solving a real constraint satisfaction model for the university

Research ArticleSolving a Real Constraint Satisfaction Model for the UniversityCourse Timetabling Problem A Case Study

Marco Antonio Cruz-Chaacutevez1 Mireya Flores-Pichardo1 Alina Martiacutenez-Oropeza1

Pedro Moreno-Bernal1 and Martiacuten H Cruz-Rosales2

1Engineering and Applied Science Research Center UAEM Avenida Universidad 1001 Colonia Chamilpa62209 Cuernavaca MOR Mexico2Faculty of Accounting Administration amp Informatics UAEM Avenida Universidad 1001 Colonia Chamilpa62209 Cuernavaca MOR Mexico

Correspondence should be addressed to Marco Antonio Cruz-Chavez mcruzuaemmx

Received 10 February 2016 Revised 12 June 2016 Accepted 20 June 2016

Academic Editor Miguel A Salido

Copyright copy 2016 Marco Antonio Cruz-Chavez et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper proposes a real mathematical constraint satisfaction model which defines the timetabling problem in the Faculty ofChemical Sciences and Engineering (FCSE) at the Autonomous University of Morelos State Mexico A Constructive ApproachAlgorithm (CAA) is used to obtain solutions in the proposed model A comparison is made between the CAArsquos results and theschedule generated by the FCSE administration Using the constraint satisfaction model it is possible to improve the allocation ofclass hours in the FCSE so that classroom use is more efficient

1 Introduction

In the field of computer science there is an area known asalgorithmic complexity which is a theoretical measure avail-able to compare different algorithmic solutions It is possibleto observe how they behave when attempting to solve acomplex problem using these tools This measure enables theselection of the most efficient algorithm for solving a prob-lemThe complexity theory classifies the universe of problemsaccording to the inherent complexity of solving them [1]Thisdepends on whether there is an exact algorithm in polyno-mial time behavior and whether there is proof of optimalityfor solving the problem in question If there is such an exactalgorithm then the problem is easy to solve and it is called aP type polynomial problem If the problem is not known todate to have exact polynomial time algorithm behavior thenthe problem is difficult to solve and it is not polynomial it isan NP type problem [2] If this is the case the best solutionfor dealing with an NP problem is to use computationalheuristics which allow for constraints in polynomial time andyield a solution very close to the global optimum [3] This

paper focuses specifically on the problem of the allocationof expensive resources (classrooms) in university schedulingIt is classified by complexity theory as an NP-hard problemwhich is themost difficult type of problem in theNP set [3 4]It is important due to the widespread application in universi-ties to manage class schedules for students and teachers

The scheduling problem consists of generating schedulesfor defined tasks searching for the best way to comply withspecific conditions or user requirements by both teachersand students These problems are very common and arefound in different types of educational environments suchas universities colleges institutes departments [5] sportsactivities [6] transportation activities [7] and other activitiesinvolving people and the use of equipment [8 9]

In the field of education the efficient establishment andplanning of schedules improve the use and organization ofstaff schedules classrooms and equipment The improveduse of resources boosts academic performance for studentsand teachers

The academic programming schedule is a specific prob-lem within the general problem of resource allocation This

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7194864 14 pageshttpdxdoiorg10115520167194864

2 Mathematical Problems in Engineering

scheduling problem is known in the scientific community asthe University Course Timetabling Problem (UCTP) [10]

The staff is responsible for organizing the schedules ofstudents and teachers during each academic period It isimportant to provide the best possible solution according toa number of restrictions which must be satisfied in order toensure that the educational programmeets quality measuresthis implies a great effort

The goal of the scheduling problem in education is toschedule the courses in an academic period (trimester quar-ter or semester) It is necessary to consider several variablesthat impact the problem such as teachers required for allcourses which students need to take the courses groups ofstudents that take a set of courses days and available periodsof the individual classrooms the number of classrooms avail-able the capacity of each room and other relevant variables

Resources that are considered critical or scarce are theclassrooms since they are not resources that are readily avail-able This is why the number characteristics and capacity ofclassrooms must be considered when scheduling as one ofthe features and restrictions of the educational institution

In the literature there is information on theoretical mod-els of UCTP timetables and algorithms to deal with thesemodels For example in [11] a complete study of a theoreticalmodel is presented which does not consider teachers as avariable This model is treated with several computationalheuristics such as genetic algorithms ant colony iteratedlocal search Simulated Annealing and Tabu Reference [12]proposes a new evolutionary algorithm-based timetablingtool for solving course scheduling problems To evaluatethe efficiency of this algorithm they work with fourteenbenchmarks of timetabling problems

The UCTP has also been addressed using real modelsapplied to universities For instance in [13] an evolutionaryalgorithm is applied in the Italian School System with thehybridization of twoheuristics based on local search In [14] agenetic algorithm is applied based on sectors reporting goodresults when applied to a real case Reference [15] presentsa study of exam scheduling in Pahang University of Malaysiawhich includes comparing a constructive heuristic with exist-ing software This case study considers four sets of hard con-straints and three sets of soft constraintsThe authors indicatethat the solutions obtained by the heuristics outperformthose obtained by the software that was implemented at theuniversity to manage all the restrictions in the case In [16]a Simulated Annealing metaheuristic was implemented andpreliminary work is presented that deals with the problem ofprogramming courses in Tamhidi University ofMalaysiaTheinstance they work with has 800 students grouped into fivestudy programs and students that attend the same coursesThe authors work with three sets of hard constraints andthree sets of soft constraints In [17] an initial solution isobtained by a mathematical programming approach basedon Lagrangian relaxation The proposed method has beentested on instances from a university in Indonesia Elevenrequirements of the problem are consideredThey are dividedinto two different categories four requirements represent aset of teacher assignment requirements while the rest arerelated to the course scheduling problem Computational

experiments were performed on two real problem datasetsfrom a university in Indonesia Reference [18] developedthree models of the University Course Timetabling Problemas used at Mkwawa University College of Education andtested using real data from the stated university It waspossible to obtain the optimal solution for real probleminstances through reformulations of models which involve amixture of binary and time-indexed variables In China [19]uses the greedy heuristic approach for UCTP Complex mul-ticonstraint conditions are considered based on the uniformteaching resources of the university which include studentsteachers and classrooms with feasible time slots respectivelyIn particular with respect to most of the graduate schools inChina there are more than 400 courses in every academicsemester A timetable should satisfy 9 basic requirementsThecharacteristics of datasets of two semesters from the graduateschool are considered in the implementation In [20] amath-ematical programming model with an approach called meta-model is proposed It can include implicit restrictions andimplicit objective functions for the assignment of a set of tem-porary classes at Toyama Prefectural University Each classduring a specific period of time has hard and soft constraintsInitially scheduling was done manually and took about 3weeks with a preparation time of 5 to 6 days To validate theproposed mathematical model algorithms apply branch andbound Reference [21] presents a linear integer programmingmodel that solves a timetabling problem in a Rio de JaneiroBrazil department The model was designed to generatesolutions that meet the preferences of the facultyrsquos managersMore specifically thismeant assigning themaximumnumberof lecturers with the highest academic title while minimizingcosts by merging courses with equivalent syllabuses

In the Research Center for Engineering and AppliedSciences (CIICAp) of the Autonomous University of MorelosState (UAEM) two projects have been carried out withrespect to scheduling university courses In [22] a schedulerwas programmed which provided a theoretical model ofschedule generation using Simulated Annealing by applyingbenchmarks of a Metaheuristic Network [11] (medium 01ndash05 hard 01 and hard 02) Feasible solutions were obtainedwith greater efficiency and effectiveness than with other algo-rithms reported in [11] for medium and large instances Thesecond project involved the development of a basic system ofadministrative schedules for the Faculty of Chemical Sciencesand Engineering (FCSE) at the UAEM with web interfaceusing a three-layer model client web service and databaseThrough this project a system of consultations of academicschedules for teachers and students via the internet wasdeveloped A basic constraint model was also proposed toavoid overlapping subjects for students in the FCSE butwithout taking into account restrictions on the allocation ofteachers for the subjects they teach [23]

