Review on Energy Resilience

Computers & Industrial Engineering 70 (2014) 183–194

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Computers & Industrial Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c a i e

Stochastic measures of resilience and their application to container terminals

http://dx.doi.org/10.1016/j.cie.2014.01.017 0360-8352/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 2012168003. E-mail addresses: [email protected], [email protected]

(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 Kingdom b School of Industrial and Systems Engineering, University of Oklahoma, United States c Engineering Management Program, System Development and Maturity Lab, School of Systems and Enterprises, Stevens Institute of Technology, United States d 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 2013 Received in revised form 29 January 2014 Accepted 31 January 2014 Available online 8 February 2014

Keywords: Resilience Infrastructure systems Vulnerability Recoverability

While early research efforts were devoted to the protection of systems against disruptive events, be they malevolent attacks, man-made accidents, or natural disasters, recent attention has been given to the resilience, 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 for uncertainty, 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 quantify the 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. The contributions herein demonstrate a starting point in the development of a resilience decision making framework.

� 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 they malevolent attacks, man-made accidents, or natural disasters, re- cent attention has been placed on preparedness, response, and recovery (PR2) from these events. This is particularly true for the nation’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 the last decade, though no standard definition or quantitative technique for the paradigm of system resilience has emerged. One approach, illustrated in Fig. 1 as described in Henry and Ramirez-Marquez (2012), describes resilience as the ability to restore a system from disrupted state, Sd, to a stable recovered state, Sf. Resilience is thus defined as the time dependent ratio of recovery over maximum loss in Eq. (1).

zðtÞ¼ RecoveryðtÞ=Maximum LossðtdÞ ð1Þ

For multi-modal transportation, as with any other CIKR, system resilience planning is important (DHS, 2009). The multi-modal transportation system plays a vital role in maintaining commodity flows across multiple industries and multiple regions. Examples of actual disruptive events that befall the transportation system in- clude the collapse of the I-40 bridge spanning the Arkansas River in Oklahoma resulting in the daily detour of 22,000 vehicles for nearly 2 months (Federal Highway Administration, 2008) and the I-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, and inland ports. Critical nodes in a transportation network (e.g., inland waterway 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 many of the same risks as coastal ports, relatively few studies have developed risk assessments of inland ports (Folga et al., 2009; MacKenzie, Barker, & Grant, 2012). Inland waterways are common in 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 the future as barge transportation represents a cheaper and environ- mentally friendlier alternative to already highly-congested truck and 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 container security and the malevolent man-made attacks that could go along with unsecured containers (GAO, 2009). Container security will be- come even more important when the planned Panama Canal expansion project opens in 2014, enabling bigger ships, and more containers, from Asian to Atlantic and Gulf coastal ports and their associated inland waterways. And as the most recent Report Card for America’s Infrastructure gave inland waterway infrastructure a 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 transportation context 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. Work described here proposes stochastic and time dependent metrics of system resilience, as applied to waterway container terminals. The contributions of the paper are twofold: (i) the deterministic met- rics described in Henry and Ramirez-Marquez (2012) are extended to the stochastic case (Time to Total System Restoration, Time to Full System Service Resilience, and Time to a%-Resilience), and (ii) these metrics are used to develop a port resilience approach that is deployed in a data-driven case study for the inland Port of Catoosa in Oklahoma. These contributions serve as a starting point in 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 port resilience framework, and Section 5 illustrates with a data-driven illustration 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 comprise our methodological approach, including previous work in measur- ing resilience and in simulating the operations at a container terminal.

2.1. General representation of resilience

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

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

Let X = (A) represent a system, where A = {i|1 6 i 6 m} is the set components comprising the system. For component i at time t, xi(t) is the state variable (real number) describing the performance of the 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 quantified with respect to an overall system performance/service measure. The service function, u(x(t)) = u(t), which can be analyzed for any possible realization of x(t), maps the system state vector into a real number system state at time t.

