Formulation and Optimization of Robust Sensor Placement Problems for Contaminant Warning systems

Abstract

The sensor placement problem (SPP) in contaminant warning system (CWS) design for water distribution networks involves maximizing the level of protection afforded by a limited number of sensors. In existing SPP formulations, the protection level is typically quantified as either the expected impact of a contamination event, weighted by occurrence probability, or the proportion of events that are detectable. In these formulations, the issue of how to mitigate against potentially high-impact events is either handled implicitly or ignored entirely. Consequently, any solutions of these formulations run the serious risk of failing to protect against any number of high-impact, 9/11-style attacks. This risk is further amplified by the fact that reliable estimation of contamination event probabilities is extremely difficult, such that existing SPP formulations may significantly discount the potential of high-impact events. In contrast, robust formulations of the SPP directly address these concerns by focusing strictly on a subset of high-impact contamination events, and placing sensors to minimize the impact of such events. We introduce several robust formulations of the SPP that are distinguished by how they quantify the potential damage due to high-impact contamination events. These include minimization of the worst-case impact, the Value at Risk (VaR), and the Tail-Conditional Expectation (TCE). The worst-case formulation is equivalent to the p-center problem in facility location theory. VaR and TCE are standard measures of robustness in the financial literature; the corresponding robust formulations of the SPP respectively minimize the (1-α)% largest impact and a weighted sum of the α% largest impacts. All formulations can be expressed as Mixed-Integer Programs (MIPs), which can be solved using both commercial MIP solvers and specialized heuristics. Additionally, we develop computational methods for exploring the performance trade-offs between robust and expectation-based SPP formulations. We use this framework to explore the nature of robust versus expectation-based solutions to the SPP on three real-world water distribution networks, ranging in size from 400 to over 10,000 junctions. We observe that robust SPP formulations are one or more orders of magnitude more difficult to solve than expectation-based SPPs. Our results indicate that simple heuristics yield optimal solutions to the smaller test problems in shorter run-times than MIP solvers, and yield higher-quality solutions for larger test problems. For realistic sensor budgets, solutions with low expected impact fail to protect against large numbers of high-impact contamination events (with impact 5-10 times larger than the expectation). In contrast, we show that solutions to robust SPPs yield 10-25number and magnitude of high-impact events. In general, our results indicate that it is possible to trade off mean impact versus high impact performance