R. Kevin Wood

Defending Critical Infrastructure

We apply new bilevel and trilevel optimization models to make critical infrastructure more resilient against terrorist attacks. Each model features an intelligent attacker (terrorists) and a defender (us), information transparency, and sequential actions by attacker and defender. We illustrate with examples of the US Strategic Petroleum Reserve, the US Border Patrol at Yuma, Arizona, and an electrical transmission system. We conclude by reporting insights gained from the modeling experience and many “red-team” exercises. Each exercise gathers open-source data on a real-world infrastructure system, develops an appropriate bilevel or trilevel model, and uses these to identify vulnerabilities in the system or to plan an optimal defense.

Deterministic network interdiction

Interest in network interdiction has been rekindled because of attempts to reduce the flow of drugs and precursor chemicals moving through river and road networks in South America. This paper considers a problem in which an enemy attempts to maximize flow through a capacitated network while an interdictor tries to minimize this maximum flow by interdicting (stopping flow on) network arcs using limited resources. This problem is shown to be NP-complete even when the interdiction of an arc requires exactly one unit of resource. New, flexible, integer programming models are developed for the problem and its variations and valid inequalities and a reformulation are derived to tighten the LP relaxations of some of these models. A small computational example from the literature illustrates a hybrid (partly directed and partly undirected) model and the usefulness of the valid inequalities and the reformulation.

Shortest-path network interdiction†

We study the problem of interdicting the arcs in a network in order to maximize the shortest s–t path length. “Interdiction” is an attack on an arc that destroys the arc or increases its effective length; there is a limited interdiction budget. We formulate this bilevel, max–min problem as a mixed-integer program (MIP), which can be solved directly, but we develop more efficient decomposition algorithms. One algorithm enhances Benders decomposition by adding generalized integer cutting planes, called “supervalid inequalities” (SVIs), to the master problem. A second algorithm exploits a unique set-covering master problem. Computational results demonstrate orders-of-magnitude improvements of the decomposition algorithms over direct solution of the MIP and show that SVIs also help solve the original MIP faster. Published 2002 Wiley Periodicals, Inc.

Stochastic Network Interdiction

Using limited assets, an interdictor attempts to destroy parts of a capacitated network through which an adversary will subsequently maximize flow. We formulate and solve a stochastic version of the interdictors problem: Minimize the expected maximum flow through the network when interdiction successes are binary random variables. Extensions are made to handle uncertain arc capacities and other realistic variations. These two-stage stochastic integer programs have applications to interdicting illegal drugs and to reducing the effectiveness of a military force moving materiel, troops, information, etc., through a network in wartime. Two equivalent model formulations allow Jensens inequality to be used to compute both lower and upper bounds on the objective, and these bounds are improved within a sequential approximation algorithm. Successful computational results are reported on networks with over 100 nodes, 80 interdictable arcs, and 180 total arcs.

Factoring Algorithms for Computing K-Terminal Network Reliability

Let GK denote a graph G whose edges can fail and with a set K ¿ V specified. Edge failures are independent and have known probabilities. The K-terminal reliability of GK, R(GK), is the probability that all vertices in K are connected by working edges. A factoring algorithm for computing network reliability recursively applies the formula R(GK) = piR(GK * ei) + qiR(GK - ei) where GK * ei is GK, with edge ei contracted, GK - ei is GK with ei deleted and pi ¿ 1 - qi is the reliability of edge ei. Various reliability-preserving reductions can be performed after each factoring operation in order to reduce computation. A unified framework is provided for complexity analysis and for determining optimal factoring strategies. Recent results are reviewed and extended within this framework.

A Two-Sided Optimization for Theater Ballistic Missile Defense

We describe JOINT DEFENDER, a new two-sided optimization model for planning the pre-positioning of defensive missile interceptors to counter an attack threat. In our basic model, a defender pre-positions ballistic missile defense platforms to minimize the worst-case damage an attacker can achieve; we assume that the attacker will be aware of defensive pre-positioning decisions, and that both sides have complete information as to target values, attacking-missile launch sites, weapon system capabilities, etc. Other model variants investigate the value of secrecy by restricting the attackers and/or defenders access to information. For a realistic scenario, we can evaluate a completely transparent exchange in a few minutes on a laptop computer, and can plan near-optimal secret defenses in seconds. JOINT DEFENDERs mathematical foundation and its computational efficiency complement current missile-defense planning tools that use heuristics or supercomputing. The model can also provide unique insight into the value of secrecy and deception to either side. We demonstrate with two hypothetical North Korean scenarios.

