Abstract
The scheduling and routing of small planes for fly-in safaris is a challenging planning problem. Given a fleet of planes and a set of flight requests with bounds on the earliest departure and latest arrival times, the planes must be scheduled and routed so that all demands are satisfied. Capacity restrictions on the load and fuel also must be satisfied. Moreover the refueling of the planes, which can only be done in certain locations, must be scheduled. We present a mixed-integer linear programming based formulation for this problem. For its solution we develop a primal heuristic based on randomized local search. We try to enhance the local search by using exact methods to solve subproblems that only involve a small number of planes. Using a branch-and-cut solver, the MILP formulation can be solved to proven optimality only for small instances. To achieve better dual bounds we present a set partitioning based formulation, where new columns are generated on demand by heuristics and exact methods. We also present a new formulation where the time windows are relaxed, and later reintroduced by incumbent branching. Numerical results on real-world instances show that this time-free approach gives the best results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ascheuer, N., Fischetti, M., Grötschel, M.: A polyhedral study of the asymmetric traveling salesman problem with time windows. Networks 36(2), 69–79 (2000)
Ascheuer, N., Fischetti, M., Grötschel, M.: Solving the asymmetric travelling salesman problem with time windows by branch-and-cut. Math. Program., Ser. A 90(3), 475–506 (2001)
Barnhart, C., Shen, S.: Logistics service network design for time-critical delivery. Lect. Notes Comput. Sci. 3616, 86–105 (2005)
Barnhart, C., Johnson, E., Nemhauser, G., Savelsbergh, M., Vance, P.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998)
Borndörfer, R., Grötschel, M., Pfetsch, M.: A column-generation approach to line planning in public transport. Transp. Sci. 41(1), 123–132 (2007)
Cornuejols, G.: Valid inequalities for mixed integer linear programs. Math. Program., Ser. B 112, 3–44 (2008)
Dantzig, G., Fulkerson, D., Johnson, S.: Solution of a large scale traveling salesman problem. Oper. Res. 2(4), 393–410 (1954)
Erera, A., Hewitt, M., Savelsbergh, M., Zhang, Y.: Improved load plan design through integer programming based local search. Technical report 3357, Optimization Online (2012)
Espinoza, D., Garcia, R., Goycoolea, M., Nemhauser, G., Savelsbergh, M.: Per-seat, on-demand air transportation part I: problem description and an integer multicommodity flow model. Transp. Sci. 42(3), 263–278 (2008)
Espinoza, D., Garcia, R., Goycoolea, M., Nemhauser, G., Savelsbergh, M.: Per-seat, on-demand air transportation part II: local search. Transp. Sci. 42(3), 279–291 (2008)
Fischer, F., Helmberg, C.: Dynamic graph generation for large scale operational train timetabling. Technical report 2011-10, Fakultät für Mathematik, Technische Universität Chemnitz (2011)
Ford, L., Fulkerson, D.: Constructing maximal dynamic flows from static flows. Oper. Res. 6(4), 419–433 (1958)
Fügenschuh, A., Homfeld, H., Schülldorf, H.: Single car routing in rail freight transport. In: Barnhart, C., Clausen, U., Lauther, U., Möhring, R. (eds.) Dagstuhl Seminar Proceedings 09261 (2009)
Gavish, B., Graves, S.: The traveling salesman problem and related problems. Technical report OR 078-78, Operations Research Center, Massachusetts Institute of Technology (1978)
Grötschel, M.: Der Satz von Frobenius für konvergente Potenzreihen. Master’s thesis, Ruhr-Universität Bochum (1973)
Grötschel, M.: Polyedrische Charakterisierungen Kombinatorischer Optimierungsprobleme. Mathematical Systems in Economics, vol. 36. Verlag Anton Hain, Meisenheim am Glan (1977)
Grötschel, M.: On the symmetric travelling salesman problem: solution of a 120-city problem. Math. Program. Stud. 12, 61–77 (1980)
Grötschel, M., Holland, O.: Solution of large-scale symmetric travelling salesman problems. Math. Program., Ser. A 51(2), 141–202 (1991)
Helmberg, C., Röhl, S.: A case study of joint online truck scheduling and inventory management for multiple warehouses. Oper. Res. 55(4), 733–752 (2007)
Kennington, J., Nicholson, C.: The uncapacitated time-space fixed-charge network flow problem: an empirical investigation of procedures for arc capacity assignment. ORSA J. Comput. 22(2), 326–337 (2010)
Kotnyek, B.: An annotated overview of dynamic network flows. Technical report 4936, Institut National de Recherche en Informatique et en Automatique (INRIA) (2003)
Maffioli, F., Sciomachen, A.: A mixed-integer model for solving ordering problems with side constraints. Ann. Oper. Res. 69, 277–297 (1997)
Marchand, H., Wolsey, L.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49(3), 363–371 (2001)
Marchand, H., Martin, A., Weismantel, R., Wolsey, L.: Cutting planes in integer and mixed integer programming. Discrete Appl. Math. 123, 397–446 (2002)
Nemhauser, G.: Column generation for linear and integer programming. In: Grötschel, M. (ed.) Optimization Stories, pp. 65–73. Deutschen Mathematiker Vereinigung, Bielefeld (2012)
Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)
UN World Tourism Organization: Tourism facts & figures (2012). www.unwto.org/facts/. Accessed 18 Sep 2012
van Eijl, C.: A polyhedral approach to the delivery man problem. Technical report 95-19, Department of Mathematics and Computer Science, Eindhoven University of Technology (1995). alexandria.tue.nl/repository/books/440440.pdf. Accessed 31 Jan 2013
Wieberneit, N.: Service network design for freight transportation: a review. OR Spektrum 30, 77–112 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fügenschuh, A., Nemhauser, G., Zeng, Y. (2013). Scheduling and Routing of Fly-in Safari Planes Using a Flow-over-Flow Model. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-38189-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38188-1
Online ISBN: 978-3-642-38189-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)