As described in Fig. 1, a system can operate in three distinct states: (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 not be 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) system recovery, taking the system from Sd to Sf. While Fig. 1 provides a broad description of the process of resilience, it does not include key entities related to resilience that are provided in the detailed representation in Fig. 2. According to Henry and Ramirez-Marquez (2012), resilience of a system at time t, is exhibited if and only if there is an external disruptive event, ej, that affects the original system state (depicted in Fig. 2 as S0) at time te. Set D¼fejj1 6 j 6 Jg describes the set of possible external disruptive events that could affect the system at time te.

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

jÞ¼ V ji where V j i 2 ½0; 1�. V

j i essentially refers to a com-

ponent’s vulnerability, or its lack of ability to maintain performance after ej. As such, the effect of ej on the state variable of component i is provided in Eq. (2). The decreasing system per- formance due to the disruptive event is seen in its response until time td when the new system state is measured. Note that a com- plete reduction in the functionality of the link occurs when V ji ¼ 1. The vector quantifying the disruptive effects of ej for all compo- nents is Vj ¼ðV j1; . . . ; V

j i; . . . ; V

j mÞ.

xiðtdÞ¼ 1 � V ji � �

xiðt0Þ ð2Þ

Note that in this work we do not focus on the trajectory of the decrease in system performance (i.e., linearly or non-linearly over td–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 be linearly 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 implications on resilience are areas of future work.

The effect of individual component vulnerabilities Vj ¼ðV j1; . . . ; V ji; . . . ; V

j mÞ on the entire system is assessed by quantifying the

damage to the system service function u(t). For example, if the sys- tem under study is a manufacturing facility, u(t) could measure throughput. After a period of degradation of length (td–te) the sys- tem service function is damaged from its original state, S0 (with corresponding 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 of the 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) and u(tf). System restoration depends upon external planning efforts planned 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 zuðtrjejÞ evaluated at time tr e (td, tf).

zuðtrjejÞ¼ ½uðtr ejÞ� uðtdjejÞ� ½uðt0Þ� uðtdjejÞ�

8ej 2D ð3Þ

2.2. Stochastic resilience measures and planning

Parameter V ji is considered stochastic due to the uncertainty associated with the nature of event ej and the subsequent reaction of component i to that event. Eq. (4) governs the behavior of V ji ly- ing in [a, b] e [0, 1].

P a < V ji 6 b � �

¼ Z b

a f v ji � �

dv ji 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 a component. Naturally, recovery time for the ith component would be a function of the initial effect of ej on the component, or UjiðV

j iðe

jÞÞ¼ UjiðV j iÞ. Similar to the initial impact, recovery is also

uncertain, therefore UjiðV j iÞ 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 xiðtrÞ¼ xiðtdÞ until recovery occurs, suggest- ing a step function to model repair. This assumption could be relaxed with a known trajectory (e.g., linear, convex, concave) describing the behavior of xi(t), t e [ts, tf].

P ts < U j iðV

j iÞ6 tr

� � ¼ Z tr

ts

f uji V j i

� �� � dv ji ð5Þ

We can devise a strategy, ðejÞ¼ ðsj1; s j 2; . . . ; s

j mÞ, which is a vector of

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

sðejÞ¼ ðoðejÞ; UðVjðejÞÞÞ

¼ oj1; U j 1

� � ; . . . ; oji; U

j i

� � ; . . . ; ojm; U

j m

� �� � 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 V ji > 00 otherwise (

ð7Þ

As such, Eqs. (8) and (9) describe the ðoji; U j iÞ duple in more detail,

respectively (Z+ refers to the set of positive integers). For example, component i might be repaired with recovery activity sji ¼ð4; UNIð4; 9ÞÞ, suggesting that component i would be recovered fourth out of the m components and the recovery time for that activity would be uniformly distributed between 4 and 9 time units.

ojiðe jÞ¼ ojijo

j i ¼ h; h 2 Z

þ ; X

i

oji ¼ X

i

eV ji ( )

ð8Þ

Ujiðe jÞ¼ UjijP ts < U

j i V

j i

� � 6 tr

� � ¼ Z tr

ts

f uji V j i

� �� � dv ji

� � ð9Þ

If the recovery orders are known and the probability distributions for the components recoveries are given, then we can devise the schedule for recovery. The set Ah ¼fsjijo

j i 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 are planned in parallel, while the different sets show the series plan- ning of the overall recovery activities. For example, consider the network 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(e j), and

within each Ah, parallel recovery sets are described (e.g., repair of links 1, 4, 5, and 10 occurs in parallel).

