imeabling Communiies Demands for an Effecive Examinaion imeabling Using Ineger Linear Programming N. F. Jamaluddin, N. A. H. Aizam Absrac his paper explains he educaional imeabling problem, a ype of scheduling problem ha is considered as one of he mos challenging problem in opimizaion and operaional research. he universiy examinaion imeabling problem (UEP), which involves assigning a se number of exams ino a se number of imeslos whils fulfilling all required condiions, has been widely invesigaed. he limiaion of available imeslos and resources wih he increasing number of examinaions are he main reasons in he difficuly of solving his problem. Dynamical change in he examinaion scheduling sysem adds up he complicaion paricularly in coping up wih he demand and new requiremens by he communiies. Our objecive is o invesigae hese demands and requiremens wih subjecs aken from Universii Malaysia erengganu (UM), hrough quesionnaires. Ineger linear programming model which reflecs he preferences obained o produce an effecive examinaion imeabling was formed. Keywords Demands, educaional imeabling, ineger linear programming, scheduling, universiy examinaion imeabling problem. I. INRODUCION IMILAR o he oher scheduling problem faced by many Sinsiuions, UEP is one of he ough assignmen problems ha resolve around educaional secor. I differs from Universiy Course imeabling Problem (UCP) [2] as UEP is scheduled o ake place a final of each semeser and all exam subjecs are scheduled simulaneously in he limied period [9]. Addiionally, each exam is only allowed o be scheduled in a limied number of available rooms prepared by he insiuions. his resricion added on he complexiy of finding he bes feasible soluion o he problem. Wih many condiions and resricions applied during he scheduling process, solving i manually is almos impossible since i ofen ime consuming and difficul edious. I is already difficul o find he feasible soluion considering he necessary consrain and condiion se by insiuions, if he oher consrains associaed wih he desire of he communiy were o be included, he feasible soluion will be cerainly hard o find [17]. Due o his complexiy, numerous researchers are sill ye N. F. Jamaluddin is wih he School of Informaics and Applied Mahemaics, Universii Malaysia erengganu, 21030 Kuala erengganu, erengganu, Malaysia (phone: +60134088628; e-mail: nfarahin01@ymail.com). N. A. H. Aizam is wih he Marine Managemen Sciences Research Group, School of Informaics and Applied Mahemaics, Universii Malaysia erengganu, 21030 Kuala erengganu, erengganu, Malaysia (corresponding auhor; phone: +6096683979/+60146272356; e-mail: aidya@ um.edu.my). aemping o analyze and develop he bes soluion o solve he problem. In searching for he soluion o his problem, scheduler ends o ignore he mos imporan feaures ha cerify he qualiy of he schedule. Sof consrains, especially he consrains relaed o he human needs such as preferring exams in he hall over smaller classroom and requiring aking only one exam in a day are he guidelines for he consrucion of a high qualiy examinaion imeable. hese ypes of preferences are ofen seen as insignifican requiremens due o difficuly in solving i wihou geing infeasible soluion as he resul. hus, i is undersandable why he scheduler ignored hese consrains. However, rying o solve as many sof consrains as possible is much beer opion in order o produce he bes qualiy soluion for his problem [1]. his paper presens a real world examinaion imeabling problem faced by UM communiies. In he nex secion, we describe he examinaion imeabling problem and he lieraure behind he problem. In Secion III, survey mehod and modeling of general ineger programming model ha is used in his paper are presened. Secion IV describes he resul of he survey and newly designed model. In he las secion, we summarize he conribuion and conclude our research. II. LIERAURE REVIEW A. Examinaion imeabling Problem Generally, in educaional imeabling problem, he problem is classified ino hree main classes; school, course and examinaion imeabling. School imeabling problem can be similar o he course imeabling problem as i is assigned weekly wih purpose of prevening overlapping of eacher or lecurer from having confliced schedule [20]. In his paper, he only focus is on examinaion imeabling problem. Examinaion imeabling problem is defined as follows: he assigning of examinaions o a limied number of available imes periods in such a way ha here are no conflics or clashes" [12]. Solving imeabling problem is even more complicaed in any insiuion of higher educaion due o he number of sudens, courses, and subjecs a he insiuion especially when planning he examinaion imeable for all sudens a every end of he semeser. Being differen from course imeable assignmen, he exams imeable is assigned wih he purpose o avoid he suden from having exam a consecuive 263
