours as compared to the lower position lecturer (Grade 2). The Grade 1 lecturers have a load of four to eight teaching hours per week, while Grade 2 lecturers are given a load of 8 to 16 teaching hours per week. In addition to this, Grade 1 lecturer will be paid for RM150 for each session of class, while Grade 2 lecturer will be paid at RM100.

The model used in this research was adapted from Mabini (2015) which was modified to accommodate all the restriction set by the faculty. The model sets for this study are listed as:

Li : Set of lecturer, i = 1, 2, 3,…, m Cj : Set of courses, j = 1, 2, 3,…, n Sk : Set of time block, k = 1, 2, 3,…, p

Table 1 depicts the representation of the time block used in this study.

Table 1: The representation of the time block

Day,d Time block, Sk Description

1 1 Monday 8 a.m -10 a.m

1 2 Monday 10 a.m -12 p.m

1 3 Monday 2 p.m - 4 p.m

1 4 Monday 4 p.m -6 p.m

2 5 Tuesday 8 a.m -10 a.m

2 6 Tuesday 10 a.m -12 p.m

2 7 Tuesday 2 p.m -4 p.m

2 8 Tuesday 4 p.m -6 p.m

3 9 Wednesday 8 a.m -10 a.m

3 10 Wednesday 10 a.m -12 p.m

3 11 Wednesday 2 p.m -4 p.m

3 12 Wednesday 4 p.m -6 p.m

4 13 Thursday 8 a.m -10 a.m

4 14 Thursday 10 a.m -12 p.m

4 15 Thursday 2 p.m -4 p.m

4 16 Thursday 4 p.m -6 p.m

5 17 Friday 8 a.m -10 a.m

5 18 Friday 10 a.m -12 p.m

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The model parameters for this study are defined as below:

Loadi : The teaching hours assigned to lecturer i.

Ui : The upper limit on the teaching hours can be assigned to each lecturer i Li : The lower limit on the teaching hours can be assigned to each lecturer i aj : The number of hours for course j for each class

nj : The number of groups taking course j cij : The cost of lecturer i teaching course j

aj : The number of hours of each session of course j

This study only incorporates two types of upper and lower limit of teaching hours as there are only two grades of lecturer involved. Grade 1 lecturers have the upper limit of 8 hours and the lower limit of 4 teaching hours per week. Meanwhile, Grade 2 lecturers have a maximum load of 16 hours and the minimum load of 12 hours weekly. The cost of lecturer teaching a course is based on the grade of the lecturer. The cost of paying Grade 1 lecturers is higher than Grade 2 lecturers.

There is no cost difference between courses.

Stage 1

The modified model in this study will be solved in two stages. In the first stage, the objective is to assign all groups of course to all the lecturers available. The decision variable xij presents the decision if a lecturer i is assigned to teach course j.

xij = 1, if faculty member, i is assigned to course j 0, otherwise

The objective function (1) for this stage is to allocate the courses to each of the lecturers, such that the cost is minimized.

Minimize Z = ∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 𝑐_𝑖𝑗𝑥_𝑖𝑗 (1)

subject to

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 𝑎_𝑗𝑥_𝑖𝑗 ≤ 𝑈_𝑖 , for all i (2)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 𝑎_𝑗𝑥_𝑖𝑗 ≥ 𝐿_𝑖 , for all i (3)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 𝑥_𝑖𝑗 = 𝑛_𝑗 , for all j (4)

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Constraint (2) controls the allocation of the lecturers’ teaching load. It sets the boundary for the maximum load to be allocated to every lecturer, ensuring that they will not be assigned more than the maximum hour as set by the faculty. Constraint (3) restricts the number of teaching hours the lecturers will be assigned to, so that it will not be lower than the minimum teaching hours set by the faculty. Constraint (4) is needed to ensure that the assignment of lecturers to a course must satisfy the number of groups taking the course.

Stage 2

The second stage model included new set of constraints to assign the lecturers and their respective courses to a set of time blocks. The solution of the first stage will be used to run the model in the second stage. The decision variable wijk presents the decision if a lecturer i is assigned to teach course j in the time block k.

wijk = 1, if faculty member i is assigned to course j with time block k 0, otherwise

The objective function (5) for this stage is to allocate the assigned lecturers and courses to the respective time blocks, such that the cost is minimized.

