2003/9 Vol. J86 D I No. 9 GA GA [8] [10] GA GA GA SGA GA SGA2 SA TS GA C1: C2: C3: 1 C4: C5: 692

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1 Comparisons of Genetic Algorithms for Timetabling Problems Hiroaki UEDA, Daisuke OUCHI, Kenichi TAKAHASHI, and Tetsuhiro MIYAHARA GA GA GA GA GA SGA GA SGA2SA TS 6 SGA2 GA GA SA 1. GA [1] [12] GA Faculty of Information Sciences, Hiroshima City University, Hiroshima-shi, Japan Compaq Computer K.K, Utsunomiya-shi, Japan GA GA GA [3] GA [4] D I Vol. J86 D I No. 9 pp

2 2003/9 Vol. J86 D I No. 9 GA GA [8] [10] GA GA GA SGA GA SGA2 SA TS GA C1: C2: C3: 1 C4: C5: 692

3 4 C6: C7: C8: C9: C6 C7 2 C8 C9 4 R1: R2: R3: R4: 9 4 GA GA SGA SGA GA 3. 1 GA 2 GA [8] [10] GA 1 SI ID C i R j 2 GA GA 2 (a) The genotype for class scheduling. (b) The genotype for room allocation. 1 Fig. 1 The genotype. 693

4 2003/9 Vol. J86 D I No. 9 Table 1 1 An example of a timetable before the repair operation. Class Period 1 Period 2 Period 3 Period 4 Period 5 Subject-1 Subject-5 Subject-6 A-1 Staff-1 Staff-5 Staff-5 Subject-2 Subject-7 B-1 Staff-2 Staff-6 and Staff-7 Subject-3 Subject-1 C-1 Staff-3 Staff-1 Subject-1 Subject-4 Subject-2 Subject-8 D-1 Staff-1 Staff-4 Staff-2 Staff-3 2 GA Fig. 2 A flow diagram of TGA. 1 <C i,r j > 2 1 C2 C2 3 SI SI (S 1) S 1 SI (S 2) S 2 C3 S 2 S 1 SI S 1 1 D-1 Subject-1 S 1 Subject-2 D-1 Subject-4 Subject-8 S 2 Subject-2 S 1 SI Subject-1 Subject-4 Subject-2 4 C2 1. n c C nc + c S {p 1,p 2,..., p c} 694

5 2 Table 2 An example of a timetable after the repair operation. Class Period 1 Period 2 Period 3 Period 4 Period 5 Subject-5 Subject-1 Subject-6 A-1 Staff-5 Staff-1 Staff-5 Subject-2 Subject-7 B-1 Staff-2 Staff-6 and Staff-7 Subject-3 Subject-1 C-1 Staff-3 Staff-1 Subject-2 Subject-1 Subject-4 Subject-8 D-1 Staff-2 Staff-1 Staff-4 Staff-3 4. p i S p i p i S S 1 Subject-1 Subject-2 Subject- 1 Subject = = 10 Subject-2 Subject-2 Period1 2 / Subject-1 Period2 Subject-2 Subject-1 Period3 2 5 C i CC(C i) CC(C i) C1-3 C6 C8 R1-3 C i 6 7 R j RC(R j) RC(R j) R j C4 C5 C9 R4 RCC(C i,r j) RCC(C i,r j) CC(C i) RC(R j) 1. CC(C i) N RC(R j) M 2. 1 RCC(C i,r j) 3. (1) (2) CC(C i) (3) fitness(c i) CC(C i) CC(C i) + min(rcc(c i,r j)) (1) j CC(C i) CC(C i) + average(min(rcc(c k,r j))) k j + std(min(rcc(c k,r j))) (2) k j fitness(c i) = max(cc(c k )) CC(C i) (3) k (1) 1 C i (2) C i min() average() std() max() CC(C i)+rc(r j) 0 <C i,r j > 3. 2 GA GA SGA SGA GA 695

6 2003/9 Vol. J86 D I No. 9 (4) Cost(i) i fitness(i) = max(cost(k)) Cost(i) (4) k 3. 3 GA GA GA GA 1. C1-3 C6 C8 R1-3 CC(C i) (3) CC(C i) RC (R j) 1 R j C4 C5 C9 R4 RC (R j) GA SGA SGA2 SGA