period or having o ake exams wice in a day [15]. Mos insiuions face he similar problem when assigning imeable for heir suden s examinaion. Some consrains mus be solved wihou any violaion and some consrains need o be solved as long as hey give feasible soluion o he problem. hese consrains are known as hard and sof consrain. Hard consrain canno be violaed and he imeable is considered feasible when all hard consrains are compleely solved. An example of hard consrain is ha all exams mus be scheduled o a imeslo (compleeness). While sof consrain is a consrain ha is no necessary o be solved, if scheduler wishes o consruc he bes qualiy imeable, i is beer o solve he maximum number of sof consrain. One can say ha he sof consrain is he medium o evaluae he qualiy of he soluion for he imeable [17]. Sof consrains are he preference of communiies such as desired gap or res day beween each exam. More examples on hard and sof consrain can be found in [16], [3], [8]. B. Relaed Work 1. Survey Among he researches done in he lieraure, only few have came ou wih a survey research ha discusses on general consrains ha can be used o produce a soluion or model which can be applied o solve his problem as a whole. In he oher words, no survey has been conduced o idenify he consrains ha can be used o build one model which is applicable and compaible o all educaional insiuions. his is due o he difference in preference and oher facors such as religions which have prevened researcher o find one fix soluion o he problem. hus, researcher has conduced a research o solve his difference by surveying he relaed communiies o know he differen consrain ha needed o be saisfied as o find he feasible soluion a he sudied insiuion. Reference [3] has conduced a survey quesionnaire on he preference of Universiy Malaysia erengganu communiies on heir examinaion imeable. he survey inquires he communiies abou heir preferred requiremens and opinions on he crieria of a good examinaion imeable. he survey was disribued o he mahemaics sudens and lecures from School of Informaics and Applied Mahemaics (SIAM) in UM. As a resul, we managed o idenify eigh facors ha affec he scheduling of examinaion imeable based on he communiies preference. We have also lised down he crieria of a good imeable based on he communiies preference which should be included in developing he bes model o solve he examinaion imeabling problem in UM. Oher researchers ha have conduced a survey on examinaion imeabling problem can be found in [11], [13], [14]. 2. Modelling Many mahemaical programming and heurisic-based approaches have been proposed for solving a variey of imeabling problem. he same goes o he examinaion imeabling problem. However, here is sill ye a model or soluion ha can solve his problem generally due o he individual specificaion and requiremens by he insiuions. he need or consrain varies from one insiuion o anoher as differen insiuion has differen requiremen and number of enrolmen as well as differen ype of course offered. herefore, he bes way o consruc examinaion imeable for an insiuion is by considering he needs and requiremen of he individual insiues. For more informaion, [19] discuss on he varian of he problem. Ineger Linear Programming (LP) mehod is a widely used mahemaical programming by he previous researchers in solving he imeabling problem. Reference [6] did a research using a mixed ineger programming approach o a class imeabling problem, reference [10] has done a research on Sudoku problem using IP model, and reference [21] uses a formulaion of binary ineger programming for scheduling in marke-driven foundries. In UEP problem, researchers used his mehod o generae a conflic-free exams imeable and in a decision making problem o produce he bes qualiy imeable. Reference [18] repors he research on he maximizaion of paper spread in predefined examinaion imeable wih he purpose o increase he amoun of sudy ime beween exams for each suden using he ineger programming model. Consrain used in his model is he consrain and requiremen ha can be found in mos academic insiuions. As example he conflic ype consrain such as