The present work proposes a constraint satisfactionmodel for the problemof programming university courses forthe Faculty of Chemical Sciences and Engineering (FCSE) Itincreases the number of actual model restrictions for teach-ers aiming to create a complete model for the slot allocationin FCSE The administration of the Faculty of Engineeringcreates a timetable of courses manually Usually there are

Mathematical Problems in Engineering 3

overlaps in the classes taken by students which leaves themwith the dilemma of which course to take In addition thestudents usually attend classes in a discontinuous scheduleThe considerations described above detract in several waysfrom the quality of student learning in the FCSE

Given the problem of schedule assignment in the FCSE aconstraint model is proposed which considers the real caseof allowing the FCSE to offer courses during each periodsatisfying in its entirety the restrictions of the proposedmodel The contributions of this paper are the mathematicformulation of a real problem and the results related to a realapplication

This paper is divided into six sections Section 1 is anintroduction to the University Course Timetabling Problemand its application in universities Section 2 describes theproblem of allocation of slots in the Faculty of ChemicalSciences and Engineering (FCSE) in the city of CuernavacaMorelos Mexico Section 3 presents the model of the con-straint satisfaction problem for the FCSE Section 4 definesthe instance solved in the allocation problem for the FCSEschedules It also explains why the decision was made to usea Constructive Approach Algorithm (CAA) for generatingfeasible solutions that meet the constraints of the totalallocation model for the scheduling algorithm of the FCSESection 5 shows the experimental results of feasible solutionsobtained by the CAA and compares the solution obtained bythe administration of the FCSE Finally Section 6 presents theconclusions

2 University Course Timetabling Problemin the Faculty of Chemical Sciences andEngineering

The Faculty of Chemical Sciences and Engineering at theAutonomous University of Morelos State started in 1977 aspart of the solution to growing needs of industries in theMorelos area Currently the FCSE offers majors in indus-trial chemistry chemical engineering industrial engineeringmechanical engineering and electrical engineering

The FCSE has access to 41 spaces including the following

(i) 21 classrooms located in the main building of theFCSE

(ii) 1 academic computing center (located on the top floorof the FCSE building)

(iii) 2 industrial analysis laboratories (Chemistry Labora-tories 3 and 5 both located on the top floor of theFCSE building)

(iv) 2 Chemistry Laboratories (Chemistry Labs 1 and 2both located on the ground floor of the FCSE build-ing)

(v) Basic multidisciplinary workshop (TAMULBA)which includes 1 physics lab and 10 classrooms

(vi) Unit Operations Laboratory (LOU) 1 Unit Opera-tions Lab and 3 classrooms

The completion of the degrees takes 10 semesters studentsare permitted a maximum of 12 semesters Students attend

classes either in themorning or in the afternoon shift Duringthe first two semesters the students are in rigid groups andthe classes are determined by the department administrationAfter those two semesters the groups are flexible and studentsadvance based on their ability From the third semester oneach student can choose the subjects they wish to take inwhich semester they take them and with the professor oftheir choice Students are admitted to the FCSE each year andbegin study in September or February

The enrollment of students in the FCSE is 1500 on averagebut varies slightly each semester To teach the courses offeredevery semester the FCSE uses an average of 150 tenuredteachers plus others that are not tenured

The FCSE performs manual scheduling attempting totake into account the capacity of the classrooms and facilitiesavailability of full time professors days and existing periodsthe classes that students can choose and the restrictions onchoice of subjects

The FCSE administration considers the following whengenerating class schedules

(i) A subject is formed by a subset of 4 or 6 events Thenumber of events that forms a subject depends onthe subject itself The events that are scheduled ina classroom must have the necessary infrastructureand the required facilities provided by the classroom(television projectors and other necessary electricalconnections)

(ii) A search is done to find the subjects and their eventsthat can be scheduled in the periods and classroomsavailable for the week

(iii) An attempt is made to prevent classrooms from beingunderutilized Classrooms are assigned according tothe size of the student group A small group of stu-dents needs a small space not a large lecture hall Theclassroom assigned must have the necessary capacityfor the event

(iv) Tenured teachers can decide the day and periodthey teach their subjects Because the assignment oftenured teachers is known and is chosen by themtheir class schedules are knownThe only assignmentfor tenured teachers is the classroom where they willteachThe use of theword tenured at theAutonomousUniversity of Morelos State warrants definition as itvaries from itsmost commonusage In this universitytenure means that a particular teacherprofessor isa permanent member of the FCSE This person hasalso earned the rights to a certain course or coursessimilar to being the owner of the course No otherperson can teach the subject unless the tenuredteacher requests a temporary leave allowing someoneelse to teach the course only during the tenuredteacherrsquos absence Upon return the tenured teacherreclaims rights to the course

(v) Students should have continuous schedules theyshould not attend classes in the morning and late inthe day The events of the same subject must be pre-sented in consecutive periods if they appear on the


same day The subjects taught are composed of 4 or 6eventsThese events should be scheduled consideringthat half of the events that correspond to a subjectshould be taught on the same day in consecutiveperiods For example if a subject is composed of 4events and event 1 is scheduled on day 1 in period1 then event 2 must be scheduled consecutively inperiod 2 of day 1 If event 3 is scheduled in period4 then event 4 must be scheduled consecutively inperiod 5 of the same day

(vi) A search checks for classroom allocation in whichoverlap of classes for students is avoided

(vii) The search for a classroom allocation attempts toavoid schedule overlap for tenured teachers Thesubjects of nontenured teachersmust be scheduled ondays and periods that are free after programming thetenured teachers

It has never been possible to generate an adequate allocationschedule in the FCSE which satisfies all the considerationsdefined in the allocation problem This paper proposesa model of constraint satisfaction with a proposed solu-tion using a Constructive Approach Algorithm (CAA) forscheduling resources in the FCSEThis allows class schedulesto be generated which satisfy the toughest considerationscurrently taken into account which the model calls restric-tions The schedules generated are proposed solutions to thetimetabling problem in the FCSE

The results of this work improve the process of gener-ating class schedules and attempt to eliminate the currentproblems with compliance to restrictions It also aims todecrease the process time it takes to schedule resourcesmanual scheduling requires up to two weeks to completeIn addition there is an attempt to utilize the resources ofthe FCSE more efficiently which in turn promotes studentachievement

3 Constraint Satisfaction Model forthe Faculty of Chemical Sciences andEngineering

In the constraint model several sets of variables are takeninto account such as the set of events 119864 = 1 2 3 119899

119864

the set of events that require laboratory equipment 119864119897=

1 2 3 119899119864119897 the set of time slots 119879 = 1 2 3 119899

119879

the days 119863 = 1 2 3 4 119899119863 the set of periods 119875 =

1 2 3 119899119875 the set of classrooms 119877 = 1 2 3 119899

119877

the set of classrooms that have laboratory equipment 119877119897=

1 2 3 119899119877119897 student enrollment 119878 = 1 2 3 119899

119878

teacher enrollment 119872 = 119872119879119872119873 which is composed of

two subsets the enrollment 119872119879 119872119879= 1 2 3 119898

119879

of tenured teachers and 119872119873 119872119873= 1 2 3 119898

119873 of

nontenured teachers the set of classes 119862 = 1 2 3 119899119862

the subset of events in each class 119888119894 119888119894= 119890119894 119890119895 and finally

the set of facilities 119865 = 1 2 3 119899119865

The constraint satisfaction model is as follows119899119879

sum

119905=1

119899119877

sum

119903=1

119883 (119905 119877 (119903 119890)) = 1 forall119890 (1)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883 (119890 Φ (119905 119903) 119877 (119903 119890)) = 2 forall119890 (2)

119899119864

sum

119890=1

119883 (119890 Φ (119905 119903)) le 1 forall (119905 119903) (3)

119899119878

sum

119904=1

119885 (119904 119883 (119890 Φ (119905 119903))) le Capacity (119903)

forall (119890 119905 119903)

(4)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883(119890tt Φ (119905 119903) 119877 (119903 119890tt)) = 2 forall119890tt (5)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119904 119883 (119890 Φ (119905 119903))) le 1 forall (119904 119905) (6)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119898119883 (119890 Φ (119905 119903))) le 1 forall (119898 119905) (7)

