An analytical approach to the protection planning of a rail intermodal terminal network

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Production, Manufacturing and Logistics

An analytical approach to the protection planning of a rail intermodal terminal network Hassan Sarhadi a, David M. Tulett a, Manish Verma b,∗ a b

Faculty of Business Administration, Memorial University of Newfoundland, Canada DeGroote School of Business, McMaster University, Canada

a r t i c l e

i n f o

Article history: Received 5 January 2015 Accepted 16 July 2016 Available online xxx Keywords: (O) Transportation (T) Logistics Mixed-integer program Fortification Decomposition heuristic

a b s t r a c t Rail-truck intermodal transportation has experienced remarkable growth over the past three decades, and plays a vital role in the freight transportation system in North America. Hence, a crucial issue is to guarantee continuity of service and to minimize the adverse impacts following disruption, natural or man-made. We make the first attempt to develop an analytical framework that could be used by rail intermodal owners to determine the best fortification plan in order to minimize the impact of a worstcase attack. The complexity of the resulting tri-level mathematical model motivated the development of a decomposition-based heuristic solution technique, and the resulting analytical approach was used to solve and analyze problem instances generated using the realistic infrastructure of a railroad operator. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Intermodal transportation, defined as the transportation of goods by a sequence of at least two different modes, continues to be one of the dominant segments of the transportation industry. This has been attributed to the competitive pressures on global supply chains (Szyliowicz, 2003), the increasing demand for new service patterns driven by ocean carriers (Stone, 1997), as well as the globalization of industry (Rondinelli & Berry, 20 0 0). Rail-truck intermodal transportation, which exploits the accessibility advantage of trucks and the scale economies of railroads, has experienced phenomenal growth over the past three decades. The most recent statistics indicate that railroad intermodal traffic, measured in ton-miles, increased by 254 percent between 1993 and 2007 (DOT, 2010), and became the largest revenue segment for the railroad industry (AAR, 2014a; Hatch, 2014). Rail-truck combination is attractive, in part, for three reasons: first, the significant reduction in both delivery and lead-time uncertainty because of the schedule-based operation of intermodal trains (Nozick and Morlok, 1997); second, a more efficient and cost-effective overall movement ensured by combining the best attributes of the two modes (AAR, 2014b); and, third increase in fuel costs have undermined the competitiveness of long-haul trucking (Jennings & Holcomb, 2007). Furthermore, rail-truck intermodal is being promoted as the preferred transportation medium because of its role in alleviating ∗

Corresponding author. E-mail addresses: [email protected] (H. Sarhadi), [email protected] (D.M. Tulett), [email protected] (M. Verma).

highway congestion (Bryan, Weisbrod, & Martland, 2007), and in reducing carbon emission (Kim & Van Wee, 2014). A significant volume of traffic transits the rail-truck intermodal transportation network, which is crucial to the economic growth of North America, and thus the associated infrastructure could be deemed critical, i.e., systems and assets whose destruction (or disruption) would have a crippling effect on security, economy, public health, and safety (US DHS, 2014). Disruptions could be induced by nature such as hurricane Katrina in 20 05 (Mouawad, 20 05), or man-made threats such as the 9/11 terrorist attacks in the United States (Scaparra & Church, 2012). One of the ways to mitigate the impact of disruption is to design supply chain infrastructure so that it operates efficiently (i.e., low cost) both normally and when a disruption occurs (Snyder, Scaparra, Daskin, & Church, 2006). Alternatively, one could ascertain the vulnerability of a critical infrastructure to failure and then develop strategies to preclude it. This paper falls under the latter domain, and proposes an analytical framework to preserve the functionality of a rail-truck intermodal transportation system given a worst-case disruption perpetuated by an intentional attack (or by natural disasters). More specifically we focus on intentional attacks on transportation system, a very real and pertinent issue reflected in the global terrorism database containing over six-thousand attacks on the transportation infrastructure, including numerous on the railroad and the highway networks (START, 2015). Furthermore, the National Counterterrorism Center, a United States government organization responsible for national and international counterterrorism efforts, notes that the proportion of accidents on transportation infrastructure has increased 34 percent since 1998 (NCTC, 2015). In an effort

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to conduct a focused investigation, we consider disruption only at the terminals and identify those crucial to the intermodal infrastructure, and then discuss different strategies to fortify them. A tri-level defender-attacker-defender (DAD) approach is proposed, where the outermost problem belongs to the network operator with a limited budget to protect or harden some of the terminals, the middle level to the attacker with enough resources to interdict some of the un-protected terminals, and the innermost to the intermodal operator who attempts to meet demand on a reduced network. It is pertinent that intentional disruption domain received increased engagements from academics and practitioners over the last decade, starting with the work of Brown, Carlyle, Salmeron, and Wood (2005), and the subsequent contributions mostly focused on fortifying fixed facilities (such as Scaparra & Church, 2008a, 2008b). The authors made the first attempt to extend the discussion about intentional disruption of fixed facilities within a transportation context (Sarhadi, Tulett, & Verma, 2015). It is important to note that, unlike the small problem size in Sarhadi et al. (2015) that could be solved via a commercial solver, in here we are aiming to solve realistic size problem instances that challenge the capability of the existing optimization packages, and thus also propose an efficient decomposition-based solution technique. The resulting analytical framework (i.e., mixed-integer programming model and the heuristic solution technique) was used to study the rail-truck intermodal transportation system of a Class I railroad operator in North America, and the resulting analysis led to the following conclusions. First, finite resources should be spent appropriately if the post-interdiction connectivity of the rail-truck intermodal network needs to be preserved. Second, focusing on just the critical terminals will not result in optimal fortification. The rest of the paper is organized as follows. Section 2 reviews the relevant literature, followed by the problem description and assumptions in Section 3. The analytical framework, i.e., a tri-level mixed-integer programming model and the decomposition-based solution technique, is developed in Section 4, followed by an outline of parameter estimation in Section 5. Solution and analyses of the realistic size problem instances are discussed in Section 6. Finally, conclusions, contributions and directions for future research are outlined in Section 7. 2. Literature review Given the focus of this work on fortification and interdiction of rail-truck intermodal terminals, the relevant papers can be organized under two streams: protection and fortification planning; and, rail-truck intermodal transportation systems. Protection and fortification planning is an enormous exercise especially given the complexity of a typical intermodal infrastructure, the interdependencies among various components (Liberatore, Scaparra, & Daskin, 2012), and the prohibitive cost. As alluded earlier, this emerging area has started receiving increased attention from researchers over the past decade, and we organize the efforts under three sub-streams: redesign of the network; protection of an existing system; and, uncertainty in protection and interdiction. The first sub-stream focused on developing protection strategies by embarking on a full redesign of the network so that the system is robust to attacks. Snyder and Daskin (2005) extended the classical p-median and un-capacitated fixed charge location problems to account for failures of the facilities. More recently, O’Hanley and Church (2011) proposed a resilient design problem for coveragebased service systems that aims to locate a set of facilities such that the combination of initial demand coverage and the minimum coverage following a loss is maximized. Finally, Peng, Snyder, Lim, and Liu (2011) proposed a mathematical model for designing a logistics network that can perform well in pre- and post-disruption conditions.

