Introduction

Intermodal transportation, which capitalizes on the synergy between the strengths of more than one transport mode, has experienced phenomenal growth over the past two decades. 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 as well as the globalization of industry (Rondinelli and Berry 2000). Rail-truck intermodal transportation, which combines the accessibility advantage of road networks with scale economies associated with railroads, is attractive to shippers for two 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); and, second a more efficient and cost-effective overall movement ensured by combining the best attributes of the two modes (AAR 2010). The most recent statistics indicate that rail-truck intermodal traffic, measured in ton-miles, increased by 254 % between 1993 and 2007 (US DOT 2010), and became the largest revenue segment for the railroad industry (Hatch 2014).

It should be evident that intermodal transportation plays a vital role in the economic growth of North America, and hence 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 Katrina and Rita in 2005 that could cripple the nation’s oil refining capacity (Mouawad 2005), or man-made threats such as the 9/11 terrorist attacks in the United States (Scaparra and Church 2012). One of the ways to mitigate the impact of disruption is to design supply chain infrastructure, including dramatic and expensive changes to the initial system configuration, so that it operates efficiently (i.e., at low cost) both normally and when a disruption occurs (Snyder et al. 2006). Alternatively, one could attempt to improve the reliability of existing infrastructure by using fortification models to identify optimal strategies for allocating limited resources. This paper makes use of the latter technique to preserve the functionality of the rail-truck intermodal transportation system. More specifically, we consider disruption only at the intermodal terminals and formulate it as a tri-level problem, in which the defender (i.e., network owner) has a limited budget to protect or harden some of the terminals, an attacker has enough resources to interdict some of the un-protected terminals, and finally the defender (i.e., the intermodal operators) attempts to meet demand on a reduced intermodal network.

The rest of the paper is organized as follows. Literature review section reviews the relevant literature, followed by the problem description and assumptions in Problem description section. The tri-level mixed-integer programming model developed in Defender-attacker-defender framework section in applied to a realistic size problem instance in Case study section, which is solved using three distinct solution techniques and then analyzed to provide insights. Conclusions, contributions and directions of future research are outlined in Conclusion section.

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

It should be evident that fortification planning is an enormous exercise especially given the complexity of a typical intermodal infrastructure, the interdependencies among various components (Liberatore et al. 2012), and the prohibitive cost. This exciting area has started receiving increased attention from researchers, whose efforts are summarized next.

A majority of the works have approached the fortification problem, within the facility location domain, as a leader-follower game (Stackelberg 1952), in which the defender is the leader and the interdictor the follower, and modeled as bi-level programming problems (Dempe 2002). Furthermore, it is assumed that the attacker is going to make the most disruptive decision, and hence worst-case scenarios are modeled (Scaparra and Church 2008a). To ease the complexity of the bi-level programming problems, Church and Scaparra (2007) and Scaparra and Church (2008b) proposed single-level formulations, and an explicit enumeration technique for solving the problems. Uncertainty associated with the attacks has been incorporated by attaching a probability of successful attacks on facilities (Church and Scaparra 2007), and by making use of a probability distribution for estimating the number of facilities that could be attacked (Liberatore et al. 2011). While the concept of investing protection measures to reduce the recovery time of the system has been investigated in Losada et al. (2012), fortification within a system of capacitated facilities has been recently investigated by Scaparra and Church (2012).

Peer-reviewed works on the disruption of a networked system, on the other hand, have primarily focused on the analysis of risk (i.e., identifying the most critical components of a system) through the development of interdiction models. The effect of interdiction on the maximum flow through a network is studied by Wood (1993), while Lim and Smith (2007) made use of a variant of multicommodity shortest path problem to investigate the impact on revenue from arc interdictions. The concept of fortification against worst-case losses for network models has been discussed in Brown et al. (2005, 2006), wherein a tri-level optimization model to represent fortification, interdiction, and network flow decisions (i.e., defender-attacker-defender) was developed. Finally, a number of applications making use of this framework appeared in the literature such as power grid (Salmeron et al. 2004; Alguacil et al. 2014), and water supply (Qiao et al. 2007).

