Review on Energy Resilience

Component Importance Measures for Multi-Industry Vulnerability of a Freight Transportation Network

Mohamad Darayi1 & Kash Barker1 & Joost R. Santos2

# Springer Science+Business Media, LLC 2017

Abstract The multi-modal freight transportation network plays an important role in the economic vitality of states, regions, and the broader country. The functionality of this network is threatened by disruptive events that can disable the capacity of the network to enable flows of commodities in portions of nodes and links. This work integrates a multi-commodity network flow formulation with an economic interdepen- dency model to quantify the multi-industry impacts of a disruption in the transportation network to ultimately measure and assess the importance of network components. The framework developed here can be used to measure the efficacy of strategies to reduce network vulnerability from the unique perspective of multi-industry impacts. The framework is illustrated with a case study considering the multi-modal freight trans- portation network consisting of inland waterways, railways, and interstate highways that connect the state of Oklahoma to surrounding states.

Keywords Freight transportation . Vulnerability. Importance measure . Economic impact . Interdependencies

1 Introduction

In response to the growing vulnerability of critical infrastructure given their exposure to natural hazards, malevolent attacks, and the challenges of aging, the Presidential Policy Directive on Critical Infrastructure Security and Resilience (PPD-21) (White House

Netw Spat Econ DOI 10.1007/s11067-017-9359-9

* Kash Barker [email protected]

1 School of Industrial and Systems Engineering, University of Oklahoma, 202 W. Boyd St., Room 124, Norman, OK 73019, USA

2 Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA

2013) was established to focus national efforts to enhance the critical infrastructure network resilience.

The Nation’s critical infrastructure provides the essential services that underpin American society. Proactive and coordinated efforts are necessary to strengthen and maintain secure, functioning, and resilient critical infrastructure – including assets, networks, and systems – that are vital to public confidence and the Nation’s safety, prosperity, and well-being.

– Presidential Policy Directive/PPD-21: Critical Infrastructure Security and Resilience (The White House 2013)

Among the critical infrastructures defined by the US government are transpor- tation networks, which are vital to a society and support many economic activities including commerce and tourism. Disruptions triggered by natural hazards, human-made events, or common failures can severely compromise a region’s ability to move people and commodities, consequently leading to irrecoverable economic losses as well as public safety concerns. Many recent large-scale examples highlight the growing need to deal with disruptions: Hurricane Sandy that affected multiple infrastructure networks, including downed power lines and massive flooding on New York and New Jersey roadways and one million cubic yards of debris that impeded transportation networks (Lipton 2013); the August 2003 US electric power blackout that caused transportation network disruptions (Minkel 2008); and Hurricane Isabel that adversely impacted the transportation system of the Hampton Roads, VA region in 2003 and overwhelmed emergency response (Smith and Graffeo 2005). The current state of disrepair of the US transportation network (e.g., roads given an American Society of Civil Engineers Infrastructure Report Card grade of D, bridges a C+, inland waterways a D- (ASCE 2013a)) could make the network especially vulnerable to a disruptive event. The situation is no better for the state of Oklahoma, where bridges in particular received a lower letter grade of D+, which by definition is interpreted as a Bpoorly performing^ infrastructure (ASCE 2013b). Recent US planning docu- ments focus on transportation network preparedness (The House Committee on Transportation and Infrastructure 2013; US Department of Transportation 2014; Yusta et al. 2011), emphasizing Bsecuring and managing flows of people and goods^ along transportation networks (DHS 2014).

The physical freight transportation network of the US, the largest in the world, consists of four million miles of public roads, 140,000 miles of railroad tracks, 11,000 miles of navigable waterways, and a network of airports with the combined ability of shipping almost 68,000 tons of cargo per year (U.S. Department of Transportation 2013). Furthermore, the same document highlights the importance of the US transportation network in facilitating the convenient movement of resources among suppliers, manufacturers, wholesalers, and customers, with more than 300 million people and 7.5 million organizations across 3.8 million square miles being served. The vital role the freight network plays in transporting raw materials and final products between manufacturers and consumers highlights its position in commerce. The functionality of this network is threatened by disruptive events that can disable the capacity of the network to enable flows of commodities and cause an interruption of

Darayi M., Barker K., Santos J. R.

economic productivity across multiple industries. That is, the ultimate usefulness of understanding transportation network disruptions is not just a descriptor of physical damage, but of economic interruption due to infrastructure inoperability (Tierney 1997, Webb et al. 2000). As such, discussions of transportation network vulnerability should account for multi-industry impacts.

This work focuses on the freight transportation network, particularly on its role of enabling the flow of commodities and facilitating economic productivity, and thus a methodological approach to measure network vulnerability in the context of multi- industry impacts is sought. That is, this work seeks to answer: if a transportation node or link is disrupted, what is the effect on local industries? This research addresses (i) measuring the vulnerability of a multimodal freight transportation network with multi- industry impacts in mind, and (ii) using this vulnerability analysis to develop a measure of importance for each network component.

This paper is arranged as follows. Section 2 offers some background literature regarding the vulnerability and economic impacts of transportation networks. Section 3 describes the proposed methodology for freight transportation network vulnerability analysis integrating a multi-commodity network flow formulation with a risk-based economic interdependency model. Also, a new network component impor- tance measure based on multi-industry vulnerability analysis is developed. Section 4 presents an illustrative example based on a partial freight transportation network within the state of Oklahoma consisting of three important business economic areas and the multi-modal freight network infrastructure that facilitates trade with centers out of the state. Section 5 provides concluding remarks and future research avenues of this work.

2 Background and Literature Review

Presidential Policy Directive 21 states that critical infrastructure Bmust be secure and able to withstand and rapidly recover from all hazards^ (The White House 2013). This combination of the ability to (i) withstand the effects of a disruption and (ii) recover timely from the disruption is often referred to as resilience (Hosseini et al. 2016). Figure 1 highlights these two dimensions of resilience: vulnerability and recoverability (Henry and Ramirez-Marquez 2012; Pant et al. 2014). The network service function φ(t) describes the behavior or performance of the network at time t (e.g., φ(t) could describe traffic or commodity flow in a transportation network). The vulnerability dimension of resilience is the focus of this work.

In a freight transportation network, vulnerability is considered to be a problem of interrupted serviceability or accessibility of network components, leading to reduced system functionality (Berdica 2002; Chen et al. 2007). O’Kelly (O’Kelly 2014) classifies network vulnerability into link vulnerability, or the reduction of a network’s capability after losing a link, and nodal vulnerability, or the extent to which a node plays a critical role in the operation of the whole network. From the network interdic- tion literature, where network components (nodes or arcs) are disabled intentionally, there are three approaches to evaluate network vulnerability (Murray et al. 2008): (i) scenario-specific evaluation, where the potential consequences of a specific disruptive scenario or set of scenarios is evaluated (e.g., studying the impact of losing a bridge, a road segment, or a hub on network performance (Jenelius and Mattsson 2012,

Component Importance Measures for Multi-Industry...