WðejÞ¼ A1; A2; . . . ; Al; l 6 X

i

eV ji ( )

ð10Þ

Three resilience metrics are of interest in this paper. First, the met- ric Time to Total System Restoration, TT(e

j), measures the total time spent from the point when recovery activities commence at time ts, up to the time when all recovery activities are finalized irrespec- tive of whether components are repaired in series or parallel. From a recovery planning perspective this metric gives an idea of the man-hours required to repair each component individually. This is represented mathematically in Eq. (11). Based on TT(e

j), one can cal- culate the probability that total system restoration is finished be- fore mission time t as PRðtÞ¼ PðT TðejÞ6 tÞ.

T TðejÞ¼ X

W

Uji ð11Þ

The second metric, Time to Full System Service Resilience, Tuðt0Þðe jÞ,

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., zuðtfjejÞ¼ 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 PFðtfÞ¼ PðTuðt0Þðe

jÞ6 tfÞ; noting that T TðejÞ P Tuðt0Þðe

jÞ:

Tuðt0Þðe jÞ¼

X Ah

max Uj

i 2Ah ½Uji�

! ð12Þ

Finally, the metric Time to a � 100%-Resilience, Ta(ej), measures the total 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., zuðtajejÞ¼ 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 time ta as PaðtaÞ¼ PðTaðejÞ6 taÞ:

While not within the scope of this paper, these metrics can guide the resilience planning and decision making effort, wherein investments are made to strengthen resilience (e.g., improve recovery time).

3. Model of container terminal operations

Several modeling approaches have been applied in the trans- portation studies for different types of transfer facilities (Lee et 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 departure of commodities for exports and the arrival of commodities for import.

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 m components of the system. The port model that follows continues with this notation.

A discrete time model describes the four port operations (Pant et al., 2011; Simao & Powell, 1992). It is assumed that commodities arrive 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, there are m parallel queueing systems in operation. For a time increment of Dt, the discrete time model captures the evolution of a queueing model at all times t (=0, Dt, 2Dt,. . .). Random variables required for quantifying different elements of normal port operations include the 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 at time 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 i that can be loaded from the yard to trucks or trains in the time interval (t, t + Dt];

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

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

The arrival of commodities, Yi(t), the service capabilities of cranes, Mi(t), and the import loading process, Wi(t), would likely be known from port data sources. As such, the remaining random variables are functions of Yi(t), Mi(t), and Wi(t), and they are calcu- lated differently depending on the nature of the arriving and departing commodity (export or import) and whether there are disruptions at the port.

3.1. Export operations

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

Niðt þ DtÞ¼ max½0; NiðtÞ� MiðtÞ�þ Y iðt þ tÞ ð13Þ

CiðtÞ¼ min½NiðtÞ; MiðtÞ� ð14Þ

3.2. Import operations

Imported commodities arrive at the docks and are transferred from 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) calculates the number of units of commodity i at the yard as the sum of the units remaining and the units transferred. Under normal port oper- ations, the number of units of commodity i departing the port are equal to the units transferred by the crane, as shown in Eq. (17).