Rail-truck intermodal transportation systems

Although rail-truck intermodal transportation has been an active research area over the past two decades (Macharis and Bontekoning 2004), the discussion about disruption is still in its infancy. To the best of our knowledge, there is no peer-reviewed work that deals with intentional attacks within the rail-truck intermodal transportation setting. We invite the reader to refer to Bontekoning et al. (2004) for an excellent discussion on intermodal transportation, and SteadieSeifi et al. (2014) for the state of the art review.

To sum, the posed problem makes use of a defender-attacker-defender framework to investigate the optimal protection strategy for rail-truck intermodal terminals, and thus draws from the works of Scaparra and Church (2012) and Brown et al. (2005, 2006).

Problem description

In this section, we provide a formal statement of the problem, emphasize its complexity, and then state the modeling assumptions.

The protection planning of rail-truck intermodal transportation problem entails interaction amongst three players, i.e., network owner, the interdictor, and the network operator –wherein each is in a different hierarchy (Fig. 1). At the highest level, the network owner tries to minimize the cost of using the system 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 network operator’s problem. Finally, following interdiction, the network operator has to meet customer demand using the available intermodal terminals and (reduced) train services.

Fig. 1
figure 1

Hierarchical structure for the protection planning

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For expositional reasons, we note that a rail-truck intermodal transportation system comprises three processes: (i) inbound drayage, (ii) rail-haul, and (iii) outbound drayage. Thus, the network operator endeavours 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 trains. In an effort to ensure feasible solutions (i.e., demand is satisfied), direct trucking is permitted between each shipper-receiver. Note that the central question about allocating limited resources such that post-disruption functionality of the intermodal infrastructure is preserved is fairly complex, in large part due to the interaction amongst the three players.

In an effort to explain the complexity of the problem, we reproduce a portion of the intermodal service chain network introduced in Verma et al. (2012), which is represented via a geographical information system (GIS) model using ArcView (ESRI 2008). Figure 2 depicts the 18 intermodal terminals, which are the access points for 399 demand pairs (i.e., shipper-receiver). 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 % faster. Finally, the network owner has resources to fortify a limited number of terminals, the interdictor has resources of destroy/ disrupt a limited number of (un-protected) terminals, and the network operator has to meet demand using the reduced intermodal network.

Fig. 2
figure 2

Intermodal rail services (source: Verma et al. 2012)

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We now turn to our modeling assumptions: first, a protected terminal cannot be interdicted; second, each intermodal terminal has finite traffic handling capacity; third, delivery dates are specified when placing the order, and a penalty 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.

Defender-attacker-defender framework

In this section, we provide the a tri-level mathematical formulation for the managerial problem and then discuss the estimation of the basic parameters of the model in Estimation of parameters section.

Mathematical model

Our notation and the model (P) is provided below.

Sets

I :

Set of shippers, indexed by i

J :

Set of receivers, indexed by j

M ij :

Set of intermodal paths between shipper i and receiver j, indexed by m

K :

Set of intermodal terminals in the network, indexed by k

M k ij :

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

R v :

Set of service legs for train service v, indexed by r

S r,v :

Set of intermodal paths using service leg r of train service v

Variables

x m ij :

Number of containers using intermodal path m between shipper i and receiver j

xt ij :

Number of containers using direct trucking service between shipper i and receiver j

n v :

Number of trains of type v

$$ {z}_k = \left\{\begin{array}{c}\hfill 1\ if\ terminal\ k\ is\ protected\hfill \\ {}\hfill 0\ otherwise\hfill \end{array}\right. $$
$$ {y}_k = \left\{\begin{array}{c}\hfill 1\ if\ terminal\ k\ is\ interdicted\hfill \\ {}\hfill 0\ otherwise\hfill \end{array}\right. $$

Parameters

w :

