Options and Challenges of a Resilience-Based, Network-Focused Port Security Grant Program

Prescribing Post-Attack Performance

We must first establish baseline resilience values for supply chain networks, since we want to measure changes in resilience based on hypothetical PSGP resilience investments (allocations). We document desired supply chain network performance, whether it be the aggregate throughput of petrochemical product of each node, or some derivative metric such as aggregate of node profit from moving product.³ We can treat this desired supply chain performance as a consequence so as to maintain conceptual ties to the legacy PSGP risk approach. If the network fails, there will be a consequence, or lack of performance, so we show it as Con^l(max), the maximum consequence of a lth supply chain network. Alderson et al. (2011, op. cit.) leverage a similar principle: they aggregate metrics such as commuter time for all bridge and road pathways into a transportation network metric, and minimize that aggregated metric as their objective function.

Network Expected Consequence

We then leverage the concept of expected network performance or output, and we compare that metric to desired performance Con^l(max), to assess network resilience. The expected network performance can be rewritten as expected consequence, or lost throughput, after a network disruption. This expected consequence would be the maximum consequence (loss of all throughput) modified by the probability that throughput would be lost. This probability would be the likelihood of node failure to restore productivity. This departs conceptually from Alderson et al.’s model by introducing the probability of a failure, which differs from the binary fail/do not fail approach.

To explain further, each node has an organic probability of failing to restore productivity after an attack on the maritime infrastructure supplier nodes. This probability is a function of excess capacity raw product stored on site. We model this organic “failure susceptibility” as an exponential function of allocation to ensure excess capacity at a node:

In Equation 5, represents existing allocation to ensure excess capacity, as compared to is elimination cost, the maximum amount of allocation necessary to ensure that node stores enough excess capacity to restore desired productivity after an attack, without help from other nodes.⁴

Each node also has an inherited susceptibility of failure to restore productivity. These susceptibilities are not shown explicitly in the proposed approach. Instead, they are implicitly reflected by upstream node degree g_i. Intuitively it makes sense to use degree, or the number of links to other nodes, as a proxy for node interdependency and thus as a proxy for inherited failure susceptibility. The more highly connected a node is, the more interdependent it is with its customers. This means a higher potential for cascading failure because many customers would potentially suffer from one node’s failure. Thus, those downstream customers have higher inherited failure susceptibility than they would have if the upstream node had less connectivity, or lower degree.

We show how degree is generally applied to lth network expected consequence Con^l(exp) where represents node maximum consequence:

This equation in expanded form accounts for nuances of supplier node failure susceptibility. We model supplier node failure susceptibility as a function of: (1) organic failure susceptibility or lack of raw product on site, (2) the probability of failure to rebuild after an attack, and (3) the probability that maritime infrastructure will fail to leverage redundant suppliers such as other ports. We show this:

The expansion of Equation 6 to account for Equation 7 will be shown in Equation 9.

Node failure susceptibility and maximum consequence data can be entered into MBRA. Since we have related network performance to consequence, more specifically expected consequence which reflects probabilistic modification of maximum consequence, we have now shown convergence between (1) the concepts of risk from network science and PRA, and (2) the concept of performance from OR/engineering. We can now convert expected consequence to conditional risk, so that the proposed approach will include a risk-based metric, not just a consequence-based metric.

Network Conditional Risk

We use the concept of “conditional risk” and exclude attacker intent probabilities, so as to leave room for adding in intelligent adversary considerations later. For now we show what a conditional network risk equation would look like:

This shows that conditional risk is expected consequence if there is a successful attack, but conditional upon two probabilities. The first is the probability that the attacker has the capability Cap_S to attack a supplier node. The second is the vulnerability V_s, or probability that a supplier node will fail upon attack and thus precipitate a cascading failure throughout the network. Equation 8 is closer to risk in the traditional sense of Equation 1. Capability and supplier node vulnerability data can be entered into MBRA. Equation 8 can now be expanded, leveraging Equations 6 and 7, to show organic and inherited failure susceptibilities of all nodes:

Port Conditional Risk

We would first disburse PSGP resilience funding amongst ports to be consistent with the current approach. Thus we have to first aggregate conditional risk to all supply chains in each port, in order to create a port risk value. An equation for port risk is:

In this approach, port risk only reflects the specific consequence of economic loss caused by an attack on a supply chain’s maritime infrastructure. This is slightly different than the current PSGP approach of compiling population, economic effects, infrastructure risk, and military presence as four separate consequence metrics.

We want to write Equation 10 to reflect the nuances of Equation 9. First, we isolate terms that contain rebuilding funding This actually simplifies the substitution of Equation 9 into Equation 10.

The expanded versions of terms A, B, C, D and E are available from the author. Notice that the superscripts l and m have been added to the allocation and elimination terms; this is to distinguish allocation to the lth network from the allocation to the mth network.