Minimize Z = ∑ ∑ 𝑐^𝑚_𝑖 ^𝑛_𝑗 _𝑖𝑗 ∑ 𝑤^𝑝_𝑘 _𝑖𝑗𝑘 (5)

subject to ∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑎¹⁸₁ _𝑗𝑤_𝑖𝑗𝑘 = 𝐿𝑜𝑎𝑑_𝑖 , for all i (6)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤¹⁸₁ _𝑖𝑗𝑘 ≤ 1 for all i and k (7)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤¹⁸₁ _𝑖𝑗𝑘 = 2 , for all 𝑥_𝑖𝑗 = 1 (8)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤⁴₁ _𝑖𝑗𝑘 ≤ 1 , for all i and j (9)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤⁸₅ _𝑖𝑗𝑘 ≤ 1 , for all i and j (10)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤¹²₉ _𝑖𝑗𝑘 ≤ 1 , for all i and j (11)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤¹⁶₁₃ _𝑖𝑗𝑘 ≤ 1 , for all i and j (12)

∑^𝑖=20_𝑖=1 ∑^𝑗=14_𝑗=1 ∑ 𝑤¹⁸₁₇ _𝑖𝑗𝑘 ≤ 1 , for all i and j (13)

Constraint (6) is applied to the mathematical model for the second stage of the research to ensure the load of each lecturer achieved from the first stage holds. Constraint (7) ensures that each lecturer will only teach at most one course at any of the time block. Constraint (8) ensures that each course assigned to a lecturer must be scheduled for a meeting twice a week. In addition, Constraint (9) up to Constraint (13) are introduced to ensure the two course meetings assigned to a lecturer will not be scheduled on the same day.

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4. Results and Discussion

4.1 Result Analysis for Stage 1 Model

The first stage of the experiment produced the list of courses that lecturers are to be assigned to teach for the academic session. The model has the objective function of minimizing the cost function and the constraint used for this stage are the course load and lecturer maximum and minimum load. Upon solving the first stage of the mathematical model, the result is shown in Table 2 that listed the courses to be taught by lecturers. Note that there are some lecturers will be teaching more than one course.

Table 2: List of lecturers assigned to each course Course Lecturer Assigned to course, c

C1 L7 , L12 C2 L2 , L13

C3 L5 , L10 , L13 , L16

C4 L6 , L8 , L10 , L12 , L18 C5 L3 , L4 , L7 , L14 , L19 C6 L2 , L7 , L9 , L14 , L16 C7 L1 , L5 , L6 , L13 , L15

C8 L1 , L7 , L10 , L11 , L12 , L19 , L20 C9 L1 , L3 , L4 , L8 , L9 , L17 , L20 C10 L2 , L3 , L4 , L5 , L6 , L11 , L14 C11 L4 , L8

C12 L2 , L8 , L9, L11 C13 L1 , L9 , L17 , L18 C14 L3 , L5 , L6 , L15

All fourteen courses take four hours lecture each week. Therefore, the result obtained also indicates the total course load and the number of groups for each course. For example, Course C1 has two groups of students and will be taught by two lecturers, L4 and L8. On the other hand, course C8 has the largest number of groups which are 7 groups. The result obtained also agreed with the faculty policy of the number of teaching hours that can be assigned to each lecturer in a week. For example, lecturer L20 who is a Grade 1 lecturer is assigned two groups which consist of 8 hours teaching per week. In contrast, Lecturer L1 who is a Grade 2 lecturer is assigned four groups of students which took 16 teaching hours per week. Both samples of loading satisfy the limit of teaching hours as set by the faculty.

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The loading hours of each lecturer are presented in Table 3. The optimum load is generated by solving the first stage of the mathematical model. Among the Grade 2 lecturers, there are 9 lecturers will be assigned to a maximum of 16 teaching hours per week, while 5 lecturers will be assigned to 12 hours. On the other hand, all the Grade 1 lecturers will be teaching a maximum of 8 hours weekly.

Table 3: The lecturers’ load hours set by the faculty and the optimum load assigned by model

Lecturer

Load i Optimum

load generated

from the model

Lecturer

Load i Optimum

load generated

from the model

Minimum Maximum Minimum Maximum

L1 12 16 16 L11 12 16 12

L2 12 16 16 L12 12 16 12

L3 12 16 16 L13 12 16 12

L4 12 16 16 L14 12 16 12

L5 12 16 16 L15 4 8 8

L6 12 16 16 L16 4 8 8

L7 12 16 16 L17 4 8 8

L8 12 16 16 L18 4 8 8

L9 12 16 16 L19 4 8 8

L10 12 16 12 L20 4 8 8

4.2 Result Analysis for Stage 2 Model

The result obtained from the first stage will be used in the second stage. The modification of the model takes place in the second stage in which the lecturers are assigned to a time block, at the same time satisfying the faculty policies regarding the time and frequency of the class meetings.

