Review on Energy Resilience

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International Journal of Production Economics

journal homepage: www.elsevier.com/locate/ijpe

A multi-industry economic impact perspective on adaptive capacity planning in a freight transportation network Mohamad Darayia, Kash Barkerb,∗, Charles D. Nicholsonb a Systems Engineering, Great Valley School of Graduate Professional Studies, The Pennsylvania State University, USA bSchool of Industrial and Systems Engineering, University of Oklahoma, USA

A R T I C L E I N F O

Keywords: Adaptive capacity Multi-industry impact Freight transportation Network resilience

A B S T R A C T

The multi-modal freight transportation network plays a vital role in maintaining commodity flows across multiple industries and multiple regions. As such, the effects of large-scale disruptive events could result in the closure of key transportation nodes and links, causing disruptions in commodity flows and larger disruptions to industries requiring those commodities for economic productivity. This work integrates a multi-commodity network flow formulation with an economic interdependency model to quantify the multi-industry impacts of a disrupted transportation network to devise contingent rerouting plans to strengthen the network's adaptive capacity. The formulation proposed here is illustrated with a freight transportation planning case study in the state ofOklahoma, consideringdisruptive scenarios inwhichanetwork component is lost andhow theproposed approach improves total economic productivity following a disruption.

1. Introduction and motivation

The US has defined a number of critical infrastructures, the dis- ruption of which “would have a debilitating impact on security, na- tional economic security, national public health or safety, or any combination of those matters” [White House 2013]. Among these cri- tical infrastructures are transportation networks, which enable the flow of people and commodities, and recent reports suggest that many highways, bridges, and other transit assets in the US fall short of a state of good repair, potentially threatening theefficiencyof thenetwork [US Department of Transportation, 2013]. In 2013, 55 million tons of goods valued at more than $49.3 billion

traversed the US freight transportation system each day, and freight tonnage and monetary value rose by 6.3 and 8.0 percent, respectively, over 2007 levels [US Department of Transportation, 2015]. Over the next 30 years, transportation's contribution to the US gross domestic product is expected to grow to approximately $1.6 trillion [US Department of Transportation, 2015]. Given the potential for disrup- tionbymalevolent attacks, natural disasters, human-madeaccidents, or common failures, recent US planning documents focus on the criticality of transportation network preparedness [The House Committee on Transportation and Infrastructure 2013; US Department of Transportation, 2014; Yusta et al., 2011]. Emphasis has been placed on “securing and managing flows of people and goods” along

transportation networks [DHS, 2014]. The consequences of disruptions to critical infrastructures highlight

the need to better understand resilience, or the ability to withstand the effects of and recover timely from a disruption. Particularly for critical infrastructures, The Infrastructure Security Partnership (2011) noted that a resilient infrastructure sector would “prepare for, prevent, pro- tect against, respond or mitigate any anticipated or unexpected sig- nificant threat or event” and “rapidly recover and reconstitute critical assets, operations, and serviceswithminimumdamageanddisruption.” As with any other critical infrastructure, resilience planning is im- portant formulti-modal transportationnetworks due to their role in the economic vitality of states, regions, and the broader country. The functionality of this network is threatenedbydisruptive events that can disable the capacity of the network to enable flows of commodities in portions of nodes and links [Kengpol et al., 2012; Miller-Hooks et al., 2012; Lee and Kim 2010]. Transportation network disruptions lead not only to physical damage, but also to an interruption of economic pro- ductivity across multiple industries due to infrastructure inoperability [Tierney 1997; Webb et al., 2000, Ham et al., 2005; Pant et al., 2011; Park et al., 2011]. As such, a comprehensive discussion of transporta- tion network resilience should account for multi-industry impacts. The use of the term “resilience” has increased substantially in the

literature in recent years [Mattson and Jenelius, 2015; Hosseini et al., 2016; Kamalahmadi and Parast, 2016], recognizing a shift in planning

https://doi.org/10.1016/j.ijpe.2018.12.008 Received 28 June 2016; Received in revised form 5 December 2018; Accepted 8 December 2018

∗ Corresponding author. School of Industrial and Systems Engineering, University of Oklahoma, 202 W. Boyd St., Room 124, Norman, OK 73019, USA. E-mail address: [email protected] (K. Barker).

International Journal of Production Economics 208 (2019) 356–368

Available online 14 December 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.

T

from prevention and protection to preparing for the inevitability of disruption. Several qualitative and quantitative frameworks being proposed to describe the resilience of a system (e.g., Patterson et al., (2006); Zobel (2011);Sarre et al., (2014);Patterson et al., (2006); Zobel (2011); Sarre et al., (2014)). In particular, the paradigm proposed by Henry and Ramirez-Marquez (2012), along with several applications and extensions [Barker et al., 2013; Pant et al., 2014a; Baroud et al., 2014], quantifies system resilience as a function of time. Fig. 1 depicts system performance, generally quantified with function t( ), before, during, and after a disruptive event (e.g., t( ) could describe traffic or commodity flow in a transportation network over time). Fig. 1 high- lights two dimensions of resilience: vulnerability, or the extent to which performancedegradesafter adisruption [Zioet al., 2008, Jonssonet al., 2008, Zhang et al., 2011], and recoverability, or the ability to return to a stable, desired level of performance [Barker et al., 2013; Pant et al., 2014a]. Similarly, Vugrin and Camphouse (2011) suggest that the resilience

capacity of a system is a function of three components: (i) absorptive capacity, or the ability of a system to absorb or withstand a disruption with essentially no change in performance, (ii) adaptive capacity, or a short-term means to quickly regain a desired performance, and (iii) restorative capacity, or the long-term repair of physical damage. Vugrin and Camphouse (2011) pose absorptive, adaptive, and restorative ca- pacities as first, second, and third “lines of defense,” where the next is engaged if the previous fails. In a transportation network context, (i) absorptive capacity may describe the physical characteristics of, say, a bridge to withstand the shock of an earthquake, (ii) adaptive capacity may include alternate paths in the network that could be engaged quickly to work around damaged areas, and (iii) restorative capacity may describe the long-term bridge reconstruction activities required to restore the transportation network. Relative to Fig. 1, the collection of absorptive and adaptive capacities may reduce vulnerability, while restorative capacity would improve recoverability. While most definitions of resilience recognize the time-dependent

nature of withstanding and recovering from a disruption, Rose (2004) defined static resilience as “the ability of an entity or system to maintain function when shocked.” This is depicted in Fig. 2, where %ΔDYmax

represents the maximum percentage change given the worst-case level of performance following a disruptive event, and %ΔDY represents the actual percentage change in the performance of the system (assuming the implementation of a mitigation strategy) [Rose 2009]. The original application of static resilience, as well as several subsequent studies (e.g., Rose (2007,2009),RoseandWei (2013),Hallegatte (2014), [Pant et al., 2014a, b], Baghersad and Zobel (2015)), deal with economic disruption. Mathematically, static resilience ismeasured in terms of the maximum potential drop in system performance and the estimated performance drop, as shown in Eq. (1). This quantitative approach is used in this study to define a performance measure for post-disaster rerouting, though we prefer the term adaptive capacity rather than static resilience.