119899119879

sum

119905=1

119899119877119897

sum

119903119897=1

119883(119890119897 Φ (119905 119903

119897) 119877119897(119903119897 119890119897)) = 2 forall119890

119897

(119905 119903 119890 119904 119898) isin 119873

(8)

The model of constraint satisfaction is subject to eight sets ofhard constraints which have to be satisfied so the instancesolution obtained is feasible

A feasible solution is found when events are assigned totime slots Each available room has a set of time slots in thiscase 119899

119879time slots A time slot 119905 is defined in a 2D space as

a function of coordinates (119889 119901) Therefore 119879 = 119905 isin 1198732 |119905 = 119891(119889 119901) 119905 = 1 2 3 119899

119879 The full assignment of events

takes place in a 3D space as a function ofΦ(119905 119903) where 119903 isin 119877Then for eachΦ(119905 119903)

Φ (119905 119903) =

1 If assigned to an event

0 Otherwise(9)

Classroom 119903 isin 119877 must satisfy the needs of event 119890 isin 119864 thatoccurs in the time slot 119905 isin 119879

The set of constraints in (1) implies that classroom 119903 isin 119877must satisfy event facilities 119890 isin 119864 that occur in the time slot119905 isin 119879 For each classroom 119903 isin 119877 and event 119890 isin 119864 the followingis defined

119877 (119903 119890) =

1 If 119903 satisfies facilities of 119890

0 Otherwise(10)

Each and every one of the events 119890 isin 119864 has to be scheduledEach event must be scheduled in a time slot 119905 isin 119879 in aclassroom


The set of constraints in (2) require that each and everyone of the events must be programmed Each event mustbe scheduled in a time slot of a classroom with adequatefacilities

119883(119890 Φ (119905 119903) 119877 (119903 119890))

=

1 If Φ (119905 119903) + 119877 (119903 119890) = 2

0 Otherwise

(11)

Classroom 119903 isin 119877may allowonly one event 119890 isin 119864 for each timeslot 119905 isin 119879 This implies that in time slot 119905 isin 119879 for classroom119903 isin 119877 a maximum of one event may be programmed

The set of constraints in (3) requires that a room admitonly one event per time slot This implies that for time slot119905 isin 119879 in classroom 119903 isin 119877 a maximum of one event can beprogrammed

Classroom 119903 isin 119877 must have enough capacity to hold thegroup of students 119904 isin 119878 participating in the event 119890 isin 119864 Thecapacity of each classroom 119903 isin 119877 is input data

To find when student 119904 attends an event the function pre-sented in (12) is defined This function indicates that student119904 attends the event that occurs in time slot 119905 in classroom 119903The restrictions in (4) must be met the classroom must havesufficient capacity to serve the group of students who attendthe event 119890 isin 119864The capacity of each classroom 119903 is input data

119885 (119904 119883 (119890 Φ (119905 119903)))

=

1 If 119883 (119890 Φ (119905 119903)) = 1 Then 119904 serves 119890

0 Otherwise

(12)

Each and every one of the events given by MT teachers willbe scheduled in the first instance as a partial assignment inthe requested periods

In constraint set (5) the events of the tenured teachers(tt) are programmed The assigned classroom must havethe facilities required by the teacher This gives a partialscheduling

A student 119904 isin 119878must not have overlapping eventsThe set of constraints (6) avoids the overlap of events

attended by the student 119904 isin 119878 As indicated by function(12) if the student attends event 119890 in classroom 119903 then thisstudent will not attend another event 1198901015840 during the same timeslot 119905 Therefore 119885(119904 119883(1198901015840 Φ(119905 119903))) = 0 forall119890

1015840isin 119864 with

1198901015840= 119890 Restriction (6) indicates that student 119904 may not have

overlapping events

119885 (119898119883 (119890 Φ (119905 119903)))

=

1 119878119894 119883 (119890 Φ (119905 119903)) = 1 Then 119898 attends 119890

0 Otherwise

(13)

A teacher119898 isin 119872must not have overlapping eventsThe set of constraints (7) prohibits overlap of events

taught by teacher119898 isin 119872 Function (13) ensures that if teacherm attends event 119890 in classroom 119903 then the teacher cannotattend another event 1198901015840 in the same time slot 119905 This means

Table 1 Symbolic representation of a timetable

(119889 119901)Monday Tuesday Wednesday Thursday Friday Saturday1 1199051 1199052 1199053 1199054 1199055 1199056

2 1199057 1199058 1199059 11990510 11990511 11990512

sdot sdot sdot sdot sdot sdot



15 11990585 11990586 11990587 11990588 11990589 11990590

that119885(119898119883(1198901015840 Φ(119905 119903))) = 0 forall1198901015840 isin 119864with 1198901015840 = 119890 Restriction(7) indicates that teacher119898 cannot be present in overlappingevents

In the set of constraints (8) events 119890119897which correspond

to lab classes must meet in classrooms with laboratoryequipment the assigned classroom must meet the needs ofthe teacher (10)

The values that the variables 119905 119903 119890 119904 and119898 can take mustbe natural numbers

The set of teachers is divided into two subsets119872 = 119872119879cup

119872119873 The set of time slots is also divided into two subsets

119879 = 119879119879cup 119879119873 The time slots 119905 that form part of 119879

119879are as-

signed to tenured teachers 119872119879 who teach classes at their

request Each tenured teacher requests a specific time slot 119905 inwhich to teach his or her class119872

119879= 119898 | forall119898 exist119905 isin 119879

119879 119905 =

Asigned(119898) The assignment of 119905 to 119898 is input data For alltime slots 119879

119873 which will be assigned to nontenured teachers

119872119873 the specific assignment of 119905 for each nontenured teacher

119898119899is not known Given the above the model of the FCSE

begins with a feasible partial solution because the allocationof 119879119879is known and is selected by all tenured teachers119872

119879in

the timetable of classes for the semester All that remains toassign to119872

119879is the classroom 119903 where the event will be held

The objective is to find an instance solution of theconstraint satisfaction model of the Faculty of ChemicalSciences and Engineering (FCSE) that minimizes the valueof the objective function An instance of the problem occursin each semester of the school year The instance is generatedbased on the supply of subjects subjects selected by eachstudent and the enrollment of students and teachers

31 Symbolic Representation of the Problem Instance for theFCSE A representation of the University Course TimetablingProblem is constructed using a 3DmatrixThe process beginswith a 2D matrix which represents an available classroomand is comprised of a set of time slots 90 slots in this case Atime slot 119905 is defined as a 2D space based on the coordinates(day period) as seen in Table 1

The scheduling of events for a classroom should utilizeone of the 90 time slots available so there is a 2D array foreach of the 41 classrooms (including laboratories) available inthe FCSE Table 1 shows the symbolic representation for eachclassroom

In Table 1 it is possible to observe the periods of timewhen there should be events programmed in a classroomThetwo dimensions observed are made up of days 119889 and periods


of time 119901 in which the FCSE works Since the instance has 15periods and 6 days there are 90 time slots in each classroomnumbered 1199051 to 11990590 as shown in Table 1

Table 2 shows an example of an eventmap for a classroomthat includes the variables in the FCSE casewith 6 days (Mon-day Tuesday WednesdayThursday Friday and Saturday) 15periods (119901 = 1 to 119901 = 15) and the assignment of 18 eventscorresponding to eight subjects identified with ID (15 29 4950 120 212 332 and 335) As the events are scheduled inthe time slots the identifier numbers ID of the events areplaced in the fields corresponding to the assigned periods Forexample in Table 2 the field with ID 120 consists of 4 eventsThe time slot spaces which are blank in Table 2 represent thetime periods available for scheduling other events

When making the event schedule for the different class-rooms the variable 119903 (classrooms) creates a third dimensionto form the set of tables corresponding to the set of availableclassrooms as shown in Figure 1 Therefore with the setof classrooms a tridimensional structure is obtained withcoordinates for each time slot 119905 = (119889 119901 119903) To allocate thetotal set of FCSE events 119864 in slots available 119905 in the 3Dmatrixall of the hard constraints are satisfied The hard constraints(1 to 8) were presented in this section in the constraintsatisfaction model Then a solution for the UCTP for FCSEis generated