Burgholzer et al. 2013, Rupi et al. 2014, Fotuhi and Huynh 2017)), (ii) strategy-specific assessment, where vulnerability is assessed with respect to a hypothesized sequence or strategy of disruptions targeting components perceived to be important (e.g., Erath et al. 2010, Park et al. 2011, Knoop et al. 2012), and (iii) mathematical modeling assessment (e.g., Sullivan et al. 2010; Jenelius et al. 2010), using game-theoretical techniques to find worst-case scenarios. In our work, to analyze network vulnerability and define a measure of importance for network components, a scenario-specific approach is taken by analyzing the proportional effect on the flow of commodities given the removal of one node/link at time.

Most work in network vulnerability focuses on network behavior after a disruption in terms of graph theoretic measures, such as average shortest dis- tance, network diameter, average edge betweenness, and cluster efficiency (e.g., Albert and Barabasi 2002; Jonsson et al. 2008; Chen et al. 2010; Mishkovski et al. 2011; Johansson et al. 2013), which describe what is commonly referred to as structural vulnerability. This is different from functional vulnerability, where operational characteristics (e.g., network flow) of different components are taken into consideration (Ouyang et al. 2009). To capture the functional aspects of network vulnerability, a measure of importance for network components was introduced by Nagurney and Qiang (Nagurney and Qiang 2008; Nagurney and Qiang 2007a; b) based on network performance/efficiency considering demands, costs, and flows, as well as behavior of the users of the network. Following the lead of Nagurney and Qiang, the emphasis of this paper deals with describing network vulnerability with respect to flow along the network, a more tangible approach than focusing solely on topological features of the network and ame- nable to an analysis of multi-industry economic impacts. That is, φ(t) is used to describe the flow along the transportation network. Further literature describing network component importance based on flow measures is sparse (Rocco et al. 2010; Nicholson et al. 2016), and, to the authors’ knowledge, the methodology proposed in this paper for pinpointing the contribution and importance of individual transportation network components to multi-industry economic im- pacts is an area that has not been previously pursued in the literature.

Fig. 1 System performance, φ(t), trajectory following a disruptive event (source: Henry and Ramirez- Marquez 2012)

Darayi M., Barker K., Santos J. R.

This paper considers network vulnerability as a relative drop in the commodity flows along the network after the removal of a particular node or link. And a drop in the flow of commodities would generate subsequent impacts on multiple industries relying on those commodities. While several approaches have been proposed to capture interde- pendencies among infrastructure and industries (Pederson et al. 2006; Haimes 2009; Rose et al. 2012; Ouyang 2014), this work makes use of an economic input-output model extension that quantifies the propagation of multi-industry inoperability (the extent to which industry output will not be produced) caused by perturbations in supply and/or demand.

3 Research Methodology

Despite the excellent use of network-based models in representing interdependencies which consider various aspects of network vulnerability (Holden et al. 2013; Miller- Hooks et al. 2012; Pederson et al. 2006), there exists a need to integrate parts of these models with multi-industry impacts to address freight transportation functionality as enabling the flow of commodities and facilitating economic productivity. This need is addressed with a four-step methodology, as illustrated in Fig. 2, which then culminates in a transportation network component importance measure.

3.1 Step 1. Baseline Network Flow

Freight transportation planning models have been classified at strategic, tactical, and operational levels (Crainic and Laporte 1997). At the strategic level, long-term deci- sions include the design of the physical transportation network and the location of main facilities (e.g., rail yards, multi-modal platforms). At the tactical level, medium-term

Fig. 2 Four step approach to assessing transportation component importance with multi-industry impacts

Component Importance Measures for Multi-Industry...

decisions are made, such as the design of the service network (i.e., route choice and type of service to operate, aggregate scheduling). The operational level includes shorter-term decisions, including crew or container scheduling. It is at the operational level of planning that routing of different types of commodities in an existing multi- modal transportation network is sought. The multi-modal freight transportation network of interest in this work will be modeled with a typical multi-commodity network flow problem. Multi-commodity network flow (MCNF) problems, which minimize the cost of the flow of multiple commodities across a capacitated network of supply and demand nodes, arise in a wide variety of applications, including telecommunications (Minoux 2006), warehousing (Ahuja et al. 1993), and multi-modal transportation networks (Liotta et al. 2015; Ham et al. 2005), among others.

To study the vulnerability of a multi-modal freight transportation network, which serves as a facilitator of n interacting industries, the topology of the network and corresponding supply and demand nodes must be extracted. The conventional MCNF problem for a network, G(N, L) with a set of nodes, N, a set of links, L, and a number K of commodities, is formulated in model M1. The flow of commodity k on link (i, j) is represented with f kij, and the cost of shipment for commodity k on link (i, j) is w

k ij. The

capacity of link (i, j) is represented with uij, and the supply/demand of commodity k at node i is represented with bki , defining the Bbundle^ and Bmass balance^ constraints in model M1, respectively. Note that bki is positive for supply nodes, negative for demand nodes, and zero for transshipment (or intermediate) nodes. The capacity of each link is considered as a shared constraint for all commodities flowing on the link.

min ∑ i; jð Þ∈L

∑ k wkij f

k ij

s:t: ∑ k f kij ≤uij∀ i; jð Þ∈ L

∑ i; jð Þ∈ L

f kij − ∑ j;ið Þ∈ L

f kji ¼ bki ∀∈ N;k ¼ 1;…;K

f kij > 0; ∀ i; jð Þ∈ L;k ¼ 1;…;K

ðM1Þ

In fact, a generic MCNF model provides a means to formulate the supply-demand network in which a multi-modal freight transportation network connects industries and enables trading relationships and interactions. From a tactical point of view, the integration of (i) business economic sectors and (ii) their supply capabilities or demand requirements together with (iii) the structure of the transportation network can result in a minimum cost MCNF model that can route the commodities from suppliers to the demand nodes via f kij, collectively representing the flow of commodities on the links of a baseline (undisrupted) network.

3.2 Step 2. Network Disruption

A common theme in the analysis and evaluation of network vulnerability is interdiction (Gedik et al. 2014; Murray et al. 2008; Murray et al. 2007), in which scenario-based

Darayi M., Barker K., Santos J. R.

removal of network components is assumed to represent the effects of a disrup- tive event. The consequences of a targeted attack, accident, or natural disaster are simulated as disruptions in the flow of valuable goods or services through the network caused by disabling network components. The network is analyzed to determine how vulnerable it is to a disruption, and which nodes or links, if lost, result in the most damage to network performance. Further, the temporal and spatial scales at which analysis is conducted, as well as the duration of the disruptive event, affect the disruption analysis.