wo exams in he same imeslo, wo exams in he same day and wo exams in wo consecuive imeslos. he approach is hen applied on he real world examinaion imeabling problem. Oher example can be found in [7]. here are papers which focus on he general model ha can be applied o solve he general imeabling problem in various areas. Reference [8] wroe an aricle abou he general model for imeabling problem using ineger programming approach. I is well known ha here is sill ye a model ha can be used o solve imeabling problem in general. herefore, hey analyzed he basic model ha is generally used in various fields ha are relaed o each oher as he base o design he model. Using he basic model, hey develop a general ineger programming model which is applicable in he field ha hey have sudied and used a se of random daa o es he model. he model developed is modifiable o all ypes of scheduling problems which have he same basic consrains. Anoher example of research on he same opic is [4]. III. MEHODOLOGY A. Survey Mehod he presen research invesigaes he imeabling communiies in UM. Sudens from eigh schools in UM are chosen randomly from various program and year of sudy. As he major user of he imeable, hey were given he chance o express heir opinions regarding o he scheduling of examinaions a UM. We will find ou heir views and opinions regarding o he curren examinaion imeable a UM using a survey quesionnaire which is disribued by 264
hand o all respondens. Following his sep was o creae a imeable according o heir sandards and o fulfill he requiremens of he universiy a he same ime. We used he resuls obained from he survey as our guidelines o undersand heir preferences, and hus we made a model ha fis he preferences. Quesions in he survey were design by referring o he previous references mainly [11] and [1], also from inerviews and own observaions. Consrains lised by hose references and informaion are reconsruced in form of quesion. Since his survey is conduced prior o developmen of a new model, only consrains ha are compaible o he naure of exams in UM chosen as quesion in he survey. For example, UM has a major number of Muslim sudens enrolled for almos all exam subjecs, hus hey need o include a resricion of imeslo for hose sudens in order o refrain from having o ake exam during Jumaa prayers ime which is from 12.00 p. m. o 2.00 p.m. in every Friday. his is because his ime can be said as forbidden ime for any unrelaed aciviy for all Muslims and his mus be respeced. hus, o avoid he conflic, a new consrain is added for UM schedule so ha no exam is scheduled during he prayers ime. B. Modelling Mehod We wan o develop a new mahemaical model for UM communiies so ha we can build a new examinaion imeable ha is closer o he communiies' preferences. hus, we will use Binary Ineger Programming Mehod (BIP) o develop he model and solve he problem by using he laes developed sofware in opimizaion and operaional research, AIMMS sofware. Binary Ineger Programming (BIP) or 0-1 ineger programming mehod is commonly used in modelling of opimizaion and operaional research problem, especially in solving imeabling problem such as [4] and [5]. We formulae a basic model for he problem wih he consrains from he survey analysis resul, and he consrains will be convered ino he form of equaion, relaionship, and mahemaical formula. he following expressions are he sandard formulaions of IP problem: Maximize Subjec o: (i = 1, 2,, m) 0 (j = 1, 2,, n) ineger (for some or all j = 1, 2,, n) where: = cos coefficien, = variable, = echnological coefficien, = consrains limi, m = number of consrains, n = number of variables. From he problem above, all variables are resriced o 0 and 1 as we have decided o use binary ineger programming o solve our problem. he difference beween BIP and oher IP mehod is ha if some of he variables are resriced o be ineger values and some are real values, i can be a mixed ineger programming or if all variables are ineger, i is known as pure ineger. IV. RESUL A. Survey Resul he resul in his secion shows he frequency resul of responden preference scale for he quesion. 