Robust flows with losses and improvability in evacuation planning
Introduction
Network flow with losses (and also gains) is a well-known and fruitful object of study, see, e.g., Oldham (2001), Radzik (1998) and Wayne (1999). Applications can typically be found, e.g., in telecommunication networks, electrical networks, exchange markets, machine loading, lot-sizing or the Boolean satisfiability problem.
Also the field of evacuation planning has seen rising interest in the application of operations research models to help the decision maker assessing a critical situation and making the right choices to potentially safe lives, see, e.g., Opasanon and Miller-Hooks (2009) and Lämmel et al. (2011). For general overviews, we refer to Hamacher and Tjandra (2001) and Altay and Green (2006). Also, network flows are a standard modeling technique in the field, see, e.g., Chalmet et al. (1982), Yamada (1996) and Choi et al. (1988).
In this paper, we extend the concept of flows with losses to also include improvability (i.e., the amount of loss can be reduced at a limited number of nodes) as well as robustness (i.e., the exact amount of loss is not known exactly). Both model extensions are motivated by applications in evacuation planning.
As an example, we consider the situation that several evacuees need to leave an endangered region on foot (e.g., after an earthquake or a flooding strikes an urban area). Depending on the path they choose, they face different estimated degrees of dangerousness, which lead to the potential death of evacuees. Such a situation can be captured as a network flow with losses, where the number of evacuees reaching safety is to be maximized (see Ndiaye et al. 2014).
From a short-term perspective, security forces can be used to reduce the risk in the network (e.g., by extinguishing fires with the help of airplanes, or by removing debris) during an evacuation. From a long-term perspective, the structural safety of an endangered area can be improved (e.g., by stabilizing buildings with a relatively high probability of collapse during an earthquake), which will also result in a reduced risk value during an emergency.
Furthermore, as risk values are only an estimation, they are considered as being uncertain. We present a robust optimization approach with network improvability to include both these points.
In the following, we present some further literature in which related aspects are analyzed.
The basic idea of the well-known “Contraflow”-setting is to make better use of the given infrastructure in the case of an evacuation. As an example, if a highway has two lanes entering the endangered area, and two lanes leaving the endangered area, it makes sense to reverse at least one of the two entering lanes to facilitate the outgoing flow. The contraflow problem has been considered by Xie et al. (2010) and Xie and Turnquist (2011) and many others. Similar to our setting, the lane reversal may be interpreted as the distribution of improvements in the network, where the number of such improvements is bounded. However, in our setting, improvements are not on arc capacities, but on vertex safety instead.
There are several papers considering network improvement problems (such as Schwarz and Krumke 1998; Krumke et al. 1998; Demgensky et al. 2000; Ordez and Zhao 2007; Dilkina et al. 2011; Lin and Mouratidis 2013; Campbell et al. 2006). However, the problem of improving vertex safety has not been considered yet.
Regarding the field of robust optimization in general as a means to handle optimization problems affected by uncertainty, we refer to the surveys Kouvelis and Yu (1997), Aissi et al. (2009), Bertsimas and Sim (2004), Bertsimas and Sim (2003), Ben-Tal et al. (2009) and Goerigk and Schöbel (2015). In our setting, we follow a two-stage adjustable robust approach (see Ben-Tal et al. 2004). As the resulting optimization problem is too large to be solved directly, we employ an iterative scenario-generation method (see also Agra et al. 2013; Billionnet et al. 2014; Gabrel et al. 2014; Zeng and Zhao 2013).
Contributions and overview In Sect. 2, we introduce the nominal (i.e., non-uncertain) max flow problem with losses and improvements, which we use to model pedestrian movements during an evacuation. This model is extended to include uncertainty in Sect. 3, where we also present an iterative solution algorithm. As already the nominal problem is NP-hard, we also consider heuristic solution approaches in Sect. 4 and compare these algorithms in a computational study in Sect. 5. Section 6 concludes the paper and points out further research directions.
Section snippets
Flow with losses
Let a directed graph be given. We start with considering the maximum flow problem, where in each vertex , the flow leaving node i is multiplied with a fixed factor . Given a bound on the value of outflow of a source, the flow with losses (FL) problem is to find the largest possible amount of flow entering a sink. The restriction to a single sink and a single source is without loss of generality, as multiple sources and sinks can be collected to a super-source and -sink,
Robust flows
We now extend the FLI problem to include uncertainty in the loss values p. The motivation is to better model that these values can only be estimates and will never reflect the actual risk on a vertex. Instead, we assume to know only an uncertainty set that contains all possible realizations of p.
We have to decide where to put our improvement resources before we know the realization of p. Then, the actual scenario becomes revealed and the evacuees take the best possible route with respect
Scenario generation
Using formulation (33–42) for WC(z), we can evaluate the objective value for a choice of improvements z and also produce a scenario w where this objective value is attained. In the following, this is used as part of a solution algorithm.
We start with any finite scenario set , e.g., or . Solving RFLI() yields some solution for the improvements and an objective value OBJ. Solving WC determines a new scenario given by and an objective value WC. Setting
Experiments
We present two sets of experiments. In the first one, we use randomly generated grid graphs to evaluate the algorithms presented in this paper. In the second experiment, we use realistic data based on the city of Nice, France, to compare our heuristic approaches.
All experiments were conducted on a computer with 96 GB RAM and a 16-core Intel Xeon E5-2670 processor, running at 2.60 GHz with 20 MB cache, and Ubuntu 12.04. Mixed-integer programs were solved using Cplex v. 12.6. and C programs
Conclusion
In this paper we contributed to the current literature on network flow evacuation planning models, by introducing a new evacuation model that includes losses along nodes, improvability, and uncertainty. As the scenario set and thus the model is of exponential size, it cannot be solved directly using a mixed-integer programming solver. We, therefore, developed an algorithm that iteratively increases the problem size by finding the worst-case scenario for the current (partial) solution along with
Acknowledgments
We thank the anonymous referees for their helpful comments that helped shaping this paper.
This work was partially supported within the project DSS_Evac_Logistic, by the Federal Ministry of Education and Research Germany as FKZ 13N12229, and by the French National Research Agency as ANR-11-SECU-002-01 (CSOSG 2011).
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