Upon characterizing the variables of the case study FCSEthere are 6 days (M TUW TH F and SA) 15 periods (Period1 to Period 15) and 41 classrooms (119903 = 1 to 119903 = 41)This yieldsan area of 3060 (6times15times34) time slots available in classroomsand 630 (6times15times7) time slots available in laboratories whereonly classes requiring laboratory equipment are taught

Figure 1 presents an instance of 3 events and 2 classroomsfor the University Course Timetabling Problem using abipartite graph model The graph consists of the set of eventson one side and the assignments 119905 = (119903 119889 119901) which canoccupy events on the other side In this instance the problemonly has 2 days 119889 and two periods 119901 Each classroom 119903 has acapacity119860(119896) and can be assigned a single event in each 119905Theevent 119890

119894that is assigned a t indicates that the event will be held

in classroom 119903 on day 119889 and during period 119901 The objective isto determine 119905 for each 119890

119894in the bipartite graph taking into

account the capacity 119860(119896) of classroom 119903 Figure 2 presents asolution to the problem presented in Figure 1 For exampleevent 119890

1is assigned on day 119889

1during period 119901

1 to classroom

1199031 which has a capacity 119860(1) Naturally to assign events to

classrooms the constraints present in the constraint satisfac-tionmodel presented in (1) to (8) must be taken into account

4 Definition of the Problem Instance forthe Faculty of Chemical Science andEngineering (FCSE)

The parameter values for the instance of the FCSE problemare defined according to a scheduling assignment by the FCSEadministrationThey are applied as a solution in an academicsemester Naturally this applied solution is not feasible forthat semester given the amount of overlap found in the sched-ules of the students who attended The considered instance

Table 2 Abstraction of a timetable FCSE case

(119889 119901)Monday Tuesday Wednesday Thursday Friday Saturday1 49 332 502 49 15 332 503 15 29 3354 15 335

12 120 21213 120 21214 12015 120

Table 3 Parameter instance for fall semester (August to December)2015 in the FCSE

Parameters ValueEvents (119899

119864+ 119899119897) 1514

Classrooms (119899119877+ 119899119877119897) 41 (classrooms and laboratories)

Students 1426Days 6Periods 15Time slots 90Facilities 7Tenured teachers 156

Table 4 Classification of instances

Concept Small Medium LargeEvents 100 400 400Classrooms 5 10 10Features 5 5 10Students 80 200 400MNSE 20 50 100

corresponds to the schedule for the AugustndashDecember 2015semester The referred values were obtained from the fieldstudy in the FCSE with information provided by the personresponsible for the administrative scheduling of the FCSEThe definition of the instance is presented in Table 3

Based on their research and theoretical evidence ofUTTPinstances the Metaheuristic Network group [11] provideda ranking of the instances or benchmarks collected inthree classes depending on the degree of difficulty and sizeAccording to the number of resources in the program theinstances are classified as small medium and large [11] Thecharacteristics of the three classes of instances are shown inTable 4 where MNSE is the maximum number of studentsper event

The instance of the FCSE has 1514 events 41 classrooms7 features and 1426 students In addition it has 156 tenuredteachers who teach events that should be scheduled in thefirst instance Given this information it can be observed thatthe instance of the FCSE is a size that exceeds the largestclassification of the Metaheuristic Network The number of


A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

Figure 1 Bipartite graph model for University Course Timetabling Problem

Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

Figure 2 A solution to the University Course Timetabling Problempresented in Figure 1

hard constraints in the case study is greater than the numberof hard constraints contemplated by [11]

41 Format of the Problem Instance For the execution ofthe FCSE problem instance use of the CAA (or any otherheuristic or metaheuristic) that obtains feasible solutionsrequires that the input data or instance of the problemmust be stored in text files in the proper format so theyare readable The same format is generated and then appliedto the benchmarks presented in [11] Since the number ofinstance data is very large the instancemust be stored inmorethan one file The instance data of the problem is organizedinto 6 files with the format shown below

Text File 1 (Defining the Problem)

(i) First line for the number of events number ofclassrooms number of features number of studentsall numbers are integers

(ii) One line for each classroom each line indicates thecapacity of a classroomTable 5 shows an example of the FCSE instance Inthis example it can be observed that the first lineshows the number of events number of classrooms

number of features and number of students in thatorder From line 2 to line 42 the capacity of each ofthe 41 classrooms of the FCSE is presented The onlyinformation that appears in text file 1 is the coding

Text File 2 (Assigning Events per Student)

(i) One row for each studentevent coding is done witha zero or a one A zero indicates that the student doesnot attend the event a one indicates that the studentattends the event Table 6 shows an example for 2students and 4 scheduled events In this example itcan be seen that the first student attends event numbertwo and the second student attends events one andtwo All that appears in text file 2 is the coding

Text File 3 (Assigning Events per Teacher)

(i) One line for each teacherevent Coding is done with azero or a one Zero indicates that the teacher does notteach the event one indicates that the teacher doesteach the event Table 7 presents an example with 2teachers and 4 scheduled events In this example itcan be observed that the first teacher attends eventnumber two and the second teacher attends eventsone and two All that appears in text file 3 is thecoding

Text File 4 (Tenured Teachersrsquo Hours)

(i) One line for each scheduleevent This coding isfour columns containing integers Table 8 presents thecoding for assigning events to each tenured teacherThe first column indicates the ID of the tenuredteacher who requests to teach a specific class in theschedule The second and third columns indicate theday (Monday = 1 Friday = 5) and period (period =1 period = 15) in which a tenured teacher requeststo teach the event Numbers other than zero indicatethat a tenured teacher has requested to teach the


Table 5 Coding to define the problem of the FCSE

Line Classroom capacity Event number Classroom number Feature number Student numberCoding

1 mdash 1514 41 7 14262 10 mdash mdash mdash mdash

mdash mdash mdash mdash42 60 mdash mdash mdash mdash

Table 6 Coding for assigning events per student

Student Coding Event number1 0 11 1 21 0 31 0 42 1 12 1 22 0 32 0 4

Table 7 Coding for assigning events per teacher

Teacher Coding Event number1 0 11 1 21 0 31 0 42 1 12 1 22 0 32 0 4

Table 8 Coding for assigning events to each tenured teacher

ID teacher Day Period Subject IDCoding

1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2

event at the scheduled time A zero indicates thatthe tenured teacher has not requested to teach theevent The fourth column shows the ID of the field towhich the event belongs Table 8 presents an exampleof eight events scheduled for the tenured teacher withID Teacher = 1The teacher asks to teach two subjectsone with subject ID = 1 and the other with subjectID = 2 each subject has four events The teacher asks

Table 9 Coding for the assignment of characteristics for eachclassroom

Classroom ID CodingCharacteristic

1 01 11 01 02 12 12 02 0

that two events of subject ID = 1 be scheduled onTuesday in the first and second period The teacheralso requests that the other two events of subject ID =1 be on Thursday in the first and second period Forsubject ID = 2 the teacher does not require a specialschedule and adjusts to the schedule assigned Onlythe coding appears in text file 4

Text File 5 (Facilities by Classroom)

(i) One line for each classfeature Table 9 presentsthe coding for the assignment of features for eachclassroom The first column indicates the ID of theclassroomThe second column shows the characteris-tics of a classroom A zero means the classroom doesnot satisfy the characteristic a one means it doesFor example in Table 9 column one has three class-rooms (ID = 1 2) In column two it is shown thatthere are four characteristics that the classroom couldsatisfy In the case of classroom ID = 1 it only satisfiescharacteristic number 2 Classroom ID = 2 satisfiescharacteristics 1 and 2Only the coding appears in textfile 5

Text File 6 (Requirements by Event)

(i) One line for each eventneed Table 10 presents thecoding required to allocate for each subject based onthe events that comprise the subject The first columnindicates the ID of the subject The second columnindicates what the class events need as related to thecharacteristics of a classroom A zero is coded if theevent does not need the characteristic a one is codedif the event does need the characteristic For example