36 quesions were asked o he responden, and he quesion is designed based on he consrains found in previous lieraure and hey were lised based on he observaion on he naure of examinaion in UM and from inerview wih he imeabling communiies. he resul is described on he oal number of responden by choosing he scale ha hey preferred. Based on he saisical resuls in able I, we have included he consrains ha are basic o he consrucion of UM examinaion imeable. For he modelling secion of his paper, we will only use he basic consrain o build a new mahemaical model by using he presen mehod. B. Basic Consrain in UM Sysems From our inerview wih he scheduler in UM, we have idenified five consrains used for designing he examinaion imeable in UM. he five consrains are used as he basic consrain ha is assigned firs before he oher changes due o he requess made by he communiies. he following iems are he basic consrains used by UM s scheduler o design he examinaion imeable in UM: 1) All exams mus be scheduled and are scheduled once in he imeable (Compleeness) 2) No sudens are assigning o wo exams a he same ime. (Conflic) 3) Exam wih he mos sudens mus be scheduled a he earlies imeslos. (his consrain is purposed o give lecurers ha each subjec wih a large number of sudens a longer ime o mark he paper so ha hey can finish grading before he given due dae.) 4) Exam A mus be scheduled before Exam B. (Precedence), (his consrain allows exams such as srucural exam or essay o be scheduled before objecive exams because srucural exam needs longer ime for marking and grading han objecive exams) 5) No sudens are scheduled ino wo consecuive exams wheher in imeslos or days. (Consecuiveness) Based on he above consrains, we hen formulaed a mahemaical model ha represens each requiremen. he problem of UEP in UM consiss of a se of: 1. Noaion a. Ses: E, Se of exams, e;, Se of available imeslos, ; S, Se of sudens, s; E s, Exams aken by he same sudens; E Large, Exams wih large number of sudens; Early, imeslos a he earlies ime of exams; 265
ABLE I HE PREFERENCE OF HE COMMUNIIES No. Consrain Disagree Neural Agree 1. All courses mus be scheduled ino examinaion imeable. (Compleeness) 2. Suden canno be scheduled ino wo exams a he same ime. (Conflic) 3. Same exams mus be scheduled a he same imeslo. 4. Exam wih large number of sudens mus be scheduled early in he imeable. 5. Exam mus be spread evenly hroughou he imeable. 6. Special reamen should be provided for handicapped sudens. 7. he number of invigilaor mus sui he number of suden. 8. Lecurers in charge of he subjec are scheduled o be in charged in invigilaing 92 70 208 24.9% 18.9% 56.2% 14 15 341 3.8% 4.1% 92.2% 124 66 180 33.5% 17.8% 48.6% 33 59 278 8.9% 15.9% 75.1% 6 19 345 1.6% 5.1% 93.2% 13 22 335 3.5% 5.9% 90.5% 34 47 289 9.2% 12.7% 78.1% 18 27 325 4.9% 7.3% 87.8% he exams. 9. Sudens o have only one exam in a day. 6 16 348 1.6% 4.3% 94.1% 10. No wo exams consecuively in a day. 45 46 279 12.2% 12.4% 75.4% 11. No wo exams in he same day. 90 64 216 24.3% 17.3% 58.4% 12. No core subjec exam is scheduled wice in a day. 13. No sudens are scheduled o exams in wo days consecuively. 14. Suden can be schedule o finish all exams lae 15. Sudens have one-day gap (o res) before he nex exams. 16. Maximum wo-day gap for sudens before he nex exams. 98 63 209 26.5% 17% 56.5% 56 77 237 15.1% 20.8% 64.1% 74 87 209 20% 23.5% 56.5% 26 35 309 7% 9.5% 83.5% 27 53 290 7.3% 14.3% 78.4% 17. hree-day gap beween exams is provided. 76 72 222 20.5% 19.5% 60% 18. Exam canno be scheduled a 8.00 A.M. 111 79 180 30% 21.4% 48.6% 19. Exam can be scheduled a 9.00 A.M. 50 65 255 13.5% 17.6% 68.9% 20. No exams are scheduled during lunch hour (1-2 pm). 21. No suden can be scheduled for a nigh exam from 8 pm 10 pm. 22. All sudens mus be scheduled in roaion beween morning, afernoon and evening exams. 23. No exams can be scheduled during weekend. 24. During weekend, exam canno be scheduled in Saurday only. 25. Exams during religious aciviy such as Friday prayer ime should be avoided. 26. Sudens canno be assigned o exams for any subjec during holiday. 27. Sudens canno be assigned o exams for any subjec during holiday. 28. Exams are scheduled in a room ha can only fi one course. 49 45 276 13.2% 12.2% 74.6% 46 35 289 12.4% 9.5% 78.1% 138 94 138 37.3% 25.4% 37.3% 133 56 181 35.9% 15.1% 48.9% 142 66 162 38.4% 17.8% 43.8% 34 22 314 9.2% 5.9% 84.9% 39 28 303 10.5% 7.9% 81.9% 76 57 237 20.5% 15.4% 64.1% 78 95 197 21.1% 25.7% 53.2% 29. Same exams aken by differen program 100 83 187 No. Consrain Disagree Neural Agree mus be scheduled in he same room. 27% 22.4% 50.5% 30. Suden can be schedule in he same room wih oher courses for he same exams if he room is large (DSM). 