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1 DOCUMENT RESUME ED EA AUTHOR de Werra, D. TITLE Constructon of School Tmetables by Flow Methods. INSTITUTION Waterloo Unv. (Ontaro). Dept. of Manaaement Scences. REPORT NO PUS DATE 14P-11: [70] NOTE' 2u p. EDRS PRICE' DESCRIPTORc FDPS?rce MF-S0.65 PC-$3.20 Computer Pxograms, Flexble Schedulng, *Mathematcal Models, *ScheMulna, *School Schedules, Tme PlocYs AEST1ACT In ths paper, a heurstc algorthm for constructng school tmetables s descrbed. The algorthm s based on an exact method that apples to a famly of partcular tmetable problems. The Procedure has been used to construct tmetables for Swss schools havng about!o classes, PO teachers, and weekly perods. Less than fve percent of class/teacher meetngs could not be scheduled at some perod n the week. (Author)

2 CONSTRUCTION OF SCHOOL TIMETABLES BY FLOW METHODS by D. de Werra Workng Paper No. lr/v U.S. DEPARTMENT OF HEALTH, EDUCATION E WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS DEER REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING 11. POINTS OF YlEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE Of EDUCATION POSITION OR POLICY. DEPARTMENT OP MANAGEMENT SCIENCES UNIVERSITY OP WATERLOO WATERLOO, ONTARIO

3 CONSTRUCTION OF SCHOOL TIMETABLES BY FLOW METHODS D. de Werra Department of Management Scences Unversty of Waterloo Waterloo, Ontaro

4 3. State ent of The Problem We consder that a school s characterzed by: (a) a set of classes C = {c ; = 1,..., (1) Wasetofteaaers G 1, be., p) (c) a set of subjects B = {b1; 1 = 1, sl (d) a set of perods N = {nk; k = 1,..., n} (e) a set of meetngs A whch have to occur durng the week. Each meetng lasts on,..! hour (or oae perod) and nvolves one or more classes, one or more teachers and one or more subjects. A set of meetngs nvolvng the same elements (.e. the same class(es), teacher(s) and subject(s)) s called a combnaton. A tmetable conssts then of an allocaton of every meetng n 9, to a perod of the week; n other wo.:ds t s a partton of 66 nto n subsets H ' H n where H k s the subset of all meetngs occurrng at perod nk (or more smply at perod k). Ths allocaton has to satsfy certan condtons specfed later. Problem Let us frst consder a partcular case of the tmetable problem. Suppose that every meetng ofrt nvolves exactly one teacher raj and one class c ' and neglect the subjects.

5 4. We can assocate wt; the seta an array A (pxq) wth nonnegatventegerentresauwhere a j referrng to class c and teacher m. s the number of meetngs Every combnaton s now characterzed by c., and a..; we shall denote a combnaton by ) (c, mj, a j ) or smply c -m j matrx.. A 6 usually called the requrement If the only restrcton to be consdered durng the constructon s that no teacher or cl-jas s requred at two places at once, we can solve the problem easly. sj Let a be the total number of meetngs nvolvng f no teacher and no class s nvolved n more than n maetngs, n other words f: a n a 1, q (1) J < n j S 1, p then the problem has at 1.C6St one soluton and we can construct a tmetable. We consder a bpartte multgraph G wth vertces cl, cq; ml, mp. Every combnaton c-mj s represented by a j parallel edges jonng c and m a (or P ) s the degree of vertex c (or m ),.e. the number of edges ncdent wth ths vertex. A subset of meetngs whch can occur smultaneously s represented by a matchng H n G (t s by defnton a set d of edges such that no two edges of H are adjacent). A tmetable conssts then of a partton of the edges of C nto n matchngs H1, Hn.

6 5. The determnaton of these matclngs s a flow problem; more precsely t s a sequence of n-1 problems of flow wth mnmal cost Problem II In a real school we fnd many other requrements and we have to take them nto account durng the preparaton of a tmetable. We frst ntroduce unavalablty constrants n the prevous basc model of problem I. Let us suppose that teachers and classes are not necessarly free durng n perods, but some of them are unavalable for certan perods (they may be ether absent or nvolved n preassgned meetngs). These constrants are descrbed by arrays E and D. E (ek) e 1,..., q; k 1, n D (d jk ) j 1, p; k a 1,., n where elk (or d jk ) s 1 f c1 (or m j ) s unavalable at perod k and s 0 otherwse. For all we defne the subset M of 14 as Cr.e subset contanng all teachers who have to meet class c and for all j, Cj s defned as the subset of all classes havng to meet teacher m.