The second sub-stream, seeking to avoid the huge investments associated with complete redesign of the network, focuses on protecting the pre-established systems and has witnessed most of the academic effort. A majority of the works have approached the fortification problem, within the facility location domain, as a leaderfollower game (Stackelberg, 1952), in which the defender is the leader and the interdictor the follower, and are modeled as bi-level programming problems (Dempe, 2002). For expositional reasons, we review those efforts under two threads: ascertaining criticality; and fortification. The question of ascertaining critical elements can be traced to military planning, wherein the objective was to identify the best place to disrupt or interdict an enemy’s supply line. While the first peer reviewed effort was by Wollmer (1964), subsequent works have investigated the impact of interdiction of arcs in a network to minimize flow capacity (Wood, 1993) and to maximize the shortest path between a given OD pair (Israeli & Wood, 2002), and made use of a variant of the multicommodity shortest path problem to investigate the impact on revenue (Lim & Smith, 2007). Finally, Salmeron, Wood, and Baldick (2004) used a bi-level approach to identify critical components of an electrical supply system, whereas Church, Scaparra, and Middleton (2004) studied the impact of interdicting supply and emergency facilities. The idea of finding the optimal protection plan, and not just protecting the most critical assets, has been introduced by Church and Scaparra (2007). The authors extended their median-based interdiction model by adding a layer to incorporate fortification, and then proposed solution techniques for solving the resulting bi-level programs (Scaparra & Church, 2008a, 2008b). Some recent works have considered fortification within a system of capacitated facilities (Aksen, Piyade, & Aras, 2010; Scaparra & Church, 2012). The concept of fortification against worst-case losses for infrastructure systems has been conceptually introduced in Brown et al. (2005, 2006), which uses bi-level models to represent fortification and interdiction decisions (i.e. defender-attacker framework) and tri-level models to represent fortification, interdiction, and system operating decisions, like network flow decisions (i.e., defender-attackerdefender framework). A number of applications of the proposed framework appeared in the literature such as power grid (Alguacil, Delgadillo, & Arroyo, 2014), water supply (Qiao et al., 2007), and railway infrastructure when the protection resources become available overtime (Starita & Scaparra, 2016). The last work has modeled the protection problem as a bi-level mixed integer program, and proposed two different decomposition techniques to solve them. Finally, under the third sub-stream, uncertainty associated with the attacks was incorporated by attaching a probability of successful attacks on facilities (Church & Scaparra, 2007), and by making use of a probability distribution for estimating the number of facilities that could be attacked (Liberatore, Scaparra, & Daskin, 2011). Losada, Scaparra, and O’Hanley (2012) explores investment in protection measures to reduce the recovery time of the system, whereas Zhang, Zheng, Zhu, and Cai (2014) attached a probability of success measure to investigate vulnerability of a protected facility. Subsequently, the impact of imperfect information between the defender and the attacker is discussed in Zhu, Zheng, Zhang, and Cai (2013), while the need for an all hazards approach to incorporate the possibility of worst-case and random attacks simultaneously is considered in Zhuang and Bier (2007). Rail-truck intermodal transportation systems: Although rail-truck intermodal transportation has been an active research area over the last two decades (Macharis & Bontekoning, 2004), the discussion about disruption is still in its infancy (Sarhadi et al., 2015). We invite the reader to refer to Bontekoning, Macharis, and Trip (2004) for an excellent discussion on intermodal

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transportation, and SteadieSeifi, Dellaert, Nuijten, Van Woensel, and Raoufi (2014) for the state-of-the-art review. To sum, the posed problem makes use of a defender-attackerdefender framework to investigate the optimal protection strategy for rail-truck intermodal terminals and then proposes an efficient solution technique to solve it. Thus, it draws from the works of (Brown et al., 20 05, 20 06, Scaparra and Church 2008a), and the rail-truck intermodal transportation domain (Macharis & Bontekoning, 2004).

3

Table 1 Terminals legend. Terminal

Legend

Terminal

Legend

Terminal

Legend

Chicago Fort Wayne Detroit Cleveland New York Indianapolis

Chi FoW Det Cle NY Ind

Columbus Pittsburg Philadelphia Cincinnati Roanoke Richmond

Col Pit Phi Cin Roa Ric

Norfolk Knoxville Charlotte Atlanta Macon Jacksonville

Nor Kno Cha Atl Mac Jac

3. Problem description In this section, we provide a formal statement of the problem, emphasize its complexity, and then state the modeling assumptions. Fig. 1 depicts the realistic rail-truck intermodal infrastructure of a Class I railroad operator in the United States, and represents the scope of the managerial problem of interest. It represents a por-

Fig. 1. Snapshot of the case study (Adapted from Verma et al., 2012).

tion of the intermodal transportation network developed in Verma, Verter, and Zufferey (2012), and was recreated via a geographical information system (GIS) using ArcView (ESRI, 2008). Fig. 1a depicts the 37 shippers/receivers locations that yield 399 OD pairs, which access the rail-haul component via the 18 intermodal terminals depicted in Fig. 1b. For expositional reasons, we also introduce legends for these terminals (Table 1). These terminals are connected by a total of 62 types of intermodal train services differentiated by route and intermediate stops, i.e., 31 trains of regular type, and another 31 of express type that is 25 percent faster. Thus, the managerial problem is to determine the optimal investment of the finite resources to fortify a certain number of rail intermodal terminals such that the functionality of the network, as much as possible, is preserved in the event of the worst-case disruption. To understand the complexity of the resulting problem, it is important to note that the rail-truck intermodal transportation fortification problem entails hierarchical and sequential decisions amongst three players, i.e., the network operator, the interdictor, and the intermodal operator (Fig. 2). Based on the classification presented in Macharis and Bontekoning (2004), the network operator is the owner of the rail portion of the intermodal infrastructure such as terminals, tracks, and locomotives. At the highest level, the network operator attempts to minimize the cost of the worst-case disruption by fortifying a limited number of intermodal terminals. Note that this is possible only if the owner knows the cost of the worst-case attack by the interdictor, and hence the latter’s problem is a part of the former’s. Next, the interdictor wants to maximize the cost of using the system by attacking a limited number of (unprotected) terminals, which is achieved through complete information about the intermodal operator’s problem. Finally, following interdiction, the intermodal operator makes use of the available resources on the reduced intermodal network (and possibly fewer train services) to meet customer demand at minimum cost. Hence, the indicated interaction amongst the three players can be modeled as a tri-level fortification planning problem using the defender-attacker-defender framework proposed in Brown, Carlyle, Salmerón, and Wood (2006). It is important to note that the same interaction, in large part, makes determining the optimum allocation of finite resources, such that post-disruption functionality of the intermodal infrastructure is preserved, fairly complex. We elaborate more in Section 4. It should be noted that a rail-truck intermodal transportation system comprises three processes: (i) inbound drayage, i.e., trucking service from the shipper location to the origin intermodal terminal; (ii) rail-haul between the intermodal terminals; and (iii) outbound drayage, i.e., trucking service from the destination intermodal terminal to the receiver location. Thus, the intermodal operator focuses on the rail-haul portion of the shipment routing, and endeavors to find the minimum-cost way to satisfy customer demand, given the connections between the available intermodal terminals and shippers/ receivers, and the pre-defined intermodal train services. In addition, it is important to mention that three other pertinent factors must be considered in making the routing decisions. First, since each intermodal terminal has a finite capacity, an interdiction may result in a situation where the remaining