Table 10 Coding for allocating needs for each class

Class ID CodingNeed

1 01 11 01 02 12 12 02 0

in Table 10 column one has two subjects (ID = 1 2) Incolumn two it is shown that there are 4 characteristicswhichmay be required of a subject For the events thatcomprise subject ID = 1 only characteristic numbertwo is needed For the events that comprise subjectID = 2 characteristics 1 and 2 are needed Only thecoding appears in text file 6

The Constructive Approach Algorithm was programmed togenerate feasible solutions which satisfy the hard constraints(presented in Section 3) in themodel of the Faculty of Chem-ical Science and Engineering The constraint satisfactionmodel can be solved with academic tools (solvers) includingadapted solvers designed to solve optimization models Inthis case the proposed model does not include the objectivefunction that permits optimization For simplicityrsquos sake inthis work the Constructive Approach Algorithm was used toobtain solutions to the proposed model and to compare withproposed solutions to the FSCE administration In Appendixthe Constructive Approach Algorithm for generating feasiblesolutions is presented

5 Experimental Results

Experimental tests of the Constructive Approach Algorithmwere performed on a personal computer with Intel Core i722 GHz CPU 8GB RAM SO Windows 7 Home PremiumMicrosoft Visual Studio 2008 Ver 90210228 For eachexperimental trial 30 executions of the CAA algorithmfeasible solutions generator were performed

The efficiency analysis of the algorithm to generate feasi-ble solutions is shown in Figure 3The times are shown that ittook the algorithm to obtain each of the 12 feasible solutionsand run 30 tests The shortest time required to find a feasiblesolution was 3 hours and 33 minutes and the longest was 6hours and 22 minutes There was a feasible solution obtainedin 40 of the executions In the cases where no feasiblesolution was obtained the time it took to perform the totalnumber of iterations before arriving at the stop criteria for thealgorithm ranged from 4 to 8 hours The path of the CAA isalways different because it is a nondeterministic algorithmTherefore the time it takes to perform operations to findpartial solutions will be different mainly due to the iterativesearch which checks for overlaps of students and teachers

1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution

Feasibleinfeasible solutions

FeasibleInfeasible

Figure 3 Solution generation time for 30 executions of the Con-structive Approach Algorithm

70

60

50

40

30

20

10

0

OA 65capacity 70 Average classroom occupation

OA 13capacity 15

Classrooms

lowastOA occupation average

Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513

Figure 4 Average occupancy periods per classroom

51 Frequency Allocation of Events in Classrooms Figure 4presents the average occupancy per classroom period of the41 classrooms that are considered in the instance of the FCSEThe graph shows that no classroom is occupied in all 90 avail-able periodsThe highest average occupancy was classroom 2with 65 of the 90 periods occupied that is 778 This class-room has a capacity of 70 students It was observed that class-rooms with greater capacity have higher average occupancy

The blue bars displayed in Figure 4 identify classroomsthat are laboratories They always have the same numberof busy periods ranging between 14 and 42 Only subjectsthat require laboratory equipment can be taught in theseclassrooms they cannot be employed to deliver a lecture asthey lack the infrastructure for such an activity

The existence of seven laboratories and several small-capacity classrooms decreases the number of periods avail-able for event scheduling The seven laboratories have 630(7 lowast 90) available periods but were only occupied in 203periods because they can only be occupied when the classrequires a laboratory There are 18 small-capacity meetingclassrooms which hold between 15 and 30 students (seeFigure 7) For classrooms with small capacity occupancyaverages between 13 and 38 periods leaving between 77 and52 unoccupied periods For example in Figure 4 it can beobserved that classroom 24 with a capacity of 15 has only 13out of 90 periods scheduledThe 18 small-capacity classrooms


0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events

Figure 5 Average weekly events assigned to 70-student capacityclassrooms in 12 feasible executions

0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events


are almost unusable because most of the events requireclassrooms with capacities larger than what they can provide

Figure 5 presents the Pareto chart for implementing theCAA for the 12 feasible solutions achieved in 30 executionsThe Pareto chart shows the average frequency of event pro-gramming for the 9 larger classroomswhich have amaximumcapacity of 70 students It is observed that on average foreach large classroom the CAA programs approximately 58events per week in two executions 59 events per week intwo executions 60 events per week in two executions 63events per week in two executions and 61 events per weekin four executions Each scheduled event in a large classroomcannot exceed the maximum capacity of 70 students Sincethemaximumnumber of events that can be scheduled in eachclassroom is 90 per week (see Table 1) for the nine 70-studentcapacity classrooms there is on average a 33 vacancy rate perweek

Figure 6 presents the Pareto diagram for implementationof the CAA for the 12 feasible solutions achieved in 30 execu-tions The Pareto chart shows the average frequency of eventprogramming in the 4 small classrooms that have amaximumcapacity of 15 students It is observed that on average for eachsmall classroom the CAA programs less than 15 events perweek in one execution less than 16 events per week in three

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

CapacityOccupied time slotOverlapping events

Classroom

Figure 7 Solution obtained for the UCTP of the FCSE FCSEadministration Semester AugustndashDecember 2015

executions less than 17 events per week in three executionsless than 18 events per week in three executions and less than20 events per week in two executions Each scheduled eventin small-capacity classrooms cannot exceed the maximumcapacity of 15 students Since themaximumnumber of eventsthat can be programmed in a classroom is 90 per week it canbe concluded that on average the 4 small classrooms withseating for 15 people have close to a 786 vacancy rate

52 Comparison with the Solution Obtained by the FCSEAdministration for the Study Period The solution generatedby the administration of FCSE has the following characteris-tics Of the 656 tenure events 3 could not be assigned to theperiods requested by tenured teachers They were assignedto other periods thus constraints set 5 was not satisfiedAmong the 858 events of nontenured teachers there were28 events assigned during busy periods so constraints set 3was not satisfied From this analysis it can be observed thatthe solution obtained by the FCSE administration was notfeasible Analyzing further it was observed that of the 1514events scheduled in this instance therewere 459 eventswhichpresented overlap (see Figure 7) Constraints set 7 was notsatisfied All other constraints sets were met If events thatoverlap are part of 4-event classes there are 114 classes withoverlap If the events that overlap are part of 6-event classesthere are 76 classes with overlap The number of overlappingareas in the solution proposed by the administration of theFCSE is between 76 and 114 classes for the semester fromAugust to December 2015 The solution obtained by FCSEallows overlap of classes though it is not possible for studentsto attend all of their classes if there is overlap

The solution obtained by the CAA algorithm presents thefollowing results 656 tenure events all of which could beassigned to periods requested by the teachers thus satisfyingconstraints set 5 Of the 858 nontenure events all eventswere assigned to unoccupied periods thus satisfying the con-straints set 3 All other sets of restrictions were also enforcedThe solution obtained by the proposed CAA for FCSE isfeasible Therefore there is no overlap of events for students

Figures 7 and 8 present the occupancy of classrooms andthe behavior of the overlap allowing a comparison between


CapacityOccupied time slot

Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

Figure 8 Solutions obtained for the UCTP of the FCSE Construc-tiveApproachAlgorithm (CAA) Semester AugustndashDecember 2015

the solutions generated by the FCSE administration versusthe CAA For the solution generated by the administration(Figure 7) a total of 459 events showed overlap for studentsWhen a period (slot) in a classroom is occupied it isunderstood that the period is occupied by an event Soin classroom 1 there are 67 (occupied time slot) eventsheld during the week of a total of 90 slots available inthe classroom Of those 67 events 39 are overlapping Thecapacity of classroom 1 is 70 students per eventThere are only13 classrooms where overlapping events do not appear in theadministrationrsquos solution (26 27 28 29 31 32 33 36 37 38 3940 and 41) It can also be seen that there are five classroomsin which events were not assigned (28 32 33 39 and 40)Thesolution generated by the Constructive Approach Algorithmdoes not have overlapping events (Figure 8) In the solutionproposed by the CAA it can be seen that all classrooms areoccupied No classroom use reached its maximum limit (90slots) The classrooms with smaller capacity have the fewestof events For example classrooms 24 25 30 and 31 have13 17 13 and 16 events scheduled respectively Each of theseclassrooms has a capacity of 15 students

6 Conclusions

Theproposedmathematicalmodel in this paper considers theperformance of the hard constraints identified in the real caseof university programming courses in the FCSE