Maximum number of terminals that the network owner can protect

q :

Maximum number of terminals that the interdictor can disrupt

c m ij :

Cost of transporting one container from shipper i to receiver j on intermodal path m

ct ij :

Cost of sending a container using truck on the shortest path from shipper i to receiver j

t m ij :

Expected travel time from shipper i to receiver j on intermodal path m

t ij :

Delivery time using truck on the shortest path from shipper i to receiver j

l ij :

Delivery due date promised by shipper i to receiver j

d ij :

Number of containers demanded by receiver j from shipper i

pc ij :

Penalty cost per container per unit time to be paid by shipper i to receiver j

b v :

Capacity of train service v

fc v :

Fixed cost of operating train service v

u k :

Capacity of intermodal terminal k

(P)

$$ Mi{n}_zC(z) $$
(1)

subject to:

$$ {\displaystyle {\sum}_{k\in K}{Z}_k}\le w $$
(2)
$$ {z}_k\in \left\{0,1\right\}\kern6.25em \forall k\in K $$
(3)

where,

$$ C(z)= Ma{x}_yC(y) $$
(4)

subject to:

$$ {\displaystyle {\sum}_{k\in K}{y}_k\le q} $$
(5)
$$ {y}_k\in \left\{0,1\right\}\kern6em \forall k\in K $$
(6)
$$ {y}_k+{z}_k\le 1\kern5.75em \forall k\in K $$
(7)

where,

$$ C(y)=Min\left[{\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{j\in J}{\displaystyle {\sum}_{m\in Mij}{c}_{ij}^m{x}_{ij}^m+{\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{j\in J}c{t}_{ij}x{t}_{ij}+{\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{j\in J}{\displaystyle {\sum}_{\begin{array}{l}m\in {M}_{ij}\\ {}{t}_{ij}^m>{l}_{ij}\end{array}}\left({t}_{ij}^m-{l}_{ij}\right)p{c}_{ij}{x}_{ij}^m+{\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{\begin{array}{l}j\in J\\ {}{t}_{ij}>{l}_{ij}\end{array}}\left({t}_{ij}-{l}_{ij}\right)p{c}_{ij}x{t}_{ij}+{\displaystyle {\sum}_{v\in V}f{c}^v{n}^v}}}}}}}}}}}\right] $$
(8)

subject to:

$$ {\displaystyle {\sum}_{m\in {M}_{ij}}{x}_{ij}^m+x{t}_{ij}={d}_{ij}}\kern6em \forall i\in I,\ \forall j\in J $$
(9)
$$ {\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{j\in J}{\displaystyle {\sum}_{m\in {M}_{ij}\cap {M}_{ij}^k}{x}_{ij}^m\le {u}_k}}}\kern2.25em \forall k\in K $$
(10)
$$ {\displaystyle {\sum}_{i\in I}{\displaystyle {\sum}_{j\in J}{\displaystyle {\sum}_{m\in {M}_{ij}\cap {S}^{r,v}}{x}_{ij}^m\le {b}^v{n}^v}}\kern1.25em }\ \forall v\in V,r\in {R}^v $$
(11)
$$ {x}_{ij}^m\le \varnothing \left(1-{y}_k\right)\kern8.25em \forall i\in I,\forall j\in J,\forall k\in K,\forall m\in {M}_{ij}\cap {M}_{ij}^k $$
(12)
$$ {n}^v\ge 0,\ integer\kern8.75em \forall v\in V $$
(13)
$$ {x}_{ij}^m\ge 0,\ integer\kern8.5em \forall i\in I,\forall j\in J,\forall m\in {M}_{ij} $$
(14)
$$ x{t}_{ij}\ge 0,\ integer\kern8.5em \forall i\in I,\forall j\in J $$
(15)

∅ is a larger positive integer

(P) depict the tri-level optimization model that could be used to make protection planning decision. The outer level problem belongs to the network owner 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 owner and the interdictor by prohibiting the disruption of fortified terminals. Finally, the inner level problem belongs to the network 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. Constraints sets (11) determine the number of intermodal trains of a specific type needed in the network. Constraints sets (12) link the interdictor’s problem with the network operators, and states that the interdicted terminals cannot be a part of different intermodal paths to meet demand. Note that ∅ is a large positive integer. Finally, the sign restrictions are imposed through constraints sets (13) to (15).

