A comparison of multi-objective spatial dispersion models for managing critical assets in urban areas

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Abstract

A diverse array of spatial optimization models dealing with protection, service, coverage, equity, and risk can potentially aid with the effective placement of critical assets. Protection of assets can be enhanced using the p-dispersion model, which locates facilities to maximize the minimum distance between any two. Dispersion, however, is rarely the only objective for a system of facilities, and the p-dispersion model is known to be difficult to solve. Therefore, this paper analyzes the trade-offs and computational times of four multi-objective models that combine the p-dispersion model with other facility location objectives relevant to siting critical assets, such as the p-median, max-cover, p-center, and p-maxian models. The multi-objective models are tested on a case study of Orlando, Florida. The dispersion/center model produced the most gradual trade-off curve, while the dispersion/maxian trade-off curve had the most pronounced “elbow.” The center and median multi-objective models were far more computationally demanding than the models using max cover and p-maxian. These findings may inform decision-makers and researchers in deciding what type of multi-objective models to use for planning dispersed networks of critical assets.

Highlights

► We develop four multi-objective spatial dispersion models. ► Pareto-frontiers and computational efficiencies are compared. ► Some models solve faster or have more pronounced trade-off curves than others. ► Results indicate which models are best suited for practice.

Introduction

Over the past decade, major disasters in the United States such as the 9/11 attacks, hurricane Katrina, and the H1N1 pandemic have prompted concern about homeland security. One of the most prominent issues in the homeland security community is how to properly manage critical assets (Critical Infrastructure Protection Program, 2006, The White House, 2001). Critical assets are the key infrastructure components that are crucial for the continuity of supplies, services, and communications. These assets are critical because their loss would have potentially devastating effects on society (Chopra & Sodhi, 2004). Consequently, the need for developing strategies for effectively managing critical assets and their locations has garnered the attention of policy makers and researchers, especially in the case of possible human sabotage (Parfomak, 2007).

In recent years, many researchers have explored methods for identifying critical infrastructure vulnerabilities and fortifying infrastructure networks (Akgun et al., 2010, Church et al., 2004, Li et al., 2009, Murray et al., 2008, Nagurney and Qiang, 2008, Snyder et al., 2006, Taylor et al., 2006). Models have been developed to minimize loss of both supply facilities and population demands in the context of natural disasters (Galindo and Batta, 2010, Rawls and Turnquist, 2010). In resilience-based research such as disaster relief management, objectives commonly involve locating and allocating emergency supplies for critical or vulnerable demands (Horner and Downs, 2010, Sathe and Miller-Hooks, 2005, Widener and Horner, 2011). One strategy for protecting critical assets involves fortifying, or allocating retrofitting resources to vulnerable components of various infrastructures (Daskin, 2008, Qiao et al., 2007, Scaparra and Church, 2008, Snyder et al., 2006). This paper focuses on an alternative strategy, which aims to protect critical assets by dispersing them from each other (Kim & O’Kelly, 2009). Specifically, the p-dispersion model locates p critical facilities to maximize the minimum distance separating any pair of facilities (Kuby, 1987). Clustering of like facilities increases vulnerability to system failure (Erkut, 1990, Goodman et al., 2007, Larson, 2005, Li et al., 2005, Liu et al., 2000, Lovins and Lovins, 1982). Therefore, dispersing facilities protects them by lessening the chance that a single attack or disaster will disable two neighboring facilities simultaneously.

Planners and managers, however, are unlikely to use the p-dispersion model as the sole criteria for planning a network of critical assets because it deals only with the distances between the facilities themselves, and not with distances from facilities to the populations they serve or targets that may threaten them. Critical assets should be available to populations and protected from harm. To date, research exploring these trade-offs has been sparse. While p-dispersion has been proposed or used as a secondary objective in a multi-objective model (Kim & O’Kelly, 2009), the literature lacks a systematic exploration of the trade-offs between dispersion and conflicting objectives such as coverage, service efficiency, equity, or risk involving distances to other kinds of nodes, especially with respect to man-made disasters such as terrorism and other acts of sabotage where attacks gravitate toward urban centers and the relatively exposed targets that lie therein. In addition, the p-dispersion model is well-known to be computationally difficult to solve for medium and large networks, and thus it is also important to understand how fast multi-objective models solve when the p-dispersion model is combined with other objectives.