Approaches to interdict a network differ based on how disruption scenarios are assessed and understood. A disruption scenario is defined by the set of network components that are impacted, the degree to which they are disabled, and the operating conditions (e.g., network activity and link/node capacities) of the network prior to the disruption regardless of the initiating event that causes the disruption. In extreme cases, an affected facility may be rendered completely inoperable by a disruption (e.g., losing a road completely due to a bridge collapse as in the case of the I-35 Mississippi River bridge failure in 2007). In other instances, a disruption may impact network activity to a lesser degree given that only some of the functionality of a facility may be lost (e.g., an accident blocking a single lane of an interstate highway segment). The identification of disruption scenarios enables an impact assessment. Impacts can range from those directly associated with network operation, such as connectivity, flow, or capacity reduction, to more complex associations, such as the economic impacts affecting the production and consumption of flows (Matisziw and Murray 2009).

The flexibility in defining scenario-specific disruptions based on historical data or other desired analysis makes it appropriate for network vulnerability studies. In partic- ular, it provides opportunities for understanding a component’s role and importance within a network. For example, one might be interested in the impact of the closure of a bridge or a road segment on network performance (e.g., the flow of commodities, the topological behavior of a post-disruption network, the multi-industry economic im- pacts). A deterministic scenario-specific approach (Murray et al. 2008), where the potential ramifications of the removal of a particular network component is evaluated, is often used to quantify network component importance measures (e.g., Nagurney and Qiang 2008; Jenelius et al. 2006). Stochasticity could be introduced to capture uncer- tainty in disruptive scenarios (e.g., Miller-Hooks et al. 2012; Burgholzer et al. 2013; Baroud et al. 2014).

This step evaluates the effect of losing a network component on freight flow through the network and resulting consequences on supply/demand nodes. Hence, a disruptive scenario is defined as the removal of a particular network component. An optimization formulation is developed to reroute commodity flows through the residual network, pursuing the maximum flow throughout the network and captur- ing failure in the form of remaining commodities at supply nodes and unmet demands at demand nodes, as formulated in model M2. Intuitively, a decision maker would likely desire to reroute commodities to take advantage of the remain- ing capacity of the residual network. Note the difference in perspective in the post- disruption MCNF developed here: prior to the disruption, model M1 minimizes the cost of transporting commodities along the capacitated network, where model M2 maximizes the flow to meet as much demand as possible given the interrupted network with an updated set of links, L′, and nodes N′. To capture undelivered

Component Importance Measures for Multi-Industry...

commodities remaining with the suppliers or unsatisfied demand at demand nodes, a slacL′,k variable Ski is defined. The magnitude of S

k i is positive, and multiplier γi

takes on a negative value for the set of demand nodes (after disruption) N 0 −, a

positive value for supply nodes (after disruption) N 0 þ, and zero for transshipment

nodes (after disruption) N 0 0. The objective function maximizes the sum of

commodity-specific flows, where f 0k ij represents the flow of commodity k across

link (i, j) which remains in the updated set of links, L′. Slack variable Ski will be used in the next step to calculate inoperability among multiple industries. Here it is assumed that each type of commodity represents an industry, and interdependent inoperability propagated through the entire regional economy caused by unsatisfactory levels of demands/supplies will be pursued in the next section.

max ∑ i; jð Þ∈L0 ∑k f

0 k

ij

s:t: ∑k f 0 k ij ≤u

0 ij∀ i; jð Þ∈L0

∑ i; jð Þ∈L0

f kij− ∑ i; jð Þ∈L0

f kji þ γiSki ¼ b 0k i ∀i∈N

0 ;k ¼ 1;…;K

γi ¼ −1 fori∈N

0 −

þ 1 fori∈N 0þ 0 fori∈N

0 0

8 >>< >>:

f kij > 0; S k i ≥∀ i; jð Þ∈L

0 ;k ¼ 1;…;K

ðM2Þ

3.3 Step 3. Multi-Industry Impact

To represent the multi-industry impacts of unmet demands at demand nodes and remaining commodities at the suppliers’ side in the MCNF, an extension of the input-output model is used. The input-output (I-O) model, for which Wassily Leontief (Leontief 1966) won a Nobel Prize, has been widely accepted as a useful model for analyzing the interdependent connections among industries (Miller and Blair 2009). Under a static equilibrium, the total output of the industry s is distributed to other industries and also satisfies external (consumer) demand. This equilibrium condition is described with xk ¼ ∑nr¼1zkr þ ck, where xk is the total output of industry k, zkr is the flow of commodities produced by industry k and used as input to production in industry r, and ck is the external demand for industry k. The flow of commodities zkr is assumed to be proportional to the output of industry r (r ∈ {1, … , K} and r ≠ k), expressed as zkr = akrxr. Further, it is assumed that each industry produces a sole commodity, such that industry k produces commodity k. The common form of the Leontief input-output model is expressed in Eq. (1), where x is the vector of industry production outputs, A is an industry-by-industry matrix of interdependency coefficients, akr, and c is a vector

Darayi M., Barker K., Santos J. R.

of final demands. The model shows that total production is made up of industry- to-industry intermediate production, Ax, and production to satisfy final demand, c. Terms zkr, xr, and ck are measured in monetary units.

x ¼ Ax þ c⇒x ¼ I−A½ �−1c ð1Þ

Despite of the I-O model’s assumption of a linear relationship of commodity flows among industries, the extensive usage of I-O models is due in part to the availability data describing the parameters of the I-O model in a number of countries (OECD 2011, Timmer et al. 2015). This includes a data collection effort by the US Bureau of Economic Analysis (BEA), which maintains input-output tables at different levels of aggregation (BEA 2010). Extending the capability of the I-O model, Santos and Haimes (Santos and Haimes 2004) propose the Inoperability Input-Output Model (IIM) to represent the propagation of inoperability, or the proportional extent to which industries are unproductive after a change in demand or a forced change in demand due to a lack of supply. The use of the IIM can model inoperability in an economic setting, or in a set of interdependent infrastructures (Setola and De Porcellinis 2008, Crowther and Haimes 2010, Oliva et al. 2014). The IIM and some extensions have been deployed in a number of contexts, including analyses of infrastructure disruptions (Anderson et al. 2007, Pant et al. 2011, Pant et al. 2015, Jonkeren et al. 2015, Mackenzie et al. 2012), workforce losses (Orsi and Santos 2010a; b), and supply chain risk (Barker and Santos 2010a; b), among others. Furthermore, the IIM has been used in multi-industry vulnerability studies (e.g., Yu et al. (Yu et al. 2014) developed a multi-perspective approach for vulnerability decomposition with the aim of prioritizing key economic sectors in the aftermath of disruptive events).