Estimation of parameters

Cost

In the United States, trucks can travel at a maximum speed of 50 miles/h, but due to lights and traffic an average speed of 40 miles/h is assumed (Verma and Verter 2010). Normally drayage is charged in terms of the amount of time the crew (driver-truck) is engaged, and an estimate of $300/hour including the estimated hourly fuel cost is used. A penalty cost of $40 per hour per container was used. As indicated there are two types of intermodal train services viz. 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/h for regular, and 36.8 miles/h for express service. Consistent with the published works, we estimated a rate of $0.875/mile for regular and $1.164/mile for express service. The hourly fixed cost of running a regular intermodal train is $500 per hour, which takes into consideration the hourly rates for a driver, an engineer, a brakeman, and an engine, which are $100, $100, $100, and $200, respectively. The express service is 50 % more expensive at $750 per hour (Verma 2012).

Due dates

Three different due dates have been defined: long, regular, and short. 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 speed of trucks. Finally, constants of 10, 15 and 20 were added to the travel time to, respectively, get the short, regular and long delivery due dates for each shipper-receiver pair.

Demand levels and terminal capacity

The inner problem belonging to the network operator was solved in CPLEX 12.1.0 (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 estimated that the terminal utilization was 80 %, and hence the terminal capacity is 1.25 times (i.e., 1 divided by 0.8) the traffic volume through each terminal, and the demand level was deemed medium. Finally, we assumed that high demand level would account for 95 % of terminal capacity and hence multiplied the medium demand by 1.1875 (i.e., 95 over 80), whereas low demand would result from 65 % terminal capacity.

Case study

(P) was applied to solve the realistic size case study outlined in Problem description section, and depicted in Fig. 2. All resulting mathematical models were solved using CPLEX 12.1.0 (IBM 2014). For expositional reasons, we abbreviate the names of the terminals as indicated in Table 1. Finally, we assume that the network owner has enough resources to protect two terminals, and that the interdictor has capability to disrupt two terminals. Though evident, it is assumed that the fortification is permanent and therefore the relevant terminals cannot be interdicted subsequently. To recall, we are attempting to find the best protection strategy for the network owner.

Table 1 Terminal legends
Full size table

Solution algorithms

In this section, we will first comment on the computational burden if all scenarios need to be generated, and then outline two efficient algorithms that achieve the same result. For expositional reasons, we will make use of the setting wherein demand is low and delivery due date is short to demonstrate the algorithm; we will compare the solutions resulting from the other eight scenarios in Numerical analysis section.

(P) can be solved using three techniques: complete enumeration; implicit enumeration proposed in Scaparra and Church (2008a); and, a traffic-based heuristic.

Complete enumeration

proceeds by determining an exhaustive combination for defending and interdicting two terminals, which for our problem instance amounts into \( \left(\begin{array}{c}\hfill 18\hfill \\ {}\hfill 2\hfill \end{array}\right)*\left(\begin{array}{c}\hfill 16\hfill \\ {}\hfill 2\hfill \end{array}\right)=18360 \) possible defense and attack strategies. For each defense and attack strategy, the network operator’s problem is solved assuming that the terminals attacked under this strategy have been disrupted. For each given defense strategy, the effect of all the ensuing attack strategies will be compared. The worst-case disruption following each defense strategy yields the total cost associated with the adoption of that defense strategy. The defense strategy with the lowest associated cost will be selected as the best defense strategy. Applying this procedure to the current case study will result in protection of intermodal terminals in Philadelphia and in Atlanta. The CPU time for this problem setting was 902.14 s, and it ranged from 241.1 to 2063.18 s for the other eight scenarios.

