The existing literature defines the basics of the DAD problem applied to infrastructure systems, but there are many unexplored topics associated with this problem. Literature provides formulations only for minimum cost flows and shortest paths. Literature also provides a nested decomposition approach with SEC to obtain an optimal solution to a DAD problem instance. This dissertation seeks to expand knowledge of DAD problems beyond the one solution approach. We address three interesting areas of the DAD problem that have not been developed in existing literature.
1. Efficient Enumeration Algorithms for the DAD Model
Chapter II explores the ability to generate multiple optimal solutions and near optimal solutions to a DAD minimum cost flow problem instance. The knowledge that equivalent optimal solutions exist can be useful information to a decision maker. For example, if an analyst has a defense budget of three units to defend a system against a worst-case attack, it could be helpful to know if there is only one possible optimal
defense or more than one equivalent optimal defenses. Many equivalent optimal solutions can present a range of equally good choices to defend a system. Similarly, a decision maker may want to know all of the defense options that are within a small fixed percentage of the optimal solution. Decision makers could then use a range of options to help weigh other factors to the system that cannot be modeled, but are still important to consider. In the real world, some defenses that appear to be equal on paper may have other considerations that make one set of defenses preferable to another set of defenses, even though they both produce the same objective function value. Traditionally, finding many optimal and near optimal solutions would require resolving the DAD problem for each additional solution that is desired. We design an algorithm that can find all of the optimal and near optimal solutions to a DAD minimum cost flow problem efficiently.
Chapter II harnesses the theory of implicit enumeration (IE) to efficiently generate all equivalent optimal solutions. IE can also find all near optimal solutions for any specified solution tolerance. Implicit enumeration works by restricting and subsequently relaxing the defense budget of the original problem instance. IE reduces the tri-level DAD problem into a group of simpler bi-level AD sub problems arranged in a tree structure in order to systematically build solutions. IE avoids the possibility of listing every possible solution by rejecting combinations of defenses and attacks that could not lead to an optimal or near optimal solution. The original application of IE to DAD optimization problems was developed by Scaparra and Church (2008). These authors present the use of IE in the context of defending and attacking the placement of warehouse facilities. Our contribution is to adapt, generalize and expand on known IE theory and apply it to the tri-level DAD minimum cost flow problem.
2. Impact of Nested Defenses on Optimality
Chapter III investigates system defense planning under uncertain budget scenarios, and how that uncertainty impacts optimality. It may be possible that a system analyst may not be certain about the budget for his or her defenses, the attack budget of the adversary, or both. When faced with budget uncertainty, the analyst may produce a list of prioritized defenses for the system. The prioritized list would include the best
defense, second best defense, third best defense, and so on. A prioritized list becomes a set of nested defenses to be chosen when the budget scenario becomes known. We use the term nested to refer to a monotonic sequence of sets, where each set of defenses for a particular budget scenario contains the set of defenses for the next smaller budget scenario. The use of nested defenses is attractive to decision makers because it simplifies the complex analysis of choosing the best defenses under uncertain budgetary scenarios into a single list.
The set of nested defenses almost always is not the same as the set of optimal defenses for any particular defense and attack budget. It is clear in existing literature that the best defenses chosen for a defense budget of two units is usually not a subset of the defenses chosen for a defense budget of three units, and so on. An example is shown in Figure 2 from Alderson et al. (2015). Figure 2 shows that if the defense budget is known, but the attack budget is unknown for the fuel network, then the elements of an optimal defense change. In Figure 2, the panels labeled (b) to (e) show that even when the defense budget is unchanged at four units, each change in the attack budget from two to five units results in a different set of optimal defense elements chosen and a different set of worst- case attacks. The optimal defenses for the network are shown as blue edge lines, and the worst-case attacks are shown as small explosions on edges. A similar effect exists when the defense budget is unknown and the attack budget is known.
Figure 2. Optimal Defenses Change as the Attack Budget Changes. Source: Alderson et al. (2015).
Chapter III uses the same network as Alderson et al. (2015) to investigate the effects of nested defenses. We examine how nested defenses differs from the optimal solution for any particular defense and attack budget scenario and, in particular, we
quantify the “cost” of requiring a set of defense plans to be monotonic. A gap exists in current literature for how to address the situation when both defense and attack budgets are uncertain, and we propose a strategy for that situation. We examine various heuristic nesting strategies and compare their performance against the known optimal solutions. In order to find the best possible nested defense, we develop a new parametric programming formulation of the DAD minimum cost flow problem that accepts uncertainty in the defense and attack budgets. This new formulation generates a nested defense that is as close as possible to the set of optimal non-nested solutions.
3. Incorporating Heuristics to Speed Up DAD Solution Algorithms
Chapter IV develops a DAD model for the constrained shortest path (CSP) problem. To the best of our knowledge, solution procedures for the DAD CSP problem do not exist in the literature. The challenge associated with the CSP problem is that the side constraint placed on a shortest path problem breaks the network structure of the problem. Loss of the network structure of the problem results in a much more difficult binary program because the path variables (Y) cannot be treated as continuous. First, we formulate and solve instances of the DAD CSP by adapting existing DAD nested decomposition techniques from the literature. We find the point where existing DAD solution techniques become either intractable or excessively time intensive to solve the problem. Our goal is to develop alternatives and improvements to the nested decomposition approach to solve DAD CSP problem instances more efficiently.
We improve the time it takes to find a solution to a DAD CSP instance by applying Lagrangian relaxation. Ahuja, Magnanti and Orlin (1993) explain the use of Lagrangian relaxation on a regular CSP problem instance involving a small test network. Fisher (1981) notes that the Lagrangian relaxation is able to find a greatest lower bound on an original problem instance. Since only a greatest lower bound can be determined, we observe Lagrangian relaxation is a heuristic approach to finding a solution.
We expand the Lagrangian relaxation techniques found in existing literature to include defending and attacking the CSP problem. We develop two different alternative algorithms for Lagrangian relaxation of DAD CSP. The first algorithm closes the gap
created by the Lagrangian lower bound by using path enumeration techniques proposed by Carlyle, Royset and Wood (2008). The second algorithm takes advantage of the fact that iterative Lagrangian relaxation searches for the greatest lower bound and the attacker model seeks to maximize the length of the shortest path. These similar goals allow us to combine the attacker problem and Lagrangian relaxation into a single mathematical model. Both algorithms are able to find a heuristic solution to DAD CSP quickly.
We also develop algorithms to combine our heuristic approaches with traditional methods to obtain provably optimal or near optimal solutions. By themselves, our improvements to Lagrangian relaxation remain a heuristic approach to the DAD CSP problem. The goal of our combined algorithms is to reduce the time required to obtain an optimal or approximately optimal solution to a DAD CSP instance when compared to regular nested decomposition. We test our algorithms on medium and large networks to show the scalability of our innovative approaches to solve the DAD CSP problem. Our results show that our new algorithms are significantly faster than regular nested decomposition to find an optimal or near optimal solution to a DAD CSP instance.