Instead of describing the connections between the interdependent industries in terms of commodity flow dollars, the IIM illustrates how normalized production losses propagate through interconnected industries, providing a different perspective from the traditional I-O model. The IIM is provided in Eq. (2), describing the relationships among K industries, resulting in matrices of size K × K and vectors of length K.

q ¼ A⋆q þ c⋆⇒q ¼ I−A⋆½ �−1c⋆ ð2Þ

Vector q is a vector of industry inoperability describing the proportional extent to which as-planned productivity or functionality is not realized following a disruptive event. Inoperability for industry k is defined in Eq. (3), where as-planned total output is represented with xk and degraded total output resulting from a disruption is represented with ~xk. An inoperability of 0 suggests that an industry is operating at normal production levels, while an inoperability of 1 suggests that the industry has become completely inoperable.

qk ¼ x̂k−~xk � �

=x̂k⟺q ¼ diag x̂ð Þ½ �−1 x̂−~x � �

ð3Þ

Component Importance Measures for Multi-Industry...

Normalized interdependency matrix A⋆ is a normalized form of the original A matrix describing the extent of interdependence among a set of industries. As stated by Eq. (4), the row elements of A⋆ indicate the proportion of additional inoperability that are contributed by a column industry to the row industry.

a⋆rk ¼ ark x̂k=x̂rð Þ⟺A⋆ ¼ diag x̂ð Þ½ �−1A diag x̂ð Þ½ � ð4Þ

Eq. (5) provides the calculation of c⋆, a vector of normalized demand reduction. The elements of c⋆ represent the difference in as-planned demand ck and perturbed demand ~ck divided by as-planned production, quantifying the reduced final demand for industry k as a proportion of total as-planned output.

c⋆k ¼ ĉk−~ck � �

=x̂k⟺c ⋆ ¼ diag x̂ð Þ½ �−1 ĉ−~c

� � ð5Þ

For the traditional economic loss metric, losses can be calculated by multiplying each industry’s production level in monetary units by its inoperability level: for industry k, Qk = xkqk, or for the entire economy of industries, Q = x

Tq. As such, planning decisions can be made with respect to inoperability or economic impact at the industry level, or with respect to economic impact across multiple industries.

When a disruption within the transportation network results in remaining commod- ities at supply nodes and/or unmet demand at demand nodes, inoperability propagates throughout industries in a region. Without loss of generality, each node within the network is considered to be either a supplier or a consumer of a particular commodity. Each commodity belongs to an industry in the economy as defined by the North American Industry Classification System (NAICS).

In industry k the amount of import is ∑i∈ N−∩Nkð Þ−b k i , where Nk represents the set of

nodes which are producers/consumers of commodity output from industry k, located in a geographical area of interest (e.g., a business economic area, county, state, entire country). Note again that it is assumed that industry k produces commodity k. The amount of import contributes to the total output, xk, and final demand, ck, of industry k as shown in Eq. (8). Thus, unmet demand, ∑i∈ N−∩Nkð ÞΔb

k i , in Eq (9) results in the loss

of output, Δxk, and final demand,Δck, representing ck−~ckð Þ .

∑i∈ N−∩Nkð Þ−b k i ¼ x̂k þ ĉk k∈ 1; …; Kf g ð6Þ

∑i∈ N−∩Nkð Þ−Δb k i ¼ Δx̂k þ Δĉk k∈ 1; …; Kf g ð7Þ

Therefore, for industry k, unmet demands cause an inoperability, qk, and demand perturbation, c⋆k , which are modeled in Eq. (10) and (11), respectively, as adapted from Pant et al. (Pant et al. 2011; Pant et al. 2015). Inoperability is a measure of the loss of

Darayi M., Barker K., Santos J. R.

production in industry k as a proportion of its original production level, and demand perturbation is a measure of the change in demand as a proportion of the original production level in industry k.

qk ¼ Δx̂k x̂k

¼ x̂k−~xk x̂k

ð8Þ

c⋆k ¼ Δĉk x̂k

ð9Þ

∑i∈ N−∩Nkð Þ−Δbi ¼ ∑i∈ N−∩Nkð ÞSki k∈ 1; …; Kf g ð10Þ

Considering Eqs. (8)–(12), for the industries experiencing difficulties only in importing their required commodities, there exists a demand perturbation, as modeled in Eq. (13).

c⋆k ¼ ∑i∈ N−∩Nkð ÞS

k i

x̂k −qk k∈ 1; …; Kf g ð11Þ

For the industries experiencing difficulties only in exporting commodities, the total amount of remaining commodities at supply nodes relating to that particular industry, Ski , will be considered as a perturbation in demand. Hence, the demand perturbation for exporting industry k is modeled with Eq. (14).

c⋆k ¼ ∑

i∈ Nþ∩Nkð Þ Ski

x̂k k∈ 1; …; Kf g ð12Þ

Eqs. (13) and (14) combined with the IIM in Eq. (4) form a complete solvable system that quantifies the inoperability and demand perturbations for the entire econ- omy of interconnected industries. For simplicity, the demand perturbations in Eqs. (13) and (14) assume failure in only demand nodes or in supply nodes within a particular industry, whereas in actual situations, some industries would likely consist of both demand and supply nodes. Therefore, the total demand perturbation for industry k in the case of having both importing and exporting roles is given in Eq. (15).

c⋆k ¼ ∑i∈ Nþ∩Nkð ÞS

k i

x̂k þ

∑i∈ N−∩Nkð ÞS k i

x̂k −qk ð13Þ

Based on the exporting or importing nature of the nodes representing each industry, either Eq. (13), Eq. (14), or Eq. (15) captures the perturbation vector,

Component Importance Measures for Multi-Industry...

c⋆, which parameterizes the interdependency model in Eq. (4). As such, qk can then be calculated to measure the proportional extent to which as-planned productivity or functionality is not realized following a transportation network disruption that results in unmet demand or commodities remaining with suppliers.

3.4 Step 4. Vulnerability Analysis and Component Importance

Network vulnerability analysis emerged from the network reliability literature, which is often interested in the probability of a desired network performance (Boesch et al. 2009) or the consequences of the failure of a network component regardless of the probability of failure (Taylor and Susilawati 2012; Ramirez- Marquez et al. 2016). This second perspective enables the calculation of com- ponent importance measures, a long-studied area in reliability engineering (Kuo and Zhu 2012), wherein network components that impact the performance of the network are identified.