=static resilience % DY % DY % DY

max

max (1)

Faturechi and Miller-Hooks (2015) thoroughly review the literature on transportation system performance considering disruptions to phy- sical infrastructure. Defining a four-phase disaster life cycle as (i) mi- tigation, (ii) preparedness, (iii) response, and (iv) recovery, they sug- gest that most work focuses on assessing the transportation system's ability to deal with disruption consequences, with less work assessing strategies to manage the system after the disruption. Further, the lit- erature that seeks rerouting strategies to mitigate the effects of dis- ruption by maintaining freight flow through a residual network is sparse [Khaled et al., 2015, Gedik et al., 2014]. And, to the author's knowledge, the approach proposed here to reroute flow and plan for adaptive capacity by considering the contribution of transportation network components to multi-industry impacts is non-existent in the literature. To address this gap in the literature, we propose an in- tegrated optimization formulation to reroute commodities through the residual network to decrease the effect on local industries requiring those commodities for production. To do so, we combine a multi- commodity network flow formulation of a multi-modal transportation networkwith a risk-basedmulti-industry impactmodel in an integrated formulation. In particular, we integrate adopt an input-output model to represent multi-industry impact, chosen because of its ease in integra- tion with an optimization formulation (discussed in detail later). An- other popular option for multi-industry impact is the computable gen- eral equilibrium (CGE) model, a multi-layer agent-based simulation meta model that simulates agents in an economy that react to price and quantity signals, and such a model does not lend itself so easily for use in a multi-commodity network flow optimization model. This paper is arranged as follows. Section 2 describes the proposed

approach to plan for adaptive capacity in a disrupted freight trans- portation network, developing a model to accommodate flow through the residual network after disruption by integrating a multi-commodity network flow formulation with a risk-based economic interdependency model. Section3presents an illustrative example, developedbasedona partial freight transportation network within the State of Oklahoma consisting of three important business economic areas and the multi- modal freight network infrastructure which facilitates trade with cen- ters out of the state. Section 4 provides concluding remarks and future research avenues of this work.

2. Methodological background

A disruption within a freight transportation network affects its vital role in transporting raw materials among manufacturers and final products between manufacturers and consumers. Such a disruption in the flow of commodities leads to economic losses across multiple

Fig. 1. System performance, φ(t), trajectory facing a disruptive event [Henry and Ramirez-Marquez 2012].

No disruption level

Expected performance level

Worst-case performance level

Sy st

em o

ut pu

t

% DYmax

% DY

Fig. 2. Theperformance components of static resilience [Rose2009; Pant et al., 2014b].

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industries. To devise an adaptive capacity strategy (i.e., post-disruption rerouting) to lessen total economic losses following a disruption, we propose an optimization framework that integrates (i) a multi-com- modity network flow model of freight movement, (ii) a risk-based in- terdependency model of multi-industry impacts, and (iii) an objective function that addresses adaptive capacity with a measure of static economic resilience [Rose 2009, 2013; Pant et al., 2014b]. The pro- posedoptimizationmodel isdeveloped followinga three-stepapproach, illustrated in Fig. 3.

2.1. Freight movement and disruption

To model a supply-demand network for a set of business economic areas consisting of different industries interacting with their suppliers and customers located outside of their region through a multi-modal freight transportation system, a typical multi-commodity network flow (MCNF) model (e.g., Ahuja et al. (1993);Ahuja et al., (1993)) is used. The goal of this model is to facilitate the commodity flows between suppliers and consumers through a capacitated transportation network while minimizing the cost of transportation. Planning decisions in a multi-modal freight transportation network is made at strategic, tac- tical, and operational levels [Crainic and Laporte 1997]. It is assumed that (i) strategic decisions determine general development policies and define the operating strategies of the system over relatively long time horizons (e.g., the location of the physical transportation network, the location of main facilities such as rail yards or multi-modal platforms [Liotta et al., 2015]), (ii) tactical plans deal mostly with medium-term decisions (e.g., route choice and type of service to operate, aggregate scheduling [Kengpol et al., 2012]), and (iii) operational level decisions aremadewhen real or near real-time response is required (e.g., crewor container scheduling [Wang and Yun, 2013]). In this work, when a disruption interrupts the movement of commodities through the net- work, a tactical contingent rerouting plan is sought, for the period of disruption, tomaintain the functionality of the supply-demandnetwork as much as possible. The topology of the multi-modal freight transportation network, as

well as corresponding supply and demand nodes, must be extracted to model and analyze the behavior of the network before and after dis- ruption. The transportation network is considered to be a facilitator of K interacting industries, where multiple supply and demand nodes of commodity k could represent a particular industry. Based on a con- ventional MCNF model, the network is defined on directed graph

=G N L( , ), where N is a set of nodes, each of which could be home to either suppliers or consumers of multiple commodities, and L is a set of links connecting nodes. For this graph, K denotes the number of com- modities in a network instance, each representing an industry. Let fij

k

denote the decision variable associated with the flow quantity of commodity …k K{1, , } on link i j L( , ) . Let parameter wijk denote the associated per-unit transportation cost. The costs differ based on link properties such as length and transportation mode (e.g., waterway, railway, highway). Let parameter uij denote the total flow capacity of link i j L( , ) . That is, the capacity of each link is a shared or “bundle” constraint for all commodities flowing on the link. The supply/demand requirement of commodity k at node i N is denoted by parameter bik. If bik is positive, thennode i is a supplynode of commodity k. Similarly, if bik is negative, then node i is a demand node for commodity k. If bik is zero, the node i is a transshipment node with respect to commodity k. Themathematical formulation for theMCNFproblem isprovided inEq. (2). Without loss of generality, each node within the network can be home to either suppliers or consumers ofmultiple commodities. The set of nodes then can be partitioned into three mutually exclusive sets:

= +N N N N( , , )0 where N denotes the set of nodes representing nodes which arehome to consumers, +N denotes which arehome to suppliers, and N0 denotes all transshipment nodes. Each commodity belongs to an industry in the economy as defined by the North American Industry Classification System (NAICS).

= = …

= …

w f

f u i j L

f f b i j L k K

f i j L k K

min

s.t. , ( , )

, ( , ) , 1, ,

0, ( , ) , 1, ,

i j L k ij k

ij k

k ij k

ij

i j L ij k

j i L ji k

i k

ij k

( , )

( , ) ( , )

(2)

From a tactical point of view, integrating (i) industries 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 commodities from suppliers to demand nodes via fij

k, collectively representing the flow of commodities on the links of a baseline (undisrupted) network. Natural hazards, human-made events, or common failures could

threaten the functionality of thenetwork components and consequently interrupt commodity flows. A scenario-based removal of network components known as interdiction [Murray et al., 2008] is a common theme in modeling and analysis of supply-demand network disruption. The consequences of a hazards, attacks, or failures are simulated as disruptions in the flow of valuable goods or services through the net- work causedbydisablingnetwork components. The functionality of the network is analyzed to determine how vulnerable it is to interdiction, and which nodes or links, if lost, result in the most damage to network performance. Interdiction analyses encompass a wide range of possible disruptions that may vary with respect to spatial scales, correlation of disruptive events, sequence of failures, and event duration. A disruption scenario is defined as the set of network components

that are impacted, the degree to which they are disabled, and the op- erating conditions (e.g., network activity, link/node capacities) of the network prior to the disruption regardless of the initiating event that causes the disruption. Different approaches to model a transportation network disruption have been offered (e.g., losing a bridge, a road segment, or ahub [Jenelius andMattson, 2012; Burgholzer et al., 2013; Rupi et al., 2014]), with most approaches considering one component being affected [Faturechi and Miller-Hooks 2015].A disruptednetwork component may be rendered completely inoperable by a disruption (e.g., losing a road completely due to a bridge collapse), or its func- tionality may drop to a lower level (e.g., an accident blocking a single lane of an interstate highway segment). Simulating the disruption sce- nario enables the evaluationof the impact of the failure. Impacts canbe considered as the direct associated failures in network operability (e.g., floworcapacity reduction)or consequential failures (e.g., theeconomic impacts affecting the production and consumption of flows) [Matisziw and Murray 2009]. It takes time to recover affected network compo- nents (e.g., afterHurricaneKatrina, it tookup to sixmonths in southern regions to recover highway networks, whereas northeast regions

Fig. 3. Three-step approach to planning for adaptive capacity with multi-in- dustry impacts.

