Optimal Allocation of Risk-Reduction Resources in Event Trees

Abstract

In this paper, we present a novel quantitative analysis for the strategic planning decision problem of allocating certain available prevention and protection resources to, respectively, reduce the failure probabilities of system safety measures and the total expected loss from a sequence of events. Using an event tree optimization approach, the resulting risk-reduction scenario problem is modeled and then reformulated as a specially structured nonconvex factorable program. We derive a tight linear programming relaxation along with related theoretical insights that serve to lay the foundation for designing a tailored branch-and-bound algorithm that is proven to converge to a global optimum. Computational experience is reported for a hypothetical case study, as well as for several realistic simulated test cases, based on different parameter settings. The results on the simulated test cases demonstrate that the proposed approach dominates the commercial software BARON v7.5 when the latter is applied to solve the original model by more robustly yielding provable optimal solutions that are at an average of 16.6% better in terms of objective function value; and it performs competitively when both models are used to solve the reformulated problem, particularly for larger test instances.risk management, risk reduction, event trees, system safety, global optimization, factorable programming, branch-and-bound

The Nested Event Tree Model with Application to Combating Terrorism

“…Then it can be verified thatp ∈ (see Sherali et al 2008 for a similar construction). Thus, ifp ∈ P , we have that x ȳ p C is feasible to NETO, and its objective value in (1) yields an upper bound on the optimal values for both problems NETO and NETO( ).…”

“…Although similar in concept to Sherali et al (2008Sherali et al ( , 2010, in which the authors reformulate the original problem as per Sherali and Wang (2001), our solution methodology exploits the particular structure of the NETO problem, and we also design different tailored procedures to solve a fundamental special case of our model. We then construct a polyhedral outer approximation to these isolated nonconvex sets and embed this in a branchand-bound algorithm, which ensures that a sequence of convex programs solved over recursively partitioned hyperrectangles will yield an optimal solution to NETO (within any prescribed -optimality tolerance).…”

“…We represent the basic attack-to-consequence phenomenon as an event tree (Sherali et al 2008), having tiers of event branches with associated probabilities that are nested (or layered) to consider multiple threats as determined by combinations of the terrorists' developed capabilities and their intent to mount attacks using these capabilities against specific sets of targets. Furthermore, we consider the application of resources across four domains of combating terrorism: capabilities, intent, vulnerability, and consequences.…”

“…Stranlund and Field (2006) formulate nonlinear models to apply constrained event-and consequencerelated resources and study the effect of uncertainty on the expected loss. Sherali et al (2008Sherali et al ( , 2010 examine mixed-integer programming formulations for the application of event-and consequence-related resources to reduce probabilities and outcome costs, respectively, to minimize the overall risk (expected loss) within a Bernoulli event tree representing a cascading sequence of occurrences following an initiating hazardous event. They employed nonlinear logit models for the probability-resource relationships and a linear model for the outcome-resource relationships to formulate a nonconvex model that was solved to global optimality by adopting suitable outer-approximating linear programming relaxations.…”

Optimal design of reliable integrated chemical production sites

“…In a later work, Goel et al (2003) extended the system effectiveness approach to incorporate the initial availability of each process in the system as a degree of freedom. Sherali et al (2008) studied the optimal allocation of risk-reduction resources using an event tree representation. Some resources are used to prevent events in the tree from happening, while others are used to minimize the loss once the events occur.…”