Robust flows with losses and improvability in evacuation planning
Tóm tắt
We consider a network flow problem, where the outgoing flow is reduced by a certain percentage in each node. Given a maximum amount of flow that can leave the source node, the aim is to find a solution that maximizes the amount of flow which arrives at the sink. Starting from this basic model, we include two new, additional aspects: On the one hand, we are able to reduce the loss at some of the nodes; on the other hand, the exact loss values are not known, but may come from a discrete uncertainty set of exponential size. Applications for problems of this type can be found in evacuation planning, where one would like to improve the safety of nodes such that the number of evacuees reaching safety is maximized. We formulate the resulting robust flow problem with losses and improvability as a two-stage mixed-integer program with uncertain recourse for finitely many scenarios and present an iterative scenario-generation procedure that avoids the inclusion of all scenarios from the beginning as well as several heuristic solution methods. In a computational study using both randomly generated instances and realistic data based on the city of Nice, France, we compare our solution algorithms.
Tài liệu tham khảo
Aissi H, Bazgan C, Vanderpooten D (2009) Minmax and minmax regret versions of combinatorial optimization problems: a survey. Eur J Oper Res 197(2):427–438
Agra A, Christiansen M, Figueiredo R, Hvattum LM, Poss M, Requejo C (2013) The robust vehicle routing problem with time windows. Comput Oper Res 40(3):856–866
Altay N, Green WG III (2006) OR/MS research in disaster operations management. Eur J Oper Res 175(1):475–493
Ayoub J, Poss M (2015) Decomposition for adjustable robust linear optimization subject to uncertainty polytope. http://www.optimization-online.org/DB_HTML/2015/11/5207.html
Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376
Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton
Bertsekas DP, Tsitsiklis JN, Wu C (1997) Rollout algorithms for combinatorial optimization. J Heuristics 3(3):245–262
Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program Ser B 98:2003
Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53
Billionnet A, Costa M-C, Poirion P-L (2014) 2-Stage robust MILP with continuous recourse variables. Discrete Appl Math 170:21–32
Büsing C, Koster AMCA, Kutschka M (2011) Recoverable robust knapsacks: \(\Gamma \)-scenarios. In: Network optimization. Springer, New York, pp 583–588
Campbell AM, Lowe TJ, Zhang L (2006) Upgrading arcs to minimize the maximum travel time in a network. Networks 47(2):72–80
Chalmet LG, Francis RL, Saunders PB (1982) Network models for building evacuation. Fire Technol 18(1):90–113
Choi W, Hamacher HW, Tufekci S (1988) Modeling of building evacuation problems by network flows with side constraints. Eur J Oper Res 35(1):98–110
Demgensky I, Noltemeier H, Wirth H-C (2004) Optimizing cost flows by edge cost and capacity upgrade. J Discrete Algorithms 2(4):407 – 423. The 26th international workshop on graph-theoretic concepts in computer science (WG 2000)
Dilkina B, Lai KJ, Gomes CP (2011) Upgrading shortest paths in networks. In: Achterberg T, Beck JC (eds) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems, vol 6697., Lecture notes in computer scienceSpringer, Berlin, pp 76–91
Gabrel V, Lacroix M, Murat C, Remli N (2014) Robust location transportation problems under uncertain demands. Discrete Appl Math 164 Part 1(0):100–111
Goerigk M, Schöbel A (2015) Algorithm engineering in robust optimization. In: LNCS state-of-the-art surveys. Springer, New York
Hamacher HW, Tjandra SA (2001) Mathematical modeling of evacuation problems: a state of the art. In: Schreckenberg M, Sharma SD (eds) Pedestrian and evacuation dynamics 1964:227–266
Hamacher HW, Heller S, Klein W, Köster G, Ruzika S (2011) A sandwich approach for evacuation time bounds. In: Peacock RD, Kuligowski ED, Averill JD (eds) Pedestrian and evacuation dynamics. Springer, New York, pp 503–513
Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, New York
Krumke SO, Marathe MV, Noltemeier H, Ravi R, Ravi SS (1998) Network improvement problems. Network design: connectivity and facilities location. AMSDIMACS Vol Ser Discrete Math Theor Comput Sci 40:247–268
Lämmel G, Klüpfel H, Nagel K (2011) Risk minimizing evacuation strategies under uncertainty. In: Peacock RD, Kuligowski ED, Averill JD (eds) Pedestrian and evacuation dynamics. Springer, New York, pp 287–296
Lemoine A et al (2014) Pligurian earthquake: seismic and tsunami scenario modeling, from hazard to risk assessment towards evacuations planning. In: Proceedings of the second European conference on earthquake engineering and seismology
Liebchen C, Lübbecke M, Möhring RH, Stiller S (2009) The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja RK, Möhring RH, Zaroliagis CD (eds) Robust and online large-scale optimization. Lecture note on computer science, vol 5868. Springer, New York, pp 1–27
Lin Y, Mouratidis K (2013) Best upgrade plans for large road networks. In: Nascimento MA, Sellis T, Cheng R, Sander J, Zheng Y, Kriegel H-P, Renz M, Sengstock C (eds) Advances in spatial and temporal databases, vol 8098., Lecture notes in computer scienceSpringer, Berlin, pp 223–240
Ndiaye IA, Neron E, Linot A, Monmarche N, Goerigk M (2014) A new model for macroscopic pedestrian evacuation planning with safety and duration criteria. Transp Res Proc 2(0):486–494. The conference on pedestrian and evacuation dynamics 2014 (PED 2014), 22–24 October 2014, Delft
Oldham JD (2001) Combinatorial approximation algorithms for generalized flow problems. J Algorithms 38(1):135–169
Opasanon S, Miller-Hooks E (2009) The safest escape problem. J Oper Res Soc 60:1749–1758
Ordez F, Zhao J (2007) Robust capacity expansion of network flows. Networks 50(2):136–145
Radzik T (1998) Faster algorithms for the generalized network flow problem. Math Oper Res 23(1):69–100
Schwarz S, Krumke SO (1998) On budget-constrained flow improvement. Inf Process Lett 66(6):291–297
Wayne KD (1999) Generalized maximum flow algorithms. PhD Thesis. Cornell University, New York
Xie C, Turnquist MA (2011) Lane-based evacuation network optimization: an integrated lagrangian relaxation and Tabu search approach. Transp Res Part C Emerg Technol 19(1):40–63
Xie C, Lin D-Y, Waller ST (2010) A dynamic evacuation network optimization problem with lane reversal and crossing elimination strategies. Transp Res Part E Logist Transp Rev 46(3):295–316
Yamada T (1996) A network flow approach to a city emergency evacuation planning. Int J Syst Sci 27(10):931–936
Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper Res Lett 41(5):457–461