In this paper, we develop and test multi-objective spatial dispersion models to explore and compare the spatial trade-offs and computational performances between critical asset dispersion and other relevant objectives. We integrate the p-dispersion problem with the maximal covering problem, the p-median problem, the p-center problem, and a variant of the p-maxian problem, and solve the resulting multi-objective models on a case study network for a major US city (Orlando, Florida). These models have the potential to aid in the management and siting of critical assets such as emergency relief supplies that contain vital goods such as food, water, batteries, first aid, anti-viral drugs and so on. An urban example is suitable because most urban areas do not currently have systems of strategic disaster stockpiles. Given that there are different ways of managing critical asset locations, it is not always clear as to which objectives are most appropriate. Comparing different multi-objective models’ resulting trade-off curves and computational efficiencies may inform decision-makers with regard to which models are best suited in practice to combine with facility dispersion.

Section snippets

Background on critical asset vulnerability and protection

A large body of work in the critical infrastructure analysis literature has sought to identify critical assets within infrastructure systems that are the most vulnerable or crucial given a loss (Amin, 2005, Church et al., 2004, Sternberg and Lee, 2006, The White House, 2003). Network infrastructures such as transportation, energy, and telecommunications systems have received substantial attention because of their interconnected nature. The vulnerabilities of interconnected networks and the

Multi-objective spatial modeling for siting critical assets

A diverse array of spatial optimization models can potentially aid with the effective placement of critical assets, which points to the need for multi-objective models that seek solutions for different and often conflicting objectives (Cohon, 1978, Ghosh and Rushton, 1987, Kuby et al., 2005, Malczewski, 1999). Multi-objective problems simultaneously optimize a set of objectives and provide a set of alternative solutions instead of a single solution (Marler & Arora, 2004). The most direct way of

Methods

This research employs the p-dispersion model, which maximizes the distances between like facilities, in a multi-objective model in combination with several desirable-facility objectives. The p-dispersion problem is formulated as follows:

MaximizeDsubject toi=1nYi=pDdij+M(2-Yi-Yj),i;j>iYi={1,0},iwhere

  • i, j = index of potential facility sites

  • n = number of potential facility sites

  • dij = shortest path distance between potential sites i and j

  • M = some large number; such that Mmaxi,j{dij}

  • p = number of

Multi-objective models

Each of the aforementioned models is relevant for siting critical assets because they all involve objectives of access or protection in different ways. Given that these models are relevant in siting critical assets, this research will explore the trade-offs and computational efficiencies between the p-dispersion/p-median, p-dispersion/max-cover, p-dispersion/p-center, and the p-dispersion/p-maxian (variant). Each of the multi-objective models explored are constructed using the weighting method

Study area and data

The study area for this research is the city of Orlando, Florida (Fig. 1). Orlando is a prime opportunity for a case study because of its recognized vulnerability and participation in the Cities Readiness Initiative (Caruson et al., 2005, Centers for Disease Control and Prevention, 2010). The municipal boundary of Orlando is nearly symmetrical, thereby delineating a clear study area to be investigated.

The data set for Orlando consists of a network of 268 nodes representing census tract

Computational results

Each model (15–18) was solved in ILOG CPLEX 11.0 using the branch-and-bound algorithm on a 1.8 GHZ processor running Microsoft Windows Vista with 2 GBs of RAM. The weights, w, varied in increments of tenths, except for the extreme weights where 0.01 was used to ensure there was at least some weight to assign demands to facility locations. We also set p = 5. A five-mile coverage standard was used for the max-cover problem. In terms of computational tractability, the p-dispersion, p-median,

Discussion and conclusions

The spatial management of critical assets has received substantial attention as a result of concerns regarding issues in homeland security and disaster preparedness. In the context of spatially managing assets for critical infrastructure protection, the use of multi-objective modeling can provide conspicuous trade-off gains among conflicting spatial objectives. However, this is also at the expense of considerable amounts of time finding Pareto-optimal solutions. Given that many facility

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