31. Same exams aken by differen program can be scheduled in differen room only if here are no enough seas. 32. Suden should no be scheduled o a room wih no enough seas. 33. Only exams wih same lengh period can be schedule in he same room. 34. Only exam of same period is scheduled in he same room. 35. Some exams may require specific room o be scheduled. 36. Larger hall (DSM) mus be assign for exams firs before considering a smaller room (BK). 35 49 286 9.5% 13.2% 77.3% 55 76 239 14.9% 20.5% 64.6% 41 86 243 11.1% 23.2% 65.7% 36 54 280 9.7% 14.6% 75.7% 36 57 277 9.7% 15.4% 74.9% 39 75 256 10.5% 20.3% 69.2% 32 8.6% 50 13.5% 288 77.8% E Pre, Se of exams wih differen feaures (e a, e b ) ha need o be scheduled firs before anoher. b. Parameers:,, he preference of he communiies. c. Decision Variable:, 1, 1,,,, 0, 2. Objecive Funcion he objecive funcion is o maximize he preference,, assigned o he examinaion such ha, is he preference of communiies as o have exams as hey desired. Maximize Subjec o: E s e, E Z P e e X e, X 1 e, 1 e. (1),, (2) X, (3) E L arg e Early e X 1, (4) e, ea, 1 eb, X X 0 (5) and 1,2,..,1 ( X X 1) 1, (6) e, e, i j X 0,1 (7) e, 266
Objecive funcion (1) is o maximize he preference of communiies for examinaion. Consrain (2) is o make sure ha all exams are compleely assigned o a imeslo. Nex, consrain (3) is he conflic consrain ha ensures ha no sudens were o ake more han one exam a he same ime. Consrain (4) is assigning he exam ha has large number of enrolmen a he earlies imeslo of exams o allow longer marking ime. For consrain (5), he preceden consrain, allows an exam o be scheduled before one anoher. Las bu no leas, consrain (6) is consecuiveness consrain which ensures ha no sudens are assigned in wo or more consecuive exams per day. Consrain (7) is he consrain ha allows he decision variable, o be binary (eiher 0 or 1). C. Discussion Based on our analysis on he survey, UM communiies have so many preferences ha he UM scheduler could consider during he scheduling process. his can be seen as almos every responden agreed o he consrain asked in he survey. Especially, he consrain abou sudens having only one exam per day and fair spreading of exams, i receives more han 90% agree rae by he respondens. his consrain may only be a sof consrain, bu i can also bring a major difference in examinaion imeable arrangemen as having only one exam in a day and equally disribued exams may help suden o become more focused and o have beer concenraion. Alhough he scheduler is allowed o only include he necessary consrain during he process, he consideraion of including more preferences can be appreciaed by he communiies. From he modeling resul, we managed o idenify he five basic consrains ha mus be included in he imeable from he discussion wih he scheduler in UM. As menioned, he five consrains are only he basic consrains ha are used during he scheduling process before considering any oher resricion. In he survey, we have idenified oher consrains or preferences of he communiies ha need o be included during he modeling process if we aim o produce a high qualiy imeable. V. CONCLUSION In his paper, we have invesigaed and discussed on he problem regarding o he examinaion imeabling as a whole and also focusing on he mahemaical modeling developmen in UM. Based on our observaion from he resuls, he imeabling communiies in UM, mosly he sudens have heir own opinion on he preferences and wha is bes for heir examinaion schedule. he universiy iself has some condiions ha need o be saisfied before considering he oher opions. hus, we have managed o develop he new basic model for he universiy as o fulfill hese condiions. he model in his paper will be he basic guidelines for our fuure work. In he fuure, we will invesigae and discuss inensely all oucomes from our survey and develop a new model ha will include mos communiies preference in our research. he consrain ha is no ye modeled will be considered in he fuure. ACKNOWLEDGMEN he basic examinaion consrains and relaed informaion used for his research are provided by he Academic Managemen Deparmen in UM. We wish o hank Mrs. Adibah Amin and Mrs. Napisah Abd Rahman from he Academic Managemen Deparmen in UM for fruiful discussion and conribuion in his research. he research has been suppored by Minisry of Higher Educaion (MOHE) under Research Acculuraion Gran Scheme (RAGS), Gran No. 57107. REFERENCES [1] Abdullah, S., Shaker, K., & Shaker, H., Invesigaing a round robin sraegy over muli algorihms in opimising he qualiy of universiy course imeable, Inernaional Journal of he Physical Science, 6, 2011, pp. 1452-1462. 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