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Fig. 2. Hierarchical structure for fortification planning.

terminals in the network do not have enough capacity to meet demand. Second, punctuality is the main-stay of intermodal shipments (Nozick and Morlok, 1997; Verma, 2012), and hence late deliveries should be penalized as a function of the time delay. Finally, in an effort to ensure feasible solutions (i.e., demand is satisfied), direct trucking is permitted between each shipper-receiver. We now turn to our modeling assumptions: first, a fortified terminal cannot be interdicted; second, each terminal has a finite traffic handling capacity; third, delivery dates are specified when placing the order, and a penalty cost per container per hour is incurred for late deliveries; fourth, there is no congestion at the terminals; fifth, if an intermediate terminal associated with an intermodal train service is interdicted, the train can still serve the remaining terminals on its route; and sixth, an interdicted terminal cannot be used as either origin or destination for any shipment. 4. Analytical framework In this section, we develop a tri-level mathematical formulation for the fortification problem introduced in the previous section, and then outline a decomposition-based solution heuristic. 4.1. Mathematical model Our notation and the model are provided below. Sets I set of shippers, indexed by i J set of receivers, indexed by j Pi j set of intermodal paths between shipper i and receiver j, indexed by p K set of intermodal terminals in the network, indexed by k Pikj set of intermodal paths between shipper i and receiver j which uses intermodal terminal k V set of intermodal train services defined on the network, indexed by v Lv set of service legs for train service v, indexed by l Sl,v set of intermodal paths using service leg l of train service v

Variables p Xi j number of containers using intermodal path p between shipper i and receiver j X Ti j number of containers using direct trucking service between shipper i and receiver j Nv frequency (or the number) of trains of type v  1, if terminal k is protected zk 0, otherwise  1, if terminal k is interdicted yk 0, otherwise Parameters W maximum number of terminals that the network operator can protect R maximum number of terminals that the interdictor can disrupt p Ci j cost of transporting one container from shipper i to receiver j on intermodal path p C Ti j cost of sending a container using truck on the shortest path from shipper i to receiver j p Ti j expected travel time from shipper i to receiver j on intermodal path p Ti j delivery time using truck on the shortest path from shipper i to receiver j T¯i j delivery due date promised by shipper i to receiver j Di j number of containers demanded by receiver j from shipper i PCi j penalty cost per container per unit time between shipper i and receiver j αv capacity of train service v F Cv fixed cost of operating train service v Uk capacity of intermodal terminal k

(P ) MinzC (z )

(1)

subject to:



k∈K

zk ≤ W

(2)

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∀k ∈ K

zk ∈ {0, 1}

5

(3)

where,

C (z ) = MaxyC (y )

(4)

subject to:



k∈K

yk ≤ R

(5)

∀k ∈ K

yk ∈ {0, 1}

∀k ∈ K

yk + zk ≤ 1 where,

(6) (7)

⎡ ⎢ 

C (y ) = Min⎣

i∈I





+

+

i∈I



 j∈J

p ∈ Pi j Ti pj > T¯i j



 j∈J Ti j > T¯i j

Cipj Xipj +



 j∈J

i∈I

p∈Pi j





i∈I

j∈J

C Ti j X Ti j



Ti pj − T¯i j PCi j Xipj

⎤ 

Ti j − T¯i j PCi j X Ti j +



⎥

v∈V

F C v N v ⎦ (8)

4.2. Decomposition-based solution technique

subject to:



p∈Pi j



Xipj + X Ti j ≥ Di j

 i∈I



 p∈Pikj

j∈J

 i∈I

p∈Pi j ∩Sl,v

N v ≥ 0, integer Xipj

≥ 0, integer

X Ti j ≥ 0, integer

∀i ∈ I, ∀ j ∈ J

Xipj ≤ Uk (1 − yk )

 j∈J

Fig. 3. Implicit Enumeration technique of Scaparra and Church (2008a).

Xipj ≤ α v N v

∀k ∈ K ∀v ∈ V, l ∈ Lv

∀v ∈ V ∀i ∈ I, ∀ j ∈ J, ∀ p ∈ Pi j ∀i ∈ I, ∀ j ∈ J

(9) (10) (11) (12) (13) (14)

(P) depicts the tri-level optimization model that could be used to make protection planning decisions. The outer level problem belongs to the network operator whose objective is to minimize total cost by fortifying a given number of intermodal terminals. Constraints sets (3) enforce the binary nature of the terminal fortification decision. The middle level problem belongs to the interdictor who intends to maximize the total cost of using the system. Constraints sets (5) depict the finite resources available for interdiction or disruption of intermodal terminals, whereas (6) represents the binary nature of the interdiction decisions. Constraints sets (7) combine the decisions of the network operator and the interdictor by prohibiting the disruption of fortified terminals. Finally, the inner level problem belongs to the intermodal operator who intends to minimize the total cost of using the system. Note that this is a variant of the multi-commodity flow problem with capacity, delivery time, and penalty cost considerations. The objective function, i.e., (8), will capture the overall cost of moving shipments using the rail-truck intermodal option, any direct trucking service if applicable, the penalty costs for late deliveries, and the fixed cost of running different intermodal trains in the network. Constraints sets (9) ensure the demand is satisfied either using the intermodal option or through direct truck service. Constraints sets (10) enforce the capacity at various terminals in the network, and that the interdicted terminals cannot be a part of different intermodal paths to meet demand. Constraints sets (11) determine the number of intermodal trains of a specific type needed in the network. Finally, the sign and integrality restrictions are imposed through constraints sets (12)–(14).

