Combining Examinations to Accelerate Timetable Construction

Transcription

1 Combnng Examnatons to Accelerate Tmetable Constructon Graham Kendall Jawe L School of Computer Scence, Unversty of Nottngham, UK {gxk wl}@cs.nott.ac.uk Abstract: In ths paper we propose a novel approach that combnes compatble examnatons n order to accelerate both the ntal tmetable constructon, as well as a later search. The condtons for combnng exams are descrbed, and we show that we are able to offer some guarantees as to the qualty of solutons that reman n the reduced search space. The approach s appled to one of the standard benchmarks n ths area; the St. Andrews83 nstance. The results verfy the effectveness of ths approach n smplfyng examnaton tmetablng problems, speedng up ntal tmetable constructon and assstng any subsequent search. Keywords: exam tmetablng, heurstcs, optmsaton, combnng exams Introducton Exam tmetablng s a dffcult combnatoral optmsaton problem that s a key task n all educatonal nsttutons. The problem s concerned wth dstrbutng a collecton of exams among a lmted number of tmeslots so that a set of constrants are satsfed. Varous approaches have been developed wth an obectve to ether obtan feasble tmetables n short tme or to acheve optmsed tmetables wth low cost (Carter and Laporte (996), Burke and Petrovc (00), Qu et al (006)). Snce a fulltree search of an NP-complete problem s generally mpractcal, t s natural for researchers to consder methodologes such as heurstcs, meta-heurstcs and decomposton. The dea of decomposton s to dvde the orgnal problem nto smaller subproblems so that each sub-problem can be handled by usng relatvely smple approaches (e.g. Carter (983), Burke and Newall (999), Qu and Burke (007)). Cluster methods adopt smlar deas. In these approaches the exams are sorted and splt nto groups and then the groups are assgned to tme perods wthn a greatly reduced search space (e.g. Whte and Chan (979), Balakrshnan et al. (99)). The man drawback of these methods s that the qualty of the solutons that reman n the reduced search space may be poor. The successful applcaton of meta-heurstc methods, tabu search for example (Glover and Laguna (993), Nonobe and Ibarak (998), Whte et al (004), Gendreau and Potvn (005)), n exam tmetablng problems shows that good solutons for a specfc tmetable have some common features. If the problems could be smplfed accordng to these common features, good solutons would reman n the reduced search space and then heurstc methods could not only mprove the speed of convergence to a feasble soluton but also the qualty of the fnal tmetable

2 could come wth some guarantees, wth respect to the best soluton that could be reached. In ths paper, we present an approach that smplfes exam tmetablng problems by combnng multple exams, so that they can be treated as a sngle exam. The condtons that dctate how examnatons are combned are presented, under whch (near) optmal tmetables wll stll be achevable. Instead of dvdng all exams nto groups as cluster methods do, the proposed approach only combnes those exams that satsfy predefned condtons. For the problem of exam tmetablng wth m exams and n tmeslots, the sze of the m search space s n. If two of the exams are combned, the sze of the search space m becomes n. Wth ths smaller space of solutons, a prevously used heurstc may ether fnd feasble solutons n shorter tme or reach a better soluton n a gven tme. In the followng sectons, we wll frst nfer the condtons for exams to be combned, and then show the applcaton of the proposed method on the St. Andrews83 benchmark nstance. The Approach Consder a smple example of exam tmetablng. Suppose that there are 0 students (A, B,, J), 6 exams (E,, E6) and 4 tmeslots (s,, s4), as shown n Fg. The hard constrants stpulate that no student should st for two exams at the same tme. Our am s to leave as much tme as possble between each exam for every student, so that the students perceve the schedule as beng far from ther perspectve (Kendall and Mohd Hussn (005) presents a typcal mathematcal formulaton). Exams Students Tmeslots E: A, B, C, D, E, F, G, H E: A, C, E, G, I, J E3: A, B E4: C, D E5: E, F E6: G, H, s s s3 s4 Fgure A smple example of exam tmetablng Although the optmal soluton can be found for ths problem, by usng a full-tree search, there s a more effcent way. Exams E3, E4, E5, and E6 can be combned and treated as one exam, as they have no students n common. In ths case, the problem s smplfed to three exams. Thus, we are faced wth the problem of defnng under what condtons two or more exams can be combned together? Of course, combnng several exams together should not volate the hard constrants. That s, there must not be any common students n the exams we are combnng. Secondly, combnng several exams together should mnmze the soft constrants as far as possble. If the task s to fnd the optmal soluton, combnng these exams should ensure that the optmal soluton(s) reman n the reduced search space. If, on the other hand, the task s to quckly fnd feasble solutons, combnng exams should ensure that more feasble solutons reman n the reduced search space.

