Optimizing dynamic investment decisions for railway systems protection

Highlights

• A bilevel model for optimizing security investments in railway systems is proposed.

• Two different decomposition approaches are used to solve the model.

• The dynamic security investment model is tested on the Kent, UK, railway network.

• Significant system flow loss reductions can be achieved with modest investments.

• The robustness of the approach to different disruption scenarios is analyzed.

Abstract

Past and recent events have shown that railway infrastructure systems are particularly vulnerable to natural catastrophes, unintentional accidents and terrorist attacks. Protection investments are instrumental in reducing economic losses and preserving public safety. A systematic approach to plan security investments is paramount to guarantee that limited protection resources are utilized in the most efficient manner. In this paper, we present an optimization model to identify the railway assets which should be protected to minimize the impact of worst case disruptions on passenger flows. We consider a dynamic investment problem where protection resources become available over a planning horizon. The problem is formulated as a bilevel mixed-integer model and solved using two different decomposition approaches. Random instances of different sizes are generated to compare the solution algorithms. The model is then tested on the Kent railway network to demonstrate how the results can be used to support efficient protection decisions.

Introduction

Nowadays the social well-being of people highly relies on the well functioning of critical interconnected infrastructures such as transportation, information, telecommunication, and electric power systems. Planning and protecting infrastructure systems is a complex task, especially because of their dimension and interdependence. Even small random disruptions can severely affect the normal functioning of one or more infrastructure. Intelligent attacks or large natural catastrophes can have even more dramatic consequences in terms of both economic and life losses. Examples of such events include the 1995 Paris metro bombing, the 2004 Madrid train bombing, the 2005 London underground suicide attacks, and the 2010 Moscow bombing. Most recently, severe floods hit some western regions of the UK and forced the Network Rail to pay £12.5m for the suppressed services and further £15m to repair the infrastructure (Wintour & Topham, 2014). It is therefore paramount to protect infrastructure systems so that continuity in service provision and safety for the users can be guaranteed, even when disruptions occur.

A critical aspect in planning infrastructure protection is the scarce availability of protection resources. Protecting all the components of an infrastructure system to targeted safety levels may infact be cost prohibitive. For example, the Kent (UK) railway system serves 179 stations and has 1094 miles of tracks. Protecting every station and all the tracks is economically impossible. Another complicating factor in protecting railways is that they are open and easily accessible systems. This renders them highly vulnerable to all kinds of disruption and requires careful identification of suitable protection measures. These may include the structural reinforcement of vulnerable parts (tunnels and bridges), video surveillance of critical areas (crosses, stations) and the use of a wide range of sensors to detect intrusions and obstacles on tracks. Since resources are limited, it is important to identify and protect the most critical assets of the infrastructure.

In recent years, several mathematical models have been developed to identify systems’ vulnerabilities and plan protection strategies for critical infrastructures. Predominantly, interdiction and protection of infrastructure systems have been modeled using multi-level optimization. Multi-level optimization models represent an effective tool to “model a complete infrastructure system and its value to society, including how losses of the system’s assets reduce that value, or how improvements in the system mitigate lost value” (Brown, Carlyle, Salmeron, & Wood, 2006). These models are also referred to as defender–attacker models since they emulate the game between two actors with opposite aims. The defender’s aim is to distribute limited defence resources so as to minimize the effects of a worst case disruption. On the other hand, the attacker’s aim is to choose the attack plan which minimizes the system’s value (or maximize the system’s cost). The attacker is an intelligent actor who has perfect knowledge of the system and is always able to inflict the maximum damage. In other words, he is a proxy to model worse-case disruptions. Clearly this kind of models are very useful to simulate terrorist attacks and intentional disruptions. Nonetheless, given the criticality of infrastructure systems, protection efforts are often guided by risk-averse decision making criteria, thus making these models extremely valuable also for problems involving natural catastrophes.

A third actor, referred to as the system user, is often used in multi-level models to evaluate the system’s value after protection and interdiction. Bilevel attacker-user models are typically used to identify the vulnerabilities of a system, by highlighting the outcomes of a worst-case interdiction. Trilevel defender–attacker-user models are typically used to identify the system’s components that should be hardened or protected. Sometimes the models mirroring the actions of the attacker and the system user can be collapsed into a single model (Church, Scaparra, Middleton, 2004, Losada, Scaparra, Church, Daskin, 2012a), so that a single level model can represent an interdiction problem whereas a bilevel model can represent a protection problem.