It is important to note that, due to the nested structure and the presence of integer variables in all three levels, (P) is a very complex and difficult problem. Thus, in order to solve (P) efficiently, a two- stage heuristic solution technique is proposed. In the first stage, an implicit enumeration technique as proposed in Scaparra and Church (2008a) would be used to break the trilevel DAD problem into a set of smaller bi-level AD subproblems (Fig. 2). In the second stage, the proposed decomposition-based solution technique would solve each resulting AD, and the set of decisions leading to the lowest cost subsequent attack will be the best fortification plan.

4.2.1. First stage Implicit Enumeration technique reduces the computational burden of the outermost level (i.e., the network operator) by implementing a rather simple but intuitive observation. It states that the optimal solution of the network operator should entail fortification of at least one of the terminals that would be interdicted in the worst-case attack (Scaparra & Church, 2008a). Fig. 3 depicts the scheme for the realistic size network introduced in Section 2, when the resources are just enough to fortify and interdict two terminals. It should be noted that the cost associated with the darkshaded circles in Fig. 3 is calculated based on the dataset developed by Verma et al. (2012). At the root node, the AD problem must be solved to know which terminals are going to be attacked and thus their removal will produce the worst-case disruption. Note that for the problem setting wherein resources are just enough to both fortify and interdict two terminals, a total of 153 (i.e.,

18 2

) different interdiction

strategies are possible, and that the intermodal operator’s problem has to be solved for each instance. The resulting evaluation led to the determination that intermodal terminals in Philadelphia (Phi) and Atlanta (Atl) are critical in the given network. This information was an input to the implicit enumeration scheme, which was coded in C#. Set Q, at each node, lists the terminals at least one of which must be protected to prevent the worst-case disruption. For instance, at node 1, terminal Phi could either be fortified or not.

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Fig. 4. Pseudo code of the implicit enumeration algorithm.

If Phi were not fortified, then the terminal at Atl would have to be considered for fortification (i.e., node 2). If even Atl is not fortified, then set Q is empty thereby implying that none of the other fortifications can prevent the worst-case disruption, and the resulting node is fathomed (i.e., grey shade). On the other hand, if Atl is fortified, then the worst-case disruption is prevented, and the elements for set Q must be updated by solving the interdiction problem with latest information (i.e., Atl is fortified). Thus, the updated set Q contains NY and Phi as elements representing the most disruptive interdiction given that Atl is fortified. At node 3, only one fortification resource is left; we continue the search process by selecting NY. If NY is not fortified, it is possible that it is disrupted together with Phi thereby resulting in a fathomed node. But if it is fortified, then the interdiction problem is solved given that NY and Atl are fortified. The updated set Q contains Ind and Phi, both of which would be interdicted thereby resulting in a cost of around 13.66 million dollars (i.e., dark shade). If Phi were fortified, the interdiction problem is solved thereby resulting in terminals Ind and Atl in the updated set Q. Arbitrarily selecting Ind, if it is fortified then the protection resources have been exhausted, and the resulting interdiction problem yields Cha and Atl terminals as the most disruptive. At the same time, no further branching is possible and the associated cost is around 13.35 million dollars. But if Ind is not fortified, then the updated set Q only contains Atl. If Atl is not fortified, then it will be attacked together with Ind thereby being fathomed. On the other hand, if Atl is fortified, then the interdiction problem is solved again to yield Chi and Ind as the terminals to be interdicted, and the associated cost is around 13.07 million dollars, which is also the solution for this problem instance.

To enhance the presentation of the implicit enumeration algorithm, we present the notations used to describe the pseudo code of this algorithm in Fig. 4. It is important to reiterate that the implicit enumeration scheme of Scaparra and Church (2008a) reduces the tri-level DAD problem to a set of bi-level AD problems, and then each bi-level AD problem needs to be solved. After solving all the generated bi-level AD problems, we can determine the best protection plan. We next propose an efficient solution technique for the bi-level AD problems.

4.2.2. Proposed methodology The prevalent technique for solving the AD problems is based on the duality theory, wherein the dual of the inner problem is combined with the outer problem to create a single problem (Wood, 2011). It is important that this approach would work only if the variables in the inner problem are all real valued, or if integers, then they can be relaxed without losing integrality. The innermost component of the AD subproblem contains train frequency variables, which are inherently integers, and hence the prevalent technique will not work. We adapt the classical Bender’s decomposition technique (Benders, 1962) to account for the integer variables in the inner problem, which is consistent with the approach in the literature (Gabriel, Shim, Conejo, de la Torre, & García-Bertrand, 2010; Losada et al., 2012). The proposed methodology breaks the bi-level AD problem into a master problem (MP) and a subproblem (SP). We next provide details on: how these two problems are formed; and, how they interact in the proposed solution methodology.

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Subproblem (SP): The subproblem is equivalent to the intermodal operator’s problem, whose solution provides the routing plan and the number of intermodal trains of different types, and where the latter assume integer values. The outputs of SP are used as parameters to build the MP. Master problem (MP): The master problem is a single-level interdictor’s problem that determines the terminals to be attacked such that worst-case results could be perpetuated, and three steps must be taken to build it. First, the intermodal operator’s problem is solved and the train frequency variables are fixed to their optimum (integer) values. Second, a number of problem instances where train frequency variables were fixed to their optimum values exhibited unimodularity, i.e., other integer variables could be relaxed without obtaining non-integer values (Wolsey, 1998). Hence, assuming unimodularity in the intermodal operator’s problem, we take its dual. To that end, we define three sets of dual variables corresponding to constraints (9a)–(11a) in (P).



p∈Pi j



∀i ∈ I, ∀ j ∈ J

Xipj + X Ti j ≥ Di j





i∈I

p∈Pikj

j∈J



 ωi j Positive

∀k ∈ K

Xipj ≤ Uk (1 − yk )

(9a)

(βk Negative ) (10a)

  i∈I

Xipj ≤ α v N v

∀v ∈ V, l ∈ Lv

(θl Negative )

j∈J p∈Pi j ∩Sl,v

(11a) It should be noted that in taking the dual of (P), both the train frequency variables (i.e., N v ) and the set of interdicted terminals (i.e., yk ) are given, and thus could be fixed. More specifically, the train frequency variables are fixed at their optimal values that are obtained by solving the intermodal operator’s problem in the absence of any interdiction (i.e., all yk = 0). If the dual of the intermodal operator’s problem is attached to the interdictor’s problem, we end up with the following objective function:





Maxyk Maxω,β ,θ

 i∈I

+

 

Di j ∗ ωi j +

j∈J

Uk ∗ βk ∗ (1 − yk )

k∈K



α v Nv θ



(15)

l

∀v∈V ∀l∈Lv

which can be simplified to:



Maxy,ω,β ,θ

 i∈I

+

Di j ∗ ωi j +

Uk ∗ βk ∗ (1 − yk )

k∈K

j∈J

 



α v Nv θ

(16)

l

∀v∈V ∀l∈Lv

Subject to the following constraints:

ωi j +



{∀k|

p∈Pikj

}

βk +



{∀l| p∈Sl,v }

θl ≤ Eipj

∀i ∈ I, ∀ j ∈ J, ∀ p ∈ Pi j (17)

ωi j ≤ Ei j  k∈K

∀i ∈ I, ∀ j ∈ J

yk ≤ R

(18) (19)

ωi j ≥ 0

∀i ∈ I, ∀ j ∈ J

(20)

βk ≤ 0

∀k ∈ K

(21)

θl ≤ 0

∀l ∈ Lv

(22)

∀k ∈ K

yk ∈ {0, 1} where,



Eipj



7

(23)



Cipj + Ti pj − T¯i j PCi j , i f Ti pj > T¯i j

=

Cipj ,

 Ei j =

otherwise





C Ti j + Ti j − T¯i j PCi j , i f Ti j > T¯i j C Ti j , otherwise

Note that (16)–(23), as a single-level problem, represents the master problem (MP) and treats frequency variables as fixed values and together with the subproblem (depicted in more detail in Fig. 5) will optimize the AD problem. In the third and final step of forming MP, we note that (16) has non-linear terms (i.e., βk and yk are decision variables), and thus (16)–(23) cannot be solved easily. Therefore, we next outline a suitable linearization scheme to facilitate solving the combined single-level MP. Linearization scheme  Each non-linear term of k∈K Uk ∗ βk ∗ (1 − yk ) can take two values depending on the value of the binary variable yk . It is Uk ∗ βk if yk = 0, and zero if yk = 1. Now, we replace each βk ∗ (1 − yk ) by a new variable ϕk , and constraints sets (24)–(26) will ensure that ϕk would assume the desired values. Note that M is a large positive number.

ϕk ≤ βk + M ∗ yk

(24)

ϕk ≥ −M ∗ (1 − yk )

(25)

ϕk ≤ 0

(26)

If yk = 0, (24) and (25) will ensure that ϕk ≤ βk and ϕk ≥ −M, respectively. Note that the intersection of the three constraints is ϕk ≤ βk , and because it is a maximization problem and βk is negative, the optimal value of ϕk would be βk . On the other hand, if yk = 1, the intersection of all the three inequalities would imply ϕk = 0. To formalize the discussion in this section, we next introduce the notations/parameters used to outline the pseudo code for the decomposition algorithm to solve the bi-level AD problems in Fig. 5. Finally, we depict the interaction between the master problem and the subproblem in Fig. 6. The pseudo code starts with parameters receiving their initial values, i.e., lower and upper bounds are set to minus and positive infinity, respectively; the iteration index is set to zero; and, the list of interdicted terminals is empty. The execution of the proposed algorithm starts by checking that neither of the two terminating conditions, namely, the proximity of lower and upper bounds, and the maximum number of iterations, has been satisfied. Inside the algorithm (i.e., at each iteration), and given the list of interdicted terminals, the subproblem must be solved and the upper bound should be updated. If the updated upper bound is not close enough to warrant algorithm termination, the train frequency variables are fixed to their optimal values (i.e., treated as parameters), and the master problem is built and then solved. The solution of the master problem will necessitate updating the lower bound, and checking the algorithm termination criteria. If the stopping condition is not met, the iteration index is updated and the counter returns to the beginning of the loop. The process continues, and on termination, the objective function of the master problem and the subproblem with the best attack plan will be reported as the outputs of the algorithm. It should be noted that the above decomposition algorithm will be employed at the root node and also all the nodes following defense decisions of the search tree (Fig. 3) and the produced optimal attack Y ∗ constitutes the set Q at each of these nodes.

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Fig. 5. Pseudo code for the decomposition algorithm.

Fig. 6. Interaction between the master problem and the subproblem.

(driver-truck) is engaged, and an estimate of 300 dollar per hour including the estimated hourly fuel cost is used. A penalty cost of 40 dollar per hour per container was used. As indicated there are two types of intermodal train services, namely, regular and express. Average intermodal train speed was calculated using the Railroad Performance Measure website (RPM, 2014), and was estimated to be 27.7 miles per hour for regular, and 36.8 miles per hour for express service. Consistent with the published works, we estimated a rate of 0.875 dollar per mile for regular and 1.164 dollar per mile for express service. The hourly fixed cost of running a regular intermodal train is 500 dollar per hour, which takes into consideration the hourly rates for a driver, an engineer, a brakeman, and an engine, which are 100 dollar, 100 dollar, 100 dollar, and 200 dollar, respectively. The express service is 50 percent more expensive at 750 dollar per hour (Verma, 2012). 5.2. Due dates

5. Estimation of parameters In this section, we provide the relevant details on the parameters used to solve the problem instance introduced in Section 2, whose solution and analyses will be presented in Section 6. These parameters were estimated using the publicly available information.

The distance (d in miles) between each shipper and each receiver was estimated in ArcView GIS (ESRI, 2008). Next, the travel time (in hours) was computed as d/40, where the denominator indicates the average speed of trucks. Finally, a constant of 15 was added to the travel time to obtain the delivery due date for each shipper-receiver pair.

5.1. Cost

5.3. Demand level and terminal capacity

In the United States, trucks can travel at a maximum speed of 50 miles per hour, but due to lights and traffic an average speed of 40 miles per hour is assumed (Verma & Verter, 2010). Normally drayage is charged in terms of the amount of time the crew

The innermost problem belonging to the intermodal operator was solved in CPLEX 12.6.0 (IBM, 2014) on the dataset used in Verma et al. (2012), and the solution was decoded to estimate the traffic volume through each intermodal terminal. It was assumed

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9

Table 2 Objective function value (million dollars) of the EX approach.

Table 3 Objective function value (million dollar) of the PH approach.

that the terminal utilization was 80 percent, and hence the terminal capacity is 1.25 times (i.e., 1 divided by 0.8) the traffic volume through each terminal. The demand level was set according to the dataset used in Verma et al. (2012). Finally, we set the optimality gap ε in the decomposition algorithm to 1 percent, and the maximum number of iterations to 10. 6. Solution and analyses This section has been organized into three sub-sections: efficiency of the proposed solution methodology; computational performance; and, managerial insights. 6.1. Algorithmic efficiency The analytical framework was implemented in C# using CPLEX 12.6.0 concert technology on a PC with Core 2 Quad, 2.4 gigahertz processor, and 4 gigabytes of RAM. For expositional reasons, and also to demonstrate the efficiency of the proposed solution methodology, we define two approaches here. The first approach, referred to as EX, finds the exact solution for the proposed DAD problem and therefore it applies the optimal fortification. It makes use of the mixture of implicit enumeration, to transform DAD problem to a set of ADs, and then uses the complete enumeration technique to find the optimal solution of each AD problem. Note that the complete enumeration solves AD problems by considering all possible ways that the interdictor can disrupt the transportation system. Thus, for each given interdiction, the corresponding intermodal operator’s problem in which the interdicted terminals are out of service is solved. Finally, the interdiction strategy that imposes the highest cost to the transportation system will be selected as the solution of the AD problem. The second approach, referred to as PH, finds a heuristic solution for the DAD problem. It makes use of the implicit enumeration to transform the DAD to ADs, and then uses the proposed decomposition-based heuristic to solve each resulting AD.