7 6. s sad to be consstent f, for all and k, d jk = 1 for alljsuchthatmem.=> e k = 1; smlarly D s consstent f for all j and k ek = 1 for all such that c e Cj => djk = 1. Ths means that a class c has to be consdered as unavalable at perod k f all teachers who are to meet c are unavalable at perod k (and smlarly for m ). In the remander of the paper we shall deal only wth consstent arrays E and D. As prevously, a bpartte multgraph G can be assocated wth the tmetable problem, but we have now an addtonal condton onlik:lkmustnotncludeanyedgeadjacenttovertexc.(or m j ) f e k (or d jk ) s equal to 1. Solublty Propostons Snce teachers and classes play equvalent roles n the problem, we shall smply call them elements. Necessary and suffcent condtons of solublty for problem II are not obtaned as easly ns for problem I. Obvously no element must be nvolved In a number of meetngs greater than the number of ts free p!tods: n - F e > k kl n n -Ed > 0 jk j 1,..., q j I, p (2)

8 7. Furthermore, t s necessary that for all combnatons c -m j the elements c and m 1 of perods not less than al: j are smultaneously free for a number n a,, < E (1 - elk) - dk) for all c - nj knl (3) Smple examples show that condtons (2) and condtons (3) are not suffcent for solublty of problem II. A thrd type of condton was used by Gotleb (4) ; these are based on condtons of Hall for the exstence of a system of dstnct representatves (6). For all classes c and all subsets M of M, number of meetngs of c wth the teachers of M must not be greater than the number of perods at whch c and one at least or tose teachers are smultaneously free: n E a < E (1-e k )(1 d,,) j:m " knl j:m (4.1) for all 1:1=14 and for all. Smlarly we have for all Zcr. C and for all j: n E a44 " < E (1-d41,)(1 elk) tch knl J' :ctfc (4.2) These condtons mply condtons (2) and (3), but generally they are only necessary condtons. They are suffcent for a problem wth only one teacher or one class.

9 8. We shall see n a followng paragraph the meanng of condtons (4.1) and (4.2). Let us call degree of freedom (DF) of c. the quantty Mc defned as Mc...n-Ee k -a' and the DF of mj ' Mm and defned as k wll be denoted Mm n-ed -a k Jk For combnaton c -m j' we can also defne a DF N j by Nj n E (1-ek)(1-djk) - aj Wth these defntons, condtons (2) and (3) expreas that all DF are nonnegatve. We gve now some results concernng the solublty of problem II. Proposton 1 A problem wth N m 0 for all combnatons c -m j j may not have more than one soluton. Proof: If t had two dstnct solutons there would be at least one combnaton, the aj meetngs of whch could be allocated n two dfferent ways to a perods of the week. Ths means that the number of perods where c and mj are both free s greater than aj. Ths combnaton has a postve DF; ths s n contradcton wth N j = 0. Proposton 2 If Nj t 0 for all combnatons c-mj, then all DF of classes and teachers are nonpostve.

10 9. Proof: Let us assocate wth problem II a bpartte graph G 1. Its vertces are (fg. 1): Ck = 1,..., q; k = 1, n Mjk j = 1, soft, p; k = 1, 0110, n W6 call t the set {Pk; k = 1,..., 4 for = 1,..., q anctenj the set fjk; k = 1, nj for j = 1, p. Ck and Mjk are joned by an edge f there s a combnaton c-mj and f both c and mj are free at perod k. (a) Snce E and D are consstent, for any perod k at whch c s free, at least one teacher n M s free. Ths mples that there s at least one edge adjacent to Ck f c s free, at perod k. Smlarly one edge at least s adjacent to Mjk f mj s free at perod k. (b) Snce Nj = 0 for all c-mj, there are exactly a edges n G1 jonng vertces of t and vertces of of (there are exactly aj perods to whch meetngs of c-mj can be allocated). So there are exactly d (or p) edges adjacent to vortces of t (oraj) n G1. (c) Suppose now that Me > 0 for some class c: ths means that the number (say cc) of porods where c s free s strctly greater than the number ct of meetngs n whch c s nvolved. Then (a) mples that there should be at least cc 7 a edges adjacent to vertces of t And (b) asserts that there are exactly ot such edges;