The main contribution of this paper is to study a dynamic network protection problem. The model we present is quite general and, albeit designed for railway infrastructures, can be applied in other contexts as well. The model aims at distributing protection resources among the assets of a railway system so as to maximize its survivability after a worst case disruption. Generally, survivability can be described as “the capability of a system to fulfill its mission, in a timely manner, in the presence of attacks, failures, or accidents” (Ellison, Fisher, Linger, Lipson, & Longstaff, 1997). According to the specific context, network survivability can be measured using different metrics. For instance, for shortest-path based problems, the length of a path is the key measure to assess whether a network is vulnerable and reliable. To evaluate survivability, we use the same metric introduced by Myung and Kim (2004), Murray, Matisziw, and Grubesic (2007), Matisziw and Murray (2009) and Scaparra, Starita, Sterle, Setola, Sforza, Vittorini, and Pragliola (2015). In these works, network survivability is measured in terms of lost or unserved system flow. In our model, the flow between an origin and a destination node is considered lost if after a disruption affecting some network components, the two nodes are no longer connected or they are connected but the post-disruption service is significantly deteriorated (i.e., alternative routes are too long from a user’s perspective). In this case, in fact, rail network users may abandon the trip or resort to another mode of transport.

An important issue that should be taken into account when modelling protection efforts is that protection resources usually become available at different times. Our model addresses this issue by including a temporal component whereby the available budget for protection is spread over a planning horizon. This choice renders the model more applicable to real situations. In fact, public expenditures to protect and modernize critical infrastructures are usually set in spending reviews that cover a number of years. For instance, the last UK spending review (HM Treasury, 2013) allocated £100bn for the modernization of the energy and transportation sectors. This budget is spread over a five-year period (2015–2020). Similarly, after the 2013–2014 floods in the UK, £130m were allocated by the government to repair flood defences. Of the whole budget, £30m were made available in 2014, the rest in 2015 (Carrington & Weaver, 2014). In addition, the UK Department for Environment, Food and Rural Affairs (DEFRA) set out a six-year programme of capital investment to improve flood defences up to 2021, of £2.3bn. Fixed capital settlements were allocated for each year, although flexibility to move funds between years was allowed for effective delivery (DEFRA, 2015). These examples demonstrate that funds availability is often time-related. Consequently, prioritizing expenditures over time is key to the development of long-term, effective strategies for improving infrastructure’s security and resiliency. To respond to the practical planning needs of railway stakeholders and operators, we therefore propose a protection model that optimizes the allocation of scarce protection resources over time.

Our model builds upon and extends the static protection model proposed by Scaparra et al. (2015) by considering dynamic investments. The model has a bilevel structure where the aim of the upper level is to find the best allocation of protection resources over a planning horizon so as to minimize the amount of disrupted flow. The lower level is used to evaluate worse case losses in each time period in response to a given protection plan. The resulting multi-period bilevel model is significantly more difficult to solve than its static counterpart. To find optimal solutions to problem instances of realistic size, we propose two decomposition approaches tailored to the dynamic structure of the model and test them on a set of new, randomly generated instances. Given that the multi-period structure of the model can lead to solve the same lower level program multiple times in our iterative approaches, we streamline the algorithms by using efficient data structures, thus avoiding recomputing solutions already found. The use of this expedient proved to be extremely efficient and on some preliminary tests reduced the overall computing time of the decomposition methods by as much as 80 percent. Finally, some practical insights, including an analysis of dynamic investments, are discussed for a real network representing the railway infrastructure of Kent (UK).

The remainder of this paper is organized as follows. Section 2 provides a review of the literature related to this work. In Section 3, the bilevel formulation of DNP is introduced. Section 4 provides a description of the decomposition algorithms. In Section 5, we report computational results on two sets of random problems, while the results on the Kent case study are analyzed in Section 6. Some conclusive remarks are discussed in Section 7.