Gerald G. Brown

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.

SCHEDULING SHORT-TERM MARINE TRANSPORT OF BULK PRODUCTS

A multinational company uses a personal computer to schedule a fleet of coastal tankers and barges transporting liquid bulk products among plants, distribution centres (tank farms), and industrial customers. A simple spreadsheet interface cloaks a sophisticated optimization-based decision support system and makes this system useable via a varity of natural languages. The dispatchers, whose native language is not English, and some of whom presumably speak no English at all, communicate via the spreadsheet, and view recommended schedules displayed in Gantt charts both internationally familiar tools. Inside the spreadsheet, a highly detailed simulation can generate every feasible alternate vessel employment schedule, and an integer linear set partitioning model selects one schedule for each vessel so that all loads and deliveries are completed at minimal cost while satisfying all operational requirements. The optimized fleet employment schedule is displyed graphically with hourly time resolution over a planning horizon of 2-3 weeks. Each vessel will customarily make several voyages and many port calls to load and unload products during this time.

Optimizing Ship Berthing

Abstract : Ship berthing plans reserve a location for inbound U.S. Navy surface vessels prior to their port entrance, or reassign ships once in port to allow them to complete, in a timely manner, reprovisioning, repair, maintenance, training, and certification tests prior to redeploying for future operational commitments. Each ship requires different services when in port, such as shore power, crane, ordnance, and fuel. Unfortunately, not all services are offered at all piers, and berth shifting is disruptive and expensive: A port operations scheduler strives to reduce unnecessary berth shifts. We present an optimization model for berth planning and demonstrate it for Norfolk Naval Station, which exhibits all the richness of berthing problem:. the Navy faces.

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.

Optimizing Submarine Berthing with a Persistence Incentive

Submarine berthing plans reserve mooring locations for inbound U.S. Navy nuclear submarines prior to their port entrance. Once in port, submarines may be shifted to different berthing locations to allow them to better receive services they require or to make way for other shifted vessels. However, submarine berth shifting is expensive, labor inten- sive, and potentially hazardous. This article presents an optimization model for submarine berth planning and demonstrates it with Naval Submarine Base, San Diego. After a berthing plan has been approved and published, changed requests for services, delays, and early arrival of inbound submarines are routine events, requiring frequent revisions. To encourage trust in the planning process, the effect on the solution of revisions in the input is kept small by incorporating a persistence incentive in the optimization model. q 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 301 - 318, 1997. Although the Cold War has ended, United States Navy submarines remain very capable and effective ships of war: A smaller number of submarines operated from fewer submarine bases will continue to play a significant role in national defense. The wise use of time and resources while submarines are in port will improve the state of readiness of a smaller fleet. While in port, a submarine completes preventive and corrective maintenance, replenishes stores, and conducts training and certification tests to maintain high material and personnel readiness. Ideally, a submarine in port should devote its time exclusively to these activities. However, submarines frequently spend time shifting berths. Some shifts are necessary and some are not. Services such as ordnance loading and the use of special maintenance equip- ment require that a submarine be moored at a specific location. During periodic maintenance upkeep, personnel from a submarine tender assist the submarine crew, and berthing near the tender is preferable. During training, inspection, and other periods, it is desirable to berth closer to shore, near squadron offices and training facilities. When conditions permit,

Optimizing Disaster Relief: Real-Time Operational and Tactical Decision Support

We introduce a real-time decision support system which uses optimization methods, simulation, and the judgement of the decision maker for operational assignment of units to tasks and for tactical allocation of units to task requirements. The system, named ARES for the Greek god of war, accommodates a high degree of detail in the logistics of unit movements during operations, yet separates the assignment and allocation activities in a fashion which naturally accommodates human intervention and judgement—ARES is designed to assist the decision maker, not to replace him. ARES is demonstrated with a hypothetical scenario constructed for 14 Engineering Battalions of the Hellenic Army which are assigned 20 tasks employing 25 resource types in repairing major damage to public works following a great earthquake. (This hypothetical data was prepared prior to the earthquake in Kalamata near Athens on 13 September, 1986, and exhibits uncanny, but coincidental, resemblance to that real situation.) ARES is designed for use in real time, and quick data preparation is aided by the provision from published sources of standard data for many foreseeable tasks; this data can be quickly accessed via visual icons on a computer screen and customized for the actual work at hand.

How Probabilistic Risk Assessment Can Mislead Terrorism Risk Analysts

Traditional probabilistic risk assessment (PRA), of the type originally developed for engineered systems, is still proposed for terrorism risk analysis. We show that such PRA applications are unjustified in general. The capacity of terrorists to seek and use information and to actively research different attack options before deciding what to do raises unique features of terrorism risk assessment that are not adequately addressed by conventional PRA for natural and engineered systems-in part because decisions based on such PRA estimates do not adequately hedge against the different probabilities that attackers may eventually act upon. These probabilities may differ from the defenders (even if the defenders experts are thoroughly trained, well calibrated, unbiased probability assessors) because they may be conditioned on different information. We illustrate the fundamental differences between PRA and terrorism risk analysis, and suggest use of robust decision analysis for risk management when attackers may know more about some attack options than we do.

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

On Random Binary Trees

A widely used class of binary trees is studied in order to provide information useful in evaluating algorithms based on this storage structure. A closed form counting formula for the number of binary trees with n nodes and height k is developed and restated as a recursion more useful computationally. A generating function for the number of nodes given height is developed and used to find the asymptotic distribution of binary trees. An asymptotic probability distribution for height given the number of nodes is derived based on equally likely binary trees. This is compared with a similar result for general trees. Random binary trees those resulting from a binary tree sorting algorithm applied to random strings of symbols are counted in terms of the mapping of permutations of n symbols to binary trees of height k. An explicit formula for this number is given with an equivalent recursive definition for computational use. A generating function is derived for the number of symbols given height. Lower and upper bounds on random binary tree height are developed and shown to approach one another asymptotically as a function of n, providing a limiting expression for the expected height. The random binary trees are examined further to provide expressions for the expectations of the number of vacancies at each level, the distribution of vacancies over all levels, the comparisons required for insertion of a new random symbol, the fraction of nodes occupied at a particular level, the number of leaves, the number of single vacancies at each level, and the number of twin vacancies at each level. A random process is defined for the number of symbols required to grow a tree exceeding any given height. Finally, an appendix is given with sample tabulations and figures of the distributions.