An application of the Threshold Accepting metaheuristic for curriculum based course timetabling

Noname manuscr ipt No. (will b e inserted b y the editor) An application of the Threshold Accepting metaheu ristic for cu rriculum base d course timetablin g Martin Josef Geiger Receiv ed: dat e / Accepted: date Abstract The article p resen ts a local search app roac h for the solution of timetabling problems in general, wi th a particular implementatio n for competition track 3 of the In- ternational Time tabling Competition 2007 (ITC 200 7). The h euristic sear ch p rocedure is based on Threshol d Accepting to o verco me local op t ima. A stochastic neigh borho o d is prop osed and implemented, randomly remo ving and reassigning even ts fro m the current solution. The ov erall concept has b een incrementally obtained from a series of exp eriments, whic h we describ e in eac h (sub)section of the pap er. In result, we successfully derived a p otential candid ate solution approac h for the finals of track 3 of the ITC 2007. Keywords Threshold Accepting · Curriculum Based Course Timetabling · In terna- tional Timetabling Comp etition ITC 2007 1 Introduction Timetabling describ es a v ariet y of notoriously difficult optimization problem with con- siderable practical imp act. I m p ortant are as within t his con text include employ ee time- tabling, sp ort timetabling, flight scheduling, and timetabling in universities and other institutions of (often higher) ed ucation [2]. T ypically , timetabling is concerned with the assignment of activities to resources. In more detail, these resources pro vide timeslots ( t ime interv als) to whic h the a ctivities ma y b e assigned sub ject to certain side constrain ts. The ov erall ob jective of the p roblem is to find a feasible assignment o f a ll events suc h t hat some desirable p roperties are present in t he fin al solution. Timetabling problems are c h allenging not only in terms o f t h eir complexity , bu t also as they often invo lve multiple conflicting ob jectives [8] and even multiple stake- holder with conflicting in terests a nd views. Universi ty timetabling problems present M. J. Geiger Unive rsity of Hohe nheim 70593 Stuttga rt, Germany E-mail: mjgeiger@uni-hohenheim.de 2 an interesting example of this problem domain. Here, compromise solutions must b e found th at eq ually meet the exp ectations of students and teachers. Numerous p ublications are devoted to problem domain of timetabling, with impor- tant wo rk b y the EURO W orking Group on Automated Ti metabling W A TT. Mem bers of th e group mai ntain a biblio graphy and collect other timetabling-related resources under http://www .asap.cs.no tt.ac.uk/watt/ . More recen tly , timetabling comp etitions stimulated th e s cientific d evelopment of the field, encouraging researc hers to propose solution approac hes for newly released b enchmark instances. By creating a competitive atmosphere for al gorithmic develo p- ment, similar to t he famous DI MAC S implementation challe nges, fresh ideas hav e b een developed. In 200 7, another timetabling competition started, and this article describes a contribution and the obtained results for it. The article is organized as follo ws. In the follo wi ng Section 2, the problem under inv estiga tion is b riefly d escribed. An approach for the construction of initial feasible so- lutions is p resented in Section 3, and ex p erimen tal results of t his constructive app roac h obtained on b enchmark instances are presented. The initially constructed solutions are then improv ed using the iterativ e lo cal searc h heu ristic giv en in Section 4. Exp erimental results of the iterative phase are rep orted. Conclusions follow in Section 5. 2 T he curriculum based timetabling problem The curriculum based timetabling problem [3] is a particular v arian t of an educational timetabling problem, describ ed in track 3 of the International Timetabling Comp etition ( http://www .cs.qub.ac. uk/itc2007/ ). It reflects the situation of many u niversi ties, where curricula describ e sets of courses such that any pair of courses of a curriculum ha ve students in common. Con trary to p ost-enrollment based timetabling problems, where students register for courses they wish to attend, s ome prior kno wledge ab out th e courses atten ded by groups of students is required here. H o w ever, as universit y faculties define the required courses that stud ents ha ve to attend , t h is information is usually k no wn. A technical description of th e problem is given in [3]. Besides some u su al hard constrain ts, four ‘soft constraints’ are relev ant that measure desirable properties of the sol utions, and it becomes clear that these desirable prop erties of timetables are b eneficial for both the stu dents as w ell as the lecturers: 1. A ro om capacit y soft constraint t ries to ensure that the num ber of students atten d- ing a lecture do es not exceed the ro om capacity . 