It was difficult to obtain solutions that complied withall the constraints of the proposed mathematical modelOnly 40 of the 30 tests performed found feasible solutionsThe solutions were found in an average of 7 hours eachwhich indicates the complexity of the hard constraints of theproposed model This is compared to the infeasible solutionobtained by the administration of FCSE after about twoweeksof work

The mathematical model proposed is the one that hasbeen implemented manually in the FCSE for over 20 yearsDue to the growth in student enrollment and additionalfaculty required the number of schedule overlaps for subjectshas increased It has been more difficult for the adminis-tration of the FCSE to find good solutions In addition thenumber of classrooms that have been built in recent years is

not proportional to the increase in demand The proposedmathematical model provides an opportunity to implementefficient scheduling of university courses without overlapin the FCSE This model can be solved by any heuristictechnique In this case because it is relatively simple and easyto analyze a Constructive Approach Algorithm was used

In the analysis of the results it is observed that the class-rooms with smaller capacity are underutilized One proposalto improve the allocation of these classrooms is the formationof various small groups of students in classes instead of thecurrent large groups The formation of small groups wouldgenerate a greater number of subjects (events) to programwhich would in turn require either a greater enrollment ofteachers or teachers who teach a greater number of subjects Itwould facilitate the allocation of small classrooms to increasethe use of these The student enrollment could then increasein the FCSE because it would still be possible to obtainfeasible solutions without investing in the construction ofadditional classrooms for the new smaller groups Anotherproposal is to use other heuristic techniques to solve theproposed model with greater efficacy This could facilitatefinding appropriate solutions according to the capacity of theclassrooms A change would need to bemade to the proposedmathematical model so that new constraints needed to beincludedThe changewould cause the heuristics to occupy theclassroomswith less capacity first and then use the classroomswith higher capacity

Appendix

Constructive Approach Algorithm GeneratesFeasible Solution (See Algorithm 1)

The description is as follows

(i) Line (1) The algorithm starts with the instruction toread the input instance The instance information isobtained from the 6 text files The coding of these 6text files is explained in Section 4

(ii) Line (2) Before performing the schedule of eventsitems are classified by type (taking into accounttenured teachers) and features (considering the needsof events)

(iii) Line (3) The cycle for 119894 = 0 to NumSoluciones doverifies the condition for each execution cycle toprocess the number of solutions (NumSoluciones)that are generated

(iv) Line (4) With the instruction initialize to zero arraysand empty structures the structures that are used tomanipulate data during the creation of a solution areinitialized Three-dimensional arrangements permitthe solutions to be saved These arrangements arenecessary due to the organization of data that is tobe saved (see Figure 1) Such information can then beused to obtain schedules for classrooms

(v) Lines (5)ndash(10) The schedule of events begins withthe performance of hard constraint (RD) number (5)of the model (see Section 3) The schedule of events


(1) Read input instance(2) Event clasification (type characteristics)(3) for 119894 = 0 to NumSolutions do(4) Initialize to 0 and empty arrays(5) while TenuredEventsNotScheduled do(6) Assign tenuredEvents in empty slots satisfying hard constraints (HC)(7) Assign unscheduled events (uE) in temporary arrays(8) if (uE)(9) Assign uE in empty slots randomly satisfying HClowast(10) endwhile(11) while 119895 ltmaxStages or allEventsScheduled do(12) Sort nontenureEvents (capacity)(13) Assign nontenureEvents in empty slots randomly satisfying RD(14) Assign unscheduled nontenureEvents (uNTE) in temporary arrays(15) while 119895 ltmaxAttempts do(16) for 119896 = 0 to uNTE do(17) Assign uNTE in empty slots without validating RD(18) endfor(19) for 119898 = 0 to events scheduled in empty slots without validating RD do(20) while 119904 ltmaxAttemptsB do(21) Select events scheduled in empty slots without validating RD(22) Select Random temporalSolution occupied or unoccupied slots(23) Swap or Insert events in empty slots if events satisfy RD(24) endwhile(25) endfor(26) Reschedule events(27) endwhile(28) endwhile(29) if temporalSolution feasible(30) Print Feasible Solution and time in seconds(31) Save solution(32) else Print Infeasible Solution and time in seconds(33) endfor

Algorithm 1 Constructive Approach Algorithm for generating feasible solutions for the FCSE pattern

that will be taught by tenured teachers (line (6)) isperformed An attempt is made to schedule events onthe day and in the period that the tenured teachersrequested to teach the events of their correspondingsubjects A random assignment of a classroom ismade and events are scheduled on the day and atthe time requested only if the hard constraints aresatisfied Otherwise (line (9)) a maximum number ofattempts are made to schedule the events into timeslots randomly chosen to satisfy hard constraintsOnce all the events of tenured teachers have beenscheduled the result is a feasible partial schedule

(vi) Lines (11)ndash(27) The rest of the events that have beenclassified as pertaining to nontenured teachers arethen scheduled Do this for a maximum number ofattempts or until a feasible solution is found which isthe stop criteria of the algorithm To find a solutionthe activities on lines (12) to (25) are performed

(vii) Line (12) The events are ordered taking into accountthe capacity (number of students attending the event)These events are arranged in decreasing order accord-ing to their capacity This is done so that the events

that are scheduled first are those that require class-rooms with greater capacity The events scheduledlastly are those which require less capacity

(viii) Line (13) The events to be scheduled are assignedrandomly in empty time slots that satisfy the hardconstraints Through a random selection of class-rooms and time slots (day period and classroom)as seen Figure 1 an attempt is made to program asmany nontenured teacher events as possible whilesatisfying the hard constraints

(ix) Line (14) The events that cannot be assigned ina maximum number of attempts are stored in thetemporary arrangement uNTE

(x) Lines (16) and (17) The events stored in uNTEare programmed in empty slots without verifyingwhether they satisfy the hard constraints This solu-tion becomes infeasible

(xi) Lines (19)ndash(25) To attempt to recover the feasibility ofthe solution a step of relocation of events takes placeThis event relocation step makes repeated attempts toreschedule events that do not meet hard constraintsLines (22) and (23) select random time slots and then


study the feasibility of the exchange If the time slotsare occupied with events already scheduled whichmeet hard constraints the situation is analyzed todetermine whether hard constraints will be met withthe exchange If the slots selected for relocation areempty the situation is analyzed to determine whetherthe events can bemoved to the empty slots and satisfythe hard restrictions The relocation of events takesplace only with nontenured teacher events

(xii) Line (26) Each time the cycle to program uNTEends it includes a step for reprogramming eventswhich consists of exchanges of events scheduled inthe solution provided they continue to satisfy thehard constraints It ensures that the events of tenuredteachers are not rescheduled

(xiii) Lines (29) through (32) Once the program cycle forthe nontenured teacher event schedule is finished thealgorithm is printed It is printed if the solution isfeasible or infeasible along with the time it took thealgorithm to obtain the solution

To validate the satisfaction of the hard constraints RD of themodel (see Section 3) in the CAA it is necessary to observeeach student (RD = 7) and teacher (RD = 8) to ensure thatthe restrictions are satisfied This means that there shouldnot be overlap To ensure that there is no overlap or otherinvalidation of the hard constraints an iterative search pertime slot in all the classrooms is required This is the mosttime consuming part of the algorithm execution To searcha combination of two algorithms binary search and searchBTREE are applied making it a more efficient search forUCTP [24]

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] C H Papadimitriou and K Steiglitz Combinatorial Optimiza-tion Algorithms and Complexity Dover New York NY USA1998

[2] S A Cook ldquoThe complexity of theorem-proving proceduresrdquoin Proceedings of the 3rd Annual ACM Symposium on Theory ofComputing (STOC rsquo71) pp 151ndash158 Association for ComputingMachinery 1971

[3] M G Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman amp CoNew York NY USA 1990

[4] S Even A Itai and A Shamir ldquoOn the complexity of timetableand multicommodity flow problemsrdquo SIAM Journal on Com-puting vol 5 no 4 pp 691ndash703 1976