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recovered much more quickly [DesRoches, 2006]; after an I-40 bridge collapsed in Oklahoma following a barge collision in 2002, traffic was rerouted for nearly two months while crews rebuilt the infrastructure [Aydin and Shen, 2012]). As such, devising an efficient and effective contingent rerouting strategy immediately after extreme events would assist the economic productivity of the disrupted region. In the case of any disruption modeled as the removal of a network

component or a set of components (or a drop in functionality of the network modeled as reduction of link capacities, uij), the consequences are sought bydeducting the commodityflowson theaffected links from thebaselineflow, as calculated inEq. (1). Let =G N L( , ) represent the network after disruption with updated sets of links, L and nodes, N . The sets N , +N , N0 denote the post disruption sets of nodes associated with home of consumers, home of suppliers, and transshipment nodes, respectively. The quantity of commodity k at node i that is either un- delivered and remaining with the suppliers, or unsatisfied demand of consumers, is reflected in the slack variable Sik. This slack variable will be used subsequently to drive the calculation of inoperability among multiple industries. It is assumed that each type of commodity re- presents theoutput of a lone industry, and interdependent inoperability propagated through a set of industries caused by unsatisfactory de- mands/supplies will be modeled in the next section.

2.2. Multi-industry impact

In this work, we use an extension of the input-output economic model, forwhichWassily Leontief (1966)wonaNobel Prize, to capture the multi-industry impacts of unmet demands at demand nodes and remaining commodities at supply nodes as the result of a disruption to components of the transportation network. The input-output (I-O) model is a widely accepted model for analyzing the interdependent connections among industries [Miller and Blair 2009], and the use of the I-O enterprise for studying disruptions was among the 10 Most Important Accomplishments in Risk Analysis: 1980–2010 [Greenberg et al., 2012]. Under a static equilibrium, the total outputof industry (or economic

sector) k is distributed to other industries and also satisfies external (consumption) demand. Under a proportionality assumption, this equilibrium condition is described with = +=x z ck r

K kr k1 , where xk is

the total output of industry k, zkr is the input of industry k to the production of industry r (intermediate consumption), and ck is the ex- ternal (final) consumption for industry k's output. The intermediate consumption, zkr, is assumed to be proportional to the output of in- dustry r ( …r K r k{1, , } and ), expressed as =z a xkr kr r. In the common form of the Leontief I-O model, industry production is mod- eled as = +Axx c, where x is the vector of industry production out- puts, A is an industry-by-industry matrix of interdependency coeffi- cients, akr (proportion of industry k's input to r, with respect to total production of industry r), and c is a vector of final consumption. The model shows that total production is made up of industry-to-industry intermediate production, Ax, and production to satisfy final con- sumption, c. The availability of data describing the parameters of the I-O model

in the US through the Bureau of Economic Analysis (BEA) (2010), as well as a number of other countries [OECD 2012], justifies the ex- tensive use of I-Omodels. Tomodel thepropagationof inoperability, or the proportional extent to which industries are unproductive after a change in final consumption or a forced change in final consumption due to a lack of supply, Santos and Haimes (2004) propose the In- operability Input-Output Model (IIM), extending the capability of the I- O model to model not only economic interdependency but inter- dependency in broader infrastructure sectors. This risk-based model is defined from two metrics [Haimes et al., 2005, Santos 2006]: (i) in- operability qk and (ii) final consumption perturbation ck , which are defined in Eqs. (4) and (6), respectively. Providing a different per- spective from the traditional I-O model, the IIM shows how normalized

production losses propagate through interconnected industries with a normalized interdependency matrix A . Describing the relationships among K industries, resulting in matrices of size ×K K and vectors of length K, Eq. (3) formulates the propagation of the inoperability in a group of interconnected industries.

= + =q A q c q I A c[ ] 1 (3)

Vector q is a vector of industry inoperability describing the pro- portional extent towhichas-plannedproductivityor functionality is not realized following a disruptive event. Inoperability for industry k is defined in Eq. (4), where as-planned total output is represented with x̂k and degraded total output resulting from a disruption is represented with x̃k. An inoperability of 0 suggests that an industry is operating at normal production levels, while an inoperability of 1 represents the situation in which an industry is completely inoperable.

= =q x x x q x x x( ˆ ˜ )/ ˆ [diag ( ˆ ) ] ( ˆ ˜ )k k k k 1 (4)

A normalized form of the original A matrix describing the extent of interdependence among a set of industries or sectors is defined as A . The row elements of A indicate the proportion of additional inoper- ability that are contributed by a column industry to the row industry, shown in Eq. (5),

= =a a x x A x A x( ˆ / ˆ ) [diag ( ˆ ) ] [diag ( ˆ ) ]rk rk r k 1 (5)

The calculation of c , a vector of normalized final consumption reduction is provided in Eq. (6), where the elements of c represent the difference in as-planned final consumption ĉk and perturbed final con- sumption c̃k divided by as-planned production, quantifying the reduced final consumption for industry k as a proportion of total as-planned output.

= =c c c x c x c c( ˆ ˜ )/ ˆ [diag ( ˆ ) ] ( ˆ ˜)k k k k 1 (6)

In addition to industry inoperability, a traditional economic loss metric can be calculated by multiplying each industry's production level, xk, in dollars, by its inoperability level: for industry k, =Q x qk k k. Such a measure can also be expressed for the collection of K industries,

=Q x qT . As such, decisions to plan for adaptive capacity can be made with respect to economic impact across multiple industries. The freight transportation network provides a platform for com-

modity flows between industries. Since the IIM models how demand- related risk in a given industry propagates to other industries due to their interdependent productivity, the multi-industry impact of a dis- ruption to a freight transportation network can be studied when net- work losses are related to final consumption reduction and inoper- ability terms as shown in subsequent subsections. The demand- reduction IIM proposed by Santos and Haimes (2004) has been suc- cessfully employed to study multi-industry impacts of perturbations in supply and demand (e.g., Resurreccion and Santos (2013);Pant et al., (2011);Haggerty et al., (2008);Lian and Haimes (2006);Resurreccion andSantos (2013); Pant et al., (2011);Haggerty et al., (2008); Lian and Haimes (2006)). However, some (e.g., Kujawski (2006);Kelly (2015); Kujawski (2006); Kelly (2015)) have questioned the usefulness (and theoretical plausibility) of supply-driven models developed from con- cepts by Ghosh (1958). Leung et al., (2007) integrated a supply-side price IIMandoutput-side IIM toaddress initiatingperturbations related to input factors (value added) and to industry output levels, though some aspects of this model may be impractical for integration with supply-demand networks as applied in our proposed approach (though may be effective in modeling disruptions to manufacturing systems, as noted by Kelly (2015)). Here, we translate a disruption in the form of remaining commodities at supply nodes and/or unmet demand at de- mand nodes into the two IIM metrics of inoperability and final con- sumption perturbation, based on a demand-reduction IIM implemented by Pant et al., (2011) in modeling supply and demand perturbation caused by a port closure. Pant et al., (2011) considered commodities remaining at suppliers after a disruption to calculate the final