A factoring algorithm using polygon‐to‐chain reductions for computing K‐terminal network reliability

Let G = (V,E) be an undirected graph whose edges may fail, and let G, denote G with a set K V specified. Edge failures are assumed to be statistically independent and to have known probabilities. The K-terminal reliability of G,, denoted R(G,), is the probability that all vertices in K are connected by working edges. Computing K-terminal reliability is an NP-hard problem not known to be in NP. A factoring algorithm for computing network reliability recursively applies the formula R(G,) = p,R(G,,*e,) + q,R(G, - e,), where Gxr*e, is G, with edge e, contracted, G, - e, is G, with e, deleted and p, = I - q, is the reliability of edge e,. Various reliability-preserving reductions may be performed after each factoring operation in order to reduce computational complexity. The complexity of a slightly restricted factoring algorithm using standard reductions, along with newly developed polygon-to-chain reductions, will be bounded below by an invariant of G, the “minimum domination.” For 2 5 (KI 5 5 or IVI - 2 5 IKI 5 IVI, this bound is always achievable. The factoring algorithm with polygonto-chain reductions will always perform as well as or better than an algorithm using only standard reductions, and for some networks, it will outperform the simpler algorithm by an exponential factor. This generalizes early results that were only valid for K = V. Removing the restriction on edge selection leaves results essentially unchanged in the upper range of IK(, but minimum domination becomes only a tight upper bound for the lower range.

Solving the Bi-Objective Maximum-Flow Network-Interdiction Problem

We describe a new algorithm for computing the efficient frontier of the “bi-objective maximum-flow network-interdiction problem.” In this problem, an “interdictor” seeks to interdict (destroy) a set of arcs in a capacitated network that are Pareto-optimal with respect to two objectives, minimizing total interdiction cost and minimizing maximum flow. The algorithm identifies these solutions through a sequence of single-objective problems solved using Lagrangian relaxation and a specialized branch-and-bound algorithm. The Lagrangian problems are simply max-flow min-cut problems, while the branch-and-bound procedure partially enumerates s-t cuts. Computational tests reveal the new algorithm to be one to two orders of magnitude faster than an algorithm that replaces the specialized branch-and-bound algorithm with a standard integer-programming solver.

Interdicting a Nuclear-Weapons Project

A “proliferator” seeks to complete a first small batch of fission weapons as quickly as possible, whereas an “interdictor” wishes to delay that completion for as long as possible. We develop and solve a max-min model that identifies resource-limited interdiction actions that maximally delay completion time of the proliferators weapons project, given that the proliferator will observe any such actions and adjust his plans to minimize that time. The model incorporates a detailed project-management (critical path method) submodel, and standard optimization software solves the model in a few minutes on a personal computer. We exploit off-the-shelf project-management software to manage a database, control the optimization, and display results. Using a range of levels for interdiction effort, we analyze a published case study that models three alternate uranium-enrichment technologies. The task of “cascade loading” appears in all technologies and turns out to be an inherent fragility for the proliferator at all levels of interdiction effort. Such insights enable policy makers to quantify the effects of interdiction options at their disposal, be they diplomatic, economic, or military.

Optimization of purchase, storage and transmission contracts for natural gas utilities

Natural gas utilities supply about a quarter of the energy needs of the United States. From wellhead to consumer, operations are governed by an astounding diversity of purchase, transport, and storage contract agreements which prepare a complex physical distribution system to meet future demands no more predictable than next years weather. We present a decision support system based on a highly detailed optimization model used by utilities to plan operations which minimize cost while satisfying regulatory agencies. Applications at Southwest Gas Corporation are presented along with a case study at Questar Pipeline Corporation. “But thou, contracted to thine own bright eyes, Feedst thy lights flame with self-substantial fuel” William Shakespeare, First Sonnet