Step 4 develops scenario-specific component importance measures based on vulnerability. The consideration of the economic impacts of a disruption of transportation network components enhances the literature on transportation network vulnerability, which have traditionally focused on flow or topological aspects of the network (e.g., connectivity and accessibility) as metrics for network performance (Reggiani et al. 2015, Mattsson and Jenelius 2015, Sun et al. 2017). As such, when we define our new importance measure, we consider the ultimate role of a freight transportation network as a facilitator of economic productivity. Such impact is calculated for different disruptive scenarios, ep, where p represents the component removed from the network (as displayed in Fig. 1). As a result, this work advances the study of network vulnerability from the perspective of network performance in terms of commodity-driven multi- industry impact rather than graph theoretic or flow importance measures. Malfunction of a freight transportation network serving a regional economy – comprised of interdependent industries – causes failure in the form of a delayed shipment of commodities at supply nodes and/or unmet demands at demand nodes. Here, the interdependent effect of the failure in multiple industries is captured by the IIM as described in Section 3.3. And finally, vulnerability is defined as the magnitude of this failure in terms of multi-industry economic impact, given the occurrence of a particular disruptive event, ep. Note, of course, that network vulnerability is highly dependent upon the type and extent of ep, which assumes complete removal of component p (though a proportional reduction could also be explored).

Two vulnerability measures, stated in Eqs. (14) and (15), are proposed. For network topology G, fixed demand/supply vector b, and disrupted (removed) component p, vulnerability is measured as the relative network efficiency, or the multi-industry economic loss Q(G − p, b), after p is removed from the network, G. Qmax is the maximum multi-industry loss caused by a shutdown in the entire network (i.e., a removal of all nodes). As such, the vulnerability measure in Eq. (14) quantifies the proportional economy-wide impact of a loss of component p relative to a loss in all components. This measure lies on [0,1], where 0 means

Darayi M., Barker K., Santos J. R.

that losing component p has no effect on the total economy and 1 means that the loss of component p is as disruptive as having a shutdown of the entire network.

ηp G; bð Þ ¼ Q G−p; bð Þ

Qmax ð14Þ

Eq. (15) similarly provides the economic vulnerability experienced by a particular industry k due to lost component 푝, providing an industry-by-industry perspective to the importance of vulnerable transportation network elements. Qkmax is the maximum loss in a particular industry k caused by a shutdown in the entire network, capturing indirect economic loss effect on each industry based on the IIM model.

ηkp G; bð Þ ¼ Qk G−p; bð Þ

Qkmax ð15Þ

Thus, Eqs. (14) and (15) provide economy-wide and industry-specific vul- nerability measures for the disruption of component p in the multi-modal transportation network. Naturally, certain industries may be more impacted by certain network components than others, which is an important consideration (e.g., a particular industry may be more critical to a state or regional economy than another).

4 Illustrative Example

The proposed transportation network vulnerability analysis methodology and component importance measures, found as a result of the four-step process in Section 3, is illustrated with a case study based on a portion of a multi-modal freight transportation network within the state of Oklahoma and surrounding states whose industries trade with Oklahoma industries. Oklahoma plays a strong role in the transport of goods via a multi-modal transportation network consisting of three important interstate highways, as well as railways and inland waterways that connect to the Mississippi River Navigation System via two ports.

4.1 Step 1. Baseline Network Flow, Illustrated

Figure 3 highlights a supply-demand network in which supply nodes are all within the state of Oklahoma: the three important business economic areas of Oklahoma City (node 1), the Port of Catoosa in Tulsa (node 2), and the Port of Muskogee (node 3) (Ingalls et al. 2002). Demand nodes consist of states external to Oklahoma that are the most important states to interact with Oklahoma industries: Texas, Louisiana, Arkansas, and Illinois. The effects of a disruption within the network on exporting industries within the state are sought, as are the consequences in the entire Oklahoma economy,

Component Importance Measures for Multi-Industry...

hence the four importing states are considered as four combined demand nodes connecting to Oklahoma’s multi-modal freight transportation network.

The nodes of the network are discussed in brief in Table 1. The Oklahoma City business economic area is connected to the north-south corridor through I-35 and east-west corridor through I-40 and I-44. In addition to the truck way facilities, Burlington Northern Santa Fe (BNSF) railroad has an intermodal rail-truck facility in Oklahoma City near the junction of I-35 and I-40. The Port of Catoosa, the largest inland port in the United States in terms of area, is located near the city of Tulsa, adjacent to I-44, US 169, and rail lines. Industries listed in Table 2 are almost the port’s largest exporters in terms of commodity flows. The Port of Muskogee is connected to the freight transportation network through Highway 165 and a rail marshalling yard. Supply nodes in Fig. 3 include Oklahoma City (node 1), the Port of Catoosa (node 2), and the Port of Muskogee (node 3). Demand nodes include Texas city, TX (node 4), New Orleans, LA (node 5), Little Rock, AR (node 6), and Chicago, IL (node 7). Node 8 represents the intermodal terminal that facilitates the movement of commodities from industries in the industrial park of Port of Catoosa to their out-of-state customers using railroad company, BNSF. Links (1,7) and (1,4) are part of the North America railroad which connects Oklahoma City, OK, with Chicago, IL, and with Texas City, TX, respectively. The Port of Catoosa is connected to the North America railroad through a local railroad represented by link (2,8). Links (2,5), (2,4), (2,6), and (2,7) are part of the inland waterway network navigated by McClellan–Kerr Arkansas River Navigation System and connect Port of Catoosa with the Port of New Orleans, the Port of Texas City, the Port of Little Rock, and the Port of Chicago, respectively. The Port of Muskogee is connected to the Port of Little Rock, the Port of Texas City, and the Port of New Orleans through the same inland waterway network represented by links (3,6), (3,4), and (3,5), respectively,

Fig. 3 Representations of (a) spatial location of multi-modal nodes in Oklahoma and surrounding states, and (b) the connected transportation network

Darayi M., Barker K., Santos J. R.

and it is linked to the North America railroad through a local railroad depicted with link (3,8). Node 9 is an intermediate node that connects the Oklahoma City business economic area to the U.S. interstate highways to the north and south

Table 1 Spatial location of multi-modal nodes in Oklahoma and surrounding states

Component Description

Node 1 Oklahoma City, a supply node for multiple industries

Node 2 Port of Catoosa, a supply node for multiple industries

Node 3 Port of Muskogee, a supply node for multiple industries

Node 4 Port of Texas City, a demand node for multiple industries

Node 5 Port of New Orleans, a demand node for multiple industries

Node 6 Port of Little Rock, a demand node for multiple industries

Node 7 Port of Chicago, a demand node for multiple industries

Node 8 Intermodal terminal, Tulsa, OK

Node 9 Transshipment node that connects the Oklahoma City, OK, business economic area to the north and south through I-35 and to the east through I-44