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consumption perturbation. And the authors considered unmet demands to calculate a “forced” demand reduction, assuming that a disruption decreases the supply of a commodity for a demand node while the final external consumption remains virtually unaffected. In such a case, the demand nodes temporarily sacrifice their internal need for that com- modity until it returns to its as-planned supply level, and a surrogate to supply reduction is calculated from the combination of “forced” in- ternal consumption and an output inoperability. In the following subsections, N represents the set of nodes within

the area of interest , and N ¯ represents the set of nodes outside of the area of interest, such that =N N N ¯. We formulate the economic consequences of a failure within a particular area of interest (e.g., a business economic area, county, state, entire country). As such, the failure in the form of remaining commodities at suppliers and unmet demand at consumers are captured only in the nodes within the area of interest and each of the economic parameters (i.e., x, c, c and q) are indicators of the industries specific to the regionof interest. To simplify thenotation, superscript is not included for these economicmetrics to avoid unnecessary indices.

2.2.1. Modeling remaining supply Transportation facilities operate as facilitators of commodity flows

across business economic areas. For a supplier of commodity k located in node i, any transportation network disruption that perturbs its de- sired export will be considered to be a reduction in final consumption. As modeled in Eq. (7), final consumption for industry k includes com- modities consumed by industry k itself internally, c( ˆ )k int, and the amount of external consumption that is exported through the network, c( ˆ )k G. It is assumed that the disruption results in losses of commodity flows only through the network, while industry production activities unrelated to the network experience no direct failure but might be af- fected indirectly by a disruption within the network (due to an inter- dependent loss of economic productivity). When industry k has diffi- culty only in exporting commodities, it experiences commodities remaining at supply nodes in the region of interest totaling

+ Si N N i

k ( )k

, where +N represents the set of nodes that are home to suppliers in the regionof interest after thedisruption, as shown inEq. (8). As such, the final consumption perturbation for industries that experience difficulties only in exporting commodities is modeled as the amount of slack divided by as-planned industry output in Eq. (9), Note that the supply-demand network may consist of suppliers and con- sumers located outside of the region of interest, yet failures to these suppliers and consumers are not accounted for in this model.

= + …c c c k nˆ ( ˆ ) ( ˆ ) , {1, , }k k int k G (7)

= … +

c c S k nˆ ~ , {1, , }k k i N N

i k

( )k (8)

= …+c S

x k n

ˆ , {1, , }k

i N N i k

k

( )k (9)

2.2.2. Modeling unmet demand As discussed by Pant et al., (2011), the amount of import (input) of

industry k at demand nodes in the supply-demand network defined as bi N N i

k ( )k

contributes toward the production activity and the internal consumptionof industry k. Thus,when industry k hasdifficulty only in importing commodities, it experiences unmet demands in the region of interest totaling Si N N i

k ( )k

. This results in the loss of output, x̂k, representing x x( ˆ ˜ )k k , and final internal consumption,

c( ˆ )k int.Here, N represents the set of nodes afterdisruption located in the geographical area of interest that are home to consumers of commodity k.

= + …S x c k nˆ ( ˆ ) , {1, , } i N N

i k

k k int ( )k (10)

Therefore, for industry k, unmetdemandcauses an inoperability, qk, measured as the loss of production in industry k as a proportion of its original production level, as shown inEq. (4)with x xˆ / ˆk k. Also, internal consumption failure, as shown in Eq. (7), causes a final consumption perturbation, ck , and is modeled as a measure of the change in thefinal consumptionasaproportionof theoriginal production level in industry k, as shown in Eq. (6) with c xˆ / ˆk k. The approach to formulate failure in the form of unmet demand is adapted from the port disruption work of Pant et al., (2011, 2015) and the transportation network vulnerability formulation of Darayi et al., (2017), in which a slack variable Sik is defined to captureunsatisfieddemandatdemandnodes (orundelivered commodities remaining with the suppliers), shown in Eq. (11). For the industries experiencing difficulties only in importing their required commodities, there exists afinal consumptionperturbation, asmodeled in Eq. (12).

= … c

x

S x

x k n

ˆ ˆ

ˆ

ˆ , {1, , }k

k

i N N i k

k

k

( )k (11)

= …c S

x q k n

ˆ , {1, , }k

i N N i k

k k

( )k (12)

Eqs. (9) and (12) combined with the IIM in Eq. (3) form a complete solvable system that quantifies the inoperability and final consumption perturbations for the collection of K interconnected industries. For simplicity, thedemandperturbations inEqs. (9) and (12)assume failure in either only demand nodes or only supply nodes within a particular industry, whereas in actual situations, some industries would likely consist of both supply and demand nodes. Therefore, the total final consumption perturbation for industry k, in the case of having both importing (demand) and exporting (supply) roles, is given in Eq. (13).

= ++c S S

q k n x̂ x̂

, {1, ..., }k i N N i

k

k

i N N i k

k k

( ) ( )k k (13)

Any of Eqs. (9), (12), or (13) captures the perturbation vector c that parameterizes the interdependency model in Eq. (3) based on the exporting or importing nature of the nodes belonging to each industry. Thus, q 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, and a contingent rerouting strategy can be devised during the period of disruption to lessen the multi-industry impact of the disruption.

2.3. Planning for adaptive capacity

Adaptive capacity is considered to be the extent to which a freight transportation network is capable of facilitating economic productivity by the (short-term) rerouting of commodities through the residual network to reduce remaining commodities at suppliers and unsatisfied demand at consumers. Inoperability in industry k is calculated with Eq. (3), and economic losses for industry k can be found by multiplying the proportional inoperability by expected production level in monetary units, =Q x qk k k. Economic losses for the entire set of industries is cal- culated with =Q x qT . As such, inoperability or economic impact at the industry level, or total economic impact at the across all industries, can be used to valuate strategies for strengthening adaptive capacity. Pro- posed in Eqs. (14) and (15) are two such metrics motivated by Eq. (1). When planning emphasis is placed on a particular industry (i.e.,

rerouting freight in the transportation network to reduce the impact to industry k), Eq. (14) is proposed to valuate a strategy to strengthen adaptive capacity. Term e

k is a proportional measure involving (i) the economic loss, Qek, experienced by a particular industry k following disruptive event e when no adaptive capacity planning is taken and (ii) the economic loss, QRk, in industry k when a strategy is taken to avoid the maximum economic loss in that particular industry.

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= Q Q

Qe k e

k R k

e k (14)

For a perspective that spans all industries, Eq. (15) provides a si- milar proportional metric, where Qe is the multi-industry economic loss caused by disruption e in the baseline case, and QR is themulti-industry losswhena rerouting strategy is taken to avoid themaximumeconomic loss.