[5] O Ben-Ayed and S Hamzaoui ldquoMultiobjective multiproductparcel distribution timetabling a real-world applicationrdquo Inter-national Transactions in Operational Research vol 19 no 4 pp613ndash629 2012

[6] C C Ribeiro ldquoSports scheduling problems and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 201ndash226 2012

[7] Y Shen J Xu and Z Zeng ldquoPublic transit planning andscheduling based on AVL data in Chinardquo International Trans-actions in Operational Research 2015

[8] S Kırıs ldquoAHP and multichoice goal programming integrationfor course planningrdquo International Transactions in OperationalResearch vol 21 no 5 pp 819ndash833 2014

[9] A Saldana-Crovo C Oliva-SanMartın and L Pradenas-RojasldquoInteger programming models for scheduling problem onuniversitiesrdquo Chilen Engineering Magazine vol 15 no 3 pp245ndash259 2007 (Spanish)

[10] K Banczyk T Boinski and H Krawczyk ldquoParallelization ofgenetic algorithms for solving university timetabling problemsrdquoin Proceedings of the International Symposium on IEEE XploreConference Parallel Computing in Electrical Engineering pp325ndash330 2006

[11] O Rossi-Doria M Sampels M Birattari M ChiarandiniM Dorigo and L M Gambardella ldquoA comparison of theperformance of different metaheuristics on the timetablingproblemrdquo in Practice and Theory of Automated Timetabling IVvol 2740 pp 329ndash351 Springer Berlin Germany 2003

[12] T Thepphakorn P Pongcharoen and C Hicks ldquoModifyingregeneration mutation and hybridising clonal selection forevolutionary algorithms based timetabling toolrdquo MathematicalProblems in Engineering vol 2015 Article ID 841748 16 pages2015

[13] C Di Stefano and A G B Tettamanzi ldquoAn evolutionaryalgorithm for solving the school time-tabling problemrdquo inApplications of Evolutionary Computing vol 2037 of LectureNotes in Computer Science pp 452ndash462 Springer New YorkNY USA 2001

[14] E Yu and K-S Sung ldquoA genetic algorithm for a universityweekly courses timetabling problemrdquo International Transac-tions in Operational Research vol 9 no 6 pp 703ndash717 2002

[15] M N M Kahar and G Kendall ldquoThe examination timetablingproblem at Universiti Malaysia Pahang comparison of a con-structive heuristic with an existing software solutionrdquo EuropeanJournal of Operational Research vol 207 no 2 pp 557ndash5652010

[16] N BasirW Ismail andN Norwawi ldquoA simulated annealing forTahmidi course timetablingrdquo Procedia Technology vol 11 pp437ndash445 2013

[17] A Gunawan K Ming Ng and K Leng Poh ldquoA hybridizedLagrangian relaxation and simulated annealing method for thecourse timetabling problemrdquoComputersampOperations Researchvol 39 no 12 pp 3074ndash3088 2012

[18] S Chacha and R M Allen ldquoOptimal solution strategy foruniversity course timetabling problemrdquo International Journal ofAdvanced Research in Computer Science vol 4 no 1 p 35 2013

[19] D Zhang S Guo W Zhang and S Yan ldquoA novel greedyheuristic algorithm for university course timetabling problemrdquoin Proceedings of the 11th World Congress on Intelligent Controland Automation (WCICA rsquo14) pp 5303ndash5308 IEEE ShenyangChina July 2014

[20] S Shimazaki K Sakakibara and T Matsumoto ldquoIterativeoptimization techniques using man-machine interaction foruniversity timetabling problemsrdquo SpringerPlus vol 4 no 1article 251 2015

[21] V Pereira and H Gomes Costa ldquoLinear integer model for thecourse timetabling problem of a faculty in Rio de JaneirordquoAdvances in Operations Research vol 2016 Article ID 75970629 pages 2016


[22] M H Cruz-Rosales The scheduling-courses problem at a uni-versity and a proposed solution [Thesis] CIICAp-UAEM 2010(Spanish)

[23] BMartınez-Bahena and Y Y Calderon-SeguraDevelopment ofamanagement system for FCQeI hours using a three-layer modelclient web service and database [Engineering Thesis] UpemorJiutepec Mexico 2008 (Spanish)

[24] M A Cruz-Chavez and A Martınez-Oropeza ldquoB-Tree algo-rithm complexity analysis to evaluate the feasibility of its appli-cation in the university course timetabling problemrdquo Journal ofArtificial Intelligence and Soft Computing Research vol 3 no 4pp 251ndash263 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


scheduling problem is known in the scientific community asthe University Course Timetabling Problem (UCTP) [10]

The staff is responsible for organizing the schedules ofstudents and teachers during each academic period It isimportant to provide the best possible solution according toa number of restrictions which must be satisfied in order toensure that the educational programmeets quality measuresthis implies a great effort

The goal of the scheduling problem in education is toschedule the courses in an academic period (trimester quar-ter or semester) It is necessary to consider several variablesthat impact the problem such as teachers required for allcourses which students need to take the courses groups ofstudents that take a set of courses days and available periodsof the individual classrooms the number of classrooms avail-able the capacity of each room and other relevant variables

Resources that are considered critical or scarce are theclassrooms since they are not resources that are readily avail-able This is why the number characteristics and capacity ofclassrooms must be considered when scheduling as one ofthe features and restrictions of the educational institution

In the literature there is information on theoretical mod-els of UCTP timetables and algorithms to deal with thesemodels For example in [11] a complete study of a theoreticalmodel is presented which does not consider teachers as avariable This model is treated with several computationalheuristics such as genetic algorithms ant colony iteratedlocal search Simulated Annealing and Tabu Reference [12]proposes a new evolutionary algorithm-based timetablingtool for solving course scheduling problems To evaluatethe efficiency of this algorithm they work with fourteenbenchmarks of timetabling problems

The UCTP has also been addressed using real modelsapplied to universities For instance in [13] an evolutionaryalgorithm is applied in the Italian School System with thehybridization of twoheuristics based on local search In [14] agenetic algorithm is applied based on sectors reporting goodresults when applied to a real case Reference [15] presentsa study of exam scheduling in Pahang University of Malaysiawhich includes comparing a constructive heuristic with exist-ing software This case study considers four sets of hard con-straints and three sets of soft constraintsThe authors indicatethat the solutions obtained by the heuristics outperformthose obtained by the software that was implemented at theuniversity to manage all the restrictions in the case In [16]a Simulated Annealing metaheuristic was implemented andpreliminary work is presented that deals with the problem ofprogramming courses in Tamhidi University ofMalaysiaTheinstance they work with has 800 students grouped into fivestudy programs and students that attend the same coursesThe authors work with three sets of hard constraints andthree sets of soft constraints In [17] an initial solution isobtained by a mathematical programming approach basedon Lagrangian relaxation The proposed method has beentested on instances from a university in Indonesia Elevenrequirements of the problem are consideredThey are dividedinto two different categories four requirements represent aset of teacher assignment requirements while the rest arerelated to the course scheduling problem Computational

experiments were performed on two real problem datasetsfrom a university in Indonesia Reference [18] developedthree models of the University Course Timetabling Problemas used at Mkwawa University College of Education andtested using real data from the stated university It waspossible to obtain the optimal solution for real probleminstances through reformulations of models which involve amixture of binary and time-indexed variables In China [19]uses the greedy heuristic approach for UCTP Complex mul-ticonstraint conditions are considered based on the uniformteaching resources of the university which include studentsteachers and classrooms with feasible time slots respectivelyIn particular with respect to most of the graduate schools inChina there are more than 400 courses in every academicsemester A timetable should satisfy 9 basic requirementsThecharacteristics of datasets of two semesters from the graduateschool are considered in the implementation In [20] amath-ematical programming model with an approach called meta-model is proposed It can include implicit restrictions andimplicit objective functions for the assignment of a set of tem-porary classes at Toyama Prefectural University Each classduring a specific period of time has hard and soft constraintsInitially scheduling was done manually and took about 3weeks with a preparation time of 5 to 6 days To validate theproposed mathematical model algorithms apply branch andbound Reference [21] presents a linear integer programmingmodel that solves a timetabling problem in a Rio de JaneiroBrazil department The model was designed to generatesolutions that meet the preferences of the facultyrsquos managersMore specifically thismeant assigning themaximumnumberof lecturers with the highest academic title while minimizingcosts by merging courses with equivalent syllabuses