Node 10 Transshipment node in Fort Smith, AR, that is a connecting point on I-40 to link Oklahoma City and Tulsa, OK to Little Rock, AR

Node 11 Transshipment node that connects the Tulsa Port of Catoosa industrial park to I-44

Link (1,7) Part of the North America railroad that connects Oklahoma City, OK with Chicago, IL

Link (8,7) and Link (8,4)

Part of the North America railroad which connects Tulsa, OK, with Chicago, IL and Texas City, TX, respectively

Link (2,8) A local railroad connecting Port of Catoosa to the North America railroad

Link (1,4) Part of the North America railroad that connects Oklahoma City, OK with Texas City, TX

Links (2,5), (2,4), (2,6), and (2,7)

Part of the inland waterway network navigated by McClellan–Kerr Arkansas River Navigation System and connect Port of Catoosa with the Port of New Orleans, the Port of Texas City, the Port of Little Rock, and the Port of Chicago, respectively

Links (3,6), (3,4), and (3,5)

Part of the inland waterway network navigated by McClellan–Kerr Arkansas River Navigation System and connect the Port of Muskogee to the Port of Little Rock, the Port of Texas City, and the Port of New Orleans, respectively

Link (9,4) The roadway that connects Oklahoma City to Texas City, TX using interstate highways I-35 and I-45

Link (9,11) Part of interstate highway I-44 that connects Oklahoma City to Tulsa.

Table 2 Names and NAICS codes for main industries using the network

Industry name NAICS code

Food and beverage and tobacco products 311

Petroleum and coal products 324

Chemical products 325

Nonmetallic mineral products 327

Machinery 333

Miscellaneous manufacturing 339

Component Importance Measures for Multi-Industry...

through I-35 and to the east through I-44. Node 10 is Fort Smith, AR which is a connecting point on I-40 to link Oklahoma City and Tulsa, OK to Little Rock, AR. Node 11 represents an intermediate node that connects the Port of Catoosa industrial park to interstate highway I-44, link (9,4) connects Oklahoma City to Texas City, TX, using interstate highways I-35 and I-45, and link (9,11) is part of interstate highway I-44 which connects Oklahoma City to Tulsa.

In total, there are 62 industries operating in Oklahoma as identified by NAICS, suggesting that the A⋆ matrix regionalized for Oklahoma is 62 × 62. In the proposed supply-demand network, six industries, listed in Table 2, are considered to be industries that primarily export commodities to out-of-state customers according to high trade figures (Bureau of Transportation Statistics 2010a). In the developed illustrative MCNF example, each commodity belongs to an industry as defined by NAICS economic sectors, and each node within the network is considered to be either a supplier or a customer of a particular commodity.

Table 3 lists the combined estimated annual supply and demand in tons for the associated industries and states, compiled from different databases (US Army Corps of Engineers 2013; Tulsa Port of Catoosa 2013; Bureau of Transportation Statistics 2010a, 2010b; Port of Muskogee 2013; Bureau of Economic Analysis 2010).

Baseline network flow in the supply-demand network is calculated with model M1, where the cost vector is computed based on the transportation mode and the mileage of the distances between nodes. The cost per ton-mile for a barge is estimated at $0.97, compared to $2.53 for rail, and $5.35 for trucking (Arkansas Waterway Commissions 2014). The capacity of each link, shown in Table 4, representing the availability of transportation facilities, is estimated from historical data as a shared constraint for all commodities flowing on the link (ODOT 2013). The baseline flow resulted in no remaining commodities at supply nodes and no unsatisfied demand at demand nodes, suggesting that supply nodes send out all the commodities and demand nodes satisfy all their demands. Based on Table 3, the total supply of commodity k is assumed to be

Table 3 Combined annual demands/supplies at supply/demand nodes connecting through the network (in thousand tons)

Food and beverage

Petroleum and coal

Chemical products

Nonmetallic mineral

Machinery mfg.

Misc. mfg.

Supply nodes in OK

Oklahoma City 4351 0 3606 2198 285 1419

Port of Catoosa 603 5459 3416 303 30 5

Port of Muskogee

0 408 0 383 0 361

Demand nodes outside of OK

Texas City, TX 1167 3804 2448 0 310 362

New Orleans, LA

604 221 0 0 3 0

Little Rock, AR 3183 1842 4574 492 2 654

Chicago, IL 0 0 0 2392 0 769

Darayi M., Barker K., Santos J. R.

equal to the total demand of the same commodity within the entire supply-demand network as depicted in Fig. 3.

4.2 Step 2. Network Disruption, Illustrated

Considering a disruptive scenario as the removal of a particular network component, the supply-demand network might experience a failure in satisfying demands in the interrupted network. The network components that were considered for disruption include: (i) three transshipment nodes within the state of Oklahoma, which have a vital role in connecting segments of high volume-freight-traffic interstate highways, (ii) some segments of the North America Railroad, (iii) a local railroad which connects industrial parks to the North America Railroad, and (iv) parts of waterway system (described in Table 1). Discussed previously in Section 3.2, a decision maker would likely desire to reroute commodities to take advantage of the remaining capacity of the residual network, as shown in model M2, by maximizing the flow to meet as much demand as possible given the interrupted network. Failure in the form of undelivered commodities remaining with the suppliers, or unsatisfied demand at demand nodes, represented by Ski , affect industry output and inoperability propagates through many of the interconnected industries. In the illustrative example, all the supply nodes are within the state of Oklahoma and the four demand nodes are located outside of Oklahoma. Table 5 reports ∑i∈ Nþ∩Nkð ÞS

k i , the sum of the slack (remaining supply) by industry type

at the supply nodes when different network components are disrupted.

4.3 Step 3. Multi-Industry Impact, Illustrated

In the case of any disruption within the network resulting in the loss of exports, there is a demand perturbation in the industries using the network, as calculated in Eq. (14). Assuming the only losses in the state economy are due to the loss of exports, the interdependent cascade of the demand perturbations causes losses to all the other state industries, as captured in Q. It is further assumed that industries not using the transportation network have zero demand perturbations, though could suffer from interdependent inoperability.

Table 4 Link capacities among the origin/destination nodes in the illustrative network (in thousand tons) (ODOT 2013)

Nodes 4 5 6 7 8 9 10 11

1 2800 2900 1700 6200

2 180 650 750 500 3400 3700 1350

3 355 185 3010 290

8 3800 300

9 1800 1700

10 12,000

11 1600 2000

Component Importance Measures for Multi-Industry...

As network component importance rankings are ultimately calculated on a relative basis, inoperability is calculated in terms of annual impact, as it is assumed that annual industry production accumulates consistently across the year (i.e., neither production nor interdependency relationships vary day-to-day, week- to-week, month-to-month). A smaller time horizon could be considered as a proportion of a year if a particular disruptive event is modeled (e.g., a two-week closure of port facilities (Pant et al. 2011)).

