Một phương pháp heuris cho việc giải quyết các bài toán đa chỉ số phân rã với giá trị nguyên

Automation and Remote Control - Tập 75 - Trang 1357-1368 - 2014
L. G. Afraimovich1
1Nizhni Novgorod State University, Nizhni Novgorod, Russia

Tóm tắt

Chúng tôi xem xét các bài toán đa chỉ số có giá trị nguyên thuộc loại vận tải, mà có tính NP-kho. Chúng tôi phân biệt một tiểu lớp các bài toán đa chỉ số có thể giải theo polynom, đó là các bài toán đa chỉ số có cấu trúc phân rã. Chúng tôi xây dựng một sơ đồ chung cho một phương pháp heuris nhằm giải quyết một số bài toán đa chỉ số phân rã NP-kho tương tự. Đối với một phiên bản thực hiện của sơ đồ này, chúng tôi ước lượng độ lệch của nó so với tối ưu. Chúng tôi minh họa các kết quả của mình bằng ví dụ thiết kế thời gian biểu lớp học.

Từ khóa

#bài toán đa chỉ số #phương pháp heuris #bài toán NP-kho #cấu trúc phân rã #vận tải

Tài liệu tham khảo

Afraimovich, L.G. and Prilutskii, M.Kh., Multiindex Optimal Production Planning Problems, Autom. Remote Control, 2010, vol. 71, no. 10, pp. 2145–2151. Afraimovich, L.G. and Prilutskii, M.Kh., Multiindex Resource Distributions for Hierarchical Systems, Autom. Remote Control, 2006, vol. 67, no. 6, pp. 1007–1016. Prilutskii, M.Kh., Multicriterial Multiindex Problems of Volume-Calendar Planning, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2007, no. 1, pp. 78–82. Prilutskii, M.Kh., Multicriterial Distribution of a Homogeneous Resource in Hierarchical Systems, Autom. Remote Control, 1996, vol. 57, no. 2, part 2, pp. 266–271. Prilutskii, M.Kh. and Vlasov, S.E., Multicriterial Volume Planning Problems. Lexicographic Schemes, Inform. Tekhnol., 2005, no. 7, pp. 61–66. Tanaev, V.S. and Shkurba, V.V., Vvedenie v teoriyu raspisanii (Introduction to Scheduling Theory), Moscow: Nauka, 1975. Tanaev, V.S., Gordon, V.S., and Shafranskii, Ya.M., Teoriya raspisanii. Odnostadiinye sistemy (Scheduling Systems. Single-Stage Systems), Moscow: Nauka, 1984. Tanaev, V.S., Sotskov, Yu.N., and Strusevich, V.A., Teoriya raspisanii. Mnogostadiinye sistemy (Scheduling Systems. Multi-Stage Systems), Moscow: Nauka, 1989. Arbib, C., Pacciarelli, D., and Smriglio, S.A., A Three-Dimensional Matching Model for Perishable Production Scheduling, Discrete Appl. Math., 1999, vol. 92, pp. 1–15. Briskorn, D., Drexl, A., and Spieksma, F.C.R., Round Robin Tournaments and Three Index Assignment, 4OR: A Quarterly J. Oper. Res., 2010, vol. 8, pp. 365–374. Daskalai, S., Birbas, T., and Housos, E., An Integer Programming Formulation for a Case Study in University Timetabling, Eur. J. Oper. Res., 2004, vol. 153, pp. 117–135. Franz, L.S. and Miller, J.L., Scheduling Medical Residents to Rotations: Solving the Large-Scale Multiperiod Staff Assignment Problem, Oper. Res., 1993, vol. 41, no. 2, pp. 269–279. Frieze, A.M. and Yadegar, J., An Algorithm for Solving 3-Dimensional Assignment Problems with Application to Scheduling a Teaching Practice, J. Oper. Res. Soc., 1981, vol. 32, no. 11, pp. 989–995. Gunawan, A., Ng, K.M., and Poh, K.L., Solving the Teacher Assignment-Course Scheduling Problem by a Hybrid Algorithm, Int. J. Comput. Inform. Eng., 2007, vol. 1, no. 2, pp. 136–141. Lim, A., Rodrigues, B., and Zhang, X., Scheduling Sports Competitions at Multiple Venues is Revisited, Eur. J. Oper. Res., 2006, vol. 175, pp. 171–186. Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979. Translated under the title Vychislitel’nye mashiny i trudnoreshaemye zadachi, Moscow: Mir, 1982. Crama, Y. and Spieksma, F.C.R., Approximation Algorithms for Three-Dimensional Assignment Problems with Triangle Inequalities, Eur. J. Oper. Res., 1992, vol. 60, pp. 273–279. Hoffman, A.J. and Kruskal, J.B., Integral Boundary Points of Convex Polyhedra, in Linear Inequalities and Related Systems, Kuhn, H.W. and Tucker, A.W., Eds., Princeton: Princeton Univ. Press, 1956, pp. 223–246. Lenstra, H.W., Jr., Integer Programming with a Fixed Number of Variables, Math. Oper. Res., 1983, vol. 8, no. 4, pp. 538–548. De Loera, J., Hemmecke, R., Onn, S., et al., N-Fold Integer Programming, Discrete Optim., 2008, vol. 5, pp. 231–241. Spieksma, F.C.R., Multi Index Assignment Problems, Complexity, Approximation, Applications, in Nonlinear Assignment Problems: Algorithms and Applications, Pardalos, P.M. and Pitsoulis, L.S., Eds., Dordrecht: Kluwer, 2000, pp. 1–11. Gimadi, E.Kh. and Serdyukov, A.I., Axial Three-Index Assignment and Travelling Salesman Problems: Fast Approximate Algorithms and Their Probabilistic Analysis, Izv. Vyssh. Uchebn. Zaved., Mat., 1999, no. 12, pp. 19–25. Balas, E. and Saltzman, M.J., An Algorithm for the Three-Index Assignment Problem, Oper. Res., 1991, vol. 39, pp. 150–161. Burkard, R.E., Rudolf, R., and Woeginger, G.J., Computational Investigation on 3-Dimensional Axial Assignment Problems, Belgian J. Oper. Res., Stat. Comput. Sci., 1992, vol. 32, pp. 85–98. Gimadi, E.Kh. and Glazkov, Yu.V., On an Asymptotically Exact Algorithm for Solving One Modification of the Three-Index Planar Assignment Problem, Diskret. Anal. Issled. Oper., 2006, vol. 13, no. 1, pp. 10–26. Dichkovskaya, S.A. and Kravtsov, M.K., A Study of Polynomial Algorithms for Solving the Three-Index Planar Problem of Choice, Zh. Vychisl. Mat. Mat. Fiz., 2006, vol. 46, no. 2, pp. 222–228. Sergeev, S.I., New Lower Bounds for the Triplanar Assignment Problem. Use of the Classical Model, Autom. Remote Control, 2008, vol. 69, no. 12, pp. 2039–2060. Magos, D., Tabu Search for the Planar Three-Index Assignment Problem, J. Global Optim., 1996, vol. 8, pp. 35–48. Sigal, I.Kh. and Ivanova, A.P., Vvedenie v prikladnoe diskretnoe programmirovanie. Modeli i vychislitel’nye algoritmy (Introduction to Applied Discrete Programming. Models and Computational Algorithms), Moscow: Fizmalit, 2007. Schrijver, A., Theory of Linear and Integer Programming, New York: Wiley, 1986. Translated under the title Teoriya lineinogo i tselochislennogo programmirovaniya, Moscow: Mir, 1991. Afraimovich, L.G., Multi-Index Transport Problems with Decomposition Structure, Autom. Remote Control, 2012, vol. 73, no. 1, pp. 118–133. Raskin, L.G. and Kirichenko, I.O., Mnogoindeksnye zadachi lineinogo programmirovaniya (Multiindex Linear Programming Problems), Moscow: Radio i Svyaz’, 1982. Orlin, J.B., A Faster Strongly Polynomial Minimum Cost Flow Algorithm, Oper. Res., 1993, vol. 41, no. 2, pp. 338–350. Sleator, D.D. and Tarjan, R.E., A Data Structure for Dynamic Trees, J. Comput. Syst. Sci., 1983, vol. 26, pp. 362–391. Afraimovich, L.G., Cyclic Reducibility of Multiindex Systems of Linear Inequalities of Transportation Type, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2010, no. 4, pp. 83–90. Kanatnikov, A.N. and Krishchenko, A.P., Lineinaya algebra (Linear Algebra), Moscow: Mosk. Gos. Tekh. Univ., 2002.