The objective function in this stage maintains the attempt to minimize the cost function. However, the decision variable involved here relates the function with time block. There are total of 18 time blocks in a week which consists of 4 time blocks per day from Monday to Thursday and 2 time blocks on Friday. Table 4 presents the optimal solution obtained by solving the model in the second stage, which indicate the optimum weekly schedule for the 20 lecturers in the department.

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Lecturer

Table 4: The assigned time slot for each lecturer in a week

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY

8 am

10 am

2 pm

4 pm

8 am

10 am

2 pm

4 pm

8 am

10 am

2 pm

4 pm

8 am

10 am

2 pm

4 pm

8 am

10 am

L1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0

L2 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1

L3 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0

L4 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0

L5 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1

L6 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1

L7 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 1

L8 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0

L9 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0

L10 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1

L11 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1

L12 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0

L13 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0

L14 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1

L15 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0

L16 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0

L17 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0

L18 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0

L19 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0

L20 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0

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The timetable produced in the computational experiment has resulted a total of 126 meeting per week involving 20 lecturers teaching 14 courses throughout the week. For instance, Lecturer L19

is assigned to teach in four time block per week which is on k= 4, 5, 8 and 11. As each meeting takes two hours long, therefore the lecturer has a load of eight teaching hours per week. The optimal solution obtained from the model satisfies all the model constraints and consequently fulfil the requirements set by the faculty policies. All lecturers are assigned with teaching load within his/her allowed range of teaching hours. In addition, there is no lecturer teach more than one course at any of the time block.

Table 5: The optimal weekly timetable for Lecturer L1

8am-10am 10am-12pm 2pm-4pm 4pm-6pm

Monday Course 7 Course 9

Tuesday Course 8 Course 7

Wednesday Course 13 Course 9

Thursday Course 13

Friday Course 8

Beside this, the optimal solution also scheduled each course to have a meeting twice a week in which both meetings are held on different day. As an example, Table 5 shows the weekly timetable scheduled for Lecturer L1.

Lecturer L1 is a Grade 2 lecturer with 16-hour maximum loading allowed. He/she is assigned to four courses weekly. Each of the courses meets on two different days which yield the total teaching hour of 16 hours. In addition, there is no overlapping of class assignment on any time block.

Table 6 shows the optimal timetable generated for Lecturer L20, who is a Grade 1 lecturer. As his/her maximum loading allowed is 8 hours weekly, hence the lecturer is assigned to two courses which are Course 8 and Course 9.

Table 6: The weekly timetable for Lecturer L20

8am-10am 10am-12pm 2pm-4pm 4pm-6pm

Monday Course 9 Course 8

Tuesday

Wednesday Course 8 Thursday Course 9 Friday

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The teaching cost payable to the lecturers is based on the number of class meetings. The teaching cost per class for Grade 1 and Grade 2 lecturer are RM150 and RM100, respectively. Table 7 shows the total teaching cost payable to the lecturers based on the optimal scheduled timetable.

Table 7: The total teaching cost payable to the assigned lecturers

Grade Cost per class Number of meetings assigned Total

Grade 1 RM150 24 10200

Grade 2 RM100 102 3600

Optimum Cost RM13800

5. Conclusion

This study has presented a two-stage approach to solve the class timetabling problem for an academic department in a local university. The first stage consists of assigning the list of courses to the lecturers, while minimizing the teaching cost payable. This involves assigning 14 courses with various groups to 20 lecturers who are categorized as Grade 1 and Grade 2 lecturers. Using data in March 2018 academic session, the first stage model generated results which successfully assigned all groups of course to all the lecturers available. The second stage model included new set of constraints to assign the lecturers and their respective courses to a set of time blocks. The novelty of the study lies at the generation of new constraints which are introduced to accommodate the faculty policies in terms of class time and frequency in a week. The mathematical model used a cost function in the objective function which favoured optimizing cost when there is different grade of lecturers in the department.

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