CiðtÞ¼ Y iðt þ DtÞ ð15Þ

Niðt þ DtÞ¼ max½0; NiðtÞ� W iðtÞ�þ CiðtÞ ð16Þ

DiðtÞ¼ min½NiðtÞ; W iðtÞ� 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 terminal can 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 resilience of container terminals to disruptive events. Particular decision making insights are made for different commodities, as well as for 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. Due to the varied nature of these commodities, different docks may handle specific commodity types. For example, a large crane at one dock within the port may be required to load and unload large items such as machinery, while a conveyor at a separate dock may enable the flow of bulk goods such as grains. As such, we refer to the ith commodity queue type, i = 1, . . . , m, where different subsets of those m commodity queues would be handled by different docks. If any of these docks were to become inoperable, it would stop the flow of the specific type of commodity handled by that dock.

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

4.2. Modeling port disruptions and recovery

Vector Vj ¼ðV j1; . . . ; V j i; . . . ; V

j mÞ 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 the operations 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 cranes are located, and (iii) the river through which barges enter and exit the port. As such based on the locations disruptions are modeled as two scenarios: (i) terminal closure due rail, road, or river disrup- tions or port disruption, and (ii) crane outage due to wear and tear to equipment. The individual modeling of each of these scenarios, and how each alters the random variables and arrival/departure processes, is described below.

To allow for dock-specific disruptions, the following scenarios are parameterized to account for specific commodity types. Note that the general definition of the degraded state of the commodity,

xiðtdÞ¼ ð1 � V jiÞxiðt0Þ, 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, V ji is allowed to vary with time to represent how the rates of commodity han- dling change as the disruptive event evolves following its onset and subsequent recovery. And commodity departures, u(t), are then represented as a function of these rates.

4.2.1. Terminal closure A 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) quantifies how the mean rates of arrival for a commodity, kjiðtÞ, would evolve over the time period examined. Assume that the as-planned arrival rate for commodity i, or the arrival rate when the terminal is in its original state, is kjiðt0Þ. The mean arrival rate is then disrupted by a time-dependent monotonically increasing factor V jiðtÞ 2 ½0; 1�s:t� V jiðt1Þ > V

j iðt2Þ8t1 > t2 following the disruption at time te. The ter-

minal is then in its disrupted state, where the as-planned arrival mean rate is degraded to kjiðt0Þð1 � V

j iðtÞÞ between times te and td

The degraded mean rate stabilizes to kjiðt0Þð1 � V j iðtdÞÞ between

times td and ts as no further degradation can take place. During the recovery transition, the mean arrival rate improves over time with kjiðtfÞð1 � r

1 i ðtÞÞ P k

j iðt0Þð1 � V

j iðtdÞÞ until time tf, when the

recovered terminal operates with the desired mean arrival rate kjiðtfÞ. The factor r

j iðtÞ 2 ½0; 1�s:t � r

j iðt1Þ < r

j iðt2Þ8t1 > t2 is monotoni-

cally decreasing the time because it signifies improved arrival rates.

kjiðtÞ¼

kjiðt0Þ t 2 ðt0; te�

kjiðt0Þ 1 � V j iðtÞ

� � t 2 ðte; td�

kjiðt0Þ 1 � V j iðtdÞ

� � t 2 ðtd; ts�

kjiðtfÞ 1 � r j iðtÞ

� � t 2 ðts; tf�

kjiðtfÞ t 2 ðtf;1Þ

8>>>>>>>>>>< >>>>>>>>>>:

ð18Þ

4.2.2. Crane outage Disruptive 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 to Eq. (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 V j iðtÞ 2 ½0; 1� s:t�

V jiðt1Þ > V j iðt2Þ8t1 > t2 after the disruptive event at time te. The dis-

rupted state is characterized with mean service rate to

ljiðt0Þð1 � V j iðtÞÞ between times te and td. When recovery is ongoing,

the service rate improves to ljiðtfÞð1 � r j iðtÞÞ P l

j iðt0Þð1 � V

j iðtdÞÞ

with rjiðtÞ 2 ½0; 1�s:t � r j iðt1Þ < r

j iðt2Þ8t1 > t2. The recovered state for

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

ljiðtÞ¼

ljiðt0Þ t 2 ðt0; te�

ljiðt0Þ 1 � V j iðtÞ

� � t 2 ðte; td�

ljiðt0Þ 1 � V j iðtdÞ

� � t 2 ðtd; ts�

ljiðtfÞ 1 � r j iðtÞ

� � t 2 ðts; tf�

ljiðtfÞ t 2 ðtf;1Þ

8>>>>>>>>>>< >>>>>>>>>>:

ð19Þ

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

Commodity description Dock Annual commodity flow

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

4.3. Developing restoration activities

The unique blend of port stakeholders can result in a complex relationship for preparedness and recovery planning. A port authority 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 the production facilities and warehouses adjacent to the port infra- structure. And finally, guidance (and investment) is provided to the port authority by a number of state and federal agencies, including the Federal Emergency Management Agency, the US Army 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 objective of maritime transportation system recovery is to ‘‘facilitate short- term national, State, local, and private sector efforts to restore basic functions and services and (maritime transportation system) infra- structure after a transportation disruption during the response phase of incident management and help set the stage for long-term recovery.’’

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

uðtÞ¼ X

i2export DiðtÞþ

X i2import

DiðtÞ ð20Þ

For recovery the strategy sðejÞ¼ fsj1; . . . ; s j i; . . . ; s

j mg designed to

improve recovery takes the specific commodity and the docks into consideration. For a planning perspective interest lies in making an 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 is the same. We can say that the order set oðejÞ¼ foj1; . . . ; o

j i; . . . ; o

j mg

maps to the set h = {1, 2, . . ., l}. The stochastic resilience metrics de- fined in Section 2 depend upon the inter-departure times of the commodities 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 deployed with an illustrative example addressing the disruption of the Port of Catoosa in Tulsa, Oklahoma. Spread over an area of approxi- mately 2500 acres, the Port of Catoosa is the largest inland port in the US in terms of area. Annual freight volume of 2.2 million tons is 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 deals with a specific commodity type. The General Dry Cargo dock han- dles large items, primarily steel, iron, and machinery. The Dry Bulk dock handles a variety of loose commodities that are moved by conveyer, such as sand, gravel, and fertilizers. The Grains dock moves 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 docks were to become inoperable, it would stop the flow of the specific type 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 specific docks handling those commodities and the 2007 US dollar value of flow, in millions, of those commodities. Roughly $937 million in commodity flow was handled at the Port in 2007.

The commodity queues depend upon the exports and imports of the 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 going through four docks of the port. As such in the port simulation and resilience 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 of Catoosa, 2012). Therefore the estimates of mean daily flow rates (kiðtÞ) of exports and imports through the port are calculated by dividing the Table 2 numbers by 250. For the simulation models and resilience analysis the time step considered is 1 day, which means that t is in units of days. Under normal operations cranes on the general dry cargo dock can service an estimated 2240 tons of inbound or outbound cargo daily. Hence, under normal opera- tions, the mean daily service rate (li(t)) for the cranes in import or export is assumed to be 2240 tons. The dry bulk dock has cranes that can handle an estimated 3200 tons of inbound or outbound cargo per day, which becomes the mean service rate of cranes on this dock. For the grains dock the estimated daily crane service rate (li(t)) for outbound cargo is 5540 tons and for inbound cargo is 3700 tons. For the liquid bulk dock transport is facilitated through

Table 2 2007 Estimates of annual tonnage of commodities divided into exports and imports through the Port of Catoosa.

Commodity description Annual commodity flow (tons)

Export Import

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

Having set the above parameters for normal port operations, we implement Eqs. (13)–(17) as stationary Poisson queueing processes and 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 of the 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 the commodity departing through the port as exports and imports respectively.

