Multi-objective Modeling for a Course Timetabling Problem

Vertic Eridani Budi Darmawan

, Yuh Wen Chen

, Aisyah Larasati

, Deni Prastyo

and Anik

Dwiastuti

Department of Industrial Engineering, Universitas Negeri Malang, Malang, Indonesia

Institute of Industrial Engineering and Management, Da Yeh University, ChangHwa, Taiwan

Keywords:

University Timetabling, Curriculum-Based Course Timetabling, Multi-objective, Optimization.

Abstract:

This paper presents a multi-objective modelling approach for Curriculum Based Course Timetabling (CB-

CTT) problem. The problem comprises optimizing weekly scheduling by assigning offered courses to class-

rooms and time-periods. The model accommodates resource utilization of classroom occupation and the

limited cost by given two objective functions of minimizing loss cost of an empty seat and minimizing the

cost to open the course. The proposed model also satisﬁes the solutions that meet the lecturers time prefer-

ence, thereby produce the applied schedule for the lecturers. In terms of response time and quality, the model

outperformed manual alternatives to accommodate minimizing cost resources. The proposed model test using

private university data in Taiwan. The computational results are favourable and interactive using CPLEX 12.9

solver builds on AMPL.

1 INTRODUCTION

Producing an excellent educational timetabling is an

important issue and challenging task for every ad-

ministrative staff in an academic institution. It’s

needed hard work to fulﬁlls the important stakeholder

(lecturers, students, and management) requirements.

Mostly, accommodating the new situation for the cur-

rent practice is to replicate the timetable in previous

years to speed up the process with some minor mod-

iﬁcation. However, recently, some changes may fre-

quently occur due to some new policy for each insti-

tution and mending of what has been developed pre-

viously take so much work. In these circumstances,

given the progress made in achieving the growth of

hardware and software systems, the scientiﬁc commu-

nity continues to investigate this issue in order to es-

tablish general and automated processes for establish-

ing effective and desirable timetables for stakeholders

(Bettinelli et al., 2015).

The curriculum-based course timetabling problem

deﬁnes as constructing the timetable by assigning the

university courses to the time-periods over the work-

ing days in a week and to the speciﬁc classroom that

suitable for the course need and the number of stu-

dents who registered the course and lecturer for each

courses in the weekly basis while satisfying various of

constraint (De Cesco et al., 2008). In every university,

the objective of producing a contented timetabling

could be a difference among others due to the dif-

ferences in preference and educational policy. How-

ever, the purpose is remaining the same to creates a

compelling and feasible timetabling by satisfying the

stakeholder’s preferences for at least to a certain ex-

tent (Kristiansen and Stidsen, 2013).

The curriculum-based course timetabling problem

has been explored extensively since 1980 by several

well-known techniques of the operational research

and the computer science ﬁeld. In the last decade,

the primary methodologies use categorized as swarm

intelligence algorithm, evolutionary algorithm, lo-

cal search algorithm, graph coloring algorithm, and

the exact methods (Kristiansen and Stidsen, 2013).

Among those methodologies, the accurate methods,

integer programming, become one of the preferable

techniques that used in solving the problem to over-

come the quality of the solution found as the advance-

ment in computer software and hardware (Caccetta

and Aizam, 2012).

Clarence H. Martin (Martin, 2004) employed this

approach for Ohio University’s College of Busi-

ness by considering the practical issue on lecturers,

courses, classrooms, and timeslots. (Daskalaki et al.,

2004) perform a novel 0-1 integer programming for-

mulation by minimizing the linear cost function to

accommodate providing a signiﬁcant number of op-

Budi Darmawan, V., Chen, Y., Larasati, A., Prastyo, D. and Dwiastuti, A.

Multi-objective Modeling for a Course Timetabling Problem.

DOI: 10.5220/0009857300100014

In Proceedings of the International Conference on Creative Economics, Tourism and Information Management (ICCETIM 2019) - Creativity and Innovation Developments for Global

Competitiveness and Sustainability, pages 10-14

ISBN: 978-989-758-451-0

erational rules and requirements for the constraints,

such as teaching periods, or days of week or class-

room speciﬁed courses using real data from Depart-

ment Electrical and Computer Science, University of

Patras. Lach and L

ubbecke (Lach and L

ubbecke,

2012) approaching the problem through integer lin-

ear programming using two stages decomposed tech-

nique using real data from the University of Udine,

Department Electrical and Computer Engineering.