7 2. C N N HCU

8 2003/9 Vol. J86 D I No. 9 3 Table 3Timetabling problems. HCU 97 NC NL NCL NR NCR Table 4 Penalty values for room allocation NC the Number of Classes NL the Number of Lectures 0.9 NCL the Number of Complex Lectures 0.4 NR the Number of Rooms 12.5 NCR the Number of Constraints or Requests 3 NR C4 1 C Subject-2 C (2 1) 2= C Sun Ultra MHz % 1% 500 GA 10 HCU 97 NC

9 5 Table 5 Experimental results. GA SGA GA SGA2 SA TS HCU (92.0s) 1.9 (187s) 1.5 (66.0s) 0.3(25.0s) 0.4 (20.2s) 14.0 (80.2s) NC 10.6 (10.4h) 15.4 (11.2h) 0.0 (2.4h) 0.0 (1.1h) 36.0 (2.7h) NL 0.0 (8.5s) 0.0 (27.6s) 0.0 (8.4s) 0.0 (3.7s) 0.0 (25.8s) 3.0 (289s) NCL 0.0 (13.5s) 0.0 (4.1s) 0.0 (26.8s) 0.0 (2.3s) 0.02 (13.0s) 7.5 (132s) NR 0.5 (235s) 2.5 (298s) 3.6 (148s) 0.0 (5.0s) 0.0 (11.6s) 4.00 (80.5s) NCR 0.18 (613s) 0.41 (611s) 0.50 (597s) 0.19 (503s) 0.10 (239s) 1.64 (352s) SA TS SA adapitve cooling reheating [11] 30 NC 5 GA 4 SA SI TS 2 + /2 30 TS GA 4 HCU 97 GA 4 TS TS SA HCU 97 0 GA 37 SGA 17 GA 12 SGA2 46 SA 45 TS 0 SGA2 SA GA 70% SGA2 SA 4 GA SGA TS 4 TS TS SGA2 GA NC GA SGA2 NC GA GA SGA SA 0 TS TS NC 5 NL TS NL GA GA SGA2 GA SGA SA NCL TS NCL / 699

10 2003/9 Vol. J86 D I No. 9 SGA SGA2 NR SGA2 SA GA C4 GA GA 50% SGA2 SA NCR 0 GA SGA2 SA NCR 0 HCU 97 0 GA SGA2 SA 70% 0 0 NCR 4. 4 GA SGA GA SGA2 SA TS SGA2 C10: C11: SGA2 5 SGA2 C10 C11 SGA2 5 SA GA GA SGA2 GA GA 700

11 [1] M. Mitchell, An Introduction to Genetic Algorithm, The MIT Press, Cambridge, [2] M.W. Carter and G. Laporte, Recent developments in practical course timetabling, ed. E. Burke and M. Carter, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1408, pp.3 19, Springer-Verlag, Berlin, [3] (D-I), vol.j82-d-i, no.6, pp , June [4] D.C. Rich, A smart genetic algorithm for university timetabling, ed. E. Burke and R. Ross, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1153, pp , Springer-Verlag, Berlin, [5] W. Erben and J. Keppler, A genetic algorithm solving a weekly course-timetabling problem, ed. E. Burke and P. Ross, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1153, pp , Springer- Verlag, Berlin, [6] E. Burke and J-P. Newall, Multi-stage evolutionary algorithm for timetable problem, IEEE Trans. Evol. Comput., pp , [7] B. Peachter, R.C. Rankin, and A. Cumming, Improving a lecture timetabling system for universitywide use, ed. E. Burke and M. Carter, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1408, pp , Springer-Verlag, Berlin, [8] AI-99-53, [9] H. Ueda, D. Ouchi, K. Takahashi, and T. Miyahara, Co-evolving timeslot/room assignment genetic algorithm technique for university timetabling, ed. E. Burke and W. Erben, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.2079, pp.48 63, Springer- Verlag, Berlin, [10] 11 pp , [11] M.A.S. Elmohamed, P. Coddington, and G. Fox, A comparison of annealing techniques for academic course scheduling, ed. E. Burke and M. Carter, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1408, pp , Springer-Verlag, Berlin, [12] K.A. Dowsland, Off-the-peg or made-to-measure? Timetabling and scheduling with SA and TS, ed. E. Burke and M. Carter, The practice and theory of automated timetabling: Selected papers. Lecture Notes in Computer Science, vol.1408, pp.37 52, Springer- Verlag, Berlin, IEEE IEEE

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