Organizing Teacher Education In German Universities

Transcription

1 An Operatons Research Approach Bob Grün Unversty of Kaserslautern Department of Mathematcs AG Optmzaton Research Semnar OptALI: Optmzaton and ts Applcatons n Learnng and Industry

2 Outlne 1 Introducton European Unversty System Study Plans for Teacher Tranng n Germany 2 Varables Constrants IP Relaxaton Computatonal Results 3 Current Problems and Further Research 2 / 32

3 Introducton European Unversty System European Unversty System (Bologna Accords) 3 years DOCTORAL STUDIES 2 years MASTER 3 years BACHELOR SECONDARY EDUCATION 3 / 32

4 Introducton Study Plans for Teacher Tranng n Germany General Bachelor Structure for Teacher Tranng educatonal scences 50 credts 1st subject 60 credts 2nd subject 60 credts practcal tranng bachelor thess 10 credts 4 / 32

5 Introducton Study Plans for Teacher Tranng n Germany Man Task The unversty of Kaserslautern wants to provde a study plan to the students: at the begnnng of ther studes durng ther studes f they have already taken some courses f they have preferences on dfferent lectures 5 / 32

6 Introducton Study Plans for Teacher Tranng n Germany Advantages: students don t loose a semester due to plannng mstakes free tme slots can be computed for new lectures Constructon Strateges untl 2008 manual constructon constructon by heurstcs wth a frst verson of StuPlan from 2012 constructon wth the computaton of an nteger program 6 / 32

7 Introducton Study Plans for Teacher Tranng n Germany 6 / 32

8 Introducton Study Plans for Teacher Tranng n Germany 6 / 32

9 Introducton Study Plans for Teacher Tranng n Germany 6 / 32

10 Introducton Study Plans for Teacher Tranng n Germany 6 / 32

11 Introducton Study Plans for Teacher Tranng n Germany Advantages: students don t loose a semester due to plannng mstakes free tme slots can be computed for new lectures Constructon Strateges untl 2008 manual constructon constructon by heurstcs wth a frst verson of StuPlan from 2012 constructon wth the computaton of an nteger program 6 / 32

12 Introducton Study Plans for Teacher Tranng n Germany Lecture data: current semester (WS 11/12) prevous semester (SS 11) assumpton that lecture tmetable does not change n the next semesters Sem 1 WS 11/12 Sem 2 Sem 3 Sem 4 Sem 5 Sem 6 SS 11 WS 11/12 SS 11 WS 11/12 SS 11 7 / 32

13 Introducton Study Plans for Teacher Tranng n Germany 8 / 32

14 Introducton Study Plans for Teacher Tranng n Germany 9 / 32

15 Varables Varables { 1 f course c s planned - course varables c C x c = 0 f course c s not planned { 1 f date d s planned - date varables d D x d = 0 f date d s not planned 10 / 32

16 Varables Course 1 (2 oblgatory lectures), offered n each semester (1...6) has: one course varable x1 c 12 date varables: xc1,s1,1 d, x c1,s1,2 d,..., x c1,s6,1 d, x c1,s6,2 d, where xc,sj,k d s the kth date of lecture n the jth semester Sem 1 Sem 6 Course 1 x d c1,s1,1 Course 1 x d c1,s6,1 8AM Course 6 Course 6 10AM Course 1 x d c1,s1,2 Course 1 x d c1,s6,2 Course 4 Course 4 1PM Course 3 Course 3 Course 6 Course 6 3PM Course 4 Course 4 Course 6 Course 6 11 / 32

17 Constrants Condton 1: Relaton of course and date varables courses c C N Obl (c) x c = d D C(d)=c where N Obl s the number of oblgatory lectures Example: Course 1 x d 2x1 c = x c1,s1,1 d + x c1,s1,2 d + xc1,s2,1 d }{{} + x c1,s2,2 d xc1,s6,1 d }{{} + x c1,s6,2 d }{{} 1st semester 2nd semester 6th semester 12 / 32

18 Constrants Condton 2: Coherent dates n the same semester courses c wth at least 2 oblgatory lectures dates d 1 of course c dates d 2 of course c and Sem(d 1 ) = Sem(d 2 ), x d1 = x d2 Example: Course 1 xc1,s1,1 d = xc1,s1,2 d 1st sem. xc1,s2,1 d = xc1,s2,2 d 2nd sem. xc1,s3,1 d = xc1,s3,2 d 3rd sem.. 8AM 10AM Sem 1 Course 1 Course 6 Course 1 Course 4 x d c1,s1,1 x d c1,s1,2 Sem 2 Course 1 Course 6 Course 1 Course 4 x d c1,s2,1 x d c1,s2,2 13 / 32

19 Constrants Condton 3: Mnmal module credts modules m c C Module(c)=m Example: Module 1 Credts(c) x c MnmalCredts(m) 4x c 1 + 2x c 2 }{{} +4x c 3 + 4x c 4 + 2x c 5 + 4x c 6 10 wth 2 credt ponts 14 / 32

20 Constrants Condton 4: Mnmal 1 course n each submodule submodules sm x c 1 c C SM(c)=sm Example: Submodule 1 x c 1 + x c 2 + x c / 32