Port Organic Resilience

We must convert our port conditional risk metrics to resilience metrics if we are going to allocate PSGP resilience funding to different ports. We do this by simply subtracting conditional port risk from maximum port performance. Maximum port performance Con^port(max) is simply the sum of desired maximum performance of each supply chain network that originates at a maritime infrastructure in that port. For our example:

Then, port organic resilience can be shown as a percentage, which represents the potential to restore performance after an attack. Leveraging Equation 12, we now have:

We now have an equation to support the first proposed definition of quantifiable resilience. We then sum organic resilience metrics for all ports in competition for funding, and show port resilience ratios The ratio is the percentage of aggregate port resilience that each port “owns”:

We then allocate PSGP resilience funding in a simple fashion, inversely proportional to port organic resilience ratios. The higher the ratio, the less PSGP funding that port receives. For example, if there is only $2 million available, one port’s resilience ratio is 55% of aggregate port resilience, and the other port’s ratio is 45%, then the former receives $900,000, and the latter receives $1,100,000. This approach avoids the need for a formal optimization of limited PSGP funding.⁵

Intelligent Adversary Considerations

Before we allocate funds to competing ports based on port organic resilience, we can consider how existing allocations influence the ports’ relative attractiveness to adversaries. At this point, we can incorporate intelligent adversary considerations, a principle espoused by the OR community.

We have only used conditional risk thus far. To incorporate intelligent adversary concerns, we now leverage the concept of intent ratios to represent unconditional port risk. This also gives us the opportunity to represent port organic resilience as a function of unconditional risk. A decision maker may prefer to characterize a port’s resilience based on not only its organic potential to rebuild its infrastructure after an attack, but also based on the extent to which such potential might deter a prospective attacker and change their intent.

We convert port conditional risk equations to expected utility equations to facilitate the calculation of intent ratios. Utility represents the gain an attacker could achieve from executing a specific attack option, and expected utility is that potential gain modified by the probability of successful attack. For simplicity in this proposal, we simply take Equation 11 and claim that attacker expected utility is the same as port conditional risk:

We do the same for other ports in competition for PSGP resilience funding. Then, we create a ratio of expected utility from attacking one port, to the aggregate expected utility of all attacker options. The ratio of expected utilities is a proxy for attacker intent (Taquechel and Lewis 2012). For port 1, the resulting intent ratio is shown:

We repeat this for other ports. We then combine each intent ratio with the expected utility for the respective attacker option, and claim that represents unconditional port risk. For port 1, unconditional risk is shown:

We now have the option to characterize port organic resilience as a function of unconditional risk:

We can thus allocate PSGP resilience funding amongst ports inversely proportional to organic resilience based on unconditional port risk, as an alternative to using Equations 13 and 14.

Supply Chain Network Organic Resilience

Once we have made a decision on how to allocate PSGP funding amongst ports, then each port must allocate their portion amongst their constituent supply chains. In practice, this would be a competition between maritime infrastructures in each port. This would be similar to the current PSGP approach, but the data supporting that competition would be resilience-based in the proposed approach.

We evaluate each supply chain network conditional risk and resulting organic resilience (before PSGP allocations are simulated) as baseline metrics. We leverage Equation 8 to do this and define the organic resilience of our lth supply chain network:

We have now created an equation to support the second definition of quantifiable resilience proposed earlier. A large value of this equation will mean the lth network is highly resilient, or there is a large difference between desired performance and conditional risk. In other words, there is a low potential for expected consequence if the network is attacked, meaning a high potential to restore network performance.

We can also consider individual supply chain network attractiveness to a potential attacker, as we did with port attractiveness. This allows us to write Equation 19 as a function of supply chain unconditional risk instead of conditional risk. We apply the same intelligent adversary technique. Attacker expected utility from attacking an lth supply chain network is the same as conditional risk, so Equation 8 is modified:

We can repeat this for the other supply chains in the port under consideration. Then, we develop intent ratios for each supply chain, which are ratios of expected utility from attacking one supply chain to aggregate expected utility from all attacker options:

We finally apply this intent value to supply chain conditional risk to get unconditional risk, and can express network organic resilience as a function of unconditional risk if desired. We developed “port resilience ratios” per Equation 14 to govern how funding is disbursed amongst ports, but in this case MBRA can simulate a more formal optimization of funding amongst supply chain MCIKR.

Optimizing Allocation of PSGP Resilience Funding to Individual MCIKR

MBRA leverages an iterative algorithm, similar to simulated annealing randomization,⁶ to optimize funding allocation and achieve an objective function. This iterative algorithm leverages the concept of emergence.

Emergence means that the model moves user-specified proportions of available budget between network nodes at random, to try and achieve the objective function. The objective function in the present approach is to maximize resilience. The model continues this until moving money from any nodes to any other nodes in the network will maximize resilience no further. Thus, achievement of the objective function emerges, representing an equilibrium of optimal allocations.