It is assumed that there could be a disruption at the port on any given day during the year, resulting in disrupted commodity flows. In this study, the disruption is assumed to occur on day 50 of port operation in the year. Based on the modeling concepts developed in the previous sections, two different types of disruption effects are studied separately: (i) the terminal closure, and (ii) the crane outage. 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 of the vector V1 are all equal to 1. While 4! = 24 dock recovery se- quences exist, our heuristic for the order of repair for the four docks is based on the dollar value and tonnage of commodity types through the dock, shown in Table 1: recovery of the General Dry Cargo 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 that initially for 2 days the port is completely closed, after which time the repair of the docks commences. Each dock repair and restora- tion of previous operational levels takes 2 days after which the next dock repair is undertaken. Repair restores the flow rate of the commodities entering the port. Eq. (18) generalized the repair schedule for each dock is constructed for resilience planning. Assuming in the simulation the time when disruption strikes is t = 1, the commodity arrival rate losses and restorations through each dock are shown in the Eqs. (21)–(24). The notation kki is used to denote mean arrival rate of commodity i at dock k, where k e {1, 2, 3, 4} represent the General Dry Cargo, Liquid Bulk, Grains, and Dry Bulk docks respectively. The as-planned mean arrival rate parameter kki ð0Þ of commodity i at dock k quantifies the flow before the disruption and is also the flow rate expected when dock oper- ations are finally restored. Once the disruption occurs, this rate goes to zero for until dock repair is started. Every repair is esti- mated to last 2 days, suggesting that the mean arrival rate during recovery is described linearly with kki ð0Þð1 � 0:5ðtf � tÞÞ.

k1i ðtÞ¼

k1i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 3� k1i ð0Þð1 � 0:5ð5 � tÞÞ t 2 ð3; 5� k1i ð0Þ t 2 ð5;1Þ

8>>>>< >>>>:

ð21Þ

k2i ðtÞ¼

k2i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 5� k2i ð0Þð1 � 0:5ð7 � tÞÞ t 2 ð5; 7� k2i ð0Þ t 2 ð7;1Þ

8>>>>< >>>>:

ð22Þ

k3i ðtÞ¼

k3i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 7� k3i ð0Þð1 � 0:5ð9 � tÞÞ t 2 ð7; 9� k3i ð0Þ t 2 ð9;1Þ

8>>>>< >>>>:

ð23Þ

k4i ðtÞ¼

k4i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 9� k4i ð0Þð1 � 0:5ð11 � tÞÞ t 2 ð9; 11� k4i ð0Þ t 2 ð11;1Þ

8>>>>< >>>>:

ð24Þ

To simulate the disruption and recovery process, we modify the Poisson 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 modified based 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 simulation runs 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 90 during a given year when the disruptive event closing the entire Port of Catoosa occurs on day 50. The trajectory of dock recovery is also depicted in Fig. 6.

Fig. 7 depicts a sample simulation result for the resilience of the overall Port of Catoosa with the previously described disruption occurring on day 50 of port operation. The resilience metric in Eq. (3) is calculated in terms of the overall port commodity flows obtained by summing up the export and import flows as shown in 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 the vicinity of day 65, suggesting that it takes roughly 15 days to re- cover after the onset of disruption. Once the recovery activities commence most of the resilience is built up when the General Dry Cargo dock, the Liquid Bulk dock, and the Grains dock are restored.

The time metrics for resilience, which are provided in Eqs. (11) and (12), illustrate another perspective on port resilience. These time metrics are calculated based on the commodity departure times 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 system restoration, the TT metric from Eq. (11), is the time when all the docks have reached recovery, occurring when for each dock the daily 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 of commodity departures of the entire port reaches or exceeds its pre-disruption levels, which could happen before the total recov- ery of the last dock because (i) the other docks are at full service and 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 recovery might result in full service resilience recovery. As depicted through one sample simulation in Fig. 7, Tuðt0Þ ¼ 15 days, we now develop a mean 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 1000 simulations 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 suggested by the distribution in Fig. 8) and Tuðt0Þ would be around 9 days (as suggested by the distribution in Fig. 9) because of the recovery schedule being implemented, there are longer recovery durations due 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 range and can thus be estimated with less uncertainty if the aim is not to have full system resilience restored. Note that since we are gener- ating many simulation runs for the Poisson process, there will be outliers in the results due to the MATLAB Poisson random number generator function used. As such the histograms in Figs. 8–10 have large variances. Also there is some difference in the histogram means and variances plot between Tuðt0Þ and T0.98 in Figs. 9 and 10 respectively because in this specific study T0.98 in most cases does not account for the recovery of the Dry Bulk dock because it contributes 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 terminal closure 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 terminal closure disruption, the notation lki denotes the mean arrival rate of commodity i at dock k (equivalent to the mean service rate of the 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 lki ð0Þ of commodity i at dock k quantifies the as-planned flow prior to the disruption, which is also the flow rate achieved when flow is finally restored.