Tables 2 and 3 depict the solutions arrived at using these two approaches, and when the budget for both fortification and interdiction is enough for a maximum of nine terminals (i.e., eighty-one problem instances). For expositional reasons, we have rounded the values. It is evident from Table 2 that EX approach could solve sixty-one problem instances, but became challenged when confronted with larger fortified-interdicted combinations. In fact, it could not solve any problem instances that involved interdiction of eight or nine terminals (i.e., shaded in grey). For both these settings, a time limit of thirteen hours was imposed on the algorithm. While we comment on the computational performance in the next sub-section, it is important that PH was unable to converge to the optimum solutions in only three fewer instances than EX, and the respective values are italicized in Table 3. But more importantly, for larger problem instances, PH returned superior solutions than those returned by EX. Note that for a given column, and in both tables, the objective function value decreases when moving down. This is because with the fixed number of attacks (i.e., a specific column), having more protection resources will decrease the effect of attacks and thus the interdictor is forced to destroy less important terminals in the network. Interestingly, an increasing trend happens by going from left to right on each row of these tables. This is due to the fact that, while the defense resource is fixed at each row of these tables (i.e., a specific row), the increase in the attack budget will eventually increase the cost imposed on the network. Consequently, by combining the two trends on rows and columns, we observe that the most costly situation happens on the top right of the table where the maximum number of attacks and minimum number of defenses happen. It is pertinent to indicate that the heuristic behavior of PH stems from the fact that the intermodal operator’s problem (i.e., Eqs. (8)–(14)) is a variant of the multi-commodity flow problem, and thus may not necessarily exhibit unimodularity to yield integer solutions if the integrality constraints are relaxed. However, for the case study presented in this paper, including

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H. Sarhadi et al. / European Journal of Operational Research 000 (2016) 1–15 Table 4 Objective function values (dollars million) for the four approaches.

constraints (9) and (10) ensured that optimum integer solutions were obtained in majority of the problem instances even when no integer restrictions were imposed (Table 3). In an effort to better comment on EX and PH, we focus on the problem instances where the number of terminals fortified equals the number interdicted, and introduce two additional defense approaches (Table 4). The first additional approach, referred to as WC, solves an AD problem to identify the critical terminals and then fortifies them. To find the solution of the AD problem, WC uses complete enumeration. Note that, as explained in Fig. 3, this is equivalent to providing the list of worst-case disruption to the root node to initialize the implicit enumeration scheme. Furthermore, for a comparative assessment, we also indicate the solution for interdiction without fortification instances in the fifth column in Table 4 and refer to them as NO. It is important to reiterate that no terminal is defended under this approach. Finally, we augment the table with additional columns to illustrate how optimal solutions found by EX and PH under assumptions of equal number of defense and attacks, would produce different solutions when the two are not equal. To this end, PH(N-1) and EX(N-1) indicate the objective function values arrived at via the previous heuristic/optimal solution under the new setting, i.e., the number of terminals attacked is one fewer than the number of terminals defended. Similarly, PH(N+1) and EX(N+1) represent the settings wherein the number of terminals attacked is one more than the number defended. The first row of Table 4 depicts the situation where no terminal is either fortified or interdicted; hereafter referred to as the Base-Case, and returns a single unique solution across all approaches. The nine subsequent rows provide a snapshot of the results when both the number of terminals fortified and interdicted are increased. Table 4 could be used to make seven points. First, all the approaches with fortification outperform the NO, and that NO exhibits increasing cost with higher numbers of interdicted terminals. The latter point should be clear since higher numbers of interdicted terminals implies increased reliance on the more expensive transportation option, i.e., direct truck service. Second, as we anticipated, WC approach does not provide a better solution than either EX or PH. This result is consistent with the observations of Scaparra and Church (2008a, 2008b) and Brown et al. (2006), who advise against using the output of the interdiction model to make fortification decisions, since such strategies are suboptimal and will never provide the best protection against worst-case disruptions. Third, the proposed decomposition heuristic is able to find the optimal solution in all but one problem instance. Fourth, for both EX and PH, compared to WC, the cost shows less fluctuations and even with the increase in the number of attacked/defended terminals it starts decreasing which is in sharp contrast with the

Fig. 7. Cost imposed to the system following each defense approach.

trend observed in NO. The cost associated with implementing each of the four defense approaches has been plotted in Fig. 7 for different numbers of attacked/defended terminals. It is obvious from this figure that PH demonstrates the best performance compared to other approaches. We postulate that the costs would decrease further if the program was allowed to naturally terminate for the eight- and nine-terminals problem instances. As alluded earlier, this is because although the number of terminals interdicted increases, the defender is also able to fortify a larger number of (more important) terminals, which in turn forces the interdiction of terminals not likely to increase the resulting cost. Fifth, for the last two rows and within the specified cut-off time, the bestencountered solutions with PH are much better than those with EX. We elaborate on this observation when discussing computational time in Section 6.2. Also, for these rows, the improving trend in PH/EX columns has been stopped, especially by going from 8 attacks/defenses to 9 attacks/defenses. This can be attributed to the fact that the fixed time allocated to solve these two problems is not enough to produce good results and to continue the decreasing trend which was observed previously in these two columns. Also the cost associated with implementing WC worsens for 8 and 9 attacked/defended terminals and produces very weak results. Sixth, the solutions returned under PH and EX (i.e., columns 2 and 3) produce better (lower) objective function values, when the actual number of attacks is one fewer than the number of defenses, i.e., PH(N-1) and EX(N-1), respectively. On the other hand, when the number of attacks is one more than the number of defenses, applying the solutions returned under PH and EX would yield higher objective function values. This incremental analysis throws light on the performance of the optimal solution under an uncertain environment, i.e., when the number of terminals attacked could be one fewer/greater than the number defended.

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H. Sarhadi et al. / European Journal of Operational Research 000 (2016) 1–15 Table 5 Rail-truck modal split percentage.

Table 6 Number of intermodal train services.