11 10. ths s a contradcton and thus all Mc are nonpostve. Clearly all Mmj are also nonpostve and th ends the proof. We know furthermore that t s always possble to assocate wth every tmetable problem P (havng 1 Mc > 0 and mm 0 for j =.1 al erld j) another proble 2 wth Mc = M mj = 0 for all and j. 2 P and P2 are equvalent n the sense that one has :t soluton f and only f the other one s soluble(1)(5). P s called a reduced 2 problem and has the followng propertes: teachers s equal to the number of classes: (1) the number of (2) at any perod k, the number of free classes n equal to the number of free teachers. So we only have to consder reduced problems. Proposton 3 A suffcent condton for solublty of a reduced problem II s N j = 0 for all c.-m.. j Proof: W free durng exactly a perods and by the consstency of E there are exactly a vertces of adjacent to at least one edge: snce there are exactly a edges adjacent to vertces of t3 (N. = 0 for all j c-mj) no vertex n 81 s adjacent to more than one edge. Smlarly any vertex n jc s adjacent to a;: most one edge, so that all edges of G 1 represent a tnetall1e: at every ferod class (or teacher) meets at most one teacher (or one class). Ths result s worthy of consderaton, because one normally thnks that the "probablty" that a problem has a soluton ncreases

12 H. when the DP are zlugma,,ad. Ths proposton shows that t s not always true: a reduced problem always has a soluton f N j = 0, but t may have none when N 0. j Some specal cases Let us consder reduced problems; the unavalablty constrants can he regarded as preassgnments because at each perod the number of unavalable classes s equal to the number of unavalable teachers (4)(5). So each element s to be nvolved n n meetngs: some of these meetngs are preassgned to fxed perods (correspondng to the unavalable perods of ths element). Proposton 5 A reduced problem such that one class only (or one teacher only) has dome preaesgned meetngs, s soluble. Proof: We consder the bpartte multgraph G assocated wth that problem; all edges representng preassgned meetngs are adjacent. Thus any matchng H n G contans at most one of these preassgned edges. So any decomposton of G nto n matcnngs wll gve a tmetable. Note that f G s not connected, the problem stll soluble when at most one class (or one teacher) n each connected component has some preassgned meetngs.

13 12. Now for a problem whch s not reduced, let us defne two famles of unavalablty constrants: (a) all classes are free at any perod, one teacher at most s not avalable at any perod. (b) one class exactly, say c, s unavalable at some perods. The unavalabltes of teachers are at any perod at whch c s unavalable, one teacher at most s unavalable. Smlar constrants outaned by permutng classes and teachers are consdered as equvalent to (a) or (b). Then w'; have the followng result, provded that a +Ee k for all and S +Ed <pfor all j k Corollary 5.1 A problem II wth unavalablty constrants of type (a) or (b) has always a soluton. Proof: In case (a) we ntroduce a spurous class c ; f a teacher s unavalable at perod %, we say he has to meet c at perod k. In case (b) f c and mj are both unavalable at perod k, we say thatcllastomeet111,3 at perod k. All these preassgned meetngs are represented by adjacent edges n G for (a) and for (b). G whose vertces have degrees not greater than n can be decomposed nto n matchngs; these matchngs represent a tmetable.

14 13. Of course the problem has stll a soluton when we have constrants (a) or (b) for any connected component of G. Now let us examne the case where condtons of Hall hold for all classes and all teachers. Ths means that for a reduced problem the preassgnments are such that t s possble to construct separate tmetables for all classes and all teachers. The problem may have no soluton; f ths happens we can keep fxed all preassgned meetngs of one class c (or of one teacher m ) and allow the remanng preassgned meetngs to be possbly allocated to some dfferent perods. Ths s equvalent to consder that we have preassgned meetngs only for one class c or for one teacher m5. From prnposton 5 such a problem always has a soluton. Thus we can formulate: Corollary 5.2 Let P be a reduced problem wth a set of preassgned meetngs. teachers. Suppose condtons of Hall hold foc all classes and Then ether P has a soluton or t can be transformed nto a soluble problem P' by keepng fxed the preassgned meetngs of any class c (or any teacher ) and possbly modfyng the remanng preassgnments. I