In the Research Center for Engineering and AppliedSciences (CIICAp) of the Autonomous University of MorelosState (UAEM) two projects have been carried out withrespect to scheduling university courses In [22] a schedulerwas programmed which provided a theoretical model ofschedule generation using Simulated Annealing by applyingbenchmarks of a Metaheuristic Network [11] (medium 01ndash05 hard 01 and hard 02) Feasible solutions were obtainedwith greater efficiency and effectiveness than with other algo-rithms reported in [11] for medium and large instances Thesecond project involved the development of a basic system ofadministrative schedules for the Faculty of Chemical Sciencesand Engineering (FCSE) at the UAEM with web interfaceusing a three-layer model client web service and databaseThrough this project a system of consultations of academicschedules for teachers and students via the internet wasdeveloped A basic constraint model was also proposed toavoid overlapping subjects for students in the FCSE butwithout taking into account restrictions on the allocation ofteachers for the subjects they teach [23]

The present work proposes a constraint satisfactionmodel for the problemof programming university courses forthe Faculty of Chemical Sciences and Engineering (FCSE) Itincreases the number of actual model restrictions for teach-ers aiming to create a complete model for the slot allocationin FCSE The administration of the Faculty of Engineeringcreates a timetable of courses manually Usually there are
































119864



119879



119877



119878



119879


119873 of





sum

119905=1

119899119877

sum

119903=1

119883 (119905 119877 (119903 119890)) = 1 forall119890 (1)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883 (119890 Φ (119905 119903) 119877 (119903 119890)) = 2 forall119890 (2)

119899119864

sum

119890=1

119883 (119890 Φ (119905 119903)) le 1 forall (119905 119903) (3)

119899119878

sum

119904=1

119885 (119904 119883 (119890 Φ (119905 119903))) le Capacity (119903)

forall (119890 119905 119903)

(4)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883(119890tt Φ (119905 119903) 119877 (119903 119890tt)) = 2 forall119890tt (5)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119904 119883 (119890 Φ (119905 119903))) le 1 forall (119904 119905) (6)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119898119883 (119890 Φ (119905 119903))) le 1 forall (119898 119905) (7)

119899119879

sum

119905=1

119899119877119897

sum

119903119897=1

119883(119890119897 Φ (119905 119903

119897) 119877119897(119903119897 119890119897)) = 2 forall119890

119897

(119905 119903 119890 119904 119898) isin 119873

(8)







Φ (119905 119903) =


0 Otherwise(9)



119877 (119903 119890) =


0 Otherwise(10)




119883(119890 Φ (119905 119903) 119877 (119903 119890))

=

1 If Φ (119905 119903) + 119877 (119903 119890) = 2

0 Otherwise

(11)





119885 (119904 119883 (119890 Φ (119905 119903)))

=

1 If 119883 (119890 Φ (119905 119903)) = 1 Then 119904 serves 119890

0 Otherwise

(12)





1015840isin 119864 with


overlapping events

119885 (119898119883 (119890 Φ (119905 119903)))

=

1 119878119894 119883 (119890 Φ (119905 119903)) = 1 Then 119898 attends 119890

0 Otherwise

(13)





2 1199057 1199058 1199059 11990510 11990511 11990512




15 11990585 11990586 11990587 11990588 11990589 11990590








119879are as-




119879 119905 =






119879in



















1 to classroom







12 120 21213 120 21214 12015 120



119864+ 119899119897) 1514









A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2


Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

























1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































119864



119879



119877



119878



119879


119873 of





sum

119905=1

119899119877

sum

119903=1

119883 (119905 119877 (119903 119890)) = 1 forall119890 (1)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883 (119890 Φ (119905 119903) 119877 (119903 119890)) = 2 forall119890 (2)

119899119864

sum

119890=1

119883 (119890 Φ (119905 119903)) le 1 forall (119905 119903) (3)

119899119878

sum

119904=1

119885 (119904 119883 (119890 Φ (119905 119903))) le Capacity (119903)

forall (119890 119905 119903)

(4)

119899119879

sum

119905=1

119899119877

sum

119903=1

119883(119890tt Φ (119905 119903) 119877 (119903 119890tt)) = 2 forall119890tt (5)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119904 119883 (119890 Φ (119905 119903))) le 1 forall (119904 119905) (6)

119899119864

sum

119890=1

119899119877

sum

119903=1

119885 (119898119883 (119890 Φ (119905 119903))) le 1 forall (119898 119905) (7)

119899119879

sum

119905=1

119899119877119897

sum

119903119897=1

119883(119890119897 Φ (119905 119903

119897) 119877119897(119903119897 119890119897)) = 2 forall119890

119897

(119905 119903 119890 119904 119898) isin 119873

(8)







Φ (119905 119903) =


0 Otherwise(9)



119877 (119903 119890) =


0 Otherwise(10)




119883(119890 Φ (119905 119903) 119877 (119903 119890))

=

1 If Φ (119905 119903) + 119877 (119903 119890) = 2

0 Otherwise

(11)





119885 (119904 119883 (119890 Φ (119905 119903)))

=

1 If 119883 (119890 Φ (119905 119903)) = 1 Then 119904 serves 119890

0 Otherwise

(12)





1015840isin 119864 with


overlapping events

119885 (119898119883 (119890 Φ (119905 119903)))

=

1 119878119894 119883 (119890 Φ (119905 119903)) = 1 Then 119898 attends 119890

0 Otherwise

(13)





2 1199057 1199058 1199059 11990510 11990511 11990512




15 11990585 11990586 11990587 11990588 11990589 11990590








119879are as-




119879 119905 =






119879in



















1 to classroom







12 120 21213 120 21214 12015 120



119864+ 119899119897) 1514









A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2


Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

























1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119883(119890 Φ (119905 119903) 119877 (119903 119890))

=

1 If Φ (119905 119903) + 119877 (119903 119890) = 2

0 Otherwise

(11)





119885 (119904 119883 (119890 Φ (119905 119903)))

=

1 If 119883 (119890 Φ (119905 119903)) = 1 Then 119904 serves 119890

0 Otherwise

(12)





1015840isin 119864 with


overlapping events

119885 (119898119883 (119890 Φ (119905 119903)))

=

1 119878119894 119883 (119890 Φ (119905 119903)) = 1 Then 119898 attends 119890

0 Otherwise

(13)





2 1199057 1199058 1199059 11990510 11990511 11990512




15 11990585 11990586 11990587 11990588 11990589 11990590








119879are as-




119879 119905 =






119879in



















1 to classroom







12 120 21213 120 21214 12015 120



119864+ 119899119897) 1514









A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2


Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

























1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















1 to classroom







12 120 21213 120 21214 12015 120



119864+ 119899119897) 1514









A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2


Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

























1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A(1)

Event

A(1)

A(1)

A(1)

A(1)

A(1)A(1)

A(1)A(2) A(2)

A(2) A(2)A(1)

A(2)A(2)A(2) A(2)

A(1)

A(1) A(2)A(2)A(2)

A(2)A(1)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2


Event

A(1) A(2)A(2)

1 2 3

2 2 22 2 12 1 11 2 21 2 11 1 21 1 1

Assignment (r d p)

2 1 2

























1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















1

2 1 12 2 14 1 14 2 10 0 20 0 20 0 20 0 2




1 01 11 01 02 12 12 02 0








Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Class ID CodingNeed

1 01 11 01 02 12 12 02 0






1 9753 11 292725232119171513

9876543210

Tim

e (ho

urs)

Execution


FeasibleInfeasible


70

60

50

40

30

20

10

0


OA 13capacity 15

Classrooms


Occ

upie

d pe

riods

1 9753 11 292725 353331 413937232119171513






0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

25

3

35

4

45

6050 5750 5850 5950 6250

Freq

uenc

y

Events


0

05

1

15

2

25

3

35

1525 1625 1725 1925 1425

Freq

uenc

y

Events





1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41


Classroom








Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Classroom1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41



6 Conclusions






Appendix
























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









solving a real constraint satisfaction model for the university

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