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when the infrastructure is functioning normally, i.e., the Base-Case. Note that the proportion of shipments using the rail-truck option decreases with the increase in the number of terminals interdicted for the NO, which is expected because fewer train services could be routed on the reduced network. Furthermore, as indicated in Table 4, WC results in a suboptimal fortification plan, since the proportion of traffic using the rail-truck option is only greater than the ones under NO. Finally, under both EX and PH, the proportion of traffic using rail-truck never drops below 60 percent (except for EX in the case of nine attacks/defenses when the allocated time is not enough to yield a good solution), which demonstrates the effectiveness of both fortification strategies in ensuring that the majority of the intermodal network is operational, irrespective of the interdiction budget. The above observations are aptly supported by the number of trains of both regular and express types being used in the four approaches (Table 6). It should be evident that with the higher number of interdiction (and fortification) faster trains are used, when possible, to avoid late deliveries. 6.2. Computational performance

Seventh, it is interesting to note that the solutions found by WC, for problems with more than two attacks/defenses, are always inferior to those found by EX or PH in an uncertain setting (i.e., the number of terminal interdicted is either one fewer or one more than the number of terminals fortified). This is a very strong result since it demonstrates that even when the actual number of terminals interdicted is greater than the number of terminals optimally fortified, EX or PH would still yield superior solutions than WC. Recall that WC would never accomplish superior fortification than either EX or PH, but now it is safe to state that the latter approaches would return better solutions even in the absence of accurate estimates about the number of interdictions and thus they are robust in this situation. Finally, we comment on the traffic split and the number of intermodal train services. Table 5 depicts the percentage of traffic assigned to the rail-truck intermodal system (the rest is assigned to the expensive truck-only option) under the different problem instances when the number of attacks equals the number of defenses. As expected, the entire traffic uses the intermodal option

In this subsection, we will comment on the computational performance of EX and PH approaches. Table 7 provides a snapshot of the computational time (in thousands of seconds) for the EX approach. For a given row, i.e., the fortification resource is fixed, the time needed for achieving an optimal defense increases sharply with the increase in the number of interdicted terminals. This is because an increase in the number of terminals interdicted would necessitate enumerating a much larger number of attack combinations. On the other hand for a given column, i.e., the number of interdicted terminals is fixed, the time needed for achieving optimal defense increases rather slowly in the first few columns, and then a bit more sharply in the rest. This is because, for all the cells in a given column, a common set of attack possibilities must be constructed, which are then used to build the tree. Once the common set has been determined, the incremental time to evaluate additional fortification plan is not significant for small trees (i.e., fewer number of interdicted terminals). On the other hand, for higher interdiction values, a much larger tree needs to be constructed, which in turns has a significant impact on the computational time. For example, the combinations towards the bottom right of the table yields huge trees, which require significantly higher computation times. It should be added that the grey cells in this table and Table 8 are instances that could not be solved to optimality in 13 hours, or 46,800 seconds, and thus we report the best encountered solution for these instances. Table 8 depicts the computational time for PH, and exhibits a trend distinct than that for EX. First, it should be noted that the number of problem instances that can be solved within a reasonable computation time is much more than those with the EX

Table 7 Computational time (’0 0 0 seconds) for EX.

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H. Sarhadi et al. / European Journal of Operational Research 000 (2016) 1–15 Table 8 Computational time (’0 0 0 seconds) for PH.

Table 9 Computational time (seconds) for the three fortification approaches. # Fortified/Interdicted

1 2 3 4 5 6 7

# Possible outcomes

18 153 816 3060 8568 18,564 31,824

approach. For a given column, a clear increasing trend is exhibited, which can be attributed to the increasing size of the resulting tree. Note that the different cells in a given column (i.e., different number of terminals fortified), unlike Table 7, are not using a common set of attack possibilities. Thus, at each node of the resulting tree, a separate AD problem must be solved to find the worst-case attack. Also, in each row, the size of the AD problem increases with the increase in the number of interdicted terminals, which in turn will increase the computational time. However, like in Table 7, an increasing trend is witnessed when going across and down the table, and that the combinations in the bottom right corner require much higher computational time. Also, more problems can be solved to optimally by PH compared to EX within the limited computational time. In closing, it should be evident that in all problem instances, except for three, PH is able to find higher quality solutions in much shorter computational time than EX. Finally, in an effort to further underline the computational performance of PH, we focus on the problem instances where the number of terminals fortified equals the number interdicted (i.e., Table 4). Table 9 depicts the snapshot, and it is clear that the number of possible outcomes requiring evaluation increases exponentially thereby impacting the computational time. Although the computational time for both EX and WC is rather comparable, the latter results in suboptimal solutions and hence is not of much interest. PH is able to return as good solutions as EX, except for three problem instances (Table 3), in much shorter times. It is easy to see that this is possible because while both EX and PH are benefiting from the implicit enumeration scheme to convert the tri-level problem into a set of bi-level AD problems, in PH each resulting AD is solved efficiently by the proposed decomposition heuristic. In EX, on the other hand, each bi-level AD is solved by embarking on the complete enumeration of all the possible attack possibilities. In an effort to further highlight the effectiveness of the decomposition component of PH, we have listed, under the title HD, the total computational time required for executing the decomposition heuristic, described in pseudo code of Fig. 5. It should be clear from this table that while both WC and the proposed decomposition scheme are solving bi-level AD

EX

76.87 285.53 1570.21 5746.96 15,449.98 32,271.47 47,394.38

PH

WC

Total

HD

3.75 28.96 86.00 364.12 1206.07 3361.16 13,304.43

3.09 2.92 2.12 2.32 2.01 1.96 2.02

76.86 285.50 1570.10 5746.02 15,435.20 32,117.05 46,314.38

problems to find the worst-case attack, the latter is able to return the solutions that WC returns, i.e. the highest quality solutions, in significantly lower computation time. Finally, it is important to note that the major part of the computational time under PH must be attributed to the first stage, i.e., the implicit enumeration scheme outlined in Section 4.2.1, since for big problems the implicit enumeration transforms the DAD problem into too many bi-level ADs. Although each of the produced AD problems can be solved very efficiently, by using the proposed decomposition heuristic, solving a lot of them will eventually increase the computational time. We are currently working on developing a metaheuristic-based tree search process, which enables us to transform the DAD problem to a fewer number of bi-level ADs so that the execution of PH can be expedited. 6.3. Insights In this section, we comment on the intermodal terminals that should be fortified, and also provide some insights on their utilization under different problem instances. The terminals which are fortified are exactly the same for both EX and PH, except in one problem instance. For the five-terminal problem instance, the intermodal terminal at Chicago was fortified under EX, and Jacksonville under PH. As indicated earlier, since the output of the interdiction model is used to make fortification decisions in WC, the list of terminals has less similarity to the other two approaches. It should be clear that since interdiction of terminals renders them unusable, relevant traffic would have to be re-routed using alternative terminals thereby impacting their utilization. Since each terminal in the network has a finite capacity, it may not always be possible to reassign traffic to the next closest available terminals. In other situations, an interdiction may result in shippers and/or receivers losing their connectivity to the intermodal network and in such cases demand would have to be met using the direct trucking service. For expositional reasons, and without losing generality, we analyze terminal capacity utilizations for three distinct problem instances: the Base-Case; and, fortification/interdiction of three-terminal and seven-terminal problem instances (Table 10).

