Review on Energy Resilience

Investing in Absorptive Capacity in Interdependent Infrastructure and Industry Sectors

Mohamad Darayi, Ph.D.1; Raghav Pant, Ph.D.2; Kash Barker, Ph.D.3; and Nazanin Morshedlou, Ph.D.4

Abstract: Freight transportation infrastructure systems facilitate commodity flows across multiple industries. The closure of key infra- structures leads to an interruption of economic productivity that propagates through a system of interconnected industries. Investing in infrastructure and key industries can reduce the vulnerability of many industries by improving their ability to maintain functionality when shocked. This work investigates how a limited budget could be allocated to multiple industries to fortify them prior to a disruption to ulti- mately enhance the economic resilience across all industries by reducing the vulnerability of the underlying infrastructure. A risk-based economic interdependency model is used to implement a new measure of absorptive capacity to examine the propagation of a failure through- out the economy given the fortification of industry sectors. Sources of uncertainty in this data-driven model are considered, and a soft-robust optimization model is proposed to devise budget allocation under uncertainty. The approach is illustrated with an inland waterway port case study. The results can provide decision makers with managerial insights about how the economic interdependency affects the industries’ share of a budget to enhance absorptive capacity and how the level of budget affects the decision-making process for allocating resources. DOI: 10.1061/(ASCE)IS.1943-555X.0000514. © 2019 American Society of Civil Engineers.

Author keywords: Economic resilience; Absorptive capacity; Infrastructure and economic sectors; Epistemic uncertainty.

Introduction

Freight transportation infrastructures, including ports, intermodal stations, interstate highways, and railways as basic structures and facilities, enable commodity flows and facilitate the productivity of industries. In the past decades, numerous disruptive events, whether natural hazards, common failures, or possibly malevolent attacks, have threatened the operation of multiple modes of this infrastruc- ture system and consequently adversely impacted economic pro- ductivity. A few examples of these disruptive events include the flooding of the Mississippi and Missouri rivers in 1993, where sev- eral railroads experienced delays and cancelations (Haefner et al. 1996); Hurricane Katrina that caused damage to the US highway system in Louisiana, Mississippi, and Alabama in 2005 (Shen and Aydin 2014); and Hurricane Sandy, as a multistorm that hit the East Coast of the US in 2012, which closed all port terminal facilities and the harbor at the Port of New York and New Jersey area (Fialkoff et al. 2017). A local disruption (i.e., port closure) can have effects that propagate through the system of interdependent infrastructure and industry sectors resulting in major reductions

in regional or nation-wide economic efficiency (Pant and Barker 2011; Arnold et al. 2006). A protective approach could include hardening key industries to lessen the shocks from disruptive events (DHS 2013) [e.g., including emergency debris removal from transportation routes and temporary reconstitution of emergency services (Bye et al. 2013)]. This paper provides an approach to measure such hardening in terms of the effectiveness of investing in resilience from the perspective of a central decision maker across several interdependent industries.

Several definitions of resilience have been proposed, including the ability to withstand, adapt to, and recover from a disruption. A definition with which many would largely agree (The White House 2011). Vugrin and Camphouse (2011) defines the resilience capac- ity of a system as a function of the following: (1) absorptive capac- ity, or the extent to which a system is able to absorb shocks from disruptive events, (2) adaptive capacity, or the extent to which a system can quickly adapt after a disruption by temporary means, and (3) restorative capacity, or the extent to which the system can recover from a disruption or be reconstructed in the long term. Barker et al. (2013) highlights that the collection of absorptive and adaptive capacities addresses vulnerability mitigation, or to what extent an infrastructure withstands a disruptive event. The re- storative capacity is analogous to recoverability, or the ability of the infrastructure to recover to a desired level of performance in a timely manner. The capacities contributing to resilience, as well as the dimensions of vulnerability and recoverability, are character- istics exhibited by resilient systems and could be measured in a number of context-specific ways.

As such, absorptive, adaptive, and restorative capacities can be viewed as first, second, and third lines of defense, respectively (Hosseini and Barker 2015, Hosseini et al. 2016). Fig. 1 highlights the temporal relationship among absorptive, adaptive, and restor- ative capacities. In this figure, we measure system performance in each time period with φðtÞ (e.g., customers with power, travel time in a transportation network, and total flow reaching to demand

1Assistant Professor, Great Valley School of Graduate Professional Studies, Pennsylvania State Univ., Malvern, PA 19355. ORCID: https:// orcid.org/0000-0002-8166-9712. Email: [email protected]

2Senior Postdoctoral Researcher, Environmental Change Institute, Univ. of Oxford, Oxford OX1 3QY, UK (corresponding author). ORCID: https:// orcid.org/0000-0003-4648-5261. Email: [email protected]

3Associate Professor, School of Industrial and Systems Engineering, Univ. of Oklahoma, Norman, OK 73019. Email: [email protected]

4Postdoctoral Scholar, School of Industrial and Systems Engineering, Univ. of Oklahoma, Norman, OK 73019. Email: nazanin.morshedlou@ ou.edu

Note. This manuscript was submitted on August 3, 2017; approved on June 3, 2019; published online on December 7, 2019. Discussion period open until May 7, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Infrastructure Systems, © ASCE, ISSN 1076-0342.

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nodes). In an example of improving absorptive capacity, Bienstock and Mattia (2007) developed an optimization model to increase powerline capacities to prevent large scale cascading blackouts in a power network. An example of adaptive capacity included robust strategies to respond to the dramatic climate change in water management systems in which simulation models of several disrup- tion scenarios assess options to ameliorate vulnerabilities in the short term (Lempert and Groves 2010). Finally, debris removal from a transportation network after a natural disaster was an exam- ple of restorative capacity (Çelik et al. 2015).

This paper focuses on reducing vulnerability via absorptive capacity. The idea of absorptive capacity has also been referred to as static resilience, or “the ability of the system to maintain func- tionality when shocked” (Rose 2007). However, this term has evolved into absorptive capacity integrating into the newly defined concept of resilience (Vugrin and Camphouse 2011). Mathemati- cally, static resilience is measured in terms of the difference be- tween the maximum potential drop in system performance and the estimated performance drop (Rose 2004). That is, no notion of recovery is considered, only the ability to withstand the initial disruption. This is depicted graphically in Fig. 2 and mathemati- cally in Eq. (1), where %ΔDY and %ΔDYmax are calculated with 100% × ½φðteÞ − φ 0ðtdÞ�=φðteÞ, and %ΔDY = actual percentage

change in the performance of the system following a disruptive event when a number of limited resources are allocated in proac- tively to fortify industry sectors before disruptions; %ΔDYmax = maximum percentage change given the worst-case level of perfor- mance (Rose 2009). This quantitative approach is used in this study to define a performance measure for the system’s ability to absorb shocks (%ΔDYmax − %ΔDY) from disruptive events, though we prefer the term absorptive capacity rather than static resilience.

While Fig. 2, from Pant et al. (2014), represents changes in sys- tem performance in a general sense, Rose (2009) provides a more specific application, where %ΔDY and %ΔDYmax refer to changes in total performance, after infrastructure fortification, and in a worst-case situation, respectively, in a set of interconnected indus- tries. In this sense, these measures are analogous to the concept of inoperability, a well-studied topic in the literature of interdependent industries and infrastructures (Santos and Haimes 2004; Barker and Haimes 2009; Barker and Santos 2010a, b). Inoperability (q) quan- tifies the proportional extent to which a system does not function in an as-planned manner. That is, where other measures quantify system performance in application-specific terms (e.g., flow capac- ity, connectivity, production output), inoperability provides a more general proportional metric of performance relative to an as- planned value. As such, we adopt %ΔDY ¼ q; %ΔDYmax ¼ qmax in this work, and we relate absorptive capacity to these inoperabil- ity measures

absorptive capacity ¼ %ΔDY max − %ΔDY

%ΔDYmax ð1Þ

This paper seeks to answer: how should limited resources are allocated to harden individual industries effectively to enhance absorptive capacity with total economic impacts in mind? These economic impacts are realized due to freight disruptions. Freight transportation infrastructure disruptions lead not only to physical damage but also to an interruption of economic productivity across multiple industries due to infrastructure inoperability (Ham et al. 2005; Park et al. 2011). Arnold et al. (2006) analyzed the economic impacts of disruptions in container traffic in the ports of Los Angeles and Long Beach, California. Pant et al. (2011), using the inoperability input-output model (IIM), and (Santos and Haimes 2004) showed how a local disruption in the Port of Catoosa in Tulsa, Oklahoma, would affect multiple industries within the state and neighboring states that trade with Oklahoma. IIM is a data-driven interdependent disruption evaluation model that has been widely used to analyze interdependent connections among in- dustry sectors (Pant et al. 2014). The model proposed a balance

Fig. 1. Relationship between the vulnerability and recoverability dimensions of resilience and the components of absorptive capacity with respect to system performance φðtÞ.

Fig. 2. Performance components of static resilience. (Reprinted from Reliability Engineering & System Safety, Vol. 125, R. Pant, K. Barker, and C. W. Zobel, “Static and dynamic metrics or economic resilience for interdepended infrastructure and industry sectors,” pp. 92–102, © 2014, with permission from Elsevier.)

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supply-demand equilibrium for interacting industries. Understand- ing the absorptive capacities of affected industries could assist in preparedness planning against disruptions. In particular, prepar- edness plans could enhance the ability of the industries to absorb shocks from the disruptive events and lessen the maximum eco- nomic inoperability that the series of interdependent industries would experience.

This paper establishes inoperability through the IIM as a means to measure absorptive capacity in interdependent industries. This research addresses (1) defining a measure of absorptive capacity to invest for resilience in an interdependent economic system; and (2) planning for absorptive capacity under uncertainty; while (3) addressing some of the uncertainties of the model.

Methodological Background

The IIM is an extension of the economic input-output model (Leontief 1986). The input-output model has been widely used in analyzing the interdependent connections among industries (Santos and Haimes 2004).

In a system of n interacting industries under a static equilibrium, the total output of the ith industry is distributed to all other indus- tries and satisfies external demand. This equilibrium condition is described with xi ¼

P n j¼1 zij þ ci, where xi = output; ci = external

demand for industry i; and zij describes the flow of commodities output from industry i and is used as input to production in indus- try j. The flow of commodities zij is assumed to be proportional to the output of industry j, expressed as zij ¼ aijxj. The common form of the Leontief input-output model is expressed in Eq. (2), where x is an n × 1 vector of industry production outputs, A is an n × n industry-by-industry matrix of interdependency coefficients, and c is a n × 1 vector of final demands. The model shows that total production is made up to satisfy industry-to-industry intermediate production (Ax) and final demands (c)

x ¼ Ax þ c ⇒ x ¼ ½I − A�−1c ð2Þ Instead of describing the connections between the interdepend-

ent industries in terms of commodity flow in monetary units (e.g., dollars), the IIM illustrates how normalized production losses propagate through all interconnected industries. The IIM is pro- vided in Eq. (3) (Santos and Haimes 2004), which describes the relationships among n infrastructure and industry sectors, resulting in matrices of size n × n and vectors of length n

q ¼ A⋆q þ c⋆ ⇒ q ¼ ½I − A⋆�−1c⋆ ð3Þ Vector q here is a vector of infrastructure and industry inoper-

abilities describing the extent to which ideal functionality is not realized following a disruptive event. Inoperability for sector i is defined in Eq. (4), where the as-planned total output is represented with x̂i and degraded total output resulting from a disruption is rep- resented with ~xi. An inoperability of 0 suggests that an industry is operating at normal production levels, while an inoperability of 1 means that the industry is not producing at all

qi ¼ ðx̂i − ~xiÞ=x̂i ⇔ q ¼ ½diagðx̂Þ�−1ðx̂ − ~xÞ ð4Þ Normalized interdependency matrix A⋆ is a modified version

of the original A matrix describing the extent of economic inter- dependence among a set of infrastructure and industry sectors. Shown in Eq. (5), the row elements of A⋆ indicate the proportions of additional inoperability that are contributed by a column sector to the row sector

a⋆ij ¼ aijðx̂j=x̂iÞ ⇔ A⋆ ¼ ½diagðx̂Þ�−1A½diagðx̂Þ� ð5Þ

Eq. (6) provides the calculation of c⋆, a vector of normalized demand reduction. The elements of c⋆ represent the difference in as-planned demand ĉi and perturbed demand ~ci divided by as- planned production, quantifying the reduced final demand for sector i as a proportion of total as-planned output

c⋆i ¼ ðĉi − ~ciÞ=x̂i ⇔ c⋆ ¼ ½diagðx̂Þ�−1ðĉ − ~cÞ ð6Þ As is evident from Eq. (3), the IIM, like the economic input-

output model of Eq. (2), is a demand-driven model. Specifically, in the IIM, disruptions are translated to demand perturbations giv- ing direct economic losses, following which the indirect economic losses can be estimated through Eq. (3). An example of a demand perturbation, as discussed subsequently in the case study, would include unsatisfied demand in the petroleum and coal products industry resulting from a closure of an inland waterway port.

Total economic losses, the combination of direct and indirect losses, can be calculated by multiplying each industry’s production level by its inoperability level: for industry i, Qi ¼ xiqi, or for the entire economy of industries, Q ¼ xTq. As such, planning deci- sions can be made with respect to inoperability or economic impact at the sector level or with respect to economic impact at the multisector level.

Criticisms of the basic IIM include its linear nature and its treat- ment of interactions of industries as constant after a disruption (Kujawski 2006). However, the linear nature of the model enables it to be easily used in an optimization formulation (e.g., relative to a nonlinear computable general equilibrium model to describe the interactions among industries) as is proposed in this paper. And the constant nature of the parameters of the model is assumed here due to the short-term nature of the analysis, as changes in the economy over time (e.g., substitution among industries) do not affect the proposed formulation. Further, while data describing the parameters of the IIM are published annually by the US Bureau of Economic Analysis (BEA) and many other countries worldwide, an obvious benefit of the IIM enterprise, we instead propose a ro- bust formulation to account for any uncertainty that may be present in these parameters. Our proposed formulation also accounts for disruptions driven by unsatisfied demand at demand nodes and residual supply at supply nodes both of which can be represented with the IIM, which is demand-driven in nature.

Absorptive Capacity Measures

As discussed previously, the interdependent impacts of a disruption are calculated using the IIM, and subsequently, a measure of ab- sorptive capacity is defined based on the concept of static economic resilience (Rose 2009; Pant et al. 2014). We propose an optimiza- tion model to devise a strategy to allocate limited budget to indus- tries to enhance absorptive capacity. Epistemic data uncertainty in the IIM is considered, and as such, decision-making under uncer- tainty is discussed.

Defining Absorptive Capacity with Inoperability

As suggested previously, the percentage change in the performance of a system (%ΔDY) is analogous to the measure of inoperability (q), which represents the proportional extent to which a system is not properly functioning. If we define qmax as the maximum pos- sible inoperability that could be experienced after a disruptive event, a measure of absorptive capacity is provided in Eq. (7). Absorptive capacity of sector i is referenced with convention ЯSi ,

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adopting the Я notation of Whitson and Ramirez-Marquez (2009) because R often refers to reliability

ЯSi ¼ %ΔDYmax − %ΔDY

%ΔDYmax ¼ qi;max − qi

qi;max ð7Þ

Using the convention D⋆ ¼ ½d⋆ij� ¼ ½I − A⋆�−1, Eq. (3) can be written as q ¼ D⋆c⋆. As such, inoperability in sector i can be represented with Eq. (8)

qi ¼ Xn j¼1

d⋆ijc⋆j ð8Þ

As shown in Eq. (9), using the demand-driven paradigm, the absorptive capacity for sector i can be written as a function of maximum and expected demand perturbation levels, c⋆j;max and c⋆j , respectively. The proportional “savings” in inoperability is mea- sured by ЯSi when a priori planning can stave off the worst-case inoperability outcome in favor of reduced inoperability

ЯSi ¼ P

n j¼1 d

⋆ ijc

⋆ j;max −

P n j¼1 d

⋆ ijc

⋆ jP

n j¼1 d

⋆ ijc

⋆ j;max

¼ P

n j¼1 d

⋆ ijðc⋆j;max − c⋆j ÞP

n j¼1 d

⋆ ijc

⋆ j;max

ð9Þ

To capture absorptive capacity across the entire set of inter- dependent infrastructures and industry sectors, a more appropriate economic resilience metric would account for the widespread ability of sectors to collectively maintain operability following a disruptive event. As such, individual sector inoperability is multi- plied by sector output in dollar terms. In summation form, this is represented with Qi ¼

P n j¼1 xid

⋆ ijc

⋆ j . The resulting absorptive

capacity metric is provided in Eq. (10)

ЯStotal ¼ P

n i¼1

P n j¼1 xid

⋆ ijc

⋆ j;max −

P n i¼1

P n j¼1 xid

⋆ ijc

⋆ jP

n i¼1

P n j¼1 xid

⋆ ijc

⋆ j;max

¼ P

n i¼1

P n j¼1 xid

⋆ ijðc⋆j;max − c⋆j ÞP

n i¼1

P n j¼1 xid

⋆ ijc

⋆ j;max

ð10Þ

Planning for Absorptive Capacity

Resource allocation requires developing strategies that reduce demand perturbations effectively, leading to economic resilience. This demand-driven model is consistent with the idea that absorp- tive capacity (i.e., to what extent the system can withstand the disruptions), along with investments on expanding the redundancy in infrastructure networks, is the efficient utilization of resources and not system repair (Rose 2007).

Assume that a disruptive event perturbs demand [perhaps di- rectly, or perhaps as a forced demand reduction because of a supply shortage (Darayi et al. 2017)] in m ≤ n sectors. The worst-case demand perturbations in each of these m sectors are given by c⋆l;max, l ¼ f1; : : : ; mg. The implementation of preparedness, or resilience-building, activities is concerned with reducing c⋆l;max through efficient resource allocation. If rl is a preparedness strategy adopted to reduce the initial sector l demand perturbation impact, the effectiveness of rl is measured in terms of the new resulting demand perturbation in Eq. (11). All sector demand perturbations are governed by Eq. (12)

c⋆l ¼ flðc⋆l;max; rlÞ ð11Þ

c⋆i ¼ � c⋆l if i ∈ l 0 otherwise

ð12Þ

Assuming a numerically higher value for rl results in a more effective preparedness strategy, some candidate graphical relation- ships between c⋆l and rl are conceptually depicted in Fig. 3 with the upper bound being c⋆l;max.

Since implementing preparedness strategies comes at a cost, there is a finite budget that governs the maximum possible values taken by rl. If glðrlÞ expresses the cost of implementing strategy rl, then this budget is an upper bound. For the entire set of inter- dependent infrastructure and industry sectors, if at most budget b is available, then Eq. (13) limits a fixed budget

Xm l¼1

glðrlÞ ≤ b ð13Þ

The collection of Eqs. (10), (11), and (13) results in the resource allocation optimization problem in Eq. (14)

max c⋆ l ;rl

P n i¼1

P m l¼1 xid

⋆ ilðc⋆l;max − c⋆l ÞP

n i¼1

P m l¼1 xid

⋆ ilc

⋆ l;max

s:t: c⋆l ¼ flðc⋆l;max; rlÞ; ∀ l ∈ f1; 2; : : : ; mg Xm l¼1

glðrlÞ ≤ b

glðrlÞ ≥ 0; ∀ l ∈ f1; 2; : : : ; mg ð14Þ Eq. (14) represents a generalized formulation of the resource

allocation to maximize absorptive capacity. The functional forms of flð·Þ and glð·Þ govern the solution to the absorptive capacity planning problem. For macrolevel planning, the rl value might de- note the amount of capital that can be invested in purchasing and substituting for the lost demand (c⋆l ). Assuming c

⋆ l;max is the maxi-

mum economic loss in industry sector l, Eq. (15) would govern how planning for absorptive capacity can improve c⋆l , where αl is a mea- sure of the effectiveness of investment rl, which also shows the return for substituting for lost demand for sector l. The inclusion of αl into the absorptive capacity enhancement calculation in Eq. (15) is motivated by Barker and Santos (2010a) and Jonkeren and Giannopoulos (2014)

c⋆l ¼ c⋆l;maxe−αlrl ð15Þ

For this special case, the absorptive capacity planning optimi- zation problem can be written as Eq. (16), given that Eq. (15) is a

Fig. 3. Candidate functional relationships between c⋆l and rl.

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strict equality constraint. Note that rl represents an investment made to improve absorptive capacity; therefore, the sum of the cost of strategy rl must satisfy the budget constraint

max rl

P n i¼1

P m l¼1 xid

⋆ ilðc⋆l;max − c⋆l;maxe−αlrlÞP

n i¼1

P m l¼1 xid

⋆ ilc

⋆ l;max

s:t: Xm l¼1

glðrlÞ ≤ b

glðrlÞ ≥ 0; ∀ l ∈ f1; 2; : : : ; mg ð16Þ

Note that the duration of disruption of the infrastructure com- ponent impacts system performance, and a lengthy duration of disruption of a rather noncritical component might result in a con- siderable economic loss. However, the focus of this paper is on optimizing the allocation of limited resources that leads to an im- proved ability to withstand a disruption prior to its occurrence, as measured with multi-industry inoperability. In this paper, the model prioritizes industry sectors according to their criticality for the entire economy considering economic interdependency. The pro- posed model captures the unsatisfied demands together with their economic value in terms of the maximum loss (demand perturba- tion) to be avoided. For example, due to a port closure, the unde- livered demand in each industry (e.g., food) multiplied by its effect on the economy of that corresponding industry could be considered as the maximum demand perturbation. In general, the duration of port closure and frequency of the closures would only affect the maximum demand perturbation proportionally for all the industries under study. As such, the model captures these two attributes of provisional disruptions. The model proposes a proactive resource allocation strategy at a strategic and tactical level, but not neces- sarily an operational level, meaning that the model allocates resour- ces to multiple industries before a disruptive event (i.e., strategic level) and not during the disruption or dynamically after the dis- ruption, which changes the allocation procedure immediately after the models sees the changes in the behavior of disruptive events (i.e., operational level). The maximum demand perturbation could be modeled by multiplying the average port closure frequency and the average duration of the closure such that the average port closure duration is sufficiently long [e.g., Pant and Barker (2011) studied a two-week closure with the IIM].

Decision-Making Under Uncertainty

Decision makers with interdependent economic impacts in mind and planning for absorptive capacity to harden industries facing disruption must consider an uncertain environment (e.g., relation- ships among industries after a disruption).

In this paper, data uncertainty in the integrated model in Eq. (16), consisting of data used to parameterize the IIM and re- source allocation, is considered. As the coefficients of the A⋆ ma- trix derived from the technical coefficient matrix A, they are subject to uncertainties arising from the interindustry data collection efforts by the BEA. The BEA collects annual input-output records for a group of 15 aggregated industries and more detailed records for 65 industries every 5 years. As the x vector is derived from the same BEA data, it is prone to the same uncertainties as A⋆. Hence, the economic input-output model, and subsequently the IIM, is prone to uncertainties arising from parameterizing interdependency coefficients matrix A (and A⋆) and x vector of total output because of statistical errors in compiling massive data bases and the variant nature of these parameters over time (Bullard and Sebald 1977;

West 1986). Assuming the values of matrix A⋆ are deterministic and time invariant, the derivation from the accurate data can cause the violation of this assumption. Therefore, Barker and Haimes (2009) developed an approach to evaluate the uncertainty in infra- structure interdependencies, minimizing the sensitivity of infra- structure interdependency parameters according to unspecified substitution changes. The inexactness in quantifying the effective- ness of investments within the resource allocation model should also be recognized (MacKenzie and Zobel 2016). Hence, the opti- mization model formulated in Eq. (16) contains epistemic data un- certainty (Pate-Cornell 1996) in the estimation of (1) the A⋆ matrix and magnitude of x vector; and (2) αl as the measure of the effectiveness of investment rl in industry l.

Bullard and Sebald (1977) studied inherent uncertainties in the coefficients of A, x, and ½I − A�−1 as bounded within a small interval of the published values. Similarly, our approach considers small random noise in the model data point uncertainties in x and D⋆ ¼ ½I − A⋆�−1, whose elements are d⋆ij. Furthermore, to model uncertainty in αl, the investment effectiveness for industry l, we propose a probabilistic treatment considering the optimistic, pessimistic, and most likely estimates of αl.

We propose a robust formulation of the optimization problem of Eq. (16). This robust formulation is shown in Eq. (17), where D, X, and Ψ are uncertainty sets that contain more anticipated real- izations of respective matrices and vectors. It is assumed that the sets D and X contain bounded random variations (e.g., �5%) of the values of D⋆ and x, respectively. A triangular distribution represents the set Ψ

max f

s:t:

�Pn i¼1

P m l¼1 xid

⋆ ilðc⋆l;max − c⋆l;maxe−αlrlÞP

n i¼1

P m l¼1 xid

⋆ ilc

⋆ l;max

� ≥ f

Xm l¼1

glðrlÞ ≤ b

glðrlÞ ≥ 0; ∀ l ∈ f1; : : : ; mg D� ∈ D; x ∈ X; α ∈ Ψ ð17Þ

The nonlinearity and stochasticity of the proposed model make it difficult to solve analytically. As such, rather than guaran- teeing a certain level of absorptive capacity, we want to make sure that the proposed model suggests a resource allocation set such that the probability that maximum absorption capacity (i.e., the disruptions have minimum effect on the system perfor- mance) is reached, Pr

��P n i¼1

P m l¼1 xid

⋆ ilðc⋆l;max − c⋆l;maxe−αlrlÞ=P

n i¼1

P m l¼1 xid

⋆ ilc

⋆ l;max

� ≥ f�, is equal or greater than 1 − ε for small ε > 0. The term ε is referred to as value-at-risk in portfolio optimization and has been widely used in “soft” robust optimiza- tion (Shapiro et al. 2009; Ben-Tal et al. 2009; Rockafellar and Uryasev 2000). The final formulation is presented in Eq. (18)

max f

s:t: Pr

�Pn i¼1

P m l¼1 xid

⋆ ilðc⋆l;max − c⋆l;maxe−αlrlÞP

n i¼1

P m l¼1 xid

⋆ ilc

⋆ l;max

� ≥ f

≥ 1 − ϵ

Xm l¼1

glðrlÞ ≤ b

glðrlÞ ≥ 0; ∀ l ∈ f1; : : : ; mg D� ∈ D; x ∈ X; α ∈ Ψ ð18Þ

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Illustrative Example: Inland Waterway Port Infrastructure Disruption

The proposed planning model for absorptive capacity is applied to a case study of the Port of Catoosa in Tulsa, Oklahoma. The Port of Catoosa is connected to the US inland waterway network through the McClellan-Kerr Arkansas River Navigation System, which is part of the Mississippi River Navigation System. There are approx- imately 70 companies using the port and an annual freight volume of 2.2 million tons is sent and received through the port (US Army Corps of Engineers 2011; Tulsa Port of Catoosa 2011). As defined by the North American Industry Classification System (NAICS), 62 industry sectors operate in Oklahoma, therefore the A⋆ matrix regionalized for Oklahoma is 62 × 62. We focus on six industry sectors, shown in Table 1, which represent the port’s largest export- ers in terms of commodity flows due to high trade figures (Tulsa Port of Catoosa 2011). This study focuses on improving the absorp- tive capacities of these six industry sectors. It is assumed that a port disruption (a two-week port closure) is sufficiently local when, without loss of generality, no other industry sectors are directly im- pacted. It is further assumed that the occurrence of natural disaster is infrequent, and the analyses do not account for expected losses given some frequency of disruption (although such could be included in future work).

The exports through the Port of Catoosa contribute to the exter- nal demand of the industries in Oklahoma. As such, in case of a disaster at the port resulting in the loss of exports, there is a demand perturbation in the industries that use the port. Because of the interdependence among industries, the cascading of the demand perturbations causes losses to all the other state industries. Assum- ing that, in the case of a disruptive event, the only losses in the state economy are because of the loss of exports through the port, we can obtain estimates of the maximum demand perturbations (c⋆l;max) for the six primary industries using the port. These are found for each industry as the ratio of that industry’s mean estimate of exports to its total economic output, which are all provided in Table 2. It is further assumed that industries not using the port have zero demand perturbations, although they could suffer from interdependent inoperability.

A policymaker caring about the economy of the state of Oklahoma [e.g., The Oklahoma Department of Transportation (ODOT)] may seek the best way to allocate a limited budget to individual industries allowing them to invest in absorptive capacity improvements to avoid the maximum demand perturbation during the port closure. Depending on the industry faced with the disrup- tive event, this resource could describe: (1) maintaining additional inventory to maintain productivity, and (2) setting short-term co- ordination contracts with distributors to be ready for alternative transportation modes other than the port.

The model assumes that allocating resources reduces the impact exponentially. As more resources are allocated to an industry, the impacts on an industry decline at a constantly decreasing rate, and investing an additional dollar to reduce risk returns less benefit than investing the first dollar. For each directly impacted industry, the exponential function, shown in Eq. (16) requires estimating an investment effectiveness parameter, αl. This parameter can be assessed if rl, the amount of resources needed to reduce the direct impacts on industry l by a fraction c⋆l =c

⋆ l;max, is known or can be

estimated, since αl ¼ −½logðc⋆l =c⋆l;maxÞ=rl�. While the value of αl is always non-zero and positive with no upper bound, it is expected that αl would be small for large-scale disruptions where millions of dollars are necessary to reduce the impact.

Table 3 lists parameter estimates for the effectiveness of invest- ments (αl) in planning for absorptive capacity in different industries considering the consequences on reducing the maximum demand perturbation by 50% (i.e., setting c⋆l =c

⋆ l;max ¼ 0.5). Food and bev-

erage products would be affected dramatically by the closure of the Port of Catoosa considering estimates for the cost per ton-mile for a barge at $0.97, compared to $2.53 for rail, and $5.35 for trucking (Arkansas Waterway Commissions 2014), together with the distances to the general customers for the products of this in- dustry, it is assumed that on average $7 million should be invested to avoid the maximum demand perturbation in this industry by 50% (MacKenzie et al. 2012; Aydin and Shen 2012; Richards and Patterson 1999). Direct impacts for petroleum and coal prod- ucts is much less than food and beverage products, but the nature of this industry’s products makes it difficult to look for alterna- tive transport modes. Also, long-term investments in increasing domestic demand by developing refining facilities, pipelines, and alternate transportation infrastructures could be effective and result in a higher absorptive capacity (Davarzani et al. 2011; Halkin et al. 2017). As such, α2 is calculated assuming that an almost $5 million investment is needed to decrease the maximum demand perturbation in petroleum industries by half (Alizadeh and Nomikos 2004; Nealer et al. 2011). The investment effec- tiveness for the other four industries are estimated considering the maximum loss in each industry and the options to increase the absorptive capacity with potential contracts for alternative transpor- tation modes.

Table 1. Six primary industries using the Port of Catoosa, along with their NAICS codes

Industry name NAICS code

Food, beverage, and tobacco products 311 Petroleum and coal products 324 Chemical products 325 Nonmetallic mineral products 327 Machinery 333 Miscellaneous manufacturing 339

Table 2. Maximum demand perturbation for the major industries using the Port of Catoosa in 2007 (output and exports given in million USD)

Industry name Exports Output c⋆l;max Food, beverage, and tobacco products 140.0 5,578.5 0.0251 Petroleum and coal products 57.0 12,644.0 0.0045 Chemical products 89.0 1,327.3 0.0671 Nonmetallic mineral products 3.0 2,026.2 0.0015 Machinery 108.0 7,174.4 0.0151 Miscellaneous manufacturing 6.0 746.6 0.0080

Table 3. Estimates for the cost-effective parameter αl (given as per million USD)

l Industry name αl

1 Food, beverage, and tobacco products 0.046 2 Petroleum and coal products 0.063 3 Chemical products 0.342 4 Nonmetallic mineral products 0.201 5 Machinery 0.084 6 Miscellaneous manufacturing 0.426

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Planning for Absorptive Capacity

The proposed model in Eq. (16) is implemented to optimize the budget allocation among the six important industries that trade through the Port of Catoosa. Four different total budget amounts are considered for allocations across the six industries: $10 million, $20 million, $30 million, and $40 million. To solve the model, we use the Frontline Solvers simulation optimization plug-in solver engine, which is an optimization and simulation application for Microsoft Excel (Olson and Wu 2013).

Table 4 shows the results from solving Eq. (16), in terms of the budgets rl allocated to individual sectors. We see that to maximize the absorptive capacity of the whole interdependent economy, cer- tain budget allocations will give the following results: (1) at the smallest budget allocation of $10 million, most of the resources are distributed to miscellaneous manufacturing (339) and chemical products (325); (2) as the budget allocation is increased toward $40 million, the resources get distributed toward food and beverage and tobacco products (311) and machinery (333); (3) petroleum and coal products (324) comparatively require some resource allo- cations when budgets are increased to $30 million and beyond; and (4) nonmetallic mineral products (327) comparatively require very little resource allocations. These results make sense because, based on Table 2 data, food and beverage and tobacco products (311) and machinery (333) are the two highest exporters through the port, so to progressively maximize the absorptive capacity of the economy most of the budget allocations will be distributed toward restoring economic flows in these sectors. Miscellaneous manufacturing (339) and chemical products (325) have high initial resource allocation values because the cost effectiveness parameters αl from Table 3 are high. But this means there is a fast stabilization of absorptive capacities in these sectors at the initial investment of $10 million, and subsequently, these sectors require smaller

increments of the resource allocations to further maximize the absorptive capacity of the economic system. Table 5 confirms this conclusion above, as indicated by the values of the absorptive capacities of individual sectors as budget allocations are increased.

Also shown in Table 5 are the values of the total economic losses avoided and the level of absorptive capacity achieved [value of the objective function of Eq. (16)] corresponding to each level of budget allocation. We see that if no budget allocations were made then, there is an economic loss of $95.2 million to the six industries, which is estimated from Table 5 by summing the economic loss values for these sectors. Overall, the whole economy has a loss of $146.6 million. These results show the interdependent effects of the IIM in Eq. (3). A budget allocation of $10 million results in decreasing economic losses by $37.9, which is a 25.8% resto- ration of absorptive capacity. Similarly, a budget allocation of $40 million decreases the economic losses by $84.8, thereby restoring absorptive capacity by 57.8%.

We note from the results in Table 5 that for every subsequent $10 million increment of budget allocation results in diminishing returns in terms of the value of economic loss avoided or absorptive capacity restored. For example, Table 5 shows that as budget allo- cation is increased from $10 million to $20 million, the changes in total loss avoided is $19.8 million, whereas an increment of budget allocation from $30 million to $40 million results in changing the amount of losses avoided by $12.2 million. Hence, the decision maker has to make a trade-off between increasing budget alloca- tions and the changing amount in losses avoided. A point to stop would be when the increment in budget allocation is more than the value by which the loss is reduced.

As shown in Table 5, investment of $30 million to harden the five industries among the six most important to the port can avoid the maximum economic loss in Oklahoma by up to 50%, and it indirectly protects nonmetallic mineral products (327), although the policy maker does not devote resources directly to this industry. Furthermore, for example, nearly $40 million in economic losses across the Oklahoma economy can be avoided with a $10 million investment in absorptive capacity in the six key industries, accord- ing to the model.

Fig. 4 shows the extent, as a ratio between 0 and 1, to which each industry and the Oklahoma economy are able to absorb shocks from a port disruption by allocating a defined amount of budget to harden the most important industries in the area. In comparison to the results of Table 5, where the magnitudes of losses based on budget allocations are shown, this result shows that the level ab- sorptive capacity achieved by the chemical products (325) sector is the highest even though its allocated budget might be lower.

Table 4. Resource allocation for absorptive capacity in different industries (in million USD)

Industry name

Resources allocated to each industry for each total budget

10 20 30 40

Food, beverage and tobacco products 0.31 5.88 10.95 14.92 Petroleum and coal products 0.00 0.00 0.92 3.81 Chemical products 3.88 4.63 5.31 5.84 Nonmetallic mineral products 0.00 0.00 0.00 0.00 Machinery 2.78 5.85 8.64 10.82 Miscellaneous manufacturing 3.03 3.63 4.18 4.61

Table 5. Economic loss under different total budget plans (budgets and losses in million USD)

Industry name Economic loss (no investment)

Economic losses for each total budget

10 20 30 40

Food, beverage, and tobacco products 25.90 25.38 19.68 15.54 12.95 Petroleum and coal products 12.59 11.71 11.46 10.70 8.94 Chemical products 16.46 4.61 3.62 2.96 2.47 Nonmetallic mineral products 1.04 0.95 0.87 0.81 0.77 Machinery 20.32 16.05 12.40 9.96 8.33 Miscellaneous manufacturing 18.88 7.74 6.23 4.91 4.15 Total loss in port sectors 95.19 66.44 54.26 44.88 37.61 Total macroeconomic loss 146.63 108.74 88.88 73.98 61.79 Total economic loss avoided — 37.90 57.70 72.60 84.80 Total absorptive capacity (ratio) — 0.26 0.39 0.49 0.58

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Decision-Making Under Uncertainty

As discussed previously, epistemic uncertainty in estimating A⋆ and αl should be accounted for in this paper. Small amounts of perturbations are considered when modeling data uncertainties in x and D⋆. It is assumed that the elements insets X and D related to the six industries contain bounded random variations to the amount of �5% of the (deterministic) values of x and D⋆, respec- tively. Furthermore, a probabilistic treatment considering the opti- mistic, pessimistic, and most likely estimates of αl with a triangular distribution is assumed to model uncertainty in the investment effectiveness for industry l. The parameters of these triangular dis- tributions are shown in Table 6. We model the uncertainty in this problem and obtain an exact solution using Frontline Solvers, an optimization and simulation application for Microsoft Excel. This application makes it easy to replicate simulation runs and includes the ability to correlate variables, expeditiously select from standard distributions, aggregate and display output, and other useful func- tions (Olson and Wu 2013). The solution is bounded by the total available budget, and both deterministic and stochastic solutions are shown in Table 7.

Table 7 and Fig. 5 show how epistemic data uncertainty simu- lated with 1,000 replications affects the total economic losses re- sulting from each total budget, which are allocated based on the solutions of Eq. (16). The differences between 5th and 95th per- centiles of simulated results is $9 and $12 million in the cases of the four different budget limits. Also, as the standard deviation shows, the amount of variation or data dispersion because of the uncertainty is at least $3 million, highlighting the importance of accounting for uncertainty in decision-making. The budget alloca- tion based on the deterministic model cannot address the epistemic uncertainty in the model. Hence, we consider the data uncertainty in decision-making for allocating the limited budget to harden the six industries within the state of Oklahoma.

Implementing the proposed soft-robust optimization model in Eq. (18), ε ¼ 0.05 is considered to guarantee an absorbability that holds with probability of (1 − ε ¼ 95%). When comparing the resource allocation resulting from the data uncertainty shown in Table 8 with the results from the deterministic model shown in Table 4, the allocation differences are generally less than 10%. Some exceptions include food and beverage and tobacco products (311), which experience a 100% decrease when the total budget is $10 million and subsequently has lesser budget allocations com- pared to the values in Table 4. Similarly, petroleum and coal prod- ucts (324) encounter decreases in allocated resources, which vary for different total budget amounts. Comparatively, chemical products (325), miscellaneous manufacturing (339), and machinery (333) see increases in budget allocations in that order. These changes, though small, show how uncertainty can alter the resource allocations.

Fig. 6 shows the 95% confidence interval estimates for total losses in the Oklahoma economy expected when each total budget is allocated across the six industries.

A measure of the effectiveness of investment αl is considered to monitor the effects of changes in the data uncertainty of budget allocation. Fig. 7 illustrates the changes in the resulting allocated budget to each industry when the interval in probabilistic treatment of αl is increased by 25%, 50%, and 75%. These three incremental levels of uncertainty are deployed by changing the upper and lower limits shown in Table 6, while the mode values are kept constant. The percentage of changes in budget allocation when compared with the results of the deterministic model for different budget lim- its are plotted. As shown in Fig. 7, the different behaviors in the percentage of the changes in allocating budget to the six industries are monitored based on the total budget limits and the magnitude of the change in the uncertainty interval. For example, significant de- creases are seen in the budget that should be allocated to food and beverage and tobacco products (311) when the total budget is less than $20 million, which is comparable to the results from the deter- ministic model. Relatively small changes are seen for the allocated budget to chemical products (325). Also, a recognizable change exists in petroleum and coal products (324) when the total budget limit is more than $20 million, where an increase in the uncertainty interval decreases the budget that should be allocated. In general, it can be seen that the larger uncertainty interval for investment effectiveness caused more changes in budget allocation for higher budget limits.

Concluding Remarks

Freight transportation infrastructure plays an important role as a facilitator of economic productivity by connecting industries of multiple regions. Large-scale disruptive events can cause failures within the system that propagate through the multiple intercon- nected industries. Investing in hardening both the infrastructure

Fig. 4. Absorptive capacity at different budget limits.

Table 6. Probabilistic treatment estimating the cost-effective parameter αl

l Industry name

αl

Min Mode Max

1 Food, beverage, and tobacco products 0.024 0.046 0.057 2 Petroleum and coal products 0.040 0.063 0.070 3 Chemical products 0.100 0.342 0.400 4 Nonmetallic mineral products 0.090 0.201 0.300 5 Machinery 0.060 0.083 0.090 6 Miscellaneous manufacturing 0.200 0.426 0.500

Table 7. Total economic loss (in million USD) considering deterministic and epistemic data uncertainty for different budget amounts (in million USD)

Budget

Total economic loss

Deterministic

Epistemic uncertainty

Mean SD 5th percentile 95th percentile

10 108.74 111.49 3.00 107.21 116.95 20 88.88 93.27 3.22 88.55 99.08 30 73.98 78.93 3.60 73.50 85.45 40 61.79 67.14 3.79 61.53 73.74

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(e.g., backup equipment) and industries themselves (e.g., on hand inventory) can lessen the effects of disruptions. The interdependent nature of industries must be considered when evaluating such re- source allocation. This paper discusses modeling and analysis on the absorptive capacity of resource allocation.

The interdependent adverse effects of a disruption are measured using an interdependency model, and an exponential resource allocation model is introduced to formulate the risk reduction. Considering the three components of resilience capacity identified by Vugrin and Camphouse (2011) and the notion of static resilience proposed by Rose (2009), a measure of absorptive capacity as the ability of the system to absorb the effects of a disruption is pro- posed. Finally, in an integrated optimization model, we maximize the whole system’s absorbability by allocating a limited budget to harden different industries. Furthermore, sources of epistemic data uncertainty in the interdependency model have been considered when developing a soft-robust optimization model to help policy makers to allocate resources under uncertainty.

The proposed modeling and analysis are implemented in a case study developed from the six important industries at the Port of Catoosa that use the inland waterway to send out commodities to their consumers out of the state of Oklahoma. Results show how increasing the budget limit affects allocated budget to each industry. Although miscellaneous manufacturing (339) and chemi- cal products (325) receive the largest share with the $10 million budget, food and beverage and tobacco products (311) and machi- nery (333) receive the largest share as the budget allocation is in- creased to $40 million. We see that when considering bounded random variations to the amount of �5% of the (deterministic) val- ues of x and D⋆ together with a probabilistic treatment [i.e., incor- porating epistemic uncertainty in the calculation of Eq. (18)], the

Fig. 5. Distributions of total economic loss considering epistemic data uncertainty given total budgets of (a) $10 million; (b) $20 million; (c) $30 million; and (d) $40 million.

Table 8. Resource allocation for absorptive capacity in different industries considering uncertainty (in million USD)

Industry name

Resources allocated to each industry for each total budget

10 20 30 40

Food, beverage, and tobacco products 0.00 5.34 10.71 14.56 Petroleum and coal products 0.00 0.00 0.23 3.07 Chemical products 4.10 4.96 5.83 6.43 Nonmetallic mineral products 0.00 0.00 0.00 0.00 Machinery 2.71 5.84 8.73 10.98 Miscellaneous manufacturing 3.19 3.85 4.50 4.95

Fig. 6. 95% confidence interval estimates of total economic loss (million USD) under different budget limits (thousand USD).

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optimistic, pessimistic, and most likely estimates of αl cause dra- matic variations in the economic loss as a result of the allocated budget. Also, analysis done on the measure of effectiveness shows that when the interval in the probabilistic treatment of αl is in- creased by 25%, 50%, and 75%, the changes in the allocated budget varies in different budget limits. For example, while food and beverage and tobacco products (311) experience a significant de- crease in the allocated budget when the total budget is less than $20 million, it faces at most a 30% decrease when the total budget is $40 million.

The real-world application of this work lies in informing a cen- tral planner to invest more effectively in port resilience to maintain the continuity of service for businesses using the docks. The Port of Catoosa, where our case-study applies, is a public entity overseen by a nine-member board, representing the central planner in our case (Business View Magazine 2016). The investment into the macroeconomic sectors here refers to the investment into port dock operations that facilitate a faster continuity of service flowing through the port for the businesses associated with these sectors. For example, the freight of miscellaneous manufacturing and ma- chinery is handled at the General Dry Cargo dock, which handles the largest tonnage in the port [see Pant et al. (2015) for a variety of tonnages]. Hence, preferring these sectors for different levels of investment reflects the importance of the General Dry Cargo dock for the real-world operations of the port. Similarly, the importance of other sectors like chemical products and food and beverage and tobacco products emerges from the significance of Liquid Bulk and Grains docks, respectively. It is acknowledged here that individual businesses that depend on the port might have different preferences relative to the port operators, whose decisions are informed by the larger-scale macroeconomic impacts on the state and regional economy. The Port of Catoosa is a major industrial hub in the Tulsa metropolitan statistical area (MSA), which contributes to 33.4% of the economy of the state of Oklahoma (Tulsa Regional Chamber 2018); therefore, the options here are aimed at benefiting the wider

state and regional economy through the port. Although it can be said that for small tonnage disruptions individual businesses can use alternative suppliers and options such as road and railway trans- port, when the tonnage disruptions are substantial, their interests would align with the port’s as both can benefit from the business preference for the cheaper mode of barge transport. Evidence shows that the port received $6.25 million in 2011 through the fed- erally funded Transportation Infrastructure Generating Economic Recovery (TIGER) grant [now called Better Utilizing Investments to Leverage Development (BUILD)] (DoT 2018) and additional investments through private companies, which resulted in the port investing $13 million in renovating a 45-year old dock (Business View Magazine 2016). This suggests an interest of aligned public and private initiatives in the improvement of port operations. The analysis presented here aims to inform such spending decisions.

The proposed model can be implemented in a freight infrastruc- ture network, and the multiregional impacts of the disruption can be considered in both modeling the failure propagation and investing for absorptive capacity. Further developments of this work will explore such options.

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