The challenge in university course timetabling is

keeping the studied problem as close as to the prac-

tical problem as possible. The multi-objective opti-

mization does an inﬂuential part in this problem (Bet-

tinelli et al., 2015). In a weighted cost objective func-

tion to be minimized, a common strategy is to im-

plement constraints and penalize their violations. In

actual, some universities have some rule to consider

classroom capacity as the constraint that needs to ful-

ﬁll (Bettinelli et al., 2015) as an approach in this study

to give a real contribution and theoretical in the re-

search community ﬁeld. This paper organized as fol-

lows. The foundational problem of curriculum-based

course timetabling described. Then comes with the

explanation multi-objective model formulation, re-

sults, and discussion about the model performance

and conclusion.

2 PROPOSED APPROACH

This section describe the problem description of the

research area and the experimental design to construct

the problem.

2.1 Problem Description

In university curriculum-based course timetabling,

the problem formulated as given a set of course called

curriculum and each curriculum consists of several

lectures/courses. Each course is associated with a lec-

turer. Each course should be assigned in a classroom

at a time-period, which a time-period on particular

weekdays, without any conﬂicts. Each classroom also

has a speciﬁc size and requirement to accommodate

course needed. Fundamentally, to achieve an efﬁcient

and feasible objective, the mathematical formulation

must be satisﬁed with all the related constraints. Ev-

ery institution has the policy to deal with its timetable,

so in many cases, adjustment much likely needed to

satisfying each timetabling community. In this pa-

per, the object of the study is a private university, Da-

Yeh University which located in ChangHwa, Taiwan.

Speciﬁcally, the data from the Industrial Engineering

and Management Department undergraduate course

in the College of Engineering with the time-period

of analysis is fall semester in the last two academic

years, 2017 and 2018. Figure 1 shows the collected

data.

Figure 1: Timetable Data.

In this paper, we assume and construct the time

preference for each professor for each semester as

close as a practical problem and accommodate the

classroom used for a particular equipped class, com-

puter class. The unique attributes university appears,

for the cost, every professor depends on the academic

status (assistant professor, associate professor, and

full professor) and the time-period (morning, after-

noon, and night). The higher academic standing, the

cost per hour, is higher. The time-period divides ev-

ery three hours length based on the length of course

offered, at morning, afternoon, and night. The night

time-period value is higher than the morning and af-

ternoon time-period. To satisfy the institution point of

view, the preference in having a good classroom oc-

cupancy due to efﬁciency resources, classroom, and

cost. Based on that, the model considering the funds

to satisfy the classroom capacity constraint and get

the feasibility timetabling.

2.2 Experimental Design

Constructing the university timetable using multi-

objective programming approach and the following

notation is needed to describe the model by given pa-

rameters that building the model as the essential struc-

tural element, the decision variables, objective func-

tion, and constraint. This model is a conceptual sim-

pliﬁed cost-minimizing model.

Parameters

c : course

r : classroom

t : time-period

s : student group

l : lecturer

fr : maximal capacity of classroom r

ec : number of students enrolled in course c

cost

c,r,t

: corresponding cost when the course c is as-

signed to classroom r, at period t

Il : set of course c taught by lecturer l

Is : set of course c attended by student group s

Tl : set of period t where the lecturer l is not available

Decision Variables

c,r,t

: boolean function, the value is 1 once the course

Multi-objective Modeling for a Course Timetabling Problem

c is assigned to a classroom r, at time-period t; other-

wise, it is zero.

Objective Function

= MinΣ

(c=1)

(r=1)

(t=1)

cost

(c,r,t)

( f

− e

)

(1)

Equation (1) is used to controlling the utilization

of classroom by minimizing loss cost for each empty

seat.

= MinΣ

(c=1)

(r=1)

(t=1)

cost

(c,r,t)

(2)

Equation (2) is used to design the timetabling by

minimizing cost. The mathematical formulation of

the problem is reads as follows:

Z = w

+ w

(3)

Constraint

(r=1)

(t=1)

(c,r,t)

= 1

1 ≤ c ≤ n (4)

Equation (4) indicates that each course must be

assigned in a classroom and at a period

(c=1)

(c,r,t)

≤ 1

1 ≤ r ≤ m, 1 ≤ t ≤ p (5)

Equation (5) indicates that each course cannot be

assigned to the same classroom and period, no colli-

sion.