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Table 10 Capacity utilization of the terminals. Number Fortified

0

Number Attacked

3

0 3 7 3

7

7

Approaches

Base-case NO WC EX PH WC EX PH

Avg. Cap. Utz. (percent)

Number of terminals (utilization)

Percent loss

<25

<50

<65

<80

<90

<100

79 46 26 61 68

0 6 14 4 3

0 4 2 4 3

0 6 0 2 1

10 0 1 4 7

7 1 0 1 3

1 1 1 3 1

0 71.7 95 37 24.7

58 68

9 10

1 0

3 2

2 2

0 2

3 2

31.4 1

As expected the average capacity utilization of active terminals is the highest under the Base-Case, which results from the connectedness of all shippers/receivers and the proper functioning of all terminals and not using any direct trucking service. With no fortification, the average utilization drops from 46 percent to 26 percent when the number of terminals interdicted increases from three to seven and the loss in connectivity increases from 71 percent to 95 percent. Therefore, with no fortification, it is almost impossible to connect shippers and receivers by the rail-truck intermodal network when seven terminals have been attacked. It was interesting that the average capacity utilization under both EX and PH was always better than the average capacity utilization under WC, and with the increase in the number of fortified and attacked terminals from three to seven, while WC shows a drop in the average capacity utilization from 61 percent to 58 percent, the capacity utilization under EX and PH was stabilized at 68 percent. In terms of the loss in connectivity, with the increase in the number of fortified/attacked terminals, all the defense strategies, i.e. WC, EX and PH, show a decreasing trend. Specifically, the drop in the loss of connectivity under EX and PH is remarkable such that almost all the demands can be shipped by the rail-truck intermodal network. This highlights the importance of having suitable defense strategies in place, which will enable the rail truck network to deliver almost all the demands from the shippers to the receivers even in the tough condition when seven terminals are attacked. Finally, it is possible to say that unlike NO and WC approaches where some terminals are either over-utilized and some others are idle, the utilization is more balanced under both EX and PH. Finally, in terms of the distribution of individual terminals based on their capacity utilization, according to Table 10, in the Base-Case, most of the terminals have been fully utilized and there is no idle terminal in the network. In the NO case, a huge disparity, in terms of the capacity utilization among the terminals is visible which results in the situation in which some terminals are underutilized and some others are utilized up to their capacity. This unfairness in the capacity utilization among the terminals is also happening, to a lesser extent, in both cases of the WC. Contrary to the trend observed in the NO and the WC cases, in all cases of EX and PH, the workloads are more fairly distributed among terminals. In closing, we note that the proposed problem could also be modeled using robust optimization, a popular methodology useful to tackle uncertainties in model parameters in the absence of any relevant historical data (Ben-Tal, El Ghaoui, & Nemirovski, 2009). We note a parallel in that robust optimization applications seek to find a solution that minimizes the cost of the worst-case realization of the input data within the prescribed uncertainty set, whereas in our problem instance the network operator aims to choose a fortification plan that will yield the best result in the event of a worst-case attack. Furthermore, we believe that the more recent concepts about recoverable robustness (Liebchen, Lübbecke, Möhring, & Stiller, 2009) and adaptable robustness

(Adjiashvili & Zenklusen, 2011) where decision variables are impacted by the realized data could be applied to the network operator’s decisions in response to the interdictor’s decisions, and not in anticipation as is currently the case. It is pertinent that the similarity lends credence to the solution obtained via Fig. 3 (and subsequently), since it implies that even if the interdictor does not have precise information about the fortified terminals, the resulting damage can never be more than the one already determined by the network operator. 7. Conclusion In this paper, we develop a tri-level mathematical model to devise strategies to protect a given number of rail intermodal terminals such that the effect of disruption is minimized. The resulting complexity, and the model characteristics, motivated the development of a decomposition-based solution heuristic. The resulting analytical framework was used to solve and analyze problem instances based on the realistic infrastructure of a class I railroad operator. In addition, the computational efficiency of the proposed heuristic was highlighted in relation to the existing techniques. Through computational experiments, we can conclude three things. First, it is important to spend the finite resources judiciously to fortify a given number of intermodal terminals, since doing so would improve the post-interdiction performance of the remaining intermodal transportation system dramatically. Second, given equal number of terminals fortified and interdicted, the solutions returned by either EX or PH approaches, when applied to situations where the number of fortified terminals is not equal to the number interdicted, are able to produce superior solutions compared to those found the WC approach. Note that this observation further underscores the superiority of both EX and PH approaches to the WC approach in an uncertain setting, i.e., where the actual number of interdictions may not be known. Third, the proposed decomposition-based heuristic returns solutions comparable to the complete enumeration technique much faster. There are a number of future research directions. First, we are currently working on developing a metaheuristic-based tree search to specify the initial points for the proposed-decomposition heuristic, since that will replace the implicit enumeration scheme of Scaparra and Church (2008a) thereby expediting the computation time for much larger problem instances. Second, the terminal capacities could be endogenous to the model such that the remaining network has enough capacity to meet the demand. Third, design of intermodal train services could be explored such that loss in the connectivity, following interdictions, would not be dramatic. Note that this area of research is quite developed within the railroad domain, especially real-time recovery of passenger trains, but there is a potential for integrating the post-disruption freight transportation element into the defender-attacker-defender framework. Finally, the current model can be augmented by adding uncertainty

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in the decision making. For instance, in the absence of perfect information, the attacker may disrupt terminals other than the most important ones. Alternatively, the defender may not know the number of attacks in advance, and may have to work with an appropriate probability distribution. Acknowledgments This research was in part supported by a grant from the National Science and Engineering Research Council of Canada (OGP 312936). The third author is a member of the Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT), and acknowledges the research infrastructure provided by the Centre. Also the comments and suggestions of four anonymous referees and the Editor helped improve the paper significantly. Appendix. The methodology to find a right value for M. We need to set a value for M so that it works in two different cases: •

When yk = 0, we have three inequalities at the same time:

ϕk ≤ βk ϕk ≥ −M ϕk ≤ 0

•

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M value must be as big such that we always get ϕk = βk . This means that −M must be always smaller or equal to βk (i.e. −M ≤ βk , ∀k ∈ K). This means that −M ≤ MIN (βk ). So, we need to select M value such that this inequality is always true. In the same way, if we set yk = 1, in order to conclude ϕk = 0, we must have −M ≤ βk which means that −M ≤ MIN (βk ). So, M value must be selected such that this inequality is always true.

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Please cite this article as: H. Sarhadi et al., An analytical approach to the protection planning of a rail intermodal terminal network, European Journal of Operational Research (2016), http://dx.doi.org/10.1016/j.ejor.2016.07.036