3 E E E3 E4 E5 E6 E E E3 E4 E5 E Fgure Clash matrx (Clashes between exams are expressed by postve numbers). The clash matrx of the example exam tmetable (Fg. ) reveals some common features these combned exams should have. The entres marked zero (n the dashed box) show that there s no clash between the two respectve exams. The entres n the dotted box show those exams that clash wth other exams n a smlar way - each of these exams has two clashes wth E and one clash wth E. As wll be dscussed later n the paper, these equvalent numbers of clashes among these exams makes sure that combnng them mnmzes the penalty of a soft constrant so that good solutons can be retaned n the search space. Therefore, the condtons for combnng two or more exams can be expressed as:. There are no clashes between them.. They are equally clashed wth other exams. The optmal soluton remans after combnng two or more exams when Condtons and (above) are satsfed. We gve ths proof below. Proof: We use reducton to absurdty. Suppose that two exams, E and E, are scheduled nto dfferent tmeslots, s and s, for the optmal soluton of an exam tmetablng problem. Let P and P denote the cost of schedulng E to s and E to s respectvely. Accordng to Condton and, we can obtan a new feasble soluton by movng E from s to s. Comparng the new soluton wth the best soluton, there should be P + () P P Agan, another feasble soluton s obtaned by movng E from s to s, and we have P + () P P It s obvous that () and () cannot hold at the same tme unless P P, whch means that the cost of schedulng E and E nto the same tmeslot s equvalent to

4 the cost of the best soluton. Therefore, schedulng E and E nto same tmeslot must keep the best soluton n the search space. The cost of the tmetable obtaned s mnmzed when combnng exams that satsfy Condtons and. Ths means that we can search for the best soluton n a reduced space. However, Condton s too strct to apply snce t may be dffcult to fnd two exams that satsfy ths condton n real world exam tmetablng problems. Ths condton needs to be relaxed and then developed nto a sutable algorthm. Consder the exam tmetablng problem wth m exams ( E,, E m ) and n tmeslots ( s,, s n ). Let c denote whether or not there s clash between exams and E, wth clashes c (3) 0 wthout clash where, {, K, m} and. Compatblty s defned to measure to what degree two exams are sutable to be combned. m s ( k) f c 0 C m (4) k 0 0 f c E where,, k {, K, m},, and 0 f ck c k,or k,or k s ( k). f ck c k Values of C are ranged n [,] 0. C denotes perfect compatblty between two exams. For example, the values of compatblty between E3, E4, E5, and E6 n example of Fg. are all, and they should be scheduled nto the same tmeslot. Small values of C denote low compatblty and are therefore unsutable for schedulng these exams nto same tmeslot. In applyng the measure of compatblty, we can predefne a value C 0 and combne those exams that satsfy C > C0. Because the optmal soluton does not necessarly reman n the reduced search space after combnng two exams wth C <, there may be a trade-off between reducng computatonal tmes and the search for optmal solutons. Therefore, the value of C 0 may have a sgnfcant effect of any later search.

