規則性排程(Periodic timetabling)問題之研究，國外已行之有年，唯國內尚無相關之研究。且許多歐洲國家及日本的軌道系統，均採用規則性之班表。所以本研究藉由建立一規則性班表排程之數學模式而希望能夠對於規則性班表排程問題進行一深入之研究，以探討其特性，及其優、缺點。而後以高鐵為例子進行模式的測試與驗證，以及測試傳統之數學規劃法是否能夠有效率的求解本研究所建立之模式。
規則性排程模式與一般傳統的排程模式差別在於，規則性排程模式在求解上僅需求解一個週期內之排程問題，而傳統之排程問題則是必須求解一整天之排程，在規模上規則性排程問題遠小於傳統之排程問題。而另外則是因為規則性排程問題比起傳統排程問題要多了週期這個特性，因此在模式建立上，必須加入許多週期性之變數與限制式，所以在求解上，規則性排程問題之複雜度要比傳統之排程問題複雜的多。
本研究所建立之模式為一混合整數規劃之數學模式（Mixed Integer Problem, MIP），其假設前提是一靜態排班，單方向複線運轉之軌道系統。其中關於週期性之整數變數超過總變數數量的一半。以高鐵為例子進行模式驗證與數學規劃法求解測試，高鐵系統單一方向下行線上十一個車站，每小時發八班車的模式規模來看，程式僅需約100秒左右即可求出最佳解。結果顯示傳統之數學規劃法仍能有效率的求解規則性排程之模式。而另外在本模式之假設前提下，高鐵系統一個小時之系統容輛為10部車。

Periodic timetabling problem has already been researched several years abroad, but there is no relative research at home. There are many countries in Europe and Japan who adopt periodic timetable as their railway timetable. This research studies about the properties of periodic timetabling problem by trying to build a periodic railway timetabling model. And then take Taiwan High Speed Rail system as an example to test the model we built. Finally, test the efficiency of mathematical programming method for solving the model.
There are some differences between periodic timetabling problem and traditional timetabling problem. One of them is that the scale of periodic timetabling problem is much smaller than traditional timetabling problem because the scale of periodic timetabling problem is only about one period, like one hour, but the scale of traditional timetabling problem is about one day. The other difference between them is that periodic timetabling problem is much more complex than traditional timetabling problem because the former need lots of integer variables to describe the property of period.
The model is a mixed integer problem, and it hypothesize that it’s a static timetabling, and only consider about one-way track in multiple tracks system. We take Taiwan High Speed Rail system as the example and taking a hour as a period. We find that solving the model with scale of 8 trains and crossing 11 stations by using mathematical operation method only need about 100 seconds to find the optimal solution. This tells that the mathematical programming method still can solve the model efficiently. Finally, we test the capacity of Taiwan High Speed Rail system. The result shows that the capacity of one hour is about 10 trains.