The unconstrained objective function for this proposal would be to maximize port resilience, leveraging Equation 13. However, a constrained objective function reflects a limited PSGP resilience budget, which is more realistic.

The constrained objective function for maximizing port resilience can be shown:

This objective function means that MBRA allocates PSGP rebuilding funding back and forth between the maritime infrastructures of the two supply chains in our notional port. It does this until port resilience is maximized within the constraint of the resilience budget Budget^reb, specifically earmarked for rebuilding the maritime infrastructure after an attack. That means that when the simulation reaches equilibrium, there is an optimal funding amount allocated to the maritime infrastructure in the lth supply chain, and an optimal funding amount allocated to the maritime infrastructure in the mth supply chain. This results in enhanced port resilience, which we explain later.

The organic resilience term in Equation 22 can leverage either conditional or unconditional network risk. A decision-maker may need to decide whether we want to optimize PSGP allocation to different maritime infrastructure based on (1) that port’s relative attractiveness to other ports (resilience based on unconditional risk), or (2) excluding that factor thereby leveraging conditional risk. One consideration that may influence this decision is that we arguably invest in resilience to influence the attacker’s expected utility and intent, thus deterring an attacker. Therefore intent is the dependent variable and should be excluded from the unconstrained objective function. However, if we use unconditional risk as a basis for the resilience term in Equation 22, we are simultaneously manipulating the dependent variable and the independent variables, thereby biasing results. In that case we should use conditional risk as the basis for the resilience term.

This optimization approach is similar to what network interdiction or DAD models do: optimize performance subject to constraints. However, this approach optimizes resilience, which is treated as a function of probabilistic risk, whereas DAD models generally optimize just consequence, such as flow.

Resulting Changes in Resilience

MBRA displays the resulting network resilience once the optimization concludes. This is enhanced network resilience, shown:

is network conditional risk after an optimal allocation and can be shown in expanded form:

represents the optimal resilience allocation to the maritime supplier node (S) in the lth network after the MBRA simulation reaches equilibrium. We then repeat Equation 24 for other networks in our port. We compare each network’s enhanced resilience to its organic resilience to see the impact of optimal PSGP resilience allocations. Also, we determine port conditional risk after optimal allocation, shown:

We finally calculate enhanced port resilience given optimal allocation:

Comparing Results

We now have data to compare port and network resilience values both before and after optimal allocations. These may be valuable metrics to justify where PSGP funds are allocated.

Suboptimal Allocation of PSGP Funds, Return on Investment

Decision makers may alternatively want to know how suboptimal allocation of PSGP funding would influence network and port resilience. MBRA includes a “manual allocation” option for use after the optimization simulation concludes. Suboptimal allocation may yield different changes in resilience. Also, MBRA can show return on investment at individual nodes: the extent to which allocation at each node influences overall network resilience, divided by that allocation.

Intelligent Adversary Considerations After Resilience Allocation

We now consider intelligent adversary factors for our enhanced resilience calculations, just as we considered them for organic resilience. We determine post-resilience allocation attacker intent, to attack a port or more specifically to attack networks within a port. Then, by comparing (1) the intent to attack a specific network or port before allocation, to (2) the intent after allocation, we can quantify deterrence. We leverage the same techniques used for organic resilience intelligent adversary calculations.

Quantifying Deterrence Effects of Resilience Allocations

Taquechel and Lewis (2012, op. cit.) proposed a method to quantify deterrence effects of infrastructure security allocations. They focused on deterrent effects of allocations to decrease target vulnerability. These allocations reduced probability of attack success if an attack was attempted. Reduced probability of attack success would quantifiably deter an attacker or reflect the deterrence effectiveness of a particular allocation. Taquechel and Lewis proposed the following equation:

This equation reflects the change from (1) attacker intent before deterrence allocations I^pre to (2) attacker intent after deterrence allocations I^post. The potential resilience allocations could change attacker expected utility for the current approach. If the consequences change because we have invested to rebuild infrastructures and restore productivity after an attack, this may lower an attacker’s expected utility, measurably change intent, and thus measurably deter.

We thus substitute the intent ratios from Equations 16 and 21 into Equation 27 to quantify the deterrence effects of proposed PSGP resilience allocations upon port or supply chain resilience. We have deterred an attacker from attacking a certain port or supply chain if the output of Equation 27 is positive; if the output is negative, we have actually incentivized the attacker to attack that port or supply chain.

The deterrence quantification equation and its results are means to an end as discussed in Taquechel and Lewis (2012, op. cit.). We recommend applying new intent values after resilience allocations are simulated, to get new unconditional risk values, and then compare to pre-allocation risk values to see changes. This approach would implicitly account for quantification of deterrence, as the exact deterrence measurement Equation 27 is not applied to the conditional risk values, but new intent values are applied instead. Importantly, this deterrence quantification technique is applicable to any deliberately caused hazard, whether it is terrorism, sabotage, theft, or any act where an intelligent agent develops intent to do harm.