5 Example of Methodology We have nvestgated applyng ths approach to the Unversty of Toronto benchmarks, and found t s especally sutable for the St. Andrews83 (sta83-i) nstance. Ths data has 39 exams, 6 students, 575 enrolments, and 3 tmeslots. The usual hard constrant s defned (.e. exams wth the same students should not be assgned to the same tmeslot). A soft constrant s to mnmze an evaluaton functon whch denotes the cost of tmetables that are generated. The average cost per student s used to compare dfferent tmetables. The evaluaton functon whch calculates the cost of a tmetable s presented n formula (5) below: 5 Cost ( w N ) S (5) s s s / where w s s the weght that represents the mportance of schedulng exams wth common students s tmeslots apart, where w 6, w 8, w 4, w, w, and w s 0 for any s > 5. N s s the number of common students nvolved n the volaton of the soft constrant. S s the number of students n the problem Table Compatblty matrx between 9 exams for sta83-i benchmark E E E3 E4 E5 E6 E7 E8 E9 E E E3 E4 E5 E6 E7 E8 E The measure of compatblty s expressed n the form of matrx so that all exams could be compared wth each other. For example, the compatblty matrx between 9 exams s shown n Table. Because of zero clashes and hgh compatblty among them, these exams are sutable to be combned and scheduled nto the same tmeslot. Wth S 0 0.9, a total of 0 groups of 84 exams was combned to form 0 new larger exams, and the number of exams decreased from 39 to 65. Ths greatly smplfed the orgnal tmetablng problem. Heurstc orderng and local search methods have been appled to both the smplfed problem and the orgnal verson. The results show speed ups n both convergence to feasble solutons and searchng for optmzed tmetables. Feasble Solutons Acheved by Heurstc Orderngs Dfferent heurstc orderng methods are wdely used n obtanng feasble solutons that can be further optmzed by other meta-heurstcs. The dea of these methods s

6 that by estmatng how dffcult each exam wll be to deal wth, we can arrange the exams n a sequence from dffcult to easy, and then the exams are scheduled accordng to ther order n the sequence. Some common used heurstc orderngs nclude: Largest Degree (LD), Largest Colour degree (LC), and Saturaton Degree (SatD). The detals of these methods can be found n (e.g. Carter and Laporte (996), Burke and Newall (004)). Besdes these, two other heurstc orderngs, Largest Enrolment (LE) and Random orderng, are also used. These heurstc orderngs were appled to both the new tmetablng problem wth a reduced number of exams and the orgnal verson. Backtrackng s employed whenever there s a conflct. Methods Table Results when usng dfferent ntal orderngs Orgnal problem Smplfed problem Cost Tme (s) Cost Tme (s) LD ntal order LC ntal order SatD ntal order LE ntal order Random ntal order Average The algorthm was coded n Vsual C++ and experments were run on a PC wth dual.66ghz Intel CPU and 3.5GB RAM. The results n table show that by reducng the number of exams t takes less tme for the heurstc orderngs to obtan better solutons. We do not attempt to compare our results n terms of the computatonal tme wth others n the lterature because comparsons across dfferent platforms are mpractcal. Optmzed Soluton Acheved by Local Search In order to check the nfluence of combnng exams on meta-heurstcs, a local search algorthm was developed to mprove feasble solutons that were obtaned by usng graph orderng. The local search we use adopts a smple strategy that removes several exams from the feasble soluton and reschedules them. If a better soluton s found, the local search wll restart based on the new tmetable. Otherwse, several other exams wll be tred. Ths process wll contnue untl no further mprovement can be made. The local search adopts a two-stage structure that reschedules two exams n the frst stage and four exams n the second stage. It permts only those new tmetables wth lower cost nto the second stage n order to decrease the computatonal tme. Methods Table 3 Results when usng local search Orgnal problem Smplfed problem Tme (s) Best soluton Tme (s) Best soluton Frst stage 57, , Second stage 85, , Total 4, ,