15 14. Flow Method The method used to conf,t-ct tmetables proceeds ay successve stern; the tmetablc wll be prepared one perod at a tme; each step coassts of the allocaton of meelngs to a determned perod of the week. When we have completed the allocaton for the k frst perods we call the allocaton of the remanng meetngs to perods k+1, k+2, n the resdual problem. We have..hen to allocate meetngs at perod k n such a way that the resdual problem stll admts a soluton. As any subset H of meetngs that can occur k smultaneously s represented by a matchng n a bpartte multgraph for problem I as well as for problem II, we have to solve a flow problem at each step. To perform ths t s convenent to consder the graph G assocated wth pro'alem I; we orent any edge jonng c and mj from c to mj and gve t an nfnte capacty. We ntroduce a source S and jon t wth every c by an edge (S,c) of unt capacty; smlarly we ntroduce a snk T and edges (m.., T) of unt capactes for all m.. We (all ths graph C. Any nteger valued flow n G' from S to T corresponds to a matchng n G. At each step we have to assgn prortes to meetngs, and possbly to classes and teachers n order that the resdual problem has a soluton. These prortes can be expressed by a cost functon KF assocatng a cost KF(x,y) wth any edge (x,y) n G'. The allocaton of meetngs to a certan perod s now a mnmal cost flow problem.

16 15. For problem I the determnaton of a cost functon KF s easy because we have necessary and suffcent condtons of solublty. At any step, we can for example assgn colts equal to zero to all edges (S c ) and ( 1111,T) assocated wth c and mj whch are stll to be nvolved n a maxmum number of meetngs and postve costs to the remanng edges n G'. After each step we have to delete n G' the edges (c,m) correspondnglto meetngs whch have just been allocated; the flow has to be set equal to zero n all remanng edges before proceedng to the next step. As we do not know necessary and suffcent condtons for the solublty of problem II, we can examne dfferent cost functons correspondng to dfferent prorty crtera. I. Cost functon Fl teachers: For any perod, we calculate the DF of classes and KF(S,c) 00 f c s unavalable at that perod tc otherwse KF(m,T) 00 f m s unavalable at that perod Mm otherwse KF(c ) m any postve number. Prortes are gven to teachers and classes havng the smallest DF. F 1 s a generalzaton of the cost functon used n

17 16. problem I. Ths cost functon seems to be adapted to problems wth only few nnavalabltes. II. Cost functon F 2 At any perod, we only calculate the DF of combnatons. KF(S,c) 01 co f c s unavalable at that perod any postve number otherwse KF(TT) = co f m s unavalable at that perod any postve number otherwse KF(c m j ) = N j We try to allocate frst meetngs of combnatons havng the smallest DF. If there are no or only a few unavalabltes, we can fnd examples where F2 fals to construct a soluton whle F1 does. Consder the example n Fg. 2; such a problem obvously has a soluton snce no vertex s adjacent to more than 3 edges. At the frst step we have: KF(I,a) - KF(I,b) = KF(I,c) = 2 KF(II,a) = KF(III,b) = KF(IV,c) = 1 Wth the flow method we get H1 = {(II,a), (II1,b), (IV,c)).

18 17. The resdual problem has no soluton because class I has stll to be nvolved n 3 meetngs and there are only 2 perods left. III. Cost functon F 3 The mxed cost functon F 3 s defned Eor any perod as follows: lu(s,c) 4.311c.s unavalable at that perod Mc otherwse KF(m,T) = Go f mj s unavalable at that perod Mmj otherwse KF(c,mj)g N j Wth ths cost functon gvng prorty to elements and meetngs havng the smallest DF, we try to prevent these DF from decreasng too rapdly and f we can proceed up to the nth step wth nonnegatve DF we get a soluton for problem II. If some DF becomes negatve durng the procedure, then the resdual problem has no soluton. Ths stuaton may happen even though problem 11 s soluble because the procedure s not based on necessary and suffcent condtons for solublty. Some dfferences between problem I and problem II are to be noted now. In problem I, we could determne at each step a maxmal flow (wth mnmal cost) correspondng to a maxmal matchng (.e. contanng a maxmum number of edges). In other words we