21 Constrants Condton 5: Course precedence course requrements ( c pre, c after ) CPre dates d wth C( d) = c after N Obl (c pre ) x d d D C(d)=c pre Sem(d)<Sem(( d)) x d 16 / 32

22 Constrants Example: course 5 after course 1 2xc5,S1,1 d 0 2xc5,S2,1 d xc1,s1,1 d + x c1,s1,2 d 2x d c5,s3,1 }{{} course 5 n 3rd semester xc1,s1,1 d + x c1,s1,2 d + x c1,s2,1 d + x c1,s2,2 d }{{}. course 1 n 2nd semester 8AM 10AM 1PM Sem 1 Course 1 Course 6 Course 1 Course 4 x d c1,s1,1 x d c1,s1,2 Sem 2 Course 1 Course 6 Course 1 Course 4 Course 3 Course 3 Course 5 xc5,s1,1 d Course 5 { }} { Date of course 5 n 1st semester x d c1,s2,1 x d c1,s2,2 x d c5,s2,1 Sem 3 Course 1 Course 6 Course 1 Course 4 Course 3 Course 5 x d c5,s3,1 17 / 32

23 Constrants Condton 6: Submodule precedence submodule requrements ( sm pre dates d wth C( d) = c after x d d D SM(d)=sm pre Sem(d)<Sem( d) x d, c after ) SMPre 18 / 32

24 Constrants Condton 7: Module precedence module requrements ( m pre dates d wth C( d) = c after MnCr(m pre ) x d, c after ) MPre d D M(d)=m pre Sem(d)<Sem( d) Cr(d) x d 19 / 32

25 Constrants Condton 8: No overlappngs of dates semesters s tme blocks b = 1,..., 49 x d 1 d D,Sem(d)=s B S (d) b B E (d) Sem 1 Example: 1st semester, 8AM x d c1,s1,1 }{{} course 1 + xc2,s1,1 d }{{} course 2 + xc6,s1,1 d 1 }{{} course 6 8AM Course 1 Course 6 x d c1,s1,1 xc2,s1,1 d xc6,s1,1 d 20 / 32

26 Constrants Condton 9: Course accompanment accompanyng courses ( c, c Accom ) CAccom semesters s N Obl (c Accom ) d D C(d)=c Sem(d)=s x d N Obl (c ) Example: Course 5 accompaned by course 4 x d d D C(d)=c Accom Sem(d)=s 1st semester : 2xc5,S1,1 d 1xc4,S1,1 d + 1x c4,s1,2 d 2nd semester : 2xc5,S2,1 d 1xc4,S2,1 d + 1x c4,s2,2 d.e. c5 c4. }{{} course 5 } {{. } course 4 21 / 32

27 Constrants Condton 10: Dstrbuton of lectures semesters s S d D Sem(d)=s Example: 1st semester Cr(d) x d MnCr(m) m M S 8AM 10AM 1PM 3PM 2xc1,S1,1 d + 2x c1,s1,2 d xc6,S1,1 d }{{} + 2x c6,s1,2 d }{{} course 1 course 6 Sem 1 Course 1 Course 6 Course 1 Course 4 Course 3 Course 5 Course 4 Course 6 x d c1,s1,1 x d c2,s1,1 x d c6,s1,1 x d c1,s1,2 x d c2,s1,2 x d c4,s1,1 x d c2,s1,3 x d c3,s1,1 x d c5,s1,1 x d c2,s1,4 x d c4,s1,2 x d c6,s1,2 22 / 32

28 Constrants Intal Integer Program mn 0x d + 0x c s.t. condtons 1-10 are fulflled x c, x d {0, 1} 23 / 32

29 IP Relaxaton Relaxaton of constrants (e.g. constrant 3) modules m c C Module(c)=m Credts(c) x c MnmalCredts(m) 3 (m) 24 / 32

30 IP Relaxaton Objectve functon mnmze all relaxaton terms 3,..., 10 multobjectve scalarzaton method 25 / 32

31 IP Relaxaton mn m M (sm pre (sm pre (m pre s.t. 3 (m) +,c after ) SM Pre,c after ) SM Pre,c after ) M Pre 49 s S b=1 4 (sm) sm SM 8 (s, b) (c,c Accom ) C Accom s S semesters 10 (s) d D C(d)=c pre d D C(d)=c after d D C(d)=c after 5 (c pre, d) 6 (sm pre, d) 7 (m pre, d) 9 (c, c Accom, s) relaxed condtons 1-10 are fulflled x c, x d {0, 1} 26 / 32

32 Computatonal Results Gven data: ± 150 bnary course varables ± 1200 bnary date varables ± 1400 constrants ± 1300 nteger constrant relaxaton varables ± non zero coeffcents 27 / 32

33 Computatonal Results CPLEX computatonal results: average computaton tme 3sec ±200 dual smplex operatons (wthout presolvng) ±100 dual smplex operatons (wth presolvng) 28 / 32

34 Current Problems and Further Research Problems the tmetable nput entered by the dfferent departments s often wrong no regstraton to the lectures number of students n each lecture s NOT known 29 / 32

35 Current Problems and Further Research Further research fnd out why CPLEX s so fast multcrtera approach parametrc nteger approach (mn cx s.t. Ax b + α) 30 / 32

36 Current Problems and Further Research Overcredts Overlappngs 31 / 32

37 Thank you for your attenton!