(c,r,t)

≤ f

1 ≤ c ≤ n, 1 ≤ r ≤ m, 1 ≤ t ≤ p (6)

Equation (6) indicates the course capacity must be

satisfying classroom capacity.

(c∈I

)

nΣ

(r=1)

(c,r,t)

≤ 1

1 ≤ t ≤ p, 1 ≤ l ≤ L (7)

Equation (7) designates that each lecturer cannot

be assigned to more than one course in a given period,

no collision

(c∈I

)

(r=1)

(t∈T

)

(c,r,t)

= 0 (8)

Equation (8) indicates the courses timetable must

be satisfying the lecturer preference time.

(c∈I

)

(r=1)

(c,r,t)

≤ 1

1 ≤ t ≤ p, 1 ≤ s ≤ S (9)

Equation (9) designates that the student cannot be

assigned to more than one courses in a given period,

no collision.

The multi-objective framework considered for this

model, a weighted objective function is used to assist

the preference of decision maker to get a feasible so-

lution set (Hwang and Yoon, 1981). The subjective

weighted method use as simulated function to show

how each objective performs each other (Hwang and

Yoon, 1981).The model is designed to propose a sim-

ple input / output interface to integrate theory and user

practice (Chen, 2007).

3 RESULT AND DISCUSSION

Producing an excellent educational timetabling is an

important issue and challenging task for every admin-

istrative staff in an academic institution. It’s needed

hard work to fulﬁlls the vital issue of stakeholder

(lecturers, students, and management) requirements.

The result shows that timetable is favorably feasible,

shown in Figure 2 and Figure 3. The example of

computed timetable result applies 0.5 for each weight.

The suggested timetable shown the courses are held in

morning and afternoon time-period as a piece of ev-

idence how the model works that there is no courses

occur in the high-cost time-period.

Figure 2: Computed Timetable Academic Year 2018.

Figure 4 shows the variation value of Z in New

Taiwan Dollar (NTD) with the simulation weight be-

tween A1 and A2 (Steuer, 1986). A1 as the objective

to control the utilization of classroom to minimize the

loss cost of an empty seat. A2 as the objective to ﬁnd

the feasible classroom, course and time-period that

minimize the cost. The weight represented as subject

ICCETIM 2019 - International Conference on Creative Economics, Tourism Information Management

Figure 3: Computed Timetable Academic Year 2017.

preferences to the decision maker. The more domi-

nance penalty cost for the loss cost of an empty seat,

the cost result is getting higher.

Figure 4: Computed Result of Model Decision.

Comparing to the existing timetable, the total fund

that they need to spent to open the classroom for the

selected professor is identical. The difference is in

classroom occupancy operation; the proposed model

utilizes each classroom until reach maximum by min-

imizing loss cost of an empty seat for available class-

room and time-period for teacher except for particu-

lar equipped computer classroom, classroom H72950

and H72750, for the special equipment classroom for

special course. Meanwhile, the existing timetable

equalizes the classroom operation for each available

classroom within a time-period in the absence of the

empty seat consideration as shown in Figure 5 and

Figure 6.

Figure 5: Classroom Occupancy Academic Year 2018.

Figure 6: Classroom Occupancy Academic Year 2017.

4 CONCLUSIONS AND

RECOMMENDATIONS

Based on the results, the experimental work supports

that multi-objective programming is capable of gener-

ating the university course timetable. The developed

model considers the issue to satisﬁed the stakehold-

ers by minimizing the cost objective to accommodate

the university policy with favourable results within a

second compared to the manual process. The model

takes a penalty cost of an empty seat into considera-

tion as the way how this model can help to support the

decision maker giving broad options to optimize the

course timetable.

Interestingly, while most scholars are very famil-

iar with the concept, a few scholars consider cost in

the mathematical formulation to solve the problem

(Kristiansen and Stidsen, 2013). Since the ﬁnding

is simple, basic, and practical, this very beginning

conceptual simpliﬁed cost minimizing could be a ba-

sis for further model modiﬁcation. The exploration

of weight assessment method lead to more develop-

ment in multi-objective modeling for future directions

(Chen, 2007).

ACKNOWLEDGEMENTS

This work is a joint research conducted by Universi-

tas Negeri Malang, Indonesia and Da Yeh University,

Taiwan and ﬁnancially supported by the university re-

search fund (PNBP) 2019, the Institute of Research

and Community Services (LP2M), Universitas Negeri

Malang, Indonesia.

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Multi-objective Modeling for a Course Timetabling Problem

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