Using the remaining commodities left at supply nodes, shown in Table 5, demand perturbations were calculated with Eq. (14). In the example, the industries in Oklahoma experience difficulties in exporting commodities, individually for each of the 11 disrupted network components. The resulting industry inoperability, qk, for each disrupted component is found, as shown in Table 6 and plotted in Fig. 4. Results show that most industries are vulnerable to disruptions that affect the functionality of either rail transportation or interstate highways but less susceptible to disruptions to the inland waterway which has a smaller share (less than 5% (ODOT 2013)) in outbound freight movement in Oklahoma. As such, perhaps the external capacity in rail and truck freight transport suggest that they could serve as alternative transportation modes during a disruption, though more costly. A disruption that affects the functionality of the intermodal terminals would cause the most significant drop in the productivity of most industries. Examples of this include (i) node 8, which facilitates trade between industries located in the business economic area in Port of Catoosa, OK with their customers in Chicago, IL and Texas City, TX through the North America railroad, and (ii) nodes 9 and 11, important transshipment nodes that connect the three important business economic areas within the state of Oklahoma to their customers through interstate highways.

From a single industry point of view, it is shown that the productivity of the Petroleum and coal (324) industry is mostly vulnerable to its accessibility to the North America railroad through the intermodal terminal (node 8) in Tulsa, OK. In

Table 5 Commodities remaining at suppliers with the removal of network components (in thousand tons)

Removed component

Food and beverage

Petroleum and coal

Chemical products

Nonmetallic mineral

Machinery mfg.

Misc. mfg.

Node 9 291 0 803 0 25 2

Node 8 478 2508 623 0 0 7

Node 11 34 143 1067 0 25 7

Link (1,7) 290 0 0 2000 0 770

Link (9,11) 189 108 823 0 0 0

Link (2,5) 504 37 0 0 28 0

Link (8,4) 367 2108 923 0 0 7

Link (2,8) 189 2308 823 0 0 5

Link (2,4) 0 105 0 0 0 0

Link (3,8) 0 0 0 210 5 0

Link (8,7) 0 0 0 251 0 0

Darayi M., Barker K., Santos J. R.

general, most disruption scenarios may affect the productivity of the Chemical products (325) industry, either by a local disruption that interrupts the access of the business economic area at the Port of Catoosa through a local railroad (e.g., link (2,8) to the intermodal terminal at node 8) or a state-wide disruption that affects Oklahoma’s major trucking corridors (e.g., interstate highways I-35, I-44, and I-40). Also shown is that parts of the transportation network that are less important for most industries may be quite important to the productivity of a particular industry (e.g., the Nonmetallic and mineral products (327) industry is influenced by the malfunction of the local railroad which connects the Port of Muskogee to the North America railroad (link (3,8)) though all the other five industries are much less vulnerable to this link). Understanding these inoperability-related vulnerabilities could motivate further studies to guide investments in alternative transportation modes. The inoperability values in Table 6 may appear to be negligible at first, but these numbers are significant when linked to the concept of

Table 6 Interdependent industry inoperability resulting from network component removal

Removed component

Food and beverage

Petroleum and coal

Chemical products

Nonmetallic mineral

Machinery mfg.

Misc. mfg.

Node 9 2.98E-04 7.15E-06 4.37E-04 1.21E-05 1.81E-04 2.64E-05

Node 8 4.93E-04 1.11E-03 4.25E-04 4.03E-05 1.82E-05 9.44E-05

Node 11 3.61E-05 6.83E-05 5.71E-04 8.27E-06 1.80E-04 2.37E-05

Link (1,7) 3.02E-04 1.11E-05 2.86E-05 9.12E-04 7.71E-06 5.56E-04

Link (9,11)

1.94E-04 5.20E-05 4.43E-04 6.54E-06 2.37E-06 1.42E-05

Link (2,5) 5.16E-04 2.28E-05 2.43E-05 1.69E-05 2.05E-04 3.36E-05

Link (8,4) 3.80E-04 9.33E-04 5.66E-04 3.34E-05 1.53E-05 8.01E-05

Link (2,8) 1.98E-04 1.02E-03 5.14E-04 3.15E-05 1.53E-05 7.61E-05

Link (2,4) 1.61E-07 4.63E-05 3.39E-06 1.20E-06 6.19E-07 2.79E-06

Link (3,8) 1.41E-07 7.11E-07 2.15E-06 1.06E-04 3.80E-05 2.84E-06

Link (8,7) 7.10E-08 4.90E-07 1.36E-06 1.26E-04 2.30E-07 9.62E-07

0.0E+00

5.0E-04

1.0E-03

1.5E-03

In o p e ra b ili ty

Fig. 4 Economic inoperability across six most important industries within the state of Oklahoma

Component Importance Measures for Multi-Industry...

failure probability in the reliability or quality engineering literature (i.e., the maximum allowable failure probability for a six-sigma compliant system is 3.4E-06).

In addition to inoperability, the complementary perspective of economic losses in Table 7 can supplement the analysis. The Petroleum and coal products (324) industry is a high dollar industry in Oklahoma, and this industry would be significantly impacted by a disruption that affects the functionality of rail transportation (e.g., a local railroad such as link (2,8), part of the level-one railroad that connects Oklahoma to the North America railroad such as link (8,4), or intermodal terminal facilities such as node 8. A second prominent industry is the Food, beverage, and tobacco products (311) industry, and several transportation components contribute to the dollar volume of production in this industry, especially a part of the inland waterway network that connects Port of Catoosa with the Port of New Orleans, LA (link (2,5)), and part of the North America railroad that connects Tulsa, OK with Texas City, TX (link (8,4)). In fact, the inter- modal terminal (node 8) which facilitates freight transport at the Port of Catoosa is a prominent component in the dollar volume of several exporting industries in Oklahoma. In general, rail transportation and major trucking corridors have a high impact on the economy of most industries, though less important components (e.g., part of the inland waterway such as links (2,4) or (2,5)) may still have a large impact on a particular industry (e.g., Miscellaneous manufacturing (339) and Petroleum and coal products (324) by millions of dollars).

4.4 Step 4. Vulnerability Analysis and Component Importance, Illustrated

The component importance measures, quantifying the proportional economy-wide impact of a loss of component p relative to a loss of the whole network, are calculated with Eq. (14) and are depicted in Fig. 5. This measure lies on [0,1], where ηp = 0 means that the removal of component p doesn’t affect the whole economy, and ηp = 1 means that the particular component removal shut downs the whole economy. As shown in

Table 7 Economic losses (in 100 million USD) across the six most important industries within the state of Oklahoma

Removed component

Food and beverage

Petroleum and coal

Chemical products

Nonmetallic mineral

Machinery mfg.