2. Lectures must be spread into a minimum n umber of days, penalizing t imetables in whic h lectures app ear in too few distinct days. 3. The curricula should b e compact, meaning that isolated lectures, that is lectures without another adjacent lecture, should b e av oided. 4. All lectures of a course should b e h eld in exactly one ro om. The ov erall eva luation of the timetables is then based on a weig hted sum approac h, com bining all four criteria in a single ev aluatio n fun ction. While we adopt this approach in th e curren t article, is should b e mentioned that P areto-based approac h es m ay b e used as an alternativ e wa y to handle the multi-criteria nature of the problem. 3 3 C onstruction of feasible initial solutions 3.1 Preprocessing Prior to the computation of a first so lution, some prepro cessing is carried out. This prepro cessing is relev ant b oth for the construction of an initial solution, as well as for the follo wing improv emen t phase. In brief, some problem-specific characteristics are emplo yed, adding some additional structure to the problem. F or each given lecture L i , events E i 1 , . . . , E ie are created which are later as signed to t imeslots. The num b er of events e is giv en in the p rob lem instances. Cre ating ev ents for each lecture lea ds to a more ge neral p roblem description, and the so lution approac h only needs to concen trate on th e assignment of all even ts, one to a single timeslot, as opp osed t o keeping track of assigning a lecture to e timeslots. Second, w e catego rize for each le cture L i (and thus f or eac h ev ent b elonging to lecture L i ) the av ail able ro oms in t h ree disjunct classes R i 1 , R i 2 , R i 3 . R i 1 refers to the ro oms in whic h the lecture fits b est, that is the ro oms R k with the minim um p ositive or zero v alue of c k − s i , c k b eing t h e room capacity , s i the num b er of stud ents of lecture L i . The class R i 2 stores the ro oms in which lecture L i fits, that is s i < c k , but not b est, and R i 3 conta ins the rooms in which lecture L i does n ot fit. With resp ect to the giv en problem statement, ev en ts of lectures ma y b e assigned t o ti meslots of ro oms in R i 3 , this how ever results in a p enalty . The und erlying assumption of the classification of the rooms is th at even ts are preferably assigned to t imeslots b elonging t o a room of class R i 1 , follo w ed by R i 2 and R i 3 . I t has to be men tioned h o w ever, that this cannot be understo od as a bind ing, general ru le but rather should b e seen as a recommendation. A randomized pro cedure is therefore going to b e imp lemented when assigning even ts to timeslots (see the follo wing section), allo wing a certain deviation from the computed ro om order. 3.2 A my opic construction approach The m etho d The constructiv e phase tries to obtain a first feasible assignmen t of al l even ts to times- lots. A simple heuristic approac h is used, successiv ely assigning all even ts to timeslots, one at a time, with the giv en pseudo-co de of Algorithm 1. In this d escription, w e d e- note the set of all ev ents with E , and the set of unassigned (op en) even ts wi th E o . During the successive assignmen t pro cedure, a set of even ts that hav e been imp ossible to assign is maintained, d enoted with E u . I n cases of assigning all even ts to timeslots, E u = ∅ is returned. A greedy approac h is u sed in the assignmen t pro cedure, sel ecting in each step th e ‘most critica l’ eve nt E from E o , t h at i s the even t with the smallest num b er of timesl ots to whic h it may b e assigned. The choic e of timeslots f or the even ts reflects the initial categorization of rooms. With a probabilit y of 0.5, t imeslots of ro oms in R i 1 are preferred o v er R i 2 o ver R i 3 , and with a probability of 0.5, timeslots of R i 2 are