19 18. could allocate as many meetngs as possble at any perod; the resdual problem remaned soluble. But for problem II, t may happen that the resdual problem becomes nsoluble f we detemne an allocaton represented by a maxmal. flow (wth mnmal cost). At a gven perod we have to allocate exactly one meetng of every combnaton c -mj wth N j 0 (f c and m f we do not, the resdual problem certanly has no soluton. are both free); Hk has to nclude a subset of edges correspondng to combnatons whose DF are zero and moreover a subset of edges adjacent to all vertces c and mj (free at that perod k) wth DF equal to zero. An admssble H wll thus sometmes be a non maxmal matchng; the consequence s k that we cannot always get a tmetable wth a sequence of maxmal flows. The example n Fg. 3 llustrates ths fact. f we use F3 and determne a maxmal flow wth mnmal cost at frst step, we cannot get a soluton. Practcally we shall at any perod allocate frst meetngs of all combnatons c -mj wth N = 0 (f c and mj are free) and j then determne a maxmal flow wth mnmal cost correspondng to the meetngs nvolvng the remanng classes and teachers.

20 19. Applcaton to a Real System Many other requrements are present n real schoo.s, so that we stll have to modfy our model and ntroduce the related constrants: (1) Let B* be a subset of B; the elements of B* are called specal subjects. Meetngs nvolvng a subject be n B* have to take place n a determned room assocated wth ths subject. As the schools are supposed to nclude exactly one room for any specal subject, two meetngs wth the same specal subject be may not occur smultaneously. (2) As prevously mentoned, a meetng may nvolve several teachers, several classes and several subjects. (3) It s furthermore necessary that certan subjects are acceptably spaced throughout the week. So we want no more than one meetng of any combnaton to occur n a school day. In some partcular cases, two meetngs of the same combnaton must occur at consecutve perods. These are called double meetngs; for every combnaton the number of double meetngs s fxed. We explctly take these constrants nto account as follows. Before constructng the tmetable by the flow method, we allocate double meetngs, meetngs nvolvng several teachers and classes, and meetngs ncludng specal subjects at dfferent perods wthout volatng any constrants.

21 20. We elmnate then from J. all meetngs already assgned and consder that elements nvolved n these preassgned meetngs are unavalable at the correspondng perods. The remanng problem s now a problem II and we determne a soluton by the flow method (usually wth cost functon F3). In practce, more than 5% of meetngs are rarely preassgned, so that ths rather arbtrary preassgnment does not nfluence strongly the constructon of a soluton. The above outlned method s stll a relable procedure. A program has been wrtten n FORTRAN IV (wth a few subroutnes n MAP) for the IBM 7040 of the Centre de Calcul de l'ecole Polytechnque Fdderale de Lausanne. We used Input and Output subroutnes of a program wrtten n Germany (H.G. Genrch (3) ). A seres of experments was carred out wth the program. Because of the mposed constrants t was not always possble to avod falure n constructng the tmetables. Generally when n steps are performed, a small percentage of meetngs are left unassgned. These are advantageously allocated wth a mnmum of manual adjustments. Some results are Aurmarzed n Table 1.

22 21. TABLE 1 Classes Teachers Perods of the Week No. of Meetngs Percentage not Assgned computaton Tme mn " "

23 22. REFERENCES (1) J. Csma, Investgatons on a Tmetable Problem, Ph.D. Thess, Insttute of Computer Scence, Unversty of Toronto, (2) L.R. Ford and D.R. Fulkerson, Flows n Networks, Prnceton Unversty Press, (3) H.G. Genrch, De Automatsche Aufstellung von Stundenplanen auf relatonentheoretsche Grundlage, Forschungsbercht No. 1713, Landes Nordrhen-Westfalen, Kbln, (4) C.C. Gotleb, The Constructon of Class-Teacher 'Tmetables, Proceedngs of the IFIP Congress, North-Holland Publshng Co., Amsterdam, 1963, p. 73. (5) D. de Werra, R4soluton de problemes d'horare par la thgore des graphes, These de doctorat, Insttut de Mathgmatques Applquges, Ecole Polytechnque Fgdgrale de Lausanne, (6) P. Hall, On representatves of subsets, J. London Math. Soc. 10, 1935, p

24 23. ACKNOWLEDGEMENTS The author wshes to thank Professor Ch. Blanc (Swss Federal Insttute of Technology n Lausanne) who suggested ths research and Professor D.J. Clough (Unversty of Waterloo) for hs helpful comments on an earler verson of ths paper.

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CHAPTER 2 DECOMPOSITION OF GRAPHS

CHAPTER 2 DECOMPOSITION OF GRAPHS CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng