Computers & Industrial Engineering 70 (2014) 183–194

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Computers & Industrial Engineering

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Stochastic measures of resilience and their application to containerterminals

http://dx.doi.org/10.1016/j.cie.2014.01.0170360-8352/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 2012168003.E-mail addresses: jmarquez@stevens.edu, jose.ramirez-marquez@stevens.edu

(J.E. Ramirez-Marquez).

Raghav Pant a, Kash Barker b, Jose Emmanuel Ramirez-Marquez c,⇑, Claudio M. Rocco d

a Environmental Change Institutue, School of Geography and Environment, University of Oxford, United Kingdomb School of Industrial and Systems Engineering, University of Oklahoma, United Statesc Engineering Management Program, System Development and Maturity Lab, School of Systems and Enterprises, Stevens Institute of Technology, United Statesd Facultad de Ingenieria, Universidad Central de Venezuela, Venezuela

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 August 2013Received in revised form 29 January 2014Accepted 31 January 2014Available online 8 February 2014

Keywords:ResilienceInfrastructure systemsVulnerabilityRecoverability

While early research efforts were devoted to the protection of systems against disruptive events, be theymalevolent attacks, man-made accidents, or natural disasters, recent attention has been given to theresilience, or the ability of systems to ‘‘bounce back,’’ of these events. Discussed here is a modeling par-adigm for quantifying system resilience, primarily as a function of vulnerability (the adverse initial sys-tem impact of the disruption) and recoverability (the speed of system recovery). To account foruncertainty, stochastic measures of resilience are introduced, including Time to Total System Restoration,Time to Full System Service Resilience, and Time to a%-Resilience. These metrics are applied to quantifythe resilience of inland waterway ports, important hubs in the flow of commodities, and the port resil-ience approach is deployed in a data-driven case study for the inland Port of Catoosa in Oklahoma. Thecontributions herein demonstrate a starting point in the development of a resilience decision makingframework.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction and motivation

While early research efforts have been devoted to the protec-tion (or hardening) of systems against disruptive events, be theymalevolent attacks, man-made accidents, or natural disasters, re-cent attention has been placed on preparedness, response, andrecovery (PR2) from these events. This is particularly true for thenation’s critical infrastructure and key resources (CIKR), as DHS(2009) recently stated that ‘‘CIKR resilience may be more impor-tant than CIKR hardening.’’

Resilience research has been an emerging research area for thelast decade, though no standard definition or quantitativetechnique for the paradigm of system resilience has emerged.One approach, illustrated in Fig. 1 as described in Henry andRamirez-Marquez (2012), describes resilience as the ability torestore a system from disrupted state, Sd, to a stable recoveredstate, Sf. Resilience is thus defined as the time dependent ratio ofrecovery over maximum loss in Eq. (1).

zðtÞ ¼ RecoveryðtÞ=Maximum LossðtdÞ ð1Þ

For multi-modal transportation, as with any other CIKR, systemresilience planning is important (DHS, 2009). The multi-modaltransportation system plays a vital role in maintaining commodityflows across multiple industries and multiple regions. Examples ofactual disruptive events that befall the transportation system in-clude the collapse of the I-40 bridge spanning the Arkansas Riverin Oklahoma resulting in the daily detour of 22,000 vehicles fornearly 2 months (Federal Highway Administration, 2008) and theI-35W bridge collapse over the Mississippi River in Minnesota,which required daily rerouting of 140,000 vehicles (Zhu, Levison,Liu, & Harder, 2009).

As a result of their critical role, the effects of large-scale disrup-tive events could result in the closure of key transportation facili-ties such as rail yards, cargo terminals, airports, seaports, andinland ports. Critical nodes in a transportation network (e.g., inlandwaterway ports) are particularly susceptible to disruptions in com-modity flows (Lee, Park, & Lee, 2003; Lee & Kim, 2010; Sacone &Siri, 2009; Simao & Powell, 1992). Although inland ports face manyof the same risks as coastal ports, relatively few studies havedeveloped risk assessments of inland ports (Folga et al., 2009;MacKenzie, Barker, & Grant, 2012). Inland waterways are commonin North America and prominent in the economies of Europe(Rodrigue, Debrie, Fremont, & Gouvernal, 2010) and Asia (Xu &Zeng, 2008). The importance of the 25,000 miles of commercially

Fig. 1. System state transition with time.

184 R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194

navigable US waterways for transporting goods may grow in thefuture as barge transportation represents a cheaper and environ-mentally friendlier alternative to already highly-congested truckand train transportation. Further, an expansion of inland water-ways to deal with larger shipments (i.e., containers) has been pro-posed, leading to an increased need in addressing containersecurity and the malevolent man-made attacks that could go alongwith unsecured containers (GAO, 2009). Container security will be-come even more important when the planned Panama Canalexpansion project opens in 2014, enabling bigger ships, and morecontainers, from Asian to Atlantic and Gulf coastal ports and theirassociated inland waterways. And as the most recent Report Cardfor America’s Infrastructure gave inland waterway infrastructurea D-(ASCE, 2009), inland ports are particularly susceptible to natu-ral cause and accidental failures.

Recent explorations of resilience in transportation systems in-clude (i) a conceptual discussion of resilience with several qualita-tive definitions of resilience-related terms in a transportationcontext by Ta, Goodchild and Pitera (2009) and (ii) a graph theo-retic optimization framework for resilience by Ip and Wang(2011) that does not include an accounting for recovery time. Workdescribed here proposes stochastic and time dependent metrics ofsystem resilience, as applied to waterway container terminals. Thecontributions of the paper are twofold: (i) the deterministic met-rics described in Henry and Ramirez-Marquez (2012) are extendedto the stochastic case (Time to Total System Restoration, Time toFull System Service Resilience, and Time to a%-Resilience), and(ii) these metrics are used to develop a port resilience approachthat is deployed in a data-driven case study for the inland Port ofCatoosa in Oklahoma. These contributions serve as a starting pointin the development of a resilience decision making framework.

The remainder of this manuscript is as follows: Section 2 pro-vides the quantitative background for the resilience framework,and Section 3 integrates the work of Henry and Ramirez-Marquez(2012) and Pant, Barker, Grant, and Landers (2011) to provide sto-chastic measures of port operations. Section 4 develops the portresilience framework, and Section 5 illustrates with a data-drivenillustration from the inland Port of Catoosa in Oklahoma. Conclud-ing remarks are provided in Section 6.

2. Resilience background and methodological development

This section describes some of the modeling ideas that compriseour methodological approach, including previous work in measur-ing resilience and in simulating the operations at a containerterminal.

2.1. General representation of resilience

Several approaches to describe resilience have been proposedacross several application domains. Qualitative discussions of the‘‘resilience triangle,’’ whose area is produced by robustness (theamount of initial impact to the system) and rapidity (the speedwith which recovery takes place), is a well-studied concept in civilinfrastructure applications (Bruneau et al., 2003; Bruneau &

Reinhorn, 2007; Cimellaro, Reinhorn, & Bruneau, 2010; McDaniels,Chang, Cole, Mikawoz, & Longstaff, 2008). Zobel (2011) discusses amore quantitative decision making framework based on the resil-ience triangle, highlighting tradeoffs between robustness andrapidity for the same level of disaster resilience. MacKenzie andBarker (2012) integrate an interdependency model with regressionto quantify the resilience of electric power infrastructure disrup-tions. The quantitative measures of resilience developed in thissection are adapted from Henry and Ramirez-Marquez (2012).

Let X = (A) represent a system, where A = {i|1 6 i 6m} is the setcomponents comprising the system. For component i at time t, xi(t)is the state variable (real number) describing the performance ofthe component, possibly valuating an entity such as capacity, de-lay, or length, among others. The system state vector at time t,x(t) = (x1(t), x2(t), . . . ,xm(t)), denotes the state of all the system com-ponents at time t. The entire system performance can be quantifiedwith respect to an overall system performance/service measure.The service function, u(x(t)) = u(t), which can be analyzed forany possible realization of x(t), maps the system state vector intoa real number system state at time t.

As described in Fig. 1, a system can operate in three distinctstates: (i) its original, as-planned state, S0, (ii) its disrupted state,Sd, that results from a disruption to the system, and (iii) its recov-ered state, Sf, that results from a recovery effort. State Sf need notbe the same as S0, as the new state may reach an alternative (lower,or perhaps higher) equilibrium level (e.g., for economic systems(Rose & Liao, 2005)). Transitions between these states include (i)system disruption, taking the system from S0 to Sd, and (ii) systemrecovery, taking the system from Sd to Sf. While Fig. 1 provides abroad description of the process of resilience, it does not includekey entities related to resilience that are provided in the detailedrepresentation in Fig. 2. According to Henry and Ramirez-Marquez(2012), resilience of a system at time t, is exhibited if and only ifthere is an external disruptive event, ej, that affects the originalsystem state (depicted in Fig. 2 as S0) at time te. Set D ¼ fejj16 j 6 Jg describes the set of possible external disruptive eventsthat could affect the system at time te.

Let xi(t0) represent the as-planned state of the ith componentprior to the onset of disruptive event ej. Assume that the effect ofej is a proportional reduction in the performance of the ith compo-nent by Vj

iðejÞ ¼ Vji where Vj

i 2 ½0;1�. Vji essentially refers to a com-

ponent’s vulnerability, or its lack of ability to maintainperformance after ej. As such, the effect of ej on the state variableof component i is provided in Eq. (2). The decreasing system per-formance due to the disruptive event is seen in its response untiltime td when the new system state is measured. Note that a com-plete reduction in the functionality of the link occurs when Vj

i ¼ 1.The vector quantifying the disruptive effects of ej for all compo-nents is Vj ¼ ðVj

1; . . . ;Vji; . . . ;Vj

mÞ.

xiðtdÞ ¼ 1� Vji

� �xiðt0Þ ð2Þ

Note that in this work we do not focus on the trajectory of thedecrease in system performance (i.e., linearly or non-linearly overtd–te), but in the final decreased state until the maximum effects

Fig. 2. Detailed description of state transitions over time with respect to the system service function. Note that the disruption toward td and recovery toward tf need not belinearly increasing or decreasing.

R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194 185

of the disruption are felt. The behavior of u(t) and its implicationson resilience are areas of future work.

The effect of individual component vulnerabilities Vj ¼ ðVj1; . . . ;

Vji; . . . ;Vj

mÞ on the entire system is assessed by quantifying thedamage to the system service function u(t). For example, if the sys-tem under study is a manufacturing facility, u(t) could measurethroughput. After a period of degradation of length (td–te) the sys-tem service function is damaged from its original state, S0 (withcorresponding u(t0)), to a disrupted state, Sd (with the correspond-ing u(td)). That is, the disruptive effect of such an event is quanti-fied via the analysis of a function u(t) describing the behavior ofthe system as a function of time. After a period of time of length(ts–td), system restoration commences until it reaches a stable sys-tem state, Sf, with corresponding performance metrics x(tf) andu(tf). System restoration depends upon external planning effortsplanned in advance or following the occurrence of ej. Given a par-ticular disruptive event, ej, Eq. (3) provides a more specific quanti-fication of the value of resilience zuðtr jejÞ evaluated at time tr e (td,tf).

zuðtrjejÞ ¼ ½uðtrejÞ �uðtdjejÞ�½uðt0Þ �uðtdjejÞ� 8ej 2 D ð3Þ

2.2. Stochastic resilience measures and planning

Parameter Vji is considered stochastic due to the uncertainty

associated with the nature of event ej and the subsequent reactionof component i to that event. Eq. (4) governs the behavior of Vj

i ly-ing in [a,b] e [0,1].

P a < Vji 6 b

� �¼Z b

af v j

i

� �dv j

i ð4Þ

As recoverability in Fig. 2 refers to the speed at which the compo-nent (and ultimately, the system) recovers, recoverability can man-ifest itself as the time required to recover the functionality of acomponent. Naturally, recovery time for the ith component wouldbe a function of the initial effect of ej on the component, orUj

iðVjiðejÞÞ ¼ Uj

iðVjiÞ. Similar to the initial impact, recovery is also

uncertain, therefore UjiðV

jiÞ is a stochastic term. The probability that

link i recovers prior to time tr e (ts, tf) is found in Eq. (5). For this pa-per, it is assumed that xiðtrÞ ¼ xiðtdÞ until recovery occurs, suggest-ing a step function to model repair. This assumption could berelaxed with a known trajectory (e.g., linear, convex, concave)describing the behavior of xi(t), t e [ts, tf].

P ts < UjiðV

jiÞ 6 tr

� �¼Z tr

ts

f uji Vj

i

� �� �dv j

i ð5Þ

We can devise a strategy, ðejÞ ¼ ðsj1; s

j2; . . . ; sj

mÞ, which is a vector ofcomponent recovery activities to improve the performance of thesystem following disruptive event ej. Each element of the recoveryactivity vector s(ej) is described by (i) the order in which recoveryis performed for components, and (ii) the time required for recov-ery. This is represented with a duple, as shown in Eq. (6).

sðejÞ ¼ ðoðejÞ;UðVjðejÞÞÞ

¼ oj1;U

j1

� �; . . . ; oj

i;Uji

� �; . . . ; oj

m;Ujm

� �� �ð6Þ

We introduce eV ji in Eq. (7) to count component i among those com-

ponents that are disrupted by ej.

eV ji ¼

1 if Vji > 0

0 otherwise

(ð7Þ

As such, Eqs. (8) and (9) describe the ðoji;U

jiÞ duple in more detail,

respectively (Z+ refers to the set of positive integers). For example,component i might be repaired with recovery activitysj

i ¼ ð4;UNIð4;9ÞÞ, suggesting that component i would be recoveredfourth out of the m components and the recovery time for thatactivity would be uniformly distributed between 4 and 9 time units.

ojiðe

jÞ ¼ ojijo

ji ¼ h;h 2 Zþ;

Xi

oji ¼

Xi

eV ji

( )ð8Þ

Ujiðe

jÞ ¼ UjijP ts < Uj

i Vji

� �6 tr

� �¼Z tr

ts

f uji Vj

i

� �� �dv j

i

� �ð9Þ

If the recovery orders are known and the probability distributionsfor the components recoveries are given, then we can devise theschedule for recovery. The set Ah ¼ fsj

ijoji ¼ h;8ig is the collection

of the all those components having the same order of recovery plan-ning. The recovery planning activity schedule is thus given in Eq.(10). Each element set Ah thus shows those activities which areplanned in parallel, while the different sets show the series plan-ning of the overall recovery activities. For example, consider thenetwork of 12 links and the order of repair illustrated in Fig. 3,assuming a disruptive event impacts all 12 links in some way.According to Fig. 3, links 1, 4, 5, and 10 would be repaired first,therefore A1 = {1, 4, 5, 10}. Links 2, 7 and 11 would be repaired sec-ond, therefore A2 = {2, 7, 11}. Similarly, A3 = {3, 8, 9} and A4 = {6, 12}.

Fig. 3. Illustrative network example, adapted from Hillier and Lieberman (2009).

186 R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194

As such, four sequential sets {A1, A2, A3, A4} comprise W(ej), andwithin each Ah, parallel recovery sets are described (e.g., repair oflinks 1, 4, 5, and 10 occurs in parallel).

WðejÞ ¼ A1;A2; . . . ;Al; l 6X

i

eV ji

( )ð10Þ

Three resilience metrics are of interest in this paper. First, the met-ric Time to Total System Restoration, TT(ej), measures the total timespent from the point when recovery activities commence at timets, up to the time when all recovery activities are finalized irrespec-tive of whether components are repaired in series or parallel. Froma recovery planning perspective this metric gives an idea of theman-hours required to repair each component individually. This isrepresented mathematically in Eq. (11). Based on TT(ej), one can cal-culate the probability that total system restoration is finished be-fore mission time t as PRðtÞ ¼ PðTTðejÞ 6 tÞ.

TTðejÞ ¼X

W

Uji ð11Þ

The second metric, Time to Full System Service Resilience, Tuðt0ÞðejÞ,measures the total time spent from the point when recovery activ-ities are started, at time ts, up to the exact time, tf, when system ser-vice is completely restored (i.e., zuðtf jejÞ ¼ 1 and zuðtf � djejÞ< 18d > 0). Tuðt0Þ is formulated in Eq. (12). From Tuðt0Þ; one can de-fine the probability that system service restoration is finished be-fore mission time tf as PFðtfÞ ¼ PðTuðt0ÞðejÞ 6 tfÞ; noting thatTTðejÞP Tuðt0ÞðejÞ:

Tuðt0ÞðejÞ ¼X

Ah

maxUj

i2Ah

½Uji�

!ð12Þ

Finally, the metric Time to a � 100%-Resilience, Ta(ej), measures thetotal time spent from the point when recovery activities commence,at time ts, up to the exact time, ta, when the system service is re-stored to au(t0) (i.e., zuðtajejÞ ¼ a and zuðta � djejÞ < a8d >0;a 2 ½0;1�). From Ta(ej), one can define the probability that net-work service is restored by a � 100%, or au(t0), before mission timeta as PaðtaÞ ¼ PðTaðejÞ 6 taÞ:

While not within the scope of this paper, these metrics canguide the resilience planning and decision making effort, whereininvestments are made to strengthen resilience (e.g., improverecovery time).

3. Model of container terminal operations

Several modeling approaches have been applied in the trans-portation studies for different types of transfer facilities (Leeet al., 2003; Lee & Kim, 2010; Sacone & Siri, 2009; Simao &Powell, 1992) and have been used in analyzing transportation dis-ruptions (Wilson, 2007). We make use of a simple simulation mod-el of the operations at a container terminal proposed by Pant et al.(2011).

Port operations are divided into four components, as illus-trated in Fig. 4: (i) delivery/receipt, including the arrival of ex-port commodities and departure of imported commodities, (ii)yard operations, defined as the temporary storage of commodi-ties at the port, (iii) crane operations, used to transfer commod-ities to and from port docks, and (iv) shipment, or the departureof commodities for exports and the arrival of commodities forimport.

The general discussion of resilience in Section 2 described a sys-tem X consisting of m components. The resilience paradigm is ap-plied to a port consisting of m commodities, analogous to the mcomponents of the system. The port model that follows continueswith this notation.

A discrete time model describes the four port operations (Pantet al., 2011; Simao & Powell, 1992). It is assumed that commoditiesarrive independently of each other at the port, and each commod-ity is transported through the port operations separately. Hence,for m commodities arriving at the port via the waterway, thereare m parallel queueing systems in operation. For a time incrementof Dt, the discrete time model captures the evolution of a queueingmodel at all times t (=0, Dt, 2Dt,. . .). Random variables required forquantifying different elements of normal port operations includethe following:

1. Yi(t) is the number of units of commodity i arriving at the termi-nal in the time interval (t � Dt, t];

2. Ni(t) is the number of units of commodity i in yard storage attime t after commodities have arrived in the interval (t � Dt, t];

3. Mi(t) is the maximum units of commodity i that can be trans-ferred by the cranes to the docks in the time interval (t, t + Dt];

4. Wi(t) is the maximum number of imported units of commodity ithat can be loaded from the yard to trucks or trains in the timeinterval (t, t + Dt];

5. Ci(t) is the number of units of commodity i that are transferredto the dock for shipment in the time interval (t, t + Dt];

6. Di(t) is the number of units of commodity i departing in theinterval (t, t + Dt].

The arrival of commodities, Yi(t), the service capabilities ofcranes, Mi(t), and the import loading process, Wi(t), would likelybe known from port data sources. As such, the remaining randomvariables are functions of Yi(t), Mi(t), and Wi(t), and they are calcu-lated differently depending on the nature of the arriving anddeparting commodity (export or import) and whether there aredisruptions at the port.

3.1. Export operations

The number of units of commodity i stocked in the yard at timet + Dt is shown in Eq. (13) as the sum of the units remaining at theyard at the end of the previous time period and the units that ar-rive during the current time interval. From Eq. (14), the numberof units of commodity i transferred by cranes is the smaller ofthe number of units at the yard and crane capacity. Under normalport operations, the number of units of commodity i exported fromthe port, is equal to the number of units transferred to the docks,that is Di(t) = Ci(t).

Niðt þ DtÞ ¼ max½0;NiðtÞ �MiðtÞ� þ Yiðt þ tÞ ð13Þ

CiðtÞ ¼min½NiðtÞ;MiðtÞ� ð14Þ

3.2. Import operations

Imported commodities arrive at the docks and are transferredfrom the cranes to the yards. Eq. (15) describes the number of units

Fig. 4. General model of main inland port operations.

R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194 187

of commodity i transferred by the cranes, and Eq. (16) calculatesthe number of units of commodity i at the yard as the sum of theunits remaining and the units transferred. Under normal port oper-ations, the number of units of commodity i departing the port areequal to the units transferred by the crane, as shown in Eq. (17).

CiðtÞ ¼ Yiðt þ DtÞ ð15Þ

Niðt þ DtÞ ¼max½0;NiðtÞ �WiðtÞ� þ CiðtÞ ð16Þ

DiðtÞ ¼min½NiðtÞ;WiðtÞ� ð17Þ

Arrivals of commodities can be modeled as independent non-sta-tionary Poisson processes (with rate ki(t) for commodity i). Simi-larly, the rate of service for the crane operations at the terminalcan be modeled as a Poisson process with time-dependent rates(li(t) for commodity i).

4. Port resilience framework

This section integrates the resilience metrics and port simula-tion model to provide a framework for measuring the resilienceof container terminals to disruptive events. Particular decisionmaking insights are made for different commodities, as well asfor different docks located at the container terminal.

4.1. Describing commodity flows at a port

As defined in Section 2.1, X = (A) represents the port system,made up of m commodity queues flowing through the port. Dueto the varied nature of these commodities, different docks mayhandle specific commodity types. For example, a large crane atone dock within the port may be required to load and unload largeitems such as machinery, while a conveyor at a separate dock mayenable the flow of bulk goods such as grains. As such, we refer tothe ith commodity queue type, i = 1, . . . ,m, where different subsetsof those m commodity queues would be handled by differentdocks. If any of these docks were to become inoperable, it wouldstop the flow of the specific type of commodity handled by thatdock.

As mentioned previously, the commodity flows through theport depend upon the arrival rates and crane service rates. As suchin describing the commodity flows at the port these rates need tobe specified. Associated with each commodity its arrival rate kiðtÞand crane service rate li(t) is used to model the flows.

4.2. Modeling port disruptions and recovery

Vector Vj ¼ ðVj1; . . . ;Vj

i; . . . ;VjmÞ quantifies the extent to which

commodity i is not being serviced at the port under scenario ej. Dis-ruptive events can result in inoperability of one or more of theoperations depicted in Fig. 4. Three locations where such disrup-tions can occur are: (i) the road or rail lines that go into and out

of the port, (ii) the port terminals and docks where the cranesare located, and (iii) the river through which barges enter and exitthe port. As such based on the locations disruptions are modeled astwo scenarios: (i) terminal closure due rail, road, or river disrup-tions or port disruption, and (ii) crane outage due to wear and tearto equipment. The individual modeling of each of these scenarios,and how each alters the random variables and arrival/departureprocesses, is described below.

To allow for dock-specific disruptions, the following scenariosare parameterized to account for specific commodity types. Notethat the general definition of the degraded state of the commodity,

xiðtdÞ ¼ ð1� VjiÞxiðt0Þ, is represented here with degraded arrival

rates to the port and degraded service rates at the port, respec-

tively depending on the disruptive scenario. Further, Vji is allowed

to vary with time to represent how the rates of commodity han-dling change as the disruptive event evolves following its onsetand subsequent recovery. And commodity departures, u(t), arethen represented as a function of these rates.

4.2.1. Terminal closureA disruptive event may cause the full or partial closure of termi-

nal j for some time. For a general terminal closure with time-dependent Poisson arrival rate of commodities, Eq. (18) quantifieshow the mean rates of arrival for a commodity, kj

iðtÞ, would evolveover the time period examined. Assume that the as-planned arrivalrate for commodity i, or the arrival rate when the terminal is in itsoriginal state, is kj

iðt0Þ. The mean arrival rate is then disrupted by atime-dependent monotonically increasing factor Vj

iðtÞ 2 ½0;1�s:t�Vj

iðt1Þ > Vjiðt2Þ8t1 > t2 following the disruption at time te. The ter-

minal is then in its disrupted state, where the as-planned arrivalmean rate is degraded to kj

iðt0Þð1� VjiðtÞÞ between times te and td

The degraded mean rate stabilizes to kjiðt0Þð1� Vj

iðtdÞÞ betweentimes td and ts as no further degradation can take place. Duringthe recovery transition, the mean arrival rate improves over timewith kj

iðtf Þð1� r1i ðtÞÞP kj

iðt0Þð1� VjiðtdÞÞ until time tf, when the

recovered terminal operates with the desired mean arrival ratekj

iðtf Þ. The factor rjiðtÞ 2 ½0;1�s:t � r

jiðt1Þ < rj

iðt2Þ8t1 > t2 is monotoni-cally decreasing the time because it signifies improved arrivalrates.

kjiðtÞ ¼

kjiðt0Þ t 2 ðt0; te�

kjiðt0Þ 1� Vj

iðtÞ� �

t 2 ðte; td�

kjiðt0Þ 1� Vj

iðtdÞ� �

t 2 ðtd; ts�

kjiðtf Þ 1� rj

iðtÞ� �

t 2 ðts; tf �

kjiðtf Þ t 2 ðtf ;1Þ

8>>>>>>>>>><>>>>>>>>>>:ð18Þ

4.2.2. Crane outageDisruptive events and normal wear and tear that damage some

of the cranes (or other such equipment) may limit the number of

188 R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194

commodities that are transferred to and from the docks. Similar toEq. (18), a crane outage at terminal j could alter commodity i’s

time-dependent mean service rate of crane operations,ljiðtÞ, as

shown in Eq. (19). The impact on the crane as-planned mean ser-

vice rate, ljiðt0Þ, is measured with proportion Vj

iðtÞ 2 ½0;1� s:t�Vj

iðt1Þ > Vjiðt2Þ8t1 > t2 after the disruptive event at time te. The dis-

rupted state is characterized with mean service rate to

ljiðt0Þð1� Vj

iðtÞÞ between times te and td. When recovery is ongoing,

the service rate improves to ljiðtfÞð1� rj

iðtÞÞP ljiðt0Þð1� Vj

iðtdÞÞwith rj

iðtÞ 2 ½0;1�s:t � rjiðt1Þ < rj

iðt2Þ8t1 > t2. The recovered state for

the terminal has a crane mean service rate of ljiðtf Þ beyond time tf.

ljiðtÞ ¼

ljiðt0Þ t 2 ðt0; te�

ljiðt0Þ 1� Vj

iðtÞ� �

t 2 ðte; td�

ljiðt0Þ 1� Vj

iðtdÞ� �

t 2 ðtd; ts�

ljiðtfÞ 1� rj

iðtÞ� �

t 2 ðts; tf �

ljiðtfÞ t 2 ðtf ;1Þ

8>>>>>>>>>><>>>>>>>>>>:ð19Þ

Table 1Primary commodity descriptions, commodity-specific docks, and 2007 estimates ofannual commodity flow, in $106.

Commodity description Dock Annual commodity flow

Primary metals General Dry Cargo 313.2Machinery General Dry Cargo 107.6Fabricated metals General Dry Cargo 70.6Misc. manufacturing General Dry Cargo 6.2Minerals Dry Bulk 4.2Food and beverage products Grains 146.0Chemicals Liquid Bulk 223.5Petroleum products Liquid Bulk 66.0

4.3. Developing restoration activities

The unique blend of port stakeholders can result in a complexrelationship for preparedness and recovery planning. A portauthority often owns much of the infrastructure (e.g., cranes, piers)at the port and serves as the governing body. Private sector indus-tries own the barges arriving at the port, the rail and truck trans-port operations that import and export on land, and theproduction facilities and warehouses adjacent to the port infra-structure. And finally, guidance (and investment) is provided tothe port authority by a number of state and federal agencies,including the Federal Emergency Management Agency, the USArmy Corps of Engineers, and the Department of Transportation,among others. In the maritime transportation sector-specific plan-ning guidance document prepared by the DHS (2010), the objectiveof maritime transportation system recovery is to ‘‘facilitate short-term national, State, local, and private sector efforts to restore basicfunctions and services and (maritime transportation system) infra-structure after a transportation disruption during the responsephase of incident management and help set the stage for long-termrecovery.’’

The service function, u(t), which describes the performance ofthe system in Fig. 2, models commodity departures from the port.The ability for goods to leave the port is the ultimate measure ofhow effectively the port is operating. As such, u(t) can be ex-pressed with Eq. (20).

uðtÞ ¼X

i2export

DiðtÞ þX

i2import

DiðtÞ ð20Þ

For recovery the strategy sðejÞ ¼ fsj1; . . . ; sj

i; . . . ; sjmg designed to

improve recovery takes the specific commodity and the docks intoconsideration. For a planning perspective interest lies in makingan entire dock operable which facilitates flow recovery of all com-modities through that dock. Hence if there are l docks then the or-der of recovery for all commodities belonging to a particular dock isthe same. We can say that the order set oðejÞ ¼ foj

1; . . . ; oji; . . . ; oj

mgmaps to the set h = {1, 2, . . ., l}. The stochastic resilience metrics de-fined in Section 2 depend upon the inter-departure times of thecommodities which depend upon the mean arrival rates of Eq.(18) or the mean service rates of Eq. (19) between the times ts

and tf.

5. Illustrative example: inland port of catoosa

The resilience framework developed in this paper is deployedwith an illustrative example addressing the disruption of the Portof Catoosa in Tulsa, Oklahoma. Spread over an area of approxi-mately 2500 acres, the Port of Catoosa is the largest inland portin the US in terms of area. Annual freight volume of 2.2 million tonsis sent and received through the Port of Catoosa along the McCle-llan-Kerr Arkansas River Navigation System.

The Port of Catoosa has four main docks, each of which dealswith a specific commodity type. The General Dry Cargo dock han-dles large items, primarily steel, iron, and machinery. The Dry Bulkdock handles a variety of loose commodities that are moved byconveyer, such as sand, gravel, and fertilizers. The Grains dockmoves agricultural products such as corn, wheat, and soybeans. Fi-nally, the Liquid Bulk dock moves liquid products including chem-icals, liquid fertilizers, and even molasses. If any of these dockswere to become inoperable, it would stop the flow of the specifictype of commodity handled by that dock.

Table 1 provides the descriptions of the primary commodities(according to the North American Industry Classification System,NAICS) flowing through the Port of Catoosa, along with the specificdocks handling those commodities and the 2007 US dollar value offlow, in millions, of those commodities. Roughly $937 million incommodity flow was handled at the Port in 2007.

The commodity queues depend upon the exports and imports ofthe above commodities through the port. Table 2 provides the esti-mates of the annual tonnage of imports and export for each com-modity type through the port.

Based on the above data, there are 12 commodity queues goingthrough four docks of the port. As such in the port simulation andresilience estimation model i = {1, 2, . . ., 12} and h = {1, 2, 3, 4}.

Each of the commodity queues is quantified from Eqs. (13)–(17)depending upon whether the commodity is being exported or im-ported. It is assumed that all queues are independent non-station-ary Poisson processes, whose parameters are estimated as follows.The port is generally in operation for 5 days of the week (Monday–Friday), which cumulates to 250 days of service annually (Port ofCatoosa, 2012). Therefore the estimates of mean daily flow rates(kiðtÞ) of exports and imports through the port are calculated bydividing the Table 2 numbers by 250. For the simulation modelsand resilience analysis the time step considered is 1 day, whichmeans that t is in units of days. Under normal operations craneson the general dry cargo dock can service an estimated 2240 tonsof inbound or outbound cargo daily. Hence, under normal opera-tions, the mean daily service rate (li(t)) for the cranes in importor export is assumed to be 2240 tons. The dry bulk dock has cranesthat can handle an estimated 3200 tons of inbound or outboundcargo per day, which becomes the mean service rate of cranes onthis dock. For the grains dock the estimated daily crane service rate(li(t)) for outbound cargo is 5540 tons and for inbound cargo is3700 tons. For the liquid bulk dock transport is facilitated through

Table 22007 Estimates of annual tonnage of commodities divided into exports and importsthrough the Port of Catoosa.

Commodity description Annual commodity flow (tons)

Export Import

Primary metals 0 289,557Machinery 11,436 0Fabricated metals 0 23,485Misc. manufacturing 1600 0Minerals 38,267 7620Food and beverage products 583,743 19,183Chemicals 297,900 450,925Petroleum products 148,024 45,928

R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194 189

pipes instead of cranes, which are assumed to have very highcapacities.

Having set the above parameters for normal port operations, weimplement Eqs. (13)–(17) as stationary Poisson queueing processesand first generate the daily flow of commodities through the port.Note that the modeling approach was described as a non-station-ary Poisson process, though we have taken any seasonality out ofthe arrival and service rate parameters. Simulation code was writ-ten in MATLAB from which one instant of the Poisson random pro-cess was taken as the base case depicting normal port operations.The results of this simulation model are shown in Fig. 5 giving thecommodity departing through the port as exports and importsrespectively.

It is assumed that there could be a disruption at the port on anygiven day during the year, resulting in disrupted commodity flows.In this study, the disruption is assumed to occur on day 50 of portoperation in the year. Based on the modeling concepts developedin the previous sections, two different types of disruption effectsare studied separately: (i) the terminal closure, and (ii) the craneoutage. Assumptions and parameters are discussed subsequently.

5.1. Port of Catoosa terminal closure

It is assumed that the disruption results in a port closure, mean-ing that no exports leave or imports arrive. As such, the elements ofthe vector V1 are all equal to 1. While 4! = 24 dock recovery se-quences exist, our heuristic for the order of repair for the fourdocks is based on the dollar value and tonnage of commodity typesthrough the dock, shown in Table 1: recovery of the General DryCargo dock is carried out first, then the Liquid Bulk dock, followed

Fig. 5. Plots of simulated (a) daily exports and (b

by the Grains dock, and finally the Dry Bulk dock. It is assumed thatinitially for 2 days the port is completely closed, after which timethe repair of the docks commences. Each dock repair and restora-tion of previous operational levels takes 2 days after which thenext dock repair is undertaken. Repair restores the flow rate ofthe commodities entering the port. Eq. (18) generalized the repairschedule for each dock is constructed for resilience planning.Assuming in the simulation the time when disruption strikes ist = 1, the commodity arrival rate losses and restorations througheach dock are shown in the Eqs. (21)–(24). The notation kk

i is usedto denote mean arrival rate of commodity i at dock k, wherek e {1, 2, 3, 4} represent the General Dry Cargo, Liquid Bulk, Grains,and Dry Bulk docks respectively. The as-planned mean arrival rateparameter kk

i ð0Þ of commodity i at dock k quantifies the flow beforethe disruption and is also the flow rate expected when dock oper-ations are finally restored. Once the disruption occurs, this rategoes to zero for until dock repair is started. Every repair is esti-mated to last 2 days, suggesting that the mean arrival rate duringrecovery is described linearly with kk

i ð0Þð1� 0:5ðtf � tÞÞ.

k1i ðtÞ ¼

k1i ð0Þ t 2 ð0;1�

0 t 2 ð1;3�k1

i ð0Þð1� 0:5ð5� tÞÞ t 2 ð3;5�k1

i ð0Þ t 2 ð5;1Þ

8>>>><>>>>: ð21Þ

k2i ðtÞ ¼

k2i ð0Þ t 2 ð0;1�

0 t 2 ð1;5�k2

i ð0Þð1� 0:5ð7� tÞÞ t 2 ð5;7�k2

i ð0Þ t 2 ð7;1Þ

8>>>><>>>>: ð22Þ

k3i ðtÞ ¼

k3i ð0Þ t 2 ð0;1�

0 t 2 ð1;7�k3

i ð0Þð1� 0:5ð9� tÞÞ t 2 ð7;9�k3

i ð0Þ t 2 ð9;1Þ

8>>>><>>>>: ð23Þ

k4i ðtÞ ¼

k4i ð0Þ t 2 ð0;1�

0 t 2 ð1;9�k4

i ð0Þð1� 0:5ð11� tÞÞ t 2 ð9;11�k4

i ð0Þ t 2 ð11;1Þ

8>>>><>>>>: ð24Þ

To simulate the disruption and recovery process, we modify thePoisson queue parameters in the MATLAB simulation developed

) daily imports through the Port of Catoosa.

Fig. 7. The trajectory of resilience over time given a sample disruption scenario.

190 R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194

for the normal port operations. Here the arrival rates are modifiedbased on the specifications in Eqs. (21)–(24). Since we have as-sumed randomness in the model, there are multiple possible dis-ruption and recovery trajectories with the multiple simulationruns capturing such randomness. We have generated 1000 simula-tions for each disrupted queue. Fig. 6 depicts a snapshot of one sam-ple of simulated commodity flow by dock for days 30 through 90during a given year when the disruptive event closing the entirePort of Catoosa occurs on day 50. The trajectory of dock recoveryis also depicted in Fig. 6.

Fig. 7 depicts a sample simulation result for the resilience of theoverall Port of Catoosa with the previously described disruptionoccurring on day 50 of port operation. The resilience metric inEq. (3) is calculated in terms of the overall port commodity flowsobtained by summing up the export and import flows as shownin Eq. (20). Fig. 7 shows that the port is able to return to a pre-dis-ruption level of service, thereby reaching 100% resilience, in thevicinity of day 65, suggesting that it takes roughly 15 days to re-cover after the onset of disruption. Once the recovery activitiescommence most of the resilience is built up when the GeneralDry Cargo dock, the Liquid Bulk dock, and the Grains dock arerestored.

The time metrics for resilience, which are provided in Eqs. (11)and (12), illustrate another perspective on port resilience. Thesetime metrics are calculated based on the commodity departuretimes and the mean arrival rates of commodities. When the com-modities departing reach to their pre-disruption level then the cor-responding time signifies recovery. The time to total systemrestoration, the TT metric from Eq. (11), is the time when all thedocks have reached recovery, occurring when for each dock thedaily commodity tonnage departure is equal to or exceeds the va-lue before the disruption. The Time to Full System Service Resil-ience, Tuðt0Þ from Eq. (12), is the time when the total sum ofcommodity departures of the entire port reaches or exceeds itspre-disruption levels, which could happen before the total recov-ery of the last dock because (i) the other docks are at full serviceand their commodity flows make most of the port operational, or(ii) the data suggests that the Dry Bulk dock handles fewer com-modities relative to other docks, suggesting that partial recoverymight result in full service resilience recovery. As depicted throughone sample simulation in Fig. 7, Tuðt0Þ ¼ 15 days, we now develop amean measure for these time metrics by generating the 1000 sim-ulations for the queuing models. Figs. 8 and 9 depict the distribu-tions of the two time metrics obtained by running all 1000simulations of the port model with the given recovery strategy.

Fig. 6. Plots showing the daily (a) export and (b) import flows for the four do

While, in most instances, TT would be around 12 days (as suggestedby the distribution in Fig. 8) and Tuðt0Þ would be around 9 days (assuggested by the distribution in Fig. 9) because of the recoveryschedule being implemented, there are longer recovery durationsdue to the stochastic nature of the recovery process. Similarly,Fig. 10 depicts the distribution for the time to 98% service resil-ience, for which almost all recovery times lie in the 8–10 day rangeand can thus be estimated with less uncertainty if the aim is not tohave full system resilience restored. Note that since we are gener-ating many simulation runs for the Poisson process, there will beoutliers in the results due to the MATLAB Poisson random numbergenerator function used. As such the histograms in Figs. 8–10 havelarge variances. Also there is some difference in the histogrammeans and variances plot between Tuðt0Þ and T0.98 in Figs. 9 and10 respectively because in this specific study T0.98 in most casesdoes not account for the recovery of the Dry Bulk dock because itcontributes very little to the port exports–imports (refer Table 1),whereas Tuðt0Þ accounts for such recovery.

5.2. Port of Catoosa crane outage

The crane restoration recovery strategy follows the terminalclosure recovery strategy, where the General Dry Cargo dock is re-stored first and the Dry Goods dock last due to the value and

cks of the Port of Catoosa with the disruption and recovery of each dock.

Fig. 8. Time to total system restoration for a terminal closure/restoration expressed in (a) histogram (approximate pdf) and (b) cdf formats.

Fig. 9. Time to full service resilience for a terminal closure/restoration expressed in (a) histogram (approximate pdf) and (b) cdf formats.

Fig. 10. Time to 98% port recovery (T0.98) from terminal closure/restoration expressed in (a) histogram (approximate pdf) and (b) cdf formats.

R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194 191

amount of commodity flows through each dock. Eqs. (25)–(28)quantify linear recovery planning, where, similar to the terminalclosure disruption, the notation lk

i denotes the mean arrival rateof commodity i at dock k (equivalent to the mean service rate ofthe crane), and k e {1, 2, 3, 4} represents the General Dry Cargo, Li-quid Bulk, Grains, and Dry Bulk docks, respectively. In Eqs. (25)–(28), the mean service rate (mean arrival rate) parameter lk

i ð0Þof commodity i at dock k quantifies the as-planned flow prior tothe disruption, which is also the flow rate achieved when flow isfinally restored.

l1i ðtÞ ¼

l1i ð0Þ t 2 ð0;1�

0 t 2 ð1;3�l1

i ð0Þð1� 0:5ð5� tÞÞ t 2 ð3;5�l1

i ð0Þ t 2 ð5;1Þ

8>>><>>>: ð25Þ

l2i ðtÞ ¼

l2i ð0Þ t 2 ð0;1�

0 t 2 ð1;5�l2

i ð0Þð1� 0:5ð7� tÞÞ t 2 ð5;7�l2

i ð0Þ t 2 ð7;1Þ

8>>><>>>: ð26Þ

Fig. 11. Trajectory of port resilience over time given the crane outage scenario.

192 R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194

l3i ðtÞ ¼

l3i ð0Þ t 2 ð0;1�

0 t 2 ð1;7�l3

i ð0Þð1� 0:5ð9� tÞÞ t 2 ð7;9�l3

i ð0Þ t 2 ð9;1Þ

8>>><>>>: ð27Þ

Fig. 12. Time to total system restoration for a crane outage expre

Fig. 13. Time to full service resilience for a crane outage expres

l4i ðtÞ ¼

l4i ð0Þ t 2 ð0;1�

0 t 2 ð1;9�l4

i ð0Þð1� 0:5ð11� tÞÞ t 2 ð9;11�l4

i ð0Þ t 2 ð11;1Þ

8>>><>>>: ð28Þ

Similar simulation results to the terminal stoppage case can be ob-tained for the crane outage disruption. Again, we develop the mod-ified MATLAB code where the Poisson queue parameters from thenormal port operations are altered due to the port disruption. Craneservice rates are modified based on the specifications from Eqs.(25)–(28). We have generated 1000 simulations for each disruptedqueue to capture the randomness in the model. Fig. 11 depicts thetrajectory for one sample simulation run showing port resiliencefor the given dock recovery strategy. In general the cranes are de-signed to be very high capacity structures which are capable of han-dling far more cargo than the current flow. As such even if they arerestored to work at partial capacity the commodity flows can be re-stored and the docks can be brought to full functionality. In Fig. 11,all the docks recover by day 60 after the disruption on day 50. Assuch there is faster recovery than the sample case of terminal clo-sure as shown in Fig. 7, which is also the general trend from the100 simulation results as discussed below.

Figs. 12 and 13 show the time metrics for resilience for a simu-lation of 1000 runs, calculated based on the commodity departuretimes and quantities based on the mean arrival rates and mean ser-vice rates of the cranes. Similar to the terminal closure analysis, thetime to total system restoration, TT in Fig. 12, is the time when allthe docks have made recovery, and the Time to Full System Service

ssed in (a) histogram (approximate pdf) and (b) cdf formats.

sed in (a) histogram (approximate pdf) and (b) cdf formats.

Fig. 14. Time to 98% port recovery (T0.98) from a crane outage expressed in (a) histogram (approximate pdf) and (b) cdf formats.

Fig. 15. Input–output depiction of the resilience measurement paradigm.

R. Pant et al. / Computers & Industrial Engineering 70 (2014) 183–194 193

Resilience, Tuðt0Þ in Fig. 13, is the time when the total sum of com-modity departures of the entire port reaches or exceeds its pre-dis-ruption levels. Again, it is expected from the planning that both TT

and Tuðt0Þ would be around 9–12 days because of the recoveryschedule being implemented. Similarly, Fig. 14 shows the distribu-tion for the time to 98% service resilience. Similar to previous re-sults, there are outliers due to the randomness of the modelingand simulation process. Also the differences in the histogrammeans and variances plot between Tuðt0Þ and T0.98 in Figs. 13 and14 respectively occur due to the T0.98 not accounting for the recov-ery of the Dry Bulk dock due to its miniscule contributions to theport exports-imports (refer Table 1).

The results of the two cases discussed above show the imple-mentation of recovery activities and the trajectory of restoration/resilience times after a disruption to the port. The usefulness ofthe resilience metrics is highlighted in the analysis and these canbe used for the planners’ decisions.

6. Concluding remarks

The emphasis of risk managers and decision makers has shiftedfrom solely the prevention and protection of systems for disruptiveevents to response and recovery, highlighting the need for resil-ience for the inevitable occasion when a disruptive event occurs.This paper addresses a need in the literature by providing a generalapproach to quantifying system resilience by relating a disruptiveevent to component performance, ultimately to system perfor-mance. An input–output depiction of this general approach isfound in Fig. 15. Further, additional stochastic measures of resil-ience are proposed: Time to Total System Restoration, Time to FullSystem Service Resilience, and Time to a � 100% Resilience. Thedistinction among these three measures is particularly importantin comparing recovery strategies.

These measures have a wide range of applications in systemsengineering. Returning to the manufacturing facility example al-luded to previously, this resilience paradigm could provide a newperspective on the ability of factories to plan for machine failures,

where resilience could guide the number of maintenance crewsand the selection of machine suppliers given the rates of failureand repair. Similarly, in infrastructure networks with disruptedlinks or nodes, the prioritized order of repair and the crew size nec-essary to perform repair operations could be determined from atime-dependent value of resilience over a repair horizon. Supplychain partners (nodes in a supply chain network) could be deter-mined by the resilience that they provide to the supply chain, cal-culated from their vulnerability and recoverability, similar to theport commodity example illustrated here.

The resilience paradigm and associated measures are applied toa resilience case study for an inland waterway port, including adata-driven illustration for the inland Port of Catoosa near Tulsa,Oklahoma. Inland port and waterway systems serve an importantrole in commodity flows in the US, and their resilience is vital tothe larger multi-modal transportation system. A port simulation,introduced by Pant et al. (2011), generated zuðtrjejÞ as well as thestochastic recoverability measures. These measures can measurethe efficacy of different recovery activities, as well as risk manage-ment efforts to reduce the vulnerability of port infrastructure. Theperformance function, the flow of different commodities (in dol-lars) through the different docks, determined the order of repair:a single recovery strategy is considered here, and future work willconsider stochastic ordering of recovery strategies given the threerecoverability measures.

These contributions serve as a starting point in the develop-ment of a resilience decision making framework. Previous workhas explored firm-level tactical decision making on re-routingcommodities due to an inland port disruption (MacKenzie et al.,2012), and similar kinds of decisions could be analyzed here fromthe standpoint of resilience.

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