p referred ov er t he ones of R i 1 o ver R i 3 . Within each class , t imeslots are randomly chosen with equal probabilit y . In cases where a most-preferred class of timeslots is empty , the c hoice is made from the lesser preferred class and so on. 4 Algorithm 1 Myopic construction 1: Set E o = E 2: E u ← ∅ 3: whil e E o 6 = ∅ do 4: Select the most critical ev en t E from E o , that is the ev en t with the smallest n um ber of a v ail able timeslots 5: if E can b e assigned to at least one timeslot then 6: Select some a v ailable timeslot T f or E 7: Assign E to the timeslot T 8: else 9: E u ← E u ∪ E 10: end if 11: E o ← E o \ E 12: end w hile As mentioned abov e, timesl ots of ro oms of class R i 1 are preferable to the ones of class R i 2 and R i 3 . The randomized assignment pro cedure generally considers this aspect, how ever allowing a certain deviation from the rule. This is done as we hav e b een able to observ e t hat the assignment of events to timeslots follo wing only a single order does not lead to satisfactory results. In this case, the choice of timeslots simply is to o restrictiv e. It h as b een p ointed out in this context that the probabilit y of assigning events to timeslots of R i 1 → R i 2 → R i 3 could b e ex p ected to be greater than the one of the order R i 2 → R i 1 → R i 3 . Wh ile we generally agree with this commen t, other probabilities than 0. 5 for b oth orders ha v e not b een in vesti gated y et. Consequently , subsequent exp eriments cer tainly will hav e to examine the influence of this control parameter on the obtained results. Exp erimental r esults The constru ctive approac h has b een tested on the fi rst seven b enchmark instances of ITC 200 7 trac k 3. These are the instances t h at in itially ha v e b een made a v ailable b y the organizers of the comp etition. In F ebruary 2007, only a few wee ks b efore the sub- mission deadline, se ven mor e instances f ollo w ed ( comp08.ctt – co mp14.ctt ). Obviously , exp erimental investig ations had to start considerable earlier, and we therefore had to conclude on th e effectiveness of th e approach based on th ese early seven instances. After 1000 repetitions on eac h b enchmark instance, w e computed the n um b er of trials in whic h a ll even ts hav e su ccessfully b een assigned to timeslots, given in T able 1. T able 1 Number of tri als in which all ev en ts ha v e successfully b een assigned (out of 1,000 trials) Instance Cases with E u = ∅ comp01.ctt 1,000 comp02.ctt 354 comp03.ctt 377 comp04.ctt 1,000 comp05.ctt 0 comp06.ctt 953 comp07.ctt 827 5 The results reveal significant differences betw een the instances. While w e ha v e b een able t o alw a ys assign all even ts to t imeslots for instance comp01.ctt an d comp04.ctt , comp05.ctt t urns out to b e particularly d ifficult (constrained). After not having b een able to identify a single constructive ru n in whic h all even ts hav e been assigned to timeslots, we conclude that simply relying on more rep etitions is most probably insuf- ficient for this instance. W e rather need to adapt the constructive metho dology to t he particular instance, o v ercoming problems with the assignment of events to timeslots. 3.3 Reactiv e rep etitive reconstruction The m etho d Based on the initial constructive app roach, we prop ose a reactive proced ure th at adapts to the set of unassig ned ev ents from p rev ious run s. The logic beh ind this approac h is that the constructive procedure ‘disco ver s’ ev en ts that are difficult to assign, giving them p riorit y in successive run s. S imilar ideas have b een sketc hed by the sque aky whe el optimization approac h [6], and implemen ted in ant colony metaheuristics for examination timetabling problems [4]. In the follo wi ng, le t E p b e the set of prioritized even ts, E ¬ p the set of non-prioritized even ts, and E u the set even ts that ha v e not been assigned during the construction phase. It is requ ired that E p ⊆ E , E ¬ p ⊆ E , E p ∩ E ¬ p = ∅ , and E p ∪ E ¬ p = E . Algorithm 2 d escribes the reactive construct ion p rocedu re. Algorithm 2 R eactiv e construction 1: Set E p = ∅ , E u = ∅ , loops = 0 2: rep eat 3: E p ← E u 4: E u ← ∅ 5: E ¬ p ← E \E p 6: while E p 6 = ∅ do 7: Select the most critical ev en t E f rom E p , that is the ev en t wi th the smallest n um ber of a v ailable timesl ots 8: if E can be assigned to at least one timeslot then 9: Select some a v ail able timeslot T for E 10: Assign E to the timesl ot T 11: else 12: E u ← E u ∪ E 13: end if 14: E p ← E p \ E 15: end while 16: while E ¬ p 6 = ∅ do 17: Select the most cri tical ev en t E f rom E ¬ p , that is th e even t with the smallest n um ber of a v ailable timesl ots 18: if E can b e assigned to at least one timeslot then 19: Select some av ailable timeslot T for E 20: Assign E to the timesl ot T 21: else 22: E u ← E u ∪ E 23: end if 24: E ¬ p ← E ¬ p \ E 25: end while 26: loops ← loops + 1 27: until E u = ∅ or loops = M axloops 6 As giv en in th e pseud o-code, the construction of solutions is carried out in a lo op un- til either a feasible solution is iden tified or a maximum number of iterations M ax loops is reac hed. When constructing a solution, a set of even ts E u is k ept for whic h no times- lot has b een found. When reconstructing a solution, these ev en ts are prioritized ov er the others. In th at sense, th e constructive approach is b iased by its prev ious runs, identif ying even ts that turn out to b e d ifficult to assign. After at most a maxim um n umber of M axloops iterations, the construction pro ce- dure retu rns a solution th at is either feasible ( E u = ∅ ) or not ( E u 6 = ∅ ). It h as b een pointed out that even when ev ents are put into E p , th ey do not nec- essarily remain elemen ts of that set. Instead, they migh t b e remo v ed from E p in the subsequent loop. T o some extent, this is counterin tuitive, as the algorithm does not build u p a complete datastructure storing al l unsuccessfully assigned ev ents. Instead, the direct ‘learning’ is limited to the preceding run. It has to b e mentioned ho w ever, that some implicit information is nev ertheless tran sferred from loop to loop, as a ny loop is biased by its predecessor. It also should b e noticed that this implementation of a more limited adaptive algorithm led to satisfactory results, which is why alternativ e approac hes hav e not b een furt h er inv estiga ted yet. Exp erimental r esults In the exp eriments , w e focused on the difficult instance comp05.ctt , computing for 1000 t rials the num b er of feasible so lutions reached after a certain number of loops of the construct ive approach. The obtained results are give n in T able 2 . T able 2 F easible solutions after a certain num ber of l oops for comp05.ctt (out of 1,00 0 trials) Loops feasible solutions 1 0 2 56 3 272 4 387 5 511 6 608 7 688 8 754 9 802 10 831 The number of cases in whic h a feasible so lution has been reac hed slowly converge s to 1000, monotonically increasing with each additional loop. This indicates that the biased reconstruction in the presen ted ap p roac h succes sfully adapts to even ts which are d ifficult t o assign to timeslots. It should be noticed that the b ehavior of the approac h for the other ben chmark instances is similar. This observ ation is ho w ev er less imp ortan t, a s a rep etitive applica- tion of th e simple constructive approach wil l increase the p ercenta ge of cases in which a feas ible solution is reac hed, too. F or instance co mp05.ctt , where not a single feasible solution is found after the first lo op, th is do es not hold. 7 4 T hreshold Accepting based improv ement 4.1 Description of the approach The constructive approac h as describ ed in Section 3 only aims to identif y a first feasible assignmen t of even ts to timeslots, not taking into consideration the resulting soft con- strain t viola tions. An iterative pro cedure conti nues from here, searc hing for an optimal solution with resp ect to the soft constraints. The form ulation of the approach is rather general. One of the reason for this is th at while w e hope for a feasi ble assignment of all even ts, the constru ct ive approac h does not guara ntee it. Nev ertheless, search for impro v ed solutions needs to con tinue at some p oin t, and an approac h that is able to handle infeasible solutions is therefore required. Also, in case of an infeasible first a ssignmen t, the pro cedure should b e able to la ter identif y a feasible one. In each step of the p ro cedure, a n umber of randomly c hosen ev en ts is unassigned from the timetable and reinserted in the set E u . A reassignmen t phase follo ws. Con trary to the constructive approac h, where ev ents are selected based on whether they are critical with resp ect to the av ai lable timeslo ts, even ts are now randomly c hosen fro m E u , each even t with iden tical probab ility . The c hoice of the timeslot follo ws the logi c as described in the constru ct ive approac h, prioritizing timeslots of particular ro om classes. Again, we use the tw o possible p reference structures of ro oms, R i 1 o ver R i 2 o ver R i 3 , and R i 2 o ver R i 1 o ver R i 3 . Each of them is randomly chosen with probability 0.5. When ev aluating timetables, tw o criteria are considered. First, the number of un as- signed timeslots (distance to feas ibilit y) hc , seco nd, the total p enalty with respect to the given soft constraints sc . Comparison of solutions implies a lexicographic ordering of the h ard constrain t violations hc ov er the p enalty function sc . W e therefore accept timetables minimizing t h e distance to feasibilit y independent from the soft constraint count. This means that in ca ses in whic h the initia l construction phase is unable to as- sign all even ts to timeslots, a later assignmen t of more even ts is preferred indep endent from an increasing v alue of sc , closely follo wing the ev aluatio n of solutions as given in the ITC 200 7. In case of identical distance to feasibilit y hc , inferior solutions with respect to sc are accepted up to a threshold. This idea has been introdu ced by th e Threshold Accepting metaheuristic [5], a simplified deterministic v ariant of Sim ulated A nnealing. Previous researc h h as shown that simplifications of Sim ulated Ann ealing ma y b e v ery effective for timetabling problems [1]. The implementati on of the Threshold Accepting approach compares the quality of neighb oring solutions w ith the current b est alternative, permitting an acceptance of inferior alternatives up to the given threshold. An alternative strategy wo uld b e the comparison with the current solution instead of th e globally b est one. In this case how ever, a sub seq uent acceptance of inferior solutions ca n h app en, and for that reason, the more restrictive acceptance strategy has b een chosen. 4.2 Firs t results and comparison with other approaches Different confi gurations of th e algorithm h ave b een tested on the b enchmark data from the ITC 2007. A first implementatio n has b een made av ailable, how ev er without optimizing the co de with res p ect to sp eed and effici ency . This has b een done later, and 8 the final results for the ITC 2007, as rep orted later, are therefore significantly b etter, simply because th e final vers ion of the progra m allo w ed m uch faster computations. On an Intel Core 2 Q uad Q6600 2.4 GHz processor, equipp ed with 2 GB RAM, mounted on an ASUS motherb oard, 375 seconds of computin g time h a ve b een allow ed for each test run. Besides the d etermination of the n umber of reassigned even ts in each iteratio n, whic h has b een set to fi ve, an appropriate c hoice of the threshold needs to b e made. Three d ifferent configurations of the threshold are reported here, 0% of sc , 1%, and 2%. The follo wi ng T able 3 gives the obtained av erage v alues of th e soft constraint p enal- ties sc for three threshold configurations and compares the results to an Iterated Lo cal Searc h approach [7]. In t his context, a threshold of 0% leads to a hillcli mbing algorithm as only impro ving mov es are accepted. The Iterated Lo cal Search approac h consists of a hillclim bing algorithm (a Thresh- old Acceptin g algo rithm with threshold 0%), pertu rbing th e curren t solution af ter a num ber of non-impro ving mov es. P ertu rbations are d on e by a random reass ignment of five even ts. Con trary to the usual acceptance rule with resp ect t o th e cost funct ion sc , the p erturb ed alternative is accepted in an y case, and searc h contin ues from this new solution. Two configurations of the I terated Local Search Approac h hav e been implemented. The first vari ant, ILS 10 k, starts pertubing after 10, 000 non-improving mo ves, the oth er, ILS 3k, after 3,000 mov es. T able 3 Ave rage v alues of sc Instance T A 0% T A 1% T A 2% ILS 10k ILS 3k comp01.c tt 10 12 13 12 14 comp02.c tt 229 199 204 218 223 comp03.c tt 216 201 213 211 202 comp04.c tt 134 126 132 138 145 comp05.c tt 656 594 657 658 641 comp06.c tt 199 177 230 196 194 comp07.c tt 179 1 96 316 181 185 On the basis of the obtained results, w e conclude that a rather small threshold of 1% leads for most i nstances to th e best ave rage results. There are some i nstances in whic h the Iterated Local Search obtains goo d results, bu t T A 1% is ove rall most promising. It should b e noticed that the choice of a percentage as a threshold h as b een iden- tified after exp erimenting with other algorithmic v aria nts. The main adv an tage of this approac h appears to b e that for small va lues of sc th e algorithm b ehav es more like a hillclim bing algorithm, while for larger v alues a larger threshold is derived. 4.3 Results for the I nternatio nal Timetabling Competition ITC 2007 The initial implemen tation of the algorithm has b een optimized with resp ect to execu- tion speed, how ev er keeping the metho dological ideas as described abov e. A significant impro vemen t has b een achieve d, due in particular to a delta-ev aluation of the mov es. 9 T able 4 gives the b est results of the Threshold Accepting algorithm with a threshold of 1%. The results are based on 30 trials with different random seeds. Eac h trial w as allo w ed to run for 375 seconds on the hardw are mentioned ab o ve. The num ber of ev aluated solutions is given, to o. In contrast t he the initial experiments, we now rep ort results for 14 in stances, seve n of whic h h ad b een released a few wee ks b efore the required sub mission of th e results. T able 4 Best results and the used seeds (ou t of 30 trials) Instance seed har d constraint soft constraint ev aluat ions violations violations comp01.c tt 130 0 5 13,072,619 comp02.c tt 112 0 108 8,547,980 comp03.c tt 119 0 115 9,211,859 comp04.c tt 128 0 67 10,352,548 comp05.c tt 119 0 408 6,512,059 comp06.c tt 117 0 94 8,631,146 comp07.c tt 113 0 56 7,673,851 comp08.c tt 129 0 75 9,881,464 comp09.c tt 119 0 153 9,248,758 comp10.c tt 122 0 66 8,386,538 comp11.c tt 111 0 0 13,468,229 comp12.c tt 103 0 430 6,782,742 comp13.c tt 104 0 101 9,838,210 comp14.c tt 122 0 88 9,693,538 It can b e seen that th e approach leads to reasonable results, and that th e b est re- sults of the improv ed code are significantly bett er than the ones of the first implemen- tation. F or some instances, comp01.ctt and comp11.ctt , particularly go od solutions are found . O thers such as comp05 .ctt and comp12.ctt ha ve best found alternatives with soft constraint p enalties that are still quite large. Based on the observed imp rove- ment in comparison to the first implementation, w e can conclude that efficiency of the implementa tion plays an imp ortant role for th e fi nal results. The follo wi ng T able 5 gives the avera ge results of the top five comp etitors of I TC 2007, track 3. The columns are sorted in d escending order of the o veral l rank ing, thus sho wing the results of Thomas M¨ uller in the leftmost column. I n brief, our approach ranked 4th o vera ll. When closer analyzing th e obtained results, it becomes clear that the app roac hes of the first three finalists did indeed lea d to comparable sup erior re- sults. In relation to the approac h of Clark, Henz, and Lo ve, o ur implementation of the Threshold A ccepting algorithm t urned out t o be b etter, ho w ev er not for a ll t est instances. Unfortunately , w e do not have any informati on ab out the algorithms of t h e other fi- nalists. Consequently , the possibilities of drawing precise conclusions are limited. Nev- ertheless, w e susp ect that the top three ranked programs are substan tially b etter than our Threshold Accepting implementation, simply b ecause the ave rage results are su- p erior. This raises the question whether th e observed differences are due to a b etter (faster) implementatio n, or du e to b etter algorithmic id eas. Longer optimization runs are therefore carri ed out in the follow ing, allowi ng a b etter con ve rgence of th e meta- heuristic without the immediate pressure of terminating the searc h after on ly 375 seconds. 10 T able 5 Ave rage results of the top fiv e competitors of ITC 2007, track 3 Instance M ¨ uller Lu, Hao At suta, Geiger Clark, (USA) (F rance ) Nonobe, (German y) Henz, Lo v e Ibaraki (Singapore) (Japan) Rank: 1 2 3 4 5 comp01.c tt 5. 0 5.0 5.1 6.7 27.0 comp02.c tt 61.3 6 1.2 65.6 142.7 131.1 comp03.c tt 94.8 8 4.5 89.1 160.3 138.4 comp04.c tt 42.8 4 6.9 39.2 82.0 90.2 comp05.c tt 343.5 326.0 33 4.5 525.4 811.5 comp06.c tt 56.8 6 9.4 74.1 110.8 149.3 comp07.c tt 33.9 4 1.5 49.8 76.6 153.4 comp08.c tt 46.5 5 2.6 46.0 81.7 96.5 comp09.c tt 113.1 116.5 11 3.3 164.1 148.9 comp10.c tt 21.3 3 4.8 36.9 81.3 101.3 comp11.c tt 0. 0 0.0 0.0 0.3 5.7 comp12.c tt 351.6 360.1 36 1.6 485.1 445.3 comp13.c tt 73.9 7 9.2 76.1 110.4 122.9 comp14.c tt 61.8 6 5.9 62.3 99.0 105.9 comp15.c tt 94.8 8 4.5 89.1 160.3 138.0 comp16.c tt 41.2 4 9.1 50.2 92.6 107.3 comp17.c tt 86.6 100.7 107.3 143.4 166.6 comp18.c tt 91.7 8 0.7 73.3 129.4 126.8 comp19.c tt 68.8 6 9.5 79.6 132.8 125.4 comp20.c tt 34.3 6 0.9 65.0 97.5 179.3 comp21.c tt 108.0 124.7 138.1 185.3 185.8 4.4 Co nv ergence in longer runs In contrast to the optimization runs for the ITC 2007, w e allo w in the f ollo wing ex - p erimen ts the ev aluation of 100 million timetables b efore terminating the algorithm. Again, 30 trials have been carried out, and T able 6 gives the b est found solutions out of all test ru ns. Obviously , the Threshold Accep t ing algo rithm did not conv erge after only 375 sec- onds. R ather big improv emen ts can b e seen for most instances, sometimes impro ving the b est solution by 25% ( comp10.ctt ). F or th e instances with large v alues of sc , comp05.ctt and comp12.ctt , impro v emen ts are possible, bu t the absolute v alues re- main rath er high. W e susp ect that these instances possess properties that complicate the identification of timetables with small soft constrain t violations. R ecalling th at instance co mp05.ctt w as problema tic w ith respect to the identification of a f easible assignmen t in the initial experiments, this is how ev er not surprising. No improve ments are p ossible for instance comp01.ctt , and of course for in stance comp11.ctt . In comparison to the three top ranked finalists of ITC 2007, inferior ov erall results are found, even when allo wing the ex ecution of 100,000,00 0 ev aluations. Indep endent from the personal programming skills of the comp etitors, whic h are unknow n to u s and difficult to assess, w e susp ect that the p erformance of the approaches is mainly d ue to the algorithms as such. 11 T able 6 Best results after 100,000,000 ev aluations (out of 30 trial s ) Instance hard constrain t soft c onstraint violations vi olations comp01.c tt 0 5 comp02.c tt 0 91 comp03.c tt 0 108 comp04.c tt 0 53 comp05.c tt 0 359 comp06.c tt 0 79 comp07.c tt 0 36 comp08.c tt 0 63 comp09.c tt 0 128 comp10.c tt 0 49 comp11.c tt 0 0 comp12.c tt 0 389 comp13.c tt 0 91 comp14.c tt 0 81 5 Sum mary and conclusions The article presen ted an approac h for curriculum-based course timetabling, emplo ying the general idea of the Threshold A ccepting metaheuristic. T he meth odological con- cepts are ra ther problem-indep endent as only simple remo v als and reassignmen ts of even ts from and to the timetable are carried out d uring searc h. Initial ex p erimen ts with a first implemen tation indicated that small v alues of the threshold present a g oo d parameter setting. Comparison studies with a simple h ill- clim bing algorithm and an Iterated L o cal S earch Algorithm hav e b een carri ed out. In brief, the Threshold Accepting v ariant with a threshold of 1 % appeared to b e most promising. Comparisons of the short runs for the I nternational Timetabling Competition 2007 with long ru n s reveal th at the p rop osed algorithm d o es not conve rge within t h e given time limit. Mo re time f or computations is needed, and further impro v ements of the concept are certainly p ossible. W e are confident that a fair contribution to th e ITC 2007 has b een made. In comparison to the other participants of the I TC 2007, our approac h ranked 4th over- all. Ho w eve r, a considerable gap to t h e avera ge results of the top three con tributions b ecame ob vious, an d we are looking f orw ard to read the articles describing these ap- proac hes. Nevertheless, good solutions are found, in some cases ev en in short time. W e fi nd op t imal solutions for instance comp11.c tt , and a v ery go od one for instance comp01.ctt . Ac knowledgemen ts The author wou ld like to thank three anon ymous referees for their helpful commen ts. References 1. E. K. Burke, Y. Byko v, J. P . Newall, and S. Petro vic. A time-predefined approac h to course timetabling. Y ugoslav Journal of O p er ations R ese ar ch (YUJOR) , 13(2):139–15 1, 2003. 12 2. M . W. Carter. Timetabling. In S. Gass and C. H ar ris, edito rs, Encyclop e dia of Op er ations R e se ar ch and Management Scienc e , pages 833–836. 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