It is easy to see that the complete enumeration technique will become rather cumbersome if more than two terminals have to be considered for fortification and interdiction. For example, the number of strategies requiring evaluation for “three terminals” example would be 371,280. Thus, there is a need for a more efficient solution technique.

Implicit enumeration

Under this scheme, first, the list of the worst-case disruptions has been provided by examining all of the attack strategies. An exhaustive combination for interdicting two terminals for our problem instance will be translated into \( \left(\begin{array}{c}\hfill 18\hfill \\ {}\hfill 2\hfill \end{array}\right)=153 \) possible attacks. Then, for each attack strategy, the network operator’s problem is solved assuming that the terminals listed under this strategy have been out of service. The resulting solution gives us the total cost associated with each attack strategy, and the worst-case disruption would result from the strategy with the highest cost. The corresponding strategy called for the interdiction of intermodal terminals in Philadelphia and in Atlanta. This information will be passed to the implicit enumeration scheme proposed in Scaparra and Church (2008a), which was coded in C# and the entire search took 76.5 s. This solution algorithm is using a considerably reduced search space collectively containing only the defense strategies that will prevent the worst-case disruption. We next provide detail on how this enumeration scheme works.

As indicated above, the worst-case disruption entails interdiction of terminals in Philadelphia (Phi) and Atlanta (Atl). Hence, fortifying either or both these terminals would preclude the worst-case disruption. The implicit enumeration scheme starts at the root node, i.e., 1, by finding the worst-case disruption without fortification (i.e., Phi and Atl). Set O, at each node, lists the terminals at least one of which must be protected to prevent the worst-case. For instance, at node 1, terminal Phi could either be fortified or not.

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 O 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 O must be updated by solving the interdiction problem with latest information (i.e., Atl is fortified). Thus, the updated set O 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 O contains Atl and Ric, both of which would be interdicted thereby resulting in a cost of around $12.2mn (i.e., dark shade).

If Phi were fortified, the interdiction problem is solved thereby resulting in terminals Ind and Atl in the updated set O. 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 $12.1mn. But if Ind is not fortified, then the updated set O 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 most disruptive terminals, and the associated cost is around $11.9mn.

The CPU time for the remaining eight scenarios ranged from 16.3 to 336.24 s, which is quite good. We note that, given the definition of criticality for our problem instance, it is possible to arrive at the same solution more quickly by combining information about the traffic flow through each terminal and (an adapted version of) the implicit enumeration of Scaparra and Church (2008a).

Proposed traffic-based heuristic

The proposed heuristic works in two steps. First, the capacitated multi-commodity flow problem for the network operator is solved (i.e., the inner level problem in (P)). The resulting solution is decoded and the traffic volume through each terminal is estimated thereby generating a list, in descending order, of throughput terminal traffic. Since there are enough resources to protect two terminals, fortifying the top two candidates on the list would make the worst-case disruption impossible.

In the second step, just like Scaparra and Church (2008a), the identity of the two terminals was supplied as input at the root node 1 in Fig. 3. If Phi were fortified, then we would update set O by including the third terminal from the list generated in step one. For instance, at node 4 in Fig. 3, Ind would be selected from the list without solving the interdiction problem as in Scaparra and Church (2008a). If Ind is fortified then the protection resources are exhausted, and we select the fourth terminal on the list generated in step one, i.e., Cha, which would be interdicted along with Atl. The process continues as outlined in Fig. 3, except that we have done away with the need to solve the interdictor’s problem at each node and simply consult the list generated in step one. This luxury has a positive bearing on the computational time, which for the given problem instance was 37.86 s compared to 76.5 s using the enumeration technique of Scaparra and Church (2008a).

Fig. 3
figure 3

Tree search (adapted from: Scaparra and Church 2008a)

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It is important to reiterate that the proposed traffic-based heuristic worked well for the small problem instances, but may not for much larger instances. We are currently working on a decomposition based solution technique for much larger situations, i.e., enough resources to fortify and interdict more than two terminals.