= Q Q Qe

R e R

e (15)

Assuming a multi-industry perspective and considering a hypothe- tical decision maker interested in limiting economic losses across multiple industries, Eq. (15) serves as the objective function in the following optimization framework that integrates the multi-commodity network flow model from Section 2.1 and the Inoperability Input- Output Model from Section2.2. Following aparticular disruption e that affects a particular set of transportation links, the proposed model in Eqs. (16)–(24) seeks to optimally reroute the flow of commodities through the residual network such that a measure of static economic resilience is minimized. Here, it is assumed that the result of the model provides decision makers with a rerouting strategy across different modes. The period of disruption is assumed to be sufficiently long en- ough to employ intermodal container scheduling models (e.g., Lee and Kim (2010); Wang and Yun (2013);Lee and Kim (2010); Wang and Yun (2013)) to devise operational-level plans based on the resulted con- tingent rerouting strategy in the simplified static supply-demand net- work. Notation employed in the problem formulation is summarized as follows, noting that network variables (e.g., the sets of links and nodes) with a prime as superscript are related to the network after disruption, referred to as the residual network.

Parameters L set of links N set of nodes Nk set of nodes related to industry k uij capacity of link i j( , ) after dis-

ruption N0 set of transshipment nodes qk inoperability of industry k

N ' set of nodes that are home to consumers

N set of nodes that are home to consumers in the region of in- terest

+N set of nodes that are home to suppliers

+N set of nodes that are home to suppliers in the region of in- terest

i intermediate variable to keep the slack at node i positive

bi k mass-balance variable repre-

senting demand/supply/trans- shipment at node i after dis- ruption

µk binary coefficient with value 0 when no unsatisfied demands at demand nodes and 1 when at least one demand node with un- satisfied needs

Sik slack variable that captures undelivered commodity k re- maining with the supplier node i or unsatisfied demand at de- mand node i

ark elements of the normalized inter- dependency matrix A

ck final consumption perturbation for industry k

xk production level of industry k in monetary value

Decision variable

fij k integer variable represents the

flow of commodity k across link i j( , ) in the network after disrup- tion

Based on this notation, planning for adaptive capacity by rerouting the flow of commodities through the residual network is formulated as follows.

max eR (16)

= f u i j Ls.t. , ( , )

k

K

ij k

ij 1 (17)

+ = =f f S b k K, 1, ..., i j L

ij k

j i L ji

k i i

k i

k

( , ) ( , ) (18)

= + + i N i N i N

1 for 1 for

0 for i

0 (19)

= + =+c S

x

S

x µ q k K

ˆ ˆ , 1, ...,k

i N N i k

k

i N N i k

k k k

( ) ( )k k (20)

= M

S µ M S k K1 , 1, ..., i N N

i k

k i N N

i k

( ) ( )k k (21)

= + q

q

a a

a a

q

q

c

cK

K

K KK K K

1 11 1

1

1 1

(22)

= =

Q x qR k

K

k k 1 (23)

= =

f Z i j L k K µ k K

, ( , ) , 1, ..., {0,1}, 1, ...,

ij k

k (24)

The formulation implements the idea of planning for adaptive ca- pacity in a disrupted transportation network where the residual active network is presented by =G N L( , ), with updated sets of links, L , and nodes, N . The bundle constraint in Eq. (17) ties together the com- modities by restricting the totalflowofall the commoditiesoneach link i j( , ) to at most uij, the capacity of that particular link after disruption. In other words, all industries share the capacity of network compo- nents, resulting in competition among them for a share of undisrupted capacity. fij

k represents the flow of commodity k across link i j( , ) which remains in theupdated setof links, L . Eq. (18) representsmassbalances on each node, where bi

k captures demand/supply at each node in the residual network. A slack variable Sik is defined to capture undelivered commodities remaining with the suppliers, or unsatisfied demand at demand nodes. The magnitude of Sik is positive, and multiplier i takes on a negative value for set of demand nodes (after disruption) N , a positive value for supply nodes (after disruption) +N , and zero for transshipment nodes (after disruption) N0, as shown in Eq. (19). Eqs. (20)-(22) are constraints that translate remaining commodities at supply nodes and unsatisfied demand at demand nodes (in the geo- graphical area of interest, α) into multi-industry inoperability. Here, ck transfers remaining commodities of type k at the supplier and/or un- satisfied demands, Sik, into a final consumption reduction from Eq. (13) with respect to the total output of that particular commodity, re- presenting the total output of industry k, x̂k. Considering Nk as set of nodes related to industry k (in the residual network), which either supply or demand commodity k, in Eq. (20), qk is added to capture the consequences of unsatisfied demand at nodes within the region on the inoperability of that industry, reasoning that any disruption leading to unsatisfied demands has an impact on the output of that particular industry which needs to be taken care of in the total interdependent inoperability. As the network might connect industries within the re- gionof interest into their suppliers or customers out of thegeographical area of interest, it is desired to consider the effect of failure in terms of remaining commodities at suppliers in the region of interest re- presented by +N , and unmet demand at demand nodes within the re- gion of interest represented by N A binary coefficient, µk, in Eq. (20) takes on value 0 when there are no unsatisfied demands at demand nodes within the region under study and 1 when there is at least one demand node with unsatisfied needs. Eq. (21) requires that µk be binary, defining a sufficiently large M. Eq. (22) implements the IIM to

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capture the adverse effect of the disruption in terms of remaining commodities at supply nodes andunsatisfieddemandat demandnodes. The multi-industry economic impacts of the failure devising a rerouting strategy are captured in Eq. (23) with total economic loss QR. This equation integrates themonetaryvalueof theflowof each industry into the objective function. The objective function aims to maximize the progress in reducing economic loss, and as the structure of the network does not let the improvement in all industries flow at once, the objec- tive function prioritizes the commodities that most affect the reduction in economic loss. And the objective function is the proportional eco- nomic saving, parametrized based on Eq. (15) in which Qe, maximum economic loss experienced by the whole economy in the case of a dis- ruption when no mitigating strategy is taken, is already calculated basedonSection2.1. and2.2. Theproposed approachbenefits from the flexibility, scalability, and efficiency of the base MCNF paradigm with respect tooptimization [Ahujaet al., 1993;Manfren2012], aspracticed in modeling interdependencies in critical infrastructure networks (e.g., Lee et al., (2007);Holden et al. (2013);Lee et al., (2007); Holden et al., (2013)). The complexity of the model is n KO ( )2 , and it has KO ( ) binary

variables.Wedonote that althoughcomplexityof theproblem is linear, but the number of industries would not be drastically large, meaning that the number of constraints would be computationally manageable. However, as the network flow variables are defined as integer, nO ( )2 , the increase in the size of the network, combined with the number binary variables, complicates the calculations for large instances. In the stylized case study in the next section, the model performs well for small to medium scale problems, and the average solution time is less than 5s. Also, it is possible to enhance the performance of the for- mulation for large scale problems by relaxing the integrality of the network variables.