Misc. mfg.

Node 9 1.627 0.090 0.594 0.025 1.308 0.267

Node 8 2.688 13.974 0.578 0.082 0.132 0.954

Node 11 0.197 0.860 0.777 0.017 1.304 0.240

Link (1,7) 1.647 0.140 0.039 1.854 0.056 5.622

Link (9,11) 1.059 0.655 0.602 0.013 0.017 0.143

Link (2,5) 2.813 0.287 0.033 0.034 1.484 0.340

Link (8,4) 2.072 11.757 0.769 0.068 0.111 0.810

Link (2,8) 1.079 12.837 0.698 0.064 0.111 0.769

Link (2,4) 0.001 0.583 0.005 0.002 0.004 0.028

Link (3,8) 0.001 0.009 0.003 0.216 0.275 0.029

Link (8,7) 0.001 0.001 0.002 0.256 0.002 0.010

Darayi M., Barker K., Santos J. R.

Fig. 5, the most important components relate to the rail transportation and major trucking corridors. A main component of the rail freight transport, node 8 is the intermodal terminal facilitates the movement of commodities in the industrial park of Port of Catoosa to out-of-state customers and is the most important component in the analyzed transportation network. This facility is followed by link (8,4), a portion of railroad that connects Oklahoma to Texas City, TX, link (2,8), a local railroad that connects the Port of Catoosa to the North America railroad intermodal terminal, and link (1,7), a portion of railroad that connects Oklahoma City, OK to Chicago, IL. This suggests a further attention to the functionality of the facilities of the most important components within the network to avoid any malfunction, or in the case of any disaster which deactivates multiple components of the network, there should be priorities to recover the most important components. The framework proposed here could be used to evaluate alternative transportation modes for shipping commodities after a disruption or to guide planning for transportation investments to reduce vulnerability, and thus multi-industry impacts.

Figure 6 emphasizes component importance to individual industries, quantifying the proportional impact of a loss of component p on a particular industry relative to the

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

(1,7) LinkLinkNode 9 Node 8 Node 11

(9,11) Link (2,5)

Link (8,4)

Link (2,8)

Link (2,4)

Link (3,8)

Link (8,7)

V ul

ne ra

bi li

ty

Fig. 5 Network component importance measures across the Oklahoma economy using ηkp G; bð Þ

0.00

0.20

0.40

0.60

V u ln e ra b ili ty

Fig. 6 Network component importance measures focusing on particular Oklahoma industries using ηkp G; bð Þ

Component Importance Measures for Multi-Industry...

impact of a loss in all components on that industry, as calculated with Eq. (15). This measure lies on [0,1], where ηp

k = 0 means that the removal of component p does not affect industry k, while ηp

k = 1 suggests that the particular component removal completely shuts down industry k. As it is shown in Fig. 6, any failure that results in disconnection of link (2,5), the inland waterway connecting the Port of Catoosa to the Port of New Orleans, would have the largest impact on Food, beverages, and tobacco products (311) relative to other industries. The intermodal terminal (node 8), which connects the Port of Catoosa to its out-of-state customers through the North America railroad, has the largest impact on Petroleum and coal products (324). It also demon- strates that the malfunction of local railroads (e.g., link (2,8)) may have a high impact on the productivity of Petroleum and coal products (324). In addition, the Nonmetallic and mineral products (327) industry, primarily located in the Oklahoma City business economic area, is highly vulnerable to the functionality of the part of the North America railroad that connects Oklahoma City, OK with Chicago, IL. Note that some compo- nents are important from the perspective of a particular industry though perhaps not the entire economy, such as link (1,7), which suggests lower priority in Fig. 5 but is quite impactful for the Nonmetallic and minerals products (327) industry. Figure 5 also suggests that the Petroleum and coal products (324) industry can be impacted by the disruption of several network components, more so than any other industry.

5 Concluding Remarks

Transportation network vulnerability studies have largely attempted to quantify the reduction in system functionality, following a disruption, as (i) topological properties of the network, and (ii) flow importance measures. These structural and flow-related measures ignore a larger role that the transportation network plays in facilitating economic productivity. This work offers a broader perspective on freight transportation network vulnerability analysis with a means to measure importance of network com- ponents considering economic impacts of degradation of transportation network. In particular, this study considers a multi-modal freight network consisting of highway, railway, and waterway transportation, and implements the proposed vulnerability analysis framework to understand and rank the criticality of multi-modal transportation nodes and links.

A four-step approach (i) calculates baseline (undisrupted) multi-commodity flow according to minimum cost, (ii) measures slack at supply and demand nodes, in the form of undelivered supply and unmet demand, when individual components are removed (one-at-a-time) from the network according to a maximum flow perspective, (iii) relates slack in the network to perturbations and inoperability among interdepen- dent industries relying on commodities flowing along the network, and (iv) quantifies the importance of each component from industry-specific and overall regional economy perspectives. The primary contribution of this approach is the integration of the multi- commodity network flow representation of the multi-modal transportation network with the interdependent, multi-industry economic model and a framework to measure a transportation network component importance considering its multi-industry impact.

This approach is illustrated with a stylized case study of a multi-modal transportation network in the state of Oklahoma, where supply nodes are located within the state and

Darayi M., Barker K., Santos J. R.

demand nodes are located outside of the state. Results of the case study suggest that the Petroleum and coal products industry is particularly susceptible to disruptions in several components, and certain components can impact multiple industries. Also, analysis shows that the economy of the state and most industries are primarily vulnerable to the malfunction of the parts of the railway that connect the state to the North America railroad and major trucking corridors including interstate highways I-35, I-44, and I-40. While the application pursued in this study focused primarily on the state of Oklahoma, the base model can be applied to other freight transportation networks to identify the critical nodes/links that can instigate the largest vulnerability across interdependent sectors that uniquely vary from region to region. Hence, the proposed model and its future applications could provide significant value to homeland security preparedness planning.

Furthermore, the vulnerability analysis perspective proposed in this study can be implemented to highlight priorities in maintaining certain network components (to reduce common-cause failure), or in rerouting of commodity flows after a disruption. There also exists an opportunity to extend the base approach discussed in this work to analyze network completion strategies where capacity enhancement (e.g., link capacity) and additional transportation facilities (e.g., added links/nodes) could harden the network around the most vulnerable components. Further, longer term transportation infrastructure design plans could be informed by this kind of analysis.

Acknowledgements This work was partially supported by the National Science Foundation through award 1361116 and the Southern Plains Transportation Center under the University Transportation Center grant (DTRT13-G-UTC36) from the U.S. Department of Transportation.

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