Notional Examples

5.6.1 Two Identical Layered Networks

Consider Network 1, depicted in Figure 5.2. If no arcs are interdicted, the maximum flow of this network is 26. However, suppose there is an attacker who

wishes to minimize this maximum flow. For simplicity, assume each arc can be destroyed with 1 unit of resource, with a resource availability of 2. Using the model

in (5.1), the optimal arcs to interdict are found to be (4, 6) and (3, 6) which reduces the maximum flow to 5. Since Network 2 is identical to Network 1, if they are solved

In other words, if there is no interdependence considered, each network could

be solved separately. Of course, this would lead to a combined maximum flow (after interdiction) of 10 units (with 4 units of interdiction resource).

Figure 5.2: Two Identical Layered Networks

Now suppose that the two networks are interdependent. Specifically, assume

edges (5, 6) of both networks share a common corridor. That is, although the arcs could be interdicted separately, they could also be interdicted together with a single

resource cost of 1 unit. Specifically, this means δijkw = δ5611 = δ5621 = 1.

To account for this interdependence, the formulation in (5.7) is used. The op- timal arcs to be interdicted are (4, 6) from Network 1; (3, 6) and (4, 6) from Network

2; and interdependent arcs (5, 6). Again, this consumes all 4 units of interdiction resources, but drops the combined maximum flow to 7 (all from Network 1, as no

flow is possible in Network 2). Thus, for the same amount of resources, the flow can be further reduced accounting for interdependencies.

5.6.2 Multiple Different Layered Networks

Consider the networks depicted in Figure 5.3. If no arcs are interdicted, the

maximum flow of Network 1 is 16, Network 2 is 26, Network 3 is 33, and Network 4 is 24. Therefore, the combined maximum flow across all networks is 99 (assuming

commensurate units). From the max-flow min-cut theorem, it can be shown that the cost to cut all of these networks is also 99.

If interdependencies are included, then the cost may be reduced. Assume the

colored edges in Figure 5.3 represent common corridors. Therefore, there are three potential interdependencies. The first (represented by the blue edges) are edges

(3, 2) in Network 2 and (1, 2) in Network 3. The second (represented by orange edges) are edges (2, 3) in Network 1, (3, 7) in Network 3, (4, 3) in Network 4. The

third (represented by green edges) is (3, 4) in Network 2, and (1, 3) in Network 4. Further, assume the costs associated with cutting these interdependencies is 5, 3,

and 5; respectively.

Incorporating these interdependencies, formulation (5.5) can be used to deter- mine the minimum cost cut, which is 91. However, suppose there is an attacker

who wishes to minimize this maximum flow, but did not have enough resources (91) to completely stop the flow in all networks. This leads to the network interdiction

problem. To facilitate a comparison, assume the cost to interdict each arc is the same as the upper capacity of each arc.

As a network interdiction problem with multiple interdependent layers, For-

mulation (5.7) can be used. If a resource constraint of 91 is used, this formulation confirms that all flow through the networks can be cut. However, when a resource

level less than 91 is used, it is possible to determine which arcs should be interdicted to maximally disrupt the network flows. The graph in Figure 5.4 shows the residual

Figure 5.3: Different Layered Networks

Figure 5.4: Residual Flow versus Interdiction Resources

For example, at a interdiction resource level of 12 units, the optimal solution is to select the first and third interdependent edges (represented by the blue and

green edges). This reduces the total flow through all 4 networks from 99 to 81. However, if the interdiction resource is increased by one additional unit, then the

optimal solution changes to the first interdependent set and edge (1, 4) in the fourth network. This reduces the maximum flow (across all 4 networks) further to 79 units.

These examples demonstrate how network interdiction against layered net-

works provides alternatives and more information than traditional cut-sets and are most beneficial when an interdictor’s/attacker’s resources are limited.

This problem was also solved via the Benders’ partitioning method. However,

due to the small nature of the example, there was no discernable difference in com- putation times between the different solution methods. (Both methods result in a

GAMS reported 0.0 solution time.)

5.6.3 Nodal Interdiction Example

In the build-up to Operation Iraqi Freedom, CENTAF was concerned calcu- lating the maximum flow from its pre-positioning warehouses to locations within

Kuwait. There are several storage locations shown in Figure 5.5 and several meth- ods of transportation available to move material from there storage locations to

Kuwait.

Figure 5.5: War Reserve Material Prepositioned Locations

As shown in Figure 5.6, this scenario could be modeled and solved as a sin-

gle network. However, for demonstration purposes, these networks were modeled separately (as an airlift network, a sealift network, and a ground/road based net-

work). To determine vulnerabilities associated with these networks, it is assumed that a terrorist organization has the capabilities to stop flow from any one location,

Udeid). Disruption of flow in one location could disrupt flow in all three transporta-

tion networks as the same nodes appear in multiple networks.

Figure 5.6: Network Representation

The three network flow models were formulated as discusses in Section 5.5. Of course, the nodes appearing in multiple networks create an independency, and an

additional effect was modeled which reduced sealift to Al Udeid to zero if the Straight of Hormuz was disrupted. The optimal solution to disrupt the flow of material from

storage locations to Kuwait is stop flow from Al Udeid. The second best solution is to stop sealift to Al Udeid; however, disruption of flow from Al Udeid also stops

flow from local warehouses in Al Udeid (which would not be affected in a Straights of Hormuz disruption).