To conclude, the proposed heuristic is quicker than the other two solution techniques. Table 2 reports the relevant figures, where BC refers to the Base-Case, CE to complete enumeration, IE to the implicit enumeration scheme of Scaparra and Church (2008a), and Heuristic to the proposed traffic-based heuristic.

Table 2 Snapshot of computation time
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Numerical analysis

In this subsection, we will first provide a snapshot of the solution for the nine scenarios developed using the due date and demand level combinations as outlined in Estimation of parameters section, and then comment on terminal utilization and network connectivity.

Analysis

Following the enumeration, the resulting costs for all the leaf nodes (i.e., dark shade) were compared to conclude that the optimum strategy is to fortify Phi and Atl, which means that the interdictor would disrupt Chi and Ind and the system cost will be $11,944,726. Table 3 summarizes the results for the nine scenarios (and twenty-seven problem instances), with the first block depicting the highlights with short due date and low demand level.

Table 3 Snapshot of the nine scenarios
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It is clear from all the nine scenarios that fortification improves the performance of the transportation system, thereby resulting in lower costs vis-à-vis no protection. In fact, for the short due date setting, the performance improvement ranges from 7 % for low to 9.2 % for high demand levels. In other words, fortification has reduced the adverse effect of interdiction in each setting. It was noticed that the improvement was higher for scenarios where due dates were short and demand high versus long due date and low demand. Finally, express train service was used only with short due dates because of the pressure to deliver before the specified time (and the penalty cost).

Capacity and network connectivity

It should be clear that since interdiction of terminals render them unusable, relevant traffic would have to 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 terminal. 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 direct trucking service. In this subsection, we analyze terminal capacity utilization and connectivity (or lack of) for the twenty-seven problem instances under the nine scenarios (Table 4).

Table 4 Capacity utilization & connectivity
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Within each scenario, the Base Case has the highest average capacity utilization resulting from the connectedness of all shippers/ receivers and the proper working of all terminals, which also implies no direct truck service. Within each scenario interdiction without fortification (i.e., W/out Fort) has the lowest capacity utilization since 23 % of the customers have lost connectivity with the intermodal network, and have to make use of the direct trucking service to move shipments. It was noticed that interdiction with fortification (i.e., With Fort) yielded better capacity utilization than the without settings because the worst-case disruptions have been rendered impossible, and relatively fewer customers lose connectivity to the intermodal network. In eight of the nine scenarios, 19 % of traffic loses connectivity stemming from the fortification of Phi and Atl, and the interdiction of Chi and Ind. Finally, it was observed that for a given due date level, average capacity utilization was linearly related to the demand level.

Conclusion

In this paper, we make use of the defender-attacker-defender framework to devise strategies to protect a given number of rail-truck intermodal terminals such that the effect of disruption is minimized. A realistic size case study, based on a Class I railroad operator in the United States, was modeled as a tri-level optimization problem and solved using three distinct solution techniques, viz., complete enumeration, implicit enumeration of Scaparra and Church (2008a), and a traffic-based heuristic.

In an effort to demonstrate the significance of fortification, the proposed analytical framework was used to solve twenty-seven variations of the case study, nine each for the base-case, interdiction without fortification, and interdiction with fortification. In addition, it was demonstrated that fortification has a positive bearing on the connectivity of the intermodal network –since fewer customers are forced to use the direct-trucking option thereby resulting in more cost-efficient solution for the network and higher capacity utilization of the available terminals. Finally, it was noticed that the mix of intermodal trains used to meet demand depended on the delivery due dates, with the express trains being mostly used for short due dates.

There are a number of future research directions. First, we are currently working on devising solution techniques capable of solving larger problem instances efficiently. Second, the tri-level model could be extended to include service-design and terminal capacity decisions. Third, the assumption about perfect information for both the network owner and the interdictor could be relaxed.