l1i ðtÞ¼

l1i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 3� l1i ð0Þð1 � 0:5ð5 � tÞÞ t 2 ð3; 5� l1i ð0Þ t 2 ð5;1Þ

8>>>< >>>: ð25Þ

l2i ðtÞ¼

l2i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 5� l2i ð0Þð1 � 0:5ð7 � tÞÞ t 2 ð5; 7� l2i 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 ðtÞ¼

l3i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 7� l3i ð0Þð1 � 0:5ð9 � tÞÞ t 2 ð7; 9� l3i 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 ðtÞ¼

l4i ð0Þ t 2 ð0; 1� 0 t 2 ð1; 9� l4i ð0Þð1 � 0:5ð11 � tÞÞ t 2 ð9; 11� l4i 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 the normal port operations are altered due to the port disruption. Crane service rates are modified based on the specifications from Eqs. (25)–(28). We have generated 1000 simulations for each disrupted queue to capture the randomness in the model. Fig. 11 depicts the trajectory for one sample simulation run showing port resilience for 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 are restored 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. As such there is faster recovery than the sample case of terminal clo- sure as shown in Fig. 7, which is also the general trend from the 100 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 departure times and quantities based on the mean arrival rates and mean ser- vice rates of the cranes. Similar to the terminal closure analysis, the time to total system restoration, TT in Fig. 12, is the time when all the 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 recovery schedule 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 modeling and simulation process. Also the differences in the histogram means and variances plot between Tuðt0Þ and T0.98 in Figs. 13 and 14 respectively occur due to the T0.98 not accounting for the recov- ery of the Dry Bulk dock due to its miniscule contributions to the port 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 of the resilience metrics is highlighted in the analysis and these can be used for the planners’ decisions.

6. Concluding remarks

The emphasis of risk managers and decision makers has shifted from solely the prevention and protection of systems for disruptive events 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 general approach to quantifying system resilience by relating a disruptive event to component performance, ultimately to system perfor- mance. An input–output depiction of this general approach is found in Fig. 15. Further, additional stochastic measures of resil- ience are proposed: Time to Total System Restoration, Time to Full System Service Resilience, and Time to a � 100% Resilience. The distinction among these three measures is particularly important in comparing recovery strategies.

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

where resilience could guide the number of maintenance crews and the selection of machine suppliers given the rates of failure and repair. Similarly, in infrastructure networks with disrupted links or nodes, the prioritized order of repair and the crew size nec- essary to perform repair operations could be determined from a time-dependent value of resilience over a repair horizon. Supply chain 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 the port commodity example illustrated here.

The resilience paradigm and associated measures are applied to a resilience case study for an inland waterway port, including a data-driven illustration for the inland Port of Catoosa near Tulsa, Oklahoma. Inland port and waterway systems serve an important role in commodity flows in the US, and their resilience is vital to the larger multi-modal transportation system. A port simulation, introduced by Pant et al. (2011), generated zuðtrjejÞ as well as the stochastic recoverability measures. These measures can measure the efficacy of different recovery activities, as well as risk manage- ment efforts to reduce the vulnerability of port infrastructure. The performance 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 will consider stochastic ordering of recovery strategies given the three recoverability measures.

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

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