參考文獻
1. 交通部運輸研究所，台鐵車輛排程最適化之研究，交通部運輸研究所，民94
2. 周永暉，特殊尖峰需求下鐵路列車排程規劃之最佳化模式，國立交通大學交通運輸研究所博士論文，民89
3. 陳立文，鐵路列車排程與排點模式，國立成功大學土木工程研究所碩士論文，民90
4. 黃士哲，高速鐵路列車衝突解決與績效評估之研究，國立成功大學交通管理科學研究所碩士論文，民90
5. 謝汶進，鐵路列車排點模式之研究，國立成功大學交通管理科學研究所碩士論文，民83
6. 鍾志成，軌道運輸系統營運模式現況評估，中興工程顧問社，民94
7. A. Lova, P. Tormos, F. Barber, L. Ingolotti, M.A. Salido, and M. Abril, “Intelligent Train Scheduling on a High-Loaded Railway Network”, Universidad Politecnica de Valencia, Spain {ptormos,alova}@eio.upv.es 2 DSIC, Universidad Politecnica. http://citeseer.ist.psu.edu/731225.html.
8. A. Schrijver, “Routing and Timetabling by Topological Search”, Documenta Mathematica Extra Volume ICM 1998 III, 1998.
9. C. Liebchen, “Symmetry for Periodic Railway Timetables”, Electronic Notes in Theoretical Computer Science 92 No. 1, 2003.
10. C. Liebchen, and L. Peeters, “Some practical aspects of periodic timetabling”, In P. Chamoni et al, editor, Operations Research 2001, 2001.
11. C. Liebchen, and M. Proksch, and F. H. Wagner, “Performance of Algorithms for Periodic Timetable Optimization”, Combinational Optimization & Graph Algorithms Preprint 021-2004, 2004.
12. E.A.G. (Ello) Weits, “Railway Capacity and Timetable Complexity”, 7th International Workshop on Project Management and Scheduling, Euro, 2000.
13. J. C. Villumsen, “Construction of Timetables Based on Periodic Event Scheduling”, Kongens Lyngby 2006 IMM-Thesis-2006-52, 2006.
14. L. G. Kroon, and L. W. P. Peeters, “A Variable Trip Time Model for Cyclic Railway Timetabling”, Transportation Science c 2003 INFORMS Vol. 37, No. 2, May 2003, pp. 198–212, 2003.
15. L. G. Kroon, R. Dekker and M. J.C.M. Vromans, “Cyclic Railway Timetabling: a Stochastic Optimization Approach”, ERIM REPORT SERIES RESEARCH IN MANAGEMENT ERS-2005-051-LIS, 2005.
16. L. Kroon, D. Wagner, F. Geraets and C. Zaroliagis, “04261 Abstracts Collection Algorithmic Methods for Railway Optimization _ Dagstuhl Seminar _”, Algorithmic Methods for Railway Optimization, 2006.
17. L. Peeters, and L. Kroon, “An optimization approach to railway timetabling”, Fourth Workshop on Models and Algorithms for Planning and Scheduling Problems, Renesse, The Netherlands, 1999.
18. L. W.P. Peeters, “Cyclic railway timetable optimization”, Cyclic railway timetable optimization, 2003.
19. M. A. Odijk, “A constraint generation algorithm for the construction of periodic railway timetable”, Transpn Res.-B. Vol. 30, No.6. pp. 455-464, 1996.
20. M. A. Odijk, “Construction of periodic timetables part II: An application”, Report 94-71 Michiel A. Odijk Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics, 1994.
21. M. A. Odijk, H. Edwin Romeijn, and H. Maaren, “Generation of classes of robust periodic railway timetables”, Computers & Operations Research 33 (2006) 2283–2299, 2005.
22. N. Rangaraj, “An analysis of cyclic timetables for suburban rail services”, ABHIVYAKTI • VOLUME 16 • NO. 1 • JANUARY – MARCH, 2005.
23. R.M.P. Goverde, and M.A. Odijk, “Performance evaluation of network timetables using PETER”, Allan, J., Andersson, E., Brebbia, C.A., Hill, R.J., Sciutto, G., Sone, S., Computers in Railways VIII, WIT Press, Southampton, 2002.
24. P. Serafini, and W. Ukovich, “A mathematicl model for periodic scheduling problem”, SIAM journal on Discrete Mathematics 2:550-581, 1989.
25. A. Schrijver, and A. Steenbeek, “Dienstregelingontwikkeling voor Railned（Timetable construction for Railned）”, Report for Railned, CWI, Center for Mathematics and Informatics, Amsterdam, The Netherlands. In Dutch, 1994.

參考網頁
1. C. Liebchen, R. H. Mhring, “A case study in periodic timetabling. Electronic Notes in Theoretical Computer Science” URL: www.elsevier.nl/locate/entcs, 2002.
2. R. BARTK, On-Line Guide to Constraint Programming. URL:http://kti.ms.mff.cuni.cz/~bartak/constraints/index.html
3. Wikipedia, URL:http://en.wikipedia.org/wiki/Constraint_programming