Robust flows with losses and improvability in evacuation planning

Abstract

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.

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.