3. Illustrative example: multi-modal freight transport in Oklahoma and the surrounding region

A multi-modal freight transportation network, consisting of three important interstate highways, railways, and inland waterways that connect to the Mississippi River Navigation System via two ports, plays an important role in transporting commodities produced in thebusiness economic areas within the state of Oklahoma to consumers in neigh- boring states. A portion of this multi-modal freight transportation net- work is illustrated on a case study to implement the proposed model to improve adaptive capacity with a post-disruption rerouting strategy. A scenario-based disruption defined as the removal of a particular net- work component is considered in the illustrative example. Customers in surrounding states are considered to be four combined demand nodes connecting to Oklahoma's multi-modal freight transportation network. The multi-industry impact of the disruption within the economy of the state of Oklahoma guides the rerouting of commodities throughout the residual network as an adaptive (short-term) strategy. This illustrative network is adapted from Darayi et al., (2017). The case study has been solved using optimization software LINGO, version 15.

3.1. Supply-demand network

Fig. 4depicts a supply-demandnetworkconsidering supplynodesas the three important business economic areas within the state of Okla- homa, consisting of Oklahoma City (node 1), the Port of Catoosa in Tulsa (node2), and the Port of Muskogee (node3). Customers (demand nodes) in the most important states interacting with Oklahoma in- dustries are Texas, Louisiana, Arkansas, and Illinois [Ingalls et al., 2002]. The multi-modal freight transportation network, which enables the

commodityflows fromsupplierswithin the stateofOklahomato theout of state consumers, is discussed inbrief inTable1.Thenetworkconsists of a part of interstate highways I-35, which connects Oklahoma to the

north-south corridor, and I-40 and I-44,which enable trade through the east-west corridor. Part of US highways 169 and 165 within Oklahoma connects the Port of Catoosa and the Port of Muskogee to the interstate highway network. In addition to the truck way facilities, an intermodal rail-truck facility in Oklahoma City near the junction of I-35 and I-40, and the one in Tulsa, OK, which run by Burlington Northern Santa Fe (BNSF) railroad are considered in developing the network, as well as part of the inland waterway network navigated by McClellan–Kerr Arkansas River Navigation System which connects the Port of Catoosa and the Port of Muskogee to the Port of New Orleans, LA (node 5), the Port of Chicago, IL (node 7), the Port of Little Rock, AR (node 6), and the Port of Texas City, TX (node 4). As defined by NAICS, 62 industries operate in Oklahoma, therefore

the A matrix regionalized for Oklahoma is 62 × 62. Due to high trade figures reported by Bureau of Transportation Statistics (2010a), six industries are considered to be industries that primarily export com- modities to out-of-state customers, listed in Table 2. Discussed pre- viously, it is assumed that each commodity belongs to an industry as defined by NAICS economic sectors, and each node within the network is considered to be home to either suppliers or consumers of multiple commodities. Basedon thecombinedestimatedannual supplyanddemand in tons

for the associated industries and states compiled from different data- bases [US Army Corps of Engineers, 2013, Tulsa Port of Catoosa 2013, Bureau of Transportation Statistics 2010a,b, Port of Muskogee 2013, Bureau of Economic Analysis 2010], a list of monthly supply and de- mand is presented in Table 3 (assuming constant monthly demand, or annual demand divided by 12).

3.2. Freight movement and disruption

To parametrize the MCNF model in Eq. (1), the cost vector is computed based on the transportation mode and the mileage of the distances between nodes: the 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 monthly capacity of each link, shown in Table A1 in the appendix, is estimated from historical data as a shared constraint for all commodities flowing on the link [ODOT, 2013], representing the availability of transportation facilities. As- suming that the total supply of commodity k is equal to the total de- mand of the same commodity throughout the network, as shown in Table3, abaselineflowresulted inno remaining commodities at supply nodes and no unsatisfied demand at demand nodes when there is no disruption to the functionality of the network. In the illustrative example, disruption scenarios are defined as the

one-at-a-time removal of a single network component at a time. It is assumed that a disruption, or the removal of a particular network component, lasts for a period of one month. Assuming that annual in- dustryproductionaccumulates consistently across theyear (i.e., neither production nor interdependency relationships vary day-to-day, week- to-week, month-to-month), a smaller month-long time horizon is con- sidered here as an appropriate proportion of a year to calculate the particular disruptive event cascading effect (e.g., a two-week closure of port facilities [Pant et al., 2011]). Shown inTable 4, two transshipment nodes (nodes 9 and 11, which have a vital role in connecting segments ofhighvolume freight trafficon interstatehighways), some segmentsof the North America Railroad (node 8 and link (8-4)), a local railroad which connects industrial parks to the North America railroad (link (2–8)), and parts of the waterway system (link (2–5)) were each re- moved one-at-a-time from the network to define the disruption sce- narios. And the impact of these individual removals were measured. Focusing on the economy of the state of Oklahoma, and considering supply nodes within the state interacting with demand nodes in sur- rounding states, undelivered commodities remaining with suppliers or unsatisfied demand at demand nodes, as represented by Sik, affect in- dustry output and result in propagated inoperability through many of

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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 4 reports

+ Si N N i

k ( )k

, the sum of the slack (remaining supply) by commodity at the supply nodes when different network components are disrupted, omitting the flow on the disrupted component from the baseline flow within the network. As shown in Table 4, the Petroleum and coal industry (324) is directly vulnerable in all disruption scenarios except for the loss of link (1,7), while the Food and beverage and tobacco industry (311) would be af- fected only by the loss of link (2,5).

3.3. Multi-industry impact

As all the demand nodes are located outside of Oklahoma, failure in the form of the inability of suppliers to export commodities is modeled as a demand perturbation as calculated in Eq. (14). Other industries within the state will be affected by the interdependent effect of this failure, as capturedby qk in Eq. (3), representing the extent towhich an industry output will not be produced. And the effect of the disruption on the economy of the state is captured by Q, assuming that industries

not using the transportationnetworkhavenot experiencedanydemand perturbation. Given the remaining commodities left at supply nodes, shown in Table 4, demand perturbation is calculated with Eq. (14). Resulting industry inoperability, qk, is provided inTable5 anddepicted inFig. 5.ThePetroleum and coal industry (324) ismost vulnerable to the removal of the link (2,8), link (2,4), or node 8. The removal of these components also affect the operability of the Nonmetallic minerals in- dustry (327), though to a lesser extent than the removal of link (1,7). The productivity of the Chemical products industry (325) is highly

(a) (b)

Fig. 4. Representations of (a) spatial location of multi-modal nodes in Oklahoma and surrounding states, and (b) the connected transportation network.

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

Component Description

Node 1 Oklahoma City Node 2 Port of Catoosa Node 3 Port of Muskogee Node 4 Port of Texas City Node 5 Port of New Orleans Node 6 Port of Little Rock Node 7 Port of Chicago 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 which connects Oklahoma City, OK, with Chicago, IL. Link (2,8) A local railroad connecting Port of Catoosa to the North America railroad Link (1,4) Part of the North America railroad which 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 truck way connects Oklahoma City to Texas City, TX, using interstate highways I-35 and I-45. Link (9,11) Part of interstate highway I-44 which connects Oklahoma City to Tulsa.