7 It took approxmately fve hours for the local search to reach a tmetable wth cost (see Table 4) that was very close to the best soluton so far (a tmetable wth cost acheved by Cote et al (005)). Wthout the ntal combnng of exams, the same local search dd not reach any tmetable wth a cost below 59.0, and the computatonal tme s also much longer. Ths result shows that the proposed approach can also be effectve n acceleratng meta-heurstcs. Tmeslots Table 4 A tmetable for St. Andrews 83 benchmark wth cost Exams 5,6,7,8,30,3,3,33,34,35,36,37,39,40,4,4,43,44,45,46,47,0 38,59,60,6,6,63,64,65,66,67,68,69,70,7,3 3,,8,35, ,5,5,53,54,55,56,57,7,95,96,00 5 4,85, ,99,0,0,3, ,77,78,79,80,8,8,83,84,30,33 8 8,9,0,,,3,4,5,6,9,49, , ,86,87,88,89,90,9,9,93,94,03,04,,3,4,5,6,7,8,9 7,05,06 3,7,09,0,,,,3,4,5,6,7,8,9 3 9,0,,,3,4,5,6,73,74,75,76,07,08 Conclusons and Future Work The success of meta-heurstc approaches has provded evdence of common features among those hgh qualty solutons for a specfc exam tmetablng problem. The problem wll be greatly smplfed f these common features can be realsed so as to lmt the search space. The study n ths paper demonstrates that smplfyng the exam tmetablng problem s possble. A crteron of compatblty s defned to measure how well several exams can be combned. By combnng exams wth relatvely hgh compatblty, the qualty of solutons remanng n the reduced search space could be guaranteed. The man drawback of the measure of compatblty s that t may not be sutable for every exam tmetablng problem. There s the possblty that all compatblty measures are hgh (or low) because of very low (or hgh) densty of clashes between exams. It would therefore be dffcult to choose whch exams to combne. However, compatblty mght not be the only measure we can use to express the common features of hgh qualty solutons. Our future work wll focus on searchng for general and practcable condtons that can be used to provde a general approach to smplfyng exam tmetablng problems. Reducng the number of exams n an exam problem can also help later heurstc and meta-heurstc methods and, potentally, ths approach can also be appled n other schedulng problems (personnel tmetablng, space allocaton for example). References Balakrshnan N, Lucena A and Wong R (99) Schedulng Examnatons to Reduce Second-Order Conflcts, Computers & Operatons Research, 9:

8 Burke E and Newall J (004) Solvng Examnaton Tmetablng Problems through Adaptaton of Heurstc Orderngs Annals of Operatons Research, 9: Burke E and Newall J (999) A Mult-Stage Evolutonary Algorthm for the Tmetable Problem, IEEE Transactons on Evolutonary Computaton, 3() Burke E and Petrovc S (00) Recent Research drectons n automated tmetablng, European Journal of Operatonal Research, 40(): Carter M (983) A decomposton algorthm for practcal tmetablng problems. Techncal Paper 83-06, Department of Industral Engneerng, Unversty of Toronto. Carter M and Laporte G (996) Recent developments n practcal examnaton tmetablng. In: Burke E and Ross P (eds.) The practce and theory of automated tmetablng: Selected papers from st Internatonal Conference on the Practce and Theory of Automated Tmetablng, vol.53, pp 3- Cote P, Wong T and Sabour R. (005) Applcaton of a hybrd mult-obectve evolutonary algorthm to the uncapactated exam proxmty problem, In: E.K. Burke and M. Trck (eds). (005) Selected Papers from the 5th Internatonal Conference on the Practce and Theory of Automated Tmetablng. Sprnger Lecture Notes n Computer Scence, vol Gendreau M and Potvn J (005) Tabu Search. In: E.K. Burke and G. Kendall (eds.) (005). Introductory Tutorals n Optmsaton, Decson Support and Search Methodology. Sprnger. Chapter 6, Glover F and Laguna M (993) Tabu search. In: C.R. Reeves (ed.). Modern Heurstc Technques for Combnatoral Problems. Oxford: Scentfc Publcatons. Kendall G and Mohd Hussn N (005) A Tabu Search Hyper-heurstc Approach to the Examnaton Tmetablng Problem at the MARA Unversty of Technology, Lecture Notes n Computer Scence 366, pp Nonobe K and Ibarak T (998) A Tabu search approach to the constrant satsfacton problem as a general problem solver, European Journal of Operatonal Research, 06: Qu R and Burke E (007) Adaptve Decomposton and Constructon for Examnaton Tmetablng Problems. In Multdscplnary Internatonal Schedulng: Theory and Applcatons (MISTA 007), (Baptste P., Kendall G., Muner-Kordon A., and Sourd F., Eds.), Qu R, Burke E, McCollum B, Merlot L and Lee S (006) A Survey of Search Methodologes and Automated Approaches for Examnaton Tmetablng, Computer Scence Techncal Report No. NOTTCS-TR Whte G and Chan P (979) Towards the Constructon of Optmal Examnaton Tmetables, INFOR, Whte G, Xe B, and Zonc S (004) Usng tabu search wth longerterm memory and relaxaton to create examnaton tmetables. European Journal of Operatonal Research, 53(6): 80-9.