Table 2 Names and NAICS codes for the primary 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

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dependent on the connectivity of Tulsa and Oklahoma City through I- 44, as represented by link (9,11), as well as transshipment nodes 9 and 11. The inoperability values in Table 5 may appear to be negligible at first, but these numbers are significant when linked to the concept of failure probability in the reliability or quality engineering literature (i.e., the maximum allowable failure probability for a six sigma com- pliant system is 3.4E-06). Considering each industry's production level in monetary value and

calculating total impact of the disruption across the state's industries with Q, Table 6 and Fig. 6 provide the supplementary analysis which elaborates the magnitude of loss (in million USD) experienced by dif- ferent industries regarding the total economic loss. The interconnected nature of the industries within a region affect productivity of the other 56 industries operating in Oklahoma though individually to a much lesser extent than the six industries directly affected. Many industries are vulnerable to any sort of disruptionaffecting theoperability of node 8, the intermodal terminal facilities at the Port of Catoosa, or either of the links connecting it to nodes 2 or 4, the port itself and the state of Texas, respectively. The Petroleum and coal products industry (324) is a high dollar industry in Oklahoma affected the most by the disruption scenarios, though less vulnerable to disruptions that remove links (2,5) or (1,7) from service.

3.4. Planning for adaptive capacity

During the month-long period of disruption, the efficacy of con- tingency rerouting through the residual network is determined ac- cording to its reduction in economic productivity of Oklahoma. Respectively, Tables 7 and 8 report interdependent economic inoper- ability experienced by the six most important industries in Oklahoma and the consequential multi-industry economic losses following the contingency rerouting strategy devised from the model developed in Eqs. (16)–(24) to minimize e

R. e R is defined as a measure to lessen the

maximum potential drop in the regional economy, lies on [0,1], where = 0eR means that under a disruption scenario e, there is no way to

avoid the maximum possible loss in the economy of the region by re- routing the supply-demand network, and = 1p means that under a disruption scenario e, it is possible to maintain the full productivity of

the regional economy by rerouting commodity flows through the re- sidual network. Comparing the inoperability caused by the removal of the network component with and without devising a contingent re- routing strategy during the period of disruption, shown in Figs. 7 and 8 respectively, shows that the proposed model to plan for adaptive ca- pacity tries to facilitate the trades in high dollar industries like Petro- leum and coal products (324) and Miscellaneous manufacturing (339), while having less impact on Chemical products (325) or Food and bev- erage and tobacco (311) industries. Fig. 7 depicts how contingent rerouting would affect the maximum

loss across multiple Oklahoma industries following the removal of the particular components. And, as listed in Table 8, this strategy could lessen thevulnerability of thewhole systemwith respect to the removal of particular components like link (2,5) as part of the inland waterway network. It is also inferred that industries in Oklahoma are most vul- nerable to disruptions that cause inoperability in (i) node 8, the inter- modal terminal facilitates the movement of commodities in the in- dustrial park of Port of Catoosa to out-of-state customers, (ii) link (8,4), a portion of railroad that connects Oklahoma to Texas City, TX, or (iii) link (2,8), a local railroad that connects thePort ofCatoosa to theNorth America railroad intermodal terminal, as even rerouting cannot suffi- ciently enhance the performance of the collective industries, as mea- sured by e

R, by more than 37%. As shown in Table 8, the maximum possible loss resulting fromthe removal of anetworkcomponentwill be avoided with a contingent rerouting strategy, as in some cases system performance improved up to 85%. As a contingent rerouting strategy is sought considering the total

economic impact embedded in Eq. (15), priorities given to high-dollar industries and those with the highest interdependent impacts across industries. Though Fig. 8 shows the absolute benefit of implementing the adaptive capacity planning strategy in the case of different dis- ruption scenarios, there might be cases in which the rerouting strategy results in losses to particular industries. Because of the structure of the network in the case study, the assumption that the residual capacity in all industries should be less than or equal to its inoperability results in infeasibility as (i) the distribution of the network capacity over the network component does not allow the flow in all industries to increase at once, and (ii) the objective function tries to maximize the total economic loss in the minimum possible time immediately after dis- ruptions, andas such it focuses on rerouting theflowof those industries most affect reducing total economic loss. These results can also assist in prioritizing more important industries in after a disruption. Fig. 9 shows how contingent rerouting strategies affect different

industries (in the form of box plots generated across the eight disrup- tion scenarios). For example, the rerouting strategies taken following the eight different disruption scenarios would lessen the economic loss in Petroleum and coal products (324) industries by $25.46 million, on average, and at least $0.55 million, in the case of losing link (1,7). Overall, the Chemical products (325) and Food and beverage and tobacco (311) industries are most adversely impacted, as shown in Fig. 9, be- cause optimal contingency rerouting tends not to benefit these in- dustries in favor of the larger economy, as shown in Fig. 8.

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

Industry

311 324 325 327 333 339

Supply nodes in OK Oklahoma City 362526 0 300501 183188 23790 118242 Port of Catoosa 50244 454911 284685 25268 2470 424 Port of Muskogee 0 33962 0 31886 0 30021 Demand nodes outside of OK TX 97281 316905 204006 0 25838 30154 LA 50244 18449 0 0 267 0 AR 265245 153518 381180 41038 156 54494 IL 0 0 0 199304 0 64039

Table 4 Tons of remaining commodities at suppliers with the removal of network components.

Removed component Sum of remaining commodities at supply nodes (tons) 311 324 325 327 333 339

Node 9 0 18960 91744 0 0 19740 Node 8 0 263776 0 17509 2048 0 Node 11 0 18960 71119 0 0 0 Link (1,7) 0 0 0 177628 0 64039 Link (9,11) 0 18960 71119 0 0 0 Link (2,5) 50244 3656 0 0 267 0 Link (8,4) 0 263776 0 0 2048 0 Link (2,8) 14793 157492 88627 0 0 0

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4. Concluding remarks

With regard to the three components of resilience capacity identi- fied by Vugrin and Camphouse (2011), most freight transportation network resilience studies focus on pre-disruption prevention invest- ments via absorptive capacity or post-disaster network restoration strategies via restorative capacity. And such is typically done by de- fining system performance as a measure related to the serviceability of the system (e.g., travel time/distance, flow, throughput) or a topolo- gical measure related to the network structure (e.g. centrality, con- nectivity, betweenness). This work, however, emphasizes adaptive ca- pacity in the form of contingent rerouting strategies to manage the supply-demand network after a disruptive event to lessen the total economic impact. More specifically, this work proposes an optimization formulation

to accommodate the flow through the residual network and maintain the productivity of the economy of the desired region by (i) integrating a multi-commodity network flow model, representing a multi-modal freight transportation network, with a risk-based economic inter- dependency model, to capture the propagation of the failure in a group of interconnected industries, (ii) defining a measure of adaptive capa- city to valuate rerouting strategies, and (iii) the model incorporates the economic elements to study the disruption effects on the infrastructure networks from other perspectives. The results provide insight to deci- sion makers about the behavior of each commodity such that they may adapt policies aligned with the behavior of the model (e.g., allocating emergency warehouses for commodities whose economic loss increases

after implementing the adaptive capacity approach). Further, the for- mulation provides a means to consider the final role of a freight transportation network as the facilitator within the economy in plan- ning for adaptive capacity after a disruption. Part of a multi-modal freight transportation network connecting

Oklahoma to surrounding states has been considered to develop a sty- lized case study in which supply nodes are located in the state of Oklahoma and demand nodes are located in surrounding states. We address the efficacy of implementing the adaptive capacity planning formulation in Oklahoma when a scenario-based disruption disables a particular network component for a month. Results suggest a successful

Table 5 Industry inoperability across six most important industries within the state of Oklahoma.

Removed component Industry

Food and beverage Petroleum and coal Chemical products Nonmetallic mineral Machinery mfg. Misc. mfg.

Node 9 0 9.00E-04 4.90E-03 0 0 1.50E-03 Node 8 0 1.16E-02 9.00E-04 1.20E-03 1.60E-03 8.00E-04 Node 11 0 9.00E-04 3.80E-03 0.00E+00 0 1.00E-04 Link (1,7) 0 1.00E-04 2.00E-04 8.90E-03 1.00E-04 4.50E-03 Link (9,11) 0 9.00E-04 3.80E-03 0 0 1.00E-04 Link (2,5) 5.10E-03 2.00E-04 2.00E-04 1.00E-04 2.00E-04 2.00E-04 Link (8,4) 0 1.16E-02 9.00E-04 3.00E-04 1.60E-03 8.00E-04 Link (2,8) 4.00E-04 1.16E-02 9.00E-04 1.20E-03 1.60E-03 8.00E-04

0.000

0.004

0.008

0.012

In op

er ab

ili ty

Fig. 5. Graphical depiction of industry inoperability resulting from network component removal.

Table 6 Economic losses, inmillionUSD,across the sixmost important industrieswithin the state of Oklahoma.

Removed component

Industry Total multi- industry impact

311 324 325 327 333 339 Others

Node 9 0.12 11.04 6.67 0.09 0.20 14.77 17.59 50.47 Node 8 0.24 146.24 1.22 2.47 11.90 7.95 159.32 329.33 Node 11 0.04 10.80 5.16 0.06 0.11 0.78 12.82 29.79 Link (1,7) 0.23 0.92 0.23 18.17 0.37 45.79 22.41 88.12 Link (9,11) 0.04 10.80 5.16 0.06 0.11 0.78 12.82 29.79 Link (2,5) 28.04 2.70 0.26 0.25 1.65 2.40 23.64 58.95 Link (8,4) 0.24 146.20 1.21 0.69 11.88 7.88 158.46 326.56 Link (2,8) 2.12 146.29 1.24 2.48 11.91 8.10 160.71 332.84

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avoidance of maximum potential loss in high dollar industries such as Petroleum and coal products (324) and Miscellaneous manufacturing (339), and a consequent static resilience in the economy of the state, as the average maximum loss could be avoided by more than 50%. The ultimate usefulness of such a model could lie in (i) assisting transpor- tation planners in effective rerouting that minimizes impacts to certain industries, and (ii) assisting decision makers in those industries how certain disruptions and resulting adaptive planning may impact their companywhencertain commodities donot arrive as planned. Thougha proportion of the total economic impact has been considered to seek adaptive planning strategies in this study, further work should embed larger social and community impacts in the problem formulation. The real-world application of this work lies in informing a central

planner/policy maker to devise contingent rerouting strategies more

effectively to enhance the resilience of freight movement to maintain the continuity of service for businesses using the multi-modal trans- portation network. For example, in the case of a natural hazard that affects the Port of Catoosa, one of the most important business eco- nomic areas in Oklahoma, such a central decision maker could be re- presented by the nine-member board that oversees the port [Business View Magazine 2016]. These results suggest that it may be economic- ally beneficial for policy makers to explore ways to reroute the com- modity flow by facilitating contingent rerouting or incentivizing com- panies to move commodities through alternative transportation modes in case of a disruption to port dock operations. The proposed model helps decision makers to prioritize the affected industry sectors when devising contingent rerouting strategies to facilitate the flow of com- modities during the disruption. For example, Miscellaneous manu- facturing and Machinery freight are handled at the General Dry Cargo dock, which handles the largest tonnage in the port. Hence, these sec- tors would be vulnerable to any disruption that threatens the func- tionality of the General Dry Cargo dock for the real-world operations of the port. Similarly, any disruptions threatening the operations at Liquid Bulk and Grains docks would interrupt the flow of Chemical products and Food and beverage and tobacco products, respectively. The Port of Catoosa is a major industrial hub in the Tulsa metropolitan statistical area, which contributes to 33.4% of the State of Oklahoma's economy [Tulsa Regional Chamber 2018], hence the options here are aimed at benefiting the wider state economy through the port. The insights gained from this paper can lead to better risk management strategies to mitigate the effect of a multi-modal freight transportation disruption. This initial formulation can be further improved by accounting for

Fig. 6. Interdependent economic losses in Oklahoma due to network component removal.

Table 7 Economic inoperability caused by the disruption after devising a contingent rerouting strategy.

Removed component

Industry

311 324 325 327 333 339

Node 9 2.00E-04 0 4.80E-03 0 0 0 Node 8 1.10E-03 7.20E-03 5.00E-03 8.00E-04 1.00E-04 5.00E-04 Node 11 2.00E-04 0 4.70E-03 0 0 0 Link (1,7) 0 0 1.00E-04 7.50E-03 0 1.00E-04 Link (9,11) 2.00E-04 0 3.60E-03 0 0 0 Link (2,5) 0 1.00E-04 1.50E-03 0 2.00E-04 0 Link (8,4) 1.10E-03 7.20E-03 5.00E-03 2.00E-04 1.00E-04 5.00E-04 Link (2,8) 1.50E-03 7.00E-03 5.20E-03 2.00E-04 1.00E-04 5.00E-04

Table 8 Economic losses, in million USD, within the state of Oklahoma after planning for adaptive capacity.

Removed component

Industry Total multi- industry impact

e R

311 324 325 327 333 339 Others

Node 9 0.85 0.41 6.57 0.03 0.05 0.42 3.00 11.32 0.78 Node 8 5.98 90.15 6.80 1.66 0.78 5.17 100.68 211.22 0.36 Node 11 0.85 0.40 6.34 0.03 0.05 0.41 2.92 10.98 0.63 Link (1,7) 0.02 0.37 0.11 15.26 0.10 0.58 7.33 23.78 0.73 Link (9,11) 0.84 0.31 4.83 0.02 0.04 0.32 2.36 8.72 0.71 Link (2,5) 0.02 1.82 2.06 0.02 1.43 0.30 3.28 8.92 0.85 Link (8,4) 5.98 90.12 6.79 0.44 0.78 5.12 100.10 209.33 0.36 Link (2,8) 8.42 87.77 7.09 0.45 0.78 5.21 99.48 209.22 0.37

0.000

0.002

0.004

0.006

0.008 In

op er

ab ili

ty

Fig. 7. Economic inoperability caused by the disruption devising a contingent rerouting.

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the real-world intermodal container planning considerations and other dynamic issues. Complementary models to plan for system resilience as

a functionof absorptiveand restorative capacity, aswell as theadaptive capacity-focused formulation proposed here, could more effectively highlight the tradeoffs among different resilience capacity planning perspectives. Further, the proposed integrated framework could be extended to study the design of a resilient freight network considering uncertain disruptions of multiple components (e.g., Alderson et al., (2013);Alderson et al., (2013)).

Acknowledgments

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

Appendix

Table A1 Link capacities among the origin/destination nodes in the illustrative network (in tons) [ODOT, 2013].

Nodes 1 2 3 4 5 6 7 8 9 10 11

1 233333 241667 141667 516667 2 15000 54167 62500 41667 283333 308333 112500 3 29583 15417 250833 24167 4 5 6 7 8 316667 25